#### Precise prediction for the Higgs-boson masses in the \(\mu \nu \) SSM

Eur. Phys. J. C
Precise prediction for the Higgs-boson masses in the μν SSM
T. Biekötter 2 3
S. Heinemeyer 0 1 2
C. Muñoz 2 3
0 Instituto de Física de Cantabria (CSIC-UC) , 39005 Santander , Spain
1 Campus of International Excellence UAM
2 Instituto de Física Teórica UAM-CSIC , Cantoblanco, 28049 Madrid , Spain
3 Departamento de Física Teórica, Universidad Autónoma de Madrid (UAM) , Campus de Cantoblanco, 28049 Madrid , Spain
4 CSIC , Cantoblanco, 28049 Madrid , Spain
The μνSSM is a simple supersymmetric extension of the Standard Model (SM) capable of predicting neutrino physics in agreement with experiment. In this paper we perform the complete one-loop renormalization of the neutral scalar sector of the μνSSM with one generation of right-handed neutrinos in a mixed on-shell/DR scheme. The renormalization procedure is discussed in detail, emphasizing conceptual differences to the minimal (MSSM) and next-to-minimal (NMSSM) supersymmetric standard model regarding the field renormalization and the treatment of nonflavor-diagonal soft mass parameters, which have their origin in the breaking of R-parity in the μνSSM. We calculate the full one-loop corrections to the neutral scalar masses of the μνSSM. The one-loop contributions are supplemented by available MSSM higher-order corrections. We obtain numerical results for a SM-like Higgs boson mass consistent with experimental bounds. We compare our results to predictions in the NMSSM to obtain a measure for the significance of genuine μνSSM-like contributions. We only find minor corrections due to the smallness of the neutrino Yukawa couplings, indicating that the Higgs boson mass calculations in the μνSSM are at the same level of accuracy as in the NMSSM. Finally we show that the μνSSM can accomodate a Higgs boson that could explain an excess of γ γ events at ∼ 96 GeV as reported by CMS, as well as the 2 σ excess of bb¯ events observed at LEP at a similar mass scale.
1 Introduction
The spectacular discovery of a boson with a mass around
∼ 125 GeV by the ATLAS and CMS experiments [
1,2
] at
CERN constitutes a milestone in the quest for
understanding the physics of electroweak symmetry breaking (EWSB).
While within the present experimental uncertainties the
properties of the observed Higgs boson are compatible with the
predictions of the Standard Model (SM) [3], many other
interpretations are possible as well, in particular as a Higgs
boson of an extended Higgs sector. Consequently, any model
describing electroweak physics needs to provide a state that
can be identified with the observed signal.
One of the prime candidates for physics beyond the SM is
supersymmetry (SUSY), which doubles the particle degrees
of freedom by predicting two scalar partners for all SM
fermions, as well as fermionic partners to all bosons. The
simplest SUSY extension is the Minimal Supersymmetric
Standard Model (MSSM) [
4,5
]. In contrast to the single Higgs
doublet of the SM, the Higgs sector of the MSSM contains
two Higgs doublets, which in the CP conserving case leads
to a physical spectrum consisting of two CP-even, one
CPodd and two charged Higgs bosons. The light (or the heavy)
CP-even MSSM Higgs boson can be interpreted as the signal
discovered at ∼ 125 GeV [6].
Going beyond the MSSM, a well-motivated extension
is given by the Next-to-Minimal Supersymmetric Standard
Model (NMSSM), see e.g. [
7,8
] for reviews. In particular the
NMSSM provides a solution for the so-called “μ problem”
by naturally associating an adequate scale to the μ
parameter appearing in the MSSM superpotential [
9,10
]. In the
NMSSM a new singlet superfield is introduced, which only
couples to the Higgs- and sfermion-sectors, giving rise to
an effective μ-term, proportional to the vacuum
expectation value (vev) of the scalar singlet. Assuming CP
conservation, as we do throughout the paper, the states in the
NMSSM Higgs sector can be classified as three CP-even
Higgs bosons, hi (i = 1, 2, 3), two CP-odd Higgs bosons,
a j ( j = 1, 2), and the charged Higgs boson pair H ±. In
addition, the SUSY partner of the singlet Higgs (called the
singlino) extends the neutralino sector to a total of five
neutralinos. In the NMSSM the lightest but also the second
lightest CP-even neutral Higgs boson can be interpreted as the
signal observed at about 125 GeV, see, e.g., [
11,12
].
A natural extension of the NMSSM is the μνSSM, in
which the singlet superfield is interpreted as a right-handed
neutrino superfield [
13,14
] (see Refs. [
15–17
] for reviews).
The μνSSM is the simplest extension of the MSSM that can
provide massive neutrinos through a see-saw mechanism at
the electroweak scale. In this paper we will focus on the
μνSSM with one family of right-handed neutrino superfields,
and the case of three families will be studied in a future
publication.1 The μ problem is solved analogously to the NMSSM
by the coupling of the right-handed neutrino superfield to
the Higgs sector, and a trilinear coupling of the right-handed
neutrino generates an effective Majorana mass at the
electroweak scale. The unique feature of the μνSSM is the
introduction of a Yukawa coupling for the right-handed neutrino
of the order of the electron Yukawa coupling that induces
the explicit breaking of R-parity. One of the consequences is
that there is no lightest stable SUSY particle anymore.
Nevertheless, the model can still provide a dark matter candidate
with a gravitino that has a life time longer than the age of
the observable universe [
22–25
]. Since the lightest particle
beyond the SM is not stable, it can carry electrical charge
or even be coloured. The explicit violation of lepton number
and lepton flavor can modify the spectrum of the neutral and
charged fermions in comparison to the NMSSM. The three
families of charged leptons will mix with the chargino and the
Higgsino and form five massive charged fermions. However,
the mixing will naturally be tiny since the breaking of
Rparity is governed by the small neutrino Yukawa couplings.
In the neutral fermion sector the three left-handed neutrinos
mix with the right-handed neutrino and the four MSSM-like
neutralinos. When just one family of right-handed neutrino
is considered (as we do in this paper), the mass matrix of
the neutral fermions is of rank six, so just one light neutrino
mass is generated at tree-level, while the other two
lightneutrino masses will be generated by quantum corrections.
For the Higgs sector the breaking of R-parity has dramatic
consequences. The three left-handed and the right-handed
sneutrinos will mix with the doublet Higgses and form six
massive CP-even and five massive CP-odd states, assuming
that there is no CP-violation. Additionally, since the vacuum
of the model is not protected anymore by lepton number, the
sneutrinos will acquire a vev after spontaneous EWSB. While
the vev of the right-handed sneutrino can easily take values up
to the TeV-scale, the stability of the vacuum together with the
smallness of the neutrino Yukawa couplings force the vevs
1 The μνSSM with three families of right-handed neutrinos extends
the CP-even and CP-odd scalar sector and the neutral fermion sector
by two additional particles each, in particular allowing a more viable
reproduction of neutrino data [
13,14,18–21
].
of the left-handed sneutrinos to be several orders of
magnitude smaller [
13,14
]. As in the NMSSM, the couplings of
the doublet-like Higgses to the gauge-singlet right-handed
sneutrino provide additional contributions to the tree-level
mass of the SM-like Higgs boson, relaxing the prediction
of the MSSM, that it is bounded from above by the Z boson
mass. Still it was shown in the NMSSM [
26
] that a consistent
treatment of the quantum corrections is necessary for
accurate Higgs mass predictions (see also Refs. [
27–29
]). In this
paper we will investigate if this is also the case in the μνSSM
and if its unique couplings generate significant corrections
to the SM-like Higgs mass, that go beyond the corrections
arising in the NMSSM.
The experimental accuracy of the measured mass of the
observed Higgs boson has already reached the level of
a precision observable, with an uncertainty of less than
300 MeV [
3
]. In the MSSM the masses of the CP-even Higgs
bosons can be predicted at lowest order in terms of two
SUSY parameters characterising the MSSM Higgs sector,
e.g. tan β, the ratio of the vevs of the two doublets, and the
mass of the CP-odd Higgs boson, MA, or the charged Higgs
boson, MH± . This results in particular in an upper bound
on the mass of the light CP-even Higgs boson given by the
Z -boson mass. However, these relations receive large
higherorder corrections. Beyond the one-loop level, the dominant
two-loop corrections of O(αt αs ) [
30–35
] and O(αt2) [
36,37
]
as well as the corresponding corrections of O(αbαs ) [
38,39
]
and O(αt αb) [38] are known since more than a decade.
(Here we use α f = (Y f )2/(4π ), with Y f denoting the
fermion Yukawa coupling.) These corrections, together with
a resummation of leading and subleading logarithms from the
top/scalar top sector [
40
] (see also [
41,42
] for more details
on this type of approach), a resummation of leading
contributions from the bottom/scalar bottom sector [
38,39,43–46
]
(see also [
47,48
]) and momentum-dependent two-loop
contributions [
49,50
] (see also [
51
]) are included in the public
code FeynHiggs [
32,40,52–58
]. A (nearly) full two-loop
EP calculation, including even the leading three-loop
corrections, has also been published [
59,60
], which is,
however, not publicly available as a computer code.
Furthermore, another leading three-loop calculation of O(αt αs2),
depending on the various SUSY mass hierarchies, has been
performed [
61,62
], resulting in the code H3m and is now
available as a stand-alone code [
63
]. The theoretical
uncertainty on the lightest CP-even Higgs-boson mass within the
MSSM from unknown higher-order contributions is still at
the level of about 2−3 GeV for scalar top masses at the
TeVscale, where the actual uncertainty depends on the considered
parameter region [
40,54,64,65
].
In the NMSSM the status of the higher-order corrections to
the Higgs-boson masses (and mixings) is the following. Full
one-loop calculations including the momentum dependence
have been performed in the DR renormalization scheme in
Refs. [
66,67
], or in a mixed on-shell (OS)-DR scheme in
Refs. [
68–70
]. Two-loop corrections of O(αt αs , αt2) have
been included in the NMSSM in the leading logarithmic
approximation (LLA) in Refs. [
71,72
]. In the EP approach
at the two-loop level, the dominant O(αt αs , αbαs ) in the DR
scheme became available in Ref. [
66
]. The two-loop
corrections involving only superpotential couplings such as Yukawa
and singlet interactions were given in [
28
]. A two-loop
calculation of the O(αt αs ) corrections with the top/stop sector
renormalized in the OS scheme or in the DR scheme were
provided in Ref. [
73
]. A consistent combination of a full
one-loop calculation with all corrections beyond one-loop
in the MSSM approximation was given in Ref. [
70
], which
is included in the (private) version of FeynHiggs for the
NMSSM. A detailed comparison of the various higher-order
corrections up to the two-loop level involving a DR
renormalization was performed in Ref. [
29
], and involving an OS
renormalization of the top/stop sector for the O(αt αs )
corrections in Ref. [
74
]. Accordingly, at present the theoretical
uncertainties from unknown higher-order corrections in the
NMSSM are expected to be still larger than for the MSSM.
In this paper we go one step beyond and investigate the
scalar sector of the μνSSM, containing (mixtures of) Higgs
bosons and scalar neutrinos. As a first step we present the
renormalization at the one-loop level of the neutral scalar
sector in detail. Here a crucial point is that the NMSSM part of
the μνSSM is treated exactly in the same way as in Ref. [
70
].
Consequently, differences (at the one-loop level) appearing
for, e.g., mass relations or couplings can be directly attributed
to the richer structure of the μνSSM. As for the NMSSM in
Ref. [
70
], the full one-loop calculation is supplemented with
higher-order corrections in the MSSM limit (as provided
by FeynHiggs [
32,40,52–58
]).2 In our numerical
analysis we evaluate several “representative” scenarios using the
full one-loop results together with the MSSM-type
higherorder contributions. Differences found w.r.t. the NMSSM can
be interpreted in a twofold way. On the one hand, if
nonnegligible differences are found, they might serve as a probe
to distinguish the two models experimentally. On the other
hand, they indicate the level of theoretical uncertainties of the
Higgs-boson/scalar neutrino mass calculation in the μνSSM,
which should be brought to the same level of accuracy as in
the (N)MSSM.
The paper is organized as follows. In Sect. 2 we describe
the μνSSM, including the details for all sectors relevant in
this paper. The full one-loop renormalization of the neutral
scalar potential is presented in Sect. 3. We will establish a
convenient set of free parameters and fix their counterterms in
a mixed OS-DR scheme. The counterterms are calculated and
applied in the renormalized CP-even and CP-odd one-loop
2 A corresponding calculation using a pure DR renormalization could
in principle be performed using SARAH and SPheno [
27
].
scalar self-energies in Sect. 4. In this work we focus on the
application to the renormalized CP-even self-energies, but
the calculation of the renormalized CP-odd ones constitutes
a good additional test for the counterterms. We also describe
the incorporation of higher-order contributions taken over
from the MSSM. Our numerical analysis, including an
analysis of differences w.r.t. NMSSM, is presented in Sect. 5. We
conclude in Sect. 6.
2 The model: μνSSM with one generation of right
handed neutrinos
In the three-family notation of the μνSSM with one
generation of right-handed neutrinos the superpotential is written
as
W = ab Yiej Hˆda Lˆ ib eˆcj + Yidj Hˆda Qˆ ib dˆcj + Yiuj Hˆub Qˆ ia uˆcj
+ ab Yiν Hˆub Lˆ ia νˆ c − λ νˆ c Hˆub Hˆda
1
+ 3 κνˆ cνˆ cνˆ c. (1)
where HˆdT = (Hˆd0, Hˆd−) and HˆuT = (Hˆu+, Hˆu0) are the
MSSM-like doublet Higgs superfields, Qˆ iT = (uˆi , dˆi ) and
L T
ˆ i = (νˆi , eˆi ) are the left-chiral quark and lepton superfield
doublets, and uˆcj , dˆcj, eˆcj and νˆ c are the right-chiral quark and
lepton superfields. i and j are family indices running from
one to three and a, b = 1, 2 are indices of the fundamental
representation of SU(2) with ab the totally antisymmetric
tensor and ε12 = 1. The colour indices are undisplayed. Y u ,
Y d and Y e are the usual Yukawa couplings also present in the
MSSM. The right-handed neutrino is a gauge singlet, which
permits us to write the gauge-invariant trilinear self coupling
κ and the trilinear coupling with the Higgs doublets λ in
the second row, which are analogues to the couplings of the
singlet in the superpotential of the trilinear NMSSM. The
μterm is generated dynamically after the spontaneous EWSB,
when the right-handed sneutrino obtains a vev. The κ-term
forbids a global U(1) symmetry and we avoid the existence
of a Goldstone boson in the CP-even sector. The remarkable
difference to the NMSSM is the additional Yukawa coupling
Yiν , which induces explicit breaking of R-parity through the
λ- and κ-term, and which justifies the interpretation of the
singlet superfield as a right-handed neutrino superfield. It
should be pointed out that in this case lepton number is not
conserved anymore, and also the flavor symmetry in the
leptonic sector is broken. A more complete motivation of this
superpotential can be found in Refs. [
13,14,17
].
Working in the framework of low-energy SUSY the
corresponding soft SUSY-breaking Lagrangian can be written
as
−Lsoft = ab Tiej Hda LibL e∗j R + Tidj Hda QibL d ∗jR
equations of the left-handed sneutrinos and spoil the
electroweak seesaw mechanism that generates neutrino masses
of the correct order of magnitude. Theoretically, the absence
of these parameters mixing different fields at tree level,
(m2Hd L L )i , (m2L L )i j , (m2QL )i j , etc., can be justified by the
diagonal structure of the Kähler metric in certain
supergravity models, or when the dilaton field is the source of SUSY
breaking in string constructions [
17
]. Notice also that when
the down-type Higgs doublet superfield is interpreted as a
fourth family of leptons, the parameters m2 can be seen as
Hd L L
non-diagonal elements of m2L L [
75
]. Nevertheless, we include
them in the soft SUSY-breaking Lagrangian in this paper,
because these terms are generated at (one-)loop level, and in
our renormalization approach we need the functional
dependence of the scalar potential on m2 .
Hd L L
After the electroweak symmetry breaking the neutral
scalar fields will acquire a vev. This includes the
leftand right-handed sneutrinos, because they are not protected
by lepton number conservation as in the MSSM and the
NMSSM. We define the decomposition
H 0 1
d = √2
H 0 1
u = √2
1
νR = √
2
1
νi L = √
2
HdR + vd + i HdI ,
HuR + vu + i HuI ,
νRR + vR + i νIR ,
νiRL + vi L + i νiIL ,
+ Tiuj Hub QiaL u∗j R + h.c.
+ ab Tiν Hub LiaL ν∗R − T λ ν∗R Hda Hub
+ 31 T κ ν∗R ν∗R ν∗R + h.c.
+
+
+
1
+ 2
m2
QL i j QiaL∗ Qaj L +
m2u R i j ui∗R u j R
m2
dR i j di∗R d j R +
m2L L i j LiaL∗ Laj L
m2
Hd L L i Hda∗ LiaL + mν2R ν∗R νR + me2R i j ei∗R e j R
+ m2Hd Hda ∗ H a u
d + m2Hu Hua ∗ H a
M3 g g + M2 W W + M1 B0 B0 + h.c. .
(2)
In the first four lines the fields denote the scalar component of
the corresponding superfields. In the last line the fields denote
the fermionic superpartners of the gauge bosons. The scalar
trilinear parameters T e,ν,d,u,λ,κ correspond to the trilinear
couplings in the superpotential. The soft mass parameters
m2QL ,u R,dR,L L ,eR are hermitian 3×3 matrices in family space.
m2Hd ,Hu,νR are the soft masses of the doublet Higgs fields and
the right-handed sneutrino, and m2Hd L L is a 3-dimensional
vector in family space allowed by gauge symmetries since
the left-handed lepton fields and the down-type Higgs field
share the same quantum numbers. In the last row the
parameters M3,2,1 define Majorana masses for the gluino, wino and
bino, where the summation over the gauge-group indices in
the adjoint representation is undisplayed. While all the soft
parameters except m2Hd , m2Hu and mν2R can in general be
complex, they are assumed to be real in the following to avoid
CP-violation. Additionally, we will neglect flavor mixing
at tree-level in the squark and the quark sector, so the soft
masses will be diagonal and we write m2Qi L , mu2i R and m2di R ,
as well as for the soft trilinears Tiu = Aiu Yiu , Tid = Aid Yid ,
where the summation convention on repeated indices is not
implied, and the quark Yukawas Yiui = Yiu and Yidi = Yid
are diagonal. For the sleptons we define T e
i j = Aiej Yiej and
T ν
i = Aiν Yiν , again without summation over repeated indices.
Some care has to be taken with the parameters (m2L L )i j
contributing to the tree-level neutral scalar potential, because
these parameters cannot be set flavor-diagonal a priori. The
reason is that during the renormalization procedure (see
Sect. 3.2) the non-diagonal elements receive a counterterm.
Of course, the tree-level value of the non-diagonal elements
can and should be set to zero to avoid too large flavor
mixing. This assures that the contributions generated by virtual
corrections will always be small.
Similarly to the off-diagonal elements of the squared
sfermion mass matrices, the parameters (m2Hd L L )i are
usually not included in the tree-level Lagrangian of the μνSSM.
In the latter case because they contribute to the minimization
which is valid assuming CP-conservation, as we will do
throughout this paper.
2.1 The μνSSM Higgs potential
The neutral scalar potential VH of the μνSSM with one
generation of right-handed neutrinos is given at tree-level with
all parameters chosen to be real by the soft terms and the F
and D-term contributions of the superpotential. We find
V (0) = Vsoft + VF + VD,
with
Vsoft =
Tiν Hu0 νi L ν∗R − T λ ν∗R Hd0 Hu0
+ 31 T κ ν∗R ν∗R ν∗R + h.c.
+
m2L L i j νi∗L ν j L +
m2
Hd L L i Hd0∗νi L + mν2R ν∗R νR
+ m2Hd Hd0∗ H 0
d + m2Hu Hu0∗ Hu0,
VF = λ2 Hd0 Hd0∗ Hu0 Hu0∗ + λ2ν˜ ∗R ν˜ R Hd0 Hd0∗
1 1 1
+ 2 λvd v2R Yiν − √2 vR vu Tiν − 2 κv2R vu Yiν
+ 2 λvd vu2Yiν − 21 v2R Yiν v j L Y jν − 21 vu2Yiν v j L Y jν .
1
(9)
The tadpoles vanish in the true vacuum of the model.
During the renormalization procedure they will be treated as OS
parameters, i.e., finite corrections will be canceled by their
corresponding counterterms. This guarantees that the
vacuum is stable w.r.t. quantum corrections.
The bilinear terms
+ λ2ν˜ ∗R ν˜ R Hu0 Hu0∗ + κ2 ν˜ ∗R 2 (ν˜ R )2
− κλ ν˜ ∗R 2 Hd0∗ Hu0∗ − Yiν κν˜i L (ν˜ R )2 H 0
u
+ Yiν λν˜i L Hd0∗ Hu0∗ Hu0 + Yiν λν˜i∗L ν˜ R ν˜ ∗R Hd0 + h.c.
+ Yiν Yiν ν˜ ∗R ν˜ R Hu0 Hu0∗ + Yiν Y jν ν˜i L ν˜ ∗jL ν˜ ∗R ν˜ R
+ Yiν Y jν ν˜i ν˜ ∗j Hu0 Hu0∗,
1
VD = 8
2 2
g1 + g2
νi L νi∗L + Hd0 Hd0∗ − Hu0 Hu0∗ 2 . (10)
Using the decomposition from Eqs. (3)–(6) the linear and
bilinear terms in the fields define the tadpoles Tϕ and the
scalar CP-even and CP-odd neutral mass matrices m2ϕ and
m2σ after electroweak symmetry breaking,
VH = · · · − Tϕi ϕi + 21 ϕT m2ϕ ϕ + 21 σ T m2σ σ + · · · ,
(11)
where we collectively denote with ϕT = (HdR, HuR, νRR, νiRL )
and σ T = (HdI , HuI , νIR , νiIL ) the CP-even and CP-odd
scalar fields. The linear terms are only allowed for CP-even
fields and given by:
THdR = −m2Hd vd −
m2
Hd L L i vi L
g1 + g22 vd vd + vi L vi L − vu
2 2 2
1
− 8
1 2 2
− 2 λ vR + vu
1 1
+ √ T λvR vu + 2 κλv2R vu ,
2
λvd − vi L Yiν
1
THuR = −m2Hu vu + 8
g1 + g22 vu vd + vi L vi L − vu
2 2 2
− 21 λ2 vd2 + v2R + √12 T λvd vR
1 1
+ λvd vu vi L Yiν + 2 κλvd v2R − 2 κv2R vi L Yiν
1 1
− 2 vu vi L Yiν 2 − √ vR vi L Tiν
2
− 21 v2R vu Yiν Yiν ,
1 2 3
TνRR = −mν2R vR − √ T κ v2R − κ vR
2
+ √1 T λvd vu − 21 λ2vR vd2 + vu2
2
+ λvd vR vi L Yiν + κλvd vR vu
1
− κvR vu vi L Yiν − 2 vR vi L Yiν 2
1 1
− √ vu vi L Tiν − 2 vR vu2Yiν Yiν ,
2
TνiRL = −
m2L L i j v j L −
m2
Hd L L i vd
1
− 8
g1 + g22 vi L vd + v j L v j L − vu
2 2 2
(12)
(13)
(14)
m2
HdRνRR
m2
HuRνRR
m2
νRRνRR
m2
νiRL νRR
m2
HdRν RjL ⎞
m2 ⎟
HuRν RjL ⎟⎟ , (16)
m2
νRRν RjL ⎟⎟
m2 ⎠
νiRL ν RjL
m2
HdIνIR
m2
HuIνIR
mν2IR νIR
mν2iIL νIR
m2
HdI ν IjL ⎞
m2
mν2IR ν IjL ⎟⎟
⎠
mν2iIL ν IjL
HuI ν IjL ⎟⎟⎟ ,
are 6×6 matrices in family space whose rather lengthy entries
are given in the Appendices A.1 and A.2. We transform to
the mass eigenstate basis of the CP-even scalars through a
unitary transformation defined by the matrix U H , that
diagonalizes the mass matrix m2 ,
ϕ
(15)
(17)
(18)
(19)
(20)
⎛ m2
HdR HdR
⎜ m2
m2ϕ = ⎜⎜⎜⎜ mHνRRuRHHdRdR
⎝
m2
νiRL HdR
and
⎛ m2
HdI HdI
⎜ m2
m2σ = ⎜⎜⎜⎜ mHν2IRuIHHdIdI
⎝
mν2iIL HdI
m2
HdR HuR
m2
HuR HuR
mνRR HuR
m2
νiRL HuR
m2
HdI HuI
m2
HuI HuI
mν2IR HuI
mν2iIL HuI
U H m2ϕ U H
with
T
ϕ = U H h,
T
= m2h ,
where the hi are the CP-even scalar fields in the mass
eigenstate basis. Without CP-violation in the scalar sector the
matrix U H is real. Similarly, for the CP-odd scalar we define
the rotation matrix U A, that diagonalizes the mass matrix m2 ,
σ
U Am2σ U A
T
= m2A, with σ = U AT A.
Because of the smallness of the neutrino Yukawa couplings
Yiν , which also implies that the left-handed sneutrino vevs
vi L have to be small, so that the tadpole coefficients vanish
at tree-level [
14
], the mixing of the left-handed sneutrinos
with the doublet fields and the singlet will be small.
It is a well known fact that the quantum corrections to
the Higgs potential are highly significant in
supersymmetric models, see e.g. Refs. [
64,76,77
] for reviews. As in the
NMSSM [
7
], the upper bound on the lowest Higgs mass
squared at tree-level is relaxed through additional
contributions from the singlet [
14
];
The mass eigenstates di1 and di2 are obtained by the unitary
transformation
(21)
Nevertheless, quantum corrections were still shown to
contribute significantly especially in the prediction of the
SMlike Higgs boson mass [
26,68,70,74,78–81
]. In this paper we
will investigate how important the unique loop corrections of
the μνSSM beyond the NMSSM are in realistic scenarios.
Before that we briefly describe the other relevant sectors of
the μνSSM.
2.2 Squark sector
The numerically most important one-loop corrections to the
scalar potential are expected from the stop/top-sector,
analogous to the (N)MSSM [
79–84
] due to the huge Yukawa
coupling of the (scalar) top. The tree-level mass matrices
of the squarks differ slightly from the ones in the MSSM.
Neglecting flavor mixing in the squark sector, one finds for
the up-type squark mass matrix M ui of flavor i ,
M1u1i = m2QiL + 214 (3g22 − g12)(vd2 + v j L v j L − vu2) + 21 vu2Yiu 2
1 √
M1u2i = 2 ( 2 Aiu vu + vRY u v j L Y jν − λvd vR)
M2u2i = m2ui R + 61 g12(vd2 + v j L v j L − vu ) + 21 vu2Yiu 2.
2
It should be noted that in the non-diagonal element explicitly
appear the neutrino Yukawa couplings. This term arises in
the F-term contributions of the squark potential through the
quartic coupling of up-type quarks and one left-handed and
the right-handed sneutrino after EWSB. The mass eigenstates
ui1 and ui2 are obtained by the unitary transformation
ui1
ui2
= Uiu ui L
ui R
, UiuUiu † = 1.
Similarly, for the down-type squarks it is
M1d1i = m2Qi L − 214 (3g22 + g12)(vd2 + v j L v j L − vu2)
1 v2Y d 2
− 2 d i
M1d2i = 21 (√2 Aid vd − λvd vR )
(22)
(23)
(24)
(26)
(27)
M2d2i = m2 1 2
di R − 12 g12(vd2 + v j L v j L − vu ) + 21 vd2Yid 2. (28)
Lχ± = −(χ −)T meχ + + h.c..
(29)
(30)
(31)
(32)
(33)
(34)
(25)
which include the charged Goldstone boson H1+ = G±.
0
2.3 Charged scalar sector
Since R-parity, lepton number and lepton-flavor are broken,
the six charged left- and right-handed sleptons mix with each
other and with the two charged scalars from the Higgs
doublets. In the basis C T = (Hd−∗, Hu+, ei∗L , e∗j R ) we find the
following mass terms in the Lagrangian:
LC = −C ∗T m2H+ C,
where m2H+ assuming CP conservation is a symmetric matrix
of dimension 8,
⎛
m2
Hd− Hd−∗
⎜ m2
m2H+ = ⎜⎜⎜⎜ mHe2ui+L∗HHd−d−∗∗
⎝ m2
ei R Hd−∗
m2
Hd− Hu+
m2
Hu+∗ Hu+
m2
ei L Hu+
m2
ei R Hu+
m2
m2
m2
m2
Hd−e∗j L
Hu+∗e∗j L
ei L e∗j L
ei Re∗j L
m2
Hd−e∗j R ⎞
m2
Hu+∗e∗j R ⎟⎟⎟ .
m2
ei L e∗j R ⎟⎟
m2 ⎠
ei Re∗j R
The entries are given in Appendix A.3. The mass matrix is
diagonalized by an orthogonal matrix U +:
U +m2H+ U +
T
=
m2H+
diag
where the diagonal elements of (m2H+ )diag are the squared
masses of the mass eigenstates
H + = U + C,
2.4 Charged fermion sector
The charged leptons mix with the charged gauginos and
the charged higgsinos. Following the notation of Ref. [
17
]
we write the relevant part of the Lagrangian in terms of
two-component spinors (χ −)T = ((ei L )c∗ , W −, Hd−) and
(χ +)T = ((e j R )c, W +, Hu+):
⎛
⎛
The 5 × 5 mixing matrix me is defined by
me = ⎜⎜⎜⎜⎜
⎜⎜⎜
⎝
−gv2√RvuY223ν ⎟⎟⎟⎟⎟⎟ .
where mediag contains the masses of the charged fermions in
the mass eigenstate base
The smallness of the left-handed sneutrino vevs in
comparison to the doublet ones assures the decoupling of the three
leptons from the Higgsino and the wino.
2.5 Neutral fermion sector
The three left-handed neutrinos and the right-handed
neutrino mix with the neutral Higgsinos and gauginos. Again,
following Ref. [
17
] we write the relevant part of the
Lagrangian in terms of two-component spinors (χ 0)T =
((νi L )c∗ , B0, W 0, Hd0, Hu0, ν∗R ) as
Lχ0 = − 21 (χ 0)T mν χ 0 + h.c.,
where mν is the 8 × 8 symmetric mass matrix. The neutral
fermion mass matrix is determined by
with
χ 0 = U V † λ0,
(41)
(42)
Because of the Majorana nature of the neutral fermions
we can diagonalize mν with the help of just a single – but
complex – unitary matrix U V ,
where λ0 are the two-component spinors in the mass basis.
The eigenvalues of the diagonalized mass matrix mνdiag are the
masses of the neutral fermions in the mass eigenstate basis.
It turns out that the matrix mν is of rank six, so it can only
generate a single neutrino mass at tree-level.3 The remaining
two light neutrino masses can be generated by loop-effects.
3 Renormalization of the Higgs potential at one-loop
The first step in renormalizing the neutral scalar potential is
to choose the set of free parameters. These free parameters
will receive a counter term fixed by consistent
renormalization conditions to cancel all ultraviolet divergences that are
produced by higher-order corrections.
At tree-level the relevant part of the Higgs potential VH
is given by the tadpole coefficients Eqs. (12)–(15) and the
CP-even and CP-odd mass matrix elements in Eqs. (16) and
(17). The following parameters appear in the Higgs potential:
– Scalar soft masses: m2Hd , m2Hu , mν2R , m2
L L i j
, m2
Hd L L i
(12 parameters)
– Vacuum expectation values: vd , vu , vR , vi L
eters)
– Gauge couplings: g1, g2 (2 parameters)
– Superpotential parameters: λ, κ, Yiν (5 parameters)
– Soft trilinear couplings: T λ, T κ , Tiν (5 parameters)
(6
paramv√RY1ν
2
v√RY2ν
2
v√RY3ν
2
g1vu
2
− g22vu
λvR
− √2
0
−λvd +vi L Yiν
√2
v√uY21ν ⎞
vv√√uu0YY2232νν ⎟⎟⎟⎟⎟⎟⎟⎟ .
−0λ√v2u ⎟⎟⎟⎟
−λvd +vkL Ykν ⎟⎟
√2√κ2vR ⎠⎟
3 Including three generations of right-handed neutrinos, three light
treelevel neutrino masses are generated.
(40)
Gauge cpl.
Superpot.
Soft trilinears
The complexity of the μνSSM Higgs scalar sector becomes
evident when we compare the numbers of free parameters
(30) with the one in the real MSSM (7) [
55
] and the NMSSM
(12) [
26
]. While the number of free parameters is fixed, we
are free to replace some of the parameters by physical
parameters. We chose to make the following replacements:
The soft masses m2Hd , m2Hu , mν2R , and the diagonal
elements of the matrix m2L L will be replaced by the tadpole
coefficients. The substitution is defined by the tadpole Eqs. (12)–
(15) solved for the soft mass parameters just mentioned.
This will give us the possibility to define the
renormalization scheme in a way that the true vacuum is not spoiled by
the higher-order corrections. The Higgs doublet vevs vd and
vu will be replaced by the MSSM-like parameters tan β and
v according to
vu
tan β = vd
and v2 = vd2 + vu2 + vi L vi L .
(43)
Note that the definition of v2 differs from the one in the
MSSM by the term vi L vi L . This allows to maintain the
relations between v2 and the gauge boson masses as they are in
the MSSM. Numerically, the difference in the definition of
v2 is negligible, since the vi L are of the order of 10−4 GeV
in realistic scenarios. Analytically, however, maintaining the
functional form of tan β as it is in the (N)MSSM is
convenient to facilitate the comparison of the quantum corrections
in the μνSSM and the NMSSM. In particular, we can still
express the one-loop counterterm of tan β without having to
include the counterterms for the left-handed sneutrino vevs.
For the vev of the right-handed sneutrino we chose to make
the same substitution as was done in previous calculations in
the NMSSM [
26
]
where we make use of the fact that when the sneutrino obtains
the vev, the μ-term of the MSSM is dynamically generated.
μ = v√R λ ,
2
The gauge couplings g1 and g2 will be replaced by the gauge
boson masses MW and MZ via the definitions
1 g2v2 and
M W2 = 4 2
1
MZ2 = 4
This is reasonable because the gauge boson masses are well
measured physical observables, so we can define them as
OS parameters. Interestingly, the mass counterterm for M W2
drops out at one-loop, but it will contribute in the definition
of the counterterm for v2, so it is not a redundant
parameter. For the soft trilinear couplings we chose to adopt the
redefinitions
T λ = Aλλ, T κ = Aκ κ, Tiν = Aiν Yiν .
The reparametrization from the initial to the physical set of
independent parameters is summarized in Table 1.
In the following we will regard the entries of the
neutral scalar mass matrix as functions of the final set of
parameters,
m2ϕ = m2ϕ
MZ2 , v2, tan β, λ, . . . ,
m2σ = m2σ
MZ2 , v2, tan β, λ, . . . ,
and we define their renormalization as
m2ϕ → m2ϕ + δm2ϕ ,
m2σ → m2σ + δm2σ .
(45)
(46)
(47)
(48)
(49)
(50)
(44)
The mass counterterms δm2ϕ and δm2σ enter the
renormalized one-loop scalar self-energies. They have to be expressed
as a linear combination of the counterterms of the
independent parameters. We define their one-loop renormalization
as
THdR → THdR + δTHdR ,
THuR → THuR + δTHuR ,
TνR → TνRR + δTνR ,
R R
TνiRL → TνiRL + δTνiRL ,
m2L L i = j → m2L L i = j + δm2L L i = j
,
m2Hd L L i → m2Hd L L i + δm2Hd L L i ,
tan β → tan β + δ tan β,
v2 → v2 + δv2,
μ → μ + δμ,
vi L → vi L + δvi2L ,
2 2
M W2 → M W2 + δ M W2 ,
MZ2 → MZ2 + δ MZ2 ,
λ → λ + δλ,
κ → κ + δκ,
Y ν
i → Yiν + δYiν ,
Aλ → Aλ + δ Aλ,
Aκ → Aκ + δ Aκ ,
Aiν → Aiν + δ Aiν .
(51)
(54)
(57)
(58)
Since the μνSSM is a renormalizable theory, the
divergent parts of the counterterms are fixed to cancel the UV
divergences. The finite pieces, and thus the meaning of the
parameters have to be fixed by renormalization conditions.
We will adopt a mixed renormalization scheme, where
tadpoles and gauge boson masses are fixed OS, and the other
parameters are fixed in the DR scheme. The exact
renormalization conditions will be given in Sect. 3.2. The dependence
of the mass counterterms δm2ϕ and δm2σ on the counterterms
of the free parameters is given at one-loop by
δm2ϕ =
δm2σ =
X∈Free param.
X∈Free param.
∂∂X m2ϕ δ X,
∂ m2
∂ X σ
δ X.
(52)
In our calculation the mixing matrices are defined in a way to
diagonalize the renormalized mass matrices, so they do not
have to be renormalized, because they are defined exclusively
by renormalized quantities. The expressions for the
counterterms of the scalar mass matrices in the mass eigenstate basis
are then simply
δm2h = U H δm2ϕU H T , δm2A = U Aδm2σ U AT .
(53)
It should be noted at this point that the counterterm matrices
in the mass eigenstate basis δm2h and δm2A are not diagonal,
as they would be in a purely OS renormalization procedure,
which is often used in theories with flavor mixing [
85
].
In the following chapter we will discuss the field
renormalization, which is necessary to obtain finite scalar self-energies
at arbitrary momentum.
⎛ Hd ⎞
⎜⎝⎜ HνRu ⎟⎟⎠ →
νi L
⎛ Hd ⎞
√Z ⎜⎝⎜ HνRu ⎟⎠⎟ =
νi L
1
1 + 2 δ Z ⎜⎜⎝ νR ⎟⎠⎟ ,
⎛ Hd ⎞
Hu
νi L
where √Z and δ Z are 6 × 6 dimensional matrices and the
equal sign is valid at one-loop. It should be emphasized that
in contrast to the MSSM and the NMSSM these matrices
cannot be made diagonal even in the interaction basis. The
reason is that the μνSSM explicitly breaks lepton number
and lepton flavor, so the fields Hd and νi L share exactly the
same quantum numbers and kinetic mixing terms are already
generated at one-loop order.
For the CP-even and CP-odd neutral scalar fields the
definition in Eq. (54) implies the following field renormalization
in the mass eigenstate basis:
As renormalization conditions for the field renormalization
counterterms we chose to adopt the DR scheme. We calculate
the UV-divergent part of the derivative of the scalar CP-even
self-energies in the interaction basis and define
d
δ Zi j = − d p2 ϕi ϕ j
div
.
1
= ε − γE + ln 4π ,
Here div denotes taking the divergent part only, proportional
to ,
3.1 Field renormalization We write the renormalization of the neutral scalar-component fields as
where loop integral are solved in 4 − 2ε dimensions and
γE = 0.5772 . . . is the Euler–Mascheroni constant. Since
the field renormalization constants contribute only via
divergent parts, they do not contribute to the finite result after
Fig. 1 Generic Feynman
diagrams for the tadpoles Thi
u, d, λ±
u, d, H±
h
h
h
h
λ0
h, A
h
h
u±, uZ
W ±, Z
canceling divergences in the self-energies. As regularization
scheme we chose dimensional reduction [
86,87
], which was
shown to be SUSY conserving at one-loop [88]. In contrast
to the OS renormalization scheme our field renormalization
matrices are hermitian. This holds also true for the field
renormalization in the mass eigenstate basis, because as already
mentioned the rotations in Eqs. (18) and (20) diagonalize the
renormalized tree-level scalar mass matrices, so Eq. (56) do
not introduce non-hermitian parts into the field
renormalization, that would have to be canceled by a renormalization of
the mixing matrices U H and U A themselves.
In Appendix B.1 we list our field renormalization
counterterms δ Zi j in terms of the divergent quantity . Note that
the field counterterms mixing the down-type Higgs and the
left-handed sleptons are proportional to the neutrino Yukawa
couplings Yiν , while the counterterms mixing different flavors
of left-handed sneutrinos contain terms proportional to
nondiagonal lepton Yukawa couplings Y e and terms proportional
to Yiν Y jν . This is why their numerical impact is negligible,
but they are needed for a consistent renormalization of the
scalar self-energies.
3.2 Renormalization conditions for free parameters
In this section we describe our choice for the renormalization
conditions, where we stick to the one-loop level everywhere.
We start with the OS conditions for the gauge boson mass
parameters and the tadpole coefficients followed by our
definitions for the DR renormalized parameters.
The SM gauge boson masses are renormalized OS
requiring
T
Re ˆ Z Z
MZ2
T
= 0 and Re ˆ W W
M W2
= 0, (59)
where ˆ T stands for the transverse part of the renormalized
gauge boson self-energy. For their mass counterterms these
conditions yield
δ MZ2 = Re
δ M W2 = Re
T
Z Z
T
W W
MZ2
M W2
and
where Tϕ(i1) are the one-loop contributions to the linear terms
of the scalar potential, stemming from tadpole diagrams
shown in Fig. 1. The tadpole diagrams are calculated in the
mass eigenstate basis h. The one-loop tadpole contributions
in the interaction basis ϕ are then obtained by the rotation
Tϕ(1) = U H T Th(1).
δTϕi = −Tϕ(i1).
Accordingly we find for the one-loop tadpole counterterms
For practical purposes we decided to renormalize all
remaining parameters in the DR scheme (reflecting the fact
that there are no physical observables that could be directly
related to them). The counterterms of each parameter were
obtained by calculating the divergent parts of one-loop
corrections to different scalar and fermionic two- and three-point
functions. We state the determination of the counterterms
in the (possible) order in which they can be successively
derived. We start with the counterterms that were obtained
by renormalizing certain neutral fermion self-energies.
Renormalization of μ: The μ parameter appears isolated in
the Majorana-type mass matrix of the neutral fermions
λvR
(mν )67 = − √
= −μ,
2
(60)
(61)
(62)
(63)
(64)
χ0
χ0
χ0
u, d
u, d
ϕ, σ
χ0
χ0
χ0
χ0
χ0
χ0
χ0
χ0
W ±
W ±
ϕ, σ
χ0
(67)
which is the element mixing the down-type and the up-type
Higgsinos Hd and Hu . The entries (mν )i j get one-loop
corrections via the neutral fermion self-energies
χ˜i0χ˜ 0j , that for
χ˜i0χ˜ 0j ( p2) = /p
F0 0 ( p2) +
χ˜i χ˜ j
S0 0 ( p2).
χ˜i χ˜ j
The part F0 0 is renormalized through field renormalization
χ˜i χ˜ j
and the part
S0 0 is renormalized by both the field
renorχ˜i χ˜ j
malization and a mass counter term. Since we are interested
in the mass renormalization we focus on S0 0 and write for
χ˜i χ˜ j
the renormalized self-energy at zero momentum
S
ˆ χ˜i0χ˜ 0j (0) =
S 1
χ˜i0χ˜ 0j (0) − 2
δ Zkχi (mν )k j
+ (mν )ik δ Zkχj − δ (mν )i j .
The field renormalization constants can be obtained by
calculating the divergent part of F0 0 :
χ˜i χ˜ j
4 Left-handed components and right-handed components are the same
for Majorana fields.
δ (mν )67 = −δμ,
where we make use of the fact that there are no divergences
proportional to p2 in our case. The divergent parts of the
self-energies of the neutral fermions are calculated
diagrammatically in the interaction basis, where diagrams with mass
insertions have to be included. In Fig. 2 we show the generic
diagrams potentially contributing to the divergent part of the
self-energies. Diagrams with a scalar mass insertion or more
than one fermionic mass insertion are power-counting finite,
so we do not depict them. The diagram shown in Fig. 2 with
a mass insertion on the chargino propagator can be
divergent depending on the expressions for the couplings of the
charginos.
We checked that our results for the field renormalization
counterterms for the neutral fermions are consistent with the
one-loop anomalous dimensions γi(j1) of the corresponding
superfields, i.e.,
To extract δμ we now just have to identify
χ0
χ0
χ0
χ0
χ0
χ0
χ0
(69)
and calculate the divergent part of HSd Hu , which again is not
momentum dependent. δμ is then given by
1
δμ = 2 μ
1
− δ Z6χ6 + δ Z7χ7 + λ
+δ Z2χ6Y2ν + δ Z3χ6Y3ν
−
δ Z1χ6Y1ν
S
Hd Hu
div
,
where we made us of the fact that the matrix δ Ziχj is real and
symmetric and that components mixing left-handed
neutrinos and the down-type Higgsino are the only non-diagonal
elements contributing here.
Explicit formulas for the counterterms of the
parameters renormalized in the DR scheme are listed in the
Appendix B.2. For the DR counterterms we checked that
in the limit Yiν → 0 our results coincide with the one in the
NMSSM [
7
].
Renormalization of κ : The parameters κ appears isolated
at tree-level in the three-point vertex that couples the
righthanded neutrino to the right-handed sneutrino,
(0)
νRνRνR = −
√2κ.
ϕ, σ
νR
The divergences induced to this coupling at one-loop have to
be absorbed by the field renormalization of the right-handed
neutrino and sneutrino and the counterterm for κ, which is
the only parameter in the tree-level expression. We find
1
δκ = √2
νR νR νR (1) div
1
− 2 κ δ Z33 + 2δ Z8χ8 ,
where νR νR νR (1)|div is the divergent part of the
corresponding one-loop three-point function, and the terms containing
the field renormalization is trivial, because there is only one
singlet-like superfield so that no non-diagonal field
renormalization constants appear. The divergent one-loop
contributions to the vertex are calculated diagrammatically in the
interaction basis. The only contributing generic diagrams are
shown in Fig. 3.
All other topologies, including diagrams with one or more
mass insertion, are finite, and there are no diagrams with
Fig. 3 Potentially divergent
one-particle irreducible
diagrams contributing to the
three-point vertex between two
right-handed neutrinos and one
right-handed sneutrino
νR
νR
νR
(73)
(74)
(75)
(76)
(77)
χ0
(70)
(71)
(72)
(mν )88 =
2κμ
λ
,
where we calculated the divergent part of the right-handed
neutrino self-energy νSRνR |div diagrammatically in the
interaction basis using the diagrams already shown in Fig. 2.
Renormalization of Aκ : The counterterm for the parameter
Aκ can be extracted from the one-loop corrections to the
scalar three-point vertex of right-handed sneutrinos when δκ
is known and using the one-loop relation
gauge bosons instead of scalars in the loop, because there
are three gauge-singlet fields on the outer legs. It turns out
that the sum over the diagrams shown in Fig. 3 is also finite,
so that ν(1R)νRνR |div vanishes.
Renormalization of λ: Having calculated δμ and δκ we can
extract the counterterm for λ in the neutral fermion sector. λ
appears in the mass matrix element
δμ
μ − λ
δλ div
1
= 2 δ Z33
div
,
which was found in the NMSSM [
89
] and confirmed for this
work also in the μνSSM. For the trilinear singlet vertex we
have at tree-level
(0)
νRνRνR = −
√2κ
Aκ +
6κμ
λ
The tree-level vertex does not depend on the momentum, so
the one-loop counterterm for Aκ can be calculated through
δ Aκ = √
so we will make use of the fact that we already know the
counterterms for λ, κ and μ.
The final expression defining δ Aλ will also contain the
tree-level expressions for the couplings where the down-type
Renormalization of v2: The SM-like vev is renormalized via
the renormalization of the electromagnetic coupling in the
Thompson limit, which can be done when the counterterms
Here ν(1R)νRνR |div is the divergent part of the one-loop
corrections to the three-point vertex, which was calculated
diagrammatically in the interaction basis. The number of contributing
diagrams is rather high, so for simplicity we just show the
topologies of the diagrams contributing, that potentially lead
to divergences, in Fig. 4. In the case of the vertex νRνRνR
we can neglect the diagrams with gauge bosons, because the
right-handed sneutrinos are gauge singlets.
Renormalization of Aλ: The counterterm for the parameter
Aλ is like in the previous case extracted from the one-loop
corrections to a scalar three-point function. Here we consider
Hd HuνR , the coupling between the two doublet-type Higgses
and the right-handed sneutrino. At tree-level it is
(0) Aλλ
Hd HuνR = √2 +
√2κμ,
Higgs is replaced by one of the left-handed sneutrinos. They
are induced by the non-diagonal field renormalization of Hd
and νi L and enter the renormalization of Hd HuνR at
oneloop. We find
δ Aλ = − λ
√
2 (1)
Hd HuνR
div
1
− √2λ
(0)
δ Z11 Hd HuνR
(0) (0)
+ δ Z14 ν1L HuνR + δ Z15 ν2L HuνR
(0) (0) (0)
+ δ Z16 ν3L HuνR + δ Z22 Hd HuνR + δ Z33 Hd HuνR
2κ 2μ
− Aλλ δλ − λ δμ − λ
δκ,
(78)
with
(0)
νi L HuνR =
−Yiν Aν 2κμ
i + λ
√2
for the gauge boson masses are fixed. We follow here the
approach of Ref. [
26
] used in the NMSSM to be able to
compare the results in both models as best as possible.
The renormalization of the electromagnetic coupling is
defined by
e → e(1 + δ Ze),
and the counterterm δ Ze can be calculated via
1
δ Ze|div = 2
∂ γTγ
∂ p2 (0)
sw
+ cw MZ2 γTZ (0)
div
,
where
and
γTγ (0) is the transverse part of the photon self-energy
γTZ is the transverse part of the mixed photon-Z boson
self-energy. sw and cw are defined as sw =
cw = MW /MZ . v2 and e are related by
1 − cw2 with
v2 =
2sw2 M W2 ,
e2
so the counterterm δv2 can be obtained through
(88)
(89)
(90)
(91)
(92)
(93)
so it is necessary to have the counterterm of the gauge
coupling g1, whose renormalization we define as g1 → g1 +δg1.
We then can obtain δg1 from δ M W2 , δ MZ2 and δv2 through
the definitions of the gauge boson masses in Eq. (45),
Renormalizing the self-energies Bνi L using Eq. (66) we find
the following expression for the δvi2L :
div
Renormalization of Yiν : The counterterm for the neutrino
Yukawas Yiν can be extracted in the neutral fermion sector as
well. We decide to use the renormalization of the tree-level
masses
(mν )i7 =
μYiν ,
λ
that mix the left-handed neutrinos and the up-type Higgsino.
Since we already found δλ and δμ we can get δYiν from the
divergent part of the one-loop self-energies S ,
νi L Hu
1
δYiν = 2
δ Z1χ6λ − δ Z7χ7Yiν − δ Ziχj Y jν
−
div
.
Renormalization of tan β: We adopted the usual definition
for tan β as in the MSSM (see Eq. (43)). If we define the
renormalization for the vevs of the doublet fields as
vd2 → vd2 + δvd2, vu2 → vu2 + δvu2,
the counterterm for tan β can be written at one-loop as a linear
combination of the counterterms for the vevs of the doublet
Higgses,
(81)
(82)
(83)
(84)
(86)
(87)
1
δ tan β = 2 tan β
δvu2 δvd2
vu2 − vd2
.
Note that our renormalization of vu2 and vd2 in Eq. (92)
includes the contributions from the field renormalization
constants inside the counterterms δvu2 and δvd2. This approach is
equivalent as defining
vd →
Z11 vd + δvˆd , vu →
Z22 vu + δvˆu ,
(94)
δv2 =
where
4sw2 M W2
e2
δsw2
s2 +
w
δ M W2
M W2
− 2δ Ze
div
,
(mν )4i = −
g1vi L ,
2
sw2 → sw2 +δsw2, with δsw2 = −cw2
δ M W2
M W2
−
δ MZ2
MZ2
. (85)
Here we take only the divergent parts of the counterterms
δ MZ2 , δ M W2 and δ Ze, so that δv2 is renormalized in the DR
scheme. This implies that the counterterm δ Ze is not a free
parameter, even if we calculated it as if it would be to
determine δv2. Instead δ Ze is a dependent parameter defined by
δv2 in the DR scheme and δ MZ2 and δ M W2 in the OS scheme
through Eqs. (84) and (85),
1
δ Ze = 2sw2
cw2 δMMZ2Z2 + sw2 − cw2
δ M W2
M W2
e2
− 4M W2 δv2 .
Renormalization of vi2L: The counterterms for the three vevs
of the left-handed sneutrinos vi L can be extracted from the
divergent part of the one-loop self-energies Bνi L between
the bino and the corresponding left-handed neutrino. The
tree-level mass matrix entries we renormalize are defined by
and writing the counterterm of tan β as
1
δ tan β = 2 tan β (δ Z22 − δ Z11) + tan β
δvˆu
vu
This notation was convenient in the MSSM and the NMSSM,
because the second bracket in Eq. (95) is finite at
oneloop [
26,68,90,91
] and can be set to zero in the DR scheme,
so that δ tan β can be expressed exclusively by the field
renormalization constants. In contrast, in the μνSSM we find
δvˆu
vu
δvˆd
− vd
div
λvi L Yiν .
= − 32π 2vd
There are several possibilities to extract the counterterms δvd2
and δvu2. A convenient choice is to extract δvd2 from the
renormalization of the entry of the neutral fermion mass matrix
mixing the up-type Higgsino and the right-handed neutrino,
(mν )78 =
−λvd + vi L Yiν ,
√2
because in this case no non-diagonal field renormalization
counterterms are needed. Calculating the divergent part of
S and using the counterterms previously calculated we
HuvR
can extract δvd2 via the expression
δvd2 = − 2√λ2vd HSuvR div + vλd δ Z7χ7 + δ Z8χ8
× −vd λ + vi L Yiν
vd Y ν
+ λ i
δv2L
vL
− 2vd2 δλλ
2vd
+ λ
vi L δYiν .
i
Since all counterterms appearing in Eq. (98) are renormalized
in the DR scheme also δvd2 has no finite part. There are now
two ways to determine δvu2. Firstly, we could similarly to
δvd2 extract the counterterm δvu2 by renormalizing the up-type
Higgsino self-energy S . Alternatively, we can deduce
Hu Hu
δvu2 from the definition of v2 in Eq. (43) and simply write
δvu2 = δv2 − δvd2 − δv12L − δv22L − δv32L .
We verified that both options yield the same result, which
constitutes a consistency test for the counterterms δvi2L ,
which are unique for the μνSSM. Inserting δvd2 from Eq. (98)
and δvu2 from Eq. (99) into Eq. (93) finally gives the
counterterm for tan β. We checked that the final expression for
tan β in Eq. (172) agrees with the NMSSM result in the limit
Y ν
i → 0.
The renormalization of tan β in the DR scheme is
manifestly process-independent and has shown to give stable
numerical results in the MSSM [
92,93
] and the NMSSM
[
26,68
].
Renormalization of Aiν : The soft trilinears Aiν can be
renormalized through the calculation of the radiative corrections
to the corresponding scalar vertex in the interaction basis.
The tree-level expression for the interaction between the
uptype Higgs, one left-handed sneutrinos and the right-handed
sneutrino is given by
(0)
HuνRνi L = −
Aiν
√2 +
The renormalized one-loop corrected vertex will define the
counterterm for Aiν since the counterterms for κ, μ and λ were
already determined. We showed in Fig. 4 the topologies of
the diagrams that have to be calculated in the interaction basis
to get the divergent part of one-loop corrections (1)
HuνRνi L . As
in the case of the renormalization of Aλ the renormalization
of the scalar vertex will contain the tree-level expressions of
all the vertices with the same quantum numbers of the
external fields, because of the non-diagonal field renormalization.
Solved for δ Aiν the renormalization of the vertex leads to
√2
δ Aiν = Y ν
i
(1)
HuνRνi L
div
1
+ √2Yiν
(δ Z22 + δ Z33) (H0u)νRνi L
(0) (0)
+ δ Z1,3+i HuνR Hd + δ Z3+ j,3+i HuνRν j L
Aν
− Y νi δYiν −
i
2μ 2κ 2κμ
λ δκ − λ δμ − λYiν δYiν +
2κμ
λ2 δλ,
(101)
(102)
with
(0) λ Aλ
HuνR Hd = √2 +
√2κμ.
Renormalization of m2HdLL i: The soft scalar masses appear
in the bilinear terms of the Higgs potential. They can be
renormalized by calculating radiative corrections to scalar
self-energies. It proved to be convenient to calculate the
CPodd scalar self-energies in the mass basis, and then to rotate
the self-energies back to the interaction basis.
We find m2 at tree-level in
Hd L L i
ˆ Xi X j ( p2) =
Xi X j ( p2) + 21 p2 δ Z ji + δ Zi j
where X = (ϕ, σ ) represents either the CP-even or the
CPodd scalar fields and we made use of the fact that the field
renormalization constants δ Z and the mass matrix m2X are
real. Demanding that the renormalized self-energies ˆ Ai A j
are finite in the mass eigenstate basis we can define the
divergent parts of the mass counterterms via
δ m2A i j
div
=
Ai A j (0) div
+ m2Ai δ Z A
,
δ Z A
ji
m2A j
where the field counterterms in the mass eigenstate basis were
defined in Eq. (56) and the masses m2Ai are the eigenvalues
of the diagonal CP-odd scalar mass matrix m2A. In Fig. 5
we show the diagrams that have to be calculated to get the
quantum corrections to scalar self-energies at one-loop in the
mass eigenstate basis.
We calculated all diagrams in the ’t Hooft-Feynman gauge,
in which the Goldstone bosons A1 and H1± and the ghost
fields u± and u Z have the same masses as the corresponding
gauge bosons. Calculating the CP-odd self-energies Ai Ai
diagrammatically, we get the mass counterterms in mass
eigenstate basis through the Eq. (105). Now inverting the
rotation in Eq. (53) we can get the mass counterterms for the
CP-odd self-energies in the interaction basis via
δm2σ div = U AT δm2A div U A.
Recognizing that
δm2σ 3+i,1 = δmν2iIL HdI ,
(104)
(105)
(106)
(107)
+
−
μ2 1 2
λ + 2 λ vd + vu2 sin2 β δYiν
μ2Yiν
λ2
(110)
Renormalization of m2 : Since we neglect CP-violation
LL i j
the counterterms for the non-diagonal elements of the
hermitian matrix m2 are symmetric under the exchange of
L L i j
the indices i and j . Then we can extract the counterterms
for the non-diagonal elements in the same way as the ones
for m2 in the CP-odd scalar sector. They appear in the
Hd L L i
tree-level mass matrix in
div
− vd + vu2 Yiν Y jν cos3 β sin β δ tan β
2
+ 21 Yiν Y jν sin2 β δv12L + δv32L + δv32L .
Yiν δY jν − Y jν δYiν
− 21 Yiν Y jν sin2 β δv2
δ(m2Hd L L )i through
and that mν2iIL HdI depends on (m2Hd L L )i , we can extract
δ m2
Hd L L i =
δm2σ 3+i,1
div
+
2μYiν δμ
λ
+ λ vd + vu2 Yiν cos3 β sin β δ tan β
2
1
+ 2 λYiν sin2 β δv2
1
− 2 λ sin2 βYiν δv12L + δv22L + δv32L
FeynArts modelfile: The diagrams and their amplitudes
that had to be calculated to obtain the counterterms, as
described in this section, were generated using the
Mathematica package FeynArts [
94
] and further evaluated with the
package FormCalc [
95
]. The FeynArts model file for the
μνSSM was created with the Mathematica program SARAH
[
96
]. We modified the model file to neglect CP-violation by
choosing all relevant parameters to be real. We also neglected
flavor-mixing in the squark- and the quark-sector in this
work. The FeynArts model file can be provided by the
authors upon request. The calculation of renormalized
twoand three-point functions of the neutral scalars of the μνSSM
at one-loop accuracy is thereby fully automated. (as it is in
the MSSM [
97
]).
In Sect. 5 we will present our predictions for the Higgs
masses in the μνSSM compared to the ones of the NMSSM.
To be able to make this comparison, we had to
calculate the NMSSM-predictions in the same renormalization
scheme and using the same conventions as were used in
the μνSSM. This is why we calculated the one-loop
selfenergies in the NMSSM with our own NMSSM-modelfile
h, A
Z
h, A
h, A
W ±
H±
h/A h/A
h/A h/A
h/A h/A
h/A h/A
h/A h/A
h/A h/A
h/A h/A
h/A h/A
h/A h/A
h/A h/A
h/A h/A
h/A h/A
H±
λ±
λ±
H±
H±
Z
h, A
u±
u±
u, d
λ0
λ0
u, d
u, d
W ±
W ±
uZ
uZ
h/A
h/A h/A
h/A
for FeynArts/FormCalc created with SARAH using the
same procedure as for the μνSSM. We verified that the results
calculated in the NMSSM with our modelfile are equal to the
results calculated with the modelfile presented in Ref. [
98
],
which was a good check that the generation of the modelfiles
for the NMSSM and the μνSSM was correct.
4 Loop corrected Higgs boson masses
In the previous section we have derived an OS/DR
renormalization scheme for the μνSSM Higgs sector. This can be
applied (via the future FeynArts model file, once the
counterterms are implemented) to any higher-order correction in
the μνSSM. As a first application, we evaluate the full
oneloop corrections to the CP-even scalar sector in the μνSSM.
Due to the still missing implementation of counterterms in the
FeynArts model file, the calculation of the renormalized
scalar self-energies is done in two steps. Firstly, the
unrenormalized self-energies are calculated using FeynArts and
FormCalc, and subsequently the self-energies are
renormalized subtracting (by hand) the field renormalization and
mass counterterms, as will be described in the next
section.
W ±
u, u
d, d
W ±
H±
Z
Z
h/A
h/A
h/A
h/A
4.1 Evaluation at one-loop
Here we describe the final form of the renormalized CP-even
scalar self-energies ˆ hh and how the loop corrected physical
masses of the Higgs boson masses are evaluated.
The one-loop renormalized self-energies in the mass
eigenstate basis are given by
ˆ h(1ih) j ( p2) =
h(1ih) j ( p2) + δ ZiHj
with the field renormalization constants δ Z H and the mass
counter terms δm2h in the mass eigenstate basis defined by
the rotations in Eqs. (56) and (53). hi h j is the
unrenormalized self-energy obtained by calculating the diagrams
shown in Fig. 5 with the CP-even states h on the
external legs. The self-energies were calculated in the Feynman
gauge, so that gauge-fixing terms do not yield counterterm
contributions in the Higgs sector at one-loop. The loop
integrals were regularized using dimensional reduction [
86,87
]
and numerically evaluated for arbitrary real momentum using
LoopTools [95]. The contributions from complex values of
p2 were approximated using a Taylor expansion with respect
to the imaginary part of p2 up to first order.
In Eq. (111) we already made use of the fact that δ Z H is
real and symmetric in our renormalization scheme. The mass
counterterms are defined as functions of the counterterms
of the free parameters following Eqs. (52) and (53). They
contain finite contributions from the tadpole counterterms
and from the counterterm for the gauge boson mass MZ2 . The
matrix δm2h is real and symmetric.
The renormalized self-energies enter the inverse
propagator matrix
ˆ h = i p2 1− m2h − ˆ h p2
, with
ˆ h i j = ˆ hi h j .
(112)
The loop-corrected scalar masses squared are the zeroes of
the determinant of the inverse propagator matrix. The
determination of corrected masses has to be done numerically
when we want to account for the momentum-dependence of
the renormalized self-energies. This is done by an iterative
method that has to be carried out for each of the six squared
loop-corrected masses [
99
].
4.2 Inclusion of higher orders
In Eq. (112) we did not include the superscript (1) in the
selfenergies. Restricting the numerical evaluation to a pure
oneloop calculation would lead to very large theoretical
uncertainties. These can be avoided by the inclusion of corrections
beyond the one-loop level. Here we follow the approach of
Ref. [
70
] and supplement the μνSSM one-loop results by
higher-order corrections in the MSSM limit as provided by
FeynHiggs (version 2.13.0) [
32,40,52–56,58
]. In this way
the leading and subleading two-loop corrections are included,
as well as a resummation of large logarithmic terms, see the
discussion in Sect. 1,
ˆ h ( p2) = ˆ h(1)( p2) + ˆ h(2 ) + ˆ hresum.
(113)
(2 ) we take over the
In the partial two-loop contributions ˆ h
corrections of O(αs αt , αs αb, αt2, αt αb), assuming that the
MSSM-like corrections are also valid in the μνSSM. This
assumption is reasonable since the only difference between
the squark sector of the μνSSM in comparison to the MSSM
are the terms proportional to Yiν vi L in the non-diagonal
element of the up-type squark mass matrices (see Eq. (23)) and
the terms proportional to vi L vi L in the diagonal elements of
the up- and down-type squark mass matrices (see Eqs. (22),
(24), (26) and (28)), which numerically will always be
negligible in realistic scenarios since vi L vd , vu , vR .
Furthermore,. in Ref. [
26
] the quality of the MSSM
approximation was tested in the NMSSM, showing that the genuine
NMSSM contributions are in most cases sub-leading. The
same is expected for the contributions stemming from the
resummation of large logarithmic terms given by ˆ hresum.
5 Numerical analysis
In the following we present for the first time the full one-loop
corrections to the scalar masses in the μνSSM, with one
generation of right-handed neutrinos obtained in the
Feynmandiagrammatic approach, taking into account all parameters
of the model and the complete dependence on the
external momentum, which includes a consistent treatment of
the imaginary parts of the scalar self-energies. Our results
extend the known ones in the literature of the MSSM and
the NMSSM to a model, which has a rich and unique
phenomenology through explicit R-parity breaking. The
oneloop results are supplemented by known higher-loop results
from the MSSM (see the previous section) to reproduce the
Higgs mass value of ∼ 125 GeV [
3
]. Here the theory
uncertainty must be kept in mind. In the MSSM it is estimated to
be at the level of 2−3 GeV [
54,57
], and in extended models
it is naturally slightly larger.
We will present results in several different scenarios, in all
of which one scalar with the correct SM-like Higgs mass is
reproduced. To get an estimation of the significance of
quantum corrections to the Higgs masses that are unique for the
μνSSM, we compare the results to the corresponding ones
in the NMSSM. The results in the NMSSM are obtained by
a calculation based on Ref. [
26
], but with slightly changed
Table 2 Input parameters for
the NMSSM-like crossing point
scenario; all masses and values
for trilinear parameters are in
GeV
renormalization conditions to be as close as possible to the
calculation in the μνSSM. While Ref. [
26
] uses the mass
squared of the charged Higgs mass as input parameter and
renormalizes it as OS parameter we instead use DR
conditions for Aλ.
The benchmark points used in the following were not
tested in detail against experimental bounds including the
Rparity violating effects of the μνSSM. They have been chosen
to exemplify the potential magnitude of unique μνSSM-like
corrections. Nevertheless, the values we picked for the free
parameters should be close to realistic and experimentally
allowed scenarios: the parameters in the scalar sector are
taken over from calculations in the NMSSM [
26
], and unique
μνSSM parameters are chosen in a range to reproduce
neutrino masses of the correct order of magnitude. That means
that the neutrino Yukawas Yiν should be of the order 10−6 to
generate neutrino masses of the order less than 1 eV. For the
left-handed sneutrino vevs this directly implies vi L vd , vu
so that the tadpole coefficients vanish at tree-level [
14
]. We
will leave a more detailed discussion of numerical results
for a future publication, in which we will also include three
generations of right-handed neutrinos.
5.1 NMSSM-like crossing point scenario
The first scenario we want to analyze is one studied in
the NMSSM with a singlet becoming the LSP in the
region of λ > κ taken from Ref. [
26
]. This scenario was
tested therein against the experimental limits implemented
in HiggsBounds 4.1.3 [
100–104
]. It has the nice
feature that there is a crossing point when λ ≈ κ in the
neutral scalar sector, in which the masses of the singlet and the
SM-like Higgs become degenerate and NMSSM-like loop
corrections become significant [
70
].
In Table 2 we list the values chosen for the parameters.
The SM-like parameters from the electroweak sector and the
lepton and quark masses are given in Appendix C in Table 5.
The parameters present in the μνSSM and the NMSSM
are of course chosen equally in both models. The region
λ < 0.026 is excluded because the left-handed sneutrinos
become tachyonic at tree-level. The flavor-changing
nondiagonal elements in the slepton sector are zero. The value for
Aλ is chosen to correspond to a mass of m H± = 1000 GeV
for the charged Higgs mass in the NMSSM with m H±
renorvi L /√2
10−4
M2
300
Yiν
10−6
M3
1500
m2
QiL
15002
Aiν
−1000
malized OS and Aλ not being a free parameter. Aκ should be
chosen to be negative in our convention (when κ is positive)
to avoid false vacua [
14
] or tachyons in the pseudo-scalar
sector [
105
]. It should be kept in mind that the diagonal soft
scalar masses in the neutral sector are extracted from the
values for vi L , tan β and μ via the tadpole equations, and their
non-diagonal, flavor-violating elements are always set to zero
at tree-level. This is of crucial importance for the comparison
of the scalar masses in the μνSSM and the NMSSM, since
in the NMSSM the soft slepton masses m2 are independent
L
parameters, while in the μνSSM the diagonal elements are
dependent parameters fixed by the tadpole Eq. (15), when
the vevs are used as input. The latter strategy is
particularly convenient since the order of magnitude of the vevs
is roughly fixed through the electroweak seesaw mechanism
by demanding neutrino masses below the eV scale, while
the soft scalar masses are not directly related to any physical
observable. Consequentially, for each parameter point
calculated in the μνSSM, the corresponding values that have
to be chosen for m2 in the NMSSM have to be adjusted
L
accordingly, defined as a function of all the free parameters
appearing in the Higgs potential.
In Fig. 6 we show the resulting spectrum of the
CPeven scalars at tree-level and including the full one-loop
and two-loop contributions.5 The standard model Higgs mass
value is reproduced accurately when the quantum corrections
are included. The heavy MSSM-like Higgs H and the
lefthanded sneutrinos are at the TeV-scale and rather decoupled
from the SM-like Higgs boson. The three left-handed
sneutrinos are degenerate because the μνSSM-like parameters
are set equal for all flavors.
The singlet-like scalar mass heavily depends on λ, because
when μ is fixed, increasing λ leads to a smaller value for
vR (see Eq. (44)). As was observed in Ref. [
26
], the
loopcorrected mass of the singlet becomes smaller than the
SMlike Higgs boson mass at about λ ≈ κ. We observe
nonnegligible loop-corrections to the singlet in the region of λ
where the singlet is the lightest neutral scalar.
Due to the similarity of the Higgs sectors of the NMSSM
and the μνSSM, the masses of the doublet-like Higgs bosons
and the right-handed sneutrino will be of comparable size as
the masses predicted for the doublet-like Higgses and the
5 Here and in the following we denote with “two-loop” result the
oneloop plus partial two-loop plus resummation corrected masses.
V
e
G
1600
800
400
200
125
100
50
Fig. 6 Spectrum of CP-even scalar masses in NMSSM-like
crossing point scenario. The three left-handed sneutrinos νi L are
degenerate (All plots have been produced using ggplot2 [
106
] and
tikzDevice [
107
] in R [
108
])
singlet in the NMSSM. In Fig. 7 we show the tree-level and
the one- and two-loop corrected mass of the SM-like Higgs
boson in the crossing-point scenario. One can see that, as
expected, the two-loop corrections are crucial to predict a
SM-like Higgs mass of 125 GeV. Indeed, our analysis
confirmed that differences in the prediction of the SM-like Higgs
boson mass are negligible compared to the current
experimental uncertainty [
3
] and the anticipated experimental
accuracy of the ILC of about <∼ 50 MeV [
109
], even when there
is a substantial mixing between left-handed sneutrinos and
the SM-like Higgs at tree-level or one-loop. Apart from that,
they are clearly exceeded by the (future) parametric
uncertainties in the Higgs-boson mass calculations. Consequently,
the Higgs sector alone will not be sufficient to distinguish
the μνSSM from the NMSSM. On the other hand, we can
regard the theoretical uncertainties in the NMSSM and the
μνSSM to be at the same level of accuracy.
5.2 Light τ -sneutrino scenario
In the previous scenario we observed that, in a scenario where
the left-handed sneutrinos where practically decoupled from
the SM-like Higgs boson, the unique μνSSM-like
corrections do not account for a substantial deviation of the
SMlike Higgs mass prediction compared to the NMSSM. In this
section we will investigate a scenario in which one of the
lefthanded sneutrinos has a small mass close to SM Higgs boson
mass. The phenomenology of such a spectrum was recently
studied in detail, including a comparison of its predictions
with the LHC searches [
17,110
]. It was found that a light
left-handed sneutrino as the LSP can give rise to distinct
sigh-like
H-like
νR-like
νiL-like
Tree-level
Two-loop
140
130
125
120
eV110
G
100
90
80
Tree-level
One-loop
Two-loop
Fig. 7 Tree-level, one-loop and two-loop corrected masses of the
SMlike Higgs boson in the μνSSM in the NMSSM-like crossing point
scenario
nals for the μνSSM (for instance, final states with diphoton
plus missing energy, diphoton plus leptons and multileptons).
In Table 3 we list the relevant parameters that were chosen
to obtain a light left-handed τ -sneutrino. The parameters not
shown here are chosen to be the same as in the previous case,
shown in Table 2. One can see that the vev v3L (corresponding
to ν3L ) was increased w.r.t. the NMSSM-like scenario. The
reason for this becomes clear when one extracts the leading
terms of the diagonal tree-level mass matrix element of the
left-handed sneutrinos,
mν2iRL νiRL ≈
Yiν vR vu
2vi L
√ √2μ
− 2 Aiν − κvR + tan β
.
(114)
The tree-level masses of the left-handed sneutrinos are
roughly proportional to the inverse of their vev. We also
decreased Aν3 in comparison to the previous scenario,
keeping it negative, so that it is of order κvR and the sum in the
brackets of Eq. (114) becomes small.
In Fig. 8 we show the tree-level and loop-corrected
spectrum of the scalars in the region of λ where there are no
tachyons at tree-level. For too small λ the tree-level mass
of ν3L becomes tachyonic, because when μ = (vR λ)/√2
is fixed vR has to grow and the second term in the bracket
of Eq. (114) will grow larger than the sum of the first and
the third term. For too large λ, the tree-level mass of the
SM-like Higgs boson becomes tachyonic. In particular, it
starts to mix with the tree-level singlet mass, which becomes
tachyonic because vR decreases when λ increases. The
central value of the SM Higgs boson mass is reproduced in this
scenario up to values of λ ≤ 0.22. However, considering the
Fig. 8 CP-even scalar mass spectrum of the μνSSM in the light τ
sneutrino scenario, see Table 3. On the right side we state the dominant
composition of the mass eigenstates
theoretical uncertainty even higher values of λ can be viable.
For λ = 0.236 the prediction for the SM-like Higgs mass
decreases below mh1 ≈ 122 GeV. As discussed in the
introduction we assume a theory uncertainty of ∼ 3 GeV on the
mass evaluation, so we consider in this scenario the region
λ ≤ 0.236 to be valid regarding the SM Higgs boson mass.
An interesting observation is that the masses of light
lefthanded sneutrinos are mainly induced via quantum
corrections, while the tree-level mass approaches 0 for small values
of λ. This indicates that a consistent treatment of quantum
corrections to light sneutrino masses is of crucial importance.
The large upward shift of the left-handed sneutrino masses
through the one-loop corrections is due to the fact that in the
μνSSM the sneutrino fields are part of the Higgs potential,
each with an associated tadpole coefficient Tνi L . To ensure
the stability of the vacuum w.r.t. quantum corrections, the
tadpoles are renormalized OS, absorbing all finite
corrections into the counterterms δTνi L (see Sect. 3.2). In the mass
counterterms for the left-handed sneutrinos the finite parts
δTνfii Ln introduce the main finite contribution in the form
δTνfii nL
δmν2iRfiLnνiRL = − vi L
+ · · · ,
which is enhanced by the inverse of the vev of νi L . It is
these terms inside the counterterms of the renormalized
self(1)
energies ˆ νiRL νiRL that shift the poles of the propagator matrix
(115)
H
ν1,2L
ν3L
νR
h
and increase the masses of the left-handed sneutrinos,
especially in cases where the tree-level masses are small.
This behavior is a peculiarity of the μνSSM, meaning that
the leptonic sector and the Higgs sector are mixed through the
breaking of R-parity. The relations between the vevs vi L and
the soft masses m2 via the tadpole equations automatically
L
lead to dependences between the sneutrino masses and, for
instance, the neutrino or the Higgs sector. In the NMSSM, on
the other hand, the sneutrinos are not part of the Higgs
potential, since the fields are protected by lepton-number
conservation. There, the soft masses m2 are, without further
assump
L
tions, free parameters that can be chosen without taking into
account any leptonic observable (such as neutrino masses
and mixings). In principle, the additional dependences of the
μνSSM scalar (neutrino) masses on the neutrino sector could
be used (e.g. when all neutrino masses and mixing angles will
be known with sufficient experimental accuracy) to restrict
the possible range of m2 , and thus the possible values for
L
the left-handed sneutrino masses. However, with our current
experimental knowledge on the neutrino masses, the possible
values for the vevs vi L , and hence the possible range of
lefthanded sneutrino masses, are effectively not yet constrained.
It should be noted as well, that the soft masses m2
L
also appear in the mass matrix of the charged scalars (see
Eq. (155)) and the pseudoscalars (see Eq. (147)). In many
cases they are the dominant term in the tree-level masses of
the left-handed sleptons and sneutrinos, so the values of the
masses of charged sleptons and sneutrino of the same family
will be close. A precise treatment of quantum corrections of
the size observed in Fig. 8 is extremely important in those
cases, since they might easily change the relative sign of their
mass differences. This can result in a complete change of the
phenomenology of the corresponding benchmark point, for
instance when either the neutral (pseudo)scalar or the charged
scalar is the LSP [
17,110
].
We compare the relevant spectrum of the μνSSM to the
corresponding one in the NMSSM in Fig. 9. We show the
tree-level and one-loop corrected masses of the light scalars
in the μνSSM, and the masses of the SM-like Higgs boson
and the singlet in the NMSSM on the right, with
parameters set accordingly. We shade in grey the region of λ where
the prediction for the SM-like Higgs boson mass is below
122 GeV if two-loop corrections are included. As expected,
the SM-like Higgs-boson mass and the mass of the singlet
turn out to be equal in both models. Even in regions where
there is a substantial mixing of the SM-like Higgs boson
Page 22 of 33
500
400
V300
e
G
200
100
Tree-level
One-loop
μνSSM
ν3L
NMSSM
Fig. 9 Light τ -sneutrino scenario, see Table 3. In the shaded region the
prediction for the SM-like Higgs mass is below 122 GeV. Left: Masses
of the SM-like Higgs, the left-handed τ -sneutrino and the right-handed
sneutrino in the μνSSM at tree-level and one-loop. Right: Masses of the
SM-like Higgs and the singlet in the NMSSM at tree-level and one-loop
Fig. 10 Light τ -sneutrino scenario, see Table 3. We show the
absolute values of the mixing matrix elements at tree-level |U1Hi (0)|
(left) and |U2Hi (0)| (right), whose squared value define the
admixture of the two-lightest CP-even scalar mass eigenstate h1,2 with
the fields ϕi = (Hd , Hu , νR , ν1L , ν2L , ν3L ) in the interaction basis.
with the left-handed sneutrinos, something that cannot occur
in the NMSSM, the differences in the SM-like Higgs mass
prediction are not larger than a few keV.
It is rather surprising that the SM-like Higgs masses
coincide this precisely in both models, considering the fact that
a substantial mixing with the sneutrino is possible at
treelevel, as we show in Fig. 10. We individually plot the mixing
matrix elements of the two lightest CP -even scalars, whose
squared values define the composition of each mass
eigenstate at tree-level. In the cross-over point of the τ -sneutrino
A substantial mixing of the τ -sneutrino ν3L with the SM-like
Higgs boson h125 and with the singlet νR is present in the
narrow region where the corresponding tree-level masses are
degenerate (for example in the right plot at λ ∼ 0.20237 and λ ∼
0.29692)
and the SM-like Higgs boson the lightest scalar results to be
a mixture of ντ and the doublet-components Hu and Hd , as
one can see in the upper left plot of Fig. 10. For example,
if we fine-tune λ = 0.20237 we find that the lightest Higgs
boson is composed of approximately
Hd → |U1H1(0) 2
| ∼ 1%,
Hu → |U1H2(0) 2
| ∼ 80%,
ν3L → |U1H6(0) 2
| ∼ 19%.
(116)
(117)
(118)
Nevertheless, due to the upward shift, as explained before,
the one-loop corrections break the degeneracy and no trace
on the SM-like Higgs mass remains, which would deviate it
from the NMSSM prediction.
5.3 The μνSSM and the CMS γ γ excess at 96 GeV
In this section we will investigate a scenario in which the
SM-like Higgs boson is not the lightest CP-even scalar.
This is inspired by the reported excesses of LEP [
111
] and
CMS [
112,113
] in the mass range around ∼ 96 GeV, that
(as we will show) can be explained simultaneously by the
presence of a light scalar in this mass window. While in
the NMSSM the light scalar can be interpreted as the
CPeven scalar singlet and can accommodate both excesses at
1σ level without violating any known experimental
constraints [
114,115
],6 we will interpret the light scalar as the
CP-even right-handed sneutrino of the μνSSM. Since the
singlet of the NMSSM and the right-handed sneutrino of
the μνSSM are both gauge-singlets, they share very similar
properties. However, the explanation of the excesses in the
μνSSM avoids bounds from direct detection experiments,
because R-parity is broken in the μνSSM and the dark matter
candidate is not a neutralino as in the NMSSM but a gravitino
with a lifetime longer than the age of the universe [
16
]. This
is important because the direct detection measurements were
shown to be very constraining in the NMSSM while trying
to explain the dark matter abundance on top of the excesses
from LEP and CMS [
114
].
In Table 4 we list the values of the parameters we used to
account for the lightest CP-even scalar as the right-handed
sneutrino and the second lightest one the SM-like Higgs
boson. λ is chosen to be large to account for a sizable
mixing of the right-handed sneutrino and the doublet Higgses. In
the regime where the SM-like Higgs boson is not the
lightest scalar, one does not need large quantum corrections to
the Higgs boson mass, because the tree-level mass is already
well above 100 GeV. This is why tan β can be low and the
soft trilinears Au,d,e are set to zero. The values of Aλ and
| Aν | are chosen to be around 1 TeV to get masses for the
heavy MSSM-like Higgs and the left-handed sneutrinos of
this order, so they do not play an important role in the
following discussion. On the other hand, κ is small to bring the
mass of the right-handed sneutrino below the SM-like Higgs
boson mass. Finally, the two parameters that are varied are
μ and Aκ . By increasing μ the mixing of the right-handed
sneutrino with the SM-like Higgs boson is increased, which
is needed to couple the gauge-singlet to quarks and
gaugebosons. At the same time we used the value of Aκ to keep
6 Other possible explanations of the CMS excess were analyzed in
Refs. [
116–118
]. On the other hand, in the MSSM the CMS excess
cannot be realized [
119
].
the mass of the right-handed sneutrino in the correct range.
Accordingly, the results in this chapter will all be displayed
in the scanned Aκ –μ plane.
The process measured at LEP was the production of a
Higgs boson via Higgstrahlung associated with the Higgs
decaying to bottom-quarks:
σ e+e− → Z h1 → Z bb¯
μLEP = σ SM e+e− → Z h → Z bb¯
= 0.117 ± 0.057,
where μLEP is called the signal strength, which is the
measured cross section normalized to the standard model
expectation, with the SM Higgs boson mass at ∼ 96 GeV. The
value for μLEP was extracted in Ref. [
114
] using methods
described in Ref. [
120
]. We can find an approximate
expression for μLEP factorizing the production and the decay of the
scalar and expressing it in terms of couplings to the massive
gauge bosons Ch1V V and the up- and down-type quarks Ch1uu¯
and Ch1dd¯, respectively, normalized to the SM predictions for
the corresponding couplings (where with μν we denote the
μνSSM prediction, and is the Higgs-boson decay width):
σ μν (Z ∗ → Z h1) BRμν h1 → bb¯
μLμEνP = σ SM (Z ∗ → Z h) × BRSM h → bb¯
≈ Ch1V V 2 × SbμMb¯ν × tSμoMtν
bb¯ tot
Ch1V V
2
× Ch1dd¯
2
≈
2
Ch1dd¯ (BRbSbM¯ + BRτSτM¯) + Ch1uu¯ 2 (BRSggM + BRcScM)
¯
(119)
(120)
The SM branching ratios dependent on the Higgs boson
mass can be obtained from Ref. [
121
]. The denominator is the
ratio of the total decay width of h1 in the μνSSM and h in the
SM when all SM branching ratios larger than 1% are
considered. The off-shell decay to W and Z bosons is in principle
also possible, but the BRs are very small for a SM Higgs
boson with a mass around 95 GeV (BRSWMW ∼ 0.5% and
BRSZMZ ∼ 0.06%) [
121,122
]. It is worth noticing that although
the right-handed neutrino mass is small, mνR ∼ 62−63 GeV,
in the investigated parameter region, it is nevertheless larger
than half of the SM-like Higgs boson mass in all benchmark
points, so the decay of the Higgs to the right-handed neutrino
is kinematically forbidden and cannot spoil the properties of
the SM-like Higgs. Neglecting the vevs vi L the normalized
couplings of the scalars are given at leading order by the
admixture of the mass eigenstate hi with the doublet like
Higgs Hd and Hu via
Chi dd¯ =
UiH1 ,(2 )
cos β , Chi uu¯ =
UiH2 ,(2 )
sin β
Chi V V = UiH1 ,(2 ) cos β + UiH2 ,(2 ) sin β,
(121)
where the partial two-loop plus resummation corrected
mixing matrix elements Ui Hj,(2 ) were calculated in the
approximation of vanishing momentum, see the discussion in
Sect. 4.2. We show in Fig. 11 the masses (top row) and the
normalized couplings (|Ch1dd¯| second row, |Ch1ub¯| third row,
|Ch1V V | lowest row) of the lightest and the next-to-lightest
CP-even scalar. The lower right corner (marked in gray)
results in the right-handed sneutrino becoming tachyonic (at
tree-level). The largest mixing of the right-handed sneutrino
and the SM-like Higgs boson is achieved where μ is largest
and | Aκ | is smallest. The mass of h2 is in the allowed region
for a SM-like Higgs boson at ∼ 125 GeV if we assume a
theory uncertainty of up to 3 GeV (see the previous
subsections). The LHC measurements of the SM-like Higgs boson
couplings to fermions and massive gauge bosons are still not
very precise [
123
], with uncertainties between 10 and 20% at
the 1σ confidence level (obtained with the assumption that no
beyond-the-SM decays modify the total width of the SM-like
Higgs boson). Therefore, it would be challenging to exclude
parts of the parameter space by considering the deviations
of the normalized couplings of h2. However, possible future
lepton colliders like the ILC could measure these couplings
to a %-level [
109,124
], which could exclude (or confirm)
most of the parameter space presented here. Seen from a
more optimistic perspective, the precise measurement of the
SM-like Higgs boson couplings at future colliders could be
used to make predictions for the properties of the lighter
right-handed sneutrino in this scenario.
The CMS excess was observed in the diphoton channel
with a signal strength of [
125
]
σ (gg → h1 → γ γ )
μCMS = σ SM (gg → h → γ γ ) = 0.6 ± 0.2.
(122)
We calculate the signal strength using the approximation that
the Higgs production via gluonfusion is described at leading
order exclusively by the loop-diagram with a top quark
running in the loop, and that the diphoton decay is described
by the diagrams with W bosons or a top quark in the loop,
which is sufficient in the investigated mass range of h1. One
can then write
σ μν (gg → h1) BRγμγν
μCμMνS = σ SM (gg → h) × BRγSMγ
≈ Ch1uu¯ 2 × γSγμMγγν × tStμooMtνt
Ch1uu¯ 2 × Chef1fγ γ
2
≈
2
Ch1dd¯ (BRbSbM¯ + BRτSτM¯) + Ch1uu¯ 2 (BRSggM + BRcScM)
¯
The effective coupling of the neutral scalars to photons Chefifγ γ
has to be calculated in terms of the couplings to the W boson
and the up-type quarks. In the SM the dominant contributions
to the decay to photons can be written as [
126
]
SM Gμ α2 m3h 4
γ γ = 128 √2 π 3 3
2
A1/2 (τt ) + A1 (τW ) ,
where Gμ is the Fermi-constant and the form factors A1/2
and A1 are defined as
A1/2 (τ ) = 2 τ + (τ − 1) arcsin2 √τ τ −2,
A1 (τ ) = − 2τ 2 + 3τ + 3 (2τ − 1) arcsin2 √τ τ −2,
for τ ≤ 1, and the arguments of these functions are τt =
m2h /(4mt2) and τW = m2h /(4M W2 ). In our approximation the
only difference between the μνSSM and the SM will be that
the couplings of hi to the top quark and the W boson is
modified by the factors Chi tt¯ and Chi V V , so the effective
coupling of the Higgses to photons in the μνSSM normalized
to the SM predictions can be written as
C eff
hi γ γ
2
43 Chi tt¯ A1/2 (τt ) + Chi V V A1 (τW )
2
43 A1/2 (τt ) + A1 (τW )
2
.
Using Eqs. (120) and (123) we can calculate the two
signal strengths. The result are shown in Fig. 12, the LEP (left)
and the CMS excesses (right) in the μ– Aκ plane. While the
LEP excess is easily reproduced in the observed parameter
space, we cannot achieve the central value for μCMS, but only
slightly smaller values. As already observed in Ref. [
114
],
the reason for this is that for explaining the LEP excess a
100
Fig. 11 Properties of the
lightest (left) and
next-to-lightest (right) CP -even
scalar in the μ– Aκ plane. The
couplings are normalized to the
SM-prediction of a Higgs
particle of the same mass. The
gray area is excluded because
the right-handed sneutrino
becomes tachyonic at tree-level.
First row: two-loop masses,
second row: coupling to
down-type quarks, third row:
coupling to up-type quarks,
fourth row: coupling to massive
gauge bosons
μ
μ
μ
μ
(2)
μ
413 413.5 414 414.5 415 415.5 416 416.5 417 417.5
0.37
413 413.5 414 414.5 415 415.5 416 416.5 417 417.5
sizable coupling to the bottom quark is needed. On the
contrary, the CMS excess demands a small value for Ch1dd¯ so
that the denominator in Eq. (123) becomes small and μCMS is
enhanced. Nevertheless, considering the large experimental
uncertainties in μCMS and μLEP, the scenario presented in
this section accommodates both excesses comfortably well
(at approximately 1σ ), and it is a good motivation to keep
on searching for light Higgses in the allowed mass window
below the SM-like Higgs mass. Apart from that, this
scenario illustrates the importance of an accurate calculation of
the loop-corrected scalar masses and mixings, since already
small changes in the parameters can have a big impact on
the production and the decay modes of the CP-even Higgs
bosons.
6 Conclusion and outlook
The μνSSM is a simple SUSY extension of the SM that
is capable of predicting neutrino physics in agreement with
experimental data. As in other SUSY models, higher-order
corrections are crucial to reach a theoretical uncertainty at
the same level of (anticipated) experimental accuracy. So far,
higher-order corrections in the μνSSM had been restricted to
DR calculations, which suffer from the disadvantage that they
cannot be directly connected to (possibly future observed)
new BSM particles.
In this paper we have performed the complete one-loop
renormalization of the neutral scalar sector of the μνSSM
with one generation of right-handed neutrinos in a mixed
onshell/DR scheme. The renormalization procedure was
discussed in detail for each of the free parameters appearing
in the μνSSM Higgs sector. We have emphasized the
conceptual differences to the MSSM and the NMSSM regarding
the field renormalization and the treatment of
non-flavordiagonal soft mass parameters, which have their origin in
the breaking of R-parity in the μνSSM. However, we have
ensured that the renormalization of the relevant (N)MSSM
parts in the μνSSM are in agreement with previous
calculations in those models. Consequently, numerical differences
found can directly be attributed to the extended structure of
the μνSSM. The derived renormalization can be applied to
any higher-order correction in the μνSSM. The one-loop
counterterms derived in this paper are implemented into the
FeynArts model file, so the computation of these
corrections can be done fully automatically.
We have applied the newly derived renormalization to
the calculation of the full one-loop corrections to the
neutral scalar masses of the μνSSM, where we found that all
UV-divergences cancel. In our numerical analysis the newly
derived full one-loop contributions are supplemented by
available MSSM higher-order corrections as provided by the
code FeynHiggs (leading and subleading fixed-order
corrections as well as resummed large logarithmic contributions
obtained in an EFT approach.) We investigated various
representative scenarios, in which we obtained numerical results
for a SM-like Higgs boson mass consistent with experimental
bounds. We compared our results to predictions of the various
neutral scalars in the NMSSM to investigate the relevance of
genuine μνSSM-like contributions. We find negligible
corrections w.r.t. the NMSSM, indicating that the Higgs boson
mass calculations in the μνSSM are at the same level of
accuracy as in the NMSSM.
Finally we showed that the μνSSM can accommodate
a right-handed (CP-even) scalar neutrino in a mass regime
of ∼ 96 GeV, where the full Higgs sector is in agreement
with the Higgs-boson measurements obtained at the LHC,
as well as with the Higgs exclusion bounds obtained at LEP,
the Tevatron and the LHC. This includes in particular a
SMlike Higgs boson at ∼ 125 GeV. We have demonstrated that
the light right-handed sneutrino can explain an excess of γ γ
events at ∼ 96 GeV as reported recently by CMS in their
Run I and Run II date. It can simultaneously describe the
2 σ excess of bb¯ events observed at LEP at a similar mass
scale. We are eagerly awaiting the corresponding ATLAS
Higgs-boson search results.
Acknowledgements We thank F. Domingo for helpful discussions.
This work was supported in part by the Spanish Agencia Estatal de
Investigación through the grants FPA2016-78022-P
MINECO/FEDERUE (TB and SH) and FPA2015-65929-P MINECO/FEDER-UE (CM),
and IFT Centro de Excelencia Severo Ochoa SEV-2016-0597. The
work of TB was funded by Fundación La Caixa under ‘La
CaixaSevero Ochoa’ international predoctoral grant. We also acknowledge
the support of the MINECO’s Consolider-Ingenio 2010 Programme
under grant MultiDark CSD2009-00064.
Appendix A: Mass matrices
Here we state the entries of the following scalar mass
matrices.
A.1 CP-even scalars
In the interaction basis ϕT = (HdR, HuR, νRR, νiRL ) the mass
matrix for the CP-even scalars m2ϕ is defined by:
m2 1
HdR HdR = m2Hd + 8
m2 1
HuR HuR = m2Hu + 8
+ 21 λ2 v2R + vu2 ,
g12 + g22
3vd2 + vi L vi L − vu
2
g12 + g22
3vu2 − vd − vi L vi L
2
+ 21 λ2 v2R + vd2
+ 21 v2R Yiν Yiν − vd λvi L Yiν ,
1
− vu λvi L Yiν − √
2
T λ,
m2
HuR HdR = − 4
1
g12 + g22 vd vu − 21 v2R κλ + vd vu λ2
m2
νiRL HuR = − 4
1
g12 + g22 vu vi L + 21 v2R κYiν − vd vu λYiν
m2
νiRL ν RjL =
m2 1
νiRL νRR = −λvd vR Yiν + √ vu Tiν
2
1
+ vu v j L Y jν Yiν + √ vR Tiν ,
2
+vR vu κYiν + vR Yiν v j L Y jν ,
m2 2 1
νR HdR = −vR vu κλ + vd vR λ − vR λvi L Yiν − √ vu T λ,
2
m2 2
νR HuR = −vd vR κλ + vR vu λ + vR κvi L Yiν
m2
νiRL HdR =
+ vu κvi L Yiν − vd λvi L Yiν
(131)
(134)
(135)
(136)
(137)
(138)
(139)
(140)
(141)
(129)
(130)
A.2 CP-odd scalars
In the interaction basis σ T = (HdI , HuI , νIR , νiIL ) the mass
matrix for the CP-odd scalars m2σ is defined by:
m2 1
HdI HdI = m2Hd + 8
m2 1
HdI HdI = m2Hu + 8
+ 21 λ2 v2R + vu2 ,
g12 + g22
2 2
vd + vi L vi L − vu
g12 + g22
2 2
vu − vd − vi L vi L
+ 21 λ2 v2R + vd2 + vd λvi L Yiν
1 1
mν2IR HuI = vd vR κλ − vR κvi L Yiν + √ vi L Tiν − √ vd T λ,
2 2
mν2IR νIR = mν2 + v2R κ2 + vd vu κλ + 21 λ2 vd2 + vu2
m2Hd− Hu+ = 41 g22vd vu + 21 v2R λκ − 21 vd vu λ2
1 1
+ 2 vu λvi L Yiν + √2 vR T λ,
me2i L Hd−∗ =
m2Hd L i + 41 g22vd vi L − 21 vd Y jei Y jek vk L
− 21 v2R λYiν ,
me2i L Hu+ = 41 g22vu vi L − 21 v2R κYiν + 21 vd vu λYiν
1 1
− 2 vu v j L Y jν Yiν − √2 vR Tiν ,
me2i R Hd−∗ = − √12 v j L Tiej − 21 vR vu Yiej Y jν ,
me2i R Hu+ = − 21 vR λv j L Yiej − 21 vd vR Yiej Y jν ,
me2i L e∗j L =
m2L i j + 18 δi j g12 − g22
2 2
vd − vu + vk L vk L
me2i Re∗j R =
me2 i j + 41 δi j g12 vu2 − vd2 − vk L vk L
(156)
(157)
Appendix B: Explicit expressions for counterterms
In this section we will state the one-loop counterterms that
were calculated diagrammatically in the DR scheme and
checked against master formulas for the one-loop beta
functions and anomalous dimensions of soft SUSY breaking
parameters [
127–129
], superpotential parameters [
129,130
],
vacuum expectation values [89] and wave-functions with
kinetic mixing [
131,132
]. The master formulas were
evaluated using the mathematica package SARAH [133].
B.1 Field renormalization counterterms
We list the field renormalization counterterms defined in
Eq. (57) in the DR scheme in the interaction basis (Hd , Hu ,
νR , ν1L , ν2L , ν3L ):
λ2 + Yiej Yiej + 3 Yid Yid
, (158)
δ Z11 = − 16π 2
δ Z1,3+i = 16π 2 λYiν ,
δ Z22 = − 16π 2
δ Z33 = − 16π 2
δ Z3+i,3+ j = − 16π 2
(142)
(143)
(144)
(145)
(146)
(147)
(148)
(149)
(150)
(151)
(152)
(153)
(154)
(159)
(160)
(161)
(162)
(163)
λ2 + Yiν Yiν + 3 Yiu Yiu
,
λ2 + κ2 + Yiν Yiν ,
Ykei Ykej + Yiν Y jν .
We checked that the coefficients of the divergent part of the
field renormalization counterterms are equal to the one-loop
anomalous dimensions of the corresponding superfields γi(j1),
neglecting the terms proportional to the gauge couplings g1
and g2, and divided by the loop factor 16π 2, i.e.,
δ Zi j =
which is the same relation that holds in the (N)MSSM.
Table 5 Values for parameters
of the standard model in GeV
167.48
mτ
mb
mμ
mc
me
ms
0.003
MZ
We list the explicit form of the counterterms of the free
parameters renormalized in the DR scheme:
+ Yiej Yiek v j L vkL + 2vi L Yiν 2λvd − v j L Y jν
,
vi L
δvi2L = 32π 2
4π αvi L sw2 + 3cw2
sw2cw2
+ 2 vd λYiν
− vkL Y jei Y jek − v j L Yiν Y jν
,
δYiν = 32π 2 −
4π αYiν sw2 + 3cw2
sw2cw2
+ Yiν 3Yiu Yiu
+ 2κ2 + 4λ2 + 4Y jν Y jν + Y jei Y jek Ykν ,
+ Y jei Y jek Tkν + 2Tjei Y jek Ykν + Yiν 7Y jν Tjν
(164)
(165)
(166)
(167)
(168)
(169)
(170)
(171)
(172)
+ 6Yiuj Tiuj + 4κ2 Aκ + 7λ2 Aλ
ν
+ Ai
λ
δ m2Hd L i = − 32π 2 Yiν 2mν2 + 2 Aλ Aiν + m2Hd
λ2 + Y jν Y jν Yiν − Y jei Y jek Ykν
,
+ 2m2Hu + Y jν m2L ji ,
δ m2L i j = 32π 2 2m Hd Ykei Ykej + 2Tkei Tkej
+ Ylei Ylek m2L jk + 2Ykei Ylej me2 kl
+ m2L ki Ylek Ylej − λ m2Hd L j Yiν
+ Yiν m2L jk Ykν + m2L ki Ykν Y jν
+ Yiν Y jν 2m2Hu + 2mν2 + 2Tiν Tjν
for i = j.
The counterterms in Eqs. (164)–(173) were all calculated
diagrammatically in this form and afterwards checked to fulfill
the one-loop relation
β(1)
δ X = 3X2π 2 ,
where δ X stands for one of the counterterms just mentioned,
and β(1) is the one-loop coefficient of the beta function of
X
the parameter X , which could be obtained by the help of the
mathematica package SARAH [
133
].
On the contrary, the counterterms of the soft masses stated
in Eqs. (174) and (175) are the ones derived from the
oneloop beta function we obtained with SARAH, which were
then numerically checked to be equal to the counterterms for
(m2Hd L )i and (m2L )i j we calculated diagrammatically in the
scalar CP-odd sector.
Appendix C: Standard model values
Table 5 summarizes the values for the SM-like parameters
we chose in our calculation. The value for v corresponds
to a value for the Fermi constant of G F = 1.16638 ×
10−5GeV −2. The values for the gauge boson masses define
the cosine of the weak mixing angle to be cw = 0.881 535.
Note that since the SM leptons mix with the Higgsinos and
gauginos in the μνSSM, the lepton masses are not the real
md
v
(173)
(174)
(175)
(176)
Y1e =
2me
vd
Page 30 of 33
physical input parameters. However, the mixing is tiny, so
there will always be three mass eigenstates in the charged
fermion sector corresponding to the three standard model
leptons, having approximately the masses me, mμ and mτ .
This is why we use the values for these masses from Table 5
and then calculate the real input parameters, which are the
Yukawa couplings
2mμ
vd
, Y2e =
, Y3e =
2mτ
vd
.
(177)
Page 32 of 33
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