TC/83/21U

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

SYMMETRY PROPERTIES AND DYNAMICS IN THE GAUGE THEORIES

WITH SCALAR FIELDS

V,A. Matveev

M.E. Shaposhnikov

ATOMIC ENERGY a n d

A.N. Tavkhelidze

INTERNATIONAL: ENERGYAGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1983 Ml RAM ARE -TRIESTE

TC/83/20A

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

IHTERSATIONAL CENTRE FOR THEORETICAL PHYSICS

SYMMETRY PROPERTIES AMD DYNAMICS IN THE GAUGE THEORIES

WITH SCALAR FIELDS •

V.A. Matveev and M.E. Shaposhnikov

Institute for Nuclear Research of the Academy of Sciences of the USSR,60th October Anniversary Prospect 7a, Moscov, USSR,

.and

A.H. Tavkhelidze

International Centre for Theoretical Physics, Trieste, Italy,and

Institute for Huelear Research of the Academy of Sciences of the USSR,60tn October Anniversary Prospect 7a, Moscov, USSR.

?.«£RAMA.HE - TRIESTE

October 1983

* To be submitted for publication.

I,Introduction

The possibility of spontaneous violation of the local and global

ynn-.etrles in the gauge theories with scalar fields in the funda-

r'-Mital representation is discussed. The general arguments for She

fact that the spontaneous violation of the local gauge symmetry

is impossible are presented. It is demon st rat e d v ^ he theory may

be formulated in terms of gauge-invariant ("colourless") local

fields. Using the electro-weak theory as an example, it is shown

that the existence of short range forces, being carried by the

massive vector bosons has, nothing to do with the spontaneous

symmetry violation, but entirely due to the nonzero value of

the gauge invariant order-parameter - the vacuum average <(pi^>>» >J

(the scalar condensate). The results obtained are then generalized

to the case of QCD with massless scalar quarks. Using the methods

of operator expansion and finite energy sum rules, we come to the

conclusion that the two dynamical regimes - with strong and with

weak coupling - may exist, depending on the sign and value of the

single order-parameter h . In the case of the strong coupling

regime, which takes place at large negative values of scalar conden-

sate we find that the new family of hadrons with masses o f about a

few tens GeV may exist.

It is usually assumed that the ; •••' -v?i J.L.• of the spontaneous

symmetry breaking (SSB) forms the basis of the current gauge models

of the elementary particles interactions. For example, for the

121construction of electroweak theory and different kinds of grand

unified model? this principle is often invoked. Then it is supposed

that precisely due to the breaking of the gauge symmetry and Higgs

mechanism one may construct renormaliaed interactions of the massive

vector bosons and massless photon,

In this paper we present rather simple and general enough argu-

ments which imply that the appearence of the short-range forces

due to the exchange of the massive vector bosons in the gauge theories

have nothing to do with the spontaneous breaking of the local gauge

symmetry. The introduction into the theory of scalar fields, which

is usually argued as a necessary condition for the SSB,is indeed

the only way for the realization of some peculiar • properties of

the ground state. These properties may be called colour screening*)

and may be described by the strictly gauge-invariant order para-

meter n» <tp+if>. **)

*) Here and below by colour we mean any charge of the Abelian or

non-Abelian symmetry.

**)Let us note that similar results have been obtained in the

paper [41 . The authors of Ref. - [4 1 have shown that in

the lattice formulation of the gauge theory the order parameter

^ 4 = •i *f ^ is equal to zero and presents some arguments based on the

calculation of the instanton contribution to the quantity n*~ which

Lsprir:!-- that £? «s 0 ^ in the continuous theory. In the paper / compositeKauge invariant operators v/hich play the role of the interpolatingfield were constructed. The authors claimed [4j that one can const-ruct the effective interaction of the white states from the Greenfunctions of these operators. From our point of view we find more gene-ral arguments "for the exactness of the gauge symmetry. On the otherhand we show how to construct the effective Lagrangian for the inter-action of the colourless states in the theory with sealars. We investi-gate also the strong coupling regime which takes place for the small(or negative) values of the scalar condensate *l*-0 }•

The perturbative treatment of the system of the colourless

particles appearing in the theories with scalar fields is possible

only for g » A * , where /\_ is the inverse "<••'•!'-i^ '<«-•-• radius

in the model. This Is indeed the case for the theory of the ele-tro-

\v;-!k interaction where the small coupling regime is re-iiized.

Colour screening takes place in the theories with scalars in the

fundamental representation of the gauge group when the effective

potential for the scalar fields has its extrem™ in the point far

t.he

enough from/origin. This situation may occur when the bare mass square

of the scalar field is negative or in the models a'la Coleman-

Certainly in the nonabelian gauge models with scalar fields

the strong coupling regime (confining phase) may also exist. In this

phase the colourlessness of the physical spectrum is achieved by

the formation of the white bound states according to the confining

hypothesis.

If the scalar particles have large masses then the dynamics

of their interactions is analogous to the heavy fermion quark

dynamics in QCD. In the case of the light scalar the properties

of their bound states are determined by the scalar condensate

/£/-«' A [oj . The investigation of the bound states of light

coloured scalars has been carried out in the paper f£] in

the framework of QCD with the triplet of scalar fields. In [(,~\

we have shown that these bound states have masses which may be much

more than the masses of the usual quark-gluon bound states provided

that there exists a negative scalar condensate p^-A. In/present

paper we discuss in more detail the selfconsistency of this hypothe-

sis and continue the investigation of scalar containing hadrons.

The paper is organized as follows. In Section II the problem

of the spontaneous symmetry breaking in gauge theories is discussed;

in Section III we formulate the SU(2) gauge model in t <• ; of colour-

less variables, In Section III the electroweak theory is considered.

Sections V and VI are devoted to the investigation of the particle

spectrum in the QCD with scalar fields by the finite energy sum rulesL7J

method; in section VII we present summary of the results and conelur.ior.s.

II. The problem of spontaneous symmetry breaking in the

gaue;e theories

The principle of spontaneous symmetry violation laid as a basis

for many modern gauge theories with scalar fields stnru-tl with the

assumption of the nonzero vacuum expectation value of the scalar

field <.if*>= *j . It is implied that unlike the case of unbroken

symmetry when **= 0 and the spectrum contains (in accordancethe

with;confinement hypothesis) only colourless gauge-invariant

objects*), ™en rf t 0, colour states with finite mass may appear1

in the spectrum. In the latter case long-range forces between

bare particles become short-range ones due to the exchange of massive

vector bosons . It is worth noting however that the magnitude of

the vacuum expectation value (VEV) of the gauge non-invariant scalar

field (S cannot serve as a suitable order parameter

for the description of symmetry properties and phase states of the

*) The nonabelian gauge symmetry is under consideration. In the case

of Vfi) - group at £* =• 0 there are charged states in^spectrum

interacting by means of massless vector particles.

system. Indeed, we must inevitao.l.y fix the gauge in order to get

the value of h , otherwise the problem of determining

orphysical quantities by means of functional integral / B'eynman's

graphs seems to be unfeasible. After setting some gauge fixing

it is impossible in general to interpret unequivocally the nonzero

value of the quantity h*" as a result of the symmetry breaking.

Following electrodynamics, gauge fixing IK imposed on the gauge

fields (Coulomb, Lorentz gauges and so on). Let us call such gauges

A-type ones. The scalar fields in theory transformed

according to the nontrivial representation of symmetry group open up

the possibility to set the gauge fixing in terms of scalar fields.

We call .such gauges U-type ones.

There is a qualitative difference between these two types

of gauges. Let LIB take for example the simpliest case of Abelian

v(l) ~ gauge symmetry in the model with scalar complex

field ip (scalar electrodynamics)

A - gauge :

where the

the conditions

\ function f^ (%~}fj , introduced by Dirac, satisfies

(2.2)

T/-gauge :

(2.5)

It is easy to see that in both cases gauge fixing completely

cancels the possibility of local gauge transformation:

with d(x) vanishing at infinity. But unlike the U-gauge the

gauge constraint of A-type preserves the freedom of global gauge

transformations with oC independent of "X- which are generated

by the charge operator:

(2.5)

Thus components of spinor, scalar and vector fields in U-gauge are

gauge invariant not only under the local, but under the global

gauge transformations too. Wewiiisay that they correspond to

neutral or colourless objects. Gilbert space built by means of

such variables contains only uncharged states which are annihi-

lated by charge operator Q . The vacuum expectation^Sf the scalar

fields in U-gauge ^5T> in a gauge invariant quantity. It is clear

that the nonzero value of £ H? has nothing to do with any

sy::imetry br»cki.ig. Moreover, as it follows from (2,3) U-gauge

is a ;in~;.iL'..- one beca.;se it is not well defined in the vicinity

of zero value of invariante field JT . For this reason the theory

in such a case must be quantized in the neighbourhood of nonzero

c-number value of/"•' T- Cf-JF'} C~ £ J ? ~ <-'fo*? • This

proves to be possible for the potentials like the following:

(2.6)

The effective potential of the model, formulated in terms of "white"

variables

(2.7)

has in this case the following extremums:

(2.8)

5. t.D.

Only the first possibility corresponds to a stable minimum and

admits a consistent quantization. Quantized in such a way the theory

now has a massive vector boson, described by gauge invariant

field ~/tf ~ \$j(/lf without a hypothesis of spontaneous sym-a

metry violation. The conversion of,/long-range force into a short-

range one is now a consequence of complete screening of colour

in the vacuum with nonzero VEV ^ £ > in full agreement withtn«o£-

vanishing/charge operator:

where c = -

Notice that VEV of scalar field in

at large

(2.10)

is not invariant under the global gauge transformations and i t s

n o n z e r o v a l u e may , !n p r i n c i p l e , r:nt iri".;, spontaneous symmetry Tarea- _

king. It is known i3~\ ,however, that the corresponding Goidstones

do not • appear in the physical sector of the theory,

while the vector gauge boson acquire.-, the mass.

J[I. The gauge model with a scalar field in terms of colourless

variables

In this section we use as an example the simplest nonabelian

gauge model based on SU(2) symmetry group with doublets of scalar

and spinor f i e lds . We suppose the vulue of the scalar condensate

i>- <l^(/~? *) to be nonzero and will show then how the theory

can be formulated in terms of gauge invariant (colourless)

quantities only.The Lagrangian has the following form:

('D (3.1)

*) The product of operators in the same point needs a special

definition and may be considered in sorr.e sense as an independent

field.

-8-

where

( 3 . 2 )

C - is a set of Pauli matrices ^ -f- — £ - is

antisymmetric tensor. Let us now set a gauge fixing of U-type on the

variables

'Y W ' (3 .3)As in the case of Abelian U(1)-symmetry described above this

constraint fixes gauge freedom completely and from 4 degrees

of freedom of the scalar doublet it leaves only one mode -

jf - /p ^v which is invariant under both local and

global gauge transformations. In any set of gauge equivalent

nonzero fields (f there is but one field obeying the equation

( 3. ? ). The U-type gauge fixing can tie obtained by the suitable

gauge transformation:

( 3 . 4 )

the the

The components of /fermion doublet and/gauge vector field in U-gauge

are

vti>- ff w*' ( 3 . 5 )

i * "•

~ Cwhere /fjs — T £" /X, j

as well as J^" are gauge invariant under local and global

gauge transformations. Indeed, it is easy to verify:

X, - (?+

</>(3.6a)

(3 .6b)

is the dual conjugate scalar field,where

The set of quantities sft Xi t & M form the complete set of

colourless variables. The theory may be formulated in terms of

this field' if the VEV of an invariant order parameter Jt = 4 jF >

is nonzero. Meanwhile the Lagrangian of the model in terms of

colourless variables takes the following form:

X- -

-Q- -10-

•z (3.7)

where quantities Z^ff are-expressed through invariant vector

fields ZT^, like tensor ^/f through the gauge fields /7

The singular character of U-gauge forbidding I zero value of scalar

fields causes one important feature of the quantum system, described

by the Lagranjfian (3-7). This system cannot be quantized in the

vicinity of zero classical values of field W~ O

even when the potential Ui* J has a local pr global minimum

at T- 0

The .requirfenent of'nonzero value of invariant order parameter

h- ^j?> ?6 O , provided the potential V(If) is of a

suitable form leads to the renormalizable theory with three massive

vector bosons, the scalar boson and a pair of fermions. All the

particles mentioned are colour singlets. It is evident that the

Hilbert space of states built by means of 7/j £j, } Xi. contains

only colourless states and coincides with the space of physical

states. In order to demonstrate the vanishing of the colour (non-

abelian Charge) of physical particles one should calculate the

generator of global gauge transformations:

where

with ,

colourless variables.

It is now easy to see

(3.!?)

is a conserved current, which coincides

in U-gauge and expressed through the

using the equations of motion:

4,

v (3. • n

t h a t t h e q u a n t i t y ( 3 . 9 ) -K.vordiii.-- • •? ',hr> •-.nir.- t,:i.oi--rr, ;. >VMIU-.V

to the surface integral and vanishes provided the mass of the field

? , is nonzero:

<3.io)

Thus we have shown, that the requirement of the nonzero value of

the invariant order parameter ^a. i ff~> without a hypothesis about

spontaneous symmetry breaking leads, on the one hand, to the

appearance of short-range forces , being carried by the massive

vector bosons ;on the other hand to complete ^creenimr of Lhe colour charge

of the physical states.A s in t h e Abelian

U(l)-symmetry in the case of the SU(2)-symmetry group the u:;e of

A-type gauge fixing (covariant Lorentz gauge ^ , for

example) preserves the freedom of making the global gauge trans-

formations. So the requirement of nonzero VEV of the if in A-gauge

can be considered as a result of spontaneous violation of the

global colour symmetry. In this case one can expect the vacuum to

be noninvariant under the global gauge transformations:

where Q* are generators of global gauge transfor-

mations, J~ is the gauge dependent conserved current.

Using the fact that the generators of global gauge transfor-

mations (3.11) can be written as the surface integrals:

-1.1-

-12-

, trA = /-si . (3.1&.)

Ct is easy to show that they are expressed for the asymptotic states

( £-"> £ *" ) through the corresponding quantities (3.1Q) built in

terms of colourless variables, by means of some numerical unitary

transformation. Indeed, supposing that in the asymptotic region

JZ*/-**", 4-*±f the field 0*(&J in A-gauge goes to its

classical value

where 0 is a set of numerical phases, & /frfa/

we find the dependence of the generators of global gauge transfor-

mations (3.13) among those in U-gauge (3.10)

where 7* are the matrix of the regular representation of SU(2)

algebra, U is a unitary transformation_into the

U-gauge (3.4), dependent on the scalar fields. Therefore, the

physical sector of theory is the same both in U- and in A-gauges,

and it contains only colourless asymptotic states. Thus the require-

ment of nontrivial vacuum average of the scalar field in A-gauge

leads to just illusory symmetry violation. Specifically, the charac-

teristics for the spontaneous symmetry breaking massless Goldstoneswiiic.il

modes, / appear in Lorentz invariant gauges, ia• absent in the

physical sector of the theory*) .

*) We should emphasize that the equation (3-10) is a requirement,

satisfied by the classical configurations and in quantization scheme

in Lorentz-invariar.t A-gauges, it corresponds to the following con-

for any physical states A and &.

iy. The theory of electroweak interactions

The model described above has a methodical character. Below

we generalize the analysis to the model with SU(2)<J(1) - gauge symmetry

it]

used as a basis of electroweak theory. We are going to show

that the theory of electroweak interactions can be built without

a hypothesis of the spontaneous symmetry breaking. Then the

mechanism of mass generation of the intermediate vector bosons

and the charged leptons can be explained, using the requirement

of nonzero value of the invariant order parameter , determined by

the vacuum condensate of the scalar fields # = <(j)a'fa.>

The Lagrangian of the model with one generation of leptons

and a single doublet of scalar fields*) has the following form:

(1.1)

where

and abelian

anda r e t n e strengths of nonabelian ?J

gauge fields.

*) The introduction of quarks and other leptons adds nothing

essentially new to the picture discussed.

(Jt.tb)

( t . t c )

The coupling constants 0 and 0 are associated, as

usual, with symmetry groups SU(2) and U(l), and Y[ are

hypercharges of the corresponding components of the fermion and boson

field.

We now aim to formulate the model of electroweak interactions

in terms of variables, which are SU(2) singlets. Such variables,

corresponding to weak isospin neutral objects, we call

colourless, as before.

Let us set the gauge fixing of U-type (3.3) on the components of

scalar field. The specific feature of the model, based on thedoesn't fix the gauge

SU(2)xU(l)-group, is that tne U-gaugSsfreedom completely, but redu-

ce* it to the one^-dimensional abelian one, dependent on an

arbitrary function

(I . id)

One can sho* that these variables are expressed in terms of localand S±obal SU(2} invariant quantities upto .•,.: •.rbitrary phase XU):

-eUIZ

-iJ/2 M

(1.5c)

(4.3)

where I is hypercharge.

The variables in U-gauge are defined as follows:

_ ] • • ; _

s. rts t wamoawwu 1* 11 * !«** .iti .»"• i: .« # '!' 'I " I' J> t

i'hiri arbitrariness is associated with an abelian gau

electric charge Q :

gauge group of

(4.6)

ih-;~ quantization by means of colourless variables (4.5a-5e)

-!•..!.-t lead to the existence of massless vector field-photon,

associated with charge Q , in the physical space. Before

theconstructing a local field, corresponding to/photon, notice that

the following combination of fields:

V'2 (1.7)

does not contain arbatraTiness of type oC and corresponds

to the massive.neutral boson. The combination, orthogonal to (4.7):

2e

Where fyff* f'/f, e- f'fo* corresponds to

photon field. In order to prove this choice, write down the

Lagrangian in terms of colourless variables:

2- V(rV

(it.9)

The quantization of this system in the vicinity of nonzero value

of the quantity jfc, = £ 3f> t O leads to all thethe^

consequences of/ Glashow- Salam-Weinberg model, the gauge symmetry

and colourlessness of physical states'being preserved. We

emphasize that there is no step where we use the idea of

symmetry breaking. The explicit asyianetry in weak isospin

space definition of electric charge Q-=- 7$ * £ y admitted in

the (Jlashcv-Salam-Weinberf; model, is substituted in our approach

by the following one; Q: ^ y . This becomes possible due to

the transition to colourless states,, described by the SU(2) inva-

riant variables, for which the contribution of 7j term is

precisely aero.

-17—

V. The oura rule method

V.I. The strong coupling and small coupling regimes

In this section we villconsider the 5U(2) gauge model with/fermionone<T:.-: alar doublets which were described in the Section in.

zIf <rJT>=C. then the colourless states weakly

interact with each other by means of exchange of the vector

bosons W and ~2 with the mass Mig. = z f^^ '

In the opposite case <7> ~ A the shift X = £ + 7t'

becomes invalid and the use of the perturbation theory for

the Lagrangian ( J.? ) is illegitimate. We deal with the strong

interacting white bound states, and the role of the order pa-

rameter is played not by the quantity < 7T> but by the scalar

condensate < f* > * O (in the weak coupling regime

Let us now discuss from the qualitative point of view the

difference between the strong and small coupling phases.

The vacuum in the Higgs phase (phase of the colour screening)

can be imagined as the superconducting neutral plasma LJJ

The superconductiong property of the vacuum guarantees the screening

o-f the nonabelian magnetic fields and currents (due to •''-• M<--;.:;ri<--r

effect) and the existence of the plasma leads to the colour charge

screening. The characteristic radius of ecraniaation is of the

order of ZQ v i\C . The nonabelian •• niac: rr of the gauge

interaction implies the additional scale in the theory: the confi-

nement radius tc *" i / A . One may consider three different

situations depending on the correlation between two scales Z^

and 2Tg

(i) *IQ £< t^ (electrowsak interactions). In this case

the interaction between the colourless particles denims ji.-ntl.-.

on the energy and the quantity 2 t is practically

meaningless. Indeed, if two charges placed at the distance Z *- c

then their interaction is weak due to the colour screening. Onif.

the other hand'they are at a distance Z"- £ then the colour scree-

ning is absent but the interactions are small also due to the

asymptotic freedom. Only the particles W, X>, Z a, are

present in the physical spectrum. The potential energy behaves

like the Ukawa one •->- e / i-

(ii) Ze. ~^> 1-e. • Apparently in this case two

kinds of white particles with the same quantum numbers can be

present in the physical spectrum. The masses of '•' '•"* particles^

— 2are of the order of /tfy t^ and Aft -w e %C corres-

pondingly. The potential energy of two charges grows linearly

with distance for the 2 i ^£ and then decreasesexpoten-

tially. The particles with the mass / ^ and / ^ are the

bound states and screening objects correspondingly.

(iii) *Cg z. C*> , Only bound states are in the spectrum.

Certainly we cannot give a rigorous prove of the existence

of three kinds of regimes, However, from the experimental point

of view it is clear that the case (i) is roa]i:,s.. in electroweak

interactions and the regimes (ii) or (iii) in the strong ones.

It is not excluded that the second case does not exist in nature:

one cannot obtain it for example from the sum rules method which

gives theecrrent answers for the regimes (i) and (iii) (see below)i' a n d

At the same time the case (ii) /quite reasonablejjmay be treated

-19-

in the bag model [9]

Let us now turn to the investigation of the mass spectrum

in the strong coupling regime. To this end in the next subsection

the method of the finite energy sum rules is described.

V.2. Sum rules

Since all the physical states in gauge theories are

colour singlets the mass spectrum and the scattering amplitudes

can be derived from the bare theory { 3. i ) with the help of

the Green functions of the gauge invariant composite operators. Note,

however, that the nonperturbative terms due to the existence of

the condensates of the different fields (quark, gluon, scalar) must1*0] the

be taken into account. The correct method is to consider/operator

product expansion of the Green functionsfii].

Weviii consider the correlators of different currents in the

deep Euclidean region of the momentum space

' ' ' (5.1)

For the currents with the quantum numbers of the white quarka

vector and scalar phionium one may choose the fields X1 •

3* and jf which have the canonical dimensions. However

these fields are very inconvenient for the calculations because

they are essentially nonlinear on the bare field (6 , Therefore

we determine the currents as follows;

V

, * L j R ' • - , - , ( 5 2 )

where the factors o( and c in the currents 3^. and

TjL ensure the renorminvariance of the Green functions in the

leading log approximation.

of =(5.3)

The scalars are considered to be massless and their self

interaction ( 3(1/I/) term in the Lagrangian) is considered to be small.

The quantum numbers of these currents are as follows:

(5.1)

It is convenient to introduce the quantities which are independent

on the normalization point*)

(5.5)

takes the form (forThe operator product expansion for the

the case 3 $ ):

*) The equation (.£ 5) is written in aFor the J l 3 we mean M i ^expression Spif Fl^ / <$fij

( S. 5 ) is an exact one.

. somewhat symbolical form,

and for the H ^ the

. As for the Cljj- the equality

-22-

it

(5-7)

fa 1*1 h+ *Wr> 7 (5.8)

The standard dispersion relations for /fi{$J look as follows:

*?

u* V (5.9)

where f(^ (5) are the normalized positive spectral densities

of the Green functions. The equation ( S. 9 ) is the base for the

derivation of the sum rules of the different kinds. For example

the Borel sum rules may be obtainedQlD] :

(5.10)

A +• Vj rt vi

where H is the theoretical value for the current correlators,

AJ- (s) is the precise spectral density and ^ ia the arbitrary

parameter with the dimension of mass.

Tuc EUH rules ( 5, iD } have been used for the descricti"• - of

l.-ie bound states of quarks and gluons in £ -JCj 'ind fc ...•• :.•. -

investigation of bound states of coloured sc"..a:'-

The analysis of the sum rules proceeds in the following way: for

the function R (S) one chooses some function depending on the

small number of parameters which can be determined from the expe-

riment. Then these parameters.are selected in such a way that the left-

and right-handed parts of the sum rules are equal to each other

with sufficient accuracy. The standard anzatz is

(5.1D

where 6 is determined by the asymptotic of J\ e at G) -=> '**'_,

st is the continuum threshold and A is the parameter connected

with the matrix element <f0 \ 3" \ resonance > . The values of

At So and hi are determined by the fitting procedure of the

sum rules.

Another method for determining the resonance parameters

is the finite energy sume rules; C?. l2j '

S

This method has been exploited for the analysis of the quark-

gluon bound states in Fief.[ ±2~] • The results of Hef.

are in good agreement with experiment and with the Borel sum rule

method. We will use just the finite energy sum rules for the desc-

ription of the "mesonic" resonances because this method is the

most simple one. from the point of view of calculations. The correla-

tor of the v'.-';' '" e r'T'erita will b;v ",CTL:~id^red in some detail, for

:,V" sar., '•->•• 'I*" --.ai X, '•-- r---.:-• .:: ~,r\ \ v t h e r e s u l t s .

V.i. Positive scalar condensate

By using the eq.( S. il ) at r>- 0,1,2 the following

equations can be derived (we neglect */Jf corrections and -luon conder

sale) •

S.-

(5.13)

To find the solution of this equation knowledge of the

higher matrix'elements is needed. In the tree approximation

(which holds in the weak coupling regime)

, etc (5.11)

There are perturbative and nonperturbative correction*to the-e

expressions (see Appendix 1), However they are small at Cv>A.

In this case the solution of eq. ( 5 4 3 ) has the form

(5.15)

This solution precisely coincides with the tree formulas which

may be found from the Lagrangian ( J.? ) written in terms

of the colourless variables. The particles Jf and X i n thisout

approximation turn^to be massless as is expected. It is clear that

for' tho extraction of the mass spectrum from the correlators

(f.b-S.S) at <ift'U>-? » A 1 t h e '; • • o f s u m r u l e s i s n o t

essential. T- the one hand the sumraing up of all the orr-lre-' tree,

.-•'ai;r..., .-re. is possible due to formulas like ( 5-d4 *

the other hand, the u:,i of the colourless variables for the

correlators {5-t>- S,t ) allows us to get the answer in the

summed up form. The strong coupling regime takes place when

M^"1"^?! "»- A * ) . It is not clear in this case how to

reduce the higher matrix elements to' the scalar condensate. However,

it is evident that formulas like ( 5.14 ) are incorrect

{see Appendix I). But if the scalar condensate is negative and

<^tlf>"" - h1 knowledge of the higher matrix

elements is not necessary and the analysis of the sum rules may

be done correctly.

*) The magnitude of the scalar condensate may be changed by the

variation of the scalar self-coupling. In the one-loop approxima-

tion

/,(/>

where /) is the renorminvariant parameter. The formation of

the scalar condensate ^ J? = C takes place at the scale

where A/s&*)- O . Therefore, the

er, the variation of 4 (f*(/y . Atvariation of ya f/

^/f £. yt one loop approximation for ^£*r/ becomes

invalid and the value of itf^ify cannot be determined by the

perturbative methods.

-26-

V.4 Strong coupling regime and the negative condensate.

Consider now the negative scalar condensate £ {^^ify-^ -A .

It is clear that this case cannot be realized in the framework of

perturbation theory. Therefore we deal with the strong coupling

regime. The solution of the sum rules confirms this expectation

because the behaviour of the spectral density in the channel of,

say, vector phionium/analogous to the case of the usual quarks. Note

that the negative scalar condensate has been considered in

Ref. [ t> J by means of the Borel sum rules. We "i-see that the

finite energy sum rules lead to the same results.

Vector phionium

It is easy to see from Eq.( 5. 13 ) that at <ri^Tl^><v-' /»

the following inequality should hold.

(5.16)

(5.17)

Therefore, due to the inequality ( J.dG ), one may ignore the right

hand sides of the second and third equations of the system ( S. 13 ).

Then the solution of k 5.43 ) takes the form

The reasonable evaluation of the higher matrix elements is

*>* ^f^</V>y/2

A* 9/*"*

(5-18)

(5.19)

(5.20)

Taking into account ot/F corrections and variation; of the

quantity 4(4*(f> with the normalization point the solution

may be slightly improved

('*)*/*

/£ {*+£*& (5-21)

-27- -28-

••• m •*- * • • < • • * -

where instead of the scalar condensate ^a>tyy the renorminva-

riant quantity G~= vt *' < tfi'(^> is introduced.

Note that the solution ( 5 . 1 i ) na:; all the characteristic

features of the strong coupling regime : the continuum threshold is

lager than the mass of the resonance (as in the usual p-meson

channel) and the constant A is relatively small ( f. with ( J. J5" ))

The latter fact gives evidence for the'crumbliness''of the system*)

Scalar phionium and the white quarks

The analysis of the states T and X carried out in full

analogy with the previous case gives the results:

****** 4jfin the obvious notations. The comparison of

the follgwing mass relation (see also \b\

(5.22)

(5.23)

which has an, important phenomenological consequence for the QCD

with the scalar particles [t,i3]. Indeed, from { f. 5 3 ) it

fa/^JJ Z*) In the nonrelativistic quark model 4 •" fa/^JJ Z , where ^1)

is the wave function of the quark-antiquark system at the origin.

follows that /ft i.e. ia the narrow hadronic

resonance the decay width of z? being suppressed by the QZI

rule.

Consider now in more detail the system of the white quarks

i „ . The Lagrangian ( J. i ) is symmetric under the

* Vfd)v chiral group (the axial A

current has an anomaly). The currents XL ani^ •%•£ have the

quantum numbers ( AJ i ) and { it Jl/ ) of the left-handed and right-

handed quarks correspondingly. In the case of the negative scalar

condensate the sum rules admit only the massive resonances .

Therefore in comparison with the small coupling regime we have

doubling of the fermion degrees of freedom: instead of two

massless white quarks (with the quantum numbers (f/t i ) and ( 4/W

two massive quarks with the same quantum numbers are present in

*)the spectrum. Let us note that the role of the right/ (left)

partner for the current lf"/J - lPf^ (li''** U>+

at <U'(/?-*- A Plays the current JLg

where / ia t:ie Dirac operator. As a result of the spontaneous

breaking of the chiral symmetry the degeneracy between states {

and (/)>*' ) is lost. The mass difference between the particles

with the different parity

(5.2"4)

is of the order of quark condensate

* ) 'n t.!>-

-30-

' ' (5.25)

Therefore if W"*^,? T - A and the phenomenological

anzatz ( f.if ) is correct then the transition from the small

coupling regime to the strong coupling • has the character of the

phase transition due to the change of the particle degrees of freedom.

Note that the changing of the number of the degrees of freedom is

seen on the example of the other states too. It is evident'that at

^•>>"> /I the diquark and meson states (built from the quarks)

are absent but in the confinement phase these bound states are

present.

V. 5. The merits and drawbacks of the negative condensate.

In the case 4 ff^lf) <0 > i^+^> t--A the nontrivial

situation occurs: the mass scale of hadrons, composed of light

scalar quarks, significantly exceeds the masses of usual hadrons*).

We now discuss this situation in detail;

1. At Q2 <£. 4fX1<(f*¥> the spectral density ( S- & )

becomes negative, that is impossible, because the left-hand side

is definitely positive. At the same time at $ "• 4f!l £lp'tf*>the

the corrections of type 4 (If (?)**•? /$ **" are still small, there-

fore the operator product expansion is still reliable here. Do

•) Compare ( 5~~ &d ) with the corresponding expression for the

J> -meson in QCD: /IU i 3<^V> [10], besides the effective parame-

ter of strong interactions • •ej-fr is* ar' >-- rule,

channel of scalar quarks [ 3

in the

these arguments rule out the case ^l/*(/ >"" " A 1. in general,

this is not so. The correct ions, which are not contained in theoperator procU;; expansion could be lar;;e for Q ^ . _ g j - 2 y ^

Nevertheless, at ti {f*(fy$> A 2 such corrections are excluded ( th i s

is the case for weak coupling). In order to determine one source of such

correct ions , consider the correlator of currents corresponding to

the scalar phionium: X* if . A t the expansion of the correlator

the following quantity appears:

0\-C5.26)

It does not contribute to the operator product expansion {connected

graphs are absent). At the same time 'jP may contain important

information about the correlator in the strong coupling regime (and

may not contain it in the weak coupling regime). Indeed, at x -» O

(5.27)

(5.38)

At £ tfif> s> A Z using ( S ik ) one can obtain ty[o)-* Y U ^ ^ J

so that T*IX.) is^slowly varying function, Fourier transformation

which is localized at fl-O:

while at x -> t>o (according to cluster property [i.5j )

- < iffy?

(5.29)

So it is not necessary to take into consideration *r odt, 4.(f^

In the ease ^ if^U*} "~ J\ '';"l • C S-lh ) iocs not hold

4 /0) :£. 1f'(*J*"".) > therefore, the Fourier transformation gives non-

z'-ro contrib'.;';:iori to the c o r r e l a t o r o:' ;••;• /.r.-er.ts. The

-31-

estimates*) of tylfy) yield, that "tylfy) exceeds the terms of

operator product expansion just in that region, where the power

corrections lead to negative spectral density.

2. There is some evidence of the fact that the scalar con-

densate may be negative. (Notice, that positive definitencas of

the operator ^uiy> is spoilt by renormalization,.) First the cal-

culation of 4 [f*\f> at large masses of scalar quarks yields:

" /" S" ' ' (5-30)

where£ Q ^ "> is agluon condensate. Then we can add some arguments

in favour of the fact that <\{*y> < O '• the derivative of the energy

on the mass of scalar quark is:

(5.31)

so that

At ft}1-?D it is favourable to have <£tff(f> * &

(strong couplinel. At tx?~< 0 the state with 4\ft\ty > O

is energetically favourable (Higg's phase). At last there is a

(5.32)

phenomenological argument in favour of <. 0 • The

analysis of sum rules and experimental data for usual hadrons

yield , that the duality interval S9 in channels is usually large>*)

*) It was suggested that ip (.20 " 6(pc.-\^0 < W W > + &(\x\'fc)

*«f*V> 4 ^ "v H1 ; / % is a characteristic distance,

where the cluster property holds true^/i^ ' " " / ? " .

* * ) . It holds true, at least, in vector and tensor channels,

than the mass of the low lying resonance, which has n. :-mn.-_i

If one admits:

(5.33)

)then in the scalar case .-.n.-.-ii a. situation is possible at <iy U ) ^

as the sum rules show.

3. If t^ipy^-A implies the strong coupling regime, opposite

statement holds true, provided the equation ( 5"- 33 ) is satisfied.

If this equation does/°take place, the strong coupling regime may

be realized at /< ^! < A too. If in addition to it:

1j <:((/*"</)*•?<* Ab , then the solution of the equations

{ J~, /J ) has the following form**), characteristic for the strong

coupling regime:

VI.The scalar quarks in QCD

VI. 1 The mass spectrum of hadrons.

The analysis given above is ea.ui.iy generalized to the case of

SU(3) - group. The results are analogous to the SU(2) model, numerical

*) The analysis of the case ^ {S^ijy **+A is niade in Appendix 2,

using the hypothesis of vacuum dominance.

*•) When the ^£<fti/J*> is fixed, the duality interval

proves to be maximal just in the case <{gt\j')~ <'lW'tW)3'> = O

(at the"! negative condensate).

-33-

estimates :;re ,;-iv=r. ir: the >:~rk [l]

'tli: 1 i:; t the re ."suit a of/analysis of finite c:\-:'

for U :•...- vector phioaxmr.: /j :}i> si^, (/ •'

Of,OI,means'Borel sum rules [£,]

, which is close to the calculati

i.6.2)

theNow we estimate the mass of/baryon-like state in the case of

negative condensate. The particle analogous to baryor. is:

In the lowest order of perturbation theory the correlator of

J -current is given by the expression

(6.

< 6 . '4 .1

Borel sum rules (5.10 ) are useful for the analysis of fcarycn •v.i'.zz

spectrum in this case, because the spectral densities of Green

functions of baryon currents grow at large £ and the resuks

of finite energy sum rules are strongly dependent on the continuum

model. In this case the sum rules are of the form:

DO

•j/pr-rL:j.io:i ( f. ). The standard

(6.6)

Co7-.p:-iri."-.:j. {fab ) and ( /. i ) one can see that the baryon

aprear : So be nonjtable and decays through the channel:

( 6. 7 )

".•' The scalar containing hadrons and experiment

Does the light strongly interact ing •particles exist in nature?.with experiment .

Their exiztar.ee ' • ••'• contradictVchly when the scalar condensate

is negative and large enough 4 tf+l{ > £ - (tDO M e V) * £ t>t i3J .

The investigation of/scalar contairing hadrons is carried out in

\i~b~] , where possible decay modes, widths and other important

characteristics are listed.

-': • '• T^M. problem of the spontaneous colour symmetry breaking

We shewed that the introduction of scalar particles in the

theory r : r\;-. lead to the breaking of gauge invariance. Both strong

and weak coupling regimes may exist, depending on the model para-

meters, which are in charge of the sign and value of the scalarthe

condensate. In any case only white states are present in/spectrum.

Therefore, if the colour symmetry is "broken it is

not due to the nor.vanishing vacuum expectation value of the scalar

field. If one- consider the problem of spontaneous^symmetry

breaking equivalent to the problem of'"6l.ence of the scalar

fields with small { ) or negative mass squared, then

1lle. solution is dependent on the calculation of scalar condensate for

scalars with m*-mAz {kt#}zS/Z> and /#? ?>~> A

we get < if^ify >> A Z , which leads to the weak coupling

regime and contradicts the experiment £ lt>~) - ) •

The calculation of ^\0*ify might clear up this important

question. It is clear now that in loifenergy region the interaction

between quarks and gluons does not differ from the standard

chromodynamical one, because the scalar condensate is bounded above

<r'( +C > :£ —{faOOM'&/J , so that scalar contailing hadrons masses

are of order of tens GeV. So the processes, usually suggested

as a test to chcnie between QCD and the model with spontaneous

broken colour (radiative decay of mesons, e*e~-» hadrons and etc.)

cannot confirm or deny the existence of scalar quarks with large

negative condensate. The only experimental test Is the presence or the

absence of scalar containing hadrons.

VII.Conclusion

Thus we showed that gauge symmetry breaking is absent in gauge

theories with scalars in fundamental representation. At the same

time two essentially different phases may exist, depending on the

value of model parameters (i) - Higgs phase: massive colourless

vector bosorsare present with mass proportional to the gauge

invariant scalar condensate <{f{fj >0j <(/*(/>?> /\l , as well as

scalar particle and white fermions. The interaction between particles

is a weak one. fii) - the confinement pJi.ioe: the spectrum c:" pjrvic!--

and,has intrinsic resonance character the interact i on .be* we li'i c J .Loi.:'" ° u,?.

states is strong. The mass seals of scalar-containing hadrons is

proportional to the scalar condensate too. The strong coupling regime

is certainly realised at the negative scalar condensate ^(^V) * 0,the

1— A in this ease the masses of/scalar containingthethe/the

hadrons significantly exceed the masses of/usual ones (consisting

of fernion quarks and gluons). Nevertheless it is not excluded

that the confinement phase takes place at l£lf

Then the masses of all hadrons are of the same order of magnitude.

For this reason the calculation of t l e scalar condensate, which

would deny the existence of the light scalar quarks or would point

out the energy interval of interest, seems very important from

the phenomenological point of view. The authors are grateful to the•t he-

members of/Theory Division of the Institute for Nuclear Research

for their interest in the work and fruitful discussions.

NOTE ADDED:

One of the author, (A.S.T.) « ° ^ Hk« to thank Professor H.S. Craigie

Who has pointed out two works [18,19] vhich address a similar problem from

a different point of view.

-37-

-38-

Appendix I.

We are going to show that the formula for the reduction of

higher scalar gauge invariant operators to VEV

may hold only in the weak coupling regime at <£ ( +( > V) A 1

Let uscalculate for this purpose one instanton contribution into

the matrix element <(ijtf)n*> . Consider for simplicity SU(2)-model

without fermions with a doublet of scalar particles. It's Lagrangian

has the form:

(A1.2)

where Qu. - is a covariant derivative.

Suppose A 1 . The weak coupling regix,a is realized in

this system at P 1" A , It is known that in theory ( A 1. 2 ) the

solutions of classical Euclidean equation of motion with finite

action are absent. At the sane time one can solve the equations

of the scalar fields in an instanton field, supposing the instar.ton

configuration to be fixed. The boundary conditions, corresponding ti

the finite action are of the form

*/*+**)=

(;, i . 3)

At , where /W,y is Higg'r; boson

the solution cf equation

2 V =is knc.r: [ j7]

(A1.5)

where JJ is the instanton size (it is situated at 3f • O ).

At X. /i?1" the solution { AiA ) goes faster to F , which

leads to the finite.ness of scalar action:

(A1.6)

Gne instanton correction to the matrix element has the following form:

* (A1.7)

where li^'^e density of instantons ld7~J'

A— - i s a numerical c o n s t a n t . I t - value in AfS scheme i s/US

• T h e reg ion X Z'%. /7fy gives a , small

c o n t r i b u t i o n to ( A' i. ? ' , . .:u,v r ;/::e convergence of, in tegrat ion

over the i n s t a n t c n p o s i t i o n ( A}, t> ) . At X £ ™k , one can

o , i n t e g r a t i n g them over the g lobe of the

r - td i 13 irSu "" . A f t e r c . T l c a l a t i n g ( f{i.1 ) , one o b t a i n s

(A1.9)

use ( Aj.S' ) —:r

nt ri:.'jJ.:; i :i3tnntor.: r » /I is neglioable and

, •- i !.-lng { A i. d- ) in the

r w

where R has the formAppendix II.

A; Tiriori it is not excluded that the case ^W t\>>>O > ^if*y) •"+ A

corresponds to the strong coupling regime too. Let us show,

however, that if the vacuum dominance hypothesis for the evaluation

of the higher matrix elements is correct then the spectral density

derived from the correlators of currents has nothing in common with

the usual spectral density. For the matrix elements of the type

^({p+ld)1* > t h i s hypothesis gives

(A2.1)

1.2)

It may be shown that in this case the system of equations (£'

ha3 no physically admissible solutions because the parameter So

turns out to be negative. There are two different conclusions from

this result:

(i) The hypothesis of the vacuum dominance is incorrect. The strong

coupling regime can take place at (<'(f1'i/ J ** A f ^V/*V) ? ** A .

(see section V,5 )

(ii) Anzatz ( S~.fi ) is incorrect but ( A2,i~2 ) is true. In this

case it is possible to sum up all the tree graphs. /)~ takes the

form

y>2

(A2.3)

(A2.4)

The shape of this function has nothing in common with the usual

experimental curve. Therefore it seems that the case * if ?

A 2 cannot be realized.

-1*2-

1 1 .

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- 4 3 -

- 4 4 -