MSSMEFTHiggs (FlexibleEFTHiggs for the MSSM) is an implementation of
the Standard Model, matched to the MSSM at the SUSY scale,
M_\text{SUSY}. The matching is performed at the 1-loop level
using the FlexibleEFTHiggs approach described in [1609.00371] and
[1710.03760]. The setup of MSSMEFTHiggs is shown in the following
figure.
In MSSMEFTHiggs, the HighScale variable is set to the SUSY scale,
M_{\text{SUSY}}. At this scale the quartic Higgs coupling,
\lambda(M_\text{SUSY}), is predicted from the matching of the
Higgs pole masses of the Standard Model and the MSSM at the full
1-loop level (FlexibleEFTHiggs method):
(M_h^2)_{\text{SM}} = (M_h^2)_{\text{MSSM}}
The Higgs pole mass in the Standard Model is decomposed into a
tree-level and 1-loop part as (M_h^2)_{\text{SM}} = \lambda
v^2 + (\Delta m_h^2)_{\text{SM}} and the quartic Higgs coupling is
calculated as
\lambda(M_{\text{SUSY}}) =
\frac{1}{v^2}\Big[(M_h^2)_{\text{MSSM}} - (\Delta
m_h^2)_{\text{SM}}\Big]
This physical matching condition incorporates the
O(v^2/M_\text{SUSY}^2) suppressed terms to all orders into
\lambda(M_\text{SUSY}). Thus, MSSMEFTHiggs can correctly
predict the Higgs pole mass of the MSSM at the full 1-loop level for
both low and high SUSY scales. In other words, MSSMEFTHiggs is a
hybrid calculation, which combines a fixed-order with an EFT
calculation. See [1609.00371] for a detailed description of the
FlexibleEFTHiggs method.
The 3- and partial 4- and 5-loop renormalization group equations of
[1303.4364], [1307.3536], [1508.00912], [1508.02680],
[1604.00853], [1606.08659] are used to run
\lambda(M_\text{SUSY}) down to the electroweak scale
M_Z or M_t.
If M_{\text{SUSY}} is set to zero, M_{\text{SUSY}} =
\sqrt{m_{\tilde{t}_1}m_{\tilde{t}_2}} is used.
The LowScale is set to M_Z. At this scale, the
\overline{\text{MS}} gauge and Yukawa couplings
g_{1,2,3}(M_Z), Y_{u,d,e}(M_Z), as well as the SM
vacuum expectation value (VEV), v(M_Z), are calculated at the
full 1-loop level from the known low-energy couplings
\alpha_{\text{em}}^{\text{SM(5)}}(M_Z),
\alpha_s^{\text{SM(5)}}(M_Z), from the pole masses
M_Z, M_e, M_\mu, M_\tau, M_t
as well as from the \overline{\text{MS}} masses
m_b^{\text{SM(5)}}(m_b), m_c^{\text{SM(4)}}(m_c),
m_s(2\,\text{GeV}), m_d(2\,\text{GeV}),
m_u(2\,\text{GeV}). In addition to these 1-loop corrections,
known 2-, 3- and 4-loop corrections are taken into account, see the
following table.
See the documentation of the SLHA input parameters for a
description of the individual flags to enable/disable higher-order
threshold corrections in FlexibleSUSY.
The Higgs and W boson pole masses, M_h and M_Z are
calculated at the scale M_t, which is an input parameter.
Furthermore, the electroweak symmetry breaking condition of the
Standard Model is imposed at the scale M_t to fix the value of
the bililear Higgs coupling \mu^2(M_t) in the Standard Model.
The Higgs and W boson pole masses, M_h and M_W, are
calculated at the full 1-loop level in the Standard Model, including
potential flavour mixing and momentum dependence. Depending on the
given configuration flags, additional 2-, 3- and 4-loop corrections to
the Higgs pole mass of O(\alpha_t\alpha_s + \alpha_b\alpha_s)
[1407.4336] O((\alpha_t + \alpha_b)^2) [1205.6497] and
O(\alpha_\tau^2), as well as 3-loop corrections
O(\alpha_t^3
+ \alpha_t^2\alpha_s + \alpha_t\alpha_s^2) [1407.4336] and 4-loop
corrections O(\alpha_t\alpha_s^3) [1508.00912] can be taken
into account.
Note
Note, that the 3-loop contributions O(\alpha_t^3 +
\alpha_t^2\alpha_s) are incomplete, because the
corresponding 2-loop threshold corrections
O(\alpha_t^2 + \alpha_t\alpha_s) to the running top
Yukawa coupling are not implemented yet.
The MSSM particle masses are calculated at the full 1-loop level in
the MSSM at the SUSY scale M_{\text{SUSY}}.
MSSMEFTHiggs takes the following physics parameters as input:
| Parameter |
Description |
SLHA block/field |
Mathematica symbol |
|---|
| M_{\text{SUSY}} |
SUSY scale |
EXTPAR[0] |
MSUSY |
| M_1(M_\text{SUSY}) |
Bino mass |
EXTPAR[1] |
M1Input |
| M_2(M_\text{SUSY}) |
Wino mass |
EXTPAR[2] |
M2Input |
| M_3(M_\text{SUSY}) |
Gluino mass |
EXTPAR[3] |
M3Input |
| \mu(M_\text{SUSY}) |
\mu-parameter |
EXTPAR[4] |
MuInput |
| m_A(M_\text{SUSY}) |
running CP-odd Higgs mass |
EXTPAR[5] |
mAInput |
| \tan\beta(M_\text{SUSY}) |
\tan\beta(M_\text{SUSY})=v_u(M_\text{SUSY})/v_d(M_\text{SUSY}) |
EXTPAR[25] |
TanBeta |
| (A_u)_{ij}(M_\text{SUSY}) |
trililear up-type squark couplings |
AUIN |
AuInput |
| (A_d)_{ij}(M_\text{SUSY}) |
trililear down-type squark couplings |
ADIN |
AdInput |
| (A_e)_{ij}(M_\text{SUSY}) |
trililear down-type sfermion couplings |
AEIN |
AeInput |
| (m_{\tilde{q}}^2)_{ij}(M_\text{SUSY}) |
soft-breaking left-handed squark mass parameters |
MSQ2IN |
mq2Input |
| (m_{\tilde{u}}^2)_{ij}(M_\text{SUSY}) |
soft-breaking right-handed up-type squark mass parameters |
MSU2IN |
mu2Input |
| (m_{\tilde{d}}^2)_{ij}(M_\text{SUSY}) |
soft-breaking right-handed down-type squark mass parameters |
MSD2IN |
md2Input |
| (m_{\tilde{l}}^2)_{ij}(M_\text{SUSY}) |
soft-breaking left-handed slepton mass parameters |
MSL2IN |
ml2Input |
| (m_{\tilde{e}}^2)_{ij}(M_\text{SUSY}) |
soft-breaking right-handed down-type slepton mass parameters |
MSE2IN |
me2Input |
The MSSM parameters are defined in the \overline{\text{DR}}
scheme at the scale M_{\text{SUSY}}.
We recommend to run MSSMEFTHiggs with the following configuration flags: In
an SLHA input file we recommend to use:
Block FlexibleSUSY
0 1.0e-05 # precision goal
1 0 # max. iterations (0 = automatic)
2 0 # algorithm (0 = all, 1 = two_scale, 2 = semi_analytic)
3 1 # calculate SM pole masses
4 4 # pole mass loop order
5 4 # EWSB loop order
6 4 # beta-functions loop order
7 4 # threshold corrections loop order
8 1 # Higgs 2-loop corrections O(alpha_t alpha_s)
9 1 # Higgs 2-loop corrections O(alpha_b alpha_s)
10 1 # Higgs 2-loop corrections O(alpha_t^2 + alpha_t alpha_b + alpha_b^2)
11 1 # Higgs 2-loop corrections O(alpha_tau^2)
12 0 # force output
13 3 # Top pole mass QCD corrections (0 = 1L, 1 = 2L, 2 = 3L)
14 1.0e-11 # beta-function zero threshold
15 0 # calculate observables (a_muon, ...)
16 0 # force positive majorana masses
17 0 # pole mass renormalization scale (0 = SUSY scale)
18 0 # pole mass renormalization scale in the EFT (0 = min(SUSY scale, Mt))
19 0 # EFT matching scale (0 = SUSY scale)
20 2 # EFT loop order for upwards matching
21 1 # EFT loop order for downwards matching
22 0 # EFT index of SM-like Higgs in the BSM model
23 1 # calculate BSM pole masses
24 124111421 # individual threshold correction loop orders
25 0 # ren. scheme for Higgs 3L corrections (0 = DR, 1 = MDR)
26 1 # Higgs 3-loop corrections O(alpha_t alpha_s^2)
27 1 # Higgs 3-loop corrections O(alpha_b alpha_s^2)
28 1 # Higgs 3-loop corrections O(alpha_t^2 alpha_s)
29 1 # Higgs 3-loop corrections O(alpha_t^3)
30 1 # Higgs 4-loop corrections O(alpha_t alpha_s^3)
In the Mathematica interface we recommend to use:
handle = FSMSSMEFTHiggsOpenHandle[
fsSettings -> {
precisionGoal -> 1.*^-5, (* FlexibleSUSY[0] *)
maxIterations -> 0, (* FlexibleSUSY[1] *)
solver -> 0, (* FlexibleSUSY[2] *)
calculateStandardModelMasses -> 1, (* FlexibleSUSY[3] *)
poleMassLoopOrder -> 4, (* FlexibleSUSY[4] *)
ewsbLoopOrder -> 4, (* FlexibleSUSY[5] *)
betaFunctionLoopOrder -> 4, (* FlexibleSUSY[6] *)
thresholdCorrectionsLoopOrder -> 4,(* FlexibleSUSY[7] *)
higgs2loopCorrectionAtAs -> 1, (* FlexibleSUSY[8] *)
higgs2loopCorrectionAbAs -> 1, (* FlexibleSUSY[9] *)
higgs2loopCorrectionAtAt -> 1, (* FlexibleSUSY[10] *)
higgs2loopCorrectionAtauAtau -> 1, (* FlexibleSUSY[11] *)
forceOutput -> 0, (* FlexibleSUSY[12] *)
topPoleQCDCorrections -> 1, (* FlexibleSUSY[13] *)
betaZeroThreshold -> 1.*^-11, (* FlexibleSUSY[14] *)
forcePositiveMasses -> 0, (* FlexibleSUSY[16] *)
poleMassScale -> 0, (* FlexibleSUSY[17] *)
eftPoleMassScale -> 0, (* FlexibleSUSY[18] *)
eftMatchingScale -> 0, (* FlexibleSUSY[19] *)
eftMatchingLoopOrderUp -> 0, (* FlexibleSUSY[20] *)
eftMatchingLoopOrderDown -> 1, (* FlexibleSUSY[21] *)
eftHiggsIndex -> 0, (* FlexibleSUSY[22] *)
calculateBSMMasses -> 1, (* FlexibleSUSY[23] *)
thresholdCorrections -> 124111421, (* FlexibleSUSY[24] *)
higgs3loopCorrectionRenScheme -> 0,(* FlexibleSUSY[25] *)
higgs3loopCorrectionAtAsAs -> 1, (* FlexibleSUSY[26] *)
higgs3loopCorrectionAbAsAs -> 1, (* FlexibleSUSY[27] *)
higgs3loopCorrectionAtAtAs -> 1, (* FlexibleSUSY[28] *)
higgs3loopCorrectionAtAtAt -> 1, (* FlexibleSUSY[29] *)
higgs4loopCorrectionAtAsAsAs -> 1, (* FlexibleSUSY[30] *)
parameterOutputScale -> 0 (* MODSEL[12] *)
},
...
];
In the file
model_files/MSSMEFTHiggs/MSSMEFTHiggs_uncertainty_estimate.m
FlexibleSUSY provides the Mathematica function
CalcMSSMEFTHiggsDMh[], which calculates the Higgs pole mass with
MSSMEFTHiggs and performs an uncertainty estimate of missing higher
order corrections. Two main sources of the theory uncertainty are
taken into account:
- SM uncertainty: Missing higher order corrections in the
calculation of the running Standard Model top Yukawa coupling and
in the calculation of the Higgs pole mass. The uncertainty from
this source is estimated by (i) switching on/off the 3-loop QCD
contributions in the calculation of the running top Yukawa coupling
y_t^{\text{SM}}(M_Z) from the top pole mass and by (ii)
varying the renormalization scale at which the Higgs pole mass is
calculated within the interval [M_t/2, 2 M_t].
- SUSY uncertainty: Missing higher order corrections in the
calculation of the quartic Higgs coupling
\lambda(M_\text{SUSY}). This uncertainty is estimated by
varying the matching scale within the interval
[M_{\text{SUSY}}/2, 2 M_{\text{SUSY}}].
The following code snippet illustrates the calculation of the Higgs
pole mass calculated at the 3-loop level with MSSMEFTHiggs as a
function of the SUSY scale (red solid line), together with the
estimated uncertainty (grey band).
When this script is executed, the following figure is produced:
