The MSSM without Gluinos; an Effective Field Theory for the Stop Sector

TL;DR

This paper develops an effective field theory for the MSSM focusing on the stop sector when gluinos and heavy squarks are much heavier, highlighting how parameter running differs from the full MSSM.

Contribution

It introduces a new effective field theory framework for the MSSM that accounts for partial supersymmetry and integrates out heavy particles, providing modified parameter evolution.

Findings

Sizable numerical differences between EFT and naive MSSM calculations.

Modified running of parameters in the stop sector due to heavy gluino decoupling.

Broken SUSY relations between parameters in the effective theory.

Abstract

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 $O(\alpha_3)$ and $O(Y_{t,b}^2)$ 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.

Equations

$${\mathcal{L}}_{\mathrm{eff}}$$

$$W$$

$$V_{\mathrm{soft}}$$

$$V_D=\frac{g_3^2}{2}{\left({\tilde{q}}_L^{†}{T}^A{\tilde{q}}_L-{\tilde{t}}_R^{†}{T}^A{\tilde{t}}_R-{\tilde{b}}_R^{†}{T}^A{\tilde{b}}_R\right)}^2,$$

$${\bar{Y}}_t$$

$$\bar{\mu }$$

$${\bar{m}}_{12}^2$$

$${\lambda }_1^u$$

$${\lambda }_1^d$$

$${\lambda }_4$$

$${\lambda }_7$$

$$\Delta {\bar{m}}_2^2$$

$$\Delta {\bar{m}}_{\tilde{Q}}^2$$

$$\Delta {\bar{m}}_{\tilde{t}}^2$$

$$\Delta {\bar{m}}_{12}^2$$

$$\Delta \bar{\mu }$$

$$\Delta {\bar{A}}_t$$

$$\Delta {\bar{\mu }}_t$$

$$16{\pi }^2\frac{d}{dt}{\bar{g}}_3$$

$$16{\pi }^2\frac{d}{dt}{\bar{Y}}_t$$

$$16{\pi }^2\frac{d}{dt}{\bar{Y}}_b$$

$$16{\pi }^2\frac{d}{dt}{\bar{Y}}_{q_{3L}}$$

$$16{\pi }^2\frac{d}{dt}{\bar{Y}}_{t_R}$$

$$16{\pi }^2\frac{d}{dt}{\bar{Y}}_{b_R}$$

$$16{\pi }^2\frac{d}{dt}{\bar{m}}_2^2$$

$$16{\pi }^2\frac{d}{dt}{\bar{m}}_1^2$$

$$16{\pi }^2\frac{d}{dt}{\bar{m}}_{12}^2$$

$$16{\pi }^2\frac{d}{dt}{\bar{m}}_{\tilde{Q}}^2$$

$$16{\pi }^2\frac{d}{dt}{\bar{m}}_{\tilde{t}}^2$$

$$16{\pi }^2\frac{d}{dt}\bar{\mu }=\frac{3}{2}({\bar{Y}}_{q_{3L}}^2+{\bar{Y}}_{t_R}^2+{\bar{Y}}_{b_R}^2)\bar{\mu },$$

$$16{\pi }^2\frac{d}{dt}{\bar{A}}_t$$

$$16{\pi }^2\frac{d}{dt}{\bar{\mu }}_t$$

$$16{\pi }^2\frac{d}{dt}{\lambda }_1^u$$

$$16{\pi }^2\frac{d}{dt}{\lambda }_2^u$$

$$16{\pi }^2\frac{d}{dt}{\lambda }_3^u$$

$$16{\pi }^2\frac{d}{dt}{\lambda }_1^d$$

$$16{\pi }^2\frac{d}{dt}{\lambda }_2^d$$

$$16{\pi }^2\frac{d}{dt}{\lambda }_3^d$$

$$16{\pi }^2\frac{d}{dt}{\lambda }_4$$

$$16{\pi }^2\frac{d}{dt}{\lambda }_5$$

$$16{\pi }^2\frac{d}{dt}{\lambda }_6$$

$$16{\pi }^2\frac{d}{dt}{\lambda }_7$$

$$16{\pi }^2\frac{d}{dt}{\lambda }_8$$

Full text

††preprint:

PSI-PR-17-02 ††preprint:

TU-1043

I Introduction

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 $\lambda$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 $m_{\tilde{t}_{1}}=300\,\text{GeV}$, 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 $\tilde{t}_{1}\rightarrow t\,\tilde{\chi}^{0}_{1}$ 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 $t\bar{t}$-background Cheng:2016mcw . The three-body decay $\tilde{t}_{1}\rightarrow Wb\tilde{\chi}^{0}_{1}$ was analyzed theoretically in Porod:1996at and experimentally in ATLAS-CONF-2016-076 . Finally the decay $\tilde{t}_{1}\rightarrow c\,\tilde{\chi}^{0}_{1}$ and the less important four-body decay $\tilde{t}_{1}\rightarrow\,\tilde{\chi}^{0}_{1}d_{i}f\bar{f}^{\prime}$ 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 $800\,\text{GeV}$ 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 $\mu$ has only to be larger than $350\,\text{GeV}$ 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 $\tan\beta$ suggest that CP-odd Higgs bosons should be heavier than $800\,\text{GeV}$ 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 $M$ 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 $M$. 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 $\alpha_{3}=g_{3}^{2}/(4\pi)$, $Y_{t}^{2}$ and $Y_{b}^{2}$ (neglecting $O(g_{1}^{2})$, $O(g_{2}^{2})$ 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 $O(\alpha_{3},Y_{t,b}^{2})$ 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 $M$, 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 $SU(2)$ multiplet with the left-handed stop. However, we assume that the right-handed sbottom is heavy, with the mass of the order $M$. Therefore, we consider the following effective Lagrangian which is valid below the scale $M$,

[TABLE]

with partial supersymmetry. Here ${\cal L}_{K}$ denotes the kinetic terms and gauge interactions, and $V(H_{u},H_{d})$ denotes the quartic couplings of the Higgs doublets ($H_{u}$, $H_{d}$). For the interactions involving four squarks, the $SU(3)$ color indices are contracted within the parentheses. Similarly, the $SU(2)$ indices in the two-squark-two-Higgs interactions are contracted within the parentheses. $i,j$ are the $SU(2)$ indices and the dot denotes the contraction of $SU(2)$ indices as $A\cdot B=A_{1}B_{2}-A_{2}B_{1}$. For simplicity, we also assume that the electroweak gauginos and sleptons are heavy. However, since we neglect $O(g_{1}^{2})$, $O(g_{2}^{2})$ effects in the following, relaxing this assumption would leave our RGEs unchanged. We also ignore the non-holomprphic Higgs-quark couplings $\bar{t}_{R}H_{d}^{{\dagger}}q_{3L}$ and $\bar{b}_{R}H_{u}^{{\dagger}}q_{3L}$ 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 $M$ 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

[TABLE]

the soft SUSY breaking terms

[TABLE]

and the $D$ terms

[TABLE]

where $T^{A}$ are the generators of $SU(3)$ in the fundamental representation.

The matching conditions for the bilinear terms and the trilinear couplings are

[TABLE]

The couplings between squarks and Higgs bosons are generated by F- and D-terms in the MSSM Lagrangian. At the scale $M$, they are given by

[TABLE]

keeping only Yukawa couplings and $g_{3}$.

II.2 1-loop matching

For the matching, we need to include the one-loop effects enhanced by powers of $M$ 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 ${\mu_{R}}=M$

[TABLE]

[TABLE]

For the trilinear term the shift reads

[TABLE]

All the other parameters relevant for the stop sector are dimensionless and therefore do not receive any $M$ 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 $\mathcal{O}(\alpha_{3},Y_{t,b}^{2})$ 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 ($t\equiv\log{\mu_{R}}$, where $\mu_{R}$ denotes the renormalization scale)

[TABLE]

where the first term on the right hand side is the SM contribution. The effective quark-quark-Higgs Yukawa couplings evolve according to

[TABLE]

while the evolution of the ones entering the Higgsino-quark-squark vertex is determined by

[TABLE]

For the Higgs mass parameters we find

[TABLE]

and for the bilinear squark mass terms

[TABLE]

The Higgsino mass in the effective theory evolves as

[TABLE]

and the effective trilinear $H{\tilde{q}}{\tilde{q}}$ coupling as

[TABLE]

Finally for the quartic $HH{\tilde{q}}{\tilde{q}}$ and ${\tilde{q}}{\tilde{q}}{\tilde{q}}{\tilde{q}}$ couplings one obtains

[TABLE]

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 $M$ to the stop mass scale $m_{\tilde{t}}$, we obtain the $O(\alpha_{3},Y_{t,b})$ contributions enhanced by $\log(M/m_{\tilde{t}})$.

II.4 Stop masses

In the effective theory, the stop mass matrix in the ($\tilde{t}_{L}$, $\tilde{t}_{R}$) basis reads

[TABLE]

where $v_{u,d}=\langle H^{0}_{u,d}\rangle$ 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 $\overline{\rm MS}$ scheme. These masses are closely related to the left-handed sbottom mass

[TABLE]

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 $M=5$ TeV, the stop mass scale $m_{\tilde{t}}=700$ GeV, running top mass $m_{t}(m_{\tilde{t}})=\bar{Y}_{t}(m_{\tilde{t}})v_{u}=150$ GeV, $\alpha_{3}(m_{\tilde{t}})=0.1$, and $\tan\beta=v_{u}/v_{d}=50$. 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: $\bar{m}_{\tilde{t}}(m_{\tilde{t}})=800$ GeV, $\bar{m}_{\tilde{Q}}(m_{\tilde{t}})=900$ GeV, $\bar{A}_{t}(m_{\tilde{t}})=1200$ GeV which lead to a one-loop mass of $125$ GeV for the lightest Higgs, a light stop of $700$ GeV and a sbottom mass of about $900$ GeV.

• •

The top Yukawa coupling $Y_{t}$

In the full MSSM, the Yukawa coupling $Y_{t}$ of the superpotential enters top-top-Higgs, stop-stop-Higgs couplings as well as stop-squark-Higgsino couplings in the same way. However, in the EFT these couplings are independent quantities and they evolve differently below the scale $M$. This is depicted in Fig. 1, where the evolution of $Y_{t}$ in the naive approach using MSSM RGE is compared to those of $\bar{Y}_{t}$, $\bar{Y}_{q_{3L}}$, $\bar{Y}_{t_{R}}$ and $\bar{Y}_{\tilde{t}}\equiv\bar{\mu}_{t}/\bar{\mu}$ in the EFT. When the values of $\bar{Y}_{t}$ and $Y_{t}$ are determined at the stop mass scale to give the SM running top mass, their values at the scale $M$ are quite different. Note that these couplings are dimensionless and therefore do not depend on the choice of the parameters for $\bar{m}_{\tilde{t}},\,\bar{m}_{\tilde{Q}}$ and $\bar{A}_{t}$.

• •

The quartic coupling of right-handed stops $\lambda_{8}$

In the full MSSM the quartic coupling of right-handed stops $\lambda_{8}$ is given by $\frac{1}{6}g_{3}^{2}$ by SUSY relation and evidently also evolves in the same way as $\frac{1}{6}g_{3}^{2}$. However, in the EFT $\lambda_{8}$ and $\bar{g}_{3}^{2}$ follow different RGEs below the scale $M$, as seen in Fig. 2. The relative difference at the scale $m_{\tilde{t}}$ amounts to roughly 30%. Again, since $\lambda_{8}$ has no mass dimension, its running does not depend on $\bar{m}_{\tilde{t}},\,\bar{m}_{\tilde{Q}}$ and $\bar{A}_{t}$.

Among the quartic scalar couplings $\lambda_{1-8}$, the running of $\lambda_{8}$ in the EFT exhibits the largest deviation from the one in the full theory. This is due to symmetry factors, leading to large coefficients of the box diagrams and self-couplings which are responsible for a change in sign on the $g_{3}^{4}$-dependence. The deviations of the other couplings $\lambda_{1-7}$ from the ones in the full theory are either positive or negative, but are smaller than 20% for our parameter set. We therefore do not show the figures of their runnings here.

IV Conclusions

In this article, we constructed an effective theory of the stop sector obtained from the full MSSM by integrating out the first and second generation of squarks and the gluino (which we assume to have a common mass of the order $M$). We computed the matching effects for the dimensionful quantities which are enhanced by powers of $M$ at $O(\alpha_{3},Y_{t,b}^{2})$. In addition, we obtained the complete $O(\alpha_{3},Y_{t,b}^{2})$ RGEs of the couplings within the EFT. In the numerical analysis we highlighted that couplings which are related via SUSY identities within the full MSSM have different RGEs within the EFT, which can lead to sizable differences. We illustrated this effect for the top Yukawa couplings and the quartic coupling of right-handed stops, finding differences up to 30% between the EFT and the naive approach. Such deviations could play a role in future test of the stop-stop or stop-Higgs interactions which also enter the calculation of the Higgs mass.

Acknowledgements.

J.A. and C.G. acknowledge the support from the Swiss National Science Foundation. The work of A.C. was supported by a Marie Curie Intra-European Fellowship of the European Community’s 7th Framework Programme under contract number (PIEF-GA-2012-326948) and by an Ambizione Grant of the Swiss National Science Foundation.

Appendix

.1 RGEs of the full MSSM

Here we recall the RGEs of the parameters in the full MSSM, again taking into account $O(\alpha_{3})$ and $O(Y_{t,b}^{2})$ effects.

[TABLE]

[TABLE]