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MASS SCALES in PARTICLE PHYSICS
and FIELD" THEORY
Thesis for the Degree of
Doctor of Philosophy
by
Jacob Sonnenschein
Submitted to the Scientific Council
of the Weizmann Institute of Science
Rehovot, Israel July 1987
MASS SCALES in PARTICLE PHYSICS
and FIELD THEORY
Thesis for the Degree of
Doctor of Philosophy
by
Jacob Sonnenschein
Submitted to the Scientific Council
of the Weizmann Institute of Science
Rehovot, Israel July 1987
To Nava,Nir,Ori
and to my Parents.
A C K N O W L E D G E M E N T S
In the first part of my Phd research work Prof. A.Davidson generously devoted to
me a great deal of his time, taught me the methods for studying the subject-matters
and inspired me with his scientific enthusiasm and originality. Prof. Y.Frishman, my
supervisor in the second period of my thesis work, shared with me his impressive knowl-
edge, taught me patiently the basic tools of the field, and instructed me with his deep
and serious scientific attitude. I wish, therefore, to express my sincere gratitude to both
Prof. A.Davidson and Prof. Y . Frishman.
It is a pleasure for me to thank Prof. O.W.Greenberg, Dr. H.M.Vozmediano and
Dr. G.Date for collaborating in parts of my research work.
Very useful discussions with Dr. M.Leurer and Dr. M.Bernstein are also gratefully
acknowledged.
Rehovot, July 1 9 8 7 Jacob Sonnenschein
A B S T R A C T
Three domains of research were addressed in my PhD thesis: (i) composite models
of quarks and leptons which incorporate grand unification and supersymmetry, (ii)
cosmological Kaluza-Klein models and (iii) non-abelian bosonization and its application
to multi-flavor Q C D in two dimensions.
In the first subject matter we constructed a composite model based on S U ( 7 )
G U T symmetry which leads to three generations of composite quarks and leptons and
an effective S U L ( 2 ) interaction with an appropriate Weinberg angle. We then derived an
S U ( 1 0 ) supersymmetric unified composite model consistent with the ,t Hooft anomaly
matching condition and the persistent mass condition with four generations of quarks
and leptons. Finally we studied the massless Nambu-Goldstone (NG) superfields in
the context of models with broken S U ( N ) and S U ( N ) j x S U ( N ) ! j global symmetries,
the N G mechanism in theories sharing shift symmetry CN on top of a global U ( N )
symmetry, and the relation between broken global U y ( l ) and U R ( 1 ) symmetries and a
newly born U R ( 1 ) .
As for the the Kaluza-Klein models, we developed two cosmological models. The
first one admits "naturally" compactified extra dimensions, an open three-space with a
Big-Bang origin, and the constancy of its fine-structure constant is correlated with the
smallness of the cosmological constant. In the second model the radius of compactifica-
tion tends asymptotically to a constant. We discuss also the eras of the early universe
in these models.
־ 4 ־
The third part was devoted to the study of the non-abelian bosonization and its ap-
plication to multiflavor Q C D in two dimensions. We used the non-abelian bosonization
of colored and flavored massless Dirac fermions. We gauged the S U ( N Q ) symmetry in
the bosonic theory. We overcame the difficulty of bosonizing mass bilineaxs by using the
U ( N J T N Q ) bosonization scheme which then was applied to multifalvor massive QCD2-
We deduced the low energy effective action in the strong coupling limit. The lowest
energy soliton solutions of the classical action where then found, and a semiclassical
quantization around these solitons was performed, leading to a modified C p ( • ^ - 1 )
quantum mechanical action. The Hamiltonian of the system was shown to depend lin-
early on the second Casimir operator of the flavor group. Finally we wrote down a mass
formula for the baxyons and discussed their possible flavor representations.
- 5 -
Contents
Page
Chapter 1. Introduction 6
Chapter 2. Grand unified composite models of quarks and leptons 12
2.1 Introduction 12
2.2 ,t Hooft Anomaly matching conditions and the persistent mass condition . . . . 13
2.3 Hyper-color embedding in SU(N) GUT Models 17
2.4 The solution of the Anomaly matching condition for SUJJC{3) 19
2.5 The substructure of the Q+L , 21
2.6 Effective Weak interaction and the Weinberg angle 24
2.7 Summary and Conclusions 27
Chapter 3. Supersymmetric grand unified composite models 28
3.1 Introduction 28
3.2 Construction of SU(N) supersymmetric unified models 30
3.3 Anomaly matching condition and Persistent mass condition for the S U ( I Q ) models 33
3.4 Further features of the S U ( 1 0 ) models 38
3.5 Conclusions 40
Chapter 4. Superpotentials and Nambu-Goldstone Superfields 41
4.1 Introduction 41
4.2 NG superfields from broken global symmetry generators - general analysis . . . . 42
4.3 S U ( N ) global symmetry breaking- 45
4.4 Symmetry Breaking of 5C/־(JV)/ X S U ( N ) n X U V { 1 ) X U R ( 1 ) 47
4.5 NG superfields from broken translation invariance 51
4.6 Spontaneous breaking and new born R-Invariance 54
4.7 Summary and conclusions 56
Chapter 5. Cosmological Kaluza Klein models 57
5.1 Introduction 57
5.2 Einstein equations 58
5.3 Isometry Analysis and Cosmological Compactification 60
5.4 The Cosmological Compactification model -Features of the ordinary universe . . 62
5.5 A model with asymptotically constant radius of compactification 65
5.6 Conclusions 70
Appendix A 71
Chapter 6. Non-Abelian Bosonization of Massless Multiflavor QCD2 72
6.1 Introduction 72
6.2 Wittenss Non-abelian bosonization 73
6.3 Non-Abelian bosonization of Dirac fermions with color and flavor 76
6.4 Gauging the WZW action 80
6.5 Summary and conclusions- 83
Chapter 7. Non-Abelian Bosonization of Massive QCD2 85
7.1 Introduction 85
7.2 The bosonization of a mass bilinear of Dirac fermions 86
7.3 On the equivalence between the fermionic and bosonic n-point functions 87
7.4 Bosonization of mass bilinears of colored flavored fermions -the old prescription. . 91
7.5 Bosonization using the U ( N p X N c ) WZW action 92
7.6 Multiflavor Q C D 2 using the U ( N F X N c ) WZW action 94
7.7 Summary and conclusions 97
Chapter 8. The Baryonic Spectrum of Massive Multiflavor QCD2 98
8.1 Introduction 98
8.2 Classical Soliton solutions 99
8.3 Semi-classical quantization and the Baryons 102
8.4 The baryonic spectrum Il l
8.5 Summary and conclusions 112
References 113
Chapter 1
Introduction
M y P.h.D research work was divided into two periods. In the first period, I inves-
tigated, under the supervision of Prof. A . Davidson, first the topic of composite models
of quarks and leptons which incorporate grand unification and supersymmetry and then
cosmological Kaluza-Klein models. The second period was devoted to the non-abelian
bosonization and its application to multi-flavor Q C D in two dimensions. This part of
the research work was conducted under the supervision of Prof.Y.Frishman.
The idea that there is a further stage of compositeness in nature- namely, that the
quarks and leptons are bound states of a more elementary constituents, the preons- was
introduced 1 - 2 mainly to deal with the generation puzzle and to reduce the number
(19) of free parameters of the standard model. Unfortunately, a composite model which
is both realistic and is based on a consistent field theory could not be derived. Grand
unification theory ( G U T ) , 3 on the other hand, was suggested to cope with the relation
between the electric charges of the quarks and leptons, with the different coupling con-
stants of the fundamental forces, as well as with the superfluous number of parameters
mentioned above. Combining together the two concepts may resolve the difficulties that
each separately encounters. Unified composite models based on SU(8) G U T symmetry
were proposed by us in the past 4 - 6 . In the first stage of my PhD work we were looking
for other models which are theoretically consistent and admit realistic features. We
constructed a model based on a solution o f ' t Hooft anomaly matching c o n d i t i o n 7 - 8
with a hyper-color group related to S U ( 3 ) C S U ( 7 ) . The structure of the resulting
three generations of composite quarks and leptons was deduced as well as an effective
S U 1 I { 2 ) interaction with an appropriate Weinberg angle. This work was carried out in
collaboration with my supervisor Prof. A.Davidson and its results were published in
reference 9 .
Unified compositeness, though showing some improvements over ordinary G U T
or non-unified composite models, still suffers from several drawbacks. For instance
the limited set of preons is not capable of furnishing a solution in accordance with
all the consistency requirements, or the hierarchy problem related to the G U T . These
difficulties call upon supersymmetry (SUSY) which has a potential to solve both of them.
Two main trends were established in connection with supersymmetric compositeness:
the first is a preon scenario which is an extension of the model described above and
the second is the Nambu-Goldsote superfield mechanism. In my PhD work I studied
both these two approaches. Following the first line of reasoning, we constructed a
supersymmetric SU(10) G U T composite model with four generations of quarks and
leptons. This was done together with Prof. A.Davidson and was published as reference
10
As for the second approach, Buchmuller, Peccei, and Yanagida ( B P Y ) 1 1 proposed
the idea of using the supersymmetric analog of the Nambu-Goldstone (NG) mechanism
as a way of guaranteeing the masslessness of quarks and leptons. Although this idea was
very attractive, there were several questions, the exploration of which would facilitate
model building. Our purpose in this work was to study the question of the relation
of the set of broken generators of the original global symmetry group to the set of
massless Nambu-Goldstone superfields, mainly in the context of S U ( N ) and S U ( N ) 1 x
S U ( N ) J J models, using superpotentials constructed with elementary chiral superfields.
In addition, we clarified: (i) the N G mechanism in theories sharing shift symmetry C N
on top of a global U ( N ) symmetry, (ii) the relation between broken global J7v־(l) and
17/2(1) symmetries and a newly born U R ( 1 ) . This work was done in collaboration with
Prof. O.W Greenberg from the university of Maryland and published in r e f . 1 2 - 1 3
The attribute of local gauge invariances on some geometrical symmetries of extra
space dimensions was suggested long ago by Kaluza and Klein 1 4 . Recently this idea is
having its "Renaissance" mainly in the context of trying to unify the standard-model
interactions with gravity. The basic hypothesis of the Kaluza-Klein scheme is that
space-time has 4 + K dimensions with general covariance and that, in analogy to four-
dimensions, the curvature scalar is taken to be the Lagrangian. In addition, the "ground-
state" of the system is supposed to be partially compactified, namely M 4 x B \ where M 4
denotes the four-dimensional space-time and B k is a compact k-dimensional space. The
size of B k must be sufficiently small to render it unobservable at the currently available
energies. Most of the proposed models are taking M 4 to be a Minkovski space-time, and
to be time-independent compact space ( to account for the observed constancy of
r - 8 -
the fine-structure constant). Moreover, in many cases the M 4 x B ^ manifold is assumed
and is not shown to be some solution of the relevant Einstein equations. Our work was
challenging these two last points. We showed that one can have a Cosmological Kaluza-
Klein model which admits a "ground state5' metric that solves Einstein equations. We
have constructed two such models. The first one admits a homogeneous and isotropic
ordinary space-time and "naturally" compactified extra dimensions. In addition, it has
an open three-space with a Big-Bang origin, and the constancy of its fine-structure
constant is correlated with the smallness of the cosmological constant. In the second
model the radius of compactification tends asymptoticaly to a constant. The eras of
the early universe are discussed, including a transition from a universe governed by an
equation of state of the form P = p into a P = universe. In this work, which was
the last part of the first period of my PhD thesis, I collaborated with Prof. A.Davidson
and Dr. M.Vozmediano and the results appeared in 1 5 - 1 6 .
One of the outstanding problems in elementary particle physics is the derivation
of hadron spectroscopy from QCD-the underlying theory. In the second part of my
PhD research we have solved this problem in two space-time dimensions by using the
technique of non-abelian bosonization. Bosonization 1 7 , namely finding a bosonic field
theory which is equivalent to a given local fermionic field theory, did not exhibit non-
abelian symmetry explicitly until the discovery of non-abelian bosonization by W i t t e n 1 8 .
A bosonized theory is believed to be more adequate for computations in the strong
coupling limit of QCD2 since this limit is equivalent to the weak coupling limit for
- 9 -
the bosons. Several authors in the past applied the "old" bosonization procedures to
investigate single 1 9 and multiflavor 1 9 QCD2- The method was not successful in the
multiflavor case 21. More recently the "new'5 bosonization was used in flavor space to
find the spectrum of the two dimensional baryons 2 2 for the case of two flavors. The way
we dealt with the problem is presented in the last three chapters of my thesis. This
work was summarized in two publications: reference2 3 in collaboration with Dr. G.
Date and Prof. Y . frishman and reference2 4 together with Prof. Y . Frishman.
In the first stage, which is described in chapter 6, we formulated the bosonic theory
equivalent to multiflavor massless QCD2- This included writing down the non-abelian
bosonization of colored and flavored massless Dirac fermions, analyzing its affine Kac-
Moody and Virasoro algebras and gauging the S U ( N c ) part of it.
Then we introduced quark mass terms which are crucial in the soliton description
of baryons. The bosonization rule for the mass bilinear of N Dirac fermions was checked
to reproduce the correct n-point correlation functions. A n extension of this rule for
flavored and colored fermions turned out to be incorrect. We overcame this difficulty by
using the U ( N p N c ) bosonization scheme which then was applied to multifalvor massive
QCD2- The low energy effective action in the strong coupling limit was finally deduced.
This paxt of the work is presented in chapter 7.
Two dimensional baryons are solitons of the W Z W theory is analogous to the
conjecture that in four dimensions baryons are Skyrmions. Obviously, here we have an
advantage in the sense that the non-linear sigma model is not postulated but derived
- 10 -
from the underlying theory. In the last chapter we describe how we found the lowest
energy soliton solutions of the classical action which for static configurations turned out
to be a set of Sine-Gordon actions. Then we performed a semiclassical quantization
around these solitons. Using a special parametrization we proved that the quantum
mechanical action was that of a CpO^-P - 1 ) theory with an additional term linear in
time derivatives. By choosing an unconstrained set of variables, paying the price of
having some non-linearly realized symmetries, we avoided the necessity to use Dirac
brackets for the quantization. One gauge variable still remained. The Hamiltonian of
the system was shown to depend linearly on the second Cassimir operator of the flavor
group. Finally we wrote down a mass formula for the baryons and discussed the allowed
flavor representations in terms of their Young tableaux.
- 11 -
Chapter 2
Grand unified composite models of quarks and leptons
2.1 Introduction
Combining together the ideas of compositeness of quarks and leptons ( Q + L ) 1 and
grand unification ( G U T ) 3 , may clarify the generation puzzle, which in ordinary GUTs
is not explained, and reduce the arbitrariness of the elementary constituents of the com-
posite theory under the standard model's symmetry. In addition it obviously includes
the ordinary features of G U T .
In the past 5 we investigated models based on the symmetry group
S U ( N ) D S U ( 5 ) G G x S U H c ( N - 5 ) x U ( l ) where S U G G ( 5 ) and S U H C ( N - 5 ) are related
to Georgi-Glashow G U T and hypercolor interaction respectively. We showed that only
two models based on SU(8) fulfill our theoretical requirements. In the first stage of my
P h D work we were looking for other models which will still be theoretically consistent.
A model based on S U ( 7 ) D S U H C ( 3 ) x S U C ( 3 ) x £7/(1) x U n ( l ) was developed with
a modified low energy symmetry, namely S U ( 2 ) 1 is only an effective Symmetry.
In this chapter the derivation of the model and its predictions are described. We
first explain briefly the basic constraints on the construction of massless composite
particles. These rules, which will be used also in chapter 3, are described in the next
- 12 -
section. Section 3 is devoted to possible embeddings of the hyper-color symmetry in
G U T models based on S U ( N ) gauge symmetry. In section 4 the solution to the 't
Hooft anomaly matching condition for S U ( 3 ) C S U ( N ) is presented. Then, in section
5 the substructure of the composite Q+L in the S U ( 7 ) model is described. Next the
effective S£/k(2) interaction and the Weinberg angle are explained. Finally in section 7
we summarize the features of the model.
2.2 't Hooft Anomaly matching conditions and the persistent mass con-
dition
In his original work 7 ^ ,Hooft introduced two conditions on the construction of
massless bound states, the anomaly matching condition , and another constraint based
on the Appelquist Carazzone decoupling theorem 2 5 . The anomaly matching condition
is a statement that the anomalies associated with the chiral symmetry calculated at the
preon level should be identical to those at the compositeness level, ,t Hooft presented
a simple argument to justify this condition. Assume that the global chiral symmetries
axe changed into local symmetries with a very weak coupling (so that the hypercolor
bound states are unaffected). For consistency one is forced to add on top of the flavor
gauge fields also some hypercolor spectators in order to cancel the flavor anomalies.
Now at energies lower then that of the hypercolor confining scale, there are massless
bound states that together with the spectators have to cancel the flavor anomalies.
Obviously this means that the contributions to the anomalies of the preons and the
- 13 -
massless composites have to be the same.
Frishman et al 8 . proved rigorously the statement of above. They expressed the
three point function of the currents associated with the S U 1 ( N ) x S U R ( N ) X J7y(l)flavor
symmetry:
K % = / d * ־ e » ( * ^ + * jJ(x2)J^5(0)|0 > (2.1)(x1)״T(J|״־»> < 0
in terms of Lorentz invariant amplitudes as follows:
T<rpti(k1,k2) —A1k1eT(Tpn + A2&2 £7-<rp,i + A־ 3 k 1 p k ^ k% T e a 0 a - u + (2.2)
A A k 2 p k ^ k ^ T e a p f f l x + A 5 k l c T k f k ^ r e ^ ^ ^ + A 6 k 2 f f k f k $ T e a p p f t +
where T^3^ = <i a 6 c x To•^ with dabc are the usual d symbols. By applying Bose sym-
metry and the naive axial and vector ward identities it was found that for
k2 = fc| = k2 = 0, q — (k\ + fc2)2 = 0 the whole amplitude vanishes, which is an
unacceptable result. Thus, the vector and axial currents can not be simultaneously
conserved. Using the naive ward identities only to the discontinuities of the amplitudes,
which still admits an anomaly in one of the currents (from the A A A case it is clear that
it must be an axial anomaly), Frishman et a l . 8 found the following solution
discA3(q2, k2 = 0) = C 6 ( q 2 ) C = -27 rA1( g
2 , k2 = 0) (2.3)
The immediate conclusion from the last result is that there is a zero-mass physical
threshold in A3 with a discontinuity S ( q 2 ) a and strength C. This can be produced only
by one of the following possibilities : (1) A zero-mass pseudoscalar particle namely a
- H ־
spontaneous breakdown of chiral symmetry. (2) A zero-mass fermion-antifermion "cut"
with a £(g 2 ) discontinuity . In this case the flavor symmetry is unbroken. The massless
fermion bound states should be in representations of this symmetry such that
E d a b c = E d a b c ( 2 • 4 ) elementary composites
This last requirement is the 't Hooft anomaly matching condition.
The second condition used by 't Hooft was based, as mentioned above, on the
decoupling theorem. Let us again first follow his reasoning. Suppose that one of
the preons is taken to be massive, namely one adds to the Lagrangian a term of the
form:6£ = m ^ n ^ j n + h . c . Clearly the addition of the mass term will reduce the
original flavor symmetry G p to a smaller symmetry G ' F C G p . Now if in the limit
m —• oo, then following the decoupling theorem, all effects due to the massive preons
should disappear. In particular all bound states containing this preon should also disap-
pear by becoming themselves infinitely heavy. Only if the number of such bound states
with left chirality is equal to the number of those with right chirality, a mass term can
be assigned to it. If l'r, denotes the difference between the number of left handed and
right handed particles in a given representation r of G'p, then the condition for the ׳
decoupling is
rwithr׳Cr
This application of the decoupling theorem was criticized by S.Weinberg and J.
P resk i l l 2 6 . They pointed out that in fact one is invoking a further assumption that
the pattern of spontaneous symmetry breaking and the representation content of the
bound massless particles do not change when one of the preons is given a very large
intrinsic mass. Namely one assumes that a phase transition can not occur in this
process. To avoid this further assumption, the authors of 9 proposed an alternative
stronger constraint, the so called persistent-mass condition ( P M C ) , based on "intuitive
ideas about masses of composite systems" .This condition can be stated as follows: The
composite particles of any gauge theory must be such that when one of the elementary
preons is given any mass, the remaining unbroken chiral symmetries permit all composite
particles containing this preon to get some mass. S.Weinberg and J.Preskill however,
suggested in 2 6 not to consider the persistent mass condition as a proven constraint in
constructing composite models, in their words " we cannot with further argument rely
on the intuitive idea that massive constituents make massive composites". They got
to this conclusion after presenting an example (the Nambu Jona-Lasinio model) 2 7 with
massless composite particle build from massive building block.
The prove of the persistent mass condition was finally presented by C.Vafa and
E . W i t t e n 2 8 for the case of vector-like global symmetry and with 9 = 0. They showed
that in a massive QCD-like theory the two point function of the currents associated with
the diagonal subgroup Gy of the global group (like isospin) in a background gauge fields
< J f i ( x ) J b ( 0 ) > A has to fall down exponentially with the separation x,and in a way
which is independent on the background. Now suppose that there was a massless particle
| A > with non-zero charge under Gy, then there would presumably be an associated
- 16 -
G y current J ( x ) creating |A > from the vacuum, namely < 0|J(x)|A > ^ 0. If so, the
two-point function < J(x) J(0) > would not exhibit exponential fall-off. Therefore there
can not be massless particle composed from massive constituents.
2.3 Hyper-color embedding in SU(N) GUT Models
We start with the fermionic content of the theory at the G U T energies, namely the
representation of the basic fields, under the G U T symmetry group which is assumed to
be SU(N). The minimal anomaly-free complex representation of left-handed fermions is
given by
= + ( N - 4) x £ 7 (a, 0 , ־ ך 1 , N ) (2.6)
with \E£7׳ in the fundamental representation and \& a ^ = — \&^ a in the second antisym-
metric representation of the S U ( N ) group. In fact , for practical N , the above appears
to be the only tenable choice which would not endanger color-asymptotic freedom. After
several symmetry breaking stages, S U f f c ( n ) 1S born and becomes the first interaction
to confine. The n > 3 case was worked out intensively in my master^ thesis 4 , and
the conclusion was that there is no bound-states system that admits a generation-like
structure. So from here on in this chapter we will concentrate on n=3. In chapter
3 will come back to n > 3 for supersymmetrized models. For S U J J C { ^ ) o n e g e * s the
representation:
=[311] + [3, ( N - 3)} + [1, h N - 3 ) ( N - 4)] (2.7)
+ ( N - 4) x [ 3 ! 1] + ( N - 4) x [I, ( N - 3)*}
under S U ! J C ( 3 ) x S U ( N — 3) x £7(1). The exclusiveness of the n=3 case originates from
the fact that now only two (rather than three) types of irreducible H C representations
are involved. We will call the H C charged particles preons while the others wil l be called
spectators. Note that the content of both sectors is fixed upon us by the grand unified
group and the demand for appropriate representations. At the confinement scale of the
H C interaction, one assumes that all the other coupling constants are in comparison to
OLQ small enough to be neglected. Therefore the symmetry of the effective Lagrangian
is the following:
S U H C ( 3 ) x [ S U L ( N - 3) x S U R ( N - 3) x U ( l ) x Z 3 } G n = 3 (2.8)
where G denotes a global symmetry. Under this global symmetry the preons transform
as follows: The hypercolor 3 and 3 ! transform as
r ( J V - 3 U h f l , ( J V - 3 ) * U (2.9)
It is then believed that the Q+L are hypercolor bound states of several preons. A l l
possible composites along with their associated multiplicities ij where i stands for A ,M,S
denoting antisymmetric, symmetric and mixed symmetry representation under the Left
or Right symmetries denoted by the superscript j , are listed by:
* 1); &( r21 ,1 ) ; ' s ta m 1י)5 * ;(1 , i ) ; ' & ( י 1 E21); 'S (13!י) (2.10)
#(nn.!:1);fs *A (11פי);*(£1£2י2:1)*5
where Z i x i 2 . . . denotes the S U ( N — 3) representation whose Young tableaux consists of
i\ boxes in the first row, i<1 boxes in the second row, etc.
- 18 -
2.4 The solution of the Anomaly matching condition for S U j ! c ( 3 )
As discussed in section 1.1, indices of the composite particles must come out ELS
solutions to 't Hooft anomaly matching conditions, to guarantee the existence of un-
broken chiral symmetry. Namely the anomalies associated with the global symmetry
which one calculates at the preon level have to be the same as those at the composite
level. The possible anomalies are of the following types: *
A n o [ S U l ( N - 3)], A n o i S U f i i N - 3)], (2.11)
A n o [ S U l ( N - 3) x £7(1)], Ano[S£7|(iV - 3) x *7(1)], A n o [ U 3 ( l ) }
where A n o [ S U ^ ( N — 3)] denotes an anomaly which is related to a triangular diagram
with S U L ( N — 3) currents at the three vertices and similarly for the other expressions.
In fact because of the L R symmetry the equations associated with the S U 1 ( N — 3)
anomalies are related to those associated with the S U R ( N — 3)anomalies via 11 —•
— IR, SO altogether there are only 3 independent equations. Using the values of the
dabc symbols ,the c 2 defined by T r ( T A T B ) = < ? 8 A B and the £7(1) charges for the
representations of the various composites we get the following equations:
i ( n - 3)(n - Q ) l L
A + ( n 2 - 9)lfo + \ { n + 3)(n + 6)1% + n(n - 4)1'^ (2-12) ״ ,
+ n ( n + 4)4L + - n ( n - + - n ( n + 1 ) / ^ = 3
* In fact one has to consider also the anomaly related to a triangular diagram with
one £7(1) current and two energy momentum insertions. For U y ( l ) , as in our case, this
anomaly is automatically equal to zero
hn - 2)(n - 3)15 + (n 2 - 3)/fe + i ( n + 2)(n + 3)j| + n(n - 2)# 2 " (2.13)
+ n(n + 2 ) l ' s
L + \ n ( n - l)l'£ + 1-n{n + l ) l ' s
R = 1
hn - l ) (n - 2 ) / i + (n 2 - 1 ) & + \{n + l ) (n + 2)1% + \n{n - 1 ) #
+ I n ( n + 1 ) / ׳ / + [Z1 - l R ] = 0
where n stands for N — 3
As is well known 7 , this system of equations together with the persistent mass
condition does not have a solution (apart from as unrealistic one with N2=3־). At
the time that this model was published the necessity of fulfilling the P M C was not yet
proven (see section 1.2). Today 2 8 it is clear that for the approximation of a vector-like
theory that we are using the additional condition have to be fulfilled. Note, however,
that in our model, strictly speaking, one can not give masses to any of the preons as long
as the local U w i X ) is unbroken. It is true, however, that by taking all the couplings
,apart from the H C one,to zero, we, in fact, are using a vector-like scenario. In this
respect our model has to be taken as an exercise in deriving the low-energy properties
of a unified composite model. In chapter 3 a model which is consistent with the P M C
is presented.
There are infinitely many solutions to the set of equations (2.12-2.14). We are
interested only in solutions with no exotic low mass composite particles. Since S U c ( 3 )
is embedded in our case in S U ( 4 ) , we have to look for solutions that transform only as
totally antisymmetric representations of 5*7(4). Therefore we impose
lLM = l L S = l's = 0 (2-15)
- 20 -
There is only one solution that reflects the underlying discrete L R symmetry of the
equations. This solution is given by
Integer solutions for exist only provided N=7,8. The N=8 case was discussed inten-
sively in my master thesis 4 So from here on we concentrate only in the N=7 model.
The composite particles in terms of their properties under 5*7(4) x 5*7(4) x *7(1) axe:
1PL(composite) = 3[(4! 1) + (1,4)] + [(4,6) + (6^,4*)] (2.17)
2.5 The substructure of the Q+L
Let us now, before discussing the structure of the Q+L, elaborate more on the
5*7(7) G U T model. The original gauge group undergoes twofold spontaneous symmetry
breaking pattern:
5*7(7) - M 5*7C(3) x 5C/(4) x *7/(1) (2.18)
S U H C ( 3 ) x 5*7C(3) x x U H ( 1 )
This pattern guarantees that A J J Q > A Q - Defining the electric Q and the weak W
charges in terms of the normalized *7/(1), U ! j ( l ) charges, namely
g = ־ V ^ y / + v ^ y / / w ׳ = \ f t ^ Y 1 9 • 2 ״ ( )
- 21 -
we can identify our preons as
Tl
L = ( 3 , 3 ) g = _ 1 w = _ 1 T L = (3,1) (2.20)
= 3(3!. ,1)Q_ 0 1 1 v _1 Q = 0 , W — 1י ־ 3 ־
i = 1,2, 3 is the usual color index,- while a = 1,2,3 is the initial global 5*7(3) index.
The attached spectator set is
We want now to explore the substructure of the composite Q+L, to identify the
spectator Q+L and to discuss the masses of all the other particles in the model. For
that, it is necessary first to clarify how 5*7c(3) x U Q ( 1 ) X U W O • ) hves inside 5*7(4) x
x *7y(l). In particular observe that 5*7(4) D 5*7C(3) x *7Q(1) so that the (׳(74*5
electric charge (rather than B - L ) plays the role of the forth color. 5*74) ׳ ) on the other
hand, is the nest of the initial global SU(3), with the analogous Q being such that ׳
W = j $ ( Q + Q' — TJV). Taking into account the spectators (2.21) which automatically
absorb the 1 <3י anomalies, the combined list of H C singlets fermions is specified in the
+ ( L 2 ! . ) Q = _ | 1 W = 0
3(1, 1 ) Q = _ 1 ] W = _ 1 •
(2.21)
- 22 -
following table 1.
Internal structure W # Internal structure W #
U L e I J K T R J T R K V L A
1 1 3 T R J T R V L O . 0 3
UL t i j h f R k v L 0 1 UL spectator 0 1
UL spectator 1 1 UL f R i f R v L
1 ־ 5
1
dL
1 ־ 2 3 h 3 e I J K T 3
L T K T L 0 3
dL 0 3 spectators,! 0 3
£L 1
~1 3 &L *abcVkVf-TL 1
־ 2 3
U L 3eabcVLaVLbVLc
1 2 3 VL 3eabcVLbVLcVL 0 9
The fermions appearing in table 1 can be rearranged according to \ & L ( c o m p o s i t e ) +
\ E ׳ i ( s p e c t a t o r ) = $1 (complex) + ^ 1 ( r e a l ) under S U c ( 3 ) x U Q ( 1 ) X U W { ! • ) gauge sym-
metry. Notice that U \ y ( l ) is exclusively responsible for the presence of the complex
piece. No wonder, W ttnrns out to substitute the third component T3 of the weak
isospin. In the light of equation (2.19), however this substitution holds only at the com-
posite level. Consequently, Z is elementary, whereas its would be charged partners
are expected to be composite objects. A closer look at table 1 reveals that ^^(rea/)
consists of nine right-handed neutrinos and a set of parity doublets. The latter are such
that each spectator uniquely finds a proper conjugate composite partner. This paring
feature seems to generally accompany unified compositeness. The mass scale m of the
parity doubletes can be estimated on general grounds, m must first of all vanish on
the A J J C scale once & c , Q , W 3 X 6 switched off. On the other hand it is not protected
- 23 -
by gauge invariance, and hence, following the so-called "survival hypothesis" , pre-
sumably becomes superheavy and decouples from the effective low energy theory at the
AfjC —> oo limit. The two aspects are consistent with
m ( u R a צ ( L
c , Q , w ( ^ H c ) ^ H C (2-22)
for some power k, an order of magnitude which reminds us the Weinberg mechanism 3 0
At any rate m ~ {VR) is believed to be quite heavier than the masses of the U w O > ) P r o ־
tected fermions. \&/,( complex) exhibits exactly three ordinary Q+L generations. For the
(u1,dL,U£,e1)- families it reflects the preon replication requested for gauge anomaly
cancelation, while for the ( J^ , e^, 1/jr )-families these are the derived 't Hooft multipli-
cations. It is amusing though to watch the way the horizontal structure respects the
S U G G ( 5 ) classification. A particular attention is to be given to the left-handed leptons
which by virtue of Fermi statistics, cannot have an internal derivative-free structure.
2.6 Effective Weak interaction and the Weinberg angle
We come now to the charged weak interaction structure. The central idea is that
the symmetry of the effective low energy Lagrangian is actually larger than S U c ( 3 ) X
?7/(1) x U j ! ( l ) . This has to do with the observation that the preon content of both
( u ^ d i ) as well as [ V L ^ L ) • ! f ° r each generation separately, is precisely the same. It closes
upon an SU(3)ctjyc־singlet six preon creature.
W־ ~ e a b c e i j k V ^ V ^ r L T [ T l (2.23)
- n -
which is furthermore a GIM-singlet in the global 5*7(3) x 5*7(3) generation space. Thus,
taking into account the v.£ +-» di and *-* transitions U j j ( l ) is enlarged to 5*7^(2).
The latter symmetry is special. It induces no .4?20[(5*7(2))2*7/(l)] anomalies at the ef-
fective Q+L level. Now following 't Hoofts discussion 7 , the effective interactions for
/i <C A f j c 3 X 6 vanishingly small or alternatively renormalizable. The consequent line of
thought is therefore that at present energies it may look as if 5*7^(2) is fully gauged.
A similar conclusion could not have been derived with regard to the broken genera-
tion symmetry. It does carry Q+L anomalies, and therefore the effective horizontal
interactions are highly suppressed.
Our 5£f(7)theory is intrinsically falvor-chiral. It is therefore not a surprise that,
unlike in the Rishon model the Q+L internal structure would not provide room for
S U R ( 2 ) . Nevertheless, passing through a vector like Q H C D stage, we better remark
that L *-* R symmetry violation is switched on along with the color (but not necessarily
electro-weak) forces. After \I> 1(preon) + \&i(spectator) is S U H C , C r e a ^ but 5*7#c(3) x
5*7c(3) complex. Now, although (u,d,e,v)1 and happen to essentially match
the Rishon description, (u, d, e)# do not. Yet, we formally have ( e R t i ) = {dRd1) =
( u R u L ) * . These Q = 0 W = - \ "mass terms" suggest that *7/(1) x *7//(l) -+ *7Q(1)
should be governed by a V E V of a scalar singlet heaving the above charges if the mass
relation mw = mzcos6\v is to be effectively recovered. Using 5*7(7) language, this
can be realized via threefold antisymmetric Higgs in the 35 representation. The latter
automatically introduces the desirable charge one HC,C-singlet piece as well.
- 25 -
The weak mixing angle is our next target. To calculate 9\y, stemming from the
W = T3 effective identification, we do not need the effectieveW^ scenario. At the fully
perturbative region A J J C < fJ• < M׳ the relevant coupling constant evolve v i a 3 1
׳M M י<*H O ) = a - l + ל 4 ' מ ( ע • ) + Wri )
M • ( 2 ' 2 4 )
oij (/x) =a_! + boln( )
with bk = — ( ^ ) ( l l A : — 8). Note the relative fermion-loop contribution5 x l + l x 3 = 8.
Now from equation (2.19) we learn that
\ + x = W־״ a ״ ' a w = h 2 - 2 5 ״ ( )
Altogether, using the definition sin9y/ = we obtain
־U/ ן 77^ w = i [ l - ( - ) a / n ( — ) ] (2.26)
This result expresses a novel conception. First of all, to be contrasted with the
elementary-generation value of %,sin29w —י• \ at the unification scale. This can be
confirmed directly by evaluating jv^2 over all preons and spectators. Second, it so
happens that sin?9 is almost /j, independent. These two features seem to compensate
for each other, i.e. sin9\y starts low but would not change to much. In order to be
in accordance with the experimental value sin29\y ~ 0.21 we better have < 10 3.
On the other hand, a ^ c ( ! x ) — a־J^c(n) + ^ l n ( j j 7 ) prior to the HC-confinement and
hence י ^ - must be bound below as otherwise c t c i ^ H c ) would be too close to unity. The
compromise is achieved for ~ 10 3 meaning a characteristic "energy oasis" around
- 26 -
M 1 GeV. It is important to mention that for \1 < A ׳ ~ 0 J J Q the expressions of above
will certainly be modified by non-perturbative corrections. Therefore one can by no
means consider them as meaningful predictions of the model.
2.7 Summary and Conclusions
To summarize, we presented a unified composite model based on minimal S U ( 7 )
G U T symmetry. The model includes 3 generations of Q+L, without any light exotic
particles. Part of the Q+L are H C point-like singlets (spectators), and part are com-
posite particles which fulfill the 't Hooft anomaly matching conditions. The bound
states, though, are not consistent with the P M C . We described the substructure of the
composite particles, and discussed the masses of the heavy particles. The effective weak
interaction was shown to have a Weinberg angle in accordance with the experimental
values. In chapter 3 we will incorporate also super-symmtry in the construction of
another unified composite model.
- 27 -
Chapter 3
Supersymmetric grand unified composite models
3.1 Introduction
In the last chapter we presented a unified composite model which admits three
generations of Q+L. The significance of this model is, however, not clear since, as
explained above, it is inconsistent with the P . M . C . 2 5 In addition as any other grand
unified model, it suffers from the hierarchy problem. Supersymmetry is probably the
only possible answer to the hierarchy puzzle and it includes, in general , a larger number
of elementary constituents (like the SUSY partners of the preons) so there may be more
room for solving simultaneously the 't Hooft condition 7 and the P . M . C . Motivated by
the above, we attempt to construct a supersymmetric grand unified model with realistic
low energy multigenerational structure.
Our strategy in constructing such a model is the following:
(i) Contenting ourselves with S U ( N ) gauge theories, we first search for a matter super-
field, such that its representation R would be both complex and gauge anomaly free.
(ii) Unlike in chapter 2, we have in mind here to get the whole standard model gauge
symmetry, therefore we use the decomposition:
S U ( N ) C S U ( N - 5)Hc x S U ( 5 ) G G x *7(1)^ (3.1)
- 28 -
with the understanding that the Georgi-Glashow 3 2 S U ( 5 ) Q Q is already broken down
to S U ( 3 ) x S U ( 2 ) x U ( l ) once S U ( N — 5 ) H C makes its appearance, we require the
S U f f c ( N — 5) and also S U c ( 3 ) to be asymptotically free.
(iii) Once H C confines we assume that SUSY is still preserved, that the standard flavor-
color gauge coupling are switched off and that the G U T scale is taken to infinity. The
massless composite particles have to fulfill the 't Hooft condition and P . M . C .
(iv) At relatively low energies, where SUSY is already broken, we collect all H C fermionic
singlets both composites and spectators. The resulting set of fermions is expected to
match the pattern of Q+L. The discussion of breaking down SUSY is beyond the scope
of our models.
The construction of such supersymmetric unified models based on S U ( N = 10),
and some of their properties are presented in this chapter. In section 2 models based on
S U ( N = 9,10,11) which are gauge anomaly free and asymptotically free are discussed.
The charges of the preons under the global symmetry are written down and finally an
explanation why S U ( 1 0 ) is a unique possibility for the G U T gauge symmetry is also
presented in section 2. The following section is devoted to the search for solutions to the
't Hooft and P . M . conditions. Section 4 describes some possible solutions with gener-
ational structure and explains the different "horizontal"symmetries. Some conclusions
are brought in the last section.
- 29 -
3.2 Construction of SU(N) supersymmetric unified models
Whereas the anomaly-free constraint remains practically the same as in ordinary
nonsupersymmetric S*7(iV)local gauge theories (chapter 2), the asymptotic freedom
requirement appears to be much more restrictive 3 3 in the SUSY framework. We start
from the matter supermultiplet in the representation R = Y l i h l i t with l{ denoting the
multiplicity associated with the irreducible representation V{. Now, the asymptotically
free evolution of some intermediate S U ( k ) gauge interaction, such as S U ( N — 5) or
S U G ( 3 ) , can be realized provided
Y , h C ( r i ) < 6k (3.2)
C ( r i ) is the usual Casimir operator ,weighted by the dimension of vi, and normalized
such that C(N)=1. Supersymmetry is reflected by 6k term, to be contrasted with I l k
in the nonsupersymmetric analog case. The consequent restriction on R is so severe
that we can only live with the representation which is of the form of (2.6) namely:
R = ( j V - 4 ) x M ^ z i l . i n addition , N is such that 8 < iV < 11. N < 11 is dictated
by color-asymptotic-freedom, while N > 8, otherwise S U ( N — 5) is too small to play the
roll of H C . By decomposing R via (3.1),we identify the supersymmetric hypercolored
preons. The maximal global symmetry in the limit where g j j c 1 5 the only nonvanishing
gauge coupling constant is for N > 9
S U G G ( 5 ) x S U ( N - 4) x U H ( 1 ) x U s ( l ) x £ ^ ( 1 ) (3.3)
S U G G ( 5 ) a n d UJJ(1) correspond to the momentarily switched off gauge interactions.
- 30 -
S U ( N — 4) is related to the initial replication symmetry. U s ( l ) has been chosen in a
special way in order to commute with the generators of S U ( 5 / ( N — 4)) supergroup 3 4
whose relevance in satisfying the P . M . C will soon be clarified. U R ( 1 ) is the so-called R
symmetry 3 5characterized by the fact that it commutes with SUSY. The preon assign-
ments under the global symmetry axe explicitly given in table 1.
S U H C ( N - 5) S U G G ( 5 ) S U ( N - 4) U S ( 1 ) U R ( 1 )
(N-5) 5 (N-10) (N-7) (x,x+t)
(N-5)(N-6) -10 (N-9) (y,y+t)
(N-5)* (N-4)* 5 -(N-7) (z,z+t)
(A, A) (N-5) 2 - l 0 0 (0,t)
spec. 10 2(N-5) 0 0
spec. 5* (N-4)* -(N-5) 0 0
spec. 24 0 0 0
The U R ( l ) charges fulfill 5x+(N-7)y+(N-4)z+2(N-5)t=0 ensuring the absence of
S U H C { N - 5) 2 x U R ( 1 ) anomalies. SUSY has a threefold expression in this table due
to the existence of scalar preons and spectators, gauginos, and the U R ( 1 ) symmetry. A
note concerning the case of S U ( S ) G U T symmetry 9 is in order. It turns out that even
for this case SUSY can not resolve the problem of inconsistency with P . C . M similar to
what was discussed in chapter 2. This is why only S U ( N = 9,10,11) theories deserve
further investigation.
Next, we would like to derive the set of constraints which reproduce the preon global
anomaly, fulfills the P . M . C condition and includes only "stable" Q+L and no exotic
particles. The P . M . C can in principle be realized provided the HC-singlet composites
form representations of SE7(5 / ( JV-4 ) ) x *75(1) x *7/2(1). (The grading of the 5 * 7 ( 5 / ( 7 V -
4)) corresponds, of course, to left-handed versus right-handed rather than fermions
versus scalars). Note that the ?7(1) that accompanies 5*7GG (5) X SU(N — 4) is a
mixture of the U H ( l ) and U s ( l ) namely, (JV - 5 ) Y = (JV - 9)YH + 10YS. In the notion
"stable" we mean that any composite particle should be immune against decaying into
less complicated composites. For example, denoting by a, 0,7 the SU!JC{N — 5) indices,
the composite A ^ B p C a - f D ^ is to be regarded unstable once its ( A a B a ) + ( C p y D ^ )
decay channel is noticed. A n additional rule that we introduce into the game in this
stage is the inclusion of only superpreons and not gauginos as elementary building
blocks. The possible contribution of the gauginos will be discussed in section 4.
Let us now check what gauge groups can be in accordance with our rules. 5*7(9)
has to be ruled out on the P . M . C grounds. This has to do with the fact that the
associated (x»v) preon happens to be S U j j c i ^ ) r e a h Models based on 5*7(11), on the
other hand, are much too complicated, leading to unrealistic solutions of the 't Hooft
anomaly equations. In other words the only possible gauge symmetry is 5*7(10).
- 32 -
3.3 Anomaly matching condition and Persistent mass condition for the
S U ( 1 0 ) models
Let us now construct the composite particles of the S U ( 1 0 ) model. For the sake
of simplicity, consider first three-preon HC-singlets composite of the 5x 10 x l O type.
Taking into account the handeness degree of freedom, we may have 51 x 10 £ xlOr, and
5 i ? x !0 L x 1 0 x - Under S U G G ( 5 ) x S U ( 6 ) they transform via:(5,l) and (1,_6) respectively.
The relative multiplicities of these composites are fixed by means of the P . M . C . Namely
it is the combination
(5, l ) L - (1,6)1 = ( 5 , 1 ) L + ( L 61)1 = ( 5 , 1 ) L + (I, 6)* ~ (5/6)2:. (3.4)
This is the fundamental representation of S U ( 5 / 6 ) . Notice, however, that the internal
structure of the 5x 10 x l O composite is still not completely fixed by specifying its
S U G G ( 5 ) x S U ( 6 ) x Uff(l) x £75(1) assignments. Taking advantage of the scalar pre-
ons the composite fermion can still be constructed from 1pxX> ^1PXi 1p<p<p• ^ is, of
course, the U R ( 1 ) which will differentiate between the above bound states. The various
assignments are the following:
R ( T P L X L X L ) = z + 2y R(<PL^LXL) = x + 2y + 2t R ( 1 / j L < p L < p L ) = x + 2y + 2t
R ^ R X L X L ) = ~ z + 2y R(vivl>LXL) = - * + 2y R ( i > R V i < P L ) = ~z + 2y + 2t (3-5)
Note that it is always possible to choose z=-x, such that the U R ( 1 ) commutes with
S U ( 5 / 6 ) . This choice makes it easier to verify that, as long as the R symmetry is kept un-
broken, the combination ( * I > L X L X L ) L + ( 4 > R X L X L ) R as well as ( i p L V L V ^ L + ^ L V U P ^ R
are consistent with the P . M . C whereas {4>R^LXL)L + {4>Ri>LXL)R 1S n ° t - Such a dif-
ficulty may possibly be overcome once gaugino couplings are taken into consideration.
A, of course, is trivial under chiral 5*7(5/6) x *75(1) but has a nonvanishing R charge
t ^ 0. The P . M . C partner of {(J>R(J>LXL) then seems to be ( ^ R X L ^ > L ^ L ) R 1 both having
R = x + 27/ + 2t.
For more sophisticated composites the discussion is quite similar, only that the
nonexotic requirement is not automatic. A HC-singlet composite of the 5x 5 x lO* type
for example, may apriori transform via A / 2 or 5/2 under 5*7(5/6). A / n , 5 / n denote
fully n-fold antisymmetric and fully n-fold symmetric representations respectively. But
only A /2 = (A /2 ,1) + (A /1 ,5 /1) + (1, 5/2) includes only fully antisymmetric represen-
tations under S U Q Q ( 5 ) which would not introduce flavor or color exotic particles. It
should be emphasized that unlike nonsupersymmetric composite models here we can
have also composite fermions built from an even number of preons. The simplest ex-
ample could have been of the 5x5* type such as (1p,<f>^). This last possibility drops
off because one can not construct from it a super-representation. However, four-preon
composites will play an important roll.
Altogether, we can now list all left-handed composites which have a poten-
tial physical relevance. We give their HC-preonic structure followed by their full
S U Q G ( 5 ) X S U ( Q ) x Uff(l) x *75(1) assignments, but ignoring momentarily the U R ( 1 )
־ 84 ־
complications:
(/!) 5 x 10 x 10 ^ A / 1 = [(5,1)_4 - (1,6)_5]1
( 1 2 ) 5 x 5 x 10* - A 1[0(I,21) + ־ (5,6)1 2(10,1)] = 2/
(/i) 5 x 10* x 10* x 10* A /1 = [(5,1)6 - (1,6)5]0
(/ 3) 5 x 5 x 5 x 1 0 ^ A / ־ (10, 6)_3 + (5, 21) _4 - (1, 56)_5]2 (3.6 2-(101,1)] = 3 )
( k ) 5 x 5 x 5 x 5 x 5 ^ A / ־ (!0,56)-3 2-(101,21) + !_(6 ,51) - 0(1,1)] = 5
+ ( 3 [ 5 _ ( 1 , 2 5 2 ) - 4_(5,126
( 1 0 (10 x 1 0 x 1 0 x 1 0 x 1 0 ^ A /0 = (1,1)-10,1
The associated multiplicities /j are to be fixed by the following t' Hooft anomaly
matching equations:
S U ( 5 / 6 ) 3 :11+1[- 5l2 + 14/3 + 55/ 5 = 5
S£7(5/6) 2 x U s ( l ) : h - 3/2 + 12/3 + 45/ 5 = 3 (3.7).
U s ( l f : k - h + h - 8/3 - 27/ ־ = 5 1
The corresponding three-parameter (n,/ ,m) solution is given by
10 = 2(n + 5/ + 6Sm) /! = 3(n + 3/ + 62m)
1[ = 2(n + 6/ + 72m) / 2 = (n - 1) + 7(/ + 11m) (3.8).
h = 1 /5 = m
The integer n has a direct physical meaning. It counts the total number of low-lying
quark and lepton generations. To see that let us concentrate on energy scales! A J J C
for which the effective theory contains only HC-singlets, and switch back the standard-
model interactions. Obviously, S£7c(3) x S U 1 ( 2 ) x £7(1) invariance can only protect the
- 35 -
complex piece of 1p£ from acquiring masses. The real piece is believed to gain masses
of order 3 0 ~ ( A J J C ) P A J J C • We therefore count the corresponding multiplicities of the
available 5J7GG(5) representations and find exactly "(10 + 5!) replications:
n = ( h - 7 h - 77/ 5) + 1 = ( - / ! + 6 / 2 2 1 / 3 - 1 3 2 / 5 ) + 6 (3.9)
Notice here the nontrivial spectator contributions, 1 and 6, respectively, incorrectely
establishing a generation structure, demonstrating once again the importance of overall
grand unification. In the nonsupersymmetric models, in the absence of scalar preons
and gauginos, one is immediately led to 1[ = Z 3 = 0, since only an odd number of
fermions can form a composite fermion. For that case the only possible solution is a
generationless n = 0 solution. SUSY does .allow for n ^ 0 multigenerational solutions,
but, so far, n has not yet been fixed. It would be nice, of course, if n would have been
determined by the UR(1)anomaly equations. Unfortunately this does not work. It is
shown below that the extended set of ' t Hooft anomaly equations does not admit any
integer solution, strongly suggesting that Z7R(1) is spontaneously broken. Let us take
again z = —x, the possible R charges associated with the fermionic members of the
5*7(5/6) super-multiplets are the following:
R ( l / ) = 5y + 2k0t, s = l , R ( A / 1 ) = x + 2y + 2k!t, 5 = 1
72(A1/׳) = x-3y + (2k'1 + l)t, 5 = 0, R ( A / 2 ) = 2x - y + 2k2t, 3 = 1 ,
R ( A / 3 ) = 3x + y + (2h + l)t, 5 = 2, R ( A / 5 ) = 5x + 2k5t, 5 = 3 (3.10).
This already takes into account, through the various values of the integers k{, also
- 36 -
the possibility of gaugino composites. The corresponding multiplicities /!(fc,•) obey
Sfc,- U ( k i ) = h , with li explicitly given in eq. (3.8) It is straightforwaxd to write down
the anomaly equations involving *7^(1). They are:
5*7(5/6) x U R ( l ) : Y h ( h ) R ( h ) + I ' l i k ' j R i k [ ) - 3 l 2 ( k 2 ) R ( k 2 ) + k
^ + 6 ' 3 ( * 3 ^ ) # ( ל 3 ) + l 5 k ( h ) R ( k 5 ) = 5 x k
U s ( l ) 2 x U R ( l ) : J 2 ^ 0 ) R ( k 0 ) - h i h W h ) + l 2 ( k 2 ) R ( k 2 ) + k
J2 - 4 h ( h ) R ( h ) - 9 k ( h ) R ( k 5 ) = ( ־ 9 a : + 2 y )
k 5
U S ( 1 ) 2 x U R ( 1 ) : k ( k o ) R 2 ( k 0 ) - h ( k ! ) R 2 ( h ) - I ' l i k ^ R 2 ( ^ ) + l 2 ( k 2 ) R 2 ( k 2 ) + k
Y - h ( k 3 ) R 2 ( h ) - k ( k 5 ) R 2 ( k 5 ) = 5 ( - x 3 + 2y 3 ) + 2 4 i 3
k (3.11)
The *7#(1)3 anomaly equation is of no immediate relevance since it receives contributions
form \ £ x x X i composites as well and, thus, can always be satisfied. The main
point now is that, subject to the constraint (3.8) eq.(3.11) do not seen to have integer
solutions. This means that chiral symmetry S U G G ( 5 ) x 5*7(6) x Ufj(l) x *75(1) x U R ( 1 )
must have been spontaneously broken, at least partially. On the other hand, we have
shown that S U ( 5 / 6 ) x *75(1) 't Hooft equations do have solutions with a generation
structure. Thus, one may conclude that it is apparently the U R ( 1 ) factor which gets
spontaneously broken. In principle this can be achieved v i a 3 6 < AjrA^ > ^ 0. The
Goldstone mechanism related to this symmetry breakdown is beyond the scope of our
model. For the sake of the present work, however, we would like to infer that the
- 37 -
physics responsible for fixing the total number n of generations must have its origin
beyond the 't Hooft and P . M conditions. Note also that we did not consider here
anomalie which are related to diagrams with two energy-momentum tensors and one
U ( l ) current. Consideration of this kind of anomaly implies that the £75(1) is broken
as well.
3.4 Further features of the S U ( l t i ) models
Which (n,m,l) solution of equation (3.7) is singled out to be the physical one? In
particular how many Q+L generations exist. At the present stage we can only speculate
that the preferred solution obeys some minimization principle. W i t h regard to this we
can think of two possibilities, each of which represents quite a different philosophy in
classifying the various generations:
(1) Generation proliferation via preon replication.-
It seems reasonable to look for a solution whose multiplicities /j are low as possible.
Indeed, if we demand \l{\ < q for some integer q, then the smallest q which allows for
a multi-generational structure is q = 4. Consequently, these are the four generations
solutions:
( n , l , m ) = (±4 , +1,0) (3.12)
which become our candidates. To be somewhat more explicit, the corresponding't Hooft
multiplicities are: IQ = —2, l\ = 3, l'x = —4,l 2 = —4,1% = — 1, /5 = 0 IQ = 2, /! = — 3 , 1 [ =
4 , 1 2 = 2, Z3 = 1,Z5 = 0, respectively The 10 and 10* S£7<3G(5)representations come
- 38 -
form — 4 A / 2 - A / 3 (or 2A /2+A /3) plus one spectator. Whereas A /2 contributes just one
single 10, it is the 5*7(6) global symmetry which causes the 10* in A / 3 to come with
a seven-fold replication. Hence, allowing for the usual parity-doublet formation, the
four 10 (or 10*) net survivals originate from the nontrivial 5*7(6) representation(with
the option of one of them to being the spectator). A very similar discussion holds for
the fundamental S U Q Q ( 5 ) representations. Altogether we observe that it is primarily
the original preon replication which triggers the ultimate generation proliferation. It is
interesting to notice that the resulting Q+L, are four preons composites.
(2) *7^(1) as a hor izon ta l symmetry . -
Another economical choice is the following :
(n,m,I) = (n,0,0) (3.13),
for which h = h = 0, and hence all composite representations 10 of 5*7<3G(5) appear
to be 5*7(6) singlets. This is a new situation, where the initial preon replication has
nothing to do with the final flavor proliferation. In a nosupersymmetric model this
would have meant a "superflous replication", but here it actually opens the door for
U R ( 1 ) to serve as a broken horizontal symmetry. The Q+L are viewed now as 5 x 5 x 10*
HC-singlets. If these generations are indeed internally distinguished, they may only be
of the form
1/>i>X, י / ׳ ^ H י X • (3-14)
Hence it is reasonable to expect *2 = 3, leading to n = *2 + 1 = 4. One of the four
low-lying generations is thus characteristically elementary (at least its 10 member).
- 39 -
The total number of generations primarily reflects the number of independent ways to
construct the proper type of a three-superpreon composite fermion.
3.5 Conclusions
In this chapter we showed how by supersymmetrizing the unified composite mod-
els we could overcome one of the most serious problems of of ordinary models. SUSY
changed the picture in the following aspects: (i) Introduction of scalar preons the paxt-
ners of the normal preons. (ii) Introduction of gauginos the partners of the gauge
particles, and (iii) Introduction of additional global symmetry U R ( 1 ) . Because of (i)
and (ii) the asymptotic freedom requirement became much more restrictive, and even-
tually only 5*7(10) were left as possible candidates. In a procedure similar to the one
presented in the previous chapter, we showed that apart from the *7R(1 )symmetry, there
are solutions to the combined system of ' t Hooft and P .M. conditions. Two "minimal"
models were constructed with 4 generations of Q+L. In the first, the Q+L are 4-preon
bound states and the horizontal symmetry is related to the original preon replication. In
the second one there is one elementary generation, and three 3-preon composites, with
the broken U R ( 1 ) playing the roll of the horizontal symmetry. Topics related to another
possible mechanism for constructing supersymmetric composite models are presented
in the next chapter.
- 40 -
Chapter 4
Superpotentials and Nambu-Goldstone Superfields
4.1 Introduction
Buchmuller, Peccei, and Yanagida ( B P Y ) 1 1 proposed the idea of using the super-
symmetric analog of the Nambu-Goldstone (NG) mechanism as a way of guaranteeing
the masslessness of quaxks and leptons. Although this idea was very attractive, there
were several questions whose exploration would facilitate model building. These ques-
tions include: (1) the relation of the set of broken generators of the original global
symmetry group to the set of (massless) Nambu-Goldstone f e r m i o n s , 3 7 - 4 0 (2) the rela-
tion of the set of massless fermions which saturate the anomaly-matching constraints to
the set of N G fermions, 3 6 (3) the derivation of assumed patterns of condensates formed
from composites of the underlying theory of the fundamental constituents,(4) the possi-
ble light mass scales which can be generated due to explicit masses of the constituents,
to various condensates, to explicit gauging of part of the unbroken global symmetry,
as well as to other possible mechanisms of generating mass, and (5) the mechanism of
breaking supersymmetry, which is necessary at least to split the supersymmetric paxt-
ners of the quaxks leptons and gauge bosons but which may serve other purposes as
well. Our purpose in this work was to study the first question, mainly in the context
־ 41 ־
of S U ( N ) and S U ( N ) j x S U ( N ) J J models, using super-potentials constructed with el-
ementary chiral superfields. In addition we clarified (i) the N G mechanism in theories
sharing shift symmetry CN on top of a global U ( N ) symmetry, (ii) the relation between
broken global U y ( l ) and U R ( 1 ) symmetries and a new U R ( 1 ) .
This chapter is arranged as follows: the general formalism of deriving the N G
superfields from the broken global symmetry generators is presented in the first section.
Section 2 is devoted to the study of several models with U ( N ) and section 3 with
S U ! ( N ) x S U I J ( N ) x U y ( l ) x U R ( 1 ) global symmetries. Then the relation between the
N G sf and the broken shift symmetry is discussed in section 3 and finally section 4
deals with new U R ( 1 ) that appears when both an original U R ( 1 ) and U y ( l ) are broken.
This work was done in colaboration with Prof. O.W Greenberg from the university
of Maryland and published in references 12,13.
4.2 NG superfields from broken global symmetry generators - general
analysis
The chiral superfield (sf) has the usual expansion:
$ i = A { + V 2 9 1 P i + 9 6 F i (4.1)
To carry out our analysis we construct a superpotential (sp) W ( $ i ) , which is invariant
under a global symmetry group G, such that the vaccum expectation value (vev) <
$ i >=< A{ > = V{ of the sf breaks G to a subgroup H , the symmetry group of the
- W -
vacuum. We require that the < > belong to the supersymmetric minimum of the
scalar potential,
y = £ l § 1 2 | ־ ' ™n.V = 0 (4.2)
which requires
Kugo, Ojima and Yanagida ( K O Y ) 4 0 showed that the N G supermultiplets due to spon-
taneous breaking of global symmetry in a SUSY theory are in one-to-one correspondence
with the broken generators in the coset space where G C ( H C ) is the complexification
of G ( H ) . Let us now show their derivation:
C = j d29d2W§i + {J d 2 e W ( $ i ) + h.c} (4.4)
The equations of motion axe
- l ^ , + ^ i ) = 0 , 4 , )
Invaxiance of W under the transformation
- = [ e i e T ] { $ j (4.7)
where T is the generator of Gc and e is a complex parameter, requires
S W ־ e ( ^ l ) [ T } { ^ = 0. (4.8)
- ־ 43
Multiplying the equation of motion (4.5) by [T]^$j gives
[ T ] i $ j D 2 $ i = 0. (4.9)
Introducing the shifted fields $j = <£j — V{, and keeping terms linear in $j gives
[ T } { V J D 2 ¥ = 0. (4.10)
If T is an unbroken generator, Tv = 0, and this equation is empty. Assume T is
a broken generator so that Tv ^ 0. Finally taking the adjoint of this last equation
gives the equation for the massless N G chiral supermultiplet associated with the broken
generator
D H N G [ T ] = 0 $ N G [ T ] = v j ( T } ) ) $ i . (4.11)
In terms of components of the supermultiplets the last equation leads to the equation
of motion of the N G bosons, N G quasi bosons and N G fermions as follows:
< A * 1 > [ T \ j . d 2 A j - d 2 A * { [ T ] { < A j > = 0 N G bosons
< A** > [ T ] { d 2 A j + d 2 A * i [ T } { < A j > = 0 quasi N G bosons (4.12)
< A * 1 > [ T ] { ^ d ^ j = 0 N G fermions
- U ־
4.3 S U ( N ) global symmetry breaking-
Next we come to the the exploration of the pattern of N G fermions in S U ( N )
models. Since the main motivation for the super N G mechanism was to produce massless
fermions (quarks and leptons) an emphasis is given to the N G fermions. Consider first
the case in which G = S U ( N ) and H = S U ( K ) x S U ( N - K ) x U ( l ) . Let $ be
a complex superfield in the adjoint representation of S U ( N ) . Write the most general
invariant renormalizable superpotential which depends on $ and a singlet superfield Cl
W = (gTr§2 - A 2 ) Q , + ^ T r $ 2 + ^ T r $ 3 - c T r $ , (4.13)
where c is a Lagrange multiplier. The role of Q is to eliminate the vacuum in which
the global symmetry is unbroken. The supersymmetric minima for this superpotential
correspond to all breakings for K = l to N - l . The solutions are
. N - K _ ^A I K V K N 9 ' y (Af — K ) N g י
, ( N - 2 K ) h A 2g < Q > = -m ± , '
^ K N ( N - K ) g
For these cases, there is complete doubling of the N G bosons, i . e. there is a complex
massless scalar for each broken generator (what B P Y call the maximal case) and there
is one N G fermion for each broken generator. We cannot choose renormaliazable super-
potential which will have its only minimum at a given value of K ; however, if we allow
ourselves to consider non-renormalizable effective sp. then we can do this. Choose
W = ( g T r $ 2 - A?)f2! + (g׳Tr$3 - A l ) f i 2 + ^ T r $ 2 + ^ T r $ 3 - c T r $ , (4.15)
- ־ 45
Then the simultaneous conditions from dW/dQ,1 and dW/dQ2 pick a single value of K,
which in general is not integral, in which case supersymmetry is broken. However tuning
of the parameters of the superpotential, which is not radiatively corrected in supersym-
metry, allows an integral K and a a supersymmetric minimum. In a similar way, we
can choose a non-renormalizable sp. and tune parameters to a single supersymmetric
minimum and a single global S U ( N ) symmetry breaking pattern in other examples we
discuss below.
Next we consider breaking of the global symmetry with a superpotential built from
the superfields \&, \&' transforming as (N) and ( N * ) respectively. We write down the
following superpotential
W = ( # # # m - - ׳ r f (4.16)
Since we can form three S U ( N ) invariants from \E׳ and \I׳' we can transform an arbitrary
into < ׳&\ > and < <3י >
< <HT >= (0,0, . . . V I ) , < >= (0, 0, . . . . y J B ' - ־ ^ ^ L ) , (4.17)
The extrimum of W occurs at
g < Vn > < > = n < ft > = — . (4.18) 9
The last equation only requires that A = B and B ' are not determined. In general,
the symmetry is broken to S U ( N — 2); however for any W of the form (4.16), (4.17)
allows a special solution if B B ' = A i.e. if ׳1י and \E׳' axe parallel. In this later case,
- 46 -
the symmetry is only broken to S U ( N — 1). In the analogues non-SUSY case with an
N and N * of S U ( N ) , the generic symmetry breaking would be to S U ( N — 2) as in the
SUSY case. In the non-SUSY case, breaking to S U ( N — 1) can occur only if \& and \I>
are parallel, and in addition, if the parameters of the potential satisfy a precise condition
(which will be modified by radiative corrections). The generic N G modes for this case
are: 2(2iV — 2) N G bosons, quasi N G bosons and N G fermions in the representations
(JV — 2), (JV — 2)* and 4 singlets of S U ( N — 2) occurring twice in each case. For the
case of parallel * and # / J (2JV — 1) N G bosons, quasi N G bosons and fermions occur
in the representation ( N — 1), ( N — 1)* and 1 of S U ( N — 1) in each case.
In a similar procedure one can achieve supersymetric analogs to the various patterns
of symmetry breaking caused by other representations of the group appearing in 4 4
4.4 Symmetry Breaking of S U { N ) j x S U ( N ) n x U v ( l ) x *7̂ (1)
Consider now G = S U ( N ) j x S U ( N ) J J with the self-conjugate set of chiral super-
fields A ~ ( N , N * ) and A' ~ ( N 1 , N ) . NOW the superpotential is
W = [gTr(A׳AT) - fx]Q, + mTr(A׳AT) (4.19)
Here also the purpose of the singlet superfield Q, is to force global symmetry breaking.
For this case there is a vacuum with H = S U ( N ) x *7(1), and in this vacuum
A = c x 1, A׳ = c׳ x 1 (4.20)
Here there is also a complete doubling of the real N G bosons, and there is a massless
fermion for each broken generator. A different possible vacuum is of the following form:
A = D i a g { u . . . . u , v...v) A׳ = D i a g ( u ' . . . u , v'...v') (4.21)
where there are Ku's and ( N — K)v's will break the global symmetry down to [ S U ( K ) x
S U ( N — K ) x U ( l ) ) v with the N G fermions transforming as
( d < 2 ~ D , I) + ((iV - K f - 1,1) + (1,1) + 2(( | T , (JV - K ) ) + (K, ( N - K Y ) (4.22)
Namely the complete coset-spase. It is worth mentioning that all the symmetry breaking
patterns discussed so far led to a real (not chiral) set of N G fermions since this is the
property of the coset space.
We come now to the question of getting less fermions than the total number of
broken generators, and in particular fermions which transform in a complex represen-
tation.We find that this scenario is possible both when we construct the superpotetial
from a complex set of superfields, and also when a real set is used. This last possibility
is in contradiction to what B P Y claim. First we consider the case of
G = S U ( N ) ! x S U ( N ) n x U ( l ) v x U ( 1 ) R and H = S U ( K ) V x S U ( N - K ) V x U ( l ) v x
U ( 1 ) J I , with A and Q the only chiral superfields: Here the superfields chosen do not
form a self-conjugate set.This example mimics, in a model with elementary superfields,
the situation in the model of Barbieri, et a l . 4 1 , where the hypercolor singlet \&ft bound
states have the transformation properties of our A. For N > 3, and with only A we
cannot write a renormalizable invariant superpotential. But Since the Lagrangian of
- 48 -
the composite particles can be effective, we consider a nonrenormalizable sp. Let
W ^ ^ ^ . ^ - . t . (4.23)
As usual, there are different kinds of supersymmetric minima. The simplest case is
where the v.e.v. of A is a multiple of the unit matrix. In this case, we find N 2 — 1 N G
bosons, i V 2 — 1 quasi N G bosons and i V 2 — 1 fermions, so we have the maximal case. As
we expect, we find JV 2 complex massless bosons from direct inspection of the potential
term in the Lagrangian. This result with a chiral set of massless fermions can also be
obtained in a renormalizable model. Use the superfields \& = ( N * , l ) and \&' = ( 1 , N )
in addition to A and A ' . Let
W = srtfA*' (4.24)
There is a symmetry-breaking vacuum in which the only condensate is A with the
same form as immediately above. Since the set of N G fermions depends only on the
condensates, the massless fermions stay the same as in the model just above.
The next simplest case is when the v.e.v. of A is diagonal with two different
eigenvalues as in equation (4.21). Now H = S U ( K ) V x S U ( N - K ) V x U ( l ) v x U ( 1 ) R ,
and there are N 2 + 2 N K - 2K2 broken generators. Here, we find K 2 + ( N — K ) 2 +
2 K ( N — K ) = N 2 and a real set massless fermions, which is the maximum number
we can have in terms of the chiral superfields A. (This is not a maximal set in the
terminology of B P Y ) This counting agrees with that of Barbieri, et a l . 4 1 , who find
36 massless fermions for the case N=6. If we classify these under the now-broken
־ 49 -
global subgroupS£7(707 x S U ( N - K ) ! x S U ( K ) H x S?7(JV - K ) N , we find the set
( K , 1; 1, f JV — K ) * ) + (1, (AT — .££*,1), which is a non-self-conjugate set, as is the case
of reference .
We have just seen that a complex set of chiral superfields leads to a complex
set of massless fermions. Now we give an example (to our knowledge, the first such
example) in which a self-conjugate set of chiral superfields leads to a complex set of
massless fermions. Thus this example gives a genuine example of a right-left symmetric
supersymmetric model in which spontaneous symmetry breaking leads to a chiral model.
We take G and H as just above, but change the matter fields to the following real set:
A ~ (JV,JV1), A׳ ~ ( N 1 , N ) , $ ~ (JV 2 - 1,1), and $ 1 ׳ ~ ( , J V 2 - 1 ) . We write the most
This situation is similar to the spontaneous breaking of S U ( 2 ) 1 x S U ( 2 ) R given by
Senjanovic and Mohapatra 4 5 :there is a vacuum in which the symmetry is unbroken,
and there is (in our case) a set of vacua in which the symmetry is broken, either to
5f7(JV) x U ( l ) or to S U ( K ) x S U ( N - K ) x 17(1). The vacua in which the symmetry
is broken have as condensates either < A > or < A ' >; all other vev's vanish. Another
example of this type occurs when $ and $ are replaced by * ~ (JV, 1), * ~ (JV^, 1), ft ~
(1_, JV), and ft' ~ (1, JV*). The same type of vacua occur as before, and again there is a
complex set of massless fermions. This type of mechanism of generating a chiral set of
general renormalizable invariant superpotential:
uft3
(4.25)
50 -
massless fermions from a I-II (or left-right) symmetric Lagrangian should be of use in
model building.
Another case to study is G as above and H = S U ( K ) j x S U ( N — K ) j x *7(1)/ x
S U { K ) ! I x S U ( N - K ) j l x *7(1)// x U ( l ) v x U ( 1 ) R , as well as variants in which
various of the I and II subgroups of H are omitted symmetrically.
4.5 NG superfields from broken translation invariance
In the derivation of the N G sf in section 2 it was tacitly assumed that the symmetry
group of W is homogeneous in <£j. Unlike the situation for spontaneous symmetry
breaking via the Goldstone or Higgs mechanism in a non-SUSY theories the sp. in
SUSY theories can be invariant under inhomogeneous symmetries. This possibility
arises because W is a function only of and not of <§i and singlet spectator fields can
be introduce in SUSY theories. The generalization of the K O Y theorem to the case of
translations is straightforward. Assume W is invariant under
where A and ftj are chiral superfields. (We explicitly allow the Oj- to be complex
numbers, which are a special case of a chiral sf.) Then invariance of W requires
d W 5 W = A S l ~ = 0 (4.27).
Multiplying the equation of motion by ft; gives
Q i D 2 ^ = 0 (4.28)
- 51 -
Any choice of vev. of $; including zero, breaks translation symmetry and leads to the
N G sf.
$ N G [ T ] = < ft* > (4.29)
Since the trivial case in which the sp. is independent of one or more superfields is
invariant under translations of the absent sf.s , we suggest that translation invariance
be the symmetry whose spontaneous breaking yields N G sf.s rather than G L ( N , c ) as
suggested i n 4 0
We illustrate the translation invariance on the example first considered i n 1 1 . We
start with the following model which is invariant under *7(JV) global symmetry
C = J d29d26(^fa^a + $ ! a + $£$2a + ftft) + d29W + h.c.} (4.30)
W = g V a $ l a Q + m < l > a $ 2 a + v(tt-v)2 (4.31)
Here $ ! י 2 and ft are sf.s that belong to the (JV*, - 1 ) , (JV, 1) and (1,0) of S*7(JV) x *7(1)
respectively. As was pointed out in 4 0 , the sp. of the model is invariant under G L ( N , c ) .
The authors of 4 0 associated the N G modes with the breaking of this symmetry, in
particular in the vacuum with
< # a >= 0, < $ l a >= mCa < $ 2 a > = -gvCa, < ft >= v (4.32)
where Ca is a constant N-vector which can be brought to the form C j = (o, o , C )
by a transformation in G = S U ( N ) x *7(1). In this vacuum, as discussed by K O Y
Gc = G L ( N , c) is broken to H c whose generators have zeros in the last column. Note
- 52 -
that there is also a vacuum, with all the vev.s equal to zero, except < ft >= v in which
G L ( N , c) is unbroken.
The superpotential of this model is also invariant under the following translation
symmetry
* l a $10 *־־ + m A a , $2a $2 •י־a ־ S׳ftA a, ־־• ft^ft (4.33)
Any vev. of the fields, including zero, breaks this translation invariance. Wi th the
choise < A a >= — Ca, < X >= v w e c a n transform the vacuum of eq. (4.32) to
one in which the vev.s except that of ft vanish. Now S U ( N ) x *7(1) and G L ( N , c ) axe
unbroken. The N G sf.s axe given by
=m*$la-g*v*$2a
Under complex translations
6*"° = ( |m| 2 + \ g v \ 2 ) A a + \g\2Q׳Aa (4.34)
where ft׳ = ft - v is the shifted ft sf. Thus R e l § N G \ g = 0 and I m a § N G \ E = 0 are
the N G bosons associated with the real and imaginary parts of the broken translation
generators, respectively. We now give an example to show that breaking G L ( N , c) and
of translation symmetry does not always lead to the same N G modes. Choose
r r + l - r r £= d29d29[^ + J2 $ a $ a + + + { / d 2 9 W + h.c}, (4.35)
r r
where now we suppress the summation associated with S U ( N ) , the a 1 s label differ-
ent 517(JV) x ?7(1) multiplets of sf.s:* ~ (_N*,0),§n ~ ( K , - l ) , Q a ~ (1,0) and the
9 a , f^ar m are numerical parameters. W is invariant under G L ( N , c) just as in the B P Y -
K O Y model. Now there is the larger translftion group C R N :
r
$ a - t $ a + mAa $ r + ! -> $ r + m ^ g ־ ! a n a A a a׳ = l , . . , r . (4.37)
a=l
Choosing < $ a >= 0, a = 1, ....,r + 1 break this symmetry and produces r N N G sf.s,
which can be placed in the rN-plets fo S U ( N ) x ?7(1):
= m * $ a - g*av*a$r+1 (4.38)
These r N plets are linearly independent, and can be made orthogonal, if desired.
4.6 Spontaneous breaking and new bom R־Invariance
In the previous sections breakdown of global invariance under both unitary trans-
formations and translations were considered. A n additional global symmetry can be
invoked in supersymmetric theories the R-Invariance. This symmetry which was intro-
duces by Fayet 3 5 can be used to restrict Lagrangian densities, to obtain spontaneous
breaking of both internal symmetry and supersymmetry, to classify generations in su-
persymmetric composite models (chapter 3) and for other purposes. Here our work on
the U R ( 1 ) was related to the question of identifying a modified U R ( 1 ) symmetry once
an original U R ( 1 ) is spontaneously broken. We will now illustrate the way in which
־ 54 -
spontaneous breaking of both a U ( l ) symmetry
$ _ E ־ « $ $ _ ׳ , E 4 . 3 9 ־ ( " * $ ׳ )
and R symmetry
$ e «'"»«$( X j 0 e ־ l ' a , 0 e
i a ) $ e ׳
1 ' m ' a $ 1 ) ׳ , « e ־ ' a , 9eia) (4.40)
leads to a modified R-symmetry which remains unbroken. This is analogous to the
breaking of both breaking of both an axial 174.(1) symmetry and R symmetry by instan-
tons with a modified R symmetry which remains unbroken in supersymmetric Q C D .
The fermionic component of Q, plays the role analogous to the gluino.
Let us take the Lagrangian
£ = JV0($$ + <£׳$׳) + { J d 2 W + h.c} W = ( g $ $ ' - A ) n , (4.41)
where $ and transform as given by equation (4.42) with m + m ' = 0 and
fl-4fi, ft e 2 1 a Q ( x , 9e~ia, 9eia) (4.42)
under *7,4(1) and R-symmetry, respectively. The minimization of the sp. (4.43) leads
to the conditions:
vv' = - , < $ > = v, > = v ' < >= 0 (4.43)
Since PF is invariant under eq. (4.40 ) with a complex, i.e., under GL( l , c ) . The R-
symmetry can also be complexified, and is also broken by the vev. The modified R
symmetry with the generator
E! = m Q - R (4.44)
- 55 -
where Q and R are the generators of the transformations (4.3 9),(4.42) respectively,
remains unbroken. This broken symmetry leads to the normalized massless N G sf.
$ N G = 1 ($ _ $ 4 . 4 5 $ _ |<) 1 = ף. ( ׳ ) $ = $ - ע ) v 2 v 2
The supermultiplets and ^ ( $ + $') g e t masses 2<ן 2 ע 2 and fi2׳u2, respectively.
4.7 Summary and conclusions
We summarize now the results of this chapter: (i) We proposed superpotentials
for several symmetry breakings of global SU(N) leading to a set of N G fermions which
cover the whole coset space, (ii) Despite the self-conjugate nature of the set of broken
generators when S U ( N ) r x S U ( N ) H groups break to S U V ( K ) x S U V ( N - K ) x ?7(1),
one can obtain complex sets of N G fermions either starting from a complex or from a
real set of matter superfields. In this case, of course, one has fewer N G fermions than
broken generators, (iii) We pointed out the existence of shift symmetries which axe
broken even in the presence of vanishing vev.s of the sf.s which participate in the sp.
We showed that certain massless sf.s in models presented 1 1 are the N G sf.s associated
with broken translation invariance and not with global unitary symmetry as suggested
by B P Y and K O Y . Finally, (iv) we showed the existence of a modified ?7^(1) invariance
in models where the original ?7(1) symmetry and ?7^(1) symmetry are spontaneously
broken.
- 56 -
Chapter 5
Cosmological Kaluza Klein models
5.1 Introduction
The idea that the origin of local gauge invariances may be related to some geometri-
cal symmetries of extra space dimensions was suggested long ago by Kaluza and K l e i n 1 4 .
Recently this idea is having its "Renaissance" mainly in the context of trying to unify
the standard-model interactions with gravity. The basic hypothesis of the Kaluza-Klein
scheme is that space-time has 4 + K dimensions with general covariance and that in
analogy to four-dimensions the curvature scalar is taken to be the Lagrangian. In addi-
tion the "ground-state"of the system is supposed to be partially compactified, namely
M 4 x B K , where M 4 denotes the four-dimensional space-time and B K is a compact
k-dimensional space. The size of B K must be sufficiently small to render it unobservable
at the currently available energies. Most of the proposed models are taking M 4 to be
a Minkovski space-time and B I S L to be time-independent compact space ( to account
for the observed constancy of the fine-structure constant). Moreover, in many cases the
M 4 x B ^ manifold is assumed but is not shown to be some solution of the relevant
Einstein equations. Our work examined these last two points. We were asking ourselves
whether one can have a Cosmological Kaluza-Klein model which admits a "ground
state5' metric that solves Einstein equations, which recasts the expanding nature of our
universe and which is at the same time consistent with the fine-structure constant.
This chapter is arranged as follows. In section 2 we present Einstein equations in
five dimensions and their solutions for an empty space-time with possible cosmologi-
cal constant. Section 3 is devoted to the analysis of the isometries of the geometries
that emerge from Einstein equations. In particular the scenario of cosmological com-
pactification with a homogeneous and isotropic ordinary space-time and "naturally"
compactified extra dimension is explained. Section 4 contains a description of the or-
dinary space time associated with the cosmological compactification model. It has an
open three-space with a Big-Bang origin and the constancy of its fine-structure constant
is correlated with the smallness of the cosmological constant. We generalize the result
also to more then one extra dimension. Section 5 is devoted to another possible Kaluza-
Klein cosmological model with a radius of compactification which tends asymptotically
to a constant. The eras of the early universe are discussed, including a transition from
a universe governed by an equation of state of the form P = p into a P = %p universe.
In the last section we conclude.
5.2 Einstein equations
We start by discussing an extended five-dimensional Robertston-Walker (RW)
metric 4 6 which is three-dimensional maximally symmetric, namely homogeneous and
- 58 -
isotropic in the usual sense and independent of the extra spatial coordinate.
ds2 = -dt2 + R \ t ) ^\df + a\t)dy\ (5.1) (1 + ^ k r ^ Y
with r 2 = Y A = \ R ( t ) and a(i) are time-dependent scale factors of the three-space
and extra dimensions respectively and k is the curvature parameter. Given the spatial
3+1 split, what we want to understand is why does the y coordinate compactify? In
other words, can the geometry give us, under reasonable physical assumptions, some
clues concerning the underlying topology? Moreover, we are after solutions with a ( t )
which axe in accordance with the constancy of the fine structure constant. Notice that
the line element (5.1) is the most general one subject to our symmetry constraints. The
scale functions R ( t ) and a(i) axe determined by the following five-dimensional Einstein
equations:
R 2 + k R a 1 , . n s
R 2 + k £ n R a a , x
+ A ) . (5.4)
A denotes the cosmological constant in the 5-dim. universe, p(i),P(t)and Q(t) axe the
normalized energy density and pressures defined via
T™ ~ diag(p, -P, -P, - P , - Q ) , (5.5)
with T m n is the energy-momentum tensor of the system. Let us first solve the above
equations in an empty five-dimensional space-time with a possible cosmological constant.
- 59 -
The solution is: 3k
R 2 ( t ) = £cosh2ut + r!sinh2ut + — A > 0,
R 2 ( t ) = -kt2+£t + r) A = 0, (5.6)
3k
R 2 ( t ) = £cos2ut + T]sin2tvt + — A < 0,
where u>2 = g |A| , and £ , ן 7 are fixed by the initial conditions. Combining together
equations (5.2) and (5.4), one immediately observes that
a ( t ) ~ ±R(t) (5.7)
5.3 Isometry Analysis and Cosmological Compactification
We now analyze 2 8 the local isometries of the above solutions. For the sake of
simplicity, we momentarily content ourselves with surfaces of constant 2,, dealing with
the residual 1+1 metric
ds2 = -dt2 + a2(t)dy2 (5.8)
The form invariance of equation (5.8) is respected by the infinitesimal Kil l ing transfer-
mations y —>y+ea(y, i) t —• t + eft(y,t) provided
d f t da d f t 2da
T t = a ^ a ^ = ^ j - a W = 0 ( ־ 5 9 )
The solution can be written as a = — ( % ) Y ( y ) + T ( t ) and ft = Y׳ (y) subject to the
constraint
Y״ + (aa - a 2 ) Y - a 2 f = 0 (5.10)
- 60 -
If da — a ^constant, Y ( y ) must be y independent. Consequently, only one Kil l ing
vector survives, corresponding to trivial translations in y. But if a ( t ) is special, in the
sense that da — a =constant, the situation is completely different. Equation (5.10) then
splits into
da-d2=p-> Y״+pY-q, f = ^ (5.11)
with p and q being constant parameters, so that the metric (5.8) becomes maximally
symmetric. The crucial observation is that the p > 0 case, with its corresponding scale
a ( t ) = a0cosh(ut 4־ (j>), p = u > 2 a 2 , (5.12)
exhibits Ki l l ing vectors periodic in y. This periodicity is related to the compactification
of the extra dimension.
At this stage, we find it necessary to assume that all infinitesimal Ki l l ing vectors
integrate to give a group of finite isometries. This is how local properties acquire global
significance, allowing us to identify the underlying manifold as the surface
- u \ + u\ + u j = (5.13)
living in a flat Minkovski space (ds2 = —du2 + du2 + d u 2 ) . The parametrization asso-
dated with the proper scale (5.12) is found to be
1 1 1 u ! = —sinhmt, uo = —coshutcosyjpy, U3 = —coshujtsin-y/py. (5.14)
Attached to each cosmic time t there is a circle of radius (-^=)aQC0shut, parametrized by
the angle yfpy the 50(2) subgroup of the initial 5(9(1,2). Notice that the definition of
the cosmic time is crucial. Given the same manifold (5.13), a different parametrization,
namely
u\ = —sinhut cosh^Jp y , u2 = — sinhut sinhy/p y , u3 = —coshut . (5.15) UJ UJ OJ
gives rise to ds2 = —dt'2 + a2sinh2u>t'2dy'2. Consequently we obtain d — a 2 = — p =
—OQU)2 < 0, so that the corresponding y space turns out to be infinite for any t . In a
group-theoretical language, the space defined by the new cosmic time is associated with
the noncompact 50(1,1) of our 50(1,2) .
We return now to the complete l+(3+l ) space-time. The set of Ki l l ing vectors
is more complicated now. Adding now the infinitesimal transformation X{ —* Xj + e'ji,
the expressions for a , ft and •ji are given in Appendix A . It is important to note that
the former conclusions persist. In particular the scale factor a ( t ) must again be of the
form (5.12) if we are looking for periodicity in y. A straightforward investigation of the
solutions (5.6) and (5.7) reveals that desirable a ( t ) can only live in the A > 0 case and
furthermore requires an algebraic relation between £ and 77, namely £ ף = (^)2 — 2 2 -
We proceed now to discuss the A > 0 solution. Later on, in section 4, the A = 0 will be
described.
5.4 The Cosmological Compactification model -Features of the ordinary
universe
For A > 0 the scales R ( t ) a . n d a ( t ) have the same functional form, only with ex-
- 62 -
changed amplitudes
R ( t ) = Acoshut + Bsinhwt (5.16)
a ( t ) = c(Bcoshut + A s i n h u i t ) , (5-17)
where u i 2 = jjjA. The arbitrary parameter can be as small as desired. Note that
p = ] : A c 2 ( B 2 - A 2 ) = -kc2 (5.18) 0
so that p and k are necessary of the opposite signs. If cosmological compactification is
achieved, that is, p > 0, the ordinary space must be open (k < 0). Yet the hyperbolic
structure of a(t) is not a sufficient condition for compactification. The additional re-
quirement B 2 — A 2 > 0 wil l be shown to be related with the Big-Bang singularity of
the ordinary space.
Next we study the effective four-dimensional world. The Robertson-Walker cos-
mology can be extracted from the five dimensional Kaluza-Klein model, 4 7 be ing aware
of the fact that now the four-dimensional action is
1 f ,4. r- A Z d u a d n a .
xV9i[R± + - + 0£> /-54־־ J L ~ r - \ (5-19)
where /c4 is the Plank length, R4 is the Riemann curvature scalar and 54 is the deter-
minant of the metric
g'JJ = a ( t ) d i a g [ - l , R 2 ( t ) ^ ]. (5.20) (1 + \kr*y
From the action (5.19), an effective energy-momentum tensor T^y can be read off given
by
p ־ / / ־ f ( £ ) 2 + A (5.21) 2 a a a
- 63 -
so that the effective four-dimensional equation of state is
״ ״. A A ״ ״ N P + - = p - - (5.23) a a
In order to interpret the four-dimensional metric (5.16) in the usual Robert son-Walker
way, i.e.,
ds2 = -dr2 + R l w ( T ) . / * D * ^ (5.24) (1 + %krzy
a redefinition of the proper time is needed:
r ^ v ^ o / [cosh(ut' + <t>)]*dt, . R 2
R W = a [ t ( r ) ] R 2 [ t ( r ) ] (5.25) 0׳״
In the limit ut <C 1, r = y / a Q t and
R l w ( r ) = c [ B A 2 + ^ A ( A 2 + 2 B 2 ) + (^B(B2 + 7-A2)) (5.26) v a0 a 0 ^
As we see the function R R W grows like \/r.
We want now to examine how does the traditional Kaluza-Klein scheme fit into our
discussion. Its associated vacuum is characterized by a static radius of compactification
to account for the observed constancy of the "fine-structure constant". But a ( t ) = ao,
that is p = 0, would not support the kind of compactification we want. After all, unlike
ds2 = —dt2 + 0QC0sh2u)tdy2, only two out of the three Kil l ing vectors associated with
ds2 = —dt2 + a^dy2 can be globally respected once y is forced to compactify (the boost
is lost, of course), exhibiting no built-in periodicity. Yet the gauge-coupling constancy
־ 64 -
is a most desirable feature. For our cosmological Kaluza-Klein version we find
R a 1 A a! A . — = - A -> - ~ — 5.27 i ta 6 a H
where H is the Hubble constant, meaning that the age of the universe must be small on
the scale, hence coshut ~ 1. The observed constancy of a is thus correlated with
the fact that A is practically zero.
If we attempt to extend the idea to still higher dimensions, our conclusions are likely
to survive. Although Einstein equations axe more complicated for n > 1 (n denotes the
total number of extra dimensions), a maximally symmetric solution that generalizes eq.
(5.13) always exists and is furthermore n independent save for
2 2A (n + 3)(n + 2)־
The empty space-time limit is therefore straightforwardly traced.
(5.28)
5.5 A model with asymptotically constant radius of compactification
Extended Robertson-Walker universes with time dependent scale factors R ( t ) and
a(i) can be in accordance with the constancy of the "fine-const ant", also in a different
way than that described in the last section. This is the situation when the scale factor
a ( t ) tends asymptoticaly to a constant 1 6 , namely a(t)(_ > 0 0 —• a(oo) = constant. Such
an asymptotic constancy of a ( t ) is achievable and it actually requires
A = 0 k < 0 (5.29)
- 65 -
In other words, the currently observed static strength of electromagnetic interaction
seems to have the same theoretical origin as the experimental smallness of A and the
openness of the three-space.
Let us now establish the four-dimensional perspective. A four-dimensional observer
who is not directly awaxe of the existence of an extra dimension, perhaps because
a(oo) <C 1, naturally interprets the scale factor R ( t ) as if it is governed by some effective
energy-momentum tensor. Following the determination of the effective four-dimensional
metric (5.20), we get now:
/ ־ ־ ' + g ) (5.30) y ° ° a(oo) y / ( t * + 2 e t ) V >
9*J a ( o o ) ( l + * r 2 ) 2 " ( l + |r2)2 V ^ >
For the sake of simplicity, we have chosen the origin of time at the Big Bang singularity,
that is setting 0 = ף and with e defined by e = — ^ > 0. A straight forward calculation
of the effective density and pressure reveals that
eff = p e f f = (5.32) V ( * 2 + 2 e t ) a ( t + e) 3 ^ ^
Following the details of its derivation, the above equation of state reflects the fact that
the length scale of the extra dimension is exclusively a time-dependent quantity. In a
Schwarzschild-type five-dimensional geometry 4 8the static spherically symmetric analog
of the present case, one arrives at P = 7$p (actually T r P = p) just because the radius
of compactification is a function of r. At any rate, the equation of state P = p is known
- 66 -
to be the maximal one consistent with causality. It means that the sound velocity in a
fluid obeying such an equation of state happens to match the speed of light. It is indeed
interesting to notice that precisely this feature has led Zeldovich 4 9to consider P = p
governed evolving universe.
At this stage, if we want to recover the callsical Robertson-Walker form for the
effective four-dimensional metric, a special general coordinate transformation is in order.
The new cosmic time r must be defined to satisfy the relation
0 0 V J • Jo V ( z * + 2 e z — = = = = = d z (5.33)
Consequently, the corresponding Roberston-Walker scale factor R ? ( t ) is simply given
by
R \ r ) = 2 ^ . R \ t { r ) ) . (5.34) a(oo )
The integral (5.33) contains eliptic pieces of the first and second kind. We therefore
split our discussion, dealing first of all with the very eaxly universe and then switch on
to the asymptotic time region.
The very early universe -
For t,r —>• 0 the integration variable z is very small on the e scale, so that the
integrant diverges a la r(t, e) is given by
3 V v 2
and this implies that
^=4A7(S3 + ••• (5-35)
R \ r ) = - k d U r Y + ... p • / ־ = + * / ( ־ * ) V * + - (5.36) 3 7 ד ^ R - ^ ־ ־ 6 ( r )
- 67 -
with — k > 0 reflecting the open character of the ordinary three-space. We now claim
that the evolution of the effective early universe, even when taking into account the effect
of radiation, is still governed by the extreme equation of state P = p. The argument is
very simple :
P = p R ~ A + P = \p -> R ~ r\ + .... (5.37)
so that the r3 behavior wins for r —> 0. Thus, the existence of a dynamical extra
dimension seems to require a pre-"radiation dominated era". Starting from R ~ 73־
p ~ -gg•, we anticipate a transition t 0 j R ~ ־ 7 2 <-*.p~ J ? . ) a s ׳ r developes, to be discussed
soon. At any rate, one must notice that
a ( T ) ־ + T ־ J , (5.38)
meaning that the extra space did not have a Big Bang origin. On the contrary, it used
to be perfectly visible at the time when the usual space was not.
- 68 -
T h e r ad i a t i on domina ted era-
For t,r e, the story is completely different. To decode the associated r ( i )
behavior, it is crucial to observe that the integrant in eq (5.33) differs from unity by
°(jr) terms, as the integration variable z goes to infinity. One can therefore write
Jo {/(z' + 2 « ) J, + 2ez) ' J o
2
and hence the asymptotic expansion T = t + el — | where el is the finite value ... + ן
of the difinite integral in eq.(5.39). The reciprocal relation is then substituted into eq.
(5.34) with the final result being
R 2 ( r - ־־ ( k r 2 [ l + (I — l ) - ] 2 + O ( - ) . (5.40) T T
R ( r ) becomes a linear function of r , corresponding to an inertially expanding universe.
This happens, of course, when a(r) asymptotically reaches its constant value a(oo). One
can now qualitatively argue that, in the presence of radiation, the above is in fact a
negligible effect at the T > £ regime. Recall that P = g/j implies R 2 ( r ) = ( a r + /3)2 +7
with p ~ whereas P = p leads to R 2 ( T ) ~ ( a r + / 3 ) 2 + 0 ( ^ ) for long enough r .
The inertial case with P = p = 0 is associated with R 2 ( T ) = (a r + ft)2. Here a , ft, 7
are some constant parameters. Now, suppose the effective energy-momentum tensor
contains both P = p as well as P = j/> pieces; then, the P = p effect which is only
O(^) , dies faster than the P = effect which is 0(1). The asymptotic deviation from
R = ar + ft are, therefore, dominated by the P = j p contamination. A l l this should be
contrasted with the r —> 0 behavior discussed in eq. (5.37), indicating a transition from
- 69 -
a scalar-dominated era to the radiation-dominated era. The theory provides a natural
time-scale for this transition to take place, namely r ~ e. As mentioned earlier, the
extreme smallness of e is correlated with the asymptotic constancy of the "fine-structure
constant".
5.6 Conclusions
We presented two cosmological Kaluza-Klein models. They are derived as solutions
of Einstein equations in five-dimensions. In the first case we found an isotropic and
homogeneous metric in the usual sense which exhibits a cosmological compactification
of the extra dimension. We showed that for this case a positive cosmological constant
appears as a necessary ingredient, while our three-space must be open and develops
from a Big Bang singularity. The constancy of the "fine-structure" constant was shown
to be correlated with the smallness of the cosmological constant. A straight-forward
generalization to universes with more than one extra dimension was also presented.
In the second model we described a solution with a radius of compactification
which tends asymptotically to a (very tiny) constant. The cosmological implications
of the constancy were shown to be a vanishing cosmological constant and an open
ordinary three-space. The existence of dynamical extra dimension has been proven to
be mandatory for the evolution of the very early universe. A pre "radiation-dominated
era"was a characteristic feature with the evolution governed by Zeldovich equation of
state P = p . The transition into the "radiation-dominated" era was also clarified.
- 70 -
A P P E N D I X A
Denoting the Kill ing transformations by
y ->y + ea(y, t) t - > t + e/3(y, t) 1 j - * x; + 67; ( A l )
we derive for the metric given by
ds2 = -dt2 + R H t ) r / X \ d * * + a\t) dy\ (A2) (1 + % k r * y
the following expressions for a, ,(3 and 7:
| + 1)ג<ק/ץ מ 2 ) 4 4
(A3)
]}\k~2־—71 = ^ A i x i + C T2 ־ 1 ( ׳ ־ 2 ' a ]«w( > ^y) + [6,•־ 1) + *־ | r 2 ) T y ] 5 i n ( v ׳ p y ) } (A4) ( l + | ״ 7 i ) 4 4
־ = .7 )]}^J ך 1 + | r V \ ־ . ( A 3 x J ) x i ~ k T t X i ] c o s ( , / p y )
+ [(1 + K 2 ) B i - ^ ( B j x ^ x i - k T y X i ) s i n { J p y ) } + eijkWxk (Ab)
+ (1 - k-r2)Ti + \ ( T ^ ) X i
Where A j , Tt, Ty, c, B{, ,3?,-, and T! are the set of 15 parameters associated
with the isometry of the maximally symmetric five-dimensional space-time specified
by ( A l ) , (5.19) and (5.20). In four dimensional flat space-time Tj,Tt,3?j, and Aj are
associated with 3 space translations, time translation, 3 space rotations and space-time
boosts respectively.
- 71 -
Chapter 6
Non-Abelian Bosonization of Massless Multiflavor QCD 2
6.1 Introduction
One of the outstanding problems in elementary particle physics is the derivation
of hadron spectroscopy from QCD-the underlying theory. Although baryon properties
have been calculated in the Skyrme model, producing fair agreement with experimental
da ta 5 0 , the bridge between this phenomenological model and the physics of basic con-
stituents is still missing. However, in 1+1 dimensions, the low energy effective action
can be derived directly from QCD2 using non-abelian bosonization. Bosonization 1 7 ,
namely finding a bosonic field theory which is equivalent to a given local fermionic
field theory, did not exhibit non-abelian symmetry explicitly until the discovery of non-
abelian bosonization by W i t t e n 1 8 . A bosonized theory is believed to be more adequate
for computations in the strong coupling limit of QCD2 since this limit is equivalent to
the weak coupling one for the bosons. Several authors in the past applied the "old"
bosonization procedures to investigate single 1 9 and m u l t i f l a v o r 2 0 - 2 1 QCD2- Difficulties
were encountered in the latter case 2 1 . More recently the "new" bosonization was used
in flavor space to find the spectrum of the two dimensional baryons 2 2 for the case of two
flavors. Here we suggest a new form of the bosonized action of multiflavor QCD2, derive
- 72 -
its low energy effective action in the strong coupling limit and deduce the semiclassical
masses and charges of the low-lying baryonic states.
This chapter is devoted to the non-abelian bosonization of colored and flavored
Dirac fermions and massless multiflavor QCD2. In the next chapter we discuss the
bosonization rules for the mass-bilinears which axe relevant to massive QCD2- In the
third chapter of this topic we present the semiclassical spectrum of the low-lying baryons.
A great part of the content of chapters 6-8 was published in references 23-24.
The sections in this chapter are the following. Section 2 briefly summarize Witten's
non-abelian bosonization. Then, in section 3, we elaborate on the bosonization of colored
and flavored Dirac fermions. The next section is devoted to the gauging of the W Z W
action on S U ( N Q ) group manifold and the last section contains a summary and some
conclusions.
6.2 Witten's Non-abelian bosonization
The non-abelian bosonization that was introduced by W i t t e n 1 8 is a set of rules
which determine the bosonic action and the bosonic currents which are equivalent to
those of a theory of free fermions invariant under non-abelian symmetry. (In the next
chapter a bosonic operator equivalent to the fermionic mass bilinear will be introduced
as well.) The equivalence between the bosonic and fermionic theories emerges from the
fact that in both theories the commutators of the various currents have the same current
algebra and the energy-momentum tensor is the same, when expressed in terms of the
- 73 -
currents.
The theory of N free Majorana fermions is governed by the action
S * = \ J d2x{^.kd+<H.k + V + k d - V + k ) (6.1)
where \&+ are left and right Weyl-Majorana spinor fields, d± = ^ ( ^ 0 ± 9 ! ) and
k = 1,....JV. The corresponding bosonic action is the so called Wess-Zumino-Witten
(WZW) action:
S[u] = 1 5 M
S[u] = - ^ d / ־ 2 x T r ( d u u d " u - 1 ) + - J - / d 3 y e i j k T r ( u - 1 d i u ) ( u - 1 d j u ) ( u - 1 d k u ) 87r J 127r JB
(6.2)
where u is a bosonic field expressed as a matrix C O ( N ) . The second term, the Wess
Zumino (WZ)term, 5 1 is defined on the ball B whose boundary S is taken to be the
euclidian space-time. Now, since n 2 [ 0 ( N ) } = 0, a mapping u from S into the O ( N )
manifold can be extended to a mapping of the solid ball B into O ( N ) . The W Z term
is well-defined only modulo a constant. It was normalized so that if u is a matrix
in the fundamental representation of O ( N ) the W Z W term is well defined modulo
W Z —יי W Z + 27r. The source of the ambiguity is that 7־r[0(JV)] ~ Z namely there are
topologically inequivalent ways to extend u into a mapping from B into O ( N ) .
The bosonic action is invariant under the chiral 0 1 ( N ) x O R ( N ) symmetries just
as the fermionic action. The associated transformations are:
u - > u = Au u-*u = u B A , B C O ( N ) . (6.3)
- U׳ -
In fact, the invariance is even for transformation with A ( x + ) and B ( x - ) , leading to the
Kac־Moody algebra. The discussion of this algebra and the associated currents will be
postponed to the next section.
The action (6.2) is comformaly invariant ELS well. This property was proven by
W i t t e n 1 8 who showed that if one generalize (6.2) by taking a coupling קדץ- as a coefficient
of the first term and 5 ^ of the W Z term, the ft function associated with A is given by
d\* _ ( N - 2)A 2 \*k 2
= dlnA ~ 4 T ־ [ 1 ־ y ] ' ( ־ 6 4 )
namely eq. (6.2) is at a fixed point A .and hence exhibits conformal invariance ?ך = 2
Another way to prove the equivalence of the theories now of N free Dirac fermions
and the W Z W theory on U ( N ) group manifold is by showing that the generating func-
tional of the current Green functions of the two theories are the same. For the fermions
we have
e . w , ( A M ) = J ( ^ + ^ _ ) 6 ' 7 ^ * * 6 . 5 (ע* (
where here D» = d M + i A ^ , A ^ = A A ( \ T A ) + A ^ x 1 and ( ^ T A ) C S U ( N ) . W # ( A M )
was calculated by Polyakov and Weigmann 5 2 in a regularization scheme 5 3 which pre-
serve the global chiral S U ( N ) symmetry and the local U ( l ) diagonal symmetry leading
to
W^) = - S [ A ) - S[B] - ^ J d 2 x A ^ (6.6)
where A , B c U ( N ) , »A_ = B ' ^ d - B i A + = A ^ d + A , A , and a are the S U ( N ) and
£7(1) parts of A respectively namely: A = A e - J V ׳ ^ " a and similarly for B .
- 75 -
In the bosonic theory we calculate
(6.7)
This functional integrals can be performed exactly 5 4 leading to
WB(AA) = - S [ A ] - S[B] WB(A^) = 1
( P x A ^ (6.8) 47riV
Thus the bosonic current Green functions are identical to those of the fermionic theory
regulated in the way mentioned above.
6.3 Non-Abelian bosonization of Dirac fermions with color and flavor
In his pioneering work on non-abelian bosonization Witten 1 8 proposed a prescrip-
tion also for bosonizing Majorana fermions which carry both Np , ,flavors" as well as N Q
"colors", namely transform under the group [ 0 ( N F ) x 0 ( N C ) ] L x [ O ( N p ) x 0 ( N c ) ] R .
The action for free fermions is
where now a=l , . . . . iVc, i = l , Np are the color and flavor indices respectively. The
equivalent bosonic action is
The bosonic fields g and h take their values in O ( N p ) and O ( N Q ) respectively and S[u]
is the W Z W action given in (6.2). The bosonization dictionary for the currents was
(6.9)
S[g,h] = lNeS[g] + ±NFS[h] (6.10)
- 76 -
shown to be:
J+ij = : < S > + a i - $ + a j := — { g 1 d + g ) ^ J - i j =: tf_oi¥_aj• := • ^ ( 9 9 - 9 1 ) i j (6.11)
•7+a6 =: *+ai*+W := ^ ( h ^ d + h U J_ab =: * _ a ; t f _ W := ±-(hd-h-l)ab
(6.12)
where : : stands for normal ordering with respect to fermion creation and anihilation
operators. In terms of the complex coordinates z = + i £ 2 , z = £1 — if2 (where f!
and £2 3 X 6 complex coordinates spanning C 2 ,and the Euclidian plane and Minkowski
space-time can be obtained as appropriate real sections), one can express the currents
as
J ( z ) i j = t r J - i j = ^ { g d z g 1 ) i j J{z)ij = 7 r J + i j = 1 ( 6 . 1 3 ג ע)^• ( 3 5 ע )
and similarly for the colored currents.
In a complete analogy the theory of N p x N c Dirac fermions can be expressed in
terms of the bosonic fields g , h , e V F c now in S U ( N p ) , S U ( N c ) and U ( l ) group
manifolds respectively . 5 4 The corresponding action is now:
S [ g , h] = N c S [ g ] + N F S [ h ] + cPxd^d^ (6.14)
This action is derived simply by substituting ghe V c F instead of u in (6.2).
The bosonization rules for the color and flavor currents axe obtained from (6.11)
and (6.12) by replacing the Weyl-Majorana spinors with Weyl ones, and in addition we
- דד -
have for the U ( l ) current
\ " (6.15)
The equivalence of the bosonic and fermionic Hilbert spaces was proved by showing
that the two theories have the same current algebra (Kac-Moody algebra) and that the
energy-momentum tensor can be constructed from the currents in a Sugawara form.The
Kac-Moody algebras are given by:
[Jt J&] = H A B C j Z + m + ^ n 6 A B 8 n + m : 0 (6.16)
where JA = T r ( T A J ) , TA are the matrices of S U ( N Q ) , k = N p for the colored currents
and J(z) is expanded in a Laurent series as J(z) — J n ; similar expression will
apply for the flavor currents with T1 the matrices of S U ( N p ) , and the central charge
k = N c instead of N p . The commutation relation forJ(z) will have the same form.
Generalizing the case of S U ( N ) x J7 ( l ) 5 7 the Sugawara fo rm 5 8 of the energy mo-
mentum tensor of the W Z W action for our case is given by:
C A - b I (6.17)
+ * j < l ) ( z ) j ( l ) ( z ) :
where the dots denote normal ordering of JA, J1, and J^ 1) respectively, in the sense that
quantities with positive suffix are moved to the right of quantities with negative suffix,
and the KS are constants yet to be determined. In terms of the Kac-Moody generators
- 78 -
this can be written as oo oo
T - V • TA 7 A • 4- 1 V • T1 I 1 1•"n ~ c ) K ^ / j • •JmJn—m • ' ^.^ / • •JmJn—m •
(6.18) m=—oo m=—oo
+ — V • J ( 1 ) J ( 1 ) • m=—oo
Now, by applying the last expression on any primary field <j>\ we can get a set of infinitely
many "null vectors"of the form
0
m=n ^ m=n m=n ׳־י(6.19)
for any n < 0. Since each of these vectors must certainly be a primary field, L m x n =
JmXn — JmXn = JmXn = O for m > 0 holds. This leads to expressions for the various
KS, for the central charge c of the Virasoro Algebra and for the dimensions of the primary
fields A ; = A/+ + A / _ , in terms of N Q , N F and the group properties of the primary
fields:
K C = K F = i ( i V c + N F ) , K = N F N C (6.20)
N C ( N * - 1) , N F ( N י ן (1 - *c = דזר7־— , r ד7־ד־ + ץ— . T N + 1 = N F N Q - (6.21)
C 2 F C 2 C c2(l)
A'־ = W F + N C ) + (iv^ + iv c ) + i V c i v ^ ( 6 ' 2 2 )
where c 2 c is the eigenvalue of the S U ( N c ) second Casimir operator in the representation
of the primary field <j>1 namely, TATA = c 2 ^ I , and similarly for the flavor group.
Goddard et a l 5 9 showed that a necessary and sufficient condition for such a con-
struction of the fermionic T(2) in a theory with a symmetry group G is the existence
- 79 -
of a larger group G C G' such that G' j G is a symmetric space with the fermions trans-
forming under G just as the the tangent space to G׳/G does. Based on this theorem
they found all the fermionic theories for which an equivalent W Z W bosonic action can
be constructed. The cases stated above fit in this category. Note in passing that this
does not hold for cases where the symmetry group includes more non abelian group
factors, like for example S U ( N A ) x S U ( N F ) x S U ( N C ) x U ( l ) .
The prescription described above (6.14), for the bosonic action that is equivalent
to that of colored and flavored Dirac fermions, is by no means unique. In fact it
wil l be shown that this prescription will turn out to be inconvenient once mass terms
are introduced. Another scheme based on the W Z W theory on U ( N F N c ) will be
recommended.
6.4 Gauging the WZW action
So far we have only derived the bosonic action equivalent to that of colored and
flavored Dirac (or Majorana) fermions. Naturally, the next stage toward the bosonized
version of QCD2 is gauging the S U v ( N c ) subgroup of S U i ( N c ) x S U J I ( N C ) , where -
V stands for a vector symmetry. This can be done using a trial and error method 6 0 or
via covariantizing the current. We present here the two methods which are applicable
also for the U ( N F N c ) bosonization scheme.
( i )Tr ia l and er ror N o e t h e r me thod
The W Z W action on S U ( N Q ) group manifold is, as stated above, invariant under the
- SO -
global vector transformation h —• U h U 1 where U C S U ( N c ) • Now we want to vary
h with respect to the associated local infinitesimal transformation U = 1 + i e ( x ) =
l + 1 T A e A ( x )
8€h = i[e, h] Seh'1 = i[e, h ~ 1 } (6.23)
The variation of the action = S[h] under such a transformation is obviously
given by:
8 € S W [ h ] = J d2xTr(dfieJ») (6.24)
where the Neother vector current is given by:
e^d^h - h - {^h + hd״h^d]}.± = 7״ d ' t f ] } (6.25)
We introduce now the first correction term given by
SW = J d2xTr(A״J^) 6€S{1)[h} = J d 2 x T r [ d t i e ( J t l + (6.26)
The second part of (6.26) is the variation of S^1) which is derived by using the in-
finitesimal variation of the gauge field SA^ = —D^e = — (3Me -I- i[A^,e]). J>fi is given
by:
= ^ . { [ h ^ A ^ h + h A ^ } - e ^ A v h - u A u h ) } } (6.27)
The second iteration will be by adding where now J׳^ is replacing J ^ .
$(2) = Jd2xTr(Af,J,(i) 6€sW[h} = J d2xTr(dneJ׳i*). (6.28)
It is therefore obvious that
<5£[S(0) - S ( 1 ) + 5 ( 2 ) ] = 0 (6.29)
- 81 -
Hence S [ h , A^] = [S^ - + 5 ( 2^] is given by
5 k i ״ ] = ^ ] ־ d 2 x T r ( D a h D v t f ) + - i - / d 3 y e i j k T r ( t f d i u ) ( r f d j U ) ( r f d k u ) S7r J 127r J B
- — / d 2 xe A l v Tr[1A A t ( /1 t a1/ ׳ h - ׳ W r f + t u ^ ^ ) ] 47r j
(6.30)
which can also be written in light cone coordinates
S [ u , A + , A - } = S[u} + ^- j d 2 x T r ( A + h d - t f + A - t f d + h ) , " * J (6-31)
- — / d 2 x T r ( A + h A - h * - A - A + ) •
( i i )Gauging v i a covar ian t iza t ion of the N e o t h e r current
In D space-time dimensions the current (which contains (D- l ) derivatives) is gauged by
replacing the ordinary derivatives with covariant derivatives and by adding terms which
contain products of h with F^y and covariant derivatives D ^ h . In two dimensions,
however, there is no room for such terms in the gauge covariant current (since these
terms involve e / ׳ 1 1 l l D in D dimensions contracted with F ^ s and D ^ s , and in two
dimensions they cannot be constructed). Therefore the covariantized current is given
by:
J ^ h , A p ) = ^ { [ h ^ D ^ h + h D p h l ] - e l l v [ h ^ D u h - h D u t f ) } (6.32)
Knowing the current we deduce the action via J^ = getting straightforwardly eq.
(6.30).
Finally, we combine the gauged W Z W action on the color group manifold, the
W Z W of the flavor group manifold and the action term for the gauge fields to get the
- 82 -
bosonic form of the action of massless QCD2- The well known fermionic form of the
action is
. SF[*,A״] = J d2x{-±TrF^F^-*"[(id + A W i l ) (6-33)
where
Ffju/ — O^Ay dj/Ap -f־ i [A^, A1/\
and the bosonized one is
S[g, h , A + , A - } = N c S [ g ] + N F S [ h ]
־ 2 i f / d 2 x T r F ^ F " u • (6.34)
6.5 Summary and conclusions-
Two steps in the path from the non-abelian bosonization of N free Dirac fermions
toward the bosonized massless QCD2, were taken in this chapter. The first one was
related to the alteration of the bosonic theory once the fermions are taken to carry
both flavor and color charges. We elaborated on the equivalence by discussing the Kac-
Moody algebra of the currents, the Sugawara construction of the energy-momentum
tensor from the currents and the Virasoro algebra of the latter. The second step was
the gauging of the color degrees of freedom both by a trial and error method and by
covariantization of the vector current and deducing from it the action. The next chapt
is devoted to massive QCD2
- 84 -
Chapter 7
Non-Abelian Bosonization of Massive QCD2
7.1 Introduction
In the soliton approach to baryon formation in two dimensions it is necessary to
have a mass term. Therefore writing down a mass in terms of bosonic variables is
essential in the bosonization of QCD2- We discuss in this chapter the methods used in
the past, we point out their problems, we then use another prescription and deduce the
low energy effective action of QCD2-
The chapter is arranged as follows. The bosonization rule for the mass bilinear of
Dirac fermions is discussed in the next section. In section 3 we demonstrate the equality
between the n-point functions of fermionic mass-bilinears and those of the corresponding
bosonized operators. Then section 4 presents the "old' 5 rule for the bosonization in the
case of flavored colored fermions. In section 5 we use the new bosonization scheme. In
section 6 we write down the action for massive QCD2 using the scheme mentioned above
and deduce the low energy effective action. We conclude and summarize in section 7.
- 85 -
7.2 The bosonization of a mass bilinear of Dirac fermions
A further bosonization rule has to be invoked for the mass bilinear. For a theory
with a U(N) symmetry group the rule is the following:
V^-j = i i N ^ e - ^ * (7.1)
where Np denotes normal ordering at mass scale /i . The fermion mass term rriq^f1^{
is therefore
m ' 2 N ^ J d2xTr(g+gi) (7.2)
where m'2 = m q K n , mq is the quark mass, and K is a dimensionless constant indepen-
dent of N . It was shown that the above bosonic operator transforms correctly under
the U ( N ) 1 x U ( N ) R chiral transformations. On top of that it has the correct total
dimension
A = A f f + A ^ = ( ^ z l + i ) = l (7.3)
where A g and A ^ are the dimensions associated with the SU(N) and U ( l ) group factors
respectively. Moreover it was explicitly shown that the four point function
G ( z i , z i ) = < g(z1,z1)g~1(z2,z2)g~1(z3,zs)g(z4:,z4) > (7.4)
is given by 6 1 :
G { z i , Z i ) = [(z! - z 4)(z z ־ 2*)(4* - 1»)(3» ־ 2 3 ) ] ~ A g G ( x , x ) (7.5)
G ( x , x ) is the following function of the harmonic quotients:
x _ ( 3 1 - Z 2 ) ( Z 3 - Z 4 ) - _ ( Z l - Z 2 ) ( z 3 - Z 4 ) ( z 1 - 2 4 ) ( 2 3 - Z 2 ) ( Z 1 - Z 4 ) ( 2 3 - Z 2 )
G ( x , x ) = [ x x ( l - x ) ( l - x)]7r x [ h - + J 2 - J — ] [ / ! ! + I 2 — ^ ] , (7.6) X 1 — X X 1 — X
- 86 -
where i ! ־ •^A 3 2 י 12 ,
X 6 group invariant factors. This result for the correlation function
combined with the U ( l ) part gives an expression identical to that for the fermionic
bilinears. Moreover this result can be generalized to an n-point function. This is shown
in the next section.
7.3 On the equivalence between the fermionic and bosonic n-point func-
tions
We want to show now that the general n-point function of the fields ge~ly^־^^
in a k = 1 , U ( N ) , W Z W theory and that of the n appropriate bilinears of free Dirac
fermions axe the same, namely:
— 1 _ / 47T / 4 7 r < 9 ( z 1 , z 1 ) . . . g (zn,zn) > < e x p [ - i J — < f > ( z 1 , z 1 ) ] . . . e x p [ i J — < t > ( z n , z n ) ] >
=< ^ ( ־ 1 ) * - ( z 1 ) . ״ * + ( ־ ״ ) * L ( ־ n ) >=< * $ . ( * ! ) . . . * + ( * ״ ) X * _ ( f ! ) . . . * L ( ־ n ) >
(7.7)
We wil l prove this identity by showing that the bosonic and the fermionic correlation
functions solve the same set of partial diiferential equations (for the four-point function
it is a set of ordinary differential equations) and have the same boundary conditions.
Knizhnik et a l . 6 1 found that for a general W Z W theory the equations axe the
following:
\ - K d * i + £ ־ ־ L ־ 1 " I < 9 { * l , z i ) g(zn,zn) > = 0 (7.8) ~f. 2 i ~ Zj
where K = \{C2^ + k ) , C2^ is the second Casimir operator in the adjoint representation
and similarly with 2 —* z and TA —• TA acting on g from the right. In the S U ( N ) k = 1
- 87 -
case K = \ { N + 1). Now for the fermion Green's functions, we use the fact that the
fermion energy-momentum tensor can be expressed, actually, in terms of the currents
as
T f ( z ) = : JA(z)JA(z) : + ± : J < U ( z ) j M ( z ) : (7.9)
Thus
+ A T £ < ־ § 7 + > * ־ ( ־ 1 ) * ־ ( ־ n ) > = 0 ( 7 ' 1 0 )
+ £ ( - ^ ( * < (, , + + ד - ( ־ (n-*-(*״.) >= 0 (7 1
These equations are consequence of the Ward identity corresponding to the Kac-Moody
algebra namely. < J A ( z ) $ l ( z 1 , Z ! ) §n(zn,Zn) > =
" Tf• _ _ (7.12) V 1(21,21 ־* > ) $n(zn,zn) >
z—' z — z~ j = l 1
where <J?j is an arbitrary primary field. In particular the correlation function
< \E ׳ _ (z1 ) . . . \&_(z n ) >, where \&_(z), is a free Weil fermionic field in the fundamental
representation of U ( N ) , wil l also fulfill equation (7.12).By multiplying the left hand side
with TfcSU(N) acting on 2)_<3י,•) we get
. TA < J A ( 2 ) ^ _ ( z ! ) . . . . * _ ( z n ) > =
N 2 - 1 A T X A T ? (7-13)
I J ^ + E T ^ K ' - W ״ - < * . »
and similarily for the ?7(1 )current:
< J ( 1 ) ( z ) ^ ׳ _ ( z 1 ) . . . . ^ _ ( z n ) >= V < *_(*!) > (7.14)
- SS -
We will now insert into (7.13)and (7.14) the currents JA - . # t Q , (T A ) Q : / 3 #_p : and
j ( i ) _ ,and take the limit z —• 2!. The expressions to calculate axe, therefore : ^׳!י .
U m e ^ 0 T t
A : * l a ( z . + e ) ( T A ) a ^ _ 0 ( z i + e) : * _ ( * » )
and a similar expression for the J^ 1) current. By Wick theorem we get:
: ^ { z ^ r ^ . p i z ^ y 8 ^ . ^ ) : (7.15)
+ l i m ^ Q ( T A y \ T A r ^ ^ < * l a ( z i + e ) V - y ( z i ) >
Using ( T t
A y 6 ( T A r e = 1 ( 8 3 6 / ד
a6 ~ T?{>-f6fipa) 3• 1 1d 217 f ° r the propagator we get:
iV + 1 _ t _ _ N 2 - l i _ ,
For the U ( l ) current there will be a similar result but with 1 infront of the two terms.
Thus, in the following linear combination 1
\ + (7.16)
the normal ordered term is cancelled. While inserting this result to the left hand side
of equations (7.13) and (7.14) we notice that the singular terms in both sides of the
equations axe the same. Therefore we axe left over with an expression like (7.11) where
K = ^ j " 1 . If we insert the bosonic correlation function of (7.7) into (7.15) we will get
for the correlation function of g(z{, I{) only, the same set of partial differential equations
as (7.8)
The procedure above can be repeated, in fact, for all the fermionic theories which
axe claimed to be equivalent to certain W Z W theories. Performing that we immediately
find the following necessary conditions for the equivalence: (i) The coefficient in front
of the partial derivative should be equal to K; (ii) There should be a certain linear
combination of the form of (7.15) such that the overall normal ordered term will be
canceled out. As for the first requirement the coefficient is in general cg so that one has
to have K = cg. From eq.s (7.20)-(7.22)6 1 we know that cg - 2 A K = 0 and A = ^ י 0 ' , ad
and therefore one has to demand that
* ־ 5 ־ ? V ( ׳ 7 1 7 )
cad + "
This in fact is the same condition required in ref. 59 for matching of the fermionic and
bosonic Virasoro- algebra and hence for the equivalence of the two theories.
- 90 -
7.4 Bosonization of mass bilinears of colored flavored fermions -the old
prescription.
A natural question here is how to generalize the rule (7.1) to theories given by
(6.10) and its analog for the case of S U ( N F ) x S U ( N C ) x U ( l ) given in (6.14). It was
argued that the bosonization rule for the latter case is the following:
* ? ' 6 - ¥ ׳ ; = A ^ / I ^ ' V * ^ (7.18)
Consequently, the bosonic form of the fermion mass term 771g,lf1a\&j-a is therefore
m'2Nf, J d 2 x ( T r g T r h + T r h ^ T r g ^ ) (7.19)
m 2 = mqKfj., Once again the bosonic operator (7.18) has the correct chiral transfor-
mations and the proper dimension:
A = A , + A , + A , = . N r $ ~ + \ c ) + N J l ' + \ F ) + = 1 ( 7 ' י 2 0
Unfortunately, the explicit calculation of the four point function 6 1 reveals a discrepancy
between the fermionic and bosonic results. Since g and h axe fields defined on entirely
independent group manifolds, then (ignoring for a moment the U ( l ) factor) the analog
of (7.4) can be written as :
< 9 { z \ , z 1 ) g ~ l { z 2 , z 2 ) g ~ 1 { z Z 1 Z 3 ) 9 ( H ^ A ) > < h ( z 1 , z 1 ) h ~ 1 ( z 2 , z 2 ) h ~ 1 ( z 3 , z 3 ) h ( z i , z 4 : ) >
(7.21)
This expression differs from the corresponding fermionic Green's function in at least
two aspects: (i) It includes independent "contractions" for the g and h factors whereas
in the fermionic correlation function the flavor and color contractions are correlated,
(ii) The result (3.6) is true only for a bosonic field associated with Kac-Moody centred
charge k = l . For g and h, however, the central charges are N Q and N p respectively. For
such cases the expression for the Green's function is much more complicated(expressed
in terms of hyper-geometric functions) and does not resemble the case of free fermions.
7.5 Bosonization using the U ( N F x NQ) WZW action
It is clear from the previous discussion that the bosonization prescription for our
case needs an alteration. Apriori there can be two ways out: modifying the rule for the
bosonization of the mass bilinear or using a different bosonic theory altogether. As for
the first approach, eq. (7.18) preserves the proper chiral transformation laws as well as
the correct dimension, and therefore the number of possible modifications is very limited.
For example one might think of multiplying the expression in eq. (7.18) by an operator
which is a chiral singlet with zero dimension. We do not know of such a modification.
Therefore we are going to try a different bosonic theory than eq. (6.14). The symmetry
of the free fermionic theory is actually U 1 ( N p x N Q ) X U R ( N p x N c ) rather than
[ S U ( N p ) x S U ( N C ) x U ( 1 ) ] L x [ S U ( N p ) x S U ( N C ) x U ( 1 ) ] R . The natural bosonic
action is hence a W Z W theory of u C U ( N p N c ) & n d with k = l . The action, the currents
and the mass bilinear will now be given by(6.2), (6.11) and (7.1) respectively (with u
instead of g ) , and the indices now run from 1 to N p x N c - Clearly the requirement for
Sugawara construction of T, for proper chiral transformations of all the operators and
- 92 -
for a correct dimension for the mass bilinear are fulfilled. Since now the flavor and color
degrees of freedom are attached to the same bosonic field, the previous "contraction
problem" in the n point functions is automatically resolved. Moreover as was stated
above the four-point function, and in fact any Green's function wil l now reproduce the
results of the fermionic calculation .
The currents constructed from u obey the Kac-Moody algebra with k = l . The
color currents, for instance, are JA = T r ( T A J ) , where TA are expressed as { N Q N P ) X
( N c N p ) matrices defined by yj^—AA(g)l with X A the Gell-Mann matrices in color space
and 1 stands for a unit N p x N p matrix. The factor ensures that T r ( T A T ^ ) =
2 8 A B . If we now define J׳A = T r ( X A J ) then the central charge is k = N p . The same
arguments will apply for the flavor currents, now with k = N Q . The central charge for
the t / ( l ) current is N c N p .
To see the difference between the present theory and the previous one let us express
u in terms of ( N p N c ) x ( N p N c ) matrices g, h and I in S U ( N p ) , S U ( N C ) and the
coset-space S U ( N F x N c ) / { S U ( N p ) x S U ( N C ) x U ( l ) } respectively, through u = 4T
ghle V N C N F _ Using the formula for expressing an action of the form S[AgB we
get:
S[u] = S[ghl] + i j d 2 x c ^ ׳ 5 V L J , , (7-22))
S[ght\ = S[g) + S[t\ + S[h] + — / d2xTr(g^d+gldJf + V d + h l d - P ) 1
We can now choose / = / so that ld-ft will be spanned by the generators that are only
in S U ( N F x N C ) / { S U ( N F ) x S U ( N C ) x ?7(1)}. This can be achieved by taking for
h = h 0 1 a solution of the equation d - h h ^ = - ^ T r F [ d - u u ^ ] , and in analogy for g
with • j ^ T r c • These are also the conditions that the flavor currents be expressed in
terms of g and the color currents in terms of h . For this choice we can rewrite the action
so that the first two terms are identical to those in the action of (6.14),and the last term
(of I with g or h ) vanishes:
S[u] = N c S [ g ] + N F S [ h ] + i J d 2 x d ^ d * < $ > + S[l] (7.23))
Note that / is still an S U ( N c N F ) matrix while g and h axe expressed now as S U ( N F )
and S U ( N c ) matrices respectively.
7.6 Multiflavor Q C D 2 using the U ( N F x N c ) WZW action
We now want to apply this new prescription to the case of multi-flavor Q C D 2 ,
namely gauging the S U v ( N c ) subgroup of S U i ( N c ) x S U R ( N C ) , where V stands for
a vector symmetry. Later we will take the strong coupling limit and compare the result
to a previous calculation where (6.14) was the starting point.
Using the gauging prescription discussed in the last chapter we get first the action
where the whole S U ( N c N p ) is gauged namely:
S [ u , A + , A - } = S[u] + ־^־ / d 2 x T r ( A + u d - u i + A - u j d + u )
i f , r
( 7 ' 2 4 )
- — / d 2 x T r ( A + u A - u ^ - A - . A + ) + m 2 N ^ / d 2 x T r ( u + u f )
־ 94 -
where we have also added a mass term with m 2 = r r i q K r h . Now since we are interested
in gauging only the S U { N Q ) subgroup of U ( N p N c ) , we take to be related to the
generator TD C S U ( N C ) via A u = ecA^T^. We then add to this action the kinetic
term for the gauge fields — — 7 7 / d 2 x T r F t i U F f 1 1 ' . e'c has the appropriate form so that
after tracing in flavor space the color gauge coupling is obtained, namely: e'c = \ / j V p e c .
The resulting action is invariant under:
u - + V i x ^ V - 1 { ! ) Ap - 4 V{x){A״ + S ^ " 1 ( ! ) V ( x ) C S U V ( N C ) (7.25)
u _> WuW~l W C U ( N p ) (7.26)
The symmetry group is now S U y ( N c ) x U ( N p ) , just as for the gauged fermionic theory.
We choose the gauge A - = 0, so now the action takes the form:
S A [ u , A + ] = S[u] + 42 / d 2 x T r ( d - A + ) d / ־^ + 2 2 x T r ( A + u d - u * ) & c J J (7.27)
+ m ' 2 N r h J d 2 x T r ( u + u+)
Upon the decomposition u = ghle V c F we see that the current that couples to
A + is T r p u d - u ^ = T r p h d - h ) . Thus the coupling of the current to the gauge field
j d 2 x T r ( A + h d - h i ) .To proceed we define H { x ) by d - H = i h d - h * . We take the
boundary conditions to be H(—00, x_) = 0 and then we integrate out A + obtaining
S[u] = S[u] - (£)2 f d 2 x T r ( H 2 ) 4 7 r J (7.28)
+ m ' 2 N r h J d 2 x T r { u + u f )
- 95 -
In the strong coupling limit —> oo the fields in h which contribute to H will become
infinitely heavy. The sector gl C ^5[/(̂ )̂̂ however, wil י l not acquire mass from the
gauge interaction term. Since we are interested only in the light particles we can in the
strong coupling limit ignore the heavy fields, if we first normal order the heavy fields at
the mass scale ji = -7^= . Using the relation, for a given operator O, v27r
(4-)ANflO = NrhO (7.29) m
(7.30)
to perform the change in the scale of normal ordering, and then substituting h% =
we get for the low energy effective action:
S e f f W ] = S[g] + S[l] +l-J d 2 x d ^ < J >
+ K m i f i N - p jd2xTr(e־i\f^<t'gl + e
+ < V * ^ % t )
We can now replace the two mass scales by a single scale by normal ordering at a certain
m so the final form of the effective action becomes
Seff[u] = S[g] + S[l] +l-J d 2 x d ^ < f >
+ ̂ - N m J d2xTr{e־i^^4'~gl + e + ' V ^ V )
with m given by:
m ־ [ N c K m J ^ ^ ^ c ] ^ ^ (7.32) v27r
(7.31)
For the 1 = 1 sector, defining g = ge V ' c F Q U(Np) one gets the effective action
Seff[9'] = NcS[g'] + m2Nm J d 2 x T r F ( g + </t) (7.33)
- 96 -
which is obviously the same as the "old" effective action. (Note that there is an
error in formula (2.11) of ref. and e c should be multiplied by y / N + 1 there, where
N F = N + 1 ) .
The lightest sector, which is with baryon number N Q ( we take the quark to have baryon
number one), is obtained by / = 1 and coincides with the one whose mass we computed
in our previous work 2 3 ,
7.7 Summary and conclusions
The subject of this chapter was the bosonization rule for the mass bilinears of the
colored flavored fermions. We elaborate on the proof that the n-point functions of the
fermionic bilinear and the W Z W field given in (7.1) are identical. Then we have shown
that the problem one encounters in bosonizing the fermionic bilinear in cases where
the symmetry group is S U ( N C ) x S U ( N F ) x U { \ ) (or S O ( N c ) x S O ( N F ) ) can be
overcome by bosonizing the U ( N c N F ) ( S O ( N c N F ) ) fermionic theory instead. Using
this approach we derive the action of multi-flavor Q C D 2 - We then derive an effective
action in the strong coupling limit, which turns out to be different from the one derived
from the "old" prescription The lightest sector remains the same.
- 97 -
Chapter 8
The Baryonic Spectrum of Massive Multiflavor QCD2
8.1 Introduction
In the last chapter we derived the low energy effective action of multiflavor QCD2 in
the strong coupling limit. The idea that the two dimensional baryons might be solitons
of the W Z W theory is analogous to the conjecture that in four dimensions the baryons
are Skyrmions Obviously, here we have an advantage in the sense that the non-linear
sigma model is not postulated but derived from the underlying theory. In this chapter
we analyse this action semiclassically (a method used before in the two flavor case ) 2 2 6 2
looking for the low-lying spectrum of baryons.
This chapter is arranged as follows. In the next section we find the lowest energy
soliton solution for the I = 1 sector( see definition in 7.22) of the classical action which
for the static case turns out to be a set of Sine-Gordon actions. Section 3 is devoted
to the semiclassical quantization around these solitons. Using a special parametrization
we prove that the quantum mechanical action is that of a C P ^ N F ~ ^ theory with an
additional term linear in time derivatives. By choosing an unconstrained set of variables
(one gauge variable still remains) we avoid the necessity to use Dirac brackets, paying
the price of having some non-linearly realized symmetries. Finally we show that the
- 98 -
Hamiltonian of the system depends linearly on the second Casimir operator of the flavor
group. The possible representations of the baryons are shown to be constrained. Section
4 is a description of the low lying baryonic spectrum. We write down a mass formula
for the baryons and discuss the possible flavor representations in terms of their Young
tableaux. In section 5 we discuss the / ^ 1 sector. The last section brings several
conclusions.
8.2 Classical Soliton solutions
The effective action of QCD2 in the strong coupling limit for the / = 1 sector was
found above (7.32) to be
S e f M = N C S { g ] + m 2 N m J d 2 x T r ( g + g*) (8.1)
with
where now g C U ( N p ) .
First we look for static solutions of the classical action. For a static field configu-
ration u(x),the W Z term does not contribute. One way to see this is by noting that the
variation of the W Z term can be written as
S W Z oc J d 2 x e i j T r ( 6 u ) u ^ ( d i u ) ( d j U ^ ) • (8.3)
and for u that has only spatial dependence 6 W Z = 0 . Without loss of generality we may
take, for the lowest energy, a diagonal g ( x ) 2263
g ( x ) = e (8.4)
For this ansatz and with a redefinition of the constant term, the action density reduces
to
1=1 Y - 2 m ' ( cosx/—<fi - 1
47T (8.5)
This is a sum of standard Sine Gordon actions 6 4 For each if{ the well known solu-
tions of the associated equations of motion are:
<Pi(x) = J arctg[e (8.6)
with the corresponding classical energy,
E i = 4 m ^ < ^ - i = l , . . . . N F (8.7)
Clearly the minimum energy configuration for this class of ansatz is when only one
is nonzero, for example:
90( x ) = D i a g ( l , l , . . . . , e *V"^" 0
) (8.8)
Conserved charges, corresponding to the vector current (6.25) (but now with an
extra factor N c there), can be computed using the definition:
QA[9(x)} = \j d x T r ( J 0 T A ) , (8.9)
100 -
where T ־ 4 / 2 axe the S U ( N p ) generators and the U ( l ) baryon number is generated by
the unit matrix (instead of in the equation). This follows from = J A T , and
in the fermionic basis JA = rp׳ylxjTA1p.
In particular, for (8.6), we get charges different from zero only for QB and Qy
corresponding to baryon number and "hypercharge" respectively:
Q B - N C Q Y = ~ ^ { N P ) N C , (8.10)
these charges axe determined solely by the boundary values of tp{x), which axe:
47r <p(oo) = 27r
N , 4זל— (8.11) ־,(-00) = 0
Under a general Uy(iV/r) global transformation g 0 ( x ) — י • g 0 ( x ) = A g 0 ( x ) A ~ 1 the
energy of the soliton is obviously unchanged, but chaxges other than QB and Qy will
be turned on. Let us introduce a paxametrization of A that will be useful later,
Now
A =
\ Y1
Z ( N F - 1 ) Y { N F - \ ) Z N F J
(8.12)
<70 = 1 + (e * / 4 7r
-1 )Z (8.13)
where { 1 ) a ( 5 = Z a Z ^ and from unitaxity Y^a=i Z a Z ^ = l . The chaxges with g Q ( x ) axe
(Q°)A = ±NcTr(TAZ) (8.14)
101 -
Only the baryon number is unchanged. The discussion of the possible U ( N F ) represen-
tations is clearly irrelevant here, since we axe dealing so fax with a classical system. We
wil l return to the question of possible representations after quantizing the system.
8.3 Semi-classical quantization and the Baryons
The next step in the semiclassical analysis is to consider configurations of the form
g ( x , t ) = A ( t ) g 0 ( x ) A ־ 1 ( t ) A( i ) e U ( N F ) , (8.15)
and to derive the effective action for A(t) . Quantization of this action corresponds to
doing the functional integral over g(x,t) of the above form. The effective action for A(t)
is derived by substituting g ( x , t ) = A ( t ) g Q ( x ) A ~ ^ ( t ) in the original action. Here we use
the following property of the W Z act ion 5 2 6 5
S A g B -1 = S A B -1 + S (8.16)
where S[g] and <S[g,A] axe given by equations (6.2) and (6.30) respectively, and with the
gauge field A ^ given as:
i A + = A ~ l d + A 1 A - = B ~ l d - B A , B e U ( N F ) - 1 : (8.17)
Using the above formula for A = B , noting that S(AA 1 = 1)=0, and taking A = A ( t )
d + A = d-A = A _
V2׳' (8.18)
- 102
we get S
+
+
= A(t)g0(x)A-1(t)] - S[g0]׳
^ J d 2 x T r { [ A " 1 A.flfoltA-U.flrJ]} (8.19)
^ J d 2 x T r { ( A - l A ) ( g $ d 1 g o ) }
This action is invariant under global U ( N F ) transformations A —»• U A where
U £ G = U ( N F ) . This corresponds to the invariance of the original action under
g —• UgU _ 1 . On top of that it is also invariant under the local changes A(t) —+ A(t)V(t)
where V(t)€ H = S 1 ! ( N F - l ) x U 5 ( l ) x U y ( l ) with the last two U ( l ) factors corresponding
to baryon number and hypercharge, respectively. This subgroup H of G is nothing but
the invariance group of go(x). In terms of g 0 (x) and A(t) the charges associated with
the global U ( N F ) symmetry, eq. (8.7), have the following form:
QB ־ i j ? • J dxTr [ T B A ((<^1<70 - g 0 d 1 g l ) + [gQ, [r1^!]]) A " 1 } (8.20)
The effective action eq. (8.17) is an action for the coordinates describing the coset-
space G / H = S U ( N F ) x U B ( l ) / S U ( N F - 1) x U Y ( 1 ) x U B ( 1 )
= S U ( N F ) / S U ( N F - 1) x U Y ( l ) = CPN
To see this explicitly we define the Lie algebra valued variables qA through A - 1 A =
rpAqA j 2י־^־ n t e r m s Q f these variables (8.17) takes the form (the part that depends on
q
A ) : 2 { N F - 1 ) r 2 ( N F - l ) Y
N F
q
l . ׳ ׳ ־ r ( 1 COSi 4 % ) c / x = 3 / 2 ( ^ ) ^ ־
(8.21)
N c
- 103 -
m 7r
The sum is over those qA which correspond to G / H generators and qY is associated with
the hypercolor generator. Although the qA seem to be a "natural" choice of variables
for the action eq. (8.17), which depends only on the combination A - 1 A , they are not
a convenient choice of variables. The reason for that is the explicit dependence of the
charges (8.18) on A _ 1 ( t ) and A(t) as well as on A - 1 A ( t ) .
Instead we found that a convenient parametrization is that of (8.10). One can
rewrite the action (8.17), as well as the charges (8.18), in terms of the Z ! , ,Zyv+1
variables, which however are subject to the constraint £^a=1 Z a Z a = 1 • Thus
A ( t ) 5 o A ־ 1 ( 0 % ־־ 1 o ] = S [ Z a ( t ) , i p ( x ) \ (8.22)
where
){ZlZa/ ־־ = !!\*)S[Za{t)M^-^x{{l-cos^J2d
as follows:We can do the integral over x and rewrite (8.21)
(8.23)
S [ Z * ( t ) d / 2ע = { t [ Z * Z a ~ ( Z ; Z y ) ( Z } Z 0 ) ] - i ^ f J d t { Z * a Z a - Z * a Z a ) (8.24)
where 1 /M is defined in equation (8.19). The first term in (8.22) is the usual C P ^
quantum mechanical action, while the second term is a modification due to the W Z
term if we start from eq. (8.1). Similarly we express the U ( N p ) charges in terms of the
Z variables, using equation (8.18):
(8.25) Q a p = N c Z a Z p + • ^ j ^ [ Z a Z p ( Z * Z y - Z * Z 7 ) + Z a Z p - Z p Z a ]
- 10 ; -
Of course the symmetries of S[Z] axe the global U ( N p ) group under which
z a - + z ' a = u a f 1 Z f i u e U ( N F ) (8.26)
and a local U ( l ) subgroup of H under which:
Z a -* Z׳a = eiS^Z, a (8.27)
Constructing Noether chaxges of the U ( N p ) global invariance of (8.22) leads to an
expression identical with (8.23).
Now let us count the degrees of freedom. The local U ( l ) symmetry allows us to
of freedom, so altogether we axe left over with2JVf — 2 = 2 ( N F — 1) physical degrees
corresponding phase space should have real dimension of 4N. Naively, however, we have
a phase space of 4(iVj?) dimensions and, therefore, we expect 4 constraints. Hence we
can either work with the Z variables and use Dirac brackets for quantization or choose
another set of variables which axe not redundant. From now on we shall follow the latter
approach.
We want to choose a set of new variables so that the constraint Y ^ a = l ^01=«^״ is
automatically fulfilled. There is a standard choice of such vaxiables, namely : 6 6
take one of the Z's to be real and the constraint ^ a Z a Z * = l removes one more degree
of freedom. This is exactly the dimension of the coset-space
Z { = k i / y / T + X Zf = k ? / y / 1 T x Z N F = j x / y / r + x N (8.28)
. where »'=1
- 105 -
The k{, k* and x a r e 2 N p real variables with no constraints on them. The phase
space will now have dimension 2 { 2 N p ) and there remain two constraints. After some
straightforward algebra we can write:
S [ k , k * , X ] = J d t L ( k , k * , X )
L ( k , k * , x ) = ^ * M k ^ ־ ; ^ ~ ^ k i (8.29)
4. 1 X ,.2 , .;.f י K k i - K k j N C
^ 2 M ( 1 + X ) 2 X 2 M (1 + X ) 2 ^1 + X }
where
The local U ( l ) transformations of the Z variables transcribe into the transformations
8X = e(<); Ski = i e ( t ) k i \ 6k* = - i e ( t ) k * (8.31)
and S L = — iVc׳e just like as in terms of the Z variables. This local U ( l ) symmetry can
be made manifest by defining the covariant derivatives
D k { = ki - i x k i D k * { = fcj־ + »xfc* (8.32)
The lagrangian can then be recast in a manifestly gauge invariant form:
L ( k , r , x ) = ^ D k i h i j D k j - . • g £ ^ ־ W ' + i V c x • (8.33)
Although one can now fix the gauge % = 0 we will continue to work with (8.31). The
conjugate momenta are given by:
- ~ d L - 1 D k f r - i " ? - ^ (8.34) dki 2 M 3 3 1 2 1 + X
- 106
T* = 2 L = ± . h i • 8.35) + . *ע) 1 dk? 2 M 3 3 2 1 + X y '
dJL 1 * X = M = m { k * i h i 3 D k D ־ 3 k i h n k ^ + N c T T x ( ־ 8 3 6 )
Since h{j is invertible we can solve for D k * , D k { in term of the phase space variables:
D k * = 2 M [ * j + i ^ T ^ ] h j 1
D k i = 2MhJj^-i^J±^} (8.37)
where
h T } = (1 + X ) ( 6 i j + k i k ] ) (8.38)
Also
7rx = *(**־ *זי ^ i h ) + N c (8.39)
giving the constraint equation
ן = 7 / ג r x - - W i k i ) - N c = 0 (8.40)
The canonical Hamiltonian is given by:
H c = T T i k { + 7r*fc* + 7 r xx - 1׳
and this can be further simplified to:
H c = 2M(1 + X)[7ri7r; + ( 7 ^ ; ) ( ^ * )
- 107 -
Here H c is obtained explicitly in terms of the canonical variables k*, 7r, 7r*. The
term indicates that x ^ s o behaves as a Lagrange multiplier since following the Dirac
procedure, we should define
H T = H C + \(t)xj; (8.43)
where A is a priori an arbitrary function of t. We could absorb the x 1 n
Quantization of this Hamiltonian is now essentially straightforward. Let us first
consider the symmetry generators Qap, which in terms of the new canonical variables
take the form Q i j = i(ki7Tj - n * k j )
Qi,NF = e ־ * [ * | * i _ i « + M ; * ; ) ]
TV k* ( 8 < 4 4 )
Q N F > I = ־ 1 ' * [ - ־ ־ ^ i(7r + ־ t- + * ? * • ; * ; ) ] = Q * 1 N F
QNF,Nf = N c - i ( n i k i - 7r*fc*)
We will now show that the Hj• can be expressed in terms of the second Casimir operator
of the S U ( N F ) group.
The second U ( N F ) Casimir operator is related to charge matrix elements Qap in
the following way:
QAQ* = \Qa0Qpa (8-45)
A straight forward substitution gives:
- 108 -
(8.46)
Therefore, the Hamiltonian is:
JV 2
H T = 2 M [ Q A Q A --£] + A(*)^ (8.47)
Denoting the S U ( N p ) second Casimir operator by C2, QAQA = C2 + 2 ( J V J ? ) ( ( ? B ) 2 W E
get:
ffT = 2 M [ C 2 - i V 2 ( i V ^ r 8 . 4 8 (1־)] (
The fact that Hj• is up to a constant the second Casimir operator, is another way
to show that the charges Qap are conserved. These conserved chaxges will generate
symmetry transformations via:
Ski = [ i T r e Q , k{] 6k* = [ i T r e Q , Jfc• ] (8.49)
6X = [ i T r e Q , X ]
and similar equations for the momenta 7 ,-,זלr*, 7 r x . Here 6{j = j £ A T is the matrix of ״
parameters. The transformation laws axe derived using the constraint equation ^=0
after performing the commutator calculations. Notice that Q{j and Q!yFt1vF axe linear
in coordinates and momenta and therefore, the S U ( N p — l ) x U y ( l ) transformations
they generate axe linear. The Q N F % and Q i s j ^ F chaxges, on the other hand, have cu-
bic terms as well (quadratic in coordinates), so that the coset-space transformations of
S U ( N F — 1 ) X U ( 1 ) 3 x 6 n o n ־ l i n e a r • This is a well known property of C P n models. Substi-
tution of Qap in equation (8.45) gives:
8k\ = i [ t j i k i 6 j [ + e % x t i N F 6 u — e - i X e jv F »k i fy — e j v F N F k [ } (8.50)
- 109 -
Remember that z[fc,7r]=l. Using these transformation laws it is easy to verify the
invariance of the action. The standard Noether procedure then gives the charges Qap
in terms of the coordinates and velocities, which coincide with those given in equation
(8.40). One could also deduce these transformation laws by making the change of
variables Z a : Z * —• k{, k * , x in (8.24).
We can also verify that
[ Q A , Q B ] = i f A B C Q c (8.51)
where f A B C are the structure constants of the U ( N p ) group.
Do we have further restrictions on the physical states? We shall see now that in
fact we do have. Remember that our lagrangian (8.31) includes an auxiliary gauge field
A 0 = x which has to obey the associated Gauss law:
d L = 7 T = * x = N C - * ( 8 . 5 2 ־ ־ •,*<״ 0 ( ) 0 A o dx
Since 7rx is a linear combination of Q Q and Qy, and the first is constrained to be
Q B = N c , the Qy is restricted as well. More specifically, Qy = Qy, with
fr5־Vw--Wyc 8-53) י )
- 110 -
8.4 The baryonic spectrum
The masses of the baryons (8.5) and (8.44) and the two constraints on the mul-
tiplets of the physical states, namely Q g = N c and that the multiplets contain
Qy = Q Y = h y f ( N F - 1 ) N F ^ C I 3 X 6 ^ e m a * n r e s u i ^ s °f the last section. A l l states
of the multiplet with Q Y ^ Q Y w i H be generated from the state Q y = Q Y hy S U ( N p )
transformations as in eq. (8.12). Using the above constraints we can investigate now
what possible representations will appear in the low energy baryon sector. Considering
states with quarks only (no antiquarks), the requirement of Q g = N Q implies that
only representations described by Young tableau with N c boxes appear. The extra
constraint Q y = Q Y implies that all N c quarks are from S U ( N p — 1), not involving
the ( N p ) t h . Thus no column in the Young tableau can exceed N p — 1 boxes. For
N c ( N p ) this eliminates representations with columns of length ( N p ) . How do these
results relate to the predictions of the two dimensional quark model? We claim that for
N c = 3 and N p = 2 or N p = 3 the outcome of the two models match. In the quark model
picture the overall antisymmetric wave function will be constructed from antisymmetric
color component, totally symmetric or mixed symmetric flavor part accompanied by a
"space wave function" with the same symmetry property.
What about the masses of the baryons? The total mass of a baryons is given by
the sum of (8.5) and (8.44) namely
Since the highest eigenvalue of C2 on representations with a given number of boxes in
- I l l -
2 c 2 N F
(8.54)
the Young tableau is for the totally symmetric one it is clear that this representation is
the heaviest, while the totally antisymmetric one (when possible) wil l be the lightest.
The mixed symmetry representations are between these two limits. It is amusing to
note that for N p = 3 and N c = 3 the ratio of the decouplet to octet masses for N c =3
is 1.41 while the ratio of the experimental mean masses of the decouplet and the octet
is 1.20 (C 2(10)=6 and C 2 (8) = 3).
8.5 Summary and conclusions
In this chapter the task of deriving the low-energy baryonic spectrum basically from
the underlying QCD2 theory, has been completed. The baryons emerged by semiclassical
quantization of soliton solutions. A mass formula as well as the constraints on the
possible flavor representations of the baryons were presented.
- 112 -
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יעקב זוננשייך 21 בי ולי 1987
ר י צ ק ת
המחקר במסגרת עבודת הדוקטורט שלי עסק בשלשה תחומים:
א. מודלים של קווארקים ולפטונים מורכבים, שהם גם מודלים של תורת האח וד הגדול וכלולה בהם םופרםימטריה.
ב. מודלי Kaluza-Klein קוסמול ו ג ים. ג. בוזוניזציה לא אבלית והשמוש בר. בתורת QCDמרובת טעמים בשני מימדים.
בנושא הראשון, הרכבנו מודל של קווארקים ולפטונים המבוסם על סימטרית האחודS הכולל TU(2) במודל זה התקבלו 3 דורות וכן כוח "אפקטיבי,, מטפוס .SU(7) הגדול
ערך מתאים של זוית WEINBERG. נוסף לכך פתחנו מודל הכולל םופרםימטריה המבוסם על סימטרית האחור הגדול SU(10) התנאים של התאמת האנומליה על פי Hooft ^וכן תנאי
עקיבות המסה מולאו.
Nambu - Goldstone החלק האחרון בנושא היה מחקר אודות סופר-שדות מטפוס SU(N, המתקשרים לשבירה ספונטנית של סימטריות גלובליות מטפוס)STU(N)xsTTU(N)
M
וכן סימטרית הזזה c מתלוה לסימטרית u(N), נוסף לכך נחשף הקשר בין סימטריותu חדשה. R(l U הנשברות לבין סימטרית ( R ( l ) x UV(N)
בנושא השני פתחנו מודלי Kaluza-Klein קוםמולוגים. בראשון הדחיסה של המימדים הנוספים הינה "טבעית" והמרחב - זמן הרגיל שהתקבל ראשיתו "מפץ גדול" והוא מתאר יקום פתוח. הקביעות של קבוע הצמוד העל דק נמצאה קשורה לערך הזעיר של הקבוע
הקוסמולוגי. במודל השני שהוצע על ידנו הרדיוס של המימדים הנוספים שואף אםימפטוטית לקבוע במודל זה נחקרו השלבים השונים של ראשית היקום.
החלק השלישי בעבודת הדוקטורט הוקדש לבוזוניזציה לא אבלית והשמוש בה בכדי לחשב את ספקטרום הבריונים נובע מ- QCD מרובת טעמים בשני מימדים. המחקר בנושא זה כולל את מציאת כללי הבוזונזציה של פרמיונים מטפום Dirac הנושאים תכונות של צבע וטעם, נתוח הסימטריות האפיניות מטפ ו ס Kac-Moody ו- Virasoro של התורה, וצמודה
לשדות כיול הקשורים לדרגות החופש של הצבע.
המחקר מצביע של קושי מיוחד בבוזוניזציה של אברי-מםה ומציע שיטה להתגבר על קושי זה. מן הפעולה הכללית של התורה הבוזונית הסקנו את הפעולה האפקטיבית באנרגיות נמוכות בקרוב הצמוד החזק. הפתרונות הסוליטוניים בעלי האנרגיה הנמוכה ביותר נמצאו.
קוונטיזציה חצי קלםית סביב פתרונות אלה בוצעה. נמצא שהפעולה המתקבלת הינה מסוג מודלCasimir קוונטו־מכני. הראנו שההמילטוניאן של המערכת קשור ישירות לא ו פרט ו ר c P ( ־ N f ~ 1 ^
השני של חבורת הטעם. לבסוף חושבה נוםחאת המסה של הבריונים ונמצא באלו הצגות של חבורת הטעם הם נמצאים.