ϵ′/ϵ anomaly and neutron EDM in SU(2) L × SU(2) R × U(1)B−L model with charge symmetry
JHE
symmetry
Naoyuki Haba 0
Hiroyuki Umeeda 0
Toshifumi Yamada 0
0 Graduate School of Science and Engineering, Shimane University
The Standard Model prediction for 0= based on recent lattice QCD results exhibits a tension with the experimental data. We solve this tension through WR+ gauge boson exchange in the SU(2)L
Beyond Standard Model; CP violation

SU(2)L
HJEP05(218)
SU(2)R
U(1)B
L
SU(2)R
U(1)B L model with `charge symmetry', whose
theoretical motivation is to attribute the chiral structure of the Standard Model to the
spontaneous breaking of SU(2)R
U(1)B L gauge group and charge symmetry. We show
that MWR < 58 TeV is required to account for the 0= anomaly in this model. Next, we
make a prediction for the neutron EDM in the same model and study a correlation between
0= and the neutron EDM. We con rm that the model can solve the 0= anomaly without
con icting the current bound on the neutron EDM, and further reveal that almost all
parameter regions in which the 0= anomaly is explained will be covered by future neutron
EDM searches, which leads us to anticipate the discovery of the neutron EDM.
1 Introduction
2
3
0=
3.1
3.2
3.3
4.1
4.2
Wilson coe cients for S = 1 operators Hadronic matrix elements Numerical analysis of 0=
Wilson coe cients for operators contributing to the neutron EDM
Hadronic matrix elements 4.2.1 4.2.2
calculation of the hadronic matrix elements with lattice QCD [1, 3, 4] enables us to evaluate
the K !
decay amplitude without relying on any hadron model. On the basis of the
above calculation, the same collaboration has reported that the SM prediction is separated
from the experimental value [5{7] by 2.1 , and other groups [9, 10] have also obtained
predictions for 0= that show a discrepancy of 2:9 and 2:8 , respectively. More importantly,
the lattice result corroborates the calculation with dual QCD approach [11, 12], which
has derived a theoretical upper bound on 0= that is violated by the experimental data
and has thus claimed anomaly in this observable. (However, ref. [13] presents a di erent
calculation that claims the absence of the anomaly.) Some authors have tackled this 0=
anomaly in new physics scenarios, such as a general righthanded current [14], the Littlest
{ 1 {
Higgs model with Tparity [15], supersymmetry [16{18], nonstandard interaction with Z0
and/or Z [19, 20], vectorlike quarks [21] and SU(3)c
SU(3)L
U(1)X gauge group [22].
The SU(2)L
U(1)B L gauge extension of the SM is a wellmotivated
framework for addressing the 0= puzzle, because the
avor mixing matrix for righthanded
quarks automatically introduces new CPviolating phases, and WR+ gauge boson exchange
F = 1 processes at tree level while it contributes to
F = 2 processes at
loop levels so that other experimental constraints, in particular the constraint from Re( ),
are readily evaded. Previously, ref. [14] has shown that a general SU(2)L
ancy. However, a major theoretical motivation for the SU(2)L
by adding either the leftright parity [24] or the `charge symmetry' [27].1 The leftright
parity requires invariance of the theory under the Lorentzian parity transformation plus
the exchange of SU(2)L and SU(2)R gauge groups, while the charge symmetry requires
invariance under the charge conjugation plus the exchange of SU(2)L and SU(2)R, both of
which endow the model with a symmetric structure for the left and righthanded fermions
at high energies.
In this paper, we study 0= in the SU(2)L
symmetric matrices, which restricts the quark mixing matrix associated with WR+ to be the
complex conjugate of the SM CabibboKobayashiMaskawa (CKM) matrix multiplied by
a new CP phase factor for each quark avor. Given the above restriction, one can evaluate
0= only in terms of two new CP phases, the mass of WR+ and the ratio of two vacuum
expectation values (VEVs) of the bifundamental scalar, which leads to a speci c prediction
for the model parameters.
Our analysis on 0= proceeds as follows. By integrating out WR, WL and the top quark,
we obtain the Wilson coe cients for
S = 1 operators that contribute to K !
decay.
The anomalous dimension matrix is divided into the same two 18
18 pieces for 36
operators, for which leading order expressions are obtainable from refs. [30, 31]. The hadronic
matrix elements for currentcurrent operators are seized from the lattice results [3, 4]. We
nd that among new physics operators, (su)L(ud)R and (su)R(ud)R (each with two ways of
color contraction) both dominantly contribute to 0= . Their contributions are of the same
order because the Wilson coe cients of the (su)L(ud)R operators are suppressed by the
hierarchy of two bifundamental scalar VEVs v1=v2 = tan , which is about mb=mt if there
is no netuning in accommodating the top and bottom quark Yukawa couplings, whereas
this suppression does not enter into the Wilson coe cients of the (su)R(ud)R operators.
On the other hand, the lattice computation has con rmed that the hadronic matrix
elements for the former operators are enhanced compared to the latter. Thus, these operators
possibly equally contribute to 0= . This result is in contrast to the study of ref. [14], which
has concentrated solely on the (su)L(ud)R operators.
1The `charge symmetry' is inspired by Dparity [28] in the SO(10) grand uni cation theory. However,
the model we consider cannot be embedded in the SO(10) theory, since we assume the charge symmetry
breaking scale to be below O(100) TeV.
{ 2 {
Once the 0= anomaly is explained in the SU(2)L
charge symmetry, correlated predictions for other CP violating observables are of interest.
In particular, the neutron electric dipole moment (EDM), an observable sensitive to CP
violation in the presence of CPT invariance, receives signi cant contributions from
fourquark operators in SU(2)L
U(1)B L models [14, 34{38],2 allowing us to discuss
future detectability of the neutron EDM in relation to the 0= anomaly.
Our analysis on the neutron EDM starts by integrating out WR, WL and top quark to
obtain the Wilson coe cients for CPviolating operators. The leading order expression for
the anomalous dimension matrix is found in refs. [41, 42, 44, 45]. Regarding the hadronic
matrix elements of CPviolating operators, we reveal that the pion VEV h
by fourquark operators [33] gives the leading contribution to the neutron EDM, which
0i induced
is enhanced by the quark mass ratio ms=(mu + md) in comparison to the rest.
This
enhancement is understood as follows: since WR+WL+ mixing gives rise to CPodd and
isospinodd interactions, the pion VEV h
0i, which is isospinodd, can arise without the
mu, and thus can be directly proportional to 1=(mu + md). The pion
VEV induces a CPviolating coupling for neutron n,
baryon, and kaon K+ without
0= is shown at the end of the section. In section 4, we give the Wilson coe cients for
CPviolating operators contributing to the neutron EDM. Special care is taken in evaluating
meson condensates and their impact on the neutron EDM. The nal result is a prediction
for the neutron EDM in light of the 0= anomaly. Section 5 is devoted to summary and
discussions.
2
U(1)B L model with charge symmetry
We consider SU(3)C SU(2)L SU(2)R U(1)B L gauge theory with charge symmetry.
QiL
QcRi
L
i
L
LcRi
L
R
with La and Ra being gauge parameters for SU(2)L and SU(2)R, respectively. We demand
the theory to be invariant under the following `charge symmetry' transformation:
charge conjugation of all gauge elds;
and SU(2)L $ SU(2)R;
QL $ QcRi;
i
LL $ LcRi;
i
$
T
;
L $
R:
The part of the Lagrangian describing SU(2)L
gauge groups, respectively, and Yq and Y~q are the quark Yukawa couplings.
s denotes
the antisymmetric tensor for Lorentz spinors and g denotes that for the fundamental
representation of SU(2)L or SU(2)R. Invariance under the charge symmetry transformation
eq. (2.2) leads to the following treelevel relations:
gL = gR; (Yq)ij = (Yq)ji; (Y~q)ij = (Y~q)ji:
The SU(2)R triplet scalar
R develops a VEV, vR, to break SUR(2)
U(1)B L !
U(1)Y , and the bifundamental scalar
takes a VEV con guration,
h i = p
1
2
v sin
0
0
v cos e
i
!
;
to break SU(2)L
U(1)Y ! U(1)em, where
is the spontaneous CP phase. The VEV of
L is hereafter neglected, as it is severely constrained from
parameter. The resultant
mass matrices for WLa, WRa and X gauge bosons read
gLgR sin(2 )e i v2=4!
gR2(vR2 + v2=4)
WL+!
WR+
gLgR v2=2
2gLgX v
2
R
0
1 0WL31
2gX2 v
2
R
2gLgX vR2CA B@WR3 AC :
X
(2.2)
QcRi
(2.3)
(2.4)
(2.5)
(2.6)
{ 4 {
The mass matrix for the charged gauge bosons is diagonalized as
v and gL = gR, we have an important relation for ,
which indicates that when we assume tan
couplings are naturally derived, the WLWR mixing angle
the factor 2mb=mt
0:05.
' mb=mt so that the top and bottom Yukawa
is smaller than M W2 =M W2 0 by
The quark mass matrices are given by3
{ 5 {
(2.7)
(2.8)
(2.10)
(2.11)
which we diagonalize as Mu = VuyLdiag(mu; mc; mt)VuR and Md = VdyLdiag(md; ms; mb)VdR,
with VuL; VuR; VdL; VdR being unitary matrices. However, since Yq and Y~q are complex
symmetric matrices, so are Mu and Md, and one can most generally write
0ei u
0
0
0 ei b
Hence, the SM CKM matrix, VL = VuLVdyL, and the corresponding avor mixing matrix
for righthanded quarks, VR = VuRVdyR, are related as
3U Ri
s(U Rci) , DRi
s(DRci) .
0ei u
0ei u
0
0
0 ei t
0
0
0
e i d
0
0
e i s
0
0
e i s
0
0
0
0
e i b
0
0
e i b
1
C :
A
1
C
A
1
2
2
1
2
= p U i W +
+ p U i W 0+
Eventually, the part of the Lagrangian eq. (2.3) describing avorchanging W; W 0
interactions is recast, in the unitary gauge, into the form,
In this paper, we adopt the following convention for the quark phases and
u; c; t;
rst, we rede ne the phases of ve quarks to render the CKM matrix in the
0
s12s23
c12c13
c12s23s13ei
c12c23s13e
i
s12c13
c12c23
c12s23
s12s23s13ei
s12c23s13ei
Next, we rede ne c; t; d; s; b to set
c; t; d; s; b, and .
where operators O's are de ned in appendix A. We determine the Wilson coe cients as
follows: we approximate gR = gL by ignoring di erence in RG evolutions of gL and gR at
scales below MW 0. Also, for each Wilson coe cient, if multiple terms have an identical
phase, we only consider the one in the leading order of M W2 =M W2 0 or sin . By integrating
u = 0:
(2.13)
(2.14)
(3.1)
X (VL)is(VL )id ei( s d) 1
2 F1(yi); C30 = C50 =
13 C40; (3.4)
X (VL)is(VL )id ei( s d) 2
3 E1d(yi);
C2 = (VL )us(VL)ud cos2 ;
C2c = (VL )cs(VL)cd cos2 ;
out W 0, one obtains the following leadingorder matching conditions at a scale
(note our convention with u = 0):
(3.2)
(3.3)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
By further integrating out W and the top quark, one gains the following leadingorder
matching conditions at a scale
MW (note our convention with u = 0):
C2RL = (VL)us(VL)ud ei( s ) sin cos ;
C2LR = (VL )us(VL )ud ei( d+ ) sin cos ;
C2RcL = (VL)cs(VL)cd ei( c+ s ) sin cos ; C2LcR = (VL )cs(VL )cd ei( c d+ ) sin cos ;
C4 = C6 =
C7 = C9 =
Cg =
X (VL )is(VL)id 2 F1(xi);
X (VL )is(VL)id 3 E1d(xi);
1
2
cos2 (VL )is(VL)id F2(xi) + sin cos
cos2 (VL )is(VL)id E2d(xi) + sin cos
mi (VL)is(VL)id ei( i+ s )E3d(xi) ;
C3 = C5 =
1
mi (VL)is(VL)id ei( i+ s )F3(xi) ;
i=u;c;t
C0 = X
md (VL )is(VL)id F2(xi)+sin cos
mi (VL )is(VL )id ei( i d+ )F3(xi) ;
cos2 md (VL )is(VL)id E2d(xi)+sin cos
mi (VL )is(VL )id ei( i d+ )E3d(xi) ;
ms
ms
where loop functions F1; F2; F3 and E1d; E2d; E3d are de ned in appendix B. We are aware
that the dipole operators receive two contributions with di erent phases when W 0 is
integrated out and when W is. The two are expressed as Cg; C ; Cg0; C0 and
Cg; C ;
Cg0; C0 , respectively.
(fCjRLg) and (fCjLRg), we assume that their initial conditions at scale
by eqs. (3.10){(3.15) and solve the RG equations from
We take into account RG evolutions of the Wilson coe cients at order O( s). The
fact that four sets of operators, (fOig), (fOi0g) (i = 1; 2; : : : ; 10; 1c; 2c), (fORLg), (fOjLRg)
j
(j = 1; 2; 1c; 2c), do not mix with each other facilitates the computation. For (fCig),
= MW to the scale for which
= MW are given
ref. [31], and those for (fCiRLg) and (Cg; C ) are in ref. [30].
the lattice results are reported. For (fCi0g), we assume that their initial conditions at
= MW 0 are provided by eqs. (3.2){(3.9) and solve the RG equations from
= MW 0 to
the scale of lattice results. Finally, we compute RG evolutions of the coe cients of the
dipole operators (Cg; C ) and (Cg0; C0 ), which receive contributions from (fCi; CjRLg) and
(fCi0; CjLRg), respectively. The O( s) RG equations for (fCig) and (fCiRLg) are found in
3.2
Hadronic matrix elements
We employ the lattice calculations of hadronic matrix elements h(
1; 2; : : : ; 10 for I = 0; 2 reported by RBC/UKQCD in refs. [3, 4].
)I jOijK0i for i =
Since lattice calculations for the matrix elements of O1LR and O2LR are missing, we
estimate them from the RBC/UKQCD results using isospin symmetry. In the limit of
exact isospin symmetry, we nd, for
I = 3=2 amplitudes,
h
(
)I=2jO7jK0i = h(
)I=2j(sd)L
uu
1
2
dd
1
2
ss
R
jK0i
=
=
=
3
4
3
4
3
2
h
h
h
(
(
(
)I=2j(sd)L(uu
dd)RjK0i
)I=2jp2(su)Lp2(ud)RjK0i
)I=2jO2LRjK0i;
where we have discarded
I = 1=2 part when obtaining the second line, and when deriving
the third line, we have inserted ClebschGordan coe cients for constructing the
I = 3=2
{ 8 {
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
= h(
p
= ( 3
4
3
)I=0j
2
3
2
where in the rst line, we have separated (uu)R into
second line, we have inserted ClebschGordan coe cients for constructing the
hold between the matrix elements for O1LR and O8; O6.
The hadronic matrix elements for the chromodipole operators Og; Og0 are extracted
from the calculation based on dual QCD approach [46]. Note that the above calculation is
corroborated by the fact that it is consistent with a lattice calculation of the K hadronic
matrix element [47], which is related to the
Kone by chiral perturbation theory.
3.3
Numerical analysis of 0=
The de nition for the decay amplitudes of K0
!
is
A0ei 0 = h(
)I=0j H S=1 jK0i ;
A2ei 2 = h(
)I=2j H S=1 jK0i ;
where 0;2 represent the strong phases. In terms of the above amplitudes, one writes the
direct CP violation parameter divided by the indirect one as
0
Re
= Re
i!ei( 2 0) !
p
2
ImA2
ReA2
ImA0
ReA0
;
where ! = ReA2=ReA0 is a suppression factor due to the
I = 1=2 rule. For the strong
phases, we use the values of refs. [3, 4], 2 = 23:8
5:0 degree and 0 =
11:6
2:8 degree.
For the real parts of the decay amplitudes, we employ the experimental data [8], ReA2 =
1:479
10 8 GeV and ReA0 = 33:20
10 8 GeV, which leads to ! = 4:454
10 2. In our
analysis, we separate the SM and new physics contributions as
{ 9 {
(3.23)
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
Re
NP
0
= Re
0
SM
+ Re
0
NP
:
For the SM part, we quote the calculation in the literature Re( 0= )SM = (1:38
6:90)
10 4 [4]. It is the new physics part,
Re
0
= Re
i!ei( 2 0) !
p
2
ImA2NP
ReA2
ImA0NP
ReA0
;
that we compute in this paper. In doing so, we approximate cos2
= 1 in the Wilson
coe cients eqs. (3.10){(3.19), so that the SM contribution is separated from the new physics
one at the operator level.
4
] 40
0
1
20
10
40
4
] 20
0
40
αψd=π/3, αψs=π/4
αψd=π/2, αψs=π/3
αψd=π, αψs=π/4
20
30
40
50
60
data given by PDG [8], while model predictions with speci c choice of phases are shown by the lines.
20
40
60
PDG [8], while each red dot corresponds to the model prediction with a randomly generated set of
(
d,
s).
In the analysis, we x the ratio of the bifundamental scalar VEVs at its natural value
as tan
= mb=mt.
We have found numerically that for MW 0 > 1 TeV, the
chromodipole contribution to Re( 0= ) does not exceed O(10 4) and is hence safely neglected.4
Consequently, only two combinations of new CP phases,
mass determine the new physics contribution.
d and
s, and the W 0
First, we choose speci c values for the new CP phases in the calculation of 0= to
illustrate the model prediction. In gure 1, the prediction for Re( 0= ) is presented with
speci c choices of
d and
Next, we randomly vary
s
.
d and
parameters. In
gure 2, we show the region of Re( 0= ) obtained by varying
d and
s. One observes that MW 0 < 58 TeV is necessary for 1 explanation of the anomaly.
4When we use a calculation based on the chiral quark model in ref. [48] to evaluate the hadronic matrix
elements of the chromodipole operators, we are again lead to the result that the chromodipole contribution
to Re( 0= ) is below O(10 4) for MW 0 > 1 TeV.
s in the range [0; 2 ], since they are free
We have con rmed that among the terms of S = 1 Hamiltonian eq. (3.1), Pi=1;2 Ci0Oi0
and Pj=1;2(CjRLORL + CjLROjLR) are the leading sources of the new physics contribution.
j
Wilson coe cients for operators contributing to the neutron EDM
In the e ective QCD QED theory in which W; W 0 and the top quark are integrated
out, the part of the CPviolating Hamiltonian that contributes to the neutron EDM is
parametrized as
HJEP05(218)
HnEDM = p
GF
2
X
X
where operators O's are de ned in appendix C.
We determine the Wilson coe cients C's as follows: again, for each coe cient, if
multiple terms have an identical phase, we exclusively consider the one in the leading order
of M W2 =M W2 0 or sin . By integrating out W and the top quark, one obtains the following
leadingorder matching conditions at
C2cd = 4 sin cos Im h(VL)cd(VL)cd ei( c+ d )i ;
where loop functions F1; F2; F3 and E1d; E2d; E3d; E3u are de ned in appendix B. In
eq. (4.10) (which corresponds to the Weinberg operator [49]), we present the dominant
part proportional to mt. Terms obtained by integrating out W 0 possess the same phases
as eqs. (4.2){(4.10) and are simply suppressed by M W2 =M W2 0 compared to eqs. (4.2){(4.10).
They are therefore neglected in our analysis.
e
e
4 2 sin cos
4 2 sin cos
4 2 sin cos
4gs2 sin cos
4gs2 sin cos
4gs2 sin cos
X
X
X
i=d;s;b mu
i=d;s;b mu
i=u;c;t md
X
X
i=d;s;b mu
i=d;s;b mu
X
i=u;c;t md
mt
mb
4gs3
C3 ' (16 2)2 sin cos
with xi
mi Im h(VL)id(VL)id ei( i+ d )i E3d(xi); (d ! s); (4.6)
mi Im h(VL)ui(VL)ui ei( i )i F3(xi);
mi Im h(VL)ui(VL)ui ei( c+ i )i F3(xi);
mi Im h(VL)id(VL)id ei( i+ d )i F3(xi);
(d ! s); (4.9)
F3(xt) Im h(VL)tb(VL)tb ei( t+ b )i ;
(4.4)
(4.5)
(4.7)
(4.8)
(4.10)
The RG equations at order O( s) for the Wilson coe cients are obtainable in
refs. [41, 42, 44, 45]. We assume that the initial conditions at
tions are given by eqs. (4.2){(4.10), and solve the equations from
At the 1 GeV scale, we evaluate the hadronic matrix elements.
= MW for the RG
equa= MW to
= 1 GeV.
U(1)B L model with charge symmetry, the Wilson coe cients for
the fourquark operators O1q0q = (q0q0)(qi 5q) and O2q0q = (q0 q0 )(q i 5q ) (q 6= q0; q; q0 =
u; d; s) are particularly large. Therefore, we scrutinize how these operators contribute to
the neutron EDM. Operators O1q0q contribute in the following three ways:
HJEP05(218)
The rst one is through meson condensation [33]; O1q0q operators give rise to tadpole
terms for pseudoscalar mesons and induce their VEVs. These VEVs generate
CPviolating interactions for baryons and mesons, which contribute to the neutron EDM
through baryonmeson loop diagrams.
The second one is through hadronic matrix elements of O1q0q with baryons and
mesons, hBM jO1q0qjBi (B denotes a baryon and M a meson), which contribute to
the neutron EDM through baryonmeson loop diagrams.
The third one is directly through the hadronic matrix element of O1q0q with neutrons
and photon.
On the other hand, O2q0q operators do not yield meson condensation, but do contribute
to the neutron EDM in the latter two ways. Later, it will be shown that the contribution
from the pion VEV h
0i, which belongs to the rst category, is enhanced by the factor
ms=(mu + md) compared to the latter two. We therefore investigate how O1q0q operators
bring about meson condensation, thereby contributing to the neutron EDM.
We are aware that if PecceiQuinn mechanism [50] exists, it a ects the meson
condensation and also induces an e ective nonzero
term due to incomplete cancellation
between the genuine
term and the axion VEV. Alternatively, it is logically possible to
assume
= 0 without PecceiQuinn mechanism, by considering an unknown mechanism or
through netuning, in which case we do not need to take into account the e ect of
PecceiQuinn mechanism or that of nonzero . In this paper, we consider both cases where (i) one
has
= 0 without PecceiQuinn mechanism, and (ii) PecceiQuinn mechanism is at work.
We start from the case with
= 0 without PecceiQuinn mechanism. The meson
condensation contribution is evaluated by the following steps:
(L $ R);
X
q6=q0
with Cijkl
LRLR = Cijkl
RLLR
First, we implement P C1q0qO1q0q part of the Hamiltonian eq. (4.1) into the meson
chiral Lagrangian. To this end, we rewrite
X
q6=q0;q;q0=u;d;s
C1q0qO1q0q =
iCiLjRklLR (qiLqjR)(qkLqlR) + iCiRjkLlLR (qiRqjL)(qkLqlR)
It then becomes clear that the theory would be invariant (except for U(1)A anomaly) if
From the above transformation property and the parity invariance of QCD, the meson chiral
Lagrangian at order O(p2) plus the leading CPviolating terms is found to be (remind that
tr h(D U )yD U + (U + U y)i +
tr hU D U yi tr hU yD U
i
F 2
0
F 2
12
= 0 has been assumed)
Lmesons =
h
+ a0 tr log U
GF
+ p
F 2
4
X
2 i;j;k;l=u;d;s
where Cijkl
LRLR; Cijkl
as U ! RU Ly, and
log U yi2
n
iCiLjRklLR c1[U ]ji[U ]lk
+iCiRjkLlLR c3[U y]ji[U ]lk
c3[U ]ji[U y]lk
o
;
U = exp
2i
p6F0
0 I3 +
;
2i
F
I3
diag(1; 1; 1);
= 2B0 diag(mu; md; ms):
(4.11)
(4.12)
(4.13)
(4.15)
c1[U y]ji[U y]lk + c2[U ]li[U ]jk
c2[U y]li[U y]jk
2
p1 K
2
+
1
p
2
2
p1 K0
2
p1 K+ 1
1
RLLR have been de ned in eq. (4.12). Here, U is a nonlinear representation
of the nine NambuGoldstone bosons that transforms under U(3)L
U(3)R rotations L
R
includes the quark mass term, which are given by
[U ]ij denotes the (i; j) component of matrix U . F is the pion decay constant in the chiral
limit and F0 is the decay constant for 0, which we approximate as F0 ' F . B0 satis es
B0 ' m2 =(mu + md). The term with log U represents instanton e ects, whose expression is
exact in the large Nc limit [51], and a0 satis es 48a0=F02 ' m2 + m20
2m2K . c1, c2 and c3
are unknown low energy constants (LECs), which can be estimated by nave dimensional
analysis [53] as
c1
c2
c3
(2). The CPviolating part of the Lagrangian eq. (4.13) contains tadpole terms for
mesons, which lead to nonzero meson VEVs.
Assuming that electric charge and
strangeness are not broken spontaneously, we obtain the following potential for neutral
HJEP05(218)
mesons 0
, 8 and 0
:
V ( 0; 8; 0) = F 2B0
p
The above potential is minimized with nonzero meson VEVs, h
as we are concerned with vertices with one meson, the physical modes of 0
0i, h 8i and h 0i. Insofar
, 8 and 0
elds can be approximated as
0
phys '
0
h i
0 ;
8phys ' 8
h 8i;
0phys ' 0
h 0i:
In the SU(2)L
U(1)B L model with charge symmetry, there hold relations
C1ud '
C1du and jC1udj
jC1sqj; jC1qsj (q = u; d). When (C1ud + C1du) and C1sq; C1qs are
neglected accordingly, one nds, for small VEVs,
0
h i
' p2 (C1ud C1du) B0F 2 B0F 2mumdms + 8a0(mumd + mdms + msmu)
;
c3
B0F 2(mu + md)ms + 8a0(mu + md + 4ms)
' p2 (C1ud C1du) p
' p2 (C1ud C1du) p
c3
3B0F 2 (md
p3B20cF3 2 (md
B0F 2ms + 24a0
B0F 2ms
mu) B0F 2mumdms + 8a0(mumd +mdms +msmu)
mu) B0F 2mumdms +8a0(mumd +mdms +msmu)
(4.17)
(4.18)
(4.19)
;
:
Note that h 8i and h 0i are proportional to md
mu. This is because these VEVs are
isospin singlets and hence must be constructed from the product of isospinodd coe cient
C1ud
C1du and isospinodd mass term md
mu. In contrast, h
because this VEV is isospinviolating. Since 20a0
B0F 2ms holds empirically, we nd
0i does not contain md
mu
from eq. (4.19) that h 0i is much larger than h 8i and h 0i by the factor ms=(md
mu).
(3).
Meson condensation breaks CP symmetry (and U(3)L
U(3)R symmetry) and in
duces CPviolating interactions for baryons and mesons. To study these interactions, we
write the baryon chiral Lagrangian at order O(p2) as (terms irrelevant in the current
dis
D
2
tr B
5f ; Bg
F
2
MBBB
tr B
2
5
where B represents baryons and L; R include mesons as
i r ) R +
i r ) R
i l ) L;
i l ) L;
U = R Ly;
R = Ly:
+
0
1
p
2
is a covariant derivative for baryons,
is a combination of meson elds, and + contains
quark masses, which are de ned as
+
2B0 Ly diag(mu; md; ms) R + 2B0 Ry diag(mu; md; ms) L:
MB is the baryon mass in the chiral limit. We insert meson VEVs h
baryon chiral Lagrangian eq. (4.20) and extract CPviolating interaction terms involving
0i, h 8i, h 0i into the
neutron n. We thus obtain
Lbaryons
gnn nn p0hys + gnn8 nn 8phys + gnn0 nn 0phys + gnp (pn + + np
)
+ gn 0K0 ( 0nK0 + n 0K0) + gn
K+ ( +nK
+ n
K+)
+ gn K ( nK0 + n K0);
(4.21)
(4.22)
(4.23)
(4.24)
(4.25)
(4.26)
(4.27)
gnn =
gnn8 =
gn 0K0 =
g
n
K+ =
4
3
B0
F
4
p
2ms) + bF (md + 2ms)g hF00i ; (4.29)
gnn0 =
" p
2ms) + bF (md + 2ms)g hF8i
3 fb0(mu + md + ms) + bD(md + ms) + bF (md
gnp =
F
B0 (bD + bF )
"
p2(md
mu)
0
h i
F
p
p
3
(mu + md)
ms)g hF00i ;
F
h 8i + p2 h 0i
F0
(3md + ms)
0
h i +
F
1
where the coupling constants are given by
F
B0 4 f b0(mu + md)
(bD + bF )mdg F
+ p fb0(md
mu) + (bD + bF )mdg
4
3
p fb0(md
mu) + (bD + bF )mdg F
0
h i
F
0
h i
;
gn K =
B0 (bD + 3bF )
p (3md + ms)
0
h i +
F
1
6
(md
5ms)
h 8i
F
Note in particular that h
0i enters into the expression for gn
K+ eq. (4.33) without the
mu, which is allowed because the coupling gn
K+ violates isospin. It follows
K+ is enhanced by the factor ms=(mu + md), as it contains a term msh i
We compare the above mesonVEVinduced CPviolating couplings with those arising
from direct hadronic matrix elements of O1q0q and O2q0q. The latter are estimated by nave
dimensional analysis [53] as5 (B and M represent any baryon and meson, respectively)
gBBM jdirect
GF
p
X
X
where eq. (4.19) and the nave dimensional analysis on c3 eq. (4.16) are in use. Noting
that (bD
bF )(4 F )
1 holds numerically, we observe that the meson VEV contribution
eq. (4.36) dominates over the direct hadronic matrix element one eq. (4.35) by the factor
ms=(mu + md). This fact allows us to neglect the latter contribution in the rest of the
analysis.
(4).
The neutron EDM receives contributions from baryonmeson loop diagrams involving
a CPviolating coupling of eqs. (4.27), a CPconserving baryonmeson axialvector coupling
and a photon coupling. We refer to the loop calculation of ref. [58] performed with infrared
regularization [60, 61], from which the neutron EDM, dn, is obtained as
(4.35)
(4.36)
HJEP05(218)
dnjloop =
8 2F
e
g
n
p
2
K+ (D
2
p
gnp (D + F )
F )
1
1
1
1
log
log
m2K2 +
m2
m
2mN
mK
2mN
(m
mN )
mK
:
(4.37)
Here, the divergent part 1=
1=
E + log(4 ) and the scale
stem from dimensional
regularization in 4
2 dimension with mass parameter
. In fact, the baryon chiral
Lagrangian contains a LEC which cancels the above divergence and whose
nite part
contributes to the neutron EDM. The impact of the nite part of the LEC is assessed by
nave dimensional analysis [53] as
dnjLEC
GF
p
X
X
2 i=1;2 q;q0
jCiq0qj e (4 )2
:
4 F
(4.38)
On the other hand, from eqs. (4.19) and (4.33) and the estimate on c3 eq. (4.16), the nite
part of the loop contribution eq. (4.37) is estimated to be
dnjloop
e
8 2F
e
8 2F
g
n
p
2
GF
p2 (C1ud
F )
C1du)(bD
2ms
1, we nd that the loop contribution eq. (4.39)
dominates over the LEC one eq. (4.38) by the factor ms=(mu + md). It is thus justi able
5There are also studies in which the direct hadronic matrix elements are estimated with vacuum
saturation approximation [54{57] and with hadron models [35, 36].
to estimate dn by simply extracting the nite part of the loop contribution. We further set
= mN , since mN is a natural cuto scale, and arrive at
dn
8 2F
e
g
n
2
p
log
log
m2
m2N +
m
2mN
mm2K2N +
mK
2mN
(m
mN )
mK
Next, we study the case with PecceiQuinn mechanism. We incorporate the axion eld,
a, into the meson Lagrangian eq. (4.13) by performing U(3)A chiral rotations to remove
the gluon theta term and transform the quark elds as
where u; d; s include the axion eld a as
dL ! e i d=2dL;
dR ! ei d=2dR;
(4.41)
u =
d =
s =
mumd + mdms + msmu
mumd + mdms + msmu
mdms
msmu
mumd
mumd + mdms + msmu
a
fa
a
fa
a
fa
+
+
+
;
;
;
with fa denoting the axion decay constant and
being the genuine theta term. (With
the above choice of u; d
; s, the axion does not mix with
0 or 8.) As a result, the
axion eld is associated with the quark masses and the coe cients C1q0q, and can thus be
implemented in the meson chiral Lagrangian through these terms. Accordingly, the meson
potential eq. (4.17) is modi ed to the potential of 0
, 8
, 0 and axion a,
V ( 0; 8; 0; a) = F 2B0
2c1 (C1ud + C1du) sin
(4.42)
where it should be reminded that u; d
; s are functions of a. The minimization condition
for eq. (4.42) yields meson VEVs h
0i, h 8i, h 0i and an axion VEV hai. When only the
term (C1ud
C1du) is nonzero, these VEVs are given by
0
h i
h 8i
F
F
a
h i +
fa
GF
GF
GF
' p2 (C1ud
' p2 (C1ud
' p2 (C1ud
C1du) Bp03Fc32 (md
C1du) B20cF3 2 (md
c3
C1du) B0F 2 mumd + mdms + msmu
;
mu + md + 4ms
1
mu) mumd + mdms + msmu
;
h 0i
F0
' 0;
1
mu) mumd
:
(4.43)
The VEVs of 0 and 8 remain of the same order as the case without PecceiQuinn
mechanism, and hence they contribute to the neutron EDM in an analogous way. The axion
VEV no longer cancels the genuine
term and the leftover induces an e ective
term; we
estimate its contribution by employing the result of ref. [62] as
(4.44)
(4.45)
(4.46)
(4.47)
(4.48)
(4.49)
dnjind
=
(2:7
1:2)
a
fa
nal result is the sum of the meson VEV contribution estimated analogously to
eq. (4.40), plus eq. (4.44).
4.2.2
Other CPviolating operators O1q, O2q and O3
The contributions of the dipole operators in eq. (C.1) and the Weinberg operator in eq. (C.2)
to the neutron EDM can be obtained with the QCD sum rule. The former is calculated in
ref. [63] while the latter is in ref. [64], resulting in the following relations:
dnjquark = 0:47dd
dnjqPuQark = 0:47dd
dnjWeinberg = GpF egsC3
2
0:12du + e(0:18dcd
0:18dcu
0:008dcs);
0:12du + e(0:35dcd + 0:17dcu);
(10
30) MeV;
where r.h.s. must be evaluated at 1 GeV. In eqs. (4.45), (4.46), dq and dcq(q = u; d; s),
socalled quark EDM and quark chromoEDM, are de ned as,
dq( ) =
c
dq( ) =
GF
GF
2
p e eqC1q( )mq( );
p2 C2q( )mq( ):
Equations (4.45) and (4.46) represent the quark EDM contirbutions without and with
PecceiQuinn mechanism, respectively. For the case without PecceiQuinn mechanism, we
have taken
Numerical analysis of neutron EDM versus 0=
For numerical analysis of dn, we employ the following values: the chirallimit pion decay
constant F is obtained from a lattice calculation as F
= 86:8 MeV [65]. D; F have
been measured to be D = 0:804 and F = 0:463. For bD; bF , we quote the result of
refs. [66, 67] with a NLO calculation in Lorentz covariant baryon chiral perturbation theory
with decuplet contirbutions, which reads bD = 0:161 GeV 1 and bF =
Since the same calculation formalism, combined with experimental data
0:502 GeV 1
.
N ' 59(7) MeV,
predicts a small value of the strange quark contribution to the nucleon mass s [66], we infer
that these values of bD; bF are most robust. For the quark masses, we adopt lattice results
in ref. [68], mud(2 GeV) = 3:373 MeV and ms(2 GeV) = 92:0 MeV, and further evaluate
QCD
veloop RG evolutions to obtain the masses at 1 GeV in M S scheme, which are used
in our analysis. Also, we exploit an estimate mu=md = 0:46 [68].
The main source of uncertainty in our analysis is the unknown LEC c3 in the meson
chiral Lagrangian eq. (4.13). The other unknown LEC c1 is ine ective, because the Wilson
coe cients satisfy jC1ud
C1uj
jC1ud + C1uj; jC1sqj; jCqsj. Our calculations of loopinduced
dn eq. (4.40) and axioninduced dn eq. (4.40) are hence proportional to c3 and subject to
O(1) uncertainty originating from its nave dimensional analysis eq. (4.16). The fact that
our results depend only on one LEC c3 is good news, because it excludes the possibility
of accidental cancellation between contributions with di erent LECs. Another source of
uncertainty is the renormalization scale
in the loop calculation eq. (4.37), but this is
subdominant compared to the uncertainty of c3.
In the analysis, the ratio of the bifundamental scalar VEVs is again xed as tan
=
mb=mt. The values of the new CP phases c; t; d
; s
; b
;
are randomly generated. We
nd that the contribution of the Weinberg operator is suppressed by roughly 10 7
compared with that of the fourquark operators, and thus we neglect it in the analysis.
First, we show the numerical result for the neutron EDM without the constraint from
0= in gure 3. One observes that the contribution of the fourquark operators is dominant
over that of the quark EDMs.
As stated previously, an e ective
term is induced in the presence of PecceiQuinn
mechanism. In gure 4, we additionally show the numerical prediction based on eq. (4.44).
One nds that the induced
gives subleading contribution to the neutron EDM.
Next, the correlated prediction for jdnj and Re( 0= ) is presented in gures 5 and 6 in the
cases without and with PecceiQuinn mechanism, respectively. Here, small contributions
from the quark EDMs are neglected. The cases with and without PecceiQuinn mechanism
yield almost identical results because the induced
has a subdominant e ect, as seen in
gure 4. We observe that MW 0 = 20 TeV and 50 TeV can be consistent with the data on
Re( 0= ) at 1 level, whereas the case with MW 0 = 70 TeV cannot explain it. However, the
case with MW 0 = 20 TeV has already been excluded by the current bound on the neutron
EDM, and only MW 0 = 50 TeV can be compatible with the neutron EDM bound and
the data on Re( 0= ). Figures 5 and 6 further inform us that almost all parameter points
that account for the Re( 0= ) data will be covered by future neutron EDM searches [71].
Therefore, unless the treelevel in the case without PecceiQuinn mechanism miraculously
cancels the contribution of the model, we anticipate the discovery of the neutron EDM in
the near future.
1[102
0
106
104
]
m 102
c
6
2
e
0
1
100
[
n102
d

104
Only the contributions of fourquark operators and quarklevel EDMs including both quark EDM
and chormoEDM are shown. A dashed line represents the current bound on the neutron EDM [69],
while a dashed dotted line stands for the future bound [71].
10
20
30
50
60
70
40
MW' [TeV]
in the presence of PecceiQuinn mechanism. A gray dashed line represents the current bound of
EDM [69] while a black dashed dotted line stands for the future bound [71].
5
Summary and discussions
We have addressed the 0= anomaly in the SU(2)L
SU(2)R
U(1)B L gauge extension of
the SM with charge symmetry. Since the charge symmetry gives strong restrictions on the
mixing matrix for righthanded quarks, 0= can be evaluated only in terms of two new CP
phases
d and
s, the mass of W 0 gauge boson (mostly composed of WR), and the
bifundamental scalar VEV ratio tan . By xing tan
at its natural value mb=mt, and by
randomly varying
d and
s, we have shown that MW 0 < 58 TeV must be satis ed
to account for the experimental value of 0= at 1
level.
10 100 current bound
102
future bound
MW' =20TeV
MW' =50TeV
MW' =70TeV
104
]
10 100 current bound
102
future bound
MW' =20TeV
MW' =50TeV
MW' =70TeV
1σ
range
1σ
range
20
10
0
10
20
Re(ϵ'/ϵ) [104]
= 0 without PecceiQuinn mechanism. A gray dashed line and a black dashed
dotted line represent the current [69] and the future [71] bounds on the neutron EDM, while a cyan
band stands for the 1 range of the direct CP violation in K !
decay obtained from PDG [8].
HJEP05(218)
20
10
0
Re(ϵ'/ϵ) [104]
10
20
Next, we have made a prediction for the neutron EDM dn when the SU(2)L
U(1)B L model with charge symmetry solves the 0= anomaly. We have investigated the
contribution of meson condensates induced by fourquark operators, and revealed that
the
0 VEV dominantly contributes to the neutron EDM, whose impact is enhanced by
ms=(mu + md) compared to other contributions. This enhancement is attributable to the
isospin violating coupling of W 0 gauge boson, which allows the 0 VEV to arise without
mu. Additionally, we have found that the induced
term in the presence
of PecceiQuinn mechanism yields only a subleading e ect on dn. On the basis of the above
observations, we have shown that the 0= anomaly can be explained without con icting
the current experimental bound on dn, and that the parameter space where the 0= data
are accounted for will be almost entirely covered by future experiments [71].
We comment on the constraint from Re( ) on the model. Since W 0 gauge boson
F = 2 processes only at loop levels, for MW 0 > 20 TeV, its contribution
to Re( ) is safely below the experimental bound [73]. However, the heavy neutral scalar
particles coming from the bifundamental scalar induce
F = 2 processes at tree level. Since
their mass is of the same order as or below MW 0 if there is no
netuning in the scalar
potential, these particles may lead to a tension with the data on Re( ) [73] (constraint
from Re( ) on general leftright models is found in ref. [74], and that on the model with
leftright parity is in ref. [75]) (for early studies on the Re( ) constraint, see, e.g., ref. [76]).
Acknowledgments
The authors would like to thank Monika Blanke, Andrzej Buras, Antonio Pich and Amarjit
Soni for valuable comments. This work is partially supported by Scienti c Grants by the
Ministry of Education, Culture, Sports, Science and Technology of Japan (Nos. 24540272,
26247038, 15H01037, 16H00871, and 16H02189).
A
Operators of
e
2 q=u;d;s
8 2 mss
X (s d )L(q q )L;
X (s d )L(q q )R;
X (s d )Leq(q q )R;
X (s d )Leq(q q )L;
F
PLd;
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
ed = es =
corresponding operators.
where (qq0)L
q (1
5)q0 and (qq0)R
q (1 + 5)q0, ; are color indices, and
color summation is taken in each quark bilinear unless ;
are displayed. eu = 2=3 and
1=3. The operators Oi0, OjLR are obtained by interchanging L $ R in the
The loop functions in the main text are de ned as follows:
F1(x) =
F2(x) =
F3(x) =
E1d(x) =
E2d(x) =
E3d(x) =
E3u(x) =
12(1
x(2 + 5x
4(1
4 + x + x2
2(1
25x2
36(1
6(1
3(1
x)2 +
(1
19x3
x)3 +
5x
8x2)
12(1
2(1
18(1
x2(2
2(1
x)4 log x;
3x)
x)3 log x;
3x)
x)3 log x:
4
9
2
3
x( 18 + 11x + x2)
x2( 15 + 16x
4x2)
6(1
x)4
log x +
log x + ;
2
3
x2(6 + 2x
5x2)
4
9
log x +
log x + ;
CPviolating operators that contribute to the neutron EDM
O1q =
O3 =
2 eq mqq
1 f abc
6
O4q = qq qi 5q;
O1q0q = q0q0 qi 5q;
O3q0q = q0
q0 q
i 5q;
i 5q F
G
a G
b Ga ;
O2q =
2
gs mqq
i 5T aq Ga ;
O5q = q
q q
i 5q;
O2q0q = q0 q0 q i 5q ;
O4q0q = q0
q0 q
i 5q ;
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
where q0; q = u; d; s and q0 6= q.
each quark bilinear unless ; are displayed.
are color indices, and color summation is taken in
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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