UT774
May, 1997
DirectTransmission Models of Dynamical Supersymmetry Breaking
Izawa K.I., Y. Nomura, K. Tobe,^{†}^{†}†Fellow of the Japan Society
for the Promotion of Science. and T. Yanagida
Department of Physics, University of Tokyo,
Bunkyoku, Hongou, Tokyo 113, Japan
1 Introduction
Lowenergy dynamical supersymmetry (SUSY) breaking with gauge mediation is extremely attractive, since it may not only solve various phenomenological problems but also its dynamical nature may provide a natural explanation of the large hierarchy between the electroweak and some higher (say the Planck) scales [1]. Several mechanisms [2, 3, 4, 5] for dynamical SUSY breaking have been discovered and their applications to realistic models have been also proposed [6, 7, 8].
Structures of the proposed models [6, 7, 8] predict a relatively large SUSYbreaking scale to provide sufficiently large soft masses in the SUSY standardmodel sector. On the other hand, the unclosure condition of our universe yields a constraint on the gravitino mass as [9], which corresponds to the SUSYbreaking scale . This is not achieved in the referred models. In fact, a detailed analysis [10] on the models in Ref. [6] has shown that the gravitino is likely to be heavier than , which necessitates a latetime entropy production [10, 11] to dilute the gravitino energy density in the universe.
In this paper, we systematically construct gaugemediated models of lowenergy SUSY breaking with the structure of direct transmission (that is, without messenger gauge interactions). We obtain models in which the gravitino mass can be set smaller than . The existence of such models suggests that lowenergy dynamical SUSY breaking with gauge mediation does not necessarily require complicated nonstandard cosmology.
2 Dynamical scale generation
We first discuss a dynamics for scale generation since it is crucial for the dynamical SUSY breaking in our models. We adopt a SUSY SU(2) gauge theory with four doublet chiral superfields , where is a flavor index (). Without a superpotential, this theory has a flavor SU(4) symmetry. This SU(4) symmetry is explicitly broken down to a global SP(4) by a superpotential in our models. We add gauge singlets () which constitute a fivedimensional representation of SP(4) to obtain a treelevel superpotential
(1) 
where denote a fivedimensional representation of SP(4) given by a suitable combination of gauge invariants .
An effective superpotential [12] which describes the dynamics of the SU(2) gauge interaction may be given by
(2) 
in terms of lowenergy degrees of freedom
(3) 
where is an additional chiral superfield, is a dynamically generated scale, and a gauge invariant () denotes a singlet of SP(4) defined by
(4) 
The effective superpotential Eq.(2) implies that the singlet condenses as
(5) 
and SUSY is kept unbroken in this unique vacuum. Since the vacuum preserves the flavor SP(4) symmetry, we have no massless NambuGoldstone boson. The absence of flat direction at this stage is crucial for causing dynamical SUSY breaking as seen in the next section.
3 Dynamical SUSY breaking
Let us further introduce a singlet chiral superfield to consider a superpotential for dynamical SUSY breaking [4]:
(6) 
For a relatively large value of the coupling , we again obtain the condensation Eq.(5) with the lowenergy effective superpotential approximated by
(7) 
On the other hand, the effective Kähler potential is expected to take a form
(8) 
where is a real constant of order one.
The effective potential for the scalar (with the same notation as the superfield) is given by
(9) 
If , this implies . Otherwise we expect , since the effective potential is lifted in the large () region [4, 7, 13]. Anyway, the component of superfield has nonvanishing vacuumexpectation value, , and thus SUSY is dynamically broken in this model.
In the following analyses, we assume the latter case , which results in the breakdown of symmetry.^{1}^{1}1The spontaneous breakdown of the symmetry produces a NambuGoldstone axion. This axion is, however, cosmologically harmless, since it acquires a mass from the breaking constant term in the superpotential which is necessary to set the cosmological constant to zero[14]. Modifications for the case is touched upon in the final section.
4 Onesinglet model
Let us first consider a realistic model with one singlet for SUSY breaking which couples directly to . It is referred as a ‘multiplier’ singlet, hereafter. We introduce four pairs of massive chiral superfields , , , , , , and , which are all singlets under the strong SU(2). We assume that the , and , transform as the down quark and its antiparticle, respectively, under the standardmodel gauge group. The , and , are assumed to transform as the lepton doublet and its antiparticle, respectively. These fields are referred as messenger quarks and leptons.
The superpotential of the onesinglet model is given by
(10) 
where m’s denote mass parameters.^{2}^{2}2Dynamical generation of these mass terms will be discussed in the following sections. Mass terms for SUSYbreaking transmission were considered in Ref.[7, 15]. In the course of writing this paper, we received a paper [16] which also treated similar mass terms in SUSYbreaking models. For relatively small values of the couplings and , we have a SUSYbreaking vacuum with the vacuumexpectation values of the messenger quarks and leptons vanishing. Then the soft SUSYbreaking masses of the messenger quarks and leptons are directly generated by through the couplings .
The above SUSYbreaking vacuum is the true vacuum as long as the mass parameters are much larger than for . To find the stability condition of our vacuum, we examine the scalar potential
(11)  
The vacuum
(12) 
is stable when
(13) 
In the following analysis, we restrict ourselves to the parameter region Eq.(13).
The standardmodel gauginos acquire their masses through loops of the messenger quarks and leptons when (see Figs.12 and the Appendix). The gaugino masses are obtained as
(14)  
(15)  
(16) 
where we have adopted SU(5) GUT normalization of U(1) gauge coupling, , and , and are gauginos of the standardmodel gauge groups SU(3), SU(2), and U(1), respectively. The for are defined in the Appendix. Here, we have assumed . Notice that the leading term of in Fig.1 vanishes. Hence the GUT relation among gaugino masses , , does not hold even when all the couplings and mass parameters for messenger quarks and leptons satisfy the GUT relation at the GUT scale.
The soft SUSYbreaking masses for squarks and sleptons in the standardmodel sector are generated by twoloop diagrams shown in Fig.3. We obtain them as
(17) 
where and when is in the fundamental representation of SU(3) and SU(2), and for the gauge singlets, and denotes the U(1) hypercharge (). Here the effective scales are of order . For example, the effective scales are given by
(18) 
if the messenger quarks and leptons have a degenerate SUSYinvariant mass ,^{3}^{3}3 In the present analysis, we only discuss the sfermion masses qualitatively. A more detailed analysis will be given in Ref.[17]. which is an eigenvalue of the mass matrix
(19) 
The SUSYbreaking squark and slepton masses are proportional to . On the other hand, the gaugino masses have an extra suppression as shown in Eqs.(14)(16) since the leading term of vanishes. Thus, to avoid too low masses for the gauginos, we must take . It is interesting that this condition is necessary to have a light gravitino with mass less than as shown below.
We are now at a point to derive a constraint on the gravitino mass. The conservative constraint comes from the experimental lower bounds^{4}^{4}4 These bounds are derived assuming the GUT relation of the gaugino masses. The bound on the gluino mass assumes that the gluino is heavier than all squarks. A more detailed phenomenological analysis on the models in this paper will be given in Ref.[17]. on the masses of wino and gluino[18, 19]^{5}^{5}5 We find in Ref.[17] that even when , the constraint from the righthanded slepton mass is weaker than those from the gaugino masses.
(20) 
which yield
(21)  
(22) 
We obtain
(23)  
(24) 
The gravitino mass is given by
(25)  
(26) 
Since the has the maximal value (see the Appendix), we see that in the region of
We have found that the gravitino mass can be set smaller than if are of order the SUSYbreaking scale . In principle, the masses of the messenger quarks and leptons might be considered to arise from dynamics of another strong interaction. In that case, however, it seems accidental to have . Thus it is natural to consider a model in which the SUSYbreaking dynamics produces simultaneously the mass terms for the messenger quarks and leptons. This possibility will be discussed in section 6.
We note that there is no CP violation in this model. All the coupling constants and the mass parameters () can be taken real without loss of generality. The vacuumexpectation values and are also taken real by phase rotations of the corresponding superfields. Thus only the is a complex quantity and then all the gaugino masses have a common phase coming from the phase of . However, this phase can be eliminated by a common rotation of the gauginos.^{6}^{6}6 The rotation of the gauginos induces a complex phase in the Yukawatype gauge couplings of the gauginos. However, such a complex phase is eliminated by a rotation of the sfermions and Higgs fields and , since we have no SUSYbreaking trilinear couplings and no SUSYbreaking term at the treelevel.
5 Twosinglet model
Next we consider a realistic model with two ‘multiplier’ singlets and for SUSY breaking. We introduce two pairs of chiral superfields , and , which are all singlets under the strong SU(2).
We also introduce an additional singlet to obtain a superpotential ^{7}^{7}7We could construct a model without the additional singlet superfield [7] at the sacrifice of complete naturalness. It may manage to accommodate a light gravitino with in a strongcoupling regime.
(27) 
Without loss of generality, we may set by an appropriate redefinition of and . Then the superpotential yields a vacuum with . The masses of messenger quarks and leptons are given by
(28) 
for . Since is nonvanishing, SUSY is broken.
The soft masses of the messenger quarks and leptons stem from radiative corrections. For example, the diagrams shown in Fig.4 generate an effective Kähler potential of the form
(29) 
which gives soft mass terms of the form
(30) 
when .
Since the induced soft masses for messenger squarks and sleptons are suppressed by loop factors, the gravitino mass is expected to be much larger than in this model.
6 Threesinglet model
We finally obtain a realistic model with three ‘multiplier’ singlets , , and for SUSY breaking. The model is a combination of the one and the twosinglet models discussed in the previous sections. The masses of messenger quarks and leptons in the onesinglet model are generated by Yukawa couplings of X introduced in the twosinglet model.
The superpotential in this threesinglet model is given by
(31)  
Without loss of generality, we may set by an appropriate redefinition of , , and . For relatively small values of the couplings , , , , , and , the superpotential yields a vacuum with and the vacuum expectation values of the messenger quarks and leptons vanishing. The masses of messenger quarks and leptons in the onesinglet model are given by
(32) 
for . In this vacuum, the components of are given by
(33) 
and thus SUSY is broken. The masses of gauginos, squarks, and sleptons are generated as in the onesinglet model in section 4. We should replace in Eqs.(14)(16) by .
If , the phases of the three gauginos’ masses are different from one another. Then, the phases of the gauginos’ masses cannot be eliminated by a common rotation of the gaugino fields and thus CP is broken. However, there is no such problem in the GUT models since holds even at low energies.
We comment on the problem[6, 20]. If the superfield couples to where and are Higgs fields in the standard model, the SUSYinvariant mass for Higgs and is generated. To have the desired mass , we must choose a small coupling constant , where is defined by . This is natural in the sense of ’t Hooft. We note that no large term () is induced since the component of is very small. Hence the scale may originate from the SUSYbreaking scale in the present model.^{8}^{8}8 There has been also proposed an interesting solution to the problem in Ref.[21].
Finally, we should stress that the superpotential Eq.(31) is natural, since it has a global symmetry U(1)U(1), where U(1) is an symmetry. That is, the superpotential Eq.(31) is a general one allowed by the global U(1)U(1).^{9}^{9}9 This global symmetry may forbid mixings between the messenger quarks and the downtype quarks in the standardmodel sector. This avoids naturally the flavorchanging neutral current problem[22]. Then there exists the lightest stable particle in the messenger sector[23]. The charges for chiral superfields are given in Table 1.
7 Conclusion
We have constructed gaugemediated SUSYbreaking models with direct transmission of SUSYbreaking effects to the standardmodel sector. In our threesinglet model, the gravitino mass is expected to be smaller than naturally as required from the standard cosmology: If all the Yukawa coupling constants are of order one, the SUSYbreaking scale transmitted into the standardmodel sector is given by . Imposing GeV, we get GeV, which yields the gravitino mass keV.
In the present models, we have four gauge groups SU(3) SU(2)U(1)SU(2). It is well known that the three gauge coupling constants of the SUSY standardmodel gauge groups meet at the GUT scale . It is remarkable that in the threesinglet model, all the four gauge coupling constants meet at the scale as shown in Fig.5. Here, we have assumed that the gauge coupling constant of the strong SU(2) becomes strong () at the scale .
So far we have assumed spontaneous breakdown of symmetry in the models. If , we need to introduce breaking mass terms such as to generate the standardmodel gaugino masses. These mass terms might be induced through the symmetry breaking which is necessary for the cosmological constant to be vanishing [14].
Appendix
In this Appendix, we evaluate the standardmodel gaugino masses in our SUSYbreaking models. The superpotential which relates to the mass terms of messenger fields , , , and for is represented as
(34) 
where the mass matrix is given by
(35) 
In the onesinglet model, the mass parameters are given by
(36)  
(37)  
(38) 
and in the threesinglet model, they are given by
(39)  
(40)  
(41) 
The soft SUSYbreaking mass terms of the messenger fields are given by
(42) 
where
(43) 
in the onesinglet model and
(44) 
in the threesinglet model. Then the standardmodel gauginos acquire their masses through loops of the messenger quarks and leptons. Their masses of order are given by (see Fig.1)
(45)  
(46)  
(47) 
where the masses () denote bino, wino, and gluino masses, respectively, and we have adopted the SU(5) GUT normalization of the U(1) gauge coupling (). Because of , the above contributions vanish. However, the contributions of higher powers of do not vanish in general: We now work in a basis where the supersymmetric masses are diagonalized as
(48) 
Here the mass eigenstates are given by
(49)  
(50) 
where we have taken the mass matrices to be real, which is always possible. Then, for example, the contribution of order to the gaugino masses, which is shown in Fig.2, is represented by
(51)  
(52)  
(53) 
Here, the for are defined by
(54) 
where
(55)  
This function has the maximal value 0.1 at and . Eqs.(51)(53) imply that the socalled GUT relation of the gaugino masses does not hold in general.
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U(1)  
U(1) 