Spontaneous R-parity violation in the minimal gauged $(B-L)$ supersymmetry with a $125\;{\rm GeV}$ Higgs

0.89 MB·

by  Chao-Hsi Chang

Spontaneous R-parity violation in the minimal gauged (B L)
−
supersymmetry with a 125 GeV Higgs
Chao-Hsi Changa,b,c ,Tai-Fu Fengb,c , Yu-Li Yanb,c, Hai-Bin Zhangc,d, Shu-Min Zhaob,c
∗ † ‡
a CCAST (World Laboratory),P.O.Box 8730, Beijing 100190, China
4
1 b State Key Laboratory of Theoretical Physics (KLTP),
0
2 Institute of theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China

g
u c Department of Physics, Hebei University, Baoding, 071002, China
A
d Department of Physics, Dalian University of Technology, Dalian, 116024, China

6
1
Abstract

]
h
We precisely derive the mass squared matrices for charged and neutral (CP-odd and CP-even)
p
-
p Higgs, as well as the mass matrices for neutrino-neutralino and charged lepton-chargino in the
e
h minimal R-parity violating supersymmetry with local U(1) symmetry. In the framework the
B L
[ −

nonzeroTeVscalevacuumexpectationsofright-handedsneutrinosinducetheheavymassofneutral
2
v
6 U(1)B L gauge boson, and result in relatively large mixing between the lightest CP-even Higgs
8 −
5 and three generation right-handed sneutrinos when we include the one-loop corrections to the
4
. scalar potential. We numerically show that there is parameter space of the considered model to
1
0
4 accommodate experimental data on the newly ones of Higgs signal from LHC and experimental
1
: observations on the neutrino oscillation simultaneously.
v
i
X
r PACS numbers: 12.60.Jv,14.60.St, 14.80.Cp
a
Keywords: supersymmetry, Higgs, neutrino
∗ email:[email protected]
† email:[email protected]
‡ email:[email protected]
1
I. INTRODUCTION
A main destination of the Large Hadron Collider (LHC) is to understand the origin of
the electroweak symmetry breaking, and to study the properties of neutral Higgs predicted
by the Standard Model (SM) and its various extensions. In the year of 2012, ATLAS and
CMS reported significantly excess events in a few channels which are interpreted as the
neutral Higgs with mass m 124 126 GeV[1, 2], and CP properties and couplings of
h0 ∼ −
the particle are also being established[3–6] recently. It implies that the Higgs mechanism
to break electroweak symmetry has an experimental cornerstone now. Another important
progress of particle physics in the last year is that nonzero experimental observation on the
neutrino mixing angle θ is obtained with high precision[7], which opens several prospects
13
for neutrino physics. In this work, we investigate the constraints on parameter space of the
minimal R-parity violating supersymmetry with local U(1) symmetry from the updated
B L
−
experimental data mentioned above.
R-parity, as a discrete symmetry, is defined through R = ( 1)3(B L)+2S, where B, L and
−
−
S are baryon number, lepton number and spin respectively for a concerned field[8]. When
B L is violated by an even amount, R-parity conservation is guaranteed. However, break-
−
ing B L via nonzero vacuum expectation values (VEVs) of neutral scalar fields with odd
−
U(1) chargeswillinducetheR-parityviolationsimultaneously. Intheminimalsupersym-
B L
−
metric extension of SM (MSSM) with local U(1) symmetry, R-parity is spontaneously
B L
−
broken when left- and right-handed sneutrinos acquire nonzero VEVs[9–12]. Actually, both
spontaneously violated R-parity and broken local U(1) symmetry replicate the MSSM
B L
−
with conserving baryon number but violating lepton number. The authorsofRef.[13] further
propose an extension of the MSSM, which includes right-handed neutrino superfields and
two additional superfields Xˆ, Xˆ with even U(1) charges. When sneutrinos and scalar
′ B L
−
ˆ ˆ
components of X, X acquire non zero VEVs simultaneously, local U(1) symmetry and
′ B L
−
R-parity are broken spontaneously. To account for the neutrino oscillation experiment, tiny
neutrino masses are generated through an extended seesaw mechanism in the framework
proposed in Ref.[9–13]. Furthermore, the neutral Higgs fields H0, H0 mix with the scalar
u d
components of neutrino superfields and Xˆ, Xˆ superfields after the electroweak symmetry is
′
2
broken in those models. Assuming that the scalar components of Xˆ, Xˆ and neutral Higgs
′
fields H0, H0 acquire nonzero VEVs, Ref.[14] studies mass spectrum in the model proposed
u d
in Ref.[13].
Here we study the constraints from the observed Higgs signal and neutrino oscillation
experimental data on parameter space of the MSSM with local U(1) symmetry in the
B L
−
scenarios where sneutrinos obtain nonzero VEVs[9–12]. Since the tree level mixing between
the lightest CP-even Higgs and right-handed sneutrinos is suppressed by the tiny neutrino
masses, we include the one-loop corrections to the mixing which are mainly originated from
the third generation fermions and their supersymmetric partners. Numerically the MSSM
with local U(1) symmetry accommodates naturally the experimental data on the Higgs
B L
−
particle from ATLAS/CMS collaborations and the updated experimental observations on
the neutrino oscillation simultaneously. In addition, the model also predicts two sterile
neutrinos with sub-eV masses[15, 16], which are favored by the Big-bang nucleosynthesis
(BBN) in cosmology[17].
Certainly the deviation from unitarity of the leptonic mixing matrix intervening in
charged currents might induce a tree-level enhancement of R = Γ(P+ e+ν)/Γ(P+
P → →
µ+ν) (P+ = K+, π+) [18] because of additional mixings between the active neutrinos
and the sub-eV sterile states. Ignoring the difference between hadronic matrix elements in
P+ e+ν and that in P+ µ+ν, one finds that the experimental observations on R also
→ → P
constrain the parameter space of considered model. Furthermore, the experimental data on
Z invisible width[19] also constrain the mixings between the active neutrinos and the sub-eV
sterile ones. We will address the constraints on the mixings between the active neutrinos
and the sub-eV sterile ones from lepton flavor universality (LFU) and Z invisible width
elsewhere [20].
Our presentation is organized as follows. In section II, we briefly summarize the main
ingredients of the MSSM with local U(1) symmetry, then present the mass squared
B L
−
matricesforCP-oddandchargedHiggssectors, respectively. Weanalyzetheloopcorrections
on the mass squared matrix of CP-even Higgs in section III, and present the mass matrices
forneutrino-neutralinoandchargedlepton-charginoinsectionIVandsectionV,respectively.
Furthermore, we also present the decay widths for h0 γγ, VV , (V = Z, W) in section
∗
→
3
VI. The numerical analyses are given in section VII, and our conclusions are summarized in
section VIII.
II. THE MSSM WITH LOCAL U(1) SYMMETRY
B L
−
When U(1) is a local gauge symmetry, one can enlarge the local gauge group of
B L
−
the SM to SU(3) SU(2) U(1) U(1) . In the model proposed in Ref.[9–12], the
C ⊗ L ⊗ Y ⊗ (B−L)
exoticsuperfields arethreegenerationright-handedneutrinosNˆc (1, 1, 0, 1). Meanwhile,
i ∼
quantum numbers of the matter chiral superfields for quarks and leptons are given by
Uˆ 1 1 νˆ
Qˆ = I (3, 2, , ) , Lˆ = I (1, 2, 1, 1) ,
I  Dˆ  ∼ 3 3 I  Eˆ  ∼ − −
I I
   
  4 1 2 1
Uˆc (3, 1, , ) , Dˆc (3, 1, , ) , Eˆc (1, 1, 2, 1) , (1)
I ∼ −3 −3 I ∼ 3 −3 I ∼
with I = 1, 2, 3 denoting the index of generation. In addition, the quantum numbers of
two Higgs doublets are assigned as
Hˆ+ Hˆ0
Hˆ = u (1, 2, 1, 0) , Hˆ = d (1, 2, 1, 0) . (2)
u Hˆ0  ∼ d  Hˆ  ∼ −
−
 u   d 
   
The superpotential of the MSSM with local U(1) symmetry is written as
B L
−
= + (1) . (3)
W WMSSM W(B−L)
Here is superpotential of the MSSM, and
WMSSM
(1) = Y HˆTiσ Lˆ Nˆc . (4)
W(B−L) N IJ u 2 I J
(cid:16) (cid:17)
Correspondingly, the soft breaking terms for the MSSM with local U(1) symmetry are
B L
−
generally given as
= MSSM + (1) . (5)
Lsoft Lsoft Lsoft
Here MSSM is soft breaking terms of the MSSM, and
Lsoft
(1) = (m2 ) N˜c N˜c m λ λ +h.c. + A HTiσ L˜ N˜c +h.c. , (6)
Lsoft − N˜c IJ I∗ J − BL BL BL N IJ u 2 I J
(cid:16) (cid:17) n(cid:16) (cid:17) o
4
with λ denoting the gaugino of U(1) . After the SU(2) doublets H , H , L˜ and
BL B−L L u d I
SU(2) singlets N˜c acquire the nonzero VEVs,
L
I
H+
H = u ,
u  1 υ +H0 +iP 
 √2 u u u 
 (cid:16) (cid:17)
1 υ +H0 +iP
H = √2 d d d ,
d  (cid:16) (cid:17)
H
−
 d 
 
1 υ +ν˜ +iP
L˜ = √2 LI LI LI ,
I  (cid:16) ˜ (cid:17) 
L
−
 I 
1 
N˜c = υ +ν˜ +iP , (7)
I √2 NI RI NI
(cid:16) (cid:17)
the R-parity is broken spontaneously, and the local gauge symmetry SU(2) U(1)
L ⊗ Y ⊗
U(1) is broken down to the electromagnetic symmetry U(1) . Assuming that all pa-
(B−L) e
rameters are real, we obtain the minimization conditions at one-loop level in the model
considered here
T0 +∆T υ = 0 ,
u u u
T0 +∆T υ = 0 ,
d d d
T0 +∆T υ = 0 ,
L˜I L˜ LI
T0 +∆T υ = 0 , (8)
N˜I N˜ NI
where T0, T0, T0 , T0 denote the tree level tadpole conditions, and ∆T , ∆T , ∆T as
u d L˜I N˜I u d L˜
well as ∆T are the one-loop radiative corrections to the minimization conditions from top,
N˜
bottom, tau and their supersymmetric partners respectively, their concrete expressions are
given in the appendix.B. After the local gauge group SU(2) U(1) U(1) is broken
L ⊗ Y ⊗ (B−L)
down to the electromagnetic symmetry U(1) , the masses of neutral and charged gauge
e
bosons are respectively formulated as
1
m2 = (g2 +g2)υ2 ,
Z 4 1 2 EW
1
m2 = g2υ2 ,
W 4 2 EW
m2 = g2 υ2 +υ2 υ2 . (9)
ZBL BL N EW − SM
(cid:16) (cid:17)
5
3 3
Where υ2 = υ2 + υ2, υ2 = υ2 + υ2 + υ2 , υ2 = υ2 , and g , g , g denote the
SM u d EW u d α=1 Lα N α=1 Nα 2 1 BL
gauge couplings of SU(2) , U(1) and UP(1) , respecPtively.
L Y (B−L)
To satisfy present electroweak precision observations we assume the mass of neutral
U(1) gauge boson m > 1 TeV which implies υ > 1 TeV when g < 1, then
(B−L) ZBL N BL
we derive max((Y ) ) 10 6 and max(υ ) 10 3 GeV[12] to explain experimental data
N ij ≤ − LI ≤ −
on neutrino oscillation. Ignoring the small terms and assuming that the 3 3 matrices
×
m2, m2 are real, we simplify the minimization conditions in Eq.(8) as
L˜ N˜c
g2 +g2
υ µ2 +m2 + 1 2 2υ2 υ2 +∆T +Bµυ 0 ,
u Hu 8 u − EW u d ≃
n (cid:16) (cid:17) o
g2 +g2
υ µ2 +m2 1 2 2υ2 υ2 +∆T +Bµυ 0 ,
d Hd − 8 u − EW d u ≃
n (cid:16) (cid:17) o
3 υ 3 µυ
m2 +∆T δ υ + u A υ + dζ
L˜ Iα L˜ Iα Lα √2 N Iα Nα √2 I
αX=1h(cid:16) (cid:17) i αX=1(cid:16) (cid:17)
g2 +g2 m2
υ 1 2 2υ2 υ2 + ZBL 0 ,
− LI 8 u − EW 2 ≃
n (cid:16) (cid:17) o
3 m2
m2 +∆T δ υ + ZBLυ 0 , (10)
N˜c Iα N˜ Iα Nα 2 NI ≃
αX=1h(cid:16) (cid:17) i
3
with ζ = Y υ . Note here that the first two minimization conditions respectively
I α=1 N Iα Nα
for H0, H0Par(cid:16)e no(cid:17)t greatly modified from that in the MSSM, the third condition keeps the
u d
linear terms of υ or Y , and the last equation implies that the vector (υ , υ , υ ) is an
LI N N1 N2 N3
eigenvector of 3 3 mass squared matrix m2 with eigenvalue m2 /2 ∆T . A possible
× N˜c − ZBL − N˜
symmetric 3 3 matrix satisfying the last equation in Eq.(10) is written as
×
υ
Λ2 Λ2 , 0 , N1Λ2
N˜1c − BL −υN3 N˜1c
 υ 
m2 0 , Λ2 Λ2 , N2Λ2 (11)
N˜c ≃  N˜2c − BL υ2 Λ2 −+υυ2N3Λ2N˜2c 
 υN1Λ2 , υN2Λ2 , N1 N˜1c N2 N˜2c Λ2 
 −υN3 N˜1c −υN3 N˜2c υN23 − BL 
 
with Λ2 = m2 /2 + ∆T . In order to make our final results transparently, we further
BL ZBL N˜
assume in our following discussion
m2 m2 δ , (I, J = 1, 2, 3) , (12)
IJ
L˜ IJ ≃ L˜I
(cid:16) (cid:17)
6
then we obtain
3
4√2 υ A υ +µυ ζ
υ h u α=1(cid:16) N(cid:17)Iα Nα d Ii . (13)
LI ≃ −8(m2 +∆T ) P(g2+g2) υ2 υ2 4m2
L˜I L˜ − 1 2 u − d − ZBL
(cid:16) (cid:17)
As m 1 TeV, the condition max(υ ) 10 2 GeV requires A 0.01 GeV. This
−
L˜I ∼ LI ≤ N ∼
implies that tree level contributions to the mixing between the lightest CP-even Higgs and
right-handed sneutrinos can be ignored, leading contributions to the mixing are mainly
originated from one-loop radiative corrections.
A. The mass squared matrix for charged Higgs
Using those minimization conditions, we derive the 8 8 mass squared matrix for
×
charged Higgs
2 A
MCH 2 2 CH 2 6 , (14)
 h i × h i × 
AT M2
 CH 6 2 E˜ 6 6 
 h i × h i × 
in the interaction eigenstates HT = (H , H , L˜ , E˜c ), (I, J = 1, 2, 3). Here, elements
− − − ∗
CH u d I J
of the 2 2 matrix 2 are given as
× MCH
υ g2 1 3
2 = Bµ+∆ d 2 υ2 υ2 + υ A υ
MCH 11 odd υ − 4 EW − u √2υ Lα N αβ Nβ
h i (cid:16) (cid:17) u (cid:16) (cid:17) u Xα,β (cid:16) (cid:17)
1 3
+ υ Y Y υ ,
†
2 Lα N N αβ Lβ
Xα,β (cid:16) (cid:17)
g2
2 = Bµ+∆ 2υ υ ,
MCH 12 odd − 4 u d
h i (cid:16) (cid:17)
υ g2 µε2
2 = Bµ+∆ u + 2 υ2 υ2 υ2 + N
odd
MCH 22 υ 4 EW − SM − u √2υ
h i (cid:16) (cid:17) d (cid:16) (cid:17) d
1 3
υ Y YT υ , (15)
−2 Lα E E αβ Lβ
αX,β=1 (cid:16) (cid:17)
3
with ε2 = υ Y υ . Additionally the 3 3 matrix Y is Yukawa couplings in
N α,β=1 Lα N αβ Nβ × E
(cid:16) (cid:17)
charged leptoPn sector, and the one-loop radiative correction is written as
3g2 m2A µf(m2 ) f(m2 )
∆ = 2 t t t˜1 − t˜2
odd 32π2sin2β m2 m2 m2
W t˜1 − t˜2
7
3g2 m2A µf(m2 ) f(m2 )
+ 2 b b ˜b1 − ˜b2
32π2cos2β m2 m2 m2
W ˜b1 − ˜b2
g2 m2A µf(m2 ) f(m2 )
+ 2 τ τ τ˜1 − τ˜2 . (16)
32π2cos2β m2 m2 m2
W τ˜1 − τ˜2
Here m2 , m2 and m2 are the eigenvalues of the t˜, ˜b and τ˜ mass-squared matrices, the
t˜1,2 ˜b1,2 τ˜1,2
formfactorf(m2) = m2(ln(m2/Λ2) 1)withΛdenotingrenormalizationscale. Additionally,
−
the concrete expressions for the symmetric matrix M2 and A can be found in appendix.B.
E˜ CH
Actually, the symmetric matrix in Eq.(14) contains an eigenvector with zero eigenvalue
υ υ 3 υ
G = u H d H Lα L˜ , (17)
± ± ± ±
υ u − υ d − υ α
EW EW αX=1 EW
which corresponds to the charged Goldstone eaten by charged gauge boson as electroweak
symmetry broken spontaneously. Applying the 8 8 orthogonal matrix
×
(0) = Z(0) 1 , (18)
ZCH CH 3×3
M
we separate the charged Goldstone boson from the physical states:
2 A 0 0
(0)T MCH 2 2 CH 2 6 (0) = 1×7 . (19)
ZCH ·hAT i × hM2 i × ·ZCH 0 M2 
 CH 6 2 E˜ 6 6   7×1 H± 
 h i × h i ×   
Where the 5 5 orthogonal matrix Z(0) is given as
× CH
υu , υd , υuυLK
υ υ υ υ
Z(0) =  EυWd , υSuM, (cid:16) SMυdυELWK(cid:17)1×3  . (20)
CH  (cid:16)−−υυυELEWIW(cid:17)3×1, 0υ3S×M1, (cid:16)υυESWM δI(cid:16)K−+υSαM=3υ1EεWIK(cid:17)α1×υυ3ELWα (cid:17)3×3 
 P 
Finally, we give the 8 8 mixing matrix Z in charged Higgs sector as
× CH
1 0
1 7
Z = (0) × (21)
CH ZCH · 0 Z 
 7×1 H± 7 7 
 h i × 
with Z M2 Z = diag(m2 , ,m2 ).
H†± · H± · H± H± ··· H±
2 8
8
B. The mass squared matrix for CP-odd Higgs
In the interaction basis P0,T = (P0, P0, P0 , P0 ), (I, J = 1, 2, 3), the 8 8 mass
u d L˜I N˜J ×
matrix for neutral CP-odd scalars is
2 A(0)
MCPO 2 2 CPO 2 6 , (22)
h i × h i × 
A(0)T M2
 CPO 6 2 P 6 6 
 h i × h i × 
the elements of 2 2 mass squared matrix are
×
υ 1 3
2 = Bµ+∆ d + υ A υ ,
MCPO 11 odd υ √2υ Lα N αβ Nβ
h i (cid:16) (cid:17) u u Xα,β (cid:16) (cid:17)
2 = Bµ+∆ ,
odd
MCPO 12
h i υ µε2
2 = Bµ+∆ u + N . (23)
odd
MCPO 22 υ √2υ
h i (cid:16) (cid:17) d d
As we assign the VEVs of left-handed sneutrinos to zero, the expressions in Eq.(23) recover
the elements of mass-squared matrix for CP-odd Higgs in the MSSM. Additionally, the
concrete expressions for elements of the matrix A(0) can be found in appendix.B. Similarly,
CPO
the symmetric matrix in Eq.(22) contains two massless eigenstates which correspond to
the neutral Goldstones swallowed by neutral gauge bosons Z, Z after the symmetry
BL
SU(2) U (1) U is broken down to the electromagnetic symmetry U (1):
× Y × (B−L) e
υ υ 3 υ
G0 = u P0 d P0 Lα P0 ,
υ u − υ d − υ L˜α
EW EW αX=1 EW
υ υ 3 υ 3 υ
G0 = η uP0 η dP0 +(1 η) LαP0 NαP0 , (24)
(B−L) υt u − υt d − α=1 υt L˜α −α=1 υt N˜α
X X
υ2
with η = 1 SM , and υ2 = υ2 +ηυ2 . To separate neutral Goldstones from physical states,
−υ2 t N SM
EW
we define the 8 8 orthogonal matrix
×
(0) = Z(0) P
ZP CH ZN˜c
n M o υSMυL1, υN, υSMυL3, υSMυL2
− υEWυt υt υEWυt − υEWυt
×12×2M −−υυυυSSυυEEMMNtWWυυ,υυLLtt32,, −υυυυυSSEEυMMSEWWMυυWυυLLυttυL21t3,,, −υυυSEυMSυυEWMNtυWυυL,tυL2t1,, υυυυSSEEMMυυWWNtυυυυLLtt31 M12×2
9
1, 0, 0
  
0, 0, 1 1 , (25)
×0, 1, 0M 5×5
 
then we have   
2 A 0 0 ,
(0)T MCPO 2 2 CPO 2 6 (0) = 2×2 2×6 .
ZP ·hAT i × h M2 i × ·ZP  0 M2 
 CPO 6 2 P 6 6   6×2 P0 6 6 
 h i × h i ×   h i × 
Finally, the 8 8 mixing matrix Z in CP-odd Higgs sector is written as
× A0
1 0
2 2 2 6
Z = (0) × × (26)
A0 ZP · 0 Z 
 6×2 P 6 6 
 (cid:16) (cid:17) × 
with Z M2 Z = diag(m2 , ,m2 ).
†
P · P0 · P A0 ··· A0
3 8
Where
ω ω ω +ω
P,Tm2 P = diag(0, A − B, A B) , (27)
ZN˜c N˜cZN˜c 2υ2 2υ2
N3 N3
and the concrete expressions for ω are
A,B
ω = Λ2 υ2 υ2 +Λ2 υ2 υ2 ,
A N˜1c N − N2 N˜2c N − N1
(cid:16) (cid:17) (cid:16) (cid:17)
ω2 = ω2 4Λ2 Λ2 υ2υ2 . (28)
B A − N˜1c N˜2c N N3
Additionally the orthogonal 3 3 rotation is written as
×
P υ
ZN˜c 11 1 N1
(cid:16) P (cid:17)  = υ  ,
(cid:16)ZNP˜c(cid:17)21  υN υN2 
   
 ZN˜c 31   N3 
(cid:16) (cid:17)   
P x
ZN˜c 12 1 −
(cid:16) P (cid:17)  =  y  ,
 ZN˜c 22  x 2 + y 2+ z 2  
(cid:16) P (cid:17)  | −| | | | −|  z 
  q  
 ZN˜c 32   − 
(cid:16) (cid:17)   
P x
+
ZN˜c 13 1
(cid:16) P (cid:17)  =  y  , (29)
(cid:16)ZNP˜c(cid:17)23  |x+|2 +|y|2+|z+|2  z 
  q  + 
 ZN˜c 33   
(cid:16) (cid:17)   
10