# Left-right symmetry and electric dipole moments. A global analysis

Michael J. Ramsey-Musolf<EMAIL_ADDRESS>Tsung-Dao Lee Institute and
School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan
Road, Shanghai, 200240 China.  Amherst Center for Fundamental Interactions,
Department of Physics, University of Massachusetts, Amherst, MA 01003, USA.
Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, CA
91125 USA.  Juan Carlos Vasquez<EMAIL_ADDRESS>Amherst Center for
Fundamental Interactions, Department of Physics, University of Massachusetts,
Amherst, MA 01003, USA.

###### Abstract

We perform a global fit using results of searches for electric dipole moments
(EDM) of diamagnetic systems within the context of the minimal left-right
symmetric model. In this way, we disentangle the new “left-right” electroweak
and $\bar{\theta}$ contributions that cannot be separated using a single EDM
system. Although the fit is done for a specific model, the approach can be
applied to any particle physics model. Finally, we revisit the constraint on
the $D$ coefficient in $\beta$-decay and find that current EDM bounds do not
preclude observation of this T-violating effect in a possible next generation
$\beta$-decay experiment.

††preprint: ACFI-T20-17

##  I Introduction

The Standard Model (SM) of fundamental interactions contains two sources of CP
violation (CPV): the $\bar{\theta}$ parameter and the phase $\delta$ in the
Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix. The $\bar{\theta}$ parameter is
written as $\bar{\theta}=\theta_{0}+\text{arg det}(M_{u}M_{d})$, where
$\theta_{0}$ is the parameter governing the strength of the $G{\tilde{G}}$
interaction in the QCD Lagrangian and $M_{u},M_{d}$ are the up and down quark
mass matrices. Barring accidental cancelations, electric dipole moment (EDM)
systems are in general sensitive to any source of flavor-diagonal CP
violation. In particular $\bar{\theta}$ is the physical quantity entering in
EDM expressions Baluni (1979); Crewther  _et al._ (1979). Consequently, one
cannot disentangle the $\theta_{0}$ contribution from the $\text{arg
det}(M_{u}M_{d})$ contribution within the SM. Notice that, if the
$\bar{\theta}$ parameter is zero at the tree level, radiative corrections
within the SM turns out to be small and $\bar{\theta}\sim 10^{-19}$ Ellis and
Gaillard (1979); Khriplovich (1986), well below present experimental EDM
bound: $\theta_{exp}\sim 10^{-10}$ Graner  _et al._ (2017); Abel  _et al._
(2020) –see Ref. Ginges and Flambaum (2004); Pospelov and Ritz (2005); Engel
_et al._ (2013); Chupp  _et al._ (2019) for reviews.

New physics beyond the Standard Model (BSM) may introduce new sources of CPV,
which can then be the dominant source of CPV in hadronic, atomic, and
molecular EDM systems. To illustrate, consider the neutron EDM, which has the
general formEngel  _et al._ (2013)

$d_{n}=\alpha_{n}\,\bar{\theta}+\beta_{n}^{j}\,\left(\frac{v^{2}}{\Lambda^{2}}\right)\sum_{j}\mathrm{Im}\left(C_{j}\right)+\cdots,$
(1)

where $v=246$ GeV is the electroweak scale; $\Lambda$ is the BSM scale;
$C_{j}$ are Wilson coefficients of dimension-six ($d=6$) operators built from
SM fields 111The dimension-six sources include the quark and lepton EDMs,
quark chromo-EDMs, CPV 3-gluon operator, and a set of four fermion operators.
For a summary, see Ref. Engel  _et al._ (2013).; the coefficients $\alpha_{n}$
and $\beta_{n}^{j}$ encode the sensitivity of $d_{n}$ to $\bar{\theta}$ and
the BSM CPV sources, respectively; and the “$+\cdots$”denote contributions
from higher-dimensional operators. Analogous expressions apply to other EDM
systems, with correspondingly different sensitivities to ${\bar{\theta}}$ and
the BSM CPV sources. Notice that if only one EDM system is probed, there is no
way of disentangling the $\bar{\theta}$ and BSM contributions. In this case,
no rigorous bound can be given either to $\bar{\theta}$ or
$\mathrm{Im}(C_{j})/\Lambda$ without making additional assumptions. However,
results from a variety of EDM searches on systems with complementary
sensitivities can yield discriminatory power.

In this work, we consider the implications of EDM searches in the context of
the minimal left-right symmetric model (mLRSM) Pati and Salam (1974);
Mohapatra and Pati (1975); Senjanovic and Mohapatra (1975); Senjanovic (1979);
Mohapatra and Senjanovic (1980, 1981). In particular, we focus on diamagnetic
systems, for which the leading mLRSM contributions are ${\bar{\theta}}$ and
the $d=6$ “left-right" operator defined below (see Refs. Engel  _et al._
(2013); Seng  _et al._ (2014) for a discussion of the various $d=6$ operators
in the mLRSM). We perform a global analysis of diamagnetic EDM search results,
following the approach taken in Refs. Chupp and Ramsey-Musolf (2015); Chupp
_et al._ (2019) that considered the low-energy effective parameters relevant
to the EDM systems.

The implications of the EDM results depend decisively on whether parity
($\mathcal{P}$) or charge conjugation ($\mathcal{C}$) is adopted as the LR
symmetry. For $\mathcal{P}$, one has $\theta_{0}=0$ and
$\bar{\theta}=\text{arg det}(M_{u}M_{d})$ is calculable in terms of the mLRSM
parameters. The latter is also true for other solutions of the “strong CP
problem", such as the soft breaking of parity Mohapatra and Senjanovic (1978);
Bég and Tsao (1978); Mohapatra and Senjanovic (1983); Babu and Mohapatra
(1990); Barr  _et al._ (1991) or the soft breaking of CP symmetry Georgi
(1978); Bég and Tsao (1978). In each case, $\bar{\theta}$ is a calculable
parameter, leaving only $\mathrm{Im}(C_{j})/\Lambda^{2}$ is the a priori
unknown quantity.

This is not the case in general – including the case of the SM – since there
is no way of disentangling the $\theta_{0}$ contribution from the radiative
corrections giving rise to $\text{arg det}(M_{u}M_{d})$. For $\mathcal{C}$ as
the LR symmetry, $\theta_{0}\neq 0$, and hence no bound on
$\mathrm{Im}(C_{j})/\Lambda^{2}$ can be obtained using a single EDM system.
This is our main motivation for considering the set of diamagnetic systems.
Doing so allows us to give bounds on both $\bar{\theta}$ and
$\mathrm{Im}(C_{j})/\Lambda^{2}$ without making any additional theoretical
assumptions.

In this context, we find – unsurprisingly – that EDM limits on $\bar{\theta}$
and $\mathrm{Im}(C_{j})/\Lambda^{2}$ are significantly relaxed compared to a
“sole source" analysis in which only one of the two is assumed to be present.
Perhaps, more interesting are the implications for the possible manifestations
of CPV in other observables. In particular, we consider the T-odd correlation
neutron $\beta$-decay, which has recently been constrained by the emiT
collaboration Mumm  _et al._ (2011a, b). We show that present EDM constraints
would allow for observation of a non-zero effect in a future experiment
performed with improved sensitivity, with a magnitude larger than the
uncertainty in the pseudo-T-odd contribution from final state interactions.

Our discussion of this analysis is organized as follows. In Sec. II we review
the basic interactions and the expression for the EDM of hadronic, atomic and
molecular systems. In Sec. III we discuss the relevant interactions within the
mLRSM. In Sec. IV, we present our results for a global fit using the neutron,
Xenon (Xe), Radium (Ra), and Thallium fluoride (TlF) systems. We discuss our
results in Sec. V. Then, Sec. VI gives our analysis of the interplay between
T-reversal violation in $\beta$-decays and the EDM bounds implied by our
global fit. Finally in Sec. VII we present our conclusions.

##  II EDM of hadronic and atomic systems

In this section we review and summarize the expressions needed for computing
the EDM of hadronic, atomic and molecular systems. We focus on the EDMs of
diamagnetic systems for the following reasons. The parameter that governs the
magnitude of the CP-violating couplings is $\kappa_{LR}$ –defined below –
which arises from mixing between left- and right-handed gauge bosons,
$W_{L,R}$ and generates a dimension six four quark operator. The resulting,
leading contribution to paramagnetic systems is given by the electron EDM
($d_{e}$). In the mLRSM, $d_{e}$ arises at one-loop order, introducing a the
general one-loop suppression factor $1/16\pi^{2}$ Valle (1984); Nieves  _et
al._ (1986). Moreover, $d_{e}$ depends on the neutrino Dirac mass parameter
$M_{D}$ that does not enter the diamagnetic EDMs at an appreciable level.
Consequently, one may treat $d_{e}$ as providing a separate constraint on
$M_{D}$, while the diamagnetic systems constrain ${\bar{\theta}}$ and the
left-right operator defined below. Previous $d_{e}$ bounds require
$M_{D}\lesssim(10^{-2}-1)$ MeV Tello (2012). In addition to the above
theoretical arguments, the new experimental limit $d_{e}<10^{-29}$ e.cm Baron
_et al._ (2014) implies a corresponding $d_{e}$ contribution to the 199Hg EDM
to be $\lesssim 10^{-31}$ e.cm , below the current sensitivity
$d_{A}\left({}^{199}\mathrm{Hg}\right)=(2.20\pm 2.75(\text{ stat })\pm
1.48(\mathrm{sys}))\times 10^{-30}e\cdot\mathrm{cm}$ Graner  _et al._ (2017).
The individual quark contributions to the nucleon EDMs are suppressed due to
the small Yukawa coupling of the up and down quarks Engel  _et al._ (2013).
The resulting contributions yield a weaker sensitivity to the parameters of
the mLRSM than the effects considered below. Finally, the three-gluon Weinberg
operator gives subleading contributions to the EDMs for light quarks – as
discussed in Ref. Bertolini  _et al._ (2020).

For diamagnetic systems such as the neutron and 199Hg atom, the relevant
contributions are the EDM and chromo-EDM of the quarks, the CPV three-gluon
operator, and various four fermion operators. The quark (chromo-)EDM and
three-gluon operators also arise at one-and two-loop orders Xu  _et al._
(2010), respectively. The former carry factors of the light quark Yukawa
couplings, introducing an additional source of suppression. In contrast,
$W_{L}-W_{R}$ mixing can also yield a CPV four-quark operator via tree-level
gauge boson exchange as indicated in Fig. 1. The corresponding Wilson
coefficient is also proportional to $\kappa_{LR}$ but carries neither a loop
nor Yukawa suppression factors. Thus, one expects it to yield the leading
contribution to diamagnetic EDMs. At the hadronic level, this dimension six
operator generates a “short distance" contribution to the nucleon EDM and the
leading-order (LO) chiral contribution coming from one-loop diagrams involving
the CP-violating interaction between the pions and the nucleons. Finally, the
dimension six operator also contributes to the atomic and molecular EDMs via
the nuclear Schiff moment – see Ref. Engel  _et al._ (2013) for details.

The EDM of diamagnetic atomic or molecular system is given by Chupp and
Ramsey-Musolf (2015):

$d_{A}=\sum_{N=p,n}\rho_{Z}^{N}d_{N}+\kappa_{S}S-\left[k_{T}^{(0)}C_{T}^{(0)}+k_{T}^{(1)}C_{T}^{(1)}\right],$
(2)

where the parameters $\rho_{Z}^{N}$ give the sensitivity of the EDMs to the
individual nucleons. The parameter $\kappa_{S}$ measures the sensitivity of
the atomic or molecular system to the nuclear Schiff moment $S$. For the
mLRSM, one has

$S\simeq\frac{m_{N}g_{A}}{F_{\pi}}\left[a_{0}\bar{g}_{\pi}^{(0)}+a_{1}\bar{g}_{\pi}^{(1)}\right],$
(3)

where the couplings $\bar{g}_{\pi}^{(i)}$ parametrize the T-violating,
P-violating pion-nucleon interaction in the chiral Lagrangian

$\mathcal{L}_{\chi}^{LR}=\bar{N}\left[\bar{g}_{\pi}^{(0)}\vec{\tau}\cdot\vec{\pi}+\bar{g}_{\pi}^{(1)}\pi^{0}\right]N\
\ \ .$ (4)

It is useful to express the $\bar{g}_{\pi}^{(i)}$ in terms of the Wilson
coefficient of the four-quark operator associated with Fig. 1

$\bar{g}_{\pi}^{(i)}=\lambda_{i}\bar{\theta}+\gamma_{i}^{\varphi
ud}\frac{v^{2}}{\Lambda^{2}}\operatorname{Im}(C_{\varphi ud}),\,i=0,1,\,,$ (5)

where the dimension-six effective Lagrangian contains the term

$-i\frac{\operatorname{Im}C_{\varphi
ud}}{\Lambda^{2}}\left[\bar{d}_{L}\gamma^{\mu}u_{L}\bar{u}_{R}\gamma_{\mu}d_{R}-\bar{u}_{L}\gamma^{\mu}d_{L}\bar{d}_{R}\gamma_{\mu}u_{R}\right]\,.$
(6)

We discuss the range of numerical values for the $\lambda_{i}$ and
$\gamma_{i}$ below.

Finally, $C_{T}^{0}$, $C_{T}^{1}$ are related to the nucleon-electron
interaction of the form:

$\mathcal{L}_{eN}^{\mathrm{NSD}}=\frac{8G_{F}}{\sqrt{2}}\bar{e}\sigma_{\mu\nu}ev^{\nu}\bar{N}\left[C_{T}^{(0)}+C_{T}^{(1)}\tau_{3}\right]S^{\mu}N+\cdots.$
(7)

where $S^{\mu}$ denotes the nucleon spin, which in the nucleon rest frame
takes the value $(0,\vec{\sigma}/2)$ and $\sigma_{i}$ are the Pauli matrices.
This interaction enters in the atomic EDM due to the interaction of the closed
electron shells with the hyperfine interaction polarizing these shells along
the nuclear spin Khriplovich and Lamoreaux (1997). The constants $C_{T}^{(0)}$
and $C_{T}^{(1)}$ parametrize the nuclear-spin-dependent electron-nucleon
interaction. Notice that all the semileptonic operators at dimension six can
only be obtained by integrating out neutral currents at the tree level Engel
_et al._ (2013). The diagrams involving the SM Higgs exchange (on top of being
suppressed by small lepton and light quark Yukawa couplings) cannot give the
Lorentz structure show in Eq. (7). The same reasoning applies to the exchange
of hypothetical new heavy scalar particles and the right-handed $Z$-boson of
the mLRSM.

##  III The minimal left-right symmetric model

Figure 1:  Leading tree-level diagram contributing to the neutron EDM in the
mLRSM.

The mLRSM Senjanovic (1979); Mohapatra and Senjanovic (1980, 1981) extends the
SM gauge group to $SU(3)\times SU(2)_{R}\times SU(2)_{L}\times U(1)_{B-L}$,
where $B$ and $L$ denote the SM abelian baryon and lepton quantum numbers. In
this model the $SU(2)_{R}$ gauge group is broken once the triplet field

$\Delta_{R}=\left(\begin{array}[]{cc}\Delta^{+}/\sqrt{2}&\Delta^{++}\\\
\Delta^{0}&-\Delta^{+}/\sqrt{2}\end{array}\right)_{R}$ (8)

takes a vacuum expectation value (VEV) $v_{R}$ along with its neutral
component, breaking the above gauge group down to the SM gauge group. At the
next step of symmetry breaking, the SM gauge group is broken down to
$U(1)_{QED}$ once the bidoublet field

$\Phi=\left(\begin{array}[]{cc}\phi_{1}^{0}&\phi_{2}^{+}\\\
\phi_{1}^{-}&\phi_{2}^{0}\end{array}\right)$ (9)

takes a VEV along its neutral components
$\langle\Phi\rangle=\text{diag}\,\\{v_{1},v_{2}e^{i\alpha}\\}$. Notice the
appearance of the phase $\alpha$. This is the so-called spontanous CP phase,
which gives an EDM contribution, as we discuss below.

In the mLRSM one has, in addition to the $\bar{\theta}$ contribution to the
EDMs, a contribution from the tree-level diagram shown in Fig.1, which is due
to the interaction

$\displaystyle\mathcal{L}^{W}=$
$\displaystyle-\frac{g}{\sqrt{2}}\bar{u}_{Li}\gamma^{\mu}V^{L}_{ud}\left(\cos\xi
W_{1\mu}^{+}-\sin\xi e^{-i\alpha}W_{2\mu}^{+}\right)d_{Li}$
$\displaystyle-\frac{g}{\sqrt{2}}\bar{u}_{Ri}\gamma^{\mu}V^{R}_{ud}\left(\sin\xi
e^{i\alpha}W_{1\mu}^{+}+\cos\xi W_{2\mu}^{+}\right)d_{Ri}$
$\displaystyle+\text{ h.c. }\,,$ (10)

where $W_{1}$ is light mass eigenstate, which is mostly the SM $W$ boson,
$W_{2}$ is the heavy mass eigenstate, and $V^{L,R}$ are the CKM matrix and its
right-handed analog, respectively. By the exchange of one $W$ boson shown in
Fig. 1, this interaction generates a four-quark operator proportional to
$\kappa_{LR}$, with the Wilson coefficient suppressed by the mixing between
the SM $W$ and the $W_{R}$ bosons. The resulting CPV effective Lagrangian
involving light quarks and gluons, valid at the weak scale, is given by

$\displaystyle\mathcal{L}_{CPV}=$
$\displaystyle-\frac{g_{3}^{2}}{16\pi^{2}}\bar{\theta}\operatorname{Tr}\left(G^{\mu\nu}\tilde{G}_{\mu\nu}\right)-$
$\displaystyle
i\frac{4G_{F}}{\sqrt{2}}\kappa_{LR}\left(\bar{u}_{R}\gamma_{\mu}d_{R}\,\bar{u}_{L}\gamma_{\mu}d_{L}+h.c.\right),$
(11)

where,

$\kappa_{LR}=\sin\xi Im\left(V_{ud}^{L}V_{ud}^{R*}e^{-i\alpha}\right)\ \ \ ,$
(12)
$\tan\xi=-\frac{v_{1}v_{2}}{v_{R}^{2}}\simeq-\frac{M_{W}^{2}}{M_{W_{R}}^{2}}\sin
2\beta\ \ \ ,$ (13)

with $\tan\beta\equiv v_{2}/v_{1}$, $g_{3}$ is the strong coupling constant,
$G_{\mu\nu}$ is the gluon field strength and
$\tilde{G}_{\mu\nu}=\epsilon_{\mu\nu\alpha\beta}G^{\alpha\beta}/2$.

As emphasized above, $\bar{\theta}$ and $\kappa_{LR}$ are independent
parameters when $\mathcal{C}$ is taken as the LR symmetry. Both terms in Eq.
(III) induce the neutron EDM through (a) “long-distance” chiral loops arising
from the pion nucleon interaction in Eq. (4); and (b) “short distance” effects
encoded in the finite parts of the low energy constants of the associated
chiral effective field theory. The same operator also generates a contribution
to atomic and molecular EDMs via the nuclear Schiff moment $S$ in Eq (3).
Owing to their different chiral transformation properties, the two
interactions in Eq. (III) generate leading contributions to the two different
$\bar{g}_{\pi}^{(i)}$, viz $\bar{g}_{\pi}^{(0}\sim{\bar{\theta}}$ and
$\bar{g}_{\pi}^{(1)}\sim\kappa_{LR}$.

We take into account the RGE from the electroweak scale down to the hadronic
scale using the results of Refs. de Vries  _et al._ (2013); Dekens and de
Vries (2013); Cirigliano  _et al._ (2017). In addition to the RGE effects, the
“direct" part contributing to the finite part of the low energy constant was
studied in detail in Ref. Cirigliano  _et al._ (2017), where it induces a
roughly $50\%$ uncertainty in the $\kappa_{LR}$ dependence of the neutron EDM.
The net effect of these two issues is reflected in the estimated range for the
coefficient $\gamma_{1}^{\varphi ud}$ reported in Table 7 of Ref. Engel  _et
al._ (2013). In this work we update the range of the $\gamma_{1}^{\varphi ud}$
parameter reported in Ref. Engel  _et al._ (2013)222 The coefficient
$\gamma_{1}^{\varphi ud}$ of this paper is denoted as $\gamma_{(1)}^{\varphi
ud}$ in Ref. Engel  _et al._ (2013). . The previous range $\gamma_{1}^{\varphi
ud}\in(5-150)\times 10^{-7}$ of Ref. Engel  _et al._ (2013), once the RGE
effects and the short distance effect is taken into account changes to
$\gamma_{1}^{\varphi ud}\in(254-552)\times 10^{-7}$. We have estimated this
new range by adding in quadrature the uncertainties reported in Eqs. (31) and
(32) of Ref. Cirigliano  _et al._ (2017). The short distance contribution to
the nucleon EDM was studied in detail in Ref. Engel  _et al._ (2013) and we
use the range reported for the parameter $\beta_{n}^{\varphi ud}$ in Table 7
of Ref. Engel  _et al._ (2013).

Figure 2:  Left. Values of $\chi^{2}_{min}$ as a function of the
$\bar{\theta}$ contribution. Center. Values of $\chi^{2}_{min}$ as a function
of the LR contribution. Right. Values of $\bar{\theta}$ as a function of the
LR contribution

##  IV Global analysis using EDMs of nucleons, atoms and molecules

In this section, we perform a global fit using the most recent results for the
EDMs of the neutron, atomic and molecular systems Hg, Xe, Ra, and TlF taken
from Ref. Chupp  _et al._ (2019). We use the value of the hadronic and nuclear
parameters given in Ref. Chupp and Ramsey-Musolf (2015) 333Recently, the
values of the nuclear parameters for $\mathrm{Hg}$ and $\mathrm{Xe}$ were
calculated in Ref. Yanase and Shimizu (2020) and fall between the ranges
reported in Ref. Chupp and Ramsey-Musolf (2015).  We take the centroid for the
experimental values of the EDM from Ref. Chupp  _et al._ (2019) and references
therein. The most recent neutron EDM experimental limit was taken from Ref.
Abel  _et al._ (2020).

In order to find the sensitivity of the nucleon, atomic and molecular EDM
systems to the $\bar{\theta}$ parameter and the mLRSM contribution, we perform
a $\chi^{2}$ fit (with 2 d.o.f.) using the function:

$\chi^{2}=\sum_{i=1}^{N}\frac{\left[(d_{i})_{exp}-(d_{i})_{th}\right]^{2}}{\sigma_{i}^{2}},$
(14)

where $N$ is the number of the EDM systems, $(d_{i})_{exp}$ and $(d_{i})_{th}$
denotes the experimental centroids and the theoretical values for the EDMs and
$\sigma_{i}$ denotes the experimental error of the EDM for the system $i=n$,
Hg, Xe, Ra, TlF. To take into account the theoretical uncertainties of the
hadronic, nuclear, and atomic parameters, we perform a range fit Charles  _et
al._ (2005), which we explain in detail in what follows:

  * •

We vary the parameters $a_{0},a_{1}$ and between the best theoretical ranges
reported in Tab. VI of Ref. Chupp and Ramsey-Musolf (2015), except for the
TlF, since the ranges for these parameters are not reported in Ref. Chupp and
Ramsey-Musolf (2015). For the parameter $\kappa_{S}$, we vary it only for Ra
in the range reported in Tab. VI of Ref. Chupp and Ramsey-Musolf (2015). For
Mercury, Xenon, and TlF we use the fixed values of $\kappa_{S}$ reported in
Tab. VI of Ref. Chupp and Ramsey-Musolf (2015).

  * •

The hadronic uncertainty is taken into account by varying the parameter
$\alpha_{n}$, $\beta_{n}^{\varphi ud}$ in Eq. (1) and $\lambda_{0}$,
$\lambda_{1}$ and $\gamma_{1}^{\varphi ud}$ in Eq. (5) within the range
reported in Table 7 of Ref. Engel  _et al._ (2013). The parameter
$\gamma_{0}^{\varphi ud}=0$, since the four quark operator in Eq. (III) breaks
isospin symmetry, and hence it cannot contribute to the coupling
$\bar{g}_{\pi}^{(0)}$ in Eq.(4), which conserves isospin. The only difference
is that we update the range for the parameter $\gamma_{1}^{\varphi ud}$ and
use the new one discussed in Sec. III. Finally, using the more recent results
of Ref. de Vries  _et al._ (2015), we also update the range for the parameter
$\lambda_{0}$. The updated ranges are summarized in Tab. 1.

Hadronic parameter | Updated range
---|---
$\gamma_{1}^{\varphi ud}$ | $(254-552)\times 10^{-7}$
$\lambda_{0}$ | $0.013-0.018$
Table 1:  Updated ranges for two of the parameters reported in Table 7 of Ref.
Engel  _et al._ (2013). We used the results of Ref. Cirigliano  _et al._
(2017) and Ref. de Vries  _et al._ (2015). The coefficients
$\gamma_{i}^{\varphi ud}\,(\lambda_{i})$ in this paper are denoted as
$\gamma_{(i)}^{\varphi ud},\,(\lambda_{(i)})$ in Ref. Engel  _et al._ (2013),
for $i=0,1$.  Figure 3: Ranges represent the $68\%$ and $95\%$ confidence
level interval resulting from the best fit for the diamagnetic systems
including the neutron EDM. The blue star indicates the position of the best
fit point.

  * •

For each point in the space spanned by $a_{0},a_{1}$, $\kappa_{s}$,
$\alpha_{n}$, $\beta_{n}^{\varphi ud}$, $\lambda_{0}$, $\lambda_{1}$ and
$\gamma_{1}^{\varphi ud}$ we minimize the $\chi^{2}$ in Eq. (14) with respect
to the $\bar{\theta}$ and $\kappa_{LR}$.

  * •

From all possible values of the $\chi^{2}_{min}$ function, we choose those
values that give the most conservative bound. By the most conservative bound
we mean that choosing any other value for $a_{0},a_{1}$ and $\kappa_{S}$,
$\alpha_{n}$, $\lambda_{0}$, $\lambda_{1}$ and $\gamma_{1}^{\varphi ud}$
within the ranges reported in Ref. Chupp and Ramsey-Musolf (2015) and Tab. 1
gives a stronger bound. It turns out the values of $\chi^{2}_{min}$ giving the
more conservative bound are strongly correlated with the smallest values of
$\chi^{2}_{min}$ as can be seen from Fig. (2). Finally, notice from Fig. (2)
(right) that the largest values of $\bar{\theta}$ are also correlated with the
largest values for $\kappa_{LR}$.

  * •

Once the values of the hadronic parameters $\lambda_{0}$,
$\lambda_{1},\gamma_{1}^{\varphi ud}$, $\alpha_{n}$ and $\beta_{n}^{\varphi
ud}$ , the nuclear parameters $a_{0}$, $a_{1}$ and the atomic parameter
$\kappa_{s}$, which give the weakest bound are found, we determine the $68\%$
and $95\%$ C.L. intervals as shown in Fig. (3). In Fig. (3), we plot the
68$\%$ and 95$\%$ confidence level interval in the $(\bar{\theta}$,
$\kappa_{LR})$ plane. The $\bar{\theta}$ and $\kappa_{LR}$ parameters are
bounded to be inside the colored regions. The blue star denotes the location
of the best-fit value. This value is closer to the limit $\bar{\theta}\lesssim
4\times 10^{-10}$ ($\kappa_{LR}\lesssim 3\times 10^{-6}$) derived when
assuming the dominance of the $\bar{\theta}$ term (or the $\kappa_{LR}$
contribution). As can be seen, these sole-source bounds get relaxed by roughly
one order of magnitude when doing the global analysis.

For completeness, we show in Tab. 2 the best fit values values for the atomic
and nuclearparameters used in our global fit. In Tab. 3 we give the best fit
values for the hadronic parameter $\alpha_{n}$, $\beta_{n}$, $\lambda_{0}$,
$\lambda_{1}$ and $\gamma_{1}^{\varphi ud}$.

| Atomic and nuclear parameters
---|---
EDM System | $\kappa_{S}(\text{fm}^{-2})$ | $a_{0}$ | $a_{1}$
Mercury ($\mathrm{Hg}$) | $-2.8\times 10^{-4}$ | $0.022$ | $0.0029$
Xenon ($\mathrm{Xe}$) | $2.7\times 10^{-5}$ | -0.036 | -0.024
Radium ($\mathrm{Ra}$) | $-7.6\times 10^{-4}$ | -3.45 | 5.1
Tellure Fluoride TlF | -0.74 | -0.012 | 0.16
Table 2:  Best fit values of the Schiff moments and the dependence of the Schiff moment on $\bar{g}_{\pi}^{(0)}$ and $\bar{g}_{\pi}^{(1)}$ that give the more conservative bounds in the allowed ranges reported in Ref. Chupp and Ramsey-Musolf (2015). These are the values used to obtained the confidence level intervals shown in Fig. (3).  Hadronic parameter | Best fit value
---|---
$\alpha_{n}[e\cdot\text{fm}]$ | $0.5\times 10^{-3}$
$\beta_{n}^{\varphi ud}[e\cdot\text{fm}]$ | $8.4\times 10^{-8}$
$\lambda_{0}$ | $0.017$
$\lambda_{1}$ | $2.7\times 10^{-4}$
$\gamma_{1}^{\varphi ud}$ | $311\times 10^{-7}$
Table 3:  Best fit values of the hadronic parameters that gives the more
conservative bound for their allowed ranges reported in Table 7 of Ref. Engel
_et al._ (2013). The range of the parameter $\gamma_{1}$ has been updated to
include the RGE effects and the short distance uncertainty reported in Ref.
Cirigliano  _et al._ (2017)

In Fig. (4) and Fig. (5) of the Appendix, we show the value of
$\chi^{2}_{min}$ as a function of $a_{1}(\mathrm{Hg})$ and
$a_{1}(\mathrm{Ra})$. Notice that the best fit prefer small and positive
values of the parameter $a_{1}(\mathrm{Hg})$, whereas there is a preference
for the smallest value of $a_{1}(\mathrm{Ra})$. For the other parameter, we do
not find this correlation and hence these are not interesting to report.

##  V The Strong CP problem in the minimal LR symmetric model

The strong CP problem within the mLRSM has been recently discussed Refs.
Maiezza  _et al._ (2010); Maiezza and Nemevˇsek (2014); Senjanović and Tello
(2016); Senjanovic and Tello (2020). For the sake of completeness and to
emphasize the usefulness of our fit we highlight the main points here. As a
concrete example take the expression of the neutron EDM

$\displaystyle d_{n}\simeq\alpha_{n}\,\bar{\theta}+\beta_{n}^{\varphi
ud}\,\left(\frac{v^{2}}{\Lambda^{2}}\right)\,Im\left(C_{\varphi ud}\right),$
(15)

where

$\left(\frac{v^{2}}{\Lambda^{2}}\right)Im\left(C_{\varphi
ud}\right)=\kappa_{LR}$ (16)

and $\alpha_{n}$ and $\beta_{n}^{\varphi ud}$ are given in Tab. 3. In what
follows we discuss the situation for both $\mathcal{P}$ and $\mathcal{C}$ as
the LR symmetry.

### Parity as the left-right symmetry.

For $\mathcal{P}$, one has $V_{ud}^{L}=V_{ud}^{R}+\mathcal{O}(\tan
2\beta\sin\alpha)$ Senjanović and Tello (2015, 2016), and at leading order in
$v_{2}/v_{1}$ we have the following relation

$\displaystyle d_{n}\simeq\alpha_{n}\,\bar{\theta}-\beta_{n}^{\varphi
ud}\,\left(\frac{M_{W}^{2}}{M_{W_{R}}^{2}}\right)\,|V_{ud}^{L}|^{2}\sin
2\beta\sin\alpha+\mathcal{O}(v_{2}^{2}/v_{1}^{2})$ (17)

Under the hypothesis that $\mathcal{P}$ is an exact symmetry, $\theta_{0}=0$.
In this case $\bar{\theta}\simeq\frac{m_{t}}{2m_{b}}\tan 2\beta\sin\alpha$
Maiezza and Nemevˇsek (2014), where $m_{b}$ and $m_{t}$ are bottom and top
quark masses, respectively. Invoking an explicit breaking of $\mathcal{P}$ in
the strong sector alone, such that $\theta_{0}$ cancels the
$\frac{m_{t}}{2m_{b}}\tan 2\beta\sin\alpha$ contribution to $\bar{\theta}$
would not work. As discussed in Ref. Maiezza and Nemevˇsek (2014), by doing an
anomalous chiral rotation, a tiny explicit breaking in the strong sector would
induce a large amount of explicit breaking in the Yukawa sector, and the
relation $V^{L}\simeq V^{R}$ would be lost (this has been recently discussed
in Ref. Maiezza (2020)).

By requiring the $\bar{\theta}$ parameter to be smaller than its experimental
limit $\theta_{exp}\sim 10^{-9}$ (see Fig. 3) one gets $\tan
2\beta\sin\alpha\leq\frac{2m_{b}}{m_{t}}\theta_{exp}\sim 10^{-12}$, which
makes all the phases in $V_{R}$ effectively zero – with exception of the Dirac
phase $\delta_{CKM}$, which gives a negligible contribution to the neutron EDM
$\sim 10^{-32}\,e\cdot$cm Seng (2015). The parameter $\tan 2\beta\sin\alpha$
gauging the $\bar{\theta}$ contribution is essentially the same one entering
in the new physics corrections shown in the second term of Eq.(17). Therefore,
since $\beta_{n}^{\varphi ud}\ll\alpha_{n}$ one can write the neutron EDM as

$\displaystyle d_{n}\simeq\alpha_{n}\,\bar{\theta}.$ (18)

Whereby, if $\mathcal{P}$ is an exact symmetry, one gets the prediction that
all the EDM should be dominated by the $\bar{\theta}$ term and this would give
precise patterns for the ratios of different EDMs. Note that as argued in Ref.
de Vries  _et al._ (2019) in the context of the SM effective field theory for
dim-6 operators, if the strong CP problem is solved by ultraviolet physics,
then the low energy EDMs should be dominated by the $\bar{\theta}$ term.
Similar considerations of using EDM ratios to single out the underlying
mechanism of CP violation have also been considered in Refs. Lebedev  _et al._
(2004); Bsaisou  _et al._ (2015); Dekens  _et al._ (2014).

### Charge conjugation as the left-right symmetry.

For $\mathcal{C}$ the parameter $\bar{\theta}=\theta_{0}+\text{arg
det}(M_{u}M_{d})$ with $\theta_{0}\neq 0$. This means that both the
$\bar{\theta}$ and the new physics contributions in Eq. (15) are independent
from each other. In this case one cannot get any limit using one EDM, since
one cannot invoke the idea of parity invariance to relate the first with the
second term in Eq. (15). In Fig. 3, we show the result of the global fit using
several EDM systems. We found that the typical limit $\bar{\theta}\sim
1.5\times 10^{-10}$ Dragos  _et al._ (2019) one gets when using one EDM system
gets relaxed to $|\bar{\theta}|\simeq 2.4\times 10^{-9}$ with a 95$\%$ of
confidence level. The same reasoning applies to the $\kappa_{LR}$
contribution. Unlike the case of $\mathcal{P}$, for $\mathcal{C}$ as the LR
symmetry, the ratio between different EDM do not are necessarily dominated by
the $\bar{\theta}$ term contribution.

##  VI Interplay with time-reversal symmetry violation in $\beta$-decays

In this section, we discuss the implication of our global fit to the so called
“D" coefficient in polarized $\beta$-decays Jackson  _et al._ (1957). The
neutron differential decay rate contains the triple-vector product
$d\Gamma/d\Omega\supset
D\,\langle\mathbf{\vec{J}}\rangle\cdot\vec{p}_{e}\times\vec{p}_{\nu}$, where
$\langle\mathbf{\vec{J}}\rangle$ is the nuclear polarization of the neutron,
$\vec{p}_{e}$ and $\vec{p}_{\nu}$ are the three-momentum of the outgoing
electron and neutrino, respectively. This coefficient can be decomposed as
$D=D_{f}+D_{t}$, where $D_{f}$ denotes the T-even final state interactions and
$D_{t}$ is the T-odd contribution, which gives the fundamental violation of
the time-reversal symmetry. At the lowest order in the electroweak corrections
$D=0$ and the SM corrections due to the Kobayashi-Maskawa phase and the
$\bar{\theta}$ term have been computed in Ref. Herczeg and Khriplovich (1997).
Using the current experimental limit for $\bar{\theta}$, the contributions to
the $D$ coefficient due the CKM phase and $\bar{\theta}$ are of the order
$\sim 10^{-12}$ and $\sim 10^{-15}$, respectively. Instead, the final state
interactions have been computed in Ref. Ando  _et al._ (2009) and are at the
level of $D_{f}\sim 10^{-5}$ with a $1\%$ accuracy. Keeping in mind the
current experimental limit coming from the neutron decay Mumm  _et al._
(2011a, b)

$D_{n}=(-1.0\pm 2.1)\times 10^{-4},$ (19)

there still a window for new physics observation up to $D_{n}\sim 10^{-7}$.

This constraint was previously studied in the context of the mLRSM in Ref. Ng
and Tulin (2012), where it was argued that EDM systems already exclude the
possibility of probing CP violation within the mLRSM model in $\beta$-decays
since $D_{t}\lesssim 10^{-7}$ from the neutron EDM. In this paper, we show
that this conclusion changes in the light of two facts. The first is that the
chiral effective field theory sensitivity of $d_{n}$ to the mLRSM parameter is
smaller than the one reported in Ref. Ng and Tulin (2012), as concluded in
Refs. Engel  _et al._ (2013); Seng  _et al._ (2014). The other is that our
global fit gives an upper bound to the mLRSM parameter. It turns out that in
the mLRSM, the same dim-6 operator in Eq. (III) that generates the EDM also
contributes to the $D$ coefficient in $\beta$-decay. One may thus express (for
example) $d_{n}$ in terms of ${\bar{\theta}}$ and $D_{t}$ as

$d_{n}=\alpha_{n}\bar{\theta}+\beta_{n}^{\varphi
ud}\left(\frac{D_{t}}{\kappa}\right)\,,$ (20)

where

$\frac{D_{t}}{\kappa}=\kappa_{LR}$ (21)

and $\kappa\simeq 0.87$ for the neutron Ng and Tulin (2012). Using the result
shown in Fig. 3 it is easy to see from our global fit that
$\left|\frac{D_{t}}{\kappa}\right|\leq 2.0\times 10^{-5}$ at $95\%$ C.L, which
is well above the uncertainty associated with $D_{f}$. Thus, a future, more
sensitive probe of the T-odd, triple correlation in $\beta$-decay would probe
a portion of the mLRSM parameter space not currently constrained by EDM
searches.

##  VII Conclusions

In this paper, we performed a global fit to limit the $\bar{\theta}$ parameter
and the new physics contribution in the context of the minimal left-right
symmetric model. We found that the “sole-source" limit of $\bar{\theta}\simeq
10^{-10}$ gets relaxed by roughly one order of magnitude from our global
analysis. We then discuss the implications for both parity and charge
conjugation as the left-right symmetry. This new limit cannot be evaded by
invoking cancellations between the $\bar{\theta}$ contribution and the new
physics one. Finally, we revisit the connection with the $D$ coefficient of
$\beta$-decay in the light of our global fit and find that there is still room
for the observation of fundamental time-reversal symmetry violation in
$\beta$-decay experiments.

## Appendix

Here, we show two figures giving the correlation between $\chi^{2}$ and the
nuclear parameter $a_{1}$ for mercury and radium.

Figure 4: Values of $\chi^{2}_{min}$ of the $a_{1}(\mathrm{Hg})$ nuclear
parameter.  Figure 5: Values of $\chi^{2}_{min}$ of the $a_{1}(\mathrm{Ra})$
nuclear parameter.

## Acknowlegdements

JCV thanks Goran Senanović for encouragement and useful and illuminating
discussions. JVC would like to thank T. Chupp for correspondence about the
global fit and Jordy De Vries for many valuable discussions and a careful
reading of the manuscript. JCV was supported in part under the U.S. Department
of Energy contract DE-SC0015376. MJRM was supported in part under the U.S.
Department of Energy contract DE-SC0011095. MJRM was also supported in part
under National Science Foundation of China grant No. 19Z103010239.

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