1 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 [1, 2] 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 [3,4,5,6,7,8].

Although the measured Higgs mass of around 125 GeV [9, 10] prefers rather heavy (around the TeV scale) [11,12,13] rather than light stops in the MSSM, this is not necessarily the case in the NMSSM [14], in \(\lambda \)SUSY models [15], models with light sneutrinos [16] or in supersymmertic models with additional D-term [17] or F-term [18] contributions to the scalar potential. Also large (or even maximal [19,20,21]) 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 \({\tilde{t}}_1\rightarrow t\,{\tilde{\chi }}^0_1\) [22,23,24], the limits are quite stringent, even though for light neutralinos very light stops cannot be excluded due to the high \(t{\bar{t}}\)-background [25]. The three-body decay \({\tilde{t}}_1\rightarrow Wb{\tilde{\chi }}^0_1\) was analyzed theoretically in [26] and experimentally in [27]. Finally the decay \({\tilde{t}}_1\rightarrow c\,{\tilde{\chi }}^0_1\) and the less important four-body decay \({\tilde{t}}_1\rightarrow \,{\tilde{\chi }}^0_1d_if{\bar{f}}'\) are treated in [28,29,30] and constraints were derived by the ATLAS collaboration from the monojet analysis in  [31]. Some bounds can be avoided in kinematic boundary regions or once non-minimal flavor violation is included. However, recently efforts of closing these gaps have been made [32,33,34,35] 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 [36, 37]. For sbottom quarks LHC searches suggest masses of above \(800\,\text {GeV}\) [38, 39]. The bounds on sparticles with EW interactions only are much less stringent [40,41,42,43,44]. For example, in the case of heavy winos the Higgsino mass parameter \(\mu \) has only to be larger than \(350\,\text {GeV}\) [45]. It can be shown, however, that by changing the assumptions on the composition of charginos and neutralinos, collider limits can get even further weakened [46,47,48,49]. For the Higgs bosons, different fits [50,51,52,53,54,55,56,57,58] 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 \(800\,\text {GeV}\) [59, 60].

If the gluino (or the squarks of the first two generations [2, 61]) is much heavier than the stops, an effective theory (EFT) with partial SUSY must be constructed in which the gluino (squarks) is integrated out [62,63,64]. Such a hierarchy can for example be achieved for MSSM-like models in a Scherk–Schwarz breaking scenario [65,66,67,68]. 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 [62, 69,70,71,72,73,74,75,76] 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 Sect. 3. Finally we conclude in Sect. 4.

2 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:

$$\begin{aligned} {{\mathcal {L}}}_\mathrm{eff}= & {} {{\mathcal {L}}}_K - {\bar{m}}_2^2 H_u^{\dagger }H_u - {\bar{m}}_1^2 H_d^{\dagger }H_d - V\left( H_u, H_d\right) \nonumber \\&+ {\bar{m}}_{12}^2 H_d\cdot H_u- {\bar{\mu }}{\tilde{H}}_U\cdot {\tilde{H}}_D +(\mathrm{h.c.}) \nonumber \\&-{\bar{m}}_{{\tilde{Q}}}^2 {\tilde{q}}_L^{\dagger }{\tilde{q}}_L - {\bar{m}}_{{\tilde{t}}}^2 {\tilde{t}}^{\dagger }_R{\tilde{t}}_R \nonumber \\&-{\bar{Y}}_t {\bar{t}}_R q_{3L}\cdot H_u -{\bar{Y}}_b{\bar{b}}_R H_d \cdot q_{3L} +(\mathrm{h.c.}) \nonumber \\&-\lambda _1^u ({\tilde{q}}_L^{\dagger }{\tilde{q}}_L ) (H^{\dagger }_uH_u ) -\lambda _2^u ({\tilde{q}}_L^{\dagger }H_u ) (H^{\dagger }_u{\tilde{q}}_L )\nonumber \\&-\lambda _3^u ({\tilde{t}}_R^{\dagger }{\tilde{t}}_R ) (H_u^{\dagger }H_u ) \nonumber \\&-\lambda _1^d ({\tilde{q}}_L^{\dagger }{\tilde{q}}_L ) (H_d^{\dagger }H_d ) -\lambda _2^d ({\tilde{q}}_L^{\dagger }H_d ) (H_d^{\dagger }{\tilde{q}}_L )\nonumber \\&-\lambda _3^d ({\tilde{t}}_R^{\dagger }{\tilde{t}}_R ) (H_d^{\dagger }H_d ) \nonumber \\&-\lambda _4 ({\tilde{q}}_{Li}^{\dagger }{\tilde{q}}_{Li} ) ({\tilde{q}}_{Lj}^{\dagger }{\tilde{q}}_{Lj} ) -\lambda _5 ({\tilde{q}}_{Li}^{\dagger }{\tilde{q}}_{Lj} ) ({\tilde{q}}_{Lj}^{\dagger }{\tilde{q}}_{Li} ) \nonumber \\&-\lambda _6 ({\tilde{q}}_{Li}^{\dagger }{\tilde{q}}_{Li} ) ({\tilde{t}}_{R}^{\dagger }{\tilde{t}}_{R} ) -\lambda _7 ({\tilde{q}}_{Li}^{\dagger }{\tilde{t}}_{R} ) ({\tilde{t}}_{R}^{\dagger }{\tilde{q}}_{Li} )\nonumber \\&-\lambda _8 ({\tilde{t}}_{R}^{\dagger }{\tilde{t}}_{R} ) ({\tilde{t}}_{R}^{\dagger }{\tilde{t}}_{R} ) \nonumber \\&-{\bar{A}}_t{\tilde{t}}_R^{\dagger }{\tilde{q}}_{L}\cdot H_u + {\bar{\mu }}_t {\tilde{t}}_R^{\dag } H_d^{\dag }{\tilde{q}}_L +(\mathrm{h.c.}) \nonumber \\&-{\bar{Y}}_{q_{3L}}{\tilde{t}}_R^{\dagger }q_{3L}\cdot {\tilde{H}}_U -{\bar{Y}}_{t_R}{\bar{t}}_R{\tilde{q}}_L\cdot {\tilde{H}}_U\nonumber \\&-{\bar{Y}}_{b_R}{\bar{b}}_R {\tilde{H}}_D \cdot {\tilde{q}}_L +(\mathrm{h.c.}), \end{aligned}$$
(1)

with partial supersymmetry. Here \({{\mathcal {L}}}_K\) 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 \(A\cdot B=A_1B_2-A_2B_1\). 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-holomorphic Higgs–quark couplings \({\bar{t}}_RH_d^{\dag }q_{3L}\) and \({\bar{b}}_RH_u^{\dag }q_{3L}\), which are induced at the loop level [77,78,79,80,81,82,83,84].

2.1 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 [85,86,87,88]) which originates from the superpotential

$$\begin{aligned} W= & {} Y_t T^c Q\cdot H_u + Y_b B^c H_d \cdot Q + \mu H_u\cdot H_d , \end{aligned}$$
(2)

the soft SUSY breaking terms

$$\begin{aligned} V_\mathrm{soft}= & {} m_{{\tilde{Q}}}^2 {\tilde{q}}^{\dagger }_L{\tilde{q}}_L + m_{{\tilde{t}}}^2 {\tilde{t}}^{\dagger }_R {\tilde{t}}_R+ m_{H_d}^2 H_d^{\dagger }H_d \nonumber \\&+ m_{H_u}^2 H_u^{\dagger }H_u +m_{{\tilde{b}}_R}^2{\tilde{b}}^{\dag }_R{\tilde{b}}_R \nonumber \\&+A_t {\tilde{t}}^{\dagger }_R {\tilde{q}}_{L}\cdot H_u+A_b {\tilde{b}}_R^{\dag }H_d\cdot {\tilde{q}}_{L} \nonumber \\&- m_{H_d H_u}^2 H_d\cdot H_u + (\mathrm{h.c.}) , \end{aligned}$$
(3)

and the D terms

$$\begin{aligned} {V_D} = \frac{{{g_3^2}}}{2}{ ( {{\tilde{q}}_L^\dag {T^A}{{{\tilde{q}}}_L} - {\tilde{t}}_R^\dag {T^A}{{{\tilde{t}}}_R} - {\tilde{b}}_R^\dag {T^A}{{{\tilde{b}}}_R}} )^2}, \end{aligned}$$
(4)

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

$$\begin{aligned} {\bar{Y}}_t= & {} Y_t , \quad {\bar{Y}}_b = Y_b , \quad {\bar{Y}}_{q_{3L}} = Y_t , \nonumber \\ {\bar{Y}}_{t_R}= & {} Y_t , \quad {\bar{Y}}_{b_R} = Y_b , \quad {\bar{A}}_t = A_t , \end{aligned}$$
(5)
$$\begin{aligned} {\bar{\mu }}= & {} \mu ,\quad {\bar{\mu }}_{t} = \mu Y_t ,\quad {\bar{m}}_2^2 = m_{H_u}^2+\mu ^2 , \nonumber \\ {\bar{m}}_1^2= & {} m_{H_d}^2+\mu ^2 , \end{aligned}$$
(6)
$$\begin{aligned} {\bar{m}}_{12}^2= & {} m_{H_dH_u}^2 ,\quad {\bar{m}}_{{\tilde{Q}}}^2 =m_{{\tilde{Q}}}^2 ,\quad {\bar{m}}_{{\tilde{t}}}^2 =m_{{\tilde{t}}}^2 . \end{aligned}$$
(7)

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

$$\begin{aligned} \lambda _1^u= & {} Y_t^2 ,\quad \lambda _2^u = -Y_t^2 ,\quad \lambda _3^u = Y_t^2 , \end{aligned}$$
(8)
$$\begin{aligned} \lambda _1^d= & {} Y_b^2 , \quad \lambda _2^d = -Y_b^2 ,\quad \lambda _3^d = 0 , \end{aligned}$$
(9)
$$\begin{aligned} \lambda _4= & {} -\frac{1}{12}g_3^2 , \quad \lambda _5 = \frac{1}{4}g_3^2 ,\quad \lambda _6 = \frac{1}{6} g_3^2 , \end{aligned}$$
(10)
$$\begin{aligned} \lambda _7= & {} -\frac{1}{2} g_3^2 +Y_t^2 , \quad \lambda _8 = \frac{1}{6} g_3^2 , \end{aligned}$$
(11)

keeping only Yukawa couplings and \(g_3\).

2.2 One-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 \({\mu _R}=M\):

$$\begin{aligned} \Delta {\bar{m}}_2^2= & {} 0 , \nonumber \\ \Delta {\bar{m}}_1^2= & {} -\frac{3}{16\pi ^2} (Y_b^2m_{{\tilde{b}}_R}^2+A_b^2 ) \left( 1-\log {\left( \frac{m_{{\tilde{b}}_R}^2}{M^2}\right) }\right) , \end{aligned}$$
(12)
$$\begin{aligned} \Delta {\bar{m}}_{{\tilde{Q}}}^2= & {} -\frac{1}{16\pi ^2} (Y_b^2m_{{\tilde{b}}_R}^2+A_b^2 ) \left( 1-\log {\left( \frac{m_{{\tilde{b}}_R}^2}{M^2}\right) }\right) \nonumber \\&+\frac{\alpha _3 C_F}{\pi }m^2_{{\tilde{g}}}\left( 1-\log {\left( \frac{m_{{\tilde{g}}}^2}{M^2}\right) }\right) , \end{aligned}$$
(13)
$$\begin{aligned} \Delta {\bar{m}}_{{\tilde{t}}}^2= & {} \frac{\alpha _3 C_F}{\pi }m_{{\tilde{g}}}^2\left( 1-\log {\left( \frac{m_{{\tilde{g}}}^2}{M^2}\right) }\right) , \end{aligned}$$
(14)
$$\begin{aligned} \Delta {\bar{m}}_{12}^2= & {} -\frac{3A_b\mu Y_b}{16\pi ^2}\left( 1-\log {\left( \frac{m_{{\tilde{b}}_R}^2}{M^2}\right) }\right) , \end{aligned}$$
(15)
$$\begin{aligned} \Delta {\bar{\mu }}= & {} 0 . \end{aligned}$$
(16)

For the trilinear term the shift reads

$$\begin{aligned} \Delta {\bar{A}}_t= & {} -\frac{A_b Y_t Y_b}{16\pi ^2}\left( 1-\log {\left( \frac{m_{{\tilde{b}}_R}^2}{M^2}\right) }\right) \nonumber \\&-\frac{\alpha _3 C_F}{\pi }m_{{\tilde{g}}}Y_t\left( 1-\log {\left( \frac{m_{{\tilde{g}}}^2}{M^2}\right) }\right) , \end{aligned}$$
(17)
$$\begin{aligned} \Delta {\bar{\mu }}_t= & {} 0 . \end{aligned}$$
(18)

All the other parameters relevant for the stop sector are dimensionless and therefore do not receive any M enhanced corrections.

2.3 Renormalization group evolution

The running of the full MSSM parameters [3,4,5,6,7] is known at the two-loop level [8, 89,90,91,92]. 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)

$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t}{\bar{g}}_3= & {} \left( -7+\frac{1}{2}\right) {\bar{g}}_3^3 , \end{aligned}$$
(19)

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

$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t}{\bar{Y}}_t= & {} {\bar{Y}}_t \left[ -8{\bar{g}}_3^2 +\frac{9}{2}{\bar{Y}}_t^2 +\frac{1}{2} {\bar{Y}}_b^2 +{\bar{Y}}_{t_R}^2 + \frac{1}{2} {\bar{Y}}_{q_{3L}}^2 \right] ,\nonumber \\ \end{aligned}$$
(20)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t}{\bar{Y}}_b= & {} {\bar{Y}}_b \left[ -8{\bar{g}}_3^2 +\frac{1}{2}{\bar{Y}}_t^2 + \frac{9}{2}{\bar{Y}}_b^2 + {\bar{Y}}_{b_R}^2 + \frac{1}{2} {\bar{Y}}_{q_{3L}}^2 \right] ,\nonumber \\ \end{aligned}$$
(21)

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

$$\begin{aligned}&16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} {\bar{Y}}_{q_{3L}} \nonumber \\&\quad = {\bar{Y}}_{q_{3L}} \left[ -4{\bar{g}}_3^2 +\frac{1}{2}{\bar{Y}}_t^2+\frac{1}{2}{\bar{Y}}_b^2 +4{\bar{Y}}_{q_{3L}}^2 + \frac{3}{2} {\bar{Y}}_{t_R}^2 \right] , \end{aligned}$$
(22)
$$\begin{aligned}&16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} {\bar{Y}}_{t_R} \nonumber \\&\quad = {\bar{Y}}_{t_R} \left[ -4{\bar{g}}_3^2 +{\bar{Y}}_t^2 +\frac{3}{2}{\bar{Y}}_{q_{3L}}^2 + \frac{7}{2} {\bar{Y}}_{t_R}^2 +{\bar{Y}}_{b_R}^2 \right] , \end{aligned}$$
(23)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} {\bar{Y}}_{b_R} = {\bar{Y}}_{b_R} \left[ -4{\bar{g}}_3^2 +{\bar{Y}}_b^2 +{\bar{Y}}_{t_R}^2 + \frac{7}{2} {\bar{Y}}_{b_R}^2 \right] . \end{aligned}$$
(24)

For the Higgs mass parameters we find

$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} {\bar{m}}_2^2= & {} 6{\bar{Y}}_t^2 {\bar{m}}_2^2 +6 (2\lambda ^u_1+\lambda ^u_2 ){\bar{m}}_{{\tilde{Q}}}^2 \nonumber \\&+ 6\lambda ^u_3{\bar{m}}_{{\tilde{t}}}^2 + 6 {\bar{A}}_t^2 , \end{aligned}$$
(25)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} {\bar{m}}_1^2= & {} 6 {\bar{Y}}_b^2 {\bar{m}}_1^2+6 \left( 2\lambda ^d_1+\lambda ^d_2\right) {\bar{m}}_{{\tilde{Q}}}^2 \nonumber \\&+ 6\lambda ^d_3{\bar{m}}_{{\tilde{t}}}^2 + 6 {\bar{\mu }}_t^2 , \end{aligned}$$
(26)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t}{\bar{m}}_{12}^2= & {} 3 ({\bar{Y}}_t^2+ {\bar{Y}}_b^2 ) {\bar{m}}_{12}^2 +6 {\bar{\mu }}_t {\bar{A}}_t , \end{aligned}$$
(27)

and for the bilinear squark mass terms

$$\begin{aligned} 16\pi ^2 \frac{\mathrm{d}}{\mathrm{d}t} {\bar{m}}_{{\tilde{Q}}}^2= & {} [ -8{\bar{g}}_3^2 +2{\bar{Y}}_{t_R}^2 +2{\bar{Y}}_{b_R}^2 + 28\lambda _4 +20\lambda _5 ] {\bar{m}}_{{\tilde{Q}}}^2 \nonumber \\&+(6\lambda _6+2\lambda _7){\bar{m}}_{{\tilde{t}}}^2 \nonumber \\&+ (4\lambda _1^u+2\lambda _2^u ) {\bar{m}}_2^2 + (4\lambda ^d_1+2\lambda ^d_2 ) {\bar{m}}_1^2 \nonumber \\&+2 ({\bar{A}}_t^2+{\bar{\mu }}_t^2 ) -4 ({\bar{Y}}_{t_R}^2+{\bar{Y}}_{b_R}^2 ){\bar{\mu }}^2 , \end{aligned}$$
(28)
$$\begin{aligned} 16\pi ^2 \frac{\mathrm{d}}{\mathrm{d}t} {\bar{m}}_{{\tilde{t}}}^2= & {} [ -8{\bar{g}}_3^2 +4{\bar{Y}}_{q_{3L}}^2 +16\lambda _8 ] {\bar{m}}_{{\tilde{t}}}^2\nonumber \\&+(12\lambda _6+4\lambda _7){\bar{m}}_{{\tilde{Q}}}^2 \nonumber \\&+4\lambda ^u_3 {\bar{m}}_2 +4\lambda ^d_3 {\bar{m}}_1 +4 ({\bar{A}}_t^2+{\bar{\mu }}_t^2 )\nonumber \\&-8{\bar{Y}}_{q_{3L}}^2 {\bar{\mu }}^2 . \end{aligned}$$
(29)

The Higgsino mass in the effective theory evolves as

$$\begin{aligned} 16\pi ^2 \frac{\mathrm{d}}{\mathrm{d}t} {\bar{\mu }} = \frac{3}{2} ({\bar{Y}}_{q_{3L}}^2 + {\bar{Y}}_{t_R}^2 +{\bar{Y}}_{b_R}^2 ) {\bar{\mu }} , \end{aligned}$$
(30)

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

$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t}{\bar{A}}_t= & {} {\bar{A}}_t [ -8{\bar{g}}_3^2 + 2{\bar{Y}}_{q_{3L}}^2 +{\bar{Y}}_{t_R}^2 +{\bar{Y}}_{b_R}^2 +3{\bar{Y}}_t^2 \nonumber \\&+2\lambda ^u_1 -2\lambda ^u_2 +2\lambda ^u_3 +2\lambda _6 +6\lambda _7 ] , \end{aligned}$$
(31)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t}{\bar{\mu }}_t= & {} {\bar{\mu }}_t [ -8{\bar{g}}_3^2 + 2{\bar{Y}}_{q_{3L}}^2 +{\bar{Y}}_{t_R}^2 +{\bar{Y}}_{b_R}^2 +3{\bar{Y}}_b^2 \nonumber \\&+2\lambda ^d_1 +4\lambda ^d_2 +2\lambda ^d_3 +2\lambda _6 +6\lambda _7 ] \nonumber \\&+ 4 {\bar{Y}}_{q_{3L}}{\bar{Y}}_{b_R}{\bar{Y}}_b {\bar{\mu }}. \end{aligned}$$
(32)

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

$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} \lambda _1^u= & {} 4 (\lambda _1^u )^2 +2 (\lambda _2^u )^2 +28\lambda _1^u\lambda _4 +20\lambda _1^u\lambda _5\nonumber \\&+12\lambda _2^u\lambda _4 + 4\lambda _2^u\lambda _5 +6\lambda _3^u\lambda _6 \nonumber \\&+2\lambda _3^u\lambda _7 + (-8{\bar{g}}_3^2 +6{\bar{Y}}_t^2 + 2{\bar{Y}}_{t_R}^2 +2{\bar{Y}}_{b_R}^2 )\lambda ^u_1 \nonumber \\&-4{\bar{Y}}_{t_R}^2 {\bar{Y}}_t^2 , \end{aligned}$$
(33)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} \lambda _2^u= & {} 8\lambda _1^u \lambda _2^u +4 (\lambda _2^u )^2 +4\lambda _2^u\lambda _4 +12\lambda _2^u\lambda _5 \nonumber \\&+ (-8{\bar{g}}_3^2 +6{\bar{Y}}_t^2 + 2{\bar{Y}}_{t_R}^2 +2{\bar{Y}}_{b_R}^2 )\lambda _2^u , \end{aligned}$$
(34)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} \lambda _3^u= & {} 12\lambda _1^u \lambda _6 + 6 \lambda _2^u \lambda _6 +4 \lambda _1^u \lambda _7 +2\lambda _2^u\lambda _7 \nonumber \\&+4 (\lambda _3^u )^2 +16\lambda _3^u \lambda _8 \nonumber \\&+ (-8{\bar{g}}_3^2 +6{\bar{Y}}_t^2 + 4{\bar{Y}}_{q_{3L}}^2 )\lambda _3^u -4{\bar{Y}}_{q_{3L}}^2{\bar{Y}}_t^2 , \end{aligned}$$
(35)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} \lambda _1^d= & {} 4 (\lambda _1^d )^2 +2 (\lambda _2^d )^2 +28\lambda _1^d\lambda _4 +20\lambda _1^d\lambda _5\nonumber \\&+12\lambda _2^d\lambda _4 + 4\lambda _2^d\lambda _5 +6\lambda _3^d\lambda _6 +2\lambda _3^d\lambda _7 \nonumber \\&+ (-8{\bar{g}}_3^2 +6{\bar{Y}}_b^2 + 2{\bar{Y}}_{t_R}^2 +2{\bar{Y}}_{b_R}^2 )\lambda _1^d \nonumber \\&-4{\bar{Y}}_{b_R}^2{\bar{Y}}_b^2 , \end{aligned}$$
(36)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} \lambda _2^d= & {} 8\lambda _1^d \lambda _2^d +4 (\lambda _2^d )^2 +4\lambda _2^d\lambda _4 +12\lambda _2^d\lambda _5 \nonumber \\&+ (-8{\bar{g}}_3^2 +6{\bar{Y}}_b^2 + 2{\bar{Y}}_{t_R}^2 +2{\bar{Y}}_{b_R}^2 ) \lambda _2^d , \end{aligned}$$
(37)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} \lambda _3^d= & {} 12\lambda _1^d \lambda _6 + 6 \lambda _2^d \lambda _6 +4 \lambda _1^d \lambda _7 +2\lambda _2^d\lambda _7 \nonumber \\&+4 (\lambda _3^d )^2 +16\lambda _3^d \lambda _8 \nonumber \\&+ (-8{\bar{g}}_3^2 +6{\bar{Y}}_b^2 + 4{\bar{Y}}_{q_{3L}}^2 )\lambda ^d_3 -4{\bar{Y}}_{q_{3L}}^2{\bar{Y}}_b^2 , \end{aligned}$$
(38)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} \lambda _4= & {} 2 (\lambda _1^u )^2 +2\lambda _1^u \lambda _2^u +2 (\lambda _1^d )^2 +2\lambda _1^d \lambda _2^d \nonumber \\&+40\lambda _4^2 +40\lambda _4 \lambda _5 + 12\lambda _5^2 + 3\lambda _6^2 \nonumber \\&+2\lambda _6 \lambda _7 + (-16 {\bar{g}}_3^2 +4 ({\bar{Y}}_{t_R}^2 + {\bar{Y}}_{b_R}^2 ) ) \lambda _4 \nonumber \\&+\frac{11}{12}{\bar{g}}_3^4 , \end{aligned}$$
(39)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} \lambda _5= & {} (\lambda _2^u )^2 + (\lambda _2^d )^2 +24\lambda _4 \lambda _5 + 20\lambda _5^2 + \lambda _7^2 \nonumber \\&+ (-16 {\bar{g}}_3^2 +4 ({\bar{Y}}_{t_R}^2 + {\bar{Y}}_{b_R}^2 ) ) \lambda _5 \nonumber \\&-2 ({\bar{Y}}_{t_R}^4+{\bar{Y}}_{b_R}^4 ) +\frac{5}{4} {\bar{g}}_3^4 , \end{aligned}$$
(40)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} \lambda _6= & {} (4\lambda _1^u+2\lambda _2^u )\lambda _3^u + (4\lambda _1^d+2\lambda _2^d )\lambda _3^d+ 28\lambda _4 \lambda _6 \nonumber \\&+ 8\lambda _4 \lambda _7 +20\lambda _5\lambda _6 + 4\lambda _5\lambda _7 \nonumber \\&+4\lambda _6^2 + 2\lambda _7^2 +16 \lambda _6 \lambda _8 +4\lambda _7\lambda _8 \nonumber \\&+ (-16 {\bar{g}}_3^2 +2{\bar{Y}}_{t_R}^2 +2{\bar{Y}}_{b_R}^2 + 4{\bar{Y}}_{q_{3L}}^2 ) \lambda _6 \nonumber \\&-4{\bar{Y}}_{t_R}^2 {\bar{Y}}_{q_{3L}}^2 +\frac{11}{6}{\bar{g}}_3^4 , \end{aligned}$$
(41)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} \lambda _7= & {} 4\lambda _4 \lambda _7 +8\lambda _5\lambda _7 + 8\lambda _6\lambda _7 +6\lambda _7^2 +4\lambda _7\lambda _8 \nonumber \\&+ (-16 {\bar{g}}_3^2 +2{\bar{Y}}_{t_R}^2 +2{\bar{Y}}_{b_R}^2 + 4{\bar{Y}}_{q_{3L}}^2 ) \lambda _7 +\frac{5}{2} {\bar{g}}_3^4 , \end{aligned}$$
(42)
$$\begin{aligned} 16\pi ^2\frac{\mathrm{d}}{\mathrm{d}t} \lambda _8= & {} 2 (\lambda _3^u )^2+2 (\lambda _3^d )^2 +6\lambda _6^2 +4\lambda _6 \lambda _7 +2\lambda _7^2\nonumber \\&+28\lambda _8^2 + (-16 {\bar{g}}_3^2 +8{\bar{Y}}_{q_{3L}}^2 ) \lambda _8 \nonumber \\&-4{\bar{Y}}_{q_{3L}}^4 +\frac{13}{6} {\bar{g}}_3^4 . \end{aligned}$$
(43)

Note that in all equations above we assumed real parameters. However, all formula can easily be 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 \(\log (M/m_{{\tilde{t}}})\).

Fig. 1
figure 1

Evolution of the Yukawa coupling \(Y_t\) in the naive approach without using an EFT (green) compared to the various Higgs/Higgsino–stop/top couplings in the EFT for \(M=5\,\)TeV and \(\tan \beta =50\) as a function of the renormalization scale \({\mu _R}\). Note that the only numerically sizable impact of \(\tan \beta =50\) is the splitting between the \({\bar{Y}}_{t_R}\) and \({\bar{Y}}_{q_{3L}}\). The initial condition of the Yukawa coupling is determined by the requirement that \(v_u Y_t=m_t=150\) GeV at the stop scale which we choose here to be \(500\,\)GeV. \({\bar{Y}}_{{\tilde{t}}}={\bar{\mu }}_t/{\bar{\mu }}\) shows the evolution of the \({\tilde{t}}-{\tilde{t}}-H_d\) coupling relative to the Higgsino mass term \({\bar{\mu }}\) in the EFT. We also show the projected evolution of \(Y_t\) below the scale M (black-dashed) in the MSSM RGE for the boundary condition \(Y_t(M)={\bar{Y}}_t(M)\). Note that above the scale M SUSY is restored, so that there is only one Yukawa coupling \(Y_t\) (black)

Full size image
Fig. 2
figure 2

Evolution of the quartic coupling to right-handed stops in the naive approach with the MSSM RGE (green) compared to the EFT approach, where \(\alpha _3=0.1\) at the stop scale. Note that the SUSY relation \(\lambda _8=\frac{1}{6}g_3^2\) holds only at the scale M in the EFT. The dotted-black line shows the projected evolution of \(\lambda _8\) for the boundary condition \(\lambda _8(M)=\frac{1}{6}{\bar{g}}_3^2(M)\) with the naive RGE of the full MSSM. Note that above the scale M SUSY is restored, \(\lambda _8=1/6g_3^2\) and evolves like \(g_3^2\) in the full MSSM

Full size image

2.4 Stop masses

In the effective theory, the stop mass matrix in the (\({\tilde{t}}_L\), \({\tilde{t}}_R\)) basis reads

$$\begin{aligned} \begin{aligned} {\bar{{\mathcal {M}}}}_{{\tilde{t}}}^2&= \left( \begin{array}{ll} {\bar{m}}_{{\tilde{Q}}}^2 + v_u^2 \lambda _1^u+v_d^2 (\lambda _1^d+\lambda _2^d ) &{} v_u {\bar{A}}_t^* - v_d {\bar{\mu }}_t^* \\ v_u {\bar{A}}_t - v_d {\bar{\mu }}_t &{}{\bar{m}}_{{\tilde{t}}}^2 + v_u^2 \lambda _3^u+v_d^2 \lambda _3^d \end{array} \right) , \\ \end{aligned} \end{aligned}$$
(44)

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 \(\overline{\mathrm{MS}}\) scheme. These masses are closely related to the left-handed sbottom mass,

$$\begin{aligned} M_{{\tilde{b}}_L}^2 = {\bar{m}}_{{\tilde{Q}}}^2+ v_u^2 (\lambda _1^u +\lambda _2^u ) + v_d^2 \lambda _1^d, \end{aligned}$$
(45)

by SU(2) gauge symmetry.

3 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.

4 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 a future test of the stop–stop or stop-Higgs interactions which also enter the calculation of the Higgs mass.