Supersymmetric dynamical generation of the grand unification scale

25 September

1991

PHYSICS LETTERS B ELSEVIER

Physics Letters B 410 (1997) 45-51

Supersymmetric

dynamical generation of the grand unification scale Hsin-Chia Cheng

Fermi National Accelerator Received

Laboratory,

P.O. Box 500, Batavia, IL 60510, USA

13 February 1997; revised 5 May 1997 Editor: H. Georgi

Abstract

The grand unification scale MGuT - lOI GeV may arise from dynamical effects. With the advances in understanding of supersymmetric dynamics, we show how the grand unified group may be broken dynamically by a strong gauge group with a dynamical scale - lOI GeV. We also show how this mechanism can be combined with solutions to the doublet-triplet splitting problem. The same method could also be used for other symmetry breakings at intermediate scales as well. 0 1997 Elsevier Science B.V. PACS: 11.15.E~;

12.10.-g;

12.6O.J~; 11.30.Pb

One of the outstanding questions in particle physics is the “hierarchy problem”: why is the mass scale of ordinary particle physics so small compared to the gravitational scale? In supersymmetric (SUSY) theories, the electroweak scale is related to the supersymmetry breaking scale, the problem is then translated into why supersymmetry is broken at a scale much below the Planck mass Mp,. As Witten pointed out [l], if supersymmetry is dynamically broken by non-perturbative effects, a large hierarchy between the SUSY-breaking scale and the Planck scale can be naturally generated. Recently, Seiberg and his collaborators have made great progress in understanding non-perturbative SUSY dynamics [2]. Many more models of dynamical supersymmetry breaking have been found, providing new hope to understand how supersymmetry is broken in nature and how the electroweak scale is generated. 0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00924-6

In many extensions of the standard model (SM), there are additional intermediate scales between the electroweak scale and the Planck scale. One example is the grand unification scale. The fact that the gauge couplings of SU(3),, SU(2), and U(l), of the minimal SUSY extension of the standard model meet together at about 1016 GeV [3] gives a non-trivial indication that a SUSY GUT [4] may be realized in nature and the GUT gauge group is broken at lOI GeV, two orders of magnitude beneath Mpl. ’ There are many other examples with symmetry groups broken at some intermediate scales: In flavor theories which try to understand the fermion mass

’ The two scales are so close that there may be ways to reconcile the discrepancy, e.g. in strong coupled string theories

t51.

4.6

H.-C. Cheng / Physics Letters B 410 (1997) 45-51

hierarchy, flavor symmetries spontaneously broken at some intermediate scales are used to generate small numbers by the Froggatt-Nielsen mechanism [6]. Similar to the SUSY-breaking scale, these intermediate symmetry breaking scales may also arise from dynamical effects. Since much of the recent study of strong SUSY dynamics has been focused on the dynamics itself in different theories and finding new models of dynamical supersymmetry breaking, we shall also try to apply these strong SUSY dynamics in realistic models to break these additional symmetries dynamically at intermediate scales. In paticular, we will concentrate on breaking the GUT symmetry dynamically in this paper, because the apparent unification of the standard model gauge couplings at MGUT = 1016 GeV makes it the most possible intermediate scale to exist between M,, and M,. Our philosophy is that no explicit mass parameter (except maybe M,,)should appear in the fundamental Lagrangian. There are other ways of generating intermediate scales, for example, soft SUSY-breaking scalar mass squareds can be driven negative by Yukawa couplings [7] *. However, understanding the origin of symmetry breaking is one of the deepest problems we face. These new symmetries and their breakings, if they exist, should receive the similar attention as the electroweak and chiral symmetry breakings do in the standard model, and it is important to explore various possibilities. Before going to the discussion of GUT symmetry breaking, let us first consider a simple example of symmetry breaking in SUSY theories with the superpotential,

W=hX(&-p2),

(1)

where $,+ transform under some symmetry group, X is a singlet, and p is some mass parameter. The equation JW/aX = 0 will force $,$ to get vacuum expectation values (VEVs) and break the symmetry at the scale p. We would like to have the scale p generated dynamically instead of being put in by hand. The simplest thing one may try to do is to where &Q transform under replace E_L*b y QQ, some strong gauge group and form a condensate.

2

If the scale of the symmetry broken in this way is high, there may be some cosmological problems 181.

However, the extra coupling XGQ often disturbs the original vacuum and may generate a runaway direction in which (X) -+ ~0 and -+ 0. Adding X3 interaction removes the runaway direction but also forces ( $> = (4 > = 0. One can cure this by introducing a “messenger” singlet and couple the symmetry breaking sector to the strong dynamics sector indirectly. Consider a strong gauge group SU(N) with one flavor a, Q and a messenger singlet field S, and the following tree level superpotential, W

tree

=

h,SGQ +

;S3.

(2)

Because N > 1, a nonperturbative generated dynamically [9], A3N-1

W,,,=(N-1)

i

is

1

IN-

GQ

superpotential

1

.

(3)

Integrating out e,Q, (they get mass from the VEV of S, which is justified below,) the first term in W,,,, together with Wdynwill generate a runaway superpotential for the singlet S,

(4) It is stablized by the second term in W,,,. Solving the equation of motion we find that S gets a VEV of the order of the strong SU(N) scale, (&+A

_$

( i

3N-1.

2

(5)

Now we can use the messenger singlet to transmit the dynamical scale to the symmetry breaking sector by replacing p* by S* in (1). 3 The coupling XS2 will not destroy the original vacuum. This is one way to break a symmetry group and generate the symmetry breaking scale from strong dynamics.

3 The resulting superpotential is not the most general one. The most general superpotential without any mass parameter is W = A,SGQ ++S” ++S*X ++SX* ++X3 f&S + h,X)&+. (The XGQ coupling can be removed by redefining X and S. Planck scale mass terms are assumed to be absent or may be forbidden by some discrete symmetry. Non-renonnalizable terms suppressed by M,, are smaller and should have little effect on the vacua.) Including the nonperturbative superpotential and solving the equations of motion, one finds that both symmetry preserving and symmetry breaking vacua exist.

H.-C. Cheng/Physics

Now we turn to the discussion of the GUT symmetry breaking. The breaking of SU(5)our down to the standard model gauge group can be achieved by the simple superpotential W=

~tr_X’ + fb.2.‘,

where the 2 is the adjoint field under SU(5). Minimizing the potential one can find three different vacua which have unbroken gauge groups SU(5), SU(4) X U(l), and SU(3) X SU(2) X U(1) respectively. In the SU(5) breaking minima 2 gets a VEV of the order m/A, so m has to be of the order 1016 GeV (for h - 1) to achieve the successful gauge coupling unification. To avoid putting in this mass parameter by hand, in principle one can use the messenger field discussed above to transmit the GUT scale generated by some strong dynamics to the GUT symmetry breaking sector. However, generating the mass scale dynamically necessarily complicates the model because one has to replace a mass parameter by a whole strong gauge sector, and introducing a messenger field for each of the mass parameters in the theory may be a little bit awkward if the theory has many mass parameters. To make the theory more plausible one should try to look for the simplest models for dynamical symmetry breaking as one can, especially those models in which the strong dynamics directly breaks the symmetry instead of just generating a mass scale. The simplest possibility of the dynamical GUT symmetry breaking is to consider the adjoint field .Z itself to be part of the strong dynamics sector, i.e., it is a composite field due to the confinement of the strong dynamics, 4

z-

$89.

Letters B 410 (1997) 45-51

malizable grangian,

3LM

47

interactions

in

the

fundamental

lP' @Q)‘,$x3-&ea,3.

La-

(8)

However, the minima (with nonzero (2)) occur when these terms in the superpotential balance. One obtains

(.x)-

2.

which is For (x>- MGuT, we need A- M$/M,,,, not reasonable. One may be able to cure this by including more adjoint fields, both composite and elementary, and more complicated interactions, but the simplicity is lost then. This example shows that even though we have made great progress in understanding the strong SUSY dynamics, it is not yet totally trivial that one can write down any explicit mass terms one wants by just assuming that some of the fields are composite, because other parts of the Lagrangian (e.g., Yukawa interactions) may also be affected. So, instead of considering the adjoint field as composite, we assume that it is elementary, but coupled directly to the strong dynamics sector without introducing the messenger field to transmit the dynamical scale, so that the GUT symmetry is directly broken by the strong dynamics. Consider a model based on the gauge group SU(N) X SU(5),,, with N > 5, where SU~5)our is the ordinary grand unified group, and SU(N) is strongly coupled with scale A - MGUT. To break SU(5),,, , the model contains the fields QL, aa, which transform as (?I,5 * ) and (N *,5) under SU( N) X SU(5),,, , and s/, which transform as (1,24), where LY= 1, . . . ,N and i,j= 1, . . . ,5 are SU(N) and SU(5),,, indices respectively. The tree level superpotential is given by

where s and Q are the fundamental fields transforming under the strong gauge group. The terms in the superpotential (6) can then come from the non-renor-

Wtree

4 If the adjoint itself transforms under the snong the SU(5) coupling will blow up immediately.

These are the only terms one can write down without any explicit mass parameter. This model contains only the strong dynamics sector and an SU(5),,, adjoint field, but nothing else, and no non-renormalizable interaction is needed to supply another mass

gauge group,

=A1Q2a+

;tr(S3).

48

H.-C. Cheng/Physics

scale, so it is very simple. For the SU(N) gauge group, the number of flavors Nf = 5 is less than the number of colors, so a nonperturbative superpotential is dynamically generated [9],

where the determinant is taken on the flavor (SU(5),,,) indices. Combining it with the tree-level superpotential we can look for the vacua. There is one runaway vacuum which preserve SU(5) 5: -+ ~0. We are interested in vacua (X:>=O,<~Q>al in which ( s)# 0 and SU(5) is broken. To study them we can integrate out e,Q and obtain an effective superpotential depending only on 2, 3N-5

W,,, = NA

1

N

(h:detB)’

Solving dW,,/dz; lent solutions,

_

I2

0

(P)=&

0 0 0

+ $trX3.

(12)

= 0, we find only two inequiva-

0 0 2 0 0

0 0 0 -3 0

respectively. The GUT scale is generated by the strong SU(N) group. If all Yukawa couplings are H(l), then we have MGUT - @(A> and all particles have masses N MCUT since it is the only scale in this theory. 6 If the fields get some soft SUSY-breaking masses after supersymmetry is broken, the degeneracy among these vacua will be lifted. The runaway direction will be lifted if the soft SUSY-breaking mass squares are positive. The corresponding vacuum is expected to have higher energy and therefore disfavored. In the runaway direction, the supersymmetric contribution to the potential scales as

(15) where u is the VEV of Q and Q. It is stablized by the soft breaking terms, Vsoft - T?12U2 s )

(16)

where m, is the soft breaking mass. The minimum occurs when these two terms balance,

0’ 0 0 0 -3/

Letters B 410 (1997) 45-51

(17)

u,

Then, the VEVs of Q and Q scale as u N A( A/m )+ and the vacuum energy scales as “s su m, N A N . 7 Compared with the SU(5) breaking minima, which have energy V - rnt A2, the runaway vacuum clearly has higher energy for N > 5 (and A > m,). ’ The K%hler potential is unknown so we

1 0 0

(14) In these vacua, SU(5)cur is dynamically broken down to SU(3) X SU(2) X U(1) and SU(4) X U(1)

’ In the Higgs picture, the unbroken SU(5) may be thought as a combination of SU(5)ouT and a subgroup of SU(N)

b A worry is that when incorporated into supergravity, the strong gauge dynamics may break supersymmetry [lo], then the supersymmetry breaking scale will be too high. However, there also exist those so called “racetrack” models in which supersymmetry is not broken [l 11. How to correctly incorporate supergravity is a complicated issue and worth further investigation. Without knowing it exactly, we will assume that supersymmetry is broken by some other sector, not by this strong SU(N) gauge group. ’ For some N and m,, u can be larger than M,,, and then one may not trust this result. However, we just use these results to get a rough idea of the energy of the runaway vacuum and we will ignore this problem. * In constrast, for the superpotential (61, if only the positive soft breaking mass terms are added, the SU(5) preserving minimum will have the lowest energy. If the (not too small) trilinear and bilinear soft breaking A and B terms are included, it will depend on the A and B terms.

H.-C. Cheng/Physics

can not compare the vacuum energies at the SU(3) X SU(2) X U(1) and SU(4) X U(1) minima, but we expect that for some choice of the soft SUSY-breaking parameters, the SU(3) X SU(2) X U(1) vacuum will have the lower energy. It is interesting to see that with the minimal coupling between the adjoint and the strong gauge sector (IO), the strong dynamics can correctly break the SU(5)our down to the standard model gauge group. One of the most serious problem of the grand unified theories is the “doublet-triplet splitting problem”. Higgs doublets are responsible for the electroweak symmetry breaking and hence their masses are of the order of the weak scale. On the other hand, their color-triplet partners must have GUT scale masses in order to achieve successful gauge coupling unification and/or to avoid rapid proton decay. In the minimal SUSY SU(5), this hierarchy is obtained by an extreme fine tune of the parameters in the superpotential, which is obviously unsatisfactory. In the following we will see that some solutions to the doublet-triplet splitting problem can be easily incoporated in the dynamical GUT-breaking models. The first solution we consider is the pseudo-Goldstone boson mechanism [ 121. It is based on the gauge group SU(6). The SU(6) is broken down to the standard model gauge group by two kinds of Higgs representations, an adjoint 2 with the VEV, (Z)=

diag(l,l,l,l,-

2,-2)u,

and a fundamental-antifundamental with the VEVs (H)=

(E>

= (a,O,O,O,O,O).

If there is no cross coupling the superpotential, W=W(x)+W(H,@,

between

(18) pair H and

E

(19) 2 and H,H in

(20)

then there is an effective SU(6), X SU(6), symmetry. These two SU(6)‘s are broken down to SU(4) X SU(2) X U(1) and SU(5) respectively. By a simple counting of the Goldstone modes and the broken gauge generators, one can find there are two electroweak doublets not eaten by the gauge boson and

Letters B 410 (1997) 45-51

49

hence left massless. They are linear combinations the 2 and H,H fields,

of

(21) and they will get weak scale masses from radiative corrections. The simplest way to generate u and a dynamically is to replace the SU(5) in the dynamical GUTbreaking model discussed above by SU(6), and add the H,H fields and a singlet X. The superpotential is given by

(22) Similar vacuum

to what we discussed before, in which ( 2) takes the form

(_Z)=diag(l,l,l,l,-2,-2)u,

there

is a

(23)

and u - @(A). The last two terms in the superpotential will force H,N to get @(A) VEVs. Therefore, the pseudo-Goldstone boson mechanism can be achieved with both CZ and H,H VEVs generated dynamically, and their scales naturally tied together. The problem with this model is that it is not the most general superpotential that one can write down. In particular, there is no explanation for the absence of the E_I$H coupling. This coupling, if it exists, destroys the pseudo-Goldstone boson mechanism. Although it is technically natural to omit some superpotential interactions in SUSY theories, one may prefer to having some symmetry reason to forbid this coupling. There are no such symmetries in this model. One possibility is to generate H,ff VEVs from another sector. (Then we lose the natural link between the two scales.) For example, we can use the method with messenger singlet discussed before to generate the H,H VEVs. In this case, we can assign separate 2, symmetries to the two sectors which generate the 2 and H,H VEVs. The lowest order nonrenormalizable coupling suppressed by M,, between 2 and H,H allowed will be

(24)

50

H.-C. Cheng/

Physics Letters B 410 (1997) 45-51

The induced

masses for the light Higgs doublets will and no bigger than the weak be NxJ&%“,/&J6 scale if M,,,/M,, < l/200. Another way is to use the anomalous U(1) symmetry to generate H,H VEVs and forbid Hi$:H coupling, as discussed in

with

the following &v *,5,1,0>, ~(1,24,1,0), R&5 * ,3,1), R&5,3 * , - l>, q(1,1,3,1), @(1,5*,1,0), and q&1,3 * , - l>, H(1,5,1,0), S(l,l,l,O). The tree-level superpotential is given by SU(5),,, X SU(3)n X U(l),, field content: Q(N,5”,1,0),

u31. Another solution to the doublet-triplet splitting problem which we consider is proposed by Yanagida et al. [14-161. Let us first review the idea used in the model given in [15]. The model is based on the X SU(3)n X U(l),, with the gauge group SU(5),,, following fields, R(5 * ,3,1), R(5,3 * , - l), q&3,1), q(1,3 * , - I>, -%24,1,0), H61,0), and %*,l,O), where the numbers in brackets denote the transformation properties under SU(5),,,, SU(3)n and respectively. One can write down a superpoU(l),, tential, W=m,tr(RR) + h'mq

,

(R)=
L

We are interested in the vacuum in which (J$),( R),(R) # 0. Integrating out Q,Q, we obtain wli,,=N(h:detZ)NA

0

0 0

0 0 ir

0

0

Solving

= $diag(2,2,2,-

3,-

3).

+ ;tr(Z3) +$S3+hHRij

5%% 7,

(28)

the equations Rii)

-h&(

awe,, -= aR

of motion, + A$

= 0,

(29)

A,_.xR-h,SR=O, 3N-

(c)

N

+h,R_%-h,Str(RR)

as

0 0 1

3N-5

1

awe,, -=

as follows: 0 1

(27)

+ h’HRq. (25)

1 0 0

+h,RJ%--h,Str(RR)

+ hHRij + h’HRq.

A, + yS3

+hHRq

+hRJ%+im,tr(J?2)

which has a vacuum

w=h,QzQ+$tr(_$3)

(30)

5

(26)

the gauge symmetry SU(5)ouT X is broken down to the standard su(3), X U(l), model gauge group SU(3)c X SU(2)w X U(l),. The terms HRq and mq in the superpotential marry the color-triplet Higgses in H,H to Zj and 4 so that they obtain the GUT scale masses while the doublet Higgses remain massless. The doublet-triplet splitting is achieved by the missing partner mechanism with small matter representations. The low energy SU(3)c and U(l), are diagonal subgroups of the SU(3), U(1) in SU(5)ouT and the SU(3),, U(l), groups in this model. The gauge coupling unification is not spoiled if the gauge couplings of the SU(3)n and the U(l), are big enough. In fact, the corrections from the SU(3)u and U(l), couplings lower the prediction for the strong coupling constant caysin SUSY GUT and therefore move it in the right direction [17]. To incoporate it in the dynamical GUT breaking model, we consider the gauge group SU(N) X In this vacuum

+

h2(2’ -

+tr( 2’))

(31)

+h,(RR-ftr(RR))=O, one can find a vacuum with (2)

=diag(2,2,2,1 i0 0

(R)=(RT)=

(s)=

F”, 4

where

3,-

3)u,

0 1 0

0 0 1

0 0 0

0 0 0

(32)

H.-C. Cheng / Physics Letters B 410 (1997) 45-51

Thus, we obtain the desirable vacuum in the form of (26) and the missing partner mechanism for the doublet-triplet splitting problem can be implemented. In summary, we have shown that how the grand unified gauge group can be dynamically broken down to the standard model gauge group by some strong gauge sector without inputting the GUT scale explicitly by hand. We also showed that this mechanism can be combined with solutions to the doublet-triplet splitting problem. Although we concentrated our discussion on GUT symmetry breaking, the same method ccould also be useful for other symmetry breakings at intermediate scales as well.

Acknowledgements The author would like to thank L.J. Hall, J.D. Lykken, E. Poppitz, W. Skiba, Y.-Y. Wu, and especially N. Arkani-Hamed and S. Trivedi for valuable discussions and suggestions. Fermilab is operated by Universities Research Association, Inc., under contract DE-AC02-76CH03000 with US Department of Energy.

References [l] E. Witten, Nucl. Phys. B 185 (1981) 513. [2] See K. Intriligator and N. Seiberg, Nucl. Phys. B (Proc. Suppl.) 45BC (1996) 1, hep-th/9509066, for a review and references therein. [3] P. Langacker and M. Luo, Phys. Rev. D 44 (1991) 817; U. Amaldi, W. de Boer and H. Wrstenau, Phys. L&t. B 260 (1991) 447. [4] S. Dimopoulos and H. Georgi, Nucl. Phys. B 193 (1981) 150; N. Sakai, Z. Phys. C 11 (1981) 153.

51

[5] E. Witten. Nucl. Phys. B 471 (1996) 135, hep-th/9602070. [6] C.D. Froggatt and H.B. Nielsen, Nucl. Phys. B 147 (1979) 277. [7] Generating the GUT scale from some soft breaking mass squared running to zero is considered by H. Goldberg, preprint NUB-3154/97-Th, hep_ph/9701373. [8] K. Yamamoto, Phys. Lett. B 168 (1986) 341; K. Enqvist, D.V. Nanopoulos and M. Quiros, Phys. Lett. B 169 (1986) 343; J. Ellis, K. Enqvist, D.V. Nanopoulos and K. Olive, Phys. Lett. B 188 (1987) 415. [9] I. Affleck, M. Dine and N. Seiberg, Nucl. Phys. B 241 (1984) 493. [IO] H.P. Nilles, Int. J. Mod. Phys. A 5 (1990) 4119. [ll] N.V. Krasnikov, Phys. Lett. B 193 (1987) 37; L. Dixon, in: The Rice Meeting, B. Banner and H. Miettinen, Eds. (World Scientific, 1990); J.A. Casas, Z. LaIak, C. Munoz and G.G. Ross, Nucl. Phys. B 347 (1990) 243; T. Taylor, Phys. Lett. B 252 (1990) 59; for an extended list of references, see: B. de Carlos, J.A. Casas and C. Munoz, Nucl. Phys. B 399 (1993) 623. 1121 K. Inoue, A. Kakuto and T. Takano, Prog. Theor. Phys. 75 (1986) 664; A. Anselm and A. Johansen, Phys. Lett. B 200 (1988) 331; Z. Berezhiani and G. Dvali, Sov. Phys. Lebedev Inst. Rep. 5 (1989) 55; R. Barbieri, G. Dvali and M. Moretti, Phys. Lett. B 312 (1993) 137; Z. Berezhiani C. Csaki and L. Randall, Nucl. Phys. B 441 (1995) 61. [13] G. Dvali and S. Porkorski, preprint CERN-TH-96-286, hepph/96 1043 1. [14] T. Yanagida, Phys. Lett. B 344 (1995) 211. [15] T. Hotta, K.-I. Izawa and T. Yanagida, Phys. Rev. D 53 (1996) 3913. [16] T. Hotta, K.-I. Izawa and T. Yanagida, preprint UT-733, hep-ph/9511431; Phys. Rev. D 54 (1996) 6970, hepph/9602439. [17] N. Arkani-Hamed, H.-C. Cheng and T. Moroi, Phys. Lett. B 387 (1996) 529, hep-ph/9607463.