Precise prediction for the W boson mass in the MRSSM

# Precise prediction for the W boson mass in the MRSSM

Philip Diessner,111Former address.222Corresponding author. Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany    Georg Weiglein Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany
###### Abstract

The mass of the W boson, , plays a central role for high-precision tests of the electroweak theory. Confronting precise theoretical predictions with the accurately measured experimental value provides a high sensitivity to quantum effects of the theory entering via loop contributions. The currently most accurate prediction for the W boson mass in the Minimal R-symmetric Supersymmetric Standard Model (MRSSM) is presented. Employing the on-shell scheme, it combines all numerically relevant contributions that are known in the Standard Model (SM) in a consistent way with all MRSSM one-loop corrections. Special care is taken in the treatment of the triplet scalar vacuum expectation value that enters the prediction for already at lowest order. In the numerical analysis the decoupling properties of the supersymmetric loop contributions and the comparison with the MSSM are investigated. Potentially large numerical effects of the MRSSM-specific superpotential couplings are highlighted. The comparison with existing results in the literature is discussed.

\preprint

DESY 19-035

## 1 Introduction

Supersymmetry (SUSY) is seen as one of the most attractive extensions of the Standard Model (SM) of particle physics. It provides a solution for the hierarchy problem of the SM and a prediction for the mass of the scalar resonance discovered at the LHC ATLAS:2012gk (); Chatrchyan:2012ufa (); Aad:2015zhl () if it is appropriately identified with one of the neutral Higgs bosons of the considered supersymmetric model.

So far, no further new state has been discovered at the LHC. The search limits from the LHC and previous colliders give rise to strong constraints on the parameter space of the Minimal Supersymmetric Standard Model (MSSM). The impact of those search limits may be much smaller for models with extended SUSY sectors. In particular, the introduction of R-symmetry Kribs:2012gx () leads to models where the limits on the particle masses are significantly weaker and where accordingly the discovery of TeV-scale SUSY is still in reach of current experiments.

The LHC phenomenology of the Minimal R-symmetric Supersymmetric Standard Model (MRSSM) Kribs:2007ac () and other models with R-symmetry has recently been explored in refs. PD1 (); PD2 (); PD3 (); PD4 (); Kotlarski:2016zhv (); Beauchesne:2017jou (); Benakli:2018vqz (); Alvarado:2018rfl (); Darme:2018dvz (); Chalons:2018gez (). The signatures of the MRSSM differ from the ones of the MSSM as the model includes Dirac gauginos and higgsinos instead of Majorana ones. This goes hand in hand with additional states in the Higgs sector including an SU triplet, which may acquire a vacuum expectation value breaking the custodial symmetry of the electroweak sector. Additionally, R-symmetry forbids all soft SUSY-breaking trilinear couplings between Higgs bosons and squarks or sleptons, which removes unwanted sources of flavour violation which exist in the MSSM.

Even with no direct signals of SUSY indirect probes like electroweak precision observables can provide sensitivity to SUSY contributions above the direct experimental reach. Here we study the prediction for the mass of the W gauge-boson, . The world average combining LEP and Tevatron results Schael:2013ita (); Aaltonen:2013iut () is

 Mexp.W=80.385±0.015 GeV. (1)

It may be improved by measurements of the LHC experiments, where a first ATLAS result at  TeV reports a value Aaboud:2017svj () of

 MATLASW=80.370±0.019 GeV. (2)

The statistical combination of the ATLAS measurement (2) with the one of eq. (1) yields an updated experimental result of PDG:2018 ()

 MLEP+Tevatron+ATLASW=80.379±0.012 GeV. (3)

For a meaningful comparison to experiment a precise theory prediction is necessary. In the Standard Model (SM) the prediction includes contributions at the one-loop Sirlin:1980nh (); Marciano:1980pb () and two-loop level Djouadi:1987gn (); Djouadi:1987di (); Kniehl:1989yc (); Halzen:1990je (); Kniehl:1991gu (); Kniehl:1992dx (); Freitas:2000gg (); Freitas:2002ja (); Awramik:2002wn (); Awramik:2003ee (); Onishchenko:2002ve (); Awramik:2002vu (), as well as leading three- and four-loop corrections Avdeev:1994db (); Chetyrkin:1995ix (); Chetyrkin:1995js (); Chetyrkin:1996cf (); Faisst:2003px (); vanderBij:2000cg (); Boughezal:2004ef (); Boughezal:2006xk (); Chetyrkin:2006bj (); Schroder:2005db (). In refs. Awramik:2003rn (); Awramik:2006uz () a parametrisation of the prediction containing all known higher-order corrections in the on-shell scheme has been given. Updating the experimental results for the input values PDG:2018 ()333See section 5 for the numerical values. and identifying the scalar state discovered at the LHC with the Standard Model Higgs boson leads to a prediction for the W boson mass in the Standard Model of

 MSM,on-shellW=80.356 GeV. (4)

This result shows a slight downward shift compared to the previous calculation Stal:2015zca () due to the updated input parameters. Hence, the long-standing tension of the experimental measurement and theoretical prediction of just below remains.

The largest parametric uncertainty is induced by the top quark mass. For the experimental value

 mexp.t=173.0±0.4 GeV (5)

the quoted experimental error accounts only for the uncertainty in extracting the measured parameter. This needs to be supplemented with the systematic uncertainty arising from relating the measured mass parameter to a theoretically well-defined quantity such as the mass. This uncertainty could be reduced very significantly with a measurement of the top quark mass from the threshold at a future collider

Accurate theoretical predictions for the W boson mass have also been obtained for the most popular supersymmetric extensions of the SM, in particular for the MSSM Heinemeyer:2006px (); Heinemeyer:2013dia (); Stal:2015zca (), see also ref. Heinemeyer:2004gx () and references therein, and the NMSSM Stal:2015zca (). For the MRSSM the W boson mass has been studied together with other electroweak precision observables in ref. PD1 (). While for the quoted results in the MSSM and the NMSSM the on-shell scheme was employed, the MRSSM result of ref. PD1 () was obtained for a mixed on-shell/ scheme Degrassi:1990tu (); BPMZ () where only the gauge-boson masses and are on-shell quantities, making use and adapting the tools SARAH and SPheno Staub:2008uz (); Staub:2009bi (); Staub:2010jh (); Porod:2011nf (); Staub:2012pb (); Staub:2013tta (); Goodsell:2014bna (); Goodsell:2015ira (). Recently, a new implementation of this calculation in the program FlexibleSUSY 2.0 Athron:2017fvs () has shown a large discrepancy in the MSSM with the result that was achieved in the on-shell scheme Heinemeyer:2006px (); Heinemeyer:2013dia (); Stal:2015zca () and also a large discrepancy in the MRSSM with the result of ref. PD1 ().

In this paper we present an improved prediction for the W boson mass in the MRSSM. Employing the on-shell scheme for the SM-type parameters, we obtain the complete one-loop contributions in the MRSSM and combine them with the state-of-the-art SM-type corrections up to the four-loop level. In the calculation of the MRSSM contributions a renormalisation of the triplet scalar vacuum expectation value is needed since this parameter enters the prediction for already at lowest order. We investigate the treatment of this parameter and implement a -type renormalisation. In our numerical analysis we study the decoupling limit where the SUSY particles are heavy and verify that the SM prediction is recovered in this limit. We investigate the possible size of the different MRSSM contributions and compare the results with the MSSM case. We also discuss the comparison of our result with the existing MRSSM results.

The paper is organised as follows: In section 2 we give a brief overview of the MRSSM field content and introduce the relevant model parameters required to discuss the calculation of the prediction in this model. In section 3 details on the calculation of the higher-order corrections to the muon decay process are presented. Section 4 contains the details on the implementation of the calculation, while in section 5 we present a quantitative study of the prediction in the MRSSM parameter space and a comparison of our results to previous calculations. We conclude in section 6.

## 2 The Minimal R-symmetric Supersymmetric Standard Model

### 2.1 Model overview

The minimal R-symmetric extension of the MSSM, the MRSSM, requires the introduction of Dirac mass terms for gauginos and higgsinos, since Majorana mass terms are forbidden by R-symmetry. This leads to the need for an extended number of chiral superfields containing the necessary additional fermionic degrees of freedom.

Therefore, the field content of the MRSSM is enlarged compared to the MSSM by doublet superfields carrying R-charge under the R-symmetry as well as adjoint chiral superfields , , for each of the gauge groups. The full field content of the MRSSM including the assignment of R-charges is given in table 1.

In the following we introduce the parameters of the MRSSM. All model parameters are taken to be real for the purpose of this work. The MRSSM superpotential is given as

 W= μd^Rd⋅^Hd+μu^Ru⋅^Hu+Λd^Rd⋅^T^Hd+Λu^Ru⋅^T^Hu +λd^S^Rd⋅^Hd+λu^S^Ru⋅^Hu−Yd^d^q⋅^Hd−Ye^e^l⋅^Hd+Yu^u^q⋅^Hu, (6)

where the dot denotes the contraction with . In order to achieve canonical kinetic terms the triplet is defined as

 ^T=(^T0/√2^T+^T−−^T0/√2). (7)

The usual term of the MSSM is forbidden by R-symmetry but similar terms can be written down connecting the Higgs doublets to the inert R-Higgs fields with the parameters . Trilinear couplings and couple the doublets to the adjoint triplet and singlet field, respectively. The Yukawa couplings are the same as in the MSSM.

The soft SUSY-breaking Lagrangian contains the soft masses for all scalars as well as the usual term. Trilinear A-terms of the MSSM are forbidden by R-symmetry.444 Following the arguments in ref. PD1 (), the addition of further bilinear and trilinear holomorphic terms of the adjoint scalar fields is not considered here. This part of the Lagrangian reads

Dirac mass terms connecting the gauginos and the fermionic components of the adjoint superfields are introduced. They are generated from the R-symmetric operator involving the D-type spurion Fox:2002bu () 555Alternatively, it is also possible to generate Dirac gaugino masses via F-term breaking Martin:2015eca ().

 ∫d2θ^W′α,iMWαi^Φi∋MDi~gi~g′i, (9)

where is the mediation scale of SUSY breaking , represents the gauge superfield strength tensors, is the gaugino, and is the corresponding Dirac partner with opposite R-charge which is part of a chiral superfield , , . The mass term is generated by the spurion field strength getting a vev . Additionally, integrating out the spurion in eq. (9) generates terms coupling the D-fields to the scalar components of the chiral superfields, which leads to the appearance of Dirac masses also in the Higgs sector,

 VD= MDB(~B~S−√2DBS)+MDW(~Wa~Ta−√2DaWTa)+MDg(~ga~Oa−√2DagOa)+h.c.. (10)

During electroweak symmetry breaking (EWSB) the neutral EW scalars with no R-charge develop vacuum expectation values (vevs)

 H0d= 1√2(vd+ϕd+iσd), H0u= 1√2(vu+ϕu+iσu), T0= 1√2(vT+ϕT+iσT), S= 1√2(vS+ϕS+iσS).

After EWSB the singlet and triplet vevs effectively shift the -parameters of the model, and it is useful to define the abbreviations

 μeff,±i =μi+λivS√2±ΛivT2, μeff,0i =μi+λivS√2, i=u,d. (11)

The Higgs sector contains four CP-even and three CP-odd neutral as well as three charged Higgs bosons. The Higgs doublets with R-charge 2 stay inert and do not receive a vev such that two additional complex neutral and charged scalars are predicted by the MRSSM. The sfermion sector is the same as in the MSSM with the restriction that mixing between the left- and right-handed sfermions is forbidden by R-symmetry. In the MRSSM the number of chargino and higgsino degrees of freedom is doubled compared to the MSSM as the neutralinos are Dirac-type and there are two separated chargino sectors where the product of electric and R-charge is either 1 or .

The mass matrices of the SUSY states can be found in ref. PD1 (). The masses of the gauge bosons arise as usual with an important distinction which will be discussed in more detail in the following.

### 2.2 The W boson mass in the MRSSM

The expression for the W boson mass differs from the usual form of the MSSM and the SM due to the triplet vev . With the representation in eq. (7) the kinetic term for the scalar triplet is given as

 LT,kin.=Tr[(DμT)†(DμT)],DμT=∂μT+ıg2[W,T]. (12)

The possible contribution to the gauge-boson masses from a triplet vev arises from the quartic part as

 −(ıg2)2Tr([T†,W†][W,T]) =(g24)2Tr([τa,τb][τc,τd])T†aWbWcTd =−g222(WaWaTbT†b−WaT†aWbTb), (13)

where the are the Pauli matrices. The two terms of the last expression cancel each other for the neutral component when the triplet vev is inserted. Hence, only the mass of the charged W boson receives an additional contribution. Then, the lowest-order masses of the Z and W bosons in the MRSSM are given as

 m2Z=g21+g224v2,m2W=g224v2+g22v2T, (14)

with . The appearance of in the expression for the W boson mass is also relevant for the definition of the weak mixing angle as the introduction of spoils the accidental custodial symmetry of the SM.

Numerically the contribution due to the triplet vev is strongly constrained by the measurement of electroweak precision observables, especially which leads to a limit of  GeV. A more detailed discussion of its influence on the W boson mass is given in sec. 5.2.1.

The weak mixing angle which diagonalises the neutral vector boson mass matrix leading to the photon and Z boson is related to the gauge couplings as usual

 ~c2W≡cos2(~θW)=g22g21+g22,~s2W=1−~c2W, (15)

where we have introduced the notation , and in order to distinguish those quantitities from the ones defined in eq. (17) below. Together with the electric charge derived from the fine-structure constant the weak mixing angle can be used to replace the gauge couplings,

 g1=e~cW,g2=e~sW. (16)

Additionally, we define the ratio of the masses of the electroweak gauge bosons as

 c2W≡m2Wm2Z=~c2W+e2v2T(1−~c2W)m2Z,s2W=1−c2W=~s2W−e2v2T~s2Wm2Z. (17)

In the limit of vanishing the two quantities and coincide with each other at tree-level as in the MSSM. For the extraction of the W boson mass from the muon decay constant it is helpful to solve eq. (17) for

 ~s2W=12⎛⎜⎝s2W+ ⎷s4W+4e2v2Tm2Z⎞⎟⎠ (18)

taking the physical solution such that for holds as required.

### 2.3 Determination of the W boson mass

The Fermi constant is an experimentally very precisely measured observable that is obtained from muon decay. The comparison with the theoretical prediction for muon decay yields a relation between the Fermi constant, the W boson mass, the Z boson mass and the fine-structure constant. Therefore a common approach is to use as an input in order to derive a prediction for the W boson mass in the SM and in BSM models.

Muons decay to almost 100% via . The Fermi model describes this interaction via a four-point interaction with the coupling . It is connected to the experimentally precisely measured muon lifetime extracted by the MuLan experiment Tishchenko:2012ie () via

 1τμ=m5μG2F192π3F(m2em2μ)(1+Δq). (19)

Here, collects effects of the electron mass on the final-state phase space, and denotes the QED corrections to the Fermi model up to two loops vanRitbergen:1999fi (); Steinhauser:1999bx (); Pak:2008qt ().666In the past, conventionally a factor has often been inserted in eq. (19) in order to take into account tree-level W propagator effects even though this correction term is not part of the Fermi model. With the enhanced accuracy of the MuLan experiment this previously numerically negligible factor has to be taken into account on the side of the full SM calculation when using the experimentally extracted value of from ref. PDG:2018 (). Detailed expressions can be found for instance in Chapter 10 of the Particle Data Group report PDG:2018 ().

Equating the expression in the Fermi model with the SM prediction in the on-shell scheme yields the following relation between the Fermi constant , the fine-structure constant in the Thomson limit, and the pole masses (defined according to a Breit–Wigner shape with a running width, see below) of the Z and W bosons, and , respectively,

 (20)

where all higher-order contributions besides the ones appearing in eq. (19) are contained in the quantity . The same functional relation holds also in the MSSM and the NMSSM.777As mentioned above, tree-level effects from the longitudinal part of the W propagator and, for the case of an extended Higgs sector, of the charged Higgs boson(s) are understood to be incorporated into . As effects of this kind are insignificant for our numerical analysis, we will neglect them in the following. The weak mixing angle in eq. (20) is given by in accordance with eq. (17).

In the MRSSM the relation of eq. (20) gets modified because of the contribution of the triplet vev entering at lowest order,

where originates from the gauge coupling , see eq. (16) and eq. (17). We use the notation for the higher-order contributions in the MRSSM, where eq. (21) is expressed in terms of the quantities , , defined in the on-shell scheme as in eq. (20). Inserting eq. (18) into eq. (21) and solving for yields

 (22)

As itself depends on it is technically most convenient to determine numerically through an iteration of this relation. In the limit the result of eq. (22) yields the well-known expression

 M2W∣∣vT→0=M2Z⎛⎝12+√14−απ√2GFM2Z(1+Δr)⎞⎠ (23)

that is valid in the SM, since coincides with the usual definition of in this case.

As mentioned above, in the previous calculation PD1 () for the MRSSM a mixed scheme was used where only the gauge-boson masses and are on-shell quantities. For completeness we provide a short description of this scheme. Following refs. Degrassi:1990tu (); Degrassi:2014sxa (), the running electromagnetic coupling and the running mixing angle are used in this scheme together with the on-shell definitions of the masses. Here, denotes renormalisation via modified minimal subtraction either in dimensional regularisation or reduction. Higher-order corrections are collected in several parameters that incorporate a resummation of certain reducible higher-order contributions, in particular

 ^α¯¯¯¯¯MS/¯¯¯¯¯DR=α1−Δ^α,^c2W,¯¯¯¯¯MS/¯¯¯¯¯¯¯¯¯¯%DR=M2WM2Z^ρ,^s2W,¯¯¯¯¯MS/¯¯¯¯¯DR=1−^c2W,¯¯¯¯¯MS/¯¯¯¯¯DR, (24)

where

 Δ^α =^ΠAA(0),^ΠAA(0)≡∂^ΣAAT(p2)∂p2∣∣ ∣∣p2=0,^ρ=11−Δ^ρ0−Δ^ρ, Δ^ρ0 =4v2Tv2T+v2,Δ^ρ=R⎛⎜⎝^ΣZZT(M2Z)M2Z−^ΣWWT(M2W)M2W⎞⎟⎠. (25)

Here contains the tree-level shift arising from the triplet vev, while contains the higher-order corrections including all MRSSM one-loop and leading SM two-loop effects, denotes the transverse part of the -renormalised self-energies of the gauge bosons, and indicates the real part. Analogously to eq. (21) one can write

where the quantities defined in eqs. (24), (25) have been used, and contains additonal higher-order corrections. The described approach has been used in previous work for the MRSSM and is the one implemented in several MSSM and NMSSM mass spectrum generators following refs. Chankowski:1994ua (); BPMZ (). For an approach in a pure minimal subtraction scheme see ref. Martin:2015lxa ().

In the following, we discuss the details of the on-shell calculation in the MRSSM according to eq. (21), making use of the state-of-the-art prediction of the SM part, and compare it to previously obtained results.

## 3 Δ~r in the MRSSM

For our on-shell calculation of the W boson mass in the MRSSM we include all MRSSM SUSY effects at one-loop level and all known higher-order contributions of SM type. This follows the approach taken for the MSSM Stal:2015zca (); Heinemeyer:2006px () and the NMSSM Stal:2015zca ().

### 3.1 One-loop corrections

#### 3.1.1 General contributions

At the one-loop level, receives contributions from the W boson self-energy, from vertex and box corrections, as well as from the corresponding counterterm diagrams. It can be written as

 Δ~r(α)=ΣWWT(0)−δM2W\MW2+(Vertex and Box)+12(δZeL+δZμL+δZνeL+δZνμL)+2δe−δ~s2W~s2W. (27)

The field renormalisation of the W boson drops out as the field only appears internally. The expressions for the MRSSM vertex and box contributions are given in the appendix of ref. PD1 ().

Using the on-shell scheme, we fix the renormalisation constants of the gauge-boson masses as

 M2W/Z,0=M2W/Z+δM2W/Z,δM2W/Z=RΣWW/ZZT(M2W/Z), (28)

where is the transverse part of the unrenormalised gauge-boson self-energy taken at momentum , and as before denotes the real part. The field renormalisation constant of a massless left-handed lepton is

 lL,0=(1+12δZlL)lL,δZlL=−ΣlL(0). (29)

The electric charge is renormalised as

where the quantity is extracted from experimental data and accounts for the contributions of the five light quark flavours.

For the renormalisation of the weak mixing angle  the appearance of leads to differences compared to the SM and the MSSM. The parameter is renormalised with the renormalisation constants of the gauge-boson masses

 s2W,0=s2W+δs2W,δs2Ws2W=c2Ws2WR(ΣZZT(M2Z)M2Z−ΣWWT(M2W)M2W). (31)

The renormalisation constant of the weak mixing angle can be expressed in terms of , , and using the relation between and given in eq. (17),

 ~s2W,0=~s2W+δ~s2W,δ~s2W~s2W={δs2Ws2W−s2W−~s2Ws2W[2(δe+δvTvT)−δM2ZM2Z]}(s2W2~s2W−s2W). (32)

Expressing by the self-energies of the vector bosons gives

 δ~s2W~s2W=c2W2~s2W−s2WR(ΣZZT(M2Z)M2Z−ΣWWT(M2W)M2W)+4παv2T~s2W(2~s2W−s2W)M2Z(ΠAA(0)+2~sW~cWΣAZT(0)M2Z−RΣZZT(M2Z)M2Z+2δvTvT). (33)

If the triplet vev was absent, the on-shell renormalisation of the electroweak parameters would be the same as in the SM and the MSSM, and and would coincide. It is important to note in this context that in our renormalisation prescription is treated as a dependent parameter as specified in eq. (32). The renormalisation of the triplet vev is described in the following section. This prescription for the weak mixing angle ensures that the contributions to incorporate the typical quadratic dependence on that is induced by the contribution of the top/bottom doublet to the parameter at one-loop order. This behaviour was found to be absent in renormalisation schemes where the weak mixing angle is treated as an independent parameter that is fixed as a process-specific effective parameter , see refs. Blank:1997qa (); Czakon:1999ha (); Chen:2005jx ().

#### 3.1.2 Renormalisation of vT

The triplet vev is an additional parameter of the electroweak sector in the MRSSM compared to the MSSM. As it appears in the lowest-order relation between the muon decay constant and , eq. (22),888It also appears in the lowest-order relation between and the weak mixing angle, eq. (18). its renormalisation is required for the prediction of . In principle, in the MRSSM one could instead have used as an experimental input parameter in order to determine via eqs. (14)–(18). Among other drawbacks, in such an approach the SM limit of the MRSSM, which involves , could not be carried out. Instead, we prefer to keep as an observable that can be predicted, using in particular as an experimental input, and compared to the experimental result as it is the case in the SM and the MSSM.

For the calculation of performed in this work we renormalise such that

 (34)

Accordingly, only contains the divergent contribution (in the modified minimal subtraction scheme) with a prefactor , where the correct choice of the latter ensures that the renormalised quantities are finite. The value of is taken as input at the SUSY scale . The triplet vev is the only BSM parameter entering the tree-level expression of eq. (14). All other BSM parameters of the MRSSM only enter the loop contributions and do not need to be renormalised at the one-loop level. As is a parameter of the electroweak potential a comment on the tadpole conditions is in order. The tree-level tadpoles are given as

 td= vd[18(g21+g22)(v2d−v2u)−g1MDBvS+g2MDWvT+m2Hd+(μeff,+d)2]−vuBμ, tu= vu[18(g21+g22)(v2u−v2d)+g1MDBvS−g2MDWvT+m2Hu+(μeff,−u)2]−vdBμ, tT= 12g2MDW(v2d−v2u)+12(Λdv2dμeff,+d−Λuv2uμeff,−u)+4(MDW)2vT+m2TvT, tS= 12g1MDB(v2u−v2d)+1√2(λdv2dμ%eff,+d+λuv2uμeff,−u)+4(MDB)2vS+m2SvS, (35)

and one may trade one model parameter for each of the tadpoles. The numerical values of those parameters are fixed implicitly by the conditions so that they now are expressed as functions of all the other input parameters of the model. In our calculation of we choose the set of , , and as dependent parameters. Choosing instead as a parameter to be fixed by the tadpole equations would require a renormalisation of all parameters in these relations. This would lead to a finite counterterm which would have a complicated form but would be expected to have a numerically very small impact on the prediction of as it is suppressed in eq. (33) by a prefactor of .

Our calculation for the prediction in the MRSSM is embedded into the framework of SARAH/SPheno described in section 4 below. There, we use a different set of parameters derived from solving the tadpole equations, namely is treated as an output there and as input. The value for calculated by the SARAH/SPheno routines is then passed to our implementation of the calculation as an input. This is necessary as is much smaller than for realistic parameter points. Therefore, small variations in during the required iteration might have strong effects on the BSM mass spectrum and may lead to numerical instabilities. For example, at tree-level a chosen value might lead to physical Higgs states while tachyonic states might appear after adding the loop corrections to the mass matrices and tadpole equations. This is circumvented by using in a first step as input and keeping it positive. In the following we describe the treatment of the tadpole conditions in the SARAH/SPheno framework in more detail and illustrate that the definition that we have chosen for ensures that it is suitable as input for our calculation of satisfying eq. (34).

In the SARAH/SPheno framework free parameters are renormalised in the scheme, and the masses of the SUSY states are calculated to a considered loop order. The tadpoles are an exception in this context as they are renormalised relying on on-shell conditions. Hence, their counterterms cancel the tadpole diagrams, and the tadpole contributions do not need to be considered as subdiagrams of other loop diagrams. This means that the bare tadpole is given as

 t0,i=^ti+δti,δti=−Γi(0), (36)

where are the unrenormalised one-loop one-point functions. The renormalised tadpoles are required to vanish, corresponding to the minimum of the effective potential. The parameters chosen as dependent parameters via the tadpole conditions for the SARAH/SPheno mass spectrum calculation are , , and . As the tadpoles are renormalised via on-shell conditions, the renormalised dependent parameters respect the tree-level relations while the counterterms of these parameters (, , , ) have to contain finite parts. Therefore, considering only the finite part of all appearing quantities (i.e., counterterms that only contain a divergent contribution have been dropped) the finite parts of the counterterms are given implicitly via the following relations:

 vd[(14Λ2d~vT+12Λdμd+g2MDW)δ~vT+(12λ2dδ~vS+λdμd−g1MDB)δ~vS+δ~m2Hd] =−Γd(0), vu[(14Λ2d~vT−12Λuμu−g2MDW)δ~vT+(12λ2uδ~vS+λuμu+g1MDB)δ~vS+δ~m2Hu] =−Γu(0), 12√2(λdΛdv2d−λuΛuv2u)δ~vS+[m2T+4(MDW)2+14(Λ2dv2d+Λ2uv2u)]δ~vT =−ΓT(0), 12√2(λdΛdv2d−λuΛuv2u)δ~vT+[m2S+4(MDB)2+12(λ2dv2d+λ2uv2u)]δ~vS =−ΓS(0). (37)

Note that the terms containing and arise from the terms involving and in eq. (35). In the loop calculation one can then define parameters , , and as sum of the tree-level parameter and the finite part of the counterterm. For the triplet vev we choose (as above, a purely divergent counterterm is dropped)

 vloopT=~vT+δ~vT|finite. (38)

This definition of corresponds to a renormalisation of with a purely divergent counterterm as required in eq. (34). Therefore, as an output of this procedure is a suitable input for the our calculation of the prediction. This change in parameterisation leads to a numerical difference between the value of used for the calculation on the one hand and for the derivation of the loop-corrected SUSY mass spectrum on the other. As we consider the SUSY corrections only up to the one-loop level for the one-loop shift of the model parameter formally leads to a higher-order effect that is beyond the considered order for the SUSY loop corrections.

A compact expression for can be derived by solving the third equation of (35) for it. At tree-level, we get the following

 vT=(Λuμeff,0u+g2MDW)v2u−(Λdμeff,0d+g2MDW)v2d2(m2T+4(MDW)2)+12(Λ2dv2d+Λ2uv2u). (39)

The magnitude of can be affected in several ways. On the one hand it can become small as a consequence of large SUSY mass scales appearing in the denominator. Here, the combination is the squared tree-level mass matrix entry for the CP-even Higgs triplet. On the other hand, the numerator can become small. For the term proportional to dominates. If then is numerically close to a partial cancellation is possible leading to a reduced value for . 999It should be noted that the expressions in parenthesis in the numerator also appear in the Higgs mass matrix elements mixing the doublets and triplet. Therefore, if the triplet scalar mass parameter is not large, a certain tuning is necessary to reduce the admixture of the triplet Higgs component with the state at 125 GeV in order to ensure that the latter is sufficiently SM-like. Such a tuning would at the same time reduce the numerical value of . In the numerical scenarios studied in this paper, however, such a tuning does not occur since in our numerical analysis below the triplet mass is always much larger than the mass of the SM-like state in the parameter regions where the prediction is close to the experimental measurement.

In ref. Chankowski:2006hs () the influence of the triplet vev on the decoupling behaviour in a model where the Standard Model is extended by a real triplet was studied. It was found that non-decoupling behaviour exists when the triplet mass parameter approaches large values while the triplet-doublet-doublet trilinear coupling also grows. While a detailed investigation of this issue in the MRSSM would go beyond the scope of the present paper, we note that studying the numerical one-loop contributions to the triplet tadpole we do not find a non-decoupling effect for a comparable limit. Hence, this effect does not appear in our numerical analysis, see also the discussion in section 5.1 where we compare the MRSSM prediction with the one of the SM for the case where the SUSY mass scale is made large. Our results regarding the decoupling behaviour of the MRSSM contributions can be qualitatively understood in the following way. In an effective field theory analysis of the decoupling behaviour one would study the matching of the MRSSM to an effective SM+triplet model. There, the tree-level matching conditions would fix the quartic triplet coupling to zero as it does not appear due to R-symmetry in the MRSSM. This affects the number of free parameters of the effective model, preventing the occurrence of non-decoupling behaviour. However, if one-loop matching conditions for the quartic coupling were taken into account these contributions could in principle give rise to a non-decoupling effect. As those contributions would correspond to a two-loop effect in the MRSSM for the calculation of they are outside of the scope of the present work, and we leave the investigation of contributions of this kind to further study.

### 3.2 Higher-order contributions

For a reliable prediction of in a BSM model it is crucial to take into account SM-type corrections beyond one-loop order. Only upon the incorporation of the relevant two-loop and even higher-order SM-type loop contributions it is possible to recover the state-of-the-art SM prediction within the current experimental and theoretical uncertainties in the appropriate limit of the BSM parameters, see e.g. the discussion in ref. Stal:2015zca (). In our predictions we incorporate the complete two-loop and the numerically relevant higher-order SM-type contributions.

A further important issue in this context is the precise definition of the gauge-boson masses according to a Breit–Wigner resonance shape with running or fixed width. While the difference between the two prescriptions formally corresponds to an electroweak two-loop effect, numerically the associated shifts are about 27 MeV for and 34 MeV for .

#### 3.2.1 Breit–Wigner shape

The definition of the masses of unstable particles according to the real part of the complex pole of their propagator is gauge-independent also beyond one-loop order, while the definition according to the real pole is not. Expanding the propagator around the complex pole leads to a Breit–Wigner shape with fixed width (f.w.). Experimentally, the gauge boson masses are extracted, by definition, from a Breit–Wigner shape with a running width (r.w.). Hence, it is necessary to translate from the internally used fixed-width mass to the running-width mass at the end of the calculation

 Mr.w.W=Mf.w.W+Γ2W2Mr.% w.W, (40)

where for the decay width of the W boson, , we use the theoretical prediction parametrised by including first order QCD corrections,

 ΓW=3(Mr.w.W)3GF2√2π(1+2αs3π). (41)

For