# Matching to the SM in SPheno

## General

The parameters of a model have to be matched to the measured values of masses and couplings in the SM. The procedure to do this differs between a Low or High scale SPheno version:

1. For a high-scale version one needs to match to the running $MS$ parameters of the SM
2. For a low-scale version it is common to match to the measured (pole) values of the SM parameters

## Matching to running parameters

#### Note

The method described here was introduce in SARAH 4.9.0 and replaces previous attempts to define the matching for

### General

The SPheno versions by SARAH calculate the running values for the SM gauge (g1SM, g2SM, g3SM) and Yukawa (YeSM, YdSM, YuSM) couplings as well as for the electroweak VEV (vSM) including the full one-loop corrections in the present model. This calculation is done at the scale $M_Z$ and the input parameters which are used are

• The running electroweak and strong coupling constants ($\alpha(M_Z)$, $\alpha_S(M_Z)$)
• The measured masses of Z- and W- bosons ($M_Z$, $M_W$)
• The measured masses of all SM fermions ($m_d$,$m_u$,$m_l$)
• The entries of the CKM matrix

The exact calculation of the thresholds is explained here: SPheno_threshold_corrections.

The user can then define in the SPheno.m file the matching conditions between the running SM parameters and the parameters in the model via

DEFINITION[MatchingConditions]={
{Model parameters, SM parameter},
...
};


### Matching for a One-Higgs doublet model

The simplest matching conditions are those in which the Higgs sector is the same as in the SM:

DEFINITION[MatchingConditions]={
{Ye, YeSM},
{Yd, YdSM},
{Yu, YuSM},
{g1, g1SM},
{g2, g2SM},
{g3, g3SM},
{v, vSM}
};


One can also use as short form

DEFINITION[MatchingConditions]=Default[OHDM];


In this case, SARAH makes use of the knowledge which parameters in the model correspond to the SM parameters.

### Matching for a THDM-II-like model

The matching conditions for a model which has a Higgs sector as a THDM-II like the most common SUSY models for instance, would read with this definition

DEFINITION[MatchingConditions]={
{g1, g1SM},
{g2, g2SM},
{g3, g3SM},
{vd, vSM/Sqrt[1+TanBeta^2]},
{vu, TanBeta vd},
{Yu, vSM YuSM/vu},
{Yd, vSM YdSM/vd},
{Ye, vSM YeSM,vd}
};


One can also use as short form

DEFINITION[MatchingConditions]=Default[THDMII];


In this case, SARAH makes use of the knowledge which parameters in the model correspond to the SM parameters.

### Matching for Models with extra gauge groups

One can use these conditions also to define the matching conditions in a model with has a $SU(2)_R \times U(1)_{B-L}$ gauge group which just gets broken at the SUSY scale to $U(1)_Y$. The matching conditions in this case read

DEFINITION[MatchingConditions]={
{gR, Sqrt[5/3]*g1RBLFactor*g1SM},
{gBL, -((-g1SM^2 gRBL + gBLR gR gRBL + Sqrt[g1SM^2 (gBLR - gR)^2 (-g1SM^2 + gR^2 + gRBL^2)])/(g1SM^2 - gR^2))},
{g2, g2SM},
{g3, g3SM},
{vd, vSM/Sqrt[1 + TanBeta^2]},
{vu, VEVSM1*TanBeta},
{Yu, YuSM vSM/vu},
{Yd, YdSM vSM/vd},
{Ye, YeSM vSM/vd}
};


Here, g1RBLFactor which relates $g_R$ and $g_Y^{SM}$ is in principle a free parameter which however can be fixed by imposing gauge coupling unification.

### Matching for Models with extra Higgs doublets

Even much more complicated matching conditions can be imposed. For instance, the matching in the left-right symmetry model with two generations of bi-doublets are

\begin{aligned} v_{u,1} &= v_L \sin\beta \sin\beta_u\\ v_{d,1} &= v_L \cos\beta \sin\beta_d\\ v_{u,2} &= v_L \sin\beta \cos\beta_u \\ v_{d,2} &= v_L \cos\beta \cos\beta_d\\ Y_{Q_1}&=-\frac{Y_d\sqrt{1+\tan\beta_d^2} -Y_u\sqrt{1+ \tan\beta_u^2} }{\tan\beta_d-\tan\beta_u}\\ Y_{Q_2}&=\frac{\tan\beta_u Y_d\sqrt{1+\tan\beta_d^2} -Y_u\tan\beta_d\sqrt{1+ \tan\beta_u^2}}{\tan\beta_d-\tan\beta_u}\\ Y_{L_1}&=-\frac{Y_e\sqrt{1+\tan\beta_d^2} -Y_\nu\sqrt{1+ \tan\beta_u^2} }{\tan\beta_d-\tan\beta_u}\\ Y_{L_2}&=\frac{\tan\beta_u Y_e\sqrt{1+\tan\beta_d^2} -Y_\nu\tan\beta_d\sqrt{1+ \tan\beta_u^2}}{\tan\beta_d-\tan\beta_u} \end{aligned}

where three angles are introduced to parametrize all VEVs and the neutrino Yukawa coupling is considered to be given as input. These matching conditions together with the ones in the gauge sector are defined via

(* to get simpler expressions *)
vdAux = vSM/Sqrt[1+TanBeta^2];
vuAux = TanBeta*vSM/Sqrt[1+TanBeta^2];

YuA=Transpose[YuSM]*Sqrt[1+TanBeta^2]/TanBeta
YdA=Transpose[YdSM]*Sqrt[1+TanBeta^2]
YeA=Transpose[YeSM]*Sqrt[1+TanBeta^2]

DEFINITION[MatchingConditions]={
(* Matching for gauge couplings *)
{gR, Sqrt[5/3]*g1RBLFactor*g1SM},
{gBL, (gR g1SM)/Sqrt[gR^2 - g1SM^2]},
{g2, g2SM},
{g3, g3SM},

(* Matching for VEVs *)
{vPhiu[1] , (TanBeta*TanBetaU*vSM)/(Sqrt[1 + TanBeta^2]*Sqrt[1 + TanBetaU^2])},
{vPhiu[2] , (TanBeta*vSM)/(Sqrt[1 + TanBeta^2]*Sqrt[1 + TanBetaU^2])},
{vPhid[2] , vSM/(Sqrt[1 + TanBeta^2]*Sqrt[1 + TanBetaD^2])},

(* Matching for Yukawas *)
{YQ[1;;3,1;;3,1] , -(vuAux * vPhid[2] * YuA - vdAux * vPhiu[2] * YdA)/(vPhid[2] * vPhiu[1] - vPhid[1] * vPhiu[2])},
{YQ[1;;3,1;;3,2] ,  (vuAux * vPhid[1] * YuA - vdAux * vPhiu[1] * YdA)/(vPhid[2] * vPhiu[1] - vPhid[1] * vPhiu[2])},
{YL[1;;3,1;;3,1] ,  ( vdAux * vPhiu[2] *YeA)/(vPhid[2] * vPhiu[1] - vPhid[1] * vPhiu[2])},
{YL[1;;3,1;;3,2] , (- vdAux * vPhiu[1] * YeA)/(vPhid[2] * vPhiu[1] - vPhid[1] * vPhiu[2])}
};


## Matching to on-shell parameters

With version 4.12.0 the matching procedure has been unified for high and low-scale models by introducing a two-scale matching procedure, see 1703.03267. Thus, also for non-SUSY models the full one-loop thresholds are included by default. The definition of the matching conditions is the same as explained above and done via

DEFINITION[MatchingConditions]={
...


(The old approach of writing g1SM, etc. in BoundaryLowScaleInput is no longer recommended!)

However, under some circumstances, the user might be interested in using OS values for the input parameters. That's for instance the case when performing only a tree-level calculation of the scalar masses under the assumption that all higher order corrections can be absorbed into a set of counter-terms. In this case, one can use

Block SPhenoInput
66 0 # Two Scale Matching


in the Les Houches input file. This turns off the new two-scale matching routines. For a low-scale input, this results in a pole mass matching.

### Remark

Note, the often used approach to ignore higher order corrections in the scalar sector of non-supersymmetric models should be taken with care as discussed in these papers: