[a]Aniket Sen

# Hadronic Parity Violation from 4-quark Interactions

Marcus Petschlies  Nikolas Schlage  Carsten Urbach

###### Abstract

We present an exploratory investigation of the parity odd $\Delta I=1$ pion-
nucleon coupling $h_{\pi}^{1}$ from lattice QCD. Based on the PCAC relation,
we study the parity-conserving effective Hamiltonian and extract the coupling
by determining the nucleon mass splitting arising from the effective 4-quark
interactions using the Feynman-Hellmann theorem. We present preliminary
results of the mass shift for a $32^{3}\times 64$ ensemble of $N_{f}=2+1+1$
twisted mass fermions at pion mass $260\,\mathrm{MeV}$ and lattice spacing
$a=0.097\,\mathrm{fm}$.

##  1 Introduction

The study of parity (P) violation in the weak sector of the Standard Model
started in the 1960s, immediately following the seminal paper by Lee and Yang
[1] and the first experimental evidence by Madam Wu and collaborators [2].
Despite countless experiments over the last six decades, the study of hadronic
parity violation (HPV) is still of importance. Our focus here is on flavor-
conserving hadronic interactions. For the latter there is an overwhelming
background from strong interaction, governed by quantum chromodynamics (QCD),
and electromagnetic interaction. In particular, since the strong coupling is
significantly stronger than the weak coupling, extracting the smaller signal
contribution from P-violating hadronic weak interaction is rendered a
difficult task.

The most used theory model of HPV is based on the exchange of single light
mesons, as developed by Desplanques, Donoghue and Holstein (DDH) [3]. This
model contains seven independent coupling constants. A long-standing problem
in HPV is the determination of these P-violating coupling constants. Of
particular interest to us is the $\Delta I=1$ pion-nucleon coupling
$h_{\pi}^{1}$. This particular transition is suppressed for the weak charged
current, and therefore serves as a probe for the in this case dominant neutral
current. Since the underlying weak operators and the Wilson coefficients are
known, the only remaining work is the calculation of the hadronic matrix
elements, either through non-perturbative methods or through fits to
experimental data.

On the side of experiments, a recent measurement by the NPDGamma collaboration
produced the first experimental determination of $h_{\pi}^{1}$ [4]. On the
theoretical side, however, there has been only one study using lattice QCD
[5]. Despite the fact that this theoretical determination overlaps very nicely
with the experimental measurement, one needs to be careful when considering
this result. The study was done on a single lattice with pion mass of 390 MeV,
and contributions from the so-called "quark-loop" diagrams were not
considered. The author also used 3-quark interpolators for the $N\pi$ final
state in order to simplify Wick contractions. These issues were pointed out by
Feng, Guo and Seng (FGS) in their 2018 paper [6], where they also proposed a
different strategy for determining $h_{\pi}^{1}$. They proposed to get rid of
the $N\pi$ final state by using partially conserved axial current (PCAC)
relations and to determine the coupling constant in terms of the induced
relative mass shift between proton and neutron. Our work serves as a benchmark
for this newly proposed technique.

##  2 The 4-quark operators

According to the DDH model, the parity violating weak Lagrangian, in the
hadronic sector, is written as

$\displaystyle\mathcal{L}_{PV}^{w}=\frac{h_{\pi}^{1}}{\sqrt{2}}\,\bar{N}\,\left(\vec{\tau}\times\vec{\pi}\right)_{3}\,N+\cdots\,.$
(1)

Here $N=(p\;n)^{T}$ is the isospin doublet of nucleons and $\vec{\tau}$ are
the Pauli matrices. One then obtains the coupling constant $h_{\pi}^{1}$ from
the hadronic matrix element

$\displaystyle h_{\pi}^{1}=-\frac{i}{2m_{N}}\lim_{p_{\pi}\to 0}\langle
n\pi^{+}|\mathcal{L}_{PV}^{w}(0)|p\rangle\,,$ (2)

where $m_{N}$ is the average nucleon mass. At low energies
$\mathcal{L}_{PV}^{w}$ can be expressed in terms of seven independent P-odd
4-quark operators [7]

$\displaystyle\mathcal{L}_{PV}^{w}=-\frac{G_{F}}{\sqrt{2}}\,\frac{\sin^{2}\theta_{w}}{3}\sum_{i}\left(C_{i}^{(1)}\theta_{i}+S_{i}^{(1)}\theta_{i}^{(s)}\right)\,,$
(3)

where

$\displaystyle\theta_{1}$
$\displaystyle=\bar{q_{a}}\gamma_{\mu}q_{a}\bar{q}_{b}\gamma_{\mu}\gamma_{5}\tau_{3}q_{b},\;\;\;\;\;\;\theta_{2}=\bar{q}_{a}\gamma_{\mu}q_{b}\bar{q}_{b}\gamma_{\mu}\gamma_{5}\tau_{3}q_{a},$
(4) $\displaystyle\theta_{3}$
$\displaystyle=\bar{q}_{a}\gamma_{\mu}\gamma_{5}q_{a}\bar{q}_{b}\gamma_{\mu}\tau_{3}q_{b},\;\;\;\;\;$
$\displaystyle\theta_{1}^{(s)}$
$\displaystyle=\bar{s}_{a}\gamma_{\mu}s_{a}\bar{q}_{b}\gamma_{\mu}\gamma_{5}\tau_{3}q_{b},\;\;\;\;\;\theta_{2}^{(s)}=\bar{s}_{a}\gamma_{\mu}s_{b}\bar{q}_{b}\gamma_{\mu}\gamma_{5}\tau_{3}q_{a}\,,$
$\displaystyle\theta_{3}^{(s)}$
$\displaystyle=\bar{s}_{a}\gamma_{\mu}\gamma_{5}s_{a}\bar{q}_{b}\gamma_{\mu}\tau_{3}q_{b},\;\;\;\;\;\theta_{4}^{(s)}=\bar{s}_{a}\gamma_{\mu}\gamma_{5}s_{b}\bar{q}_{b}\gamma_{\mu}\tau_{3}q_{a}\,.$

Here, $q=(u\;d)^{T}$ is the light quark doublet, $G_{F}$ is the Fermi constant
and $\theta_{w}$ is the weak mixing angle. $C_{i}$ and $S_{i}$ are well known
Wilson coefficients. The values of these coefficients, at the scale
$\Lambda_{\chi}\approx 1\,\mathrm{GeV}$, are [13]

$\displaystyle C^{(1)}(\Lambda_{\chi})$
$\displaystyle=\begin{pmatrix}-0.055,&0.810,&-0.627\end{pmatrix}$ (5)
$\displaystyle S^{(1)}(\Lambda_{\chi})$
$\displaystyle=\begin{pmatrix}5.09,&-2.55,&4.51,&-3.36\end{pmatrix}$

The presence of a soft pion in the final state of the matrix element greatly
complicates the calculation for reasons pointed out in [8]. FGS proposed to
get rid of the soft pion by relating this matrix element to a parity
conserving counterpart, using the PCAC relation

$\displaystyle\lim_{p_{\pi}\to 0}\langle
a\pi^{i}|\hat{O}|b\rangle=\frac{i}{F_{\pi}}\langle
a|\left[\hat{O},\hat{Q}_{A}^{i}\right]|b\rangle\,,$ (6)

where $F_{\pi}=92.1\,\mathrm{MeV}$ is the pion decay constant and
$\hat{Q}_{A}^{i}$ are the axial charges. This means instead of calculating
$\langle n\pi^{+}|\mathcal{L}_{PV}^{w}|p\rangle$, we can simply compute
$\langle n|\left[\mathcal{L}_{PV}^{w},\hat{Q}_{A}^{i}\right]|p\rangle$. At the
operator level, the PCAC relation yields

$\displaystyle\left[\theta_{q},\hat{Q}^{i}_{A}\right]=i\epsilon^{3ij}\theta^{\prime}_{q(j)},\;\;\;\;\;\left[\theta_{q}^{(s)},\hat{Q}^{i}_{A}\right]=i\epsilon^{3ij}\theta_{q(j)}^{(s)^{\prime}}\,,$
(7)

where $\theta^{\prime}_{q}$ and $\theta^{(s)^{\prime}}_{q}$ are the parity
even 4 quark operators defined as

$\displaystyle\theta^{{}^{\prime}\phantom{(s)}}_{1}$
$\displaystyle=\bar{q}_{a}\gamma_{\mu}q_{a}\bar{q}_{b}\gamma_{\mu}\tau_{3}q_{b}\,,\quad$
$\displaystyle\theta^{{}^{\prime}\phantom{s}}_{2}$
$\displaystyle=\bar{q}_{a}\gamma_{\mu}q_{b}\bar{q}_{b}\gamma_{\mu}\tau_{3}q_{a}\,,$
$\displaystyle\theta^{{}^{\prime}\phantom{(s)}}_{3}$
$\displaystyle=\bar{q}_{a}\gamma_{\mu}\gamma_{5}q_{a}\bar{q}_{b}\gamma_{\mu}\gamma_{5}\tau_{3}q_{b}\,,$
(8) $\displaystyle\theta_{1}^{(s)^{\prime}}$
$\displaystyle=\bar{s}_{a}\gamma_{\mu}s_{a}\bar{q}_{b}\gamma_{\mu}\tau_{3}q_{b}\,,\quad$
$\displaystyle\theta_{2}^{(s)^{\prime}}$
$\displaystyle=\bar{s}_{a}\gamma_{\mu}s_{b}\bar{q}_{b}\gamma_{\mu}\tau_{3}q_{a}\,,$
$\displaystyle\theta_{3}^{(s)^{\prime}}$
$\displaystyle=\bar{s}_{a}\gamma_{\mu}\gamma_{5}s_{a}\bar{q}_{b}\gamma_{\mu}\gamma_{5}\tau_{3}q_{b}\,,\quad$
$\displaystyle\theta_{4}^{(s)^{\prime}}$
$\displaystyle=\bar{s}_{a}\gamma_{\mu}\gamma_{5}s_{b}\bar{q}_{b}\gamma_{\mu}\gamma_{5}\tau_{3}q_{a}\,.$
(9)

This maps the matrix elements of P-odd hadronic operators to matrix elements
to P-even ones to leading order in chiral perturbation theory

$\displaystyle\lim_{p_{\pi}\to 0}\langle
n\pi^{+}|\mathcal{L}_{PV}^{w}(0)|p\rangle\approx-\frac{\sqrt{2}i}{F_{\pi}}\langle
p|\mathcal{L}_{PC}^{w}(0)|p\rangle=\frac{\sqrt{2}i}{F_{\pi}}\langle
n|\mathcal{L}_{PC}^{w}(0)|n\rangle\,,$ (10)

where $\mathcal{L}_{PC}^{w}$ is the following auxiliary parity conserving
Lagrangian

$\displaystyle\mathcal{L}_{PC}^{w}=-\frac{G_{F}}{\sqrt{2}}\frac{\sin^{2}\theta_{W}}{3}\sum_{i}\left(C_{i}^{(1)}\theta^{\prime}_{i}+S^{(1)}_{i}\theta^{(s)^{\prime}}_{i}\right)\,.$
(11)

Since the PCAC relation holds at the operator level, the Wilson coefficients
are identical to those in $\mathcal{L}_{PV}^{w}$. The matrix element can be
determined in terms of the neutron-proton mass splitting $(\delta
m_{N})_{4q}\equiv(m_{n}-m_{p})_{4q}$ induced by the Lagrangian
$\mathcal{L}_{PC}^{w}$

$\displaystyle(\delta m_{N})_{4q}=\frac{1}{m_{N}}\langle
p|\mathcal{L}_{PC}^{w}(0)|p\rangle=-\frac{1}{m_{N}}\langle
n|\mathcal{L}_{PC}^{w}(0)|n\rangle\,.$ (12)

Combining everything, one obtains the coupling constant $h_{\pi}^{1}$ as

$\displaystyle h_{\pi}^{1}\approx-\frac{(\delta
m_{N})_{4q}}{\sqrt{2}F_{\pi}}\,.$ (13)

##  3 Mass shift from Feynman-Hellmann theorem

According to Feynman-Hellmann theorem (FHT) [9, 10], a perturbation in the
Hamiltonian of the form $H\to H+\lambda H_{\lambda}$ corresponds to a
variation in the spectrum as

$\displaystyle\frac{\partial E_{n}}{\partial\lambda}=\langle
n|H_{\lambda}|n\rangle\,.$ (14)

This straightforward relation of first order in perturbation theory can be
used to extract matrix elements in lattice QCD [11]. Considering the 2-point
nucleon-nucleon correlator in the presence of an external source $\lambda$

$\displaystyle C_{\lambda}(t)$
$\displaystyle=\langle\lambda|N(t)\bar{N}(0)|\lambda\rangle=\frac{1}{Z_{\lambda}}\int\mathcal{D}\psi\,\mathcal{D}\bar{\psi}\;N(t)\bar{N}(0)\;e^{-S-S_{\lambda}}\,,$
(15)

where the source $\lambda$ is coupled to the 4-quark current introduced in the
previous section

$\displaystyle S_{\lambda}=\lambda\int d^{4}x\;\mathcal{L}_{PC}^{w}(x)\,.$
(16)

The derivative of the perturbed correlator gives

$\displaystyle-\frac{\partial
C_{\lambda}}{\partial\lambda}\bigg{|}_{\lambda=0}$
$\displaystyle=\frac{\partial
Z_{\lambda}}{\partial\lambda}\bigg{|}_{\lambda=0}\frac{C(t)}{Z}+\frac{1}{Z}\int\mathcal{D}\psi\mathcal{D}\bar{\psi}e^{-S}\int
d^{4}x\;\mathcal{L}_{PC}^{w}(x)N(t)\bar{N}(0)$ (17) $\displaystyle=-C(t)\int
d^{4}x\;\langle\Omega|\mathcal{L}_{PC}^{w}(x)|\Omega\rangle+\int
d^{4}x\;\langle\Omega|\mathcal{T}\\{N(t)\mathcal{L}_{PC}^{w}(x)\bar{N}(0)\\}|\Omega\rangle\,.$

The first term vanishes due to lattice symmetries. The second term is
effectively $\sum_{x}\langle p|\mathcal{L}_{PC}^{w}(x)|p\rangle$. The energy
shift is then obtained from the linear response of the effective mass

$\displaystyle\frac{\partial
m_{N}}{\partial\lambda}\bigg{|}_{\lambda=0}(t,\tau)=\frac{1}{\tau}\left[\frac{\partial_{\lambda}C_{\lambda}(t)}{C(t)}-\frac{\partial_{\lambda}C_{\lambda}(t+\tau)}{C(t+\tau)}\right]\,.$
(18)

Here $C$ is the unperturbed correlator ($\lambda=0$). We compute the right-
hand side of Eq. (18), which is proportional to the mass shift $(\delta
m_{N})_{4q}$.

##  4 Lattice Setup

The determination of $\langle p|\mathcal{L}_{PC}^{w}|p\rangle$ requires
calculation of three kinds of diagrams. Figure 1 gives a visual representation
of the 3 diagrams. There are also quark-disconnected diagrams, which vanish in
the limit of exact $SU(2)$ flavor symmetry and are neglected in the present
approach.

(a) B-type

(b) D-type

(c) W-type

Figure 1: The 3 kinds of diagrams under consideration. (Note that $c_{1}$ and
$c_{2}$ shown separately for clarity. Physically they are the same point on
the lattice.) The color coding for quark propagator types is as follows: red =
point-to-all, green = stochastic and blue = sequential propagator.

We use sequential propagator techniques to calculate these diagrams. For B and
D-type, we consider one fully time-, spin-, and color-diluted stochastic
source $\xi^{(k)}$, where $k$ denotes the composite dilution index,

$\displaystyle\xi^{(t,\beta,b)}(x)_{\alpha\atop a}$
$\displaystyle=\delta_{\alpha\beta}\,\delta_{ab}\,\delta_{x_{0}t}\,\xi(\vec{x})\,,$
(19) $\displaystyle\xi(\vec{x})$
$\displaystyle\sim\frac{1}{\sqrt{2}}\,\left(Z_{2}+iZ_{2}\right)\quad\mathrm{~{}iid}\,.$
(20)

With the corresponding propagator $\phi_{f}^{(k)}=D_{f}^{-1}\,\xi^{(k)}$ for
quark flavor $f$ we use the stochastic representation of the $x_{c}\to x_{c}$
quark propagator

$\displaystyle L_{f}(x_{c})_{\alpha\beta\atop
ab}=\sum\limits_{k}\,\phi_{f}^{(k)}(x_{c})_{\alpha\atop
a}\xi^{(k)*}(x_{c})_{\beta\atop b}$ (21)

such that $E\left[L_{f}(x_{c})\right]=S_{f}(x_{c};x_{c})_{\alpha\beta\atop
ab}$, i.e. the stochastic expectation behaves like a quark loop.

Sequential sources for B and D-type diagrams are then built accordingly as

$\displaystyle\Phi_{B}(x_{c};x_{i})$
$\displaystyle:=\Gamma_{c_{2}}\,\bar{L}(x_{c})\,\Gamma_{c_{1}}\,S(x_{c};x_{i})$
(22) $\displaystyle\Phi_{D}(x_{c};x_{i})$
$\displaystyle:=\text{Tr}\left[\Gamma_{c_{2}}\,\bar{L}(x_{c})\right]\,\Gamma_{c_{1}}\,S(x_{c};x_{i})\,.$
(23)

For W-type diagrams we consider a set of scalar noise sources with the
properties

$\displaystyle E\left[\eta^{(k)}(x)\right]=0,\hskip
20.00003ptE\left[\eta^{(k)}(x)\,\eta^{(l)}(y)\right]=\delta_{kl}\,\delta_{x,y}\,,$
(24)

with $E$ the expectation value, and insert such a noise between a pair of
propagators to obtain a W-type propagator

$\displaystyle\tilde{W}^{(k)}(\Gamma_{c})=\sum_{x_{c}}S(x_{f};x_{c})\,\Gamma_{c}\,\eta^{(k)}(x_{c})\,S(x_{c};x_{i})\,.$
(25)

The expectation of a product of two such propagators is needed for the W
diagram.

For the simulation, we have used $N_{f}=2+1+1$ clover-improved twisted mass
lattice ensemble $cA211.30.32$ produced by the _ETMC_ collaboration [12].
Table 1 lists the parameters of the gauge ensemble.

$L^{3}\times T$ | $a[fm]$ | $a\mu_{l}$ | $am_{\pi}$ | $m_{\pi}L$ | $m_{\pi}[MeV]$
---|---|---|---|---|---
$32^{3}\times 64$ | $0.097$ | $0.00300$ | $0.12530\,(16)$ | $4.01$ | $261.1\,(1.1)$
Table 1: Relevant parameters of gauge ensemble $cA211.30.32$. For more details
we refer to Ref. [12].

The propagators are source and sink smeared with Wuppertal smearing. APE
smearing is also performed to smooth the gauge.

For the loops in B and D-type diagrams we were limited to only one stochastic
source due to computational cost. We tried overcoming this by calculating at 8
different source coordinates per configuration. For W-type diagrams we used 8
stochastic sources and 2 source coordinates per configuration.

##  5 Results

In this work we present results for the 3 operators in the light quark sector,
Eq. (8). Figure 2 shows the 3-point correlators for the 3 operator insertions
(note we have dropped the ′ in the operator names). The symmetry between
$\theta_{1}$ and $\theta_{3}$ is expected at least at tree level. $\theta_{2}$
has considerably more noise. We suspect that it could be due to mixing under
renormalization with the lower dimensional operator
$\bar{\psi}\,\gamma_{5}\,\psi$.

Figure 2: 3-point correlators with the 4-quark operator insertions

The mass shift is calculated according to FHT. Figure 3 shows the $t$ and
$\tau$ dependence of the shift for $\theta_{1}$. The noise becomes
prohibitively large for $t/a\geq 12$. Figure 4 shows the mass shift fitted in
range $\tau/a\in[3,7]$ for the timeslices $t/a\in[6,11]$.

Figure 3: Large $t$ and $\tau$ dependence of $\theta_{1}$ (for each timeslice
the data is shown for the range $\tau/a\in[3,8]$ )

Given the exploratory nature of this study, we do not intend to determine a
value for the coupling $h_{\pi}^{1}$ here yet. However, the discrepancy
between the value obtained by naively adding these three plateau values with
corresponding Wilson coefficients and the experimental value most likely stem
from the missing renormalization process and the missing strange
contributions. We are currently working on both.

Figure 4: Mass shifts fitted over a range of $\tau/a\in[3,7]$ for each
timeslice

##  6 Conclusion

This work represents an exploratory benchmark for a newly proposed technique
to measure the coupling $h_{\pi}^{1}$. So far the results are promising. Non-
zero signal has been observed for all diagrams with all Dirac structures. The
mass shifts for individual operators have also turned out to be non-zero. Thus
the practical simplification of signal extraction envisioned by FGS is
reachable.

Work on the strange operators and renormalization of the 4-quark lattice
operators is currently ongoing. A parallel computation with traditional
3-point technique is also underway in order to compare cost and signal quality
with the current work. Once our setup is optimized, we aim at a calculation
directly at physical pion mass to obtain a theoretical estimate for
$h_{\pi}^{1}$, which is rigorously comparable to the experimental
determination.

## Acknowledgments

This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) and the NSFC through the funds provided to the Sino-
German Collaborative Research Center CRC 110 “Symmetries and the Emergence of
Structure in QCD” (DFG Project-ID 196253076 - TRR 110, NSFC Grant No.
12070131001).

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