Color and charge breaking minima in the MSSM
arXiv:hep-ph/9607287v1 11 Jul 1996
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, USA
The scalar potential of theories with broken supersymmetry can have a number of local minima characterized by di?erent gauge groups. Symmetry properties of the physical vacuum constrain the parameters of the MSSM. We discuss these constraints, in particular those that result from the vacuum stability with respect to quantum tunneling.
A generic feature of theories with (softly broken) supersymmetry is a scalar potential, V (φ), that depends on a large number of scalar ?elds φ = (φ1 , ..., φn ). For this reason, the scalar potential of the MSSM, unlike that of the Standard Model, may have a number of local minima characterized by di?erent gauge symmetries. In particular, the supersymmetric partners of quarks, ? QL and q? , may have non-zero vev in some minR ? ima, where the tri-linear terms AH2 QL q? and R ? t are large and negative (here H1,2 de?H1 QL ?R note Higgs ?elds and A is the SUSY breaking parameter). These color and charge breaking (CCB) minima may be local, or global, depending on the values of the MSSM parameters. (Of course, there might be directions along which the e?ective potential is unbounded from below (UFB), in which case all the minima are local.) Any of these local minima may serve as the ground state for the Universe at present, provided that the lifetime of the metastable state is large in comparison to the age of the Universe. The latter is plausible [1,2] because the tunneling rate in quantum ?eld theory is naturally suppressed by the exponential of a typically large dimensionless number, the saddle point value of the Euclidean action. Are there any empirical, or general theoretical considerations that could rule out the possibility of the Universe at present being in a long-lived, but metastable, vacuum? Apparently, the answer
address: firstname.lastname@example.org; address after October 1, 1996: Theory Division, CERN, CH-1211 Geneva 23, Switzerland
is no. Di?erent minima of the scalar potential are characterized by di?erent values of the cosmological constant. However, the cosmological constant problem is just as severe in the stable vacuum as it is in a metastable one. In fact, in a large class of locally supersymmetric Uni?ed theories the cosmological constant can be ?ne-tuned to be zero in any of the local minima, but not in the global minimum . In principle, one can imagine a Gedanken experiment to determine whether the vacuum is true, or false. However, if a metastable vacuum has existed for τU = 10 billion years, then any e?ects of the metastability  would be characterized by the scale < 1/τU ? 10?33 eV, beyond any hope of being observable. Sadly, the ?rst direct evidence of the vacuum instability would be a catastrophic event. The question, however, has more than just eschatological relevance. It is very important scienti?cally, because ruling out the possibility that the Universe rests in the false vacuum would impose strong constraints on theories of fundamental interactions [1,2,5]. Clearly, only the false vacua whose lifetimes are small in comparison to the age of the Universe are ruled out empirically. We concentrate on the TeV-scale CCB minima. These generally disappear at temperatures T ? 1 TeV. Therefore, if the temperature of reheating after in?ation is 10 TeV or higher, one can assume that the electroweak symmetry is restored. Then at T = Tc ? 100 GeV the electroweak phase transition proceeds from an SU (3)×SU (2)×U (1)
lightest squark mass, GeV
111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 0000002 111111 000000 -cLT 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 2 111111 000000-cRT 111111 000000 111111 000000 111111 111 000000 2 000 111 -m t 000
11111 00000 11111 00000
100 1000 2000 3000 4000 5000
A+ ? , GeV t
Figure 1. Region of parameters (shaded) which can be ruled out by requiring stability of the SU (2)×U (1) symmetric minimum above the electroweak transition temperature. Coe?cients cL ,R are de?ned in Ref. .
Figure 2. The domains of stability (stars) and instability (boxes) of the false SML vacuum with respect to tunneling into the global CCB minimum. Light top squark and large trilinear couplings generally correspond to a lower and thinner barrier and, thus, higher probability of tunneling.
-symmetric phase to the phase of the broken symmetry. It was shown in  that for a wide range of parameters (Fig. 1) this transition favors the Standard Model-like (SML) minimum over the CCB minima. The probability of tunneling from the SML into a CCB minimum, or a UFB valley, determines the lifetime of a false SML vacuum. Tunneling rate can be evaluated using the semiclassical approx? imation  and is proportional to exp(?S[φ]), ? is the Euclidean action of the so called where S[φ] ? “bounce”, φ(x), a solution of the classical Euclidean ?eld equations. In practice, however, ?nd? ing φ(x) numerically is very di?cult (or nearly impossible), especially in the case of a potential that depends on more than one scalar ?eld. This ? is because φ(x) is an unstable solution, as it must be to be a saddle point of the functional S[φ]. An e?ective alternative to solving the equations of motion is to use the method of Ref. . The idea is to replace the action S with a di?erent func? ? tional, S, for which the same solution, φ(x), is a ? minimum, rather than a saddle point. Then φ(x)
can be found numerically using a straightforward ? relaxation technique to minimize S. Another signi?cant simpli?cation comes from the observation that the tunneling rate (in semiclassical approximation) is independent of the ? physics at the scales large in comparison to φ(0), the escape point. This non-perturbative decoupling  of high-energy physics allows one to treat the UFB valleys on the same footing with the very deep CCB minima. Essentially, one can set a cuto? at, e. g., 10 TeV and, as long as the bounce solutions found numerically do not extend beyond this limit, one can justi?ably ignore the physics at the higher energy scales. For a 10 billion year old Universe, the tunnel? ing probability is negligible if S[φ] > 400. The most “dangerous” CCB minima, i. e. those that to correspond the relatively high tunneling rates, are associated with the third generation squarks. This is because the action of the bounce is proportional  to the inverse Yukawa coupling squared. The requirement of stability of the SML vac-
3 minimum. The existence of the CCB minima of the scalar potential results in some important constraints on models with low-energy sypersymmetry. However, the commonly imposed requirement that the SML minimum be global is too strong and may overconstrain the theory. REFERENCES 1. M. Claudson, L. J. Hall and I. Hinchli?e, Nucl. Phys. B228, 501 (1983). 2. A. Kusenko, P. Langacker and G. Segre, UPR-0677-T, hep-ph/9602414 . 3. S. Weinberg, Phys.Rev.Lett. 48 (1982) 1776. 4. A. Kusenko, Phys. Lett. B377 (1996) 245 (hep-ph/9509275). 5. L. Alvarez-Gaume, J. Polchinski and M. Wise, Nucl. Phys. B221, 495 (1983); J. M. Frere, D. R. T. Jones and S. Raby, Nucl. Phys. B222, 11 (1983); M. Drees, M. Gluck and K. Grassie, Phys. Lett. B157, 164 (1985); J. F. Gunion, H. E. Haber and M. Sher, Nucl. Phys. B306, 1 (1988); H. Komatsu, Phys. Lett. B215, 323 (1988); P. Langacker and N. Polonsky, Phys. Rev. D 50, 2199 (1994); A. J. Bordner, KUNS-1351 (hepph/9506409); J. A. Casas, A. Lleyda and C. Mu? oz, FTUAM 95/11, IEM-FT-100/95 n (hep-ph/9507294). 6. S. Coleman, Phys. Rev. D15, 2929 (1977); C. G. Callan and S. Coleman, Phys. Rev. D16, 1762 (1977); A. Kusenko, Phys. Lett. B358, 47 (1995). 7. A. Kusenko, Phys. Lett. B358, 51 (1995).
A t+3 ? , GeV
Figure 3. The dotted line represents the empirical criterion for the absence of the global CCB minima: A2 + 3?2 < 3M 2 , where M 2 = m2 + t ? t
m2 . Taking into account the tunneling rates ? t
relaxes this constraint to, roughly, A2 + 3?2 < t 7.5M 2, shown as the dashed line. The scale is logarithmic.
uum constrains the parameter space of the MSSM . In particular, the trilinear terms in the potential have an upper limit that depends on the quadratic mass terms of squarks (Fig. 2). However, these bounds are not as stringent as they would have been, should one require the SML minimum to be the global minimum of the potential. We found  that for a large portion of the parameter space the presence of the global CCB minimum is irrelevant because the time required for the Universe to relax to its lowest energy state exceeds its present age . This is illustrated in Fig. 3. In summary, the color and charge conserving minimum may not be the global minimum of the MSSM potential. It is possible that the Universe rests in a false vacuum whose lifetime is large in comparison to the present age of the Universe. Under fairly general conditions, the SML vacuum is favored by the thermal evolution of the Universe, even if it does not represent the global