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MSSM scenarios after discovery of the Higgs boson
Mikhail Dubinin and Elena Petrova1 Physics Department, D. V. Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov Moscow State University Leninskie Gory 1, 119991 Moscow, Russia

Higgs boson discovery at the LHC with the mass m

H

=125 GeV, not accompanied by any signals of additional

scalar resonances and superpartners of the minimal supersymmetric standard model (MSSM), indicates that the supersymmetry breaking scale is not smaller than a few TeV. Such a situation favors the MSSM scenarios close to the decoupling limit in the Higgs sector that could apparently restrict the MSSM parameter space. We analyse, whenever possible, precise restrictions imposed on the MSSM parameter space by the observed Higgs boson mass m
H

=125 GeV and capabilities of a simplified approximations of the radiative corrections in the MSSM Higgs

sector, which could reduce the multidimensional MSSM parameter space to a simple two-dimensional parameter space with a sufficient degree of precision .

1

Introduction

Higgs boson discovery at the LHC with the mass m H = 125.09 ± 0.24 GeV [1] results in an important consequences for the minimal supersymmetric standard model (MSSM) two-doublet Higgs sector. The Higgs sector includes five physical states in the C P-conserving limit: two CP-even neutral Higgs bosons h and H , two charged scalars H ± and one CP-odd neutral Higgs boson A [2]. Large radiative corrections from the third generation quarks and quark superpartners play an important role in the generation of the effective two-doublet Higgs potential at the scale below MSUSY . If in the absence of any corrections the tree mass of the lightest state mh can not exceed the Z0 -boson mass [3] and the observables (masses and couplings) can be defined by means of the two parameters, the mixing angle tan = v2 /v1 and the charged scalar mass m H ± 2 , then the primary calculation [4] of radiative corrections to the effective potential changes significantly this simple picture. Multidimensional MSSM parameter space even in the simplified variant is described besides the two-dimensional set tan , m H ± by the three additional parameters: µ (Higgs superfield mass parameter), At = Ab (degenerate trilinear Higgs boson - third generation squarks mixing parameters) 3 , MSUSY (the quark superpartners mass scale). Radiative corrections which 'bring up' the lightest Higgs boson mass from the tree limit m Z to the observable avegage value 125.09 GeV are defined by the terms of the order of µ/ MSUSY , At,b / MSUSY , see explicit formulas in [5, 6], also [7]. In the framework of parametric scenarios it is natural to simplify as much of the Higgs sector in the five-dimensional parameter space m H ± , tan , most essential variables which demonstrate the strongest influence on the particles4 . Such parametrizations [8, 9] are based on the observation that
1 2

as one can the parametrization µ, At = Ab , MSUSY separating masses and couplings of scalar the leading and the subleading
H±

petrova@theory.sinp.msu As usual v1,2 are the vacuum expectation values of the two Higgs isodoublets. Instead of m connected by a simple formula. 3 In the general case A = A , for simplicity we assume A = A . t t b b 4 except maybe the g g h and h effective couplings induced at the one-loop level

one can use m A , two masses are

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

radiative corrections are most strongly dependent on the mh value in the regime which is determined by an essential difference of radiative corrections to the matrix elements of CP-even Higgs bosons h, H mixing matrix, M21,12 0, M22 = 0 (see Section 2.2). This circumstance allows one to reduce the number 2 1 of free parameters of the effective potential from five to only two, (m A , tan ), which is attractive from phenomenological point of view. At the same time it is known from numerous facts of the case [10] that insignificant changes of the leading radiative corrections could result in the noticeable shifts of the Higgs boson masses and couplings5 . Previous comparisons (before the Higgs boson discovery) for the three approaches FeynHiggs [11], CPsuperH [12] and the present approach (realized in the CompHEP [13] shell) can be found in the Appendix of [5]. These comparisons have shown that careful treatment of approximations on the basis of 'leading' radiative corrections is appropriate. The question of their precision deserves a careful study. In the effective field theory approach the MSSM with complete particle composition at the scale above MSUSY is matched to an effective SM Lagrangian where the MSSM particles decouple at the scale mto p . The SM couplings are fixed at MSUSY scale by supersymmetric conditions. The third generation squark interactions (also called the threshold effects in this context) are accounted for by modification of the supersymmetric matching conditions at the scale MSUSY . Our analysis is based on the calculation of radiative corrections in the MSSM Higgs sector by means of the effective potential method [5], taking into account carefully the nonleading soft supersymmetry breaking D-terms, the wave-function renormalization terms and also including the QCD and weak corrections to Yukawa couplings up to two loops [7, 14, 15].

2

Higgs sector of the 2HDM

The most general form of 2HDM potential in the generic basis is [16] U ( 1 , 2 ) = - µ 2 ( 1 ) - µ 2 ( 2 ) - µ 22 ( 2 ) - µ 2 1 1 2 1 1
2 12

( 1 ) + 2

(1 )

+ 1 ( 1 ) 2 + 2 ( 2 ) 2 + 3 ( 1 ) ( 2 ) + 4 ( 2 ) ( 1 ) + 2 2 2 1 1 1 +
5 ( 1 2 ) ( 2 ) + 5 ( 1 ) ( 1 ) + 2 2 1 2 2


+ 6 ( 1 ) ( 2 ) + 6 ( 1 ) ( 1 ) + 7 ( 2 ) ( 2 ) + 7 ( 2 ) ( 1 ) , 2 2 2 2 1 1 1 1
where SU (2)-states are j =

1 2

-i + j (v j + j + i j )

,

j = 1, 2.

(2 )

In framework of MSSM the potential (1) in the mass basis is U ( h , H , A, H ± , G 0 , G ± ) = m2 2 m2 2 m2 2 h h + HH + AA +m 2 2 2
2 H±

H + H - + I (3) + I (4),

(3 )

where I (3) and I (4) are the triple and the quartic couplings of Higgs bosons, respectively.
5

for example, triple self-couplings in the mass basis include the terms proportional to m

2 h, H , A

/v

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

Here the SU (2) states 1,2 , 1 2

1 ,2

, ,

± 1 ,2

and the mass eigenstates h, H , A, H ± , G0, G ± can be written as 6 [5, 6]
1 2

=O



h H

=O



G0 A

,



± 1 ± 2

=O



G H

±

±

,

(4 )

where the rotation matrix

OX =

cos X sin X

- sin X cos X

,

X = , .

(5 )

2.1 Masses and mixing angles
The masses in (3) are obtained as the eigenvalues of the mass matrix in the representation used in [16] m
2 1 ,2 7

=

1 2

tr M2 ± Y

( t r M 2 ) 2 - 4 det M Y

2 Y

,

(6 )

T or as diagonal elements of the mass matrix O X M2 O X after rotations of the SU (2) eigenstates parametrized Y by , [5], see also [17], where

M2 = Y

M M

2 11 2 21

M M

2 12 2 22

,

M2j = i

2 U , Yi Yj

Y = , , ± ,

i , j = 1, 2.

(7 )

For example the mixing angle and the masses of C P-even Higgs bosons have the forms m
2 H ,h

= =

tan 2 where C = 4M or m
2 h 4 12

12 ( m + m 2 + M 21 + M 22 ± 2 Z 1 2A 2 - ( m2 + m2 ) s 2 M 12 Z A 2

m4 + m4 - 2m2 m2 c Z AZ A ,

4

+ C ),

(8 )

( m2 - m2 ) c Z A

2

+ M 21 - M 1

2 22

2 + ( M 21 - M 22 ) 2 - 2 ( m 2 - m 2 ) ( M 11 - M 22 ) c 2 2 Z 1 A

2

- 4 ( m 2 + m 2 ) M 22 s 2 , 1 Z A

(9 )

=s

2 +

m2 + c Z m2 + s Z

2 - 2 -

m 2 - v 2 ( 1 s2 c 2 + 2 c 2 s A
22 s

m

2 H

- 2 ( 3 + 4 ) c c s s + Re 5 ( s =c
2 +

+c

2

+ 2 ( 3 + 4 ) c c s s + Re 5 ( c 2 s 2 + s
tan 2

m 2 - v 2 ( 1 c 2 c 2 + 2 s A
2 Z 2 2

2 2

2 c2 2 s c2

(1 0 )

) - 2c ) + 2s
2

+

(Re6 s c - Re7 c s )),
(1 1 )

+

( Re 6 c c + Re 7 s s ) ) ,
(1 2 )

=

c

s2 ( m 2 + A ( m2 - m2 ) 2 Z A

m ) + v ( ( 3 + 4 ) s

+ 2 c Re 6 + 2 s 2 Re 7 )
2

+ v 2 ( 1 c 2 - 2 s 2 - Re 5 c

+ ( Re 6 - Re 7 ) s 2 )

before and after rotation by angle , respectively. The above mentioned representations give identical results. Numerical value of the Higgs boson mass mh =125 GeV allows one to simplify the equations describing the MSSM two-doublet sector. Instead of Eq.(8) which contains an inconvenient square root the compact equation for the heavy CP-even state H is valid with good numerical precision, as a rule m
6 7

2 H

= m 2 + m 2 - m 2 + M 21 + M 22 . 2 A Z 1 h

(1 3 )

Note that the SM Higgs field is at the same time the SU (2) state and the mass state. Mixing angle of CP-even states can be numerically calculated, then explicitly used for evaluation of I (3) and I (4) vertices.

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

2.2 Mass martix corrections
Radiative corrections to the mass matrix of the fields 1 , 2 can be written as

M2 =

M M

2 11 2 12

M M
2 22

2 12 2 22

+

M M

2 11 2 12

M M

2 12 2 22

,

(1 4 )

where the tree-level matrix elements are

M

2 11

= m 2 s2 + m 2 c 2 , Z A

M

= m 2 c 2 + m 2 s2 , Z A

M

2 12

= -s c ( m2 + m2 ) Z A

(1 5 )

and Mi j , k (i, j = 1, 2, k = 1, ..., 7) in the general case include corrections of all orders of the perturbation theory M M M
2 11 2 22 2 12

= - v 2 ( 1 c 2 + Re 5 s 2 + Re 6 s 2 ) ,

= - v 2 ( 2 s 2 + Re 5 c 2 + Re 7 s 2 ) ,

(1 6 )

= - v 2 ( 34 s c + Re 6 c 2 + Re 7 s 2 ) .

2 It is demonstrated in [8] that at mh = 123 - 129 GeV the corrections M11,12 0 but M22 = 0. In other 2 words, all contributions coming from higher order corrections are contained in M22 . It is obvious from 2 (8) that M22 can be effectively accounted for using the experimentally measured value of mh [8] 2

M

2 22

=

m2 ( m2 + m2 - m2 ) - m2 m2 c Z A AZ h h m 2 c 2 + m 2 s2 - m Z A
2 h

2 2

,

(1 7 )

then (8) are simplified (approximate hMSSM scenario) [8] m
2 H

=

( m 2 + m 2 - m 2 ) ( m 2 c 2 + m 2 s2 ) - m 2 m 2 c Z Z A A AZ h
m 2 c 2 + m 2 s2 - m Z A m
2 h

2 2

,

(1 8 )

= - arctan

( m2 + m2 ) s c Z A 2 c 2 + m 2 s2 - m 2 Z A h

.

(1 9 )

3

Numerical results

In this section the M2j are estimated and where it is possible the hMSSM consequenses are compared i with predictions of approach developed in [5, 6]. In Fig. 1 mh is plotted as a function of A for three sets of supersymmetric parameters and m A equal to 300 GeV or 1 TeV. The distribution is remarkably stable for various m A (the maximal shift of value mh is about 3 GeV), so in the folowing we use the intermidiate value m A = 300 GeV. Also for numerical analysis we have taken 16 sets of supersymmetric parameters (see Table 1). For example, the set (3; 4) correspond to MSUSY = 3.5 TeV, tan = 5, A = 5 TeV and any µ from (-5; 5) TeV. For tan 5 the parameter A may be a constant8 so that the condition mh = 123 - 128 GeV is satisfied. For this type of sets the values of M2j are plotted as functions of µ, for other sets the values of M2j are i i
8 Note that there exist varios sets of ( A, µ ) for which A is not a constant and m = 123 - 128 GeV, but we want to find these h exceptional cases.

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

Figure 1: mh as function of A where MSUSY = 1 TeV, tan = 30 (green line), MSUSY = 1 TeV, tan = 5 (red line), MSUSY = 3 TeV, tan = 2.5 (blue line) and µ = 1 TeV, m A = 300 GeV (solid line) or m A = 1 TeV (dashed line).

estimated numerically. In Fig. 2 these values are presented for parametric sets with small and large tan : for intermediate and high tan the assumptions of hMSSM approach may be relevant, for tan < 5 this approach is not reliable. In Fig. 3 m H and are presented in the framework of two scenarios for three parametric sets. The agreement between hMSSM and the approach [5, 6] predictions is very good for large tan (the solid and dashed green lines coincide). As discussed above for tan < 5 the disagreement is substantial. This comes into particular prominence for (the maximal shift is about 14%). The disagreement of the scenarios for m H with small tan < 5 is observable though it is only around 1% (blue lines in Fig. 3(left)). Note that curved part of lines for hMSSM m H is an unphysical consequence of the pole in (18). tan 1 2 .5 5 30 1 2 3 .5 2 6 .5 2 .2 ( 7; 10 ) 1 .4 ( - 2; 2 ) MSUSY 1 .5 6 3 .1 4 6 .5 3 ( - 5; 5 ) 1 .7 ( - 3; 3 ) [TeV] 3 5 8 .3 5 .5 14 5 ( - 5; 5 ) 0 ( - 9; 9 )

3 .5 10 8 10 8 .5 5 ( - 5; 5 ) 0 .5 ( - 11; 11 )

Table 1: Supersymmetric parametric sets where the first number is A [TeV] ( A is such that mh = 123 - 128 GeV and Ab = At = A), the second number is µ [TeV] in each set, m A = 300 GeV.

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

Figure 2: Estimation of M2j where M21 - green, M i 1 (2; 4) (left) and (3; 2) - (4; 4) (right) (see Table 1).

2 12

- blue, M

2 22

- red for parametric sets (1; 1) -

Figure 3: Predictions of hMSSM (dashed line) and the approach [5, 6] (solid line) for m H (left) and (right) where parametric set (2;4) is blue, (3;4) is red, (4;4) is green.

4

Summary and Outlook

An analysis of constraints imposed by the condition mh =125 GeV on the five-dimensional MSSM parameter space is performed in the MSSM scenarios. The MSSM parameters are chosen to define in an adequate form the radiative corrections in the MSSM Higgs sector, which are evaluated in the effective potential approach. It is demonstrated that the restriction imposed by the condition mh =125 GeV is not strong, isolating rather limited regions of the five-dimensional MSSM parameter space. The evaluations are performed for the sixteen parametric sets including the small tan region. First of all for the parametric sets (three of them coincide with [8]) we find the real µ (Higgs superfield mass parameter) and A (trilinear soft SUSY breaking parameter) with respect to the condition mh =125 GeV and evaluate the corrections to the Higgs bosons mixing matrix M2j . For these parametric sets the corrections i are different and not stable when tan is small (< 5) and they may be in a good agreement with hMSSM assumption M21,12 0, M22 = 0 when tan is intermediate or large. The hMSSM expressions for 2 1 C P-even heavy Higgs mass m H and mixing angle are compared with complete expressions where again 6


XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

the dependence of hMSSM scenario validity on the tan variation range is demonstrated. At the same time the appropriate tan does not guarantee the validity of assumption M21,12 0, 1 M22 = 0 (as a rule the values of M2j are of the same order). Therefore hMSSM scenario is reliable 2 i in some limited regions of the parameter space. For complete and precise analysis the full five-dimensional MSSM parameter space implementation is appropriate.

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