The MSSM without Gluinos; an Effective Field Theory for the Stop Sector
In this article we study the MSSM with stops and Higgs scalars much lighter than gluinos and squarks of the first two generations. In this setup, one should use an effective field theory with partial supersymmetry in which the gluino and heavy squarks are integrated out in order to connect SUSY parameters (given at a high scale) to observables in the stop sector. In the construction of this effective theory, valid below the gluino mass scale, we take into account and effects and calculate the matching as well as the renormalization group evolution. As a result, the running of the parameters for the stop sector is modified with respect to the full MSSM and SUSY relations between parameters are broken. We show that for some couplings sizable numerical differences exist between the effective field theory approach and the naive calculation based on the MSSM running.
There are several theoretical arguments for a light stop in supersymmetric theories. Foremost, in natural supersymmetry (SUSY) light stops are required to cancel the quadratic divergence of the Higgs mass originating from the self-energy involving a top quark, while the other supersymmetric partners can be much heavier Dimopoulos:1995mi (); Giudice:2004tc () due to the smaller couplings to the Higgs. Moreover, the renormalization group equations (RGE) of the minimal supersymmetric standard model (MSSM) generically drive the bilinear mass term parameters of the third generation squarks to lower values (compared to the first two generations) due to their non-negligible Yukawa couplings Inoue:1982pi (); Inoue:1983pp (); Derendinger:1983bz (); Gato:1984ya (); Falck:1985aa (); Martin:1993zk ().
Although the measured Higgs mass of around 125 GeV Aad:2012tfa (); Chatrchyan:2012ufa () prefers rather heavy (around the TeV scale) Ellis:1990nz (); Okada:1990vk (); Haber:1990aw () rather than light stops in the MSSM, this is not necessarily the case in the NMSSM Ellwanger:2009dp (), in SUSY models Hall:2011aa (), models with light sneutrinos Chala:2017jgg () or in supersymmertic models with additional D-term Batra:2003nj () or F-term Espinosa:1998re () contributions to the scalar potential. Also large (or even maximal Djouadi:2005gj (); Brummer:2012ns (); Wymant:2012zp ()) stop mixing angles help to get the right Higgs mass with rather light stops.
LHC searches for top squarks (using simplified models) set a lower bound on its mass of around , which however heavily depends on the neutralino mass. Depending on the stop and the neutralino mass, different decay modes are studied. For the decay channel Aad:2014qaa (); ATLAS-CONF-2016-050 (); Chatrchyan:2013xna (), the limits are quite stringent, even though for light neutralinos very light stops can not be excluded due to the high -background Cheng:2016mcw (). The three-body decay was analyzed theoretically in Porod:1996at () and experimentally in ATLAS-CONF-2016-076 (). Finally the decay and the less important four-body decay are treated in Aaboud:2016tnv (); Aebischer:2014lfa (); Grober:2014aha () and constraints were derived by the ATLAS collaboration from the monojet analysis in Aad:2014nra (). Some bounds can be avoided in kinematic boundary regions or once non-minimal flavour violation is included. However, recently efforts of closing these gaps have been made Macaluso:2015wja (); Belyaev:2015gna (); Kobakhidze:2015scd (); Crivellin:2016rdu () and stops should in general not be lighter than 300 GeV. Nevertheless, the mass bound for the stop is still weaker than the strong bounds on the squark masses of the first two generations and also on the gluino mass ATLAS:2016lsr (); Sakuma:2016nxo (). For sbottom quarks LHC searches suggest masses of above ATLAS:2017qih (); Sirunyan:2017kiw (). The bounds on sparticles with EW interactions only are much less stringent Aad:2015jqa (); ATLAS:2016uwq (); Arina:2016rbb (); Khachatryan:2014mma (); CMS:2016gvu (). For example, in the case of heavy winos the Higgsino mass parameter has only to be larger than Aad:2014vma (). It can be shown however that by changing the assumptions on the composition of charginos and neutralinos, collider limits can get even further weakened Bharucha:2013epa (); Martin:2014qra (); Calibbi:2014lga (); Chakraborti:2015mra (). For the Higgs bosons, different fits ATLAS:2014kua (); Khachatryan:2014jya (); Celis:2013rcs (); Chiang:2013ixa (); Chen:2013rba (); Craig:2013hca (); Wang:2014lta (); Grinstein:2013npa (); Han:2017pfo () suggest an alignment limit, in which the lightest CP-even Higgs boson takes the role of the SM Higgs. Collider limits on non-SM Higgs bosons for large values of suggest that CP-odd Higgs bosons should be heavier than Aaboud:2016cre (); Khachatryan:2014wca ().
If the gluino (or the squarks of the first two generations Giudice:2004tc (); ArkaniHamed:2004yi ()) is much heavier than the stops, an effective theory (EFT) with partial SUSY must be constructed in which the gluino (squarks) is integrated out Muhlleitner:2008yw () (Carena:2008rt (); Giudice:2011cg ()). Such a hierarchy can for example be achieved for MSSM-like models in a Scherk-Schwarz breaking scenario Pomarol:1998sd (); Dimopoulos:2014aua (); Garcia:2015sfa (); Delgado:2016vib (). The construction of the effective theory for the stop sector is the goal of this article. Assuming a common large mass of order for the gluino and the squarks of the first two generations, we compute the matching condition between the full MSSM and the effective theory, including one-loop contributions which are enhanced by powers of . Furthermore, since some supermultiplets are partially integrated out in the effective theory, the supersymmetric relations between gauge/Yukawa couplings, gaugino/Higgsino couplings and four-scalar couplings are broken in the effective theory by radiative corrections. Therefore, these couplings in the effective theory have an independent renormalization group evolution, as discussed in Chankowski:1989du (); Hikasa:1995bw (); Nojiri:1996fp (); Cheng:1997sq (); Cheng:1997vy (); Nojiri:1997ma (); Katz:1998br (); Kiyoura:1998yt (); Muhlleitner:2008yw () mainly for the gaugino-matter couplings.
This article is structured as follows: In the next section we establish our effective theory for the stop sector and calculate the matching as well as the running of the relevant parameters at order , and (neglecting , and Higgs self-coupling effects). This section is followed by a numerical analysis in Sec. III. Finally we conclude in Sec. IV.
Ii The effective theory for the stop sector
The aim of this section is to construct the effective theory for the MSSM stop sector, including and enhanced effects. As noted before, we assume that the gluino and the squarks of the first two generations are much heavier, with masses of the order , than the stops, the Higgs scalars and the Higgsinos. The left-handed sbottom is also assumed to be light such that it remains in the effective theory, forming an multiplet with the left-handed stop. However, we assume that the right-handed sbottom is heavy, with the mass of the order . Therefore, we consider the following effective Lagrangian which is valid below the scale ,
with partial supersymmetry. Here denotes the kinetic terms and gauge interactions, and denotes the quartic couplings of the Higgs doublets (, ). For the interactions involving four squarks, the color indices are contracted within the parentheses. Similarly, the indices in the two-squark-two-Higgs interactions are contracted within the parentheses. are the indices and the dot denotes the contraction of indices as . For simplicity, we also assume that the electroweak gauginos and sleptons are heavy. However, since we neglect , effects in the following, relaxing this assumption would leave our RGEs unchanged. We also ignore the non-holomprphic Higgs-quark couplings and which are induced at the loop-level Hempfling:1993kv (); Hall:1993gn (); Carena:1994bv (); Noth:2008tw (); Hofer:2009xb (); Crivellin:2010er (); Crivellin:2011jt (); Crivellin:2012zz ().
ii.1 Tree-level matching
At the matching scale the Lagrangian of eq. (1) has to be compared to the one of the full MSSM (see for example Fayet:1976cr (); Fayet:1976et (); Haber:1984rc (); Rosiek:1995kg ()) which originates from the superpotential
the soft SUSY breaking terms
and the terms
where are the generators of in the fundamental representation.
The matching conditions for the bilinear terms and the trilinear couplings are
The couplings between squarks and Higgs bosons are generated by F- and D-terms in the MSSM Lagrangian. At the scale , they are given by
keeping only Yukawa couplings and .
ii.2 1-loop matching
For the matching, we need to include the one-loop effects enhanced by powers of since their contributions may be comparable to the tree level ones shown in the previous subsection. They can only appear in bilinear and trilinear terms, as seen by dimensional analysis. The bilinear terms receive the following shifts at the matching scale
For the trilinear term the shift reads
All the other parameters relevant for the stop sector are dimensionless and therefore do not receive any enhanced corrections.
ii.3 Renormalization group evolution
The running of the full MSSM parameters Inoue:1982pi (); Inoue:1983pp (); Derendinger:1983bz (); Gato:1984ya (); Falck:1985aa () is known at the two-loop level West:1984dg (); Jones:1984cx (); Martin:1993yx (); Yamada:1993ga (); Martin:1993zk (). Here we give the one-loop beta functions to for the parameters of our effective theory in eq. (1). The corresponding results for the full MSSM are summarized in the appendix. For the strong coupling constant we have (, where denotes the renormalization scale)
where the first term on the right hand side is the SM contribution. The effective quark-quark-Higgs Yukawa couplings evolve according to
while the evolution of the ones entering the Higgsino-quark-squark vertex is determined by
For the Higgs mass parameters we find
and for the bilinear squark mass terms
The Higgsino mass in the effective theory evolves as
and the effective trilinear coupling as
Finally for the quartic and couplings one obtains
Note that in all equations above we assumed real parameters. However, all formula can be easily generalized to the complex case by simply replacing a square by the absolute value squared.
By integrating these RGEs from to the stop mass scale , we obtain the contributions enhanced by .
ii.4 Stop masses
In the effective theory, the stop mass matrix in the (, ) basis reads
where are the vacuum expectation values of the Higgs scalars. By diagonalizing this matrix one obtains the stop masses and the stop mixing angle, both in the scheme. These masses are closely related to the left-handed sbottom mass
by SU(2) gauge symmetry.
Iii Numerical Analysis
From the previous analysis, we can see that, by integrating out the gluino and the squarks of the first two generations, parameters which were originally related via SUSY in the full MSSM, do not evolve anymore in the same way in the EFT. Let us illustrate this effect with two examples where striking differences between the EFT approach and the full MSSM emerge. Here we set the input parameters as TeV, the stop mass scale GeV, running top mass GeV, , and . Furthermore, we have chosen the massive parameters such that the collider constraints for the Higgs mass and the stop and sbottom masses are fulfilled. This can be achieved by using the values: GeV, GeV,