Shifted focus point scenario from the minimal mixed mediation of SUSY breaking
We employ both the minimal gravity- and the minimal gauge mediations of supersymmetry breaking at the grand unified theory (GUT) scale in a single supergravity framework, assuming the gaugino masses are generated dominantly by the minimal gauge mediation effects original (). In such a “minimal mixed mediation model,” a “focus point” of the soft Higgs mass parameter, emerges at - energy scale, which is exactly the stop mass scale needed for explaining the Higgs boson mass without the “-term” at the three loop level. As a result, can be quite insensitive to various trial stop masses at low energy, reducing the fine-tuning measures to be much smaller than even for a - low energy stop mass and at the GUT scale. The gluino mass is predicted to be about , which could readily be tested at LHC run2.
Although the minimal supersymmetric standard model (MSSM) has been believed the most promising theory beyond the standard model (SM), guiding the SM to a grand unified theory (GUT) or string theory book (); PR1984 (), any evidence of SUSY has not been observed yet at the large hadron collider (LHC). The mass bounds on the SUSY particles have gradually increased, and now they seem to start threatening the traditional status of SUSY as a prominent solution to the naturalness problem of the SM. Actually, a barometer of the naturalness of the MSSM is the mass of “stop.” Due to the large top quark Yukawa coupling (), the top and stop dominantly contribute to the radiative physical Higgs mass squared and also the renormalization of a soft mass squared of the Higgs () in the MSSM. The renormalization effect on would linearly be sensitive to the stop mass squared, while it depends just logarithmically on a ultraviolet (UV) cutoff book (). Since the Higgs mass parameters, and are related to the the boson mass together with the “Higgsinos” mass, book (),
should be finely tuned to yield for a given , if they are excessively large. According to the recent analysis based on the three-loop calculations, the stop mass required for explaining the Higgs boson mass LHCHiggs () without any other helps is about - 3-loop (). Thus, a fine-tuning of order or smaller looks unavoidable in the MSSM for a GUT scale cut-off.
In order to more clearly see the UV dependence of and properly discuss this “little hierarchy problem”, however, one should suppose a specific UV model and analyze its resulting full renormalization group (RG) equations. One nice idea is the “focus point (FP) scenario” FMM1 (). It is based on the minimal gravity mediation (mGrM) of SUSY breaking. So the soft mass squareds such as and those of the left handed (LH) and right handed (RH) stops, () as well as the gaugino masses () are given to be universal at the GUT scale, and . As pointed out in FMM1 (), if the soft SUSY breaking “-terms” are zero at the GUT scale and the unified gaugino mass is just a few hundred GeV, converges to a small negative value around the boson mass scale in this setup, regardless of its initial values given by at the GUT scale FMM1 (). In the RG solution of at the scale, thus,
where , () can numerically be estimated using RG equations, happens to be quite small with the above universal soft masses. Since stop masses are quite sensitive to , hence, could remain small enough even with a relatively heavy stop mass in the FP scenario in contrast to the naive expectation.
However, the experimental bound on the gluino mass has already exceeded gluinomass (). As expected from Eqs. (1) and (2), a too large needed for at low energy would require a fine-tuned large for of particularly for a relatively light stop mass () cases. When the stop mass is around -, the stop should decouple from the RG equations below -, which makes sizable in Eq. (2) KS (). Then, a much larger is necessary for EW symmetry breaking. Since the RG running interval between - and scale, to which modified RG equations should be applied, is too large, the FP behavior is seriously spoiled with such heavy SUSY particles.
The best way to rescue the FP idea is to somehow shift the FP upto the stop decoupling scale KS (): needs to be made small enough before stops are decoupled. Then at the scale can be estimated using the Coleman-Weinberg potential book (); CQW (). It is approximately given by
where denotes the stop decoupling scale. Since the dependence of stop masses would be loop-suppressed, needs to be well-focused around . Due to the additional negative contribution to below , a small positive would be more desirable. In order to push up the FP to the desired stop mass scale -, we suggest to combine the mGrM and the minimal gauge mediation (mGgM) in a single supergravity (SUGRA) framework with a common SUSY breaking source. We will call it “minimal mixed mediation.”
First, let us consider the minimal Khler potential, and a superpotential where the observable and hidden sectors are separated as in the ordinary mGrM book ():
where  denotes fields in the hidden [observable] sector. The kinetic terms of and , thus, take the canonical form. We assume non-zero vacuum expectation values (VEVs) for s PR1984 ():
where and are dimensionless numbers, while () is the reduced Planck mass. Then, or gives the gravitino mass, , and the “-terms” of () become of order . The soft terms can read from the scalar potential of SUGRA: when the cosmological constant (C.C.) is fine-tuned to be zero, renormalizable terms of it are given by PR1984 ()
where is defined as , and is identified with the gravitino mass (). The first term of Eq. (6) is the -term potential in global SUSY, the second term is the universal soft mass term, and the remaining terms are -terms, which are proportional to .
Next, let us introduce one pair of messenger superfields , which are the SU(5) fundamental representations. Through their coupling with a SUSY breaking source , which is an MSSM singlet superfield,
the soft masses of the MSSM gauginos and scalar superpartners are also radiatively generated book ():
where is the quadratic Casimir invariant for a superfield , , and () denotes the MSSM gauge couplings. and are VEVs of the scalar and -term components of the superfield . The mGgM effects would appear below the messenger scale, . Here we assume that has the same magnitude as the VEV of the SU(5) breaking Higgs : . It is possible if a GUT breaking mechanism causes . Actually, the “” and “” gauge boson masses, GUT (), where is the unified gauge coupling, can be identified with the MSSM gauge coupling unification scale.
In addition to Eq. (4), the Khler potential (and hidden local symmetries we don’t specify here) can permit
where denotes a holomorphic monomial of hidden sector fields s with VEVs of order in Eq. (5), and so it is of order . Their kinetic terms still remain canonical. The U(1) symmetry forbids in the superpotential. Then, the resulting can be by including the SUGRA corrections with . Thus, the VEV of is of order like . They should be fine-tuned for the vanishing C.C.: a precise determination of is indeed associated with the C.C. problem. Here we set . Thus, the typical size of mGgM effects is estimated as
where is set to be at the GUT scale  due to relatively heavy colored superpartners (). Even for , we will keep this value, since it is fixed by a UV model. For in Eq. (7), the messenger scale drops down below . The soft masses generated by the mGgM in Eq. (8) are non-universal for , and the beta function coefficients of the MSSM fields should be modified above the scale by the messenger fields . The boundary conditions at the GUT scale are of the universal form as seen in Eq. (6). We have additional non-universal contributions by Eq. (8). They should be imposed at a given messenger scale, and so affect the RG evolutions of MSSM parameters for .
We also suppose that the gaugino masses from the mGrM are relatively suppressed. In fact, the gaugino mass term in SUGRA is associated with the first derivative of the gauge kinetic function PR1984 (), and so a constant gauge kinetic function at tree level () can realize it. Thus, the gaugino masses by Eq. (8) dominates over them in this case. Then, a simple analytic expression for the gaugino masses at the stop mass scale is possible: . It does not depend on messenger scales, , , etc.
The fact that the mGgM effects by Eq. (8) are proportional to or are important. Moreover, -terms from Eq. (6) are also proportional to . In this setup, thus, an (extrapolated) FP of must still exist at a higher energy scale. As is converted to a member of in Eq. (2), the naturalness of and becomes gradually improved, making smaller and smaller, until the FP reaches the stop decoupling scale.
Fig. 1 displays RG evolutions of under various trial s. The straight [dotted] lines correspond to the case of (or , “Case A”) [ (or , “Case B” )]. The discontinuities of the lines by additional boundary conditions arise at the messenger scales. As seen in Fig. 1, a FP of appears always at (or ) regardless of the chosen messenger scales. Hence, the wide ranges of UV parameters can yield almost the same values of at low energy. Under this situation, one can guess that happens to be selected, yielding - stop mass, and so eventually gets responsible for the Higgs mass. In both cases of Fig. 1, the low energy gaugino masses are
for . They would be testable at LHC run2. at low energy is about for Case A and B. So the contributions of to the radiative Higgs mass are smaller than 2.3 of those by the stops.
|Case I||Case II|
|Case III||Case IV|
Table 1 lists the soft squared masses at for the LH and RH stops, and the two MSSM Higgs bosons under the various s, when , and is or . We can see the changes of are quite small  under the changes of [–], because is well-focused at . Case I-IV yield again the same low energy gauginos masses as Eq. (11). at low energy turns out to be around or smaller for , and so its contribution to the Higgs boson mass is still suppressed. By Eq. (3) s further decrease to be negative below . With Eq. (1) are determined as for Case I, II, III, and IV, respectively. In Table 1, the fine-tuning measure ( FTmeasure ()) around are also presented. They are of order . () is estimated as for Case I, II, III, and IV, respectively. When , turn out to be about . Therefore, the parameter range
allows and to be smaller than and , respectively original (). We see that a larger would be preferred for a smaller . It is basically because is not focused unlike , even if it also contributes to as seen in Eq. (1). is easily obtained e.g. from the minimal SO(10) GUT GUT ().
In conclusion, we have noticed that a FP of appears at -, when the mGrM and mGgM effects are combined at the GUT scale for a common SUSY breaking source parametrized with , and the gaugino masses are dominantly generated by the mGgM effects. Even for a - stop mass explaining the Higgs mass, thus, the fine-tuning measures significantly decrease well below for and in the minimal mixed mediation. In this range, is smaller than . The expected gluino mass is about , which could readily be tested at LHC run2.
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