Higgs data constraints on the minimal supersymmetric standard model

4.7 MB·English

by  Kingman Cheung

CNU-HEP-15-01
Higgs data constraints
on the minimal supersymmetric standard model
Kingman Cheung1,2,3, Jae Sik Lee3,4, and Po-Yan Tseng1
1 Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan
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1 2 Division of Quantum Phases and Devices, School of Physics,
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2
Konkuk University, Seoul 143-701, Republic of Korea

t
c
3 Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan
O
1 4 Department of Physics, Chonnam National University,
1
300 Yongbong-dong, Buk-gu, Gwangju, 500-757, Republic of Korea

]
h
(Dated: October 13, 2015)
p
-
p Abstract
e
h
[ We perform global fits to the most recent data (after summer 2014) on Higgs boson signal

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strengths in the framework of the minimal supersymmetric standard model (MSSM). We further
v
2
impose the existing limits on the masses of charginos, staus, stops and sbottoms together with the
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current Higgs mass constraint |M −125.5GeV| < 6 GeV. The heavy supersymmetric (SUSY)
3 H1
0
. particlessuchassquarksenterintotheloopfactorsoftheHgg andHγγ verticeswhileotherSUSY
1
0
particles such as sleptons and charginos also enter into that of the Hγγ vertex. We also take into
5
1
account the possibility of other light particles such as other Higgs bosons and neutralinos, such
:
v
Xi that the 125.5 GeV Higgs boson can decay into. We use the data from the ATLAS, CMS, and the
r
a Tevatron, with existing limits on SUSY particles, to constrain on the relevant SUSY parameters.
We obtain allowed regions in the SUSY parameter space of squark, slepton and chargino masses,
and the µ parameter. We find that |∆Sγ/Sγ | < 0.1 at 68% confidence level when M > 300
SM ∼ χ˜±
1
GeV and M > 300 GeV, irrespective of the squarks masses. Furthermore, |∆Sγ/Sγ | < 0.03
τ˜1 SM ∼
when M > 500 GeV and M > 600 GeV.
χ˜±1,τ˜1 t˜1,˜b1 ∼
PACS numbers: 12.60.Jv, 14.80.Da
1
I. INTRODUCTION
The celebrated particle observed by the ATLAS [1] and the CMS [2] Collaborations at
the Large Hadron Collider (LHC) in July 2012 is mostly consistent with the standard model
(SM) Higgs boson than any other extensions of the SM [3, 4], at least in terms of some
statistical measures. The SM Higgs boson was proposed in 1960s [5], but only received the
confirmation recently through its decays into γγ and ZZ∗ → 4(cid:96) modes.
Although the data on Higgs signal strengths are best described by the SM, the other
extensions are still viable options to explain the data. Numerous activities occurred in the
constraining the SM boson [3, 6–23], higher dimension operators of the Higgs boson [24–29],
the two-Higgs doublet models [30–43], and in the supersymmetric framework [44–53]. A
very recent update to all the data as of summer 2014 was performed in Ref. [4]. We shall
describe the most significant change to the data set in Sec. III. In this work, we perform
the fits in the framework of the minimal supersymmetric standard model (MSSM) to all the
most updated data on Higgs signal strengths as of summer 2014.
In our previous analysis of the two-Higgs-doublet model (2HDM) [40], we do not specify
which neutral Higgs boson is the observed Higgs boson, so that the whole scenario can be
described by a small set of parameters. The bottom and leptonic Yukawa couplings are
determined through the top Yukawa coupling, and the HWW coupling is determined via
tanβ and top Yukawa, so that a minimal set of parameters includes only tanβ and the
top Yukawa coupling. We can easily include the effects of the charged Higgs boson by the
loop factor in the Hγγ vertex, and include possibly very light Higgs bosons by the factor
∆Γ . Here we follow the same strategy for the global fits in the framework of MSSM, the
tot
Higgs sector of which is the same as the Type II of the 2HDM, in order to go along with a
minimal set of parameters, unless we specifically investigate the spectrum of supersymmetric
particles, e.g., the chargino mass.
In this work, we perform global fits in the MSSM under various initial conditions to the
most updated data on Higgs boson signal strengths. A few specific features are summarized
here.
1. We use a minimal set of parameters without specifying the spectrum of the SUSY
particles. For example, all up-, down- and lepton-type Yukawa couplings and the
gauge-Higgs coupling are given in terms of the top Yukawa coupling, tanβ, and κ ,
d
2
where κ is the radiative correction in the bottom Yukawa coupling defined later.
d
2. Effects of heavy SUSY particles appear in the loop factors ∆Sg and ∆Sγ of the Hgg
and Hγγ vertices, respectively.
3. Effects of additional light Higgs bosons or light neutralinos that the 125.5 GeV Higgs
boson can decay into are included by the deviation ∆Γ in the Higgs boson width.
tot
4. CP-violating effects can occur in Yukawa couplings, which are quantified by the CP-
odd part of the top-Yukawa coupling. Effects of other CP sources can appear in the
loopfactorofHgg andHγγ vertices. Welabelthemas∆Pg and∆Pγ, respectively. In
Ref. [54], we have computed all the Higgs-mediated CP-violating contributions to the
electric dipole moments (EDMs) and compared to existing constraints from the EDM
measurements of Thallium, neutron, Mercury, and Thorium monoxide. Nevertheless,
we are content with CP-conserving fits in this work.
5. We impose the existing limits of chargino and stau masses when we investigate specifi-
cally their effects on the vertex of Hγγ. The current limit on chargino and stau masses
are [55]
M > 103.5 GeV, M > 81.9 GeV .
χ˜± τ˜1
Similarly, the current limits for stop and sbottom masses quoted in PDG are [55]
M > 95.7 GeV, M > 89 GeV,
t˜1 ˜b1
which will be applied in calculating the effects in Hγγ and Hgg vertices. Note that
the current LHC limits on the stop and sbottom masses are M > 650 GeV and
t˜1
M > 600 GeV at 95% confidence level in a simplified model with M = 0 GeV [55].
˜b1 χ˜01
However, there often exist underlying assumptions of search strategies and the mass
of the lightest neutralino. Therefore, we conservatively take the above mass limits on
the stops and sbottoms in most of the analysis.
6. Since we shall try to find the implication of the current Higgs signal strength data
on the SUSY spectrum, which in practice affects the lightest Higgs boson mass, we
therefore also calculate the corresponding Higgs boson mass and impose the current
Higgs mass constraint of M ∼ 125.5±6 GeV, taking at a roughly 3-σ level.
H1
3
The organization of the work is as follows. In the next section, we describe the conven-
tion and formulas for all the couplings used in this work. In Sec. III, we describe various
CP-conserving fits and present the results. In Sec. IV, we specifically investigate the SUSY
parameter space of charginos, staus, stops, and sbottoms. We put the synopsis and conclu-
sions in Sec. V.
II. FORMALISM
For the Higgs couplings to SM particles we assume that the observed Higgs boson is a
generic CP-mixed state without carrying any definite CP-parity. We follow the conventions
and notation of CPsuperH [56].
A. Yukawa couplings
The Higgs sector of the MSSM is essentially the same as the Type II of the 2HDM. More
details of the 2HDM can be found in Ref. [40]. In the MSSM, the first Higgs doublet couples
to the down-type quarks and charged leptons while the second Higgs doublet couples to the
up-type quarks only. After both doublets take on vacuum-expectation values (VEV) we can
rotate the neutral components φ0, φ0 and a into mass eigenstates H through a mixing
1 2 1,2,3
matrix O as follows:
(φ0, φ0, a)T = O (H , H , H )T ,
1 2 α αi 1 2 3 i
with the mass ordering M ≤ M ≤ M . We do not specify which Higgs boson is the
H1 H2 H3
observed one, in fact, it can be any of the H . We have shown in Ref. [40] that the
1,2,3
bottom and lepton Yukawa couplings can be expressed in terms of the top Yukawa coupling
in general 2HDM. We can therefore afford a minimal set of input parameters.
The effective Lagrangian governing the interactions of the neutral Higgs bosons with
quarks and charged leptons is
L = − (cid:88) gmf (cid:88)3 H f¯(cid:16)gS + igP γ (cid:17)f . (1)
Hf¯f 2M i Hif¯f Hif¯f 5
f=u,d,l W i=1
At the tree level, (gS,gP) = (O /c ,−O tanβ) and (gS,gP) = (O /s ,−O cotβ) for
φ1i β ai φ2i β ai
f = ((cid:96),d) and f = u, respectively, and tanβ ≡ v /v is the ratio of the VEVs of the two
2 1
4
doublets. Threshold corrections to the down-type Yukawa couplings change the relation
between the Yukawa coupling h and mass m as 1
d d
√
2m 1
d
h = . (2)
d
vcosβ 1+κ tanβ
d
Thus, the Yukawa couplings of neutral Higgs-boson mass eigenstates H to the down-type
i
quarks are modified as
(cid:18) 1 (cid:19) O (cid:18) κ (cid:19) O
gS = Re φ1i + Re d φ2i
Hid¯d 1 + κ tanβ cosβ 1 + κ tanβ cosβ
d d
(cid:20) κ (tan2β + 1) (cid:21)
d
+Im O ,
ai
1 + κ tanβ
d
(cid:18) tanβ − κ (cid:19) (cid:18) κ tanβ (cid:19) O
gP = −Re d O + Im d φ1i
Hid¯d 1 + κ tanβ ai 1 + κ tanβ cosβ
d d
(cid:18) κ (cid:19) O
−Im d φ2i , (3)
1 + κ tanβ cosβ
d
In the MSSM, neglecting the electroweak corrections and taking the most dominant
contributions, κ can be split into [57]
b
κ = (cid:15) +(cid:15) ,
b g H
where (cid:15) and (cid:15) are the contributions from the sbottom-gluino exchange diagram and from
g H
stop-Higgsino diagram, respectively. Their explicit expressions are
2α |h |2
(cid:15) = sM∗µ∗I(m2 ,m2 ,|M |2), (cid:15) = t A∗µ∗I(m2 ,m2 ,|µ|2) ,
g 3π 3 ˜b1 ˜b2 3 H 16π2 t t˜1 t˜2
where M is the gluino mass, h and A are the top-quark Yukawa and trilinear coupling,
3 t t
respectively.
B. Couplings to gauge bosons
• Interactions of the Higgs bosons with the gauge bosons Z and W± are described by
(cid:32) (cid:33)
1
L = gM W+W−µ + Z Zµ (cid:88) g H (4)
HVV W µ 2c2 µ HiVV i
W i
where
g = c O + s O . (5)
HiVV β φ1i β φ2i
1 In general settings, κ and κ are usually the same, but κ could be very different because of the third
d s b
generation squarks. However, our main concern in this work is the third-generation Yukawa couplings.
Thus, we shall focus on κ although we are using the conventional notation κ .
b d
5
• Couplings to two photons: the amplitude for the decay process H → γγ can be
i
written as
αM2 (cid:26) 2 (cid:27)
M = − Hi Sγ(M ) ((cid:15)∗ ·(cid:15)∗ )−Pγ(M ) (cid:104)(cid:15)∗(cid:15)∗k k (cid:105) , (6)
γγHi 4πv Hi 1⊥ 2⊥ Hi M2 1 2 1 2
Hi
where k are the momenta of the two photons and (cid:15) the wave vectors of the cor-
1,2 1,2
responding photons, (cid:15)µ = (cid:15)µ − 2kµ(k · (cid:15) )/M2 , (cid:15)µ = (cid:15)µ − 2kµ(k · (cid:15) )/M2 and
1⊥ 1 1 2 1 Hi 2⊥ 2 2 1 2 Hi
(cid:104)(cid:15) (cid:15) k k (cid:105) ≡ (cid:15) (cid:15)µ(cid:15)νkρkσ. ThedecayrateofH → γγ isproportionalto|Sγ|2+|Pγ|2.
1 2 1 2 µνρσ 1 2 1 2 i
The form factors are given by
Sγ(M ) = 2 (cid:88) N Q2 gS F (τ )−g F (τ )+∆Sγ,
Hi C f Hif¯f sf f HiVV 1 W i
f=b,t,τ
Pγ(M ) = 2 (cid:88) N Q2 gP F (τ )+∆Pγ, (7)
Hi C f Hif¯f pf f i
f=b,t,τ
where τ = M2 /4m2, N = 3 for quarks and N = 1 for taus, respectively. In MSSM,
x Hi x C C
the factors ∆Sγ and ∆Pγ receive contributions from charginos, sfermion, and charged
i i
Higgs boson:
√ v
∆Sγ = 2g (cid:88) gS F (τ )
i Hif¯f m sf if
f=χ˜±,χ˜± f
1 2
v2 v2
− (cid:88) N Q2g F (τ )−g F (τ ),
f˜j=t˜1,t˜2,˜b1,˜b2,τ˜1,τ˜2 C f Hif˜j∗f˜j2m2f˜j 0 if˜j HiH+H−2MH2± 0 iH±
√ v
∆Pγ = 2g (cid:88) gP F (τ ), (8)
i Hif¯f m pf if
f=χ˜±,χ˜± f
1 2
where the couplings to charginos, sfermions, and charged Higgs are defined in the
interactions:
g (cid:18) (cid:19)
L = −√ (cid:88)H χ− gS +iγ gP χ−,
Hχ(cid:101)+χ(cid:101)− 2 k(cid:101)i Hkχ˜+i χ˜−j 5 Hkχ˜+i χ˜−j (cid:101)j
i,j,k
L = v (cid:88) g (H f˜∗f˜),
Hf˜f˜ Hif˜j∗f˜k i j k
f=u,d
3
L = v (cid:88)g H H+H−. (9)
3H HiH+H− i
i=1
We shall describe the couplings of the Higgs boson to the charginos, sfermions, and
charged Higgs boson a little later.
6
• Couplings to two gluons: similar to H → γγ, the amplitude for the decay process
H → gg can be written as
i
α M2 δab(cid:26) 2 (cid:27)
M = − s Hi Sg(M )((cid:15)∗ ·(cid:15)∗ )−Pg(M ) (cid:104)(cid:15)∗(cid:15)∗k k (cid:105) , (10)
ggHi 4πv Hi 1⊥ 2⊥ Hi M2 1 2 1 2
Hi
where a and b (a,b = 1 to 8) are indices of the eight SU(3) generators in the adjoint
representation. The decay rate of H → gg is proportional to |Sg|2 + |Pg|2. The
i
fermionic contributions and additional loop contributions from squarks in the MSSM
to the scalar and pseudoscalar form factors are given by
Sg(M ) = (cid:88) gS F (τ )+∆Sg,
Hi Hif¯f sf f i
f=b,t
Pg(M ) = (cid:88) gP F (τ )+∆Pg, (11)
Hi Hif¯f pf f i
f=b,t
with
v2
∆Sg = − (cid:88) g F (τ ),
i Hif˜j∗f˜j4m2 0 if˜j
f˜j=t˜1,t˜2,˜b1,˜b2 f˜j
∆Pg = 0, (12)
i
where the ∆Pg = 0 because there are no colored SUSY fermions in the MSSM that
can contribute to ∆Pg at one loop level.
C. Interactions of neutral Higgs bosons with charginos, sfermions, and charged
Higgs
The interactions between the Higgs bosons and charginos are described by the following
Lagrangian:
g (cid:18) (cid:19)
L = −√ (cid:88)H χ− gS +iγ gP χ−,
Hχ(cid:101)+χ(cid:101)− 2 k(cid:101)i Hkχ˜+i χ˜−j 5 Hkχ˜+i χ˜−j (cid:101)j
i,j,k
1 (cid:110) (cid:111)
gS = [(C ) (C )∗ Gφ1 +(C ) (C )∗ Gφ2]+[i ↔ j]∗ ,
Hkχ˜+i χ˜−j 2 R i1 L j2 k R i2 L j1 k
i (cid:110) (cid:111)
gP = [(C ) (C )∗ Gφ1 +(C ) (C )∗ Gφ2]−[i ↔ j]∗ , (13)
Hkχ˜+i χ˜−j 2 R i1 L j2 k R i2 L j1 k
where Gφ1 = (O −is O ), Gφ2 = (O −ic O ), i,j = 1,2, and k = 1−3. The chargino
k φ1k β ak k φ2k β ak
mass matrix in the (W˜ −,H˜−) basis
 √ 
M 2M c
2 W β
 
MC = √  , (14)
 
2M s µ
W β
7
is diagonalized by two different unitary matrices C M C† = diag{M , M }, where
R C L χ˜± χ˜±
1 2
M ≤ M . The chargino mixing matrices (C ) and (C ) relate the electroweak eigen-
χ˜± χ˜± L iα R iα
1 2
states to the mass eigenstates, via
χ˜− = (C )∗ χ˜− , χ˜− = (W˜ −,H˜−)T ,
αL L iα iL αL L
χ˜− = (C )∗ χ˜− , χ˜− = (W˜ −,H˜−)T . (15)
αR R iα iR αR R
The Higgs-sfermion-sfermion interaction can be written in terms of the sfermion mass
eigenstates as
L = v (cid:88) g (H f˜∗f˜), (16)
Hf˜f˜ Hif˜j∗f˜k i j k
f=u,d
where
vg = (Γαf˜∗f˜) O Uf˜∗Uf˜ ,
Hif˜j∗f˜k βγ αi βj γk
with α = (φ ,φ ,a) = (1,2,3), β,γ = L,R, i = (H ,H ,H ) = (1,2,3) and j,k = 1,2.
1 2 1 2 3
The expressions for the couplings Γαf˜∗f˜ are shown in [56]. The stop and sbottom mass
matrices may conveniently be written in the (q˜ ,q˜ ) basis as
L R
 √ 
M2 +m2 +c M2(Tq −Q s2 ) h∗v (A∗ −µR )/ 2
M˜2q =  Q˜3 q 2β Z z √ q W q q q q  , (17)
h v (A −µ∗R )/ 2 M2 +m2 +c M2Q s2
q q q q R˜3 q 2β Z q W
with q = t,b, R = U,D, Tt = −Tb = 1/2, Q = 2/3, Q = −1/3, v = v , v = v , R =
z z t b b 1 t 2 b
tanβ = v /v , R = cotβ, and h is the Yukawa coupling of the quark q. On the other hand,
2 1 t q
the stau mass matrix is written in the (τ˜ ,τ˜ ) basis as
L R
 √ 
M2 +m2 +c M2(s2 −1/2) h∗v (A∗ −µtanβ)/ 2
M˜2τ =  L˜3 τ 2β Z W √ τ 1 τ  . (18)
h v (A −µ∗tanβ)/ 2 M2 +m2 +c M2s2
τ 1 τ E˜3 τ 2β Z W
The2×2sfermionmassmatrixM˜2 forf = t,bandτ isdiagonalizedbyaunitarymatrixUf˜:
f
Uf˜†M˜2Uf˜= diag(m2 ,m2 ) with m2 ≤ m2 . The mixing matrix Uf˜relates the electroweak
f f˜1 f˜2 f˜1 f˜2
˜ ˜
eigenstates f to the mass eigenstates f , via
L,R 1,2
(f˜ ,f˜ )T = Uf˜(f˜,f˜)T .
L R α αi 1 2 i
InteractionsbetweentheHiggsbosonsandthechargedHiggsbosoncanbefoundinRef.[40].
8
III. DATA, FITS, AND RESULTS
A. Data
Our previous works [3, 40, 54] were performed with data of the Summer 2013. Very
recently we have also updated the model-independent fits using the data of the Summer
2014 [4]. The whole set of Higgs strength data on H → γγ, ZZ∗ → 4(cid:96), WW∗ → (cid:96)ν(cid:96)ν, ττ,
¯
andbbarelistedinRef.[4]. Themostsignificantchangessincesummer2013aretheH → γγ
data from both ATLAS and CMS. The ATLAS Collaboration updated their best-measured
value from µ = 1.6 ± 0.4 to µ = 1.17 ± 0.27 [58], while the CMS H → γγ
ggH+ttH inclusive
data entertained a very dramatic change from µ = 0.78+0.28 to µ = 1.12+0.37 [59].
untagged −0.26 ggH −0.32
Other notable differences can be found in Ref. [4]. The χ2 /d.o.f. for the SM is now at
SM
16.76/29, which corresponds to a p-value of 0.966.
B. CP-Conserving (CPC) Fits
We consider the CP-conserving MSSM and use the most updated Higgs boson signal
strengths to constrain a minimal set of parameters under various conditions. Regarding the
i-th Higgs boson H as the candidate for the 125 GeV Higgs boson, the varying parameters
i
are:
• the up-type Yukawa coupling CS ≡ gS = O /s , see Eq. (1),
u Hiu¯u φ2i β
• the ratio of the VEVs of the two Higgs doublets tanβ ≡ v /v ,
2 1
• the parameter κ (assumed real) quantifying the modification between the down-type
d
quark mass and Yukawa coupling due to radiative corrections, as shown in Eq. (2),
• ∆Sγ ≡ ∆Sγ as in Eq. (8)
i
• ∆Sg ≡ ∆Sg as in Eq. (12), and
i
• the deviation in the total decay width of the observed Higgs boson: ∆Γ .
tot
The down-type and lepton-type Yukawa and the gauge-Higgs couplings are derived as
(cid:18) O +κ O (cid:19) 1
CS ≡ gS = φ1i d φ2i ,
d Hid¯d 1 + κ tanβ cosβ
d
9
O
CS ≡ gS = φ1i ,
(cid:96) Hi(cid:96)¯(cid:96) cosβ
C ≡ g = c O + s O (19)
v HiVV β φ1i β φ2i
with
(cid:113)
O = ± 1−s2(CS)2, O = CSs . (20)
φ1i β u φ2i u β
In place of tanβ we can use C as a varying parameter, and then tanβ (t ) would be
v β
determined by
(1−C2) (1−C2)
t2 = v = v . (21)
β (CS −C )2 [(CS −1)+(1−C )]2
u v u v
We note that t = ∞ when (CS − 1) = −(1 − C ) < 0 2 while t = 1 when (CS − 1) =
β u v β u
(cid:113)
± 1−C2 − (1 − C ). Therefore t changes from ∞ to 1 when (CS − 1) deviates from
v v β u
(cid:113)
−(1−C ) by the amount of ± 1−C2. This implies that the value of t becomes more and
v v β
more sensitive to the deviation of CS from 1 as C approaches to its SM value 1.
u v
We are going to perform the following three categories of CPC fits varying the stated
parameters while keeping the others at their SM values.
• CPC.II
– CPC.II.2: CS, tanβ (κ = ∆Γ = ∆Sγ = ∆Sg = 0 )
u d tot
– CPC.II.3: CS, tanβ, κ (∆Γ = ∆Sγ = ∆Sg = 0 )
u d tot
– CPC.II.4: CS, tanβ, κ , ∆Γ (∆Sγ = ∆Sg = 0 )
u d tot
• CPC.III
– CPC.III.3: CS, tanβ, ∆Sγ (κ = ∆Γ = ∆Sg = 0 )
u d tot
– CPC.III.4: CS, tanβ, ∆Sγ, κ (∆Γ = ∆Sg = 0 )
u d tot
– CPC.III.5: CS, tanβ, ∆Sγ, κ , ∆Γ (∆Sg = 0 )
u d tot
• CPC.IV
– CPC.IV.4: CS, tanβ, ∆Sγ, ∆Sg (κ = ∆Γ = 0 )
u d tot
– CPC.IV.5: CS, tanβ, ∆Sγ, ∆Sg, κ (∆Γ = 0 )
u d tot
2 Note C ≤1 and positive definite in our convention.
v
10