Right-Handed Quark Mixings in Minimal Left-Right Symmetric Model with
  General CP Violation
Right-Handed Quark Mixings in Minimal
Left-Right Symmetric Model with General CP Violation
Yue Zhang,1, 2 Haipeng An,2 Xiangdong Ji,2, 1 and R. N. Mohapatra2
1Center for High-Energy Physics and Institute of Theoretical Physics,
Peking University, Beijing 100871, China
2Department of Physics, University of Maryland,
College Park, Maryland 20742, USA
(Dated: October 31, 2018)
Abstract
We present a systematic approach to solve analytically for the right-handed quark mixings in
the minimal left-right symmetric model which generally has both explicit and spontaneous CP
violations. The leading-order result has the same hierarchical structure as the left-handed CKM
mixing, but with additional CP phases originating from a spontaneous CP-violating phase in
the Higgs vev. We explore the phenomenology entailed by the new right-handed mixing matrix,
particularly the bounds on the mass of WR and the CP phase of the Higgs vev.
http://arxiv.org/abs/0704.1662v1
The physics beyond the standard model (SM) has been the central focus of high-energy
phenomenology for more than three decades. Many proposals, including supersymmetry,
technicolors, little Higgs, and extra dimensions, have been made and studied thoroughly in
the literature; tests are soon to be made at the Large Hadron Collider (LHC). One of the
earliest proposals, the left-right symmetric (LR) model [1], was motivated by the hypothesis
that parity is a perfect symmetry at high-energy, and is broken spontaneously at low-energy
due to an asymmetric vacuum. Asymptotic restoration of parity has a definite aesthetic
appeal [2]. This model, based on the gauge group SU(2)L × SU(2)R × U(1)B−L, has a
number of additional attractive features, including a natural explanation of the weak hyper-
change in terms of baryon and lepton numbers [3], the existence of right-handed neutrinos,
and the possibility of spontaneous CP (charge-conjugation-parity) violation (SCPV) [4].
The model can easily be constrained by low-energy physics and predict clear signatures at
colliders [5]. It so far remains a decent possibility for new physics.
The LR modes are best constrained at low-energies by flavor-changing mixings and de-
cays, particularly the CP violating observables. In making theoretical predictions, the major
uncertainty comes from the unknown right-handed quark mixing matrix, conceptually simi-
lar to the left-handed quark Cabibbo-Kobayashi-Maskawa (CKM) mixing. The new mixing
generally depends on 9 real parameters: 6 CP violation phases and 3 rotational angles. Over
the years, two limiting cases of the model have usually been studied. The first case, “mani-
fest left-right symmetry”, assumes that there is no SCPV, i.e., all Higgs vacuum expectation
values (vev) are real. The quark mass matrices are then hermitian, and the left and right-
handed quark mixings become identical, modulo the sign uncertainty from possible negative
quark masses. The reality of the Higgs vev, however, does not survive radiative corrections
which generate infinite renormalization. The second case, “pseudo-manifest left-right sym-
metry”, assumes that the CP violation comes entirely from spontaneous symmetry breaking
(SSB) and that all Yukawa couplings are real [6]. Here the quark mass matrices are complex
but symmetric, the right-handed quark mixing is related to the complex conjugate of the
CKM matrix multiplied by additional CP phases. There are few studies of the model with
general CP violation in the literature [7], with the exception of an extensive numerical study
in Ref. [8] where solutions were generated through a Monte Carlo method.
In this paper, we report a systematic approach to solve analytically for the right-handed
quark mixings in the minimal LR model with general CP violation. As is well-known,
the model has a Higgs bi-doublet whose vev’s are complex, leading to both explicit and
spontaneous CP violations. The approach is based on the fact that mt ≫ mb and hence the
ratio of the two vev’s of the Higgs bi-doublet, ξ = κ′/κ, is small. In the leading-order in
ξ, we find a linear matrix equation for the right-handed quark mixing which can readily be
solved. We present an analytical solution of this equation valid to O(λ3), where λ = sin θC
is the Cabibbo mixing parameter. The leading-order solution is very close to the left-handed
CKM matrix, apart from additional phases that are fixed by ξ, the spontaneous CP phase
α, and the quark masses. This explicit right-handed quark mixing allows definitive studies
of the neutral meson mixing and CP-violating observables. We use the experimental data
on kaon and B-meson mixings and neutron electrical dipole moment (EMD) to constrain
the mass of WR and the SCPV phase α.
The matter content of the LR model is the same as the standard model (SM), except for a
right-handed neutrino for each family which, together with the right-handed charged lepton,
forms a SU(2)R doublet. The Higgs sector contains a bi-doublet φ, which transforms like
(2,2,0) of the gauge group, and the left and right triplets ∆L,R, which transform as (3, 1, 2)
and (1, 3, 2), respectively. The gauge group is broken spontaneously into the SM group
SU(2)L × U(1)Y at scale vR through the vev of ∆R. The breaking of the SM group is
accomplished through vev’s of φ.
The most general renormalizable Higgs potential can be found in Ref. [9]. Only one of
the parameters, α2, which describes an interaction between the bi-doublet and triplet Higgs,
is complex, and induces an explicit CP violation in the Higgs potential. It is known in the
literature that when this parameter is real, SCPV does not occur if the SM group is to be
recovered in the decoupling limit vR → ∞ [9]. Without SCPV, the Yukawa couplings in the
quark sector are hermitian, and we have the manifest left-right symmetry limit. Here we
are interested in the general case when α2 is complex. A complex α2 allows spontaneous CP
violation as well, generating a finite phase α for the vevs of φ,
〈φ〉 =
0 κ′eiα
. (1)
In reference [10], a relation was derived between α and the phase δ2 of α2,
α ∼ sin−1
2|α2| sin δ2
, (2)
where α3 is another interaction parameter between the Higgs bi-doublet and triplets.
The quark masses in the model are generated from the Yukawa coupling,
LY = q̄(hφ+ h̃φ̃)q + h.c. . (3)
Parity symmetry (φ → φ†, qL → qR) constrains h and h̃ be hermitian matrices. After SSB,
the above lagrangian yields the following quark mass matrices,
Mu = κh+ κ
′e−iαh̃
Md = κ
′eiαh+ κh̃ . (4)
Because of the non-zero α, both Mu and Md are non-hermitian. And therefore, the right-
handed quark mixing can in principle very different from that of the left-hand counter part.
Since the top quark mass is much larger than that of down quark, one may assume,
without loss of generality, κ′ ≪ κ, while at the same time h̃ is at most the same order as h.
We parameterize κ′/κ = rmb/mt, where r is a parameter of order unity. As a consequence,
Mu is nearly hermitian, and one may neglect the second term to leading order in ξ. One
can account for it systematically in ξ expansion if the precision of a calculation demands.
Now h can be diagonalized by a unitary matrix Uu,
Mu = UuM̂uSU
u = κh , (5)
where M̂u is diag(mu, mc, mt), and S is a diagonal sign matrix, diag(su, sc, st), satisfying
S2 = 1. Replacing the h-matrix in Md by the above expression, one finds
eiαξM̂u + κUuh̃U
uS = VLM̂dV
R (6)
where M̂d is diag(md, ms, mb), VL is the CKM matrix and VR is the right-handed mixing
matrix that we are after. Two comments are in order. First, through redefinitions of quark
fields, one can bring VL to the standard CKM form with four parameters (3 rotations and
1 CP violating phase) and the above equation remains the same. Second, all parameters in
the unitary matrix VR are now physical, including 3 rotations and 6 CP-violating phases.
To make further progress, one uses the hermiticity condition for Uuh̃U
u, which yields the
following equation,
M̂dV̂
R − V̂RM̂d = 2iξ sinα V
LM̂uSVL (7)
where V̂R is the quotient between the left and right mixing VR = SVLV̂R. There are a total
of 9 equations above, which are sufficient to solve 9 parameters in V̂R. It is interesting to
note that if there is no SCPV, α = 0, the solution is simply VR = SVLS̃, where S̃ is another
diagonal sign matrix, diag(sd, ss, sb), satisfying S̃
2 = 1. We recover the manifest left-right
symmetry case.
The above linear equation can be solved using various methods. The simplest is to utilize
the hierarchy between down-type-quark masses. Multiplying out the left-hand side and
assuming V̂Rij and V̂
Rij are of the same order, which can be justified posteriori, the solution
is (for ρ sinα ≤ 1)
ImV̂R11 = −r sinα
sc + st
A2λ4((1− ρ)2 + η2)
ImV̂R22 = −r sinα
sc + st
ImV̂R33 = −r sinα st (10)
V̂R12 = 2ir sinα
sc + st
λ4A2(1− ρ+ iη)
V̂R13 = −2ir sinαAλ
3(1− ρ+ iη)st (12)
V̂R23 = 2ir sinαAλ
2st , (13)
where ImV̂R11, ImV̂R22, ImV̂R33, V̂R12, V̂R23, V̂R13 are on the orders of λ, λ, 1, λ
2, λ2, and λ3,
respectively. The above solution allows us to construct entirely the right-handed mixing to
order O(λ3)
VR = PUV PD , (14)
where the factors PU =diag(su, sd exp(2iθ2), st exp(2iθ3)),
PD =diag(sd exp(iθ1), ss exp(−iθ2), sb exp(−iθ3)), and
1− λ2/2 λ Aλ3(ρ− iη)
−λ 1− λ2/2 Aλ2e−i2θ2
Aλ3(1− ρ− iη) −Aλ2e2iθ2 1
 ; (15)
with θi = s̃i sin
−1 ImV̂Rii. The pseudo-manifest limit is recovered when η = 0.
A few remarks about the above result are in order. First, the hierarchical structure of
the mixing is similar to that of the CKM, namely 1-2 mixing is of order λ, 1-3 order λ3 and
2-3 order λ2. Second, every element has a significant CP phase. The elements involving the
first two families have CP phases of order λ, and the phases involving the third family are
of order 1. These phases are all related to the single SCPV phase α, and can produce rich
phenomenology for K and B meson systems as well as the neutron EDM. Third, depending
on signs of the quark masses, there are 25 = 32 discrete solutions. Finally, using the right-
handed mixing at leading order in ξ, one can construct h̃ from Eq. (6) and solve Mu with a
better approximation. The iteration yields a systematic expansion in ξ.
In the remainder of this paper, we consider the kaon and B-meson mixing as well as
the neutron EMD. We will first study the contribution to the KL − KS mass difference
∆MK and derive an improved bound on the mass of right-handed gauge boson WR, using
the updated hadronic matrix elements and strange quark mass. Then we calculate the CP
violation parameter ǫ in KL decay and the neutron EDM, deriving an independent bound on
MWR. Finally, we consider the implications of the model in the B-meson system, deriving
yet another bound on MWR.
The leading non-SM contribution to the K0 −K0 mixing comes from the WL −WR box
diagram and the tree-level flavor-changing, neutral-Higgs (FCNH) diagram[11, 12]. The
latter contribution has the same sign as the former, and inversely proportional to the square
of the FCNH boson masses. We assume large Higgs boson masses (> 20TeV) from a large
α3 in the Higgs potential so that the contribution to the mixing is negligible. Henceforth we
concentrate on the box diagram only.
Because of the strong hierarchical structure in the left and right quark mixing, the WL−
WR box contribution to the kaon mixing comes mostly from the intermediate charm quark,
H12 =
4π sin2 θW
2ηλLRc λ
c (16)
× [4(1 + ln xc) + ln η]
(d̄s)2 − (d̄γ5s)
+ h.c.
where xc = m
, η = M2WL/M
, λRLc = V
RcdVLcs, and λ
c = V
LcdVRcs. The above
result is very similar to that from the manifest-symmetry limit because the phases in VRcd
and VRcs are O(λ). Therefore, we expect a similar bound on MWR as derived in previous
work [11]. However, the rapid progress in lattice QCD calculations warrants an update.
When the QCD radiative corrections are taken into account explicitly, the above effective
hamiltonian will be multiplied by an additional factor η4. [We neglect contributions of other
operators with small coefficients.] In the leading-logarithmic approximation, η4 is about 1.4
when the the four-quark operators are defined at the scale of 2 GeV in MS scheme [13].
The hadronic matrix element of the above operator can be calculated in lattice QCD and
expressed in terms of a factorized form
〈K0|d̄(1− γ5)sd̄(1 + γ5)s|K0〉
= 2MKf
KB4(µ)
ms(µ) +md(µ)
. (17)
Using the domain-wall fermion, one finds B4 = 0.81 at µ = 2 GeV in naive dimensional
regularization (NDR) scheme [14]. In the same scheme and scale, the strange quark mass
is ms = 98(6) MeV. Using the standard assumption that the new physics contribution shall
be less than the experimental value, one finds
MWR > 2.5 TeV , (18)
which is now the bound in the model with the general CP violation. This bound is stronger
than similar ones obtained before because of the new chiral-symmetric calculation of B4 and
the updated value of the strange quark mass.
The most interesting predictions of VR are for CP-violating observables. We first study
the CP violating parameter ǫ in KL decay. When the SCPV phase α = 0, the WL−WR box
diagram still makes a significant contribution to ǫ from the phase δCKM of the CKM matrix.
The experimental data then requires WR be at least 20 TeV to suppress this contribution.
When α 6= 0, it is possible to relax the constraint by cancelations. The most significant
contribution due to α comes from the element VRcd which is naturally on the order of λ. In
the presence of α, we have an approximate expression for ǫLR
ǫLR = 0.77
1 TeV
sssd Im
g(MR, θ2, θ3)e
−i(θ1+θ2)
where the function g(MR, θ2, θ3) = −2.22+[0.076+(0.030+0.013i) cos 2(θ2−θ3)] ln
80 GeV
The required value of r sinα for cancelation depends sensitively on the sign of quark masses.
When ss = sd, there exist small r sinα solutions even when MWR is as low as 1 TeV, as
shown by solid dots in Fig. 1. However, when ss = −sd, only large r sinα solutions are
possible.
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
r sin Α
FIG. 1: Constaints on the mass of WR and the spontaneous CP violating parameter r sinα from
ǫ (red dots) and neutron EDM in two different limit: small r ∼ 0.1 (green dots) and large r ∼ 1
(blue trinagles).
An intriguing feature appears if one considers the constraint from the neutron EMD as
well. A calculation of EDM is generally complicated because of strongly interacting quarks
inside the neutron. As an estimate, one can work in the quark models by first calculating
the EDM of the constituent quarks. In our model, there is a dominant contribution from
the WL − WR boson mixing [15]. Requiring the theoretical value be below the current
experimental bound, the neutron EDM prefers small r sinα solutions as can be seen in Fig.
1. The combined constraints from ǫ and the neutron EMD (dn < 3×10
−26 e cm [16]) impose
an independent bound on MWR
MWR > (2− 6) TeV , (20)
where the lowest bound is obtained for small r ∼ 0.05 and large CP phase α = π/2.
Finally we consider the neutral B-meson mixing and CP-violating decays. In Bd − Bd
and Bs − Bs mixing, due to the heavy b-quark mass, there is no chiral enhancement in
the hadronic matrix elements of ∆B = 2 operators from WL −WR box diagram as in the
kaon case. One generally expects the constraint from neutral B-meson mass difference to
be weak. In fact, we find a lower bound on WR mass of 1-2 TeV from B-mixing. On the
other hand, CP asymmetry in decay Bd → J/ψKS, SJ/ψKS = sin 2β in the standard model,
receives a new contribution from both Bd −Bd and K0 −K0 mixing in the presence of WR
[7] and is very sensitive to the relative sign sd and ss. By demanding the modified sin 2βeff
within the experimental error bar, we find another independent bound on MWR ,
MWR > 2.4 TeV , (21)
when sd = ss as required by the neutron EDM bound.
To summarize, we have derived analytically the right-handed quark mixing in the minimal
left-right symmetric model with general CP violation. Using this and the kaon and B-meson
mixing and the neutron EMD bound, we derive new bounds on the mass of right-handed
gauge boson, consistently above 2.5 TeV. To relax this constraint, one can consider models
with different Higgs structure and/or supersymetrize the theory. A more detailed account
of the present work, including direct CP observables, will be published elsewhere [17]
This work was partially supported by the U. S. Department of Energy via grant DE-
FG02-93ER-40762. Y. Z. acknowledges the hospitality and support from the TQHN group
at University of Maryland and a partial support from NSFC grants 10421503 and 10625521.
X. J. is supported partially by a grant from NSFC and a ChangJiang Scholarship at Peking
University. R. N. M. is supported by NSF grant No. PHY-0354407.
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