some phenomenological consequences of the super higgs effect

Z. Physik C, Particles and Fields 2, 247 252 (1979) zo, ch,. Partkzles for Physik C

and Fields �9 by Springer-Verlag 1979

Some Phenomenological Consequences of the Super Higgs Effect*

Yasunori Fujii and Hitoshi Nishino Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan

Received 4 April 1979

Abstract. To the combined system of supergravity of N = 1 and the Higgs multiplet (a scalar multiplet) is coupled a complex scalar multiplet as a simplified representative of the matter. An order-of-magnitude estimate is attempted on the masses and the couplings of the gravitino and other fields.

w Introduction

If supersymmetry, global or local, has any value as a realistic approach to particle physics, the crucial question is how to break the exact symmetry. An obviously attractive idea is the spontaneous breaking of supersymmetry. In the theory of local supergravity, the question is also related to how the gravitino acquires a nonzero mass.

The super Higgs effect was first discussed by Deser and Zumino on the basis of the nonlinear realization of supersymmetry [1]. It was suggested that the gravitino mass could be of the order of ~cmu z ~ 10 -19 mN,

where m N is the nucleon mass. Cremmer et al. [2] formulated the super Higgs effect by coupling a scalar multiplet ( A , B , z ) to supergravity of N = 1 according to the local version of the tensor calculus. [3] The Majorana field Z is absorbed into the components of the massive gravitino. Requiring the absence of the cosmological term was an important ingredient. A mass formula

m~ + m 2 = 4m~ z (1)

was also derived for the masses of the bosons A, B and the gravitino mass m o.

One may ask an interesting question on how these fields A, B and the gravitino manifest themselves in any of the gravitational phenomena. In order to answer this question, however, one must include the ordinary matter fields in the formulation, and determine how the above fields couple to the matter fields. We consider in this paper a complex scalar multiplet as a simplified representative of the matter.

The Dirac field in this "matter multiplet" may be interpreted as the nucleon, the quark or the electron.

The complex scalar multiplet is coupled to the system of supergravity of N = 1 which is already coupled to another real scalar multiplet (A, B, X) of Cremmer et al; the latter multiplet is called the "Higgs multiplet," for the later convenience. The two scalar multiplets are also brought into coupling through the interaction of the structure

f ~b*,,t | ~m,, | (J~Higgs. (2)

The coupling constant f is dimensionless. After the spontaneous symmetry breaking, the scalar field A of the Higgs multiplet acquires a nonzero vacuum expectation value, which yields through the interaction (2) the mass splitting (as well as the masses themselves) among the matter fields.

We notice that the vacuum expectation value of the scalar field A of the Higgs multiplet is most likely of the order ,-~ K " - 1 ~ 1019M; M being of the order of the nucleon mass (,-~ GeV), used as the standard particle mass throughout this paper. The extremely large vacuum expectation value, as also suggested by the examples in [2], is in fact a natural consequence of the super Higgs effect 1

On this basis we can estimate the orders of magnitude of the various quantities including the spontaneously generated masses of the matter fields and the coupling constantf. We consider two extreme choices on the size o f f ; f ~ ~cM ,-~ 10 -19 a n d f ~ 1. We find that the first choice gives the results which are acceptable from an over-all view point. The gravitino mass turns out to be of the order of M. With the second choice we predict the gravitino mass as light as 10-19 M, as suggested in [-1]. This choice, however, does not seem acceptable because it predicts at the same time equally light masses of the bosons A and B which yet couple to the matter fields rather strongly.

i It is interesting to recall that the vacuum expectation value v in the Salam-Weinberg model is given by the similar relation v = G~ 1/2

0170-9739/79/0002/0247/$01.20

248

The above conclusion may be subject to a modi- fication if we include other vector gauge fields which may absorb the massless (or nearly massless) spinless fields by the usual Higgs effect. This possibility will be discussed elsewhere.

We also emphasize that the matter fermion number conservation is broken at the gravitational level as a consequence of the fact that the gravitino is a Majorana field in supergravity of N = 1.

w The Model

Cremmer et al. formulated the theory of a scalar multiplet (A ,B , z ;F ' ,G ' ) coupled to N = I supergravity (e~, ~b,, ;S, P, A,). [2] They assume a general kinetic part ~f the scalar multiplet, ~ amn CIY" T(~n),

m,n

where T stands for forming a derivative multiplct. The stir-interaction term is also assumed to be completely general, ~ b, ~b" According to the general

tensor calculus [3] "they give the Lagrangian L/' o which is invariant under both the general coordinate transformation and the local supersymmetry transformation. The result is found in Eq. (46) of [2]. The expression is considerably long. The reader is advised to refer to the original paper.

To the above system we add a complex scalar multiplet, or a set of two real scalar multiplets q0i = (A/, ~i ,~i ; ~ , G~) (i = 1, 2), as a simplified representative of the realistic matter fields. We ignore the interaction among the matter fields for simplicity. The kinetic term is assumed to be canonical. We also start with the massless matter fields.

According to the prescription of [2], the Lagran- gian of this system coupled to supergravity of N = 1 is given by

e - l~p 1

i = 1 1 -

- - 2 - ~ ,ua--i eJv i - - 2 1 ' - i

- A~ca e - l ~.~o. 6~7. 0.(yi .a . Z * - 270.7/ ,) -1

1~ ,,.r,uv q ~_* " ll, ~1 _1

q_ - 1 ,- ,oaux e - 1 ( . ~ quart (3a)

where

a.~ = [Tu, 7~]/(2 i) 2

- 1 ~ 'oaux 1 t e = 2 =+ IH,I =

i = 1

~- 61K(IMI"~,~ oi" * Z*'~ h.c.) - - ~ c 2 l T/ ,12 A , A 7-

i . + ~ K A { -- �89 -- # *0 7/,)--�89

1 - + ~'~7.~ ~,} ]. (3b)

Y. Fujii and H. Nishino : Consequences of the Super Higgs Effect

We use the following symbols:

H'i=Y' i+iG' i , u = S - i P .

The gravitational coupling constant is given by ~c=(8nG) ' /Z;D. is the covariant derivative with the g,u torsion.

We next introduce the interaction between the matter multiplet and the Higgs multiplet of the structure (2). The coupling constant f is dimensionless and is assumed to be common to ~0 i. The tensor calculus gives the Lagrangian 5r z :

e - l ~f 2 = ,~'z + �89 K ~ . Tu X 2

i K2 6#o_ ,v~ ~/v + R T ( S ' s a / 2 -}- P'~2)' (4a) +a

where 2

i = 1

+ �89 H', + h.c.) + �88 ' + b.c.)], (4b) 2

~2 ~/2 + i~'~2 1 = = - 2 f Z zZ2, (4c) i = 1

2 1 ~_2 .x

X z = - f ~ (~Z ,~ ' i+g i z) (4d) i = 1

with

z = A + i B , ~ = A + i y s B , h ' = F ' + i G ' .

To the sum s asG+5(' 1+s 2 we apply the same algorithm as that of [2]. First we eliminate the auxiliary fields u,h', H'~, and A, by using the field equations for these auxiliary fields :3

9g* L 3f z '7 / '2 K u + h ' J , = - 2~b +i=1 4r

1 [-3 h' - q5 j.z=,l~-(g,, -- ,_ - g a,~)*

+ L ( l f J , z*Z*~-~2- -3 f7~72)] i=1 3 -J *

<=yz*z7 _J,z( gJ,O Z q) q) d,zz.

A . = - ~ , c - ( r - r

3 2

: ~sGiS the Lagrangian of N = 1 supergravity 3 d~ (z, z*) = ~ a,,,. z*" z", O(z) = ~ b. z"

m,n n


with

f9 = 3 l n ( - x ~ = ) - l n ( ~ ) = J - l n ( ~ ) �9

and J,~, = O~O~,J etc. We have omitted the terms which will result in the terms trilinear or higher order in the matter fields in the final Lagrangian (7). Equations (5) are substituted into the total Lagrangian.

As in Ref. 2, we then perform the Weyl-rescaling:

m ~ m [ 3 / ( 1 ~ 2)] ~/2 e u--+e e u = - ~ 4 )+~ [Zi[

i = 1

where in the last form we have neglected the higher order terms in 7/i. The rescaling procedure is also applied to the other fields;

A i -* e-z A i, [Bi -~ e-'t Bi ' ~i ~ e-32/2 ~]~i' ~t#-~ e a/2 ~tu.

The fields ~ , and ~i are redefined through the chiral transformations. The Z field is then gauged away. The final form of the total Lagrangian is:

where 5r is the Lagrangian of N = 1 supergravity and

e - i S ~ = ~-4e-*(3 + I%1V%~,)

l I s r I -- ~ - - - -u- "~ v + K-2(ff,zz*gaV~uZ*'~v Z (7a)

- f *g l z JZ ( }a l z+J~J z)

+ %7. _1 1 2

,~=t d'~Uz'Z*c3u77~ + h.c. (7b) 3

e - l ~ = tc[~u c~u~-*" ~ - - �89 f ~t/z~uyu~:~-* ~ ] + h.c.

(7c)

with ~ = - 3/(~ z ~b). A (~) is a complex scalar (pseudoscalar) field and ~ a Dirac field defined by

1 1

1 = ~ ( ~ + i ~r~2).

In s and s we have retained only the terms bilinear in the matter fields�9 The remaining part s176 a contains more complicated interaction terms

249

which are practically negligible since they are multi- plied by ~:2 or higher powers of to. 4

w The Vacuum Expect~ion Value

We assume that the matter scalar field has no vacuum expectation value,

(~>=(~>=o.

This implies that the procedure of [-2] in minimizing the potential V, the minus of the first line of (7a), remains unaffected by adding the matter fields. Since ~c is the only dimensional constant in the gravitational world, it seems natural to assume that

4~(z,z*) : ,<26(~z, Kz*),g(z) = ,<30(,~z), (8)

where q~ and 0 are the functions of the dimensionless variables ~:z and ~cz* with the dimensionless coefficients. The potential V is then re -4 times a function of ~z and ~z*. The potential is minimized and set to zero at z = z 0 = z~. If most of the dimensionless coefficients in 4) and 0 are of order unity, it is likely that

~cz o ,-~ 1, (9)

as indeed suggested by the examples in [2]. It then follows that the vacuum expectation value ( A ) = ( z ) = z 0 is very large of the order ~:-1 ~ 1019M.

One can understand why ( A ) = ( z ) cannot be small of the order ~ M in the following way.

Before including gravitation, the potential cons- tructed from a polynomial of a scalar multiplet has the general form

V = - �89 (F '2 + G '2) ~- F'ft (A, B) + G'fz (A, B), where f l (A, B) and fz (A, B) are some polynomials of A and B. The field equations for F' and G' give F' = f l , G' =f2, and consequently V = (1/2)(F '2 + G'2). It follows that the vacuum expectation value of the Lagrangian is negative;

( L ) = - ( V ) = 1 ,2 - ~-(F ) < 0, (10) as indeed verified in the simple examples. [4] This ( L ) , when coupled to gravity, will contribute a cosmological term of the order ~ M 4, which is - 10 46

times as large as the observed upper limit. Coupling the system to supergravity according to

the tensor calculus [2, 3] yields the terms

5 ~ = e[ -- �89 2 + p2) + ~c(S~ + P ~ ) ] , (11)

where d + i ~ is the "A + iB part" of the polynomial. Notice the negative sign in front of the first two terms. By using the field equations for S and P, the right-hand side of (11) is transformed into

5r �88 z + ~2). (12)

* Some terms of ~e a are ~ ~c ~ ~c in appearance, but are higher- order than the bilinear terms in the matter fields, or effectively

~c 2 after the use of (8), (9) and 03)

250

The vacuum expectation value of (12) is positive. Due to the coefficient ~:2, however, there is no chance that ( 5 ~ cancles the ( L ) of (10) if ( A ) (and hence ( ~ r ) remains "small."

In what follows, we use (9) as the basic relation on the magnitude of the vacuum expectation value. This also implies

~2~b 0 ~ ~)(KZO,lgZO)~ 1. (13)

w Masses of the Matter Fields and the Gravitino

Consider 50b and 50~. The first term of the second line, the third line and the fourth line of (7b) yield the masses of the matter fields;

m r = - ~ / 2 f Zo, (14)

1 2 (15) ~[~ (/~) + ~ ( ~ ) ] = ~ : z ~ 0 J O' 1 2 ~[~ (;~)- ~(B)]

1 2 2 h.c. =15-~otCf g 3 z + / + ~:~o (16)

The second term of (7a) yields the gravitino mass

m o = tc -1 exp( - ~o/2) = -1 ?~3/2/~2 [ g o ] , 2 , ~ (17)

where we have used the relation: e x p ( - f f / 2 ) = 1~:3/2 K3 2~ Iol ((58) of [23).

Note that the mass formula in softly broken global supersymmetry for a scalar multiplet

1./2 (/5~) _~ 1./2 (~) = 2 m~

is observed in the present model. This may indicate how far the present model is still away from reality.

We find from (9) and (13) that

J ,-, 1, ~ ,-, 1 (18)

at z = z o . From the assumed structure of the functions (a (z, z*) and g(z) it is also reasonable to expect that

J,z ~ K, J,zz* ~ K2,

g, ~ c ( g ) , z g ~ t c - l ( g ) , (19)

at z = z 0, leaving ( g ) = g(Zo) still undetermined. Using (18) and (19) in (14)-(17) we find

my, = {[#2(A) +/~2 (B)]/2} 1/2 ~ x - i f , (20)

1 [#2 (A) - - #2 ( ~ ) ] ~ x f ( g ) , (21)

m o ~ ~c 2 < g ) . (22)

From the last two equations we also obtain

m ~ ~ toy -1 [#2 (/~) __ ],/2 ( ~ ) ] . (23)

In discussing these results we consider two extreme choices on the size o f f .

Choice I

f ~ ~cM~ 10 -19. (24)


This choice follows immediately if we demand that mr = { [/~2 (A) +/~2 (B)]/2} 1/2 calculated in (20) reproduces the masses of the ordinary particles, ~ M, at least in their orders of magnitude. If we further take # 2 ( A ) - #2 (B)~ M 2, (23) gives

mg ~ ~cf - 2 M 2 ~ M , (25)

where use has been made of (24). With m0 as heavy as the ordinary particles it is hard to expect any sizable effect of the short-range force due to the gravitino.

Choice 2

f ~ 1. (26)

This choice is allowed if we ignore, for the moment, that the "symmetric" masses given by (20) turn out to be too large by 1029 compared with the realistic values. On the other hand, demanding # 2 ( A ) - #2(~) ~ M 2 in (23) together with (26) leads to an interesting possibility that

mg-,~ ~cM 2 ,-- 10-19M, (27)

as suggested by Deser and Zumino. [1] One may combine (27) with the mass formula

(i) [2]

m 2 + m~ = am 2, (28)

which follows if one requires the canonical kinetic term for the f i na l Lagrangian of the fields A and B. It then follows that m a and m n are as small as (27). This is, however, not consistent with the experiments, as will be discussed in section 5.

w Discussions

1. Nonconserva t ion o f the Fermion N u m b e r

Consider the first term of (7c) and its hermitian conjugate which may be put into the form

tr (~. + iy s g). ~c

with ~ c = c ~ r . Use has also been made of the Majorana condition 0 " = C ~ "T. Fermion number conservation is obviously violated. To lowest order in ~:, these terms have the origin in the coupling of 0 , to the conserved current of global supersymmetry of the matter fields. Nonconservation of the fermion number is due to the fact that in N = 1 supergravity the gravitino field is a Majorana field.

In N = 2 supergravity, on the other hand, there are two Majorana gravitino fields which make up a Dirac field. [5] We can still show, however, that conservation of the matter fermion number is violated at the level of •; the phase transformation of the matter fermion fields does not match the one of the gravitino field.

The free Dirac field ~ (which can be regarded

Y. Fujii and H. Nishino: Consequences of the Super Higgs Effect

as a quark, a proton, or an electron) does not decay into the gravitino and its scalar (pseudoscalar) partner /~ (~) in the symmetric limit in which f and /~, ~3 share the same mass and the gravitino is massless. In addition to the purely kinematical reason based on the energy momentum conservation, one can explicitly verify that the matrix element vanishes.

The decay

~ gravitino + ~ or g (29)

occurs if the symmetry is broken and mo + # < M, where p is the mass of/~ or B, m o being the gravitino mass. Assuming the massive Rarita-Schwinger field, we calculate the decay rates as given by

1 K 2 ( M + mo)2 - / 2 2 = _ _ _ p3 _ -- 2

F~,~ 3 4~ rn o

where p is the center-of-mass momentum of the decay products. For the sake of illustration, let us take ~ as a proton. With the tentative choice M = 940 MeV, # = 0.5M, mg= 0.25M, we obtain the life- time of the free proton, za ,-~ 3 x 101. s, which is about 10 -3 of the age of the universe. To maintain the stability of the proton, we must impose the conditions, s

% +~(/~)> M, % + ~,(B)> M. (30)

Suppose we can somehow identify the meson B as the pion. Then the gravitino must be heavier than

800MeV. Anyway the gravitino mass m o must be comparable to the nucleon mass. 6 It m a y not be justified, however, to consider that the nucleon and the pion belong to the same scalar multiplet.

Let us then consider ~ to represent a "quark" ignoring the colour and flavour for the moment. If a quark in a nucleon undergoes the decay (29), the final state will contain the particle (or particles) composed of two quarks and the spinless partner of the quark. One must be seriously concerned with the mechanism of confining quarks and their spinless partners as well. One can expect that the particles in the final state are much heavier than the nucleon, 7 and the free decay of the nucleon is forbidden even with a very light gravitino.

The stability of the electron is also assured provided the spinless partner of the electron is sufficiently heavy.

Although one can exclude the free decay of the nucleon either by a heavy gravitino or by the absence of light states composed of & or ~, the second order

5 As an illustration we calculate # by assuming the example (b) of Ref. 2, based on (15) and (16) in the present paper. The conditions (30) are satisfied i ff / (x M) < - 0.88. For this critical value we have ~(/~)/M = 1.18, #(B)/M = 0.79 and mJM = 0.21 6 It is interesting to study the decay (29) for very small m_. The decay rate fails to vanish (although it should) unless (M 2"- #2)4 goes to zero faster than M6m2 o as mg---, 0. Choosing mg ~ x M 2 and F~<10-Sx2M a yields too small a value I M - # 1 / M < . 0.12 (xM) 1/2 ~ 10-10

7 See [6]

251

tree diagrams from the first term of (7c) give the processes like,

~ + ~'-- , /~ + /~ ,

/ ~ , + ~__, ~ c + ~ . (31)

The cross section of these nucleon-number non- conserving processes is typically of the order of x 4 ~ G z ~ 10- loo cm 2, being too small to have any consequences even in the cosmological scale. We notice, however, that for the energies as large as the Planck mass x - ~, the interaction will become "strong" because of the derivative in the coupling [the first term of (7c)]. This may have some relevance in physics in the early stage of the universe. [7]

We now turn to the second term of (7c). As in the first term we obtain the interaction ~ f f f , 7 u 2 . ~ p to lowest order in x. This term does not give the decay into the free gravitino because of the condition ~k, Tu = 0. The cross section o- of the process of the type (31) will be of the order of ( f * / 4 1 t ) v - ~ M - 2 , where v is the center-of-mass velocity of the ~ particle. The mean time z during which the process takes place is given by z = (vpa) -1, where p is the density of the quarks (or nucleons). We choose p to be that in the nuclear matter and require that r be much longer than the age of the universe [,-, 105 (x M ) - Z M - 1 ] . It then follows that t f l ~ (~:M) 1/2 "" 10-~~ An obviously natural choice is (24).

It may not be imperative, however, to impose the above requirement on z, if the states composed of ~ or Ni are sufficiently heavy. As in the free decay of the nucleon, it seems premature to draw any definite conclusions from (7c).

2. Bosons A, B o f the Higgs Mul t ip le t

Finally we discuss the properties of the fields A and B. The terms in (7b), except in the first line, show that these fields couple to the quark (or the nucleon), the fields Ai, B~ with the coupling constant of the order o f f 8

If we take the choice 2 of Sect. 4, i.e. f ~ 1, the fields A, B couple rather strongly to the matter fields; at least much more strongly than the gravitational field does. Moreover, as noticed in section 4, m A and m s are of the order of --~ 10-19 M. The presence of such strong and almost long-range forces is excluded by the experiment. This may provide a serious objection against the choice 2, unless some mechanism is invented to eliminate these fields.

On the other hand, if we take the choice 1 of Sect. 4, i.e. (24), the coupling constants are typically of the gravitational strength; there will be no large effects among ordinary particles. Some comments are in order on the consequences of this choice.

8 The second term of the second line and the last term of (7b) give no effect in the static limit of the particle ~ with the spin averaged

252

The foregoing general analyses are no t enough to determine the masses of A and B sufficiently accurately. They depend on the details of the potential. In the simplified example (a) of [2], the pseudoscalar field B is massless while the scalar field A is as heavy as the matter fields because of the mass formula (28) combined with (25). N o observable effect is expected on the field A. It appears that the massless field B has some macroscopic effects. Unfor tunate ly this is also unlikely since the one-B- exchange potential between two nucleons (or quarks), like the one-pion-exchange potential, has the static limit p ropor t iona l to a l ' a 2 , which vanishes after being averaged over the spins in macroscopic phenomena, except perhaps in the strongly polarized systems like neut ron stars.

O n the other hand observable macroscopic effects may be expected if the scalar field A is massless or nearly massless for some reason. This may happen if there is a dilatation or conformal invariance which is b roken spontaneously. If the A is strictly massless,

Y. Fujii and H. Nishino: Consequences of the Super Higgs Effect

the consequences will be much the same as those of the sca la r - tensor theory of gravitat ion of Brans and Dicke. [8] The restriction co> 6 already set by the observat ions will impose an impor tan t limita- t ion to the value off .

There is another theoretical possibility that m a ~cM 2 ~ 10-19M. [9] The corresponding force-range is 2 = m j 1 ~ 105 cm. The one-A-exchange two-nucleon potential is given by

f 2 e-r/x ~ G M 2 e-r/)" (32) r r

Note that by virtue of (24) the coefficient is of the same order of magni tude as that of the ord inary Newton ian potential to which (32) is to be added. The consequences of the possible gravitat ional potential with this modificat ion were discussed in detail. [9] Al though some bounds are already available on the parameters involved, still more precise measurements will be needed to draw any definite conclusions.

References

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Nieuwerthuizen, P. van.: Nucl. Phys. B147 105. (1979); Cremmer, E., Julia, B., Scherk, J., Nieuwenhuizen, P. van, Ferrara, S., and Girardello, L. : Phys. Lett. 79B, 231 (1978)

3. Stelle, K.S., West, P.C. : Phys. Lett. 74B, 330, (1978) ; Ferrara, S., Nieuwenhuizen, P. van, : Phys. Lett. 74B, 333 (1977); ibid 76B 404 (1978); Structure of supergravity, preprint, LPTENS 78/14;.Ferrara, S. : Massive gravitinos, preprint, TH. 2555-CERN; Scherk, J. : Proceedings of the 19th International Conference on High Energy Physics, 1978

4. Salam, A., Strathdee, J. : Phys. Lett. 49B, 465 (1974);

O'Raifeartaigh L. : Nucl. Phys. B96, 331 (1975) 5. Ferrara, S., Nieuwenhuizen, P. van. : Phys. Rev. Lett. 37, 1669

(1977); Ferrara, S., Scherk, J., Zumino, B. : Nucl. Phys. B121, 393 (1977); Zachos, C.K, : Phys. Lett. 76B, 329 (1978)

6. Farrar, G.R., Fayet, P. : Phsy. Lett. 76B, 575 (1978); Goldman, T. : Phys. Lett. 78B, 110 (1978), and other papers cited therein

7. Yoshimura, M. : Phys. Rev. Lett. 41, 381 (1978); Toussaint, D., Wilczek, F. : Phys. Lett. 81B, 238 (1979); Dimopoulous, S., Susskind, L. : Phys. Lett. 81B, 416 (1979)

8. Brans. C., Dicke, R.H. : Phys. Rev. 124, 925 (1961) 9. Fujii, Y. : Nature Phys. Sci. 234, 5 (1971); Phys. Rev. D9, 874

(1974); GRG 6, 29 (1975)

some phenomenological consequences of the super higgs effect

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