# Effective field theory in the study of long range nuclear parity violation on lattice

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## Abstract

A non-zero signal \(A_\gamma ^\mathrm {np}=(-3.0\pm 1.4\pm 0.2)\times 10^{-8}\) of the gamma-ray asymmetry in the neutron-proton capture was recently reported by the NPDGamma Collaboration which provides the first determination of the \(\Delta I=1\) parity-odd pion-nucleon coupling constant \(h_\pi ^1=(2.6\pm 1.2\pm 0.2)\times 10^{-7}\). The ability to reproduce this value from first principles serves as a direct test of our current understanding of the interplay between the strong and weak interaction at low energy. To motivate new lattice studies of \(h_\pi ^1\), we review the current status of the theoretical understanding of this coupling, which includes our recent work that relates it to a nucleon mass-splitting by a soft-pion theorem. We further investigate the possibility of calculating the mass-splitting on the lattice by providing effective field theory parameterizations of all the involved quark contraction diagrams. We show that the lattice calculations of the easier connected diagrams will provide information of the chiral logarithms in the much harder quark loop diagrams and thus help in the chiral extrapolation of the latter.

## 1 Introduction

The study of parity (P)-violation in nuclear and atomic systems has continued to be a central topic in the low-energy community despite that the P-violation in Standard Model (SM) electroweak (EW) sector is well-established and all the EW parameters are already quite precisely measured. The reason is that we are really using the hadronic weak interaction (HWI) as a tool to understand the peculiarities in the strong interaction dynamics. The non-perturbative nature of Quantum Chromodynamics (QCD) in the confinement region resembles a black box that asserts non-predictable dressings to the confined quarks in a hadron. Therefore, in order to examine its properties, the “bare” weak interaction which is well-understood serves as a probe inserted into the black box which then returns the HWI that is experimentally measured. The role of P-violation in this procedure is also obvious: as the effective strong interaction coupling is \(10^6\) times larger than the weak coupling, one relies entirely on a symmetry-violating signal to disentangle the HWI from the huge strong interaction background. It is therefore not the discovery of a non-zero P-violation signal in HWI, but the precise measurement of its value that will provide us with the opportunity of testing the SM with the interplay between weak and strong interactions.

Effects of the hadronic parity violation (HPV) are usually classified according to their isospin, and among all others the \(\Delta I=1\hbox { HPV}\) possesses a special role as a unique probe of the hadronic weak neutral current. Moreover, it is the only channel that allows for a single pion-exchange, and hence plays a dominant role in the long-range HPV. Also, the \(\Delta I=1\) P-odd pion-nucleon coupling plays a nontrivial role in the \(\vec p p\) scattering through two-pion exchange as discussed in Refs. [1, 2]. The recent observation of a P-violating 2.2 MeV gamma-ray asymmetry \(A_\gamma ^\mathrm {np}=(-3.0\pm 1.4\pm 0.2)\times 10^{-8}\) in the polarized neutron capture on hydrogen by the NPDGamma Collaboration [3] provides the first solid experimental confirmation of the isovector HPV, and is promised to create a new stir to the field that has been suffering from a “slow pace of (experimental) results since 1980” [4]. It is therefore timely to review our current knowledge of HPV and discuss how it could be improved by making the fullest use of the new experimental result.

Early attempts to describe HPV at the phenomenological level are based on isospin symmetry and perturbative expansions of small interaction energies, a strategy that is now inherited by the effective field theory (EFT) approach. A well-known example of such a kind is the work by Danilov [5] in the 60s that parameterized the P-odd nucleon-nucleon interaction at very low energy in terms of five *S*–*P* transition amplitudes with \(\Delta I=0,1,2\). The ground-breaking work by Desplanques, Donoghue and Holstein (DDH) [6] in the early 80s adopted a very different starting point, namely to describe HPV through single exchange of light mesons \(\pi ,\rho \) and \(\omega \) with seven independent nucleon-meson coupling constants. Despite being a model, its succinctness has attracted much attentions and has become the basis of many experimental analysis. The development of the EFT description of HPV [7, 8, 9, 10, 11, 12, 13, 14] signifies a switch to a model-independent framework that features pion-exchanges and contact terms, where a systematic power expansion with respect to a typical small momentum scale *p* ensures the finiteness of the number of operators needed in any given order. Translation tables, sometimes known informally as the “Rosetta stone” [4, 15], are available to connect these many different effective descriptions of the same physics [2, 12, 16] (where the cutoff dependence is also discussed for the translation). Finally, nuclear model calculations have been carried out to connect the HPV coupling strengths to the experimental observables in nuclear or atomic systems; examples in the \(\Delta I=1\) channel include Refs. [12, 17, 18],

Existing model calculations of \(h_{\pi }^{1}\) in comparison to the implied value from the NPDGamma experiment quoted in Ref. [3]

Models | \(h_{\pi }^{1}\) |
---|---|

DDH range [6] | \((0-1)\times 10^{-6}\) |

Quark model [19] | \(1.3\times 10^{-7}\) |

Quark model [20] | \(2.7\times 10^{-7}\) |

Quark model [21] | \(8.7\times 10^{-8}\) |

SU(2) Skyrme [22] | \(1.8\times 10^{-8}\) |

SU(2) Skyrme [23] | \(2\times 10^{-8}\) |

SU(3) Skyrme [24] | \((0.8-1.3)\times 10^{-7}\) |

QCD sum rule [25] | \(3\times 10^{-7}\) |

\(3.4\times 10^{-7}\) | |

NPDGamma [3] | \((2.6\pm 1.2\pm 0.2)\times 10^{-7}\) |

Lattice QCD is currently the only available approach to compute low-energy hadronic observables from the first principle with a controlled error. Unfortunately, in contrast to the steady progress made in the lattice calculation of \(\Delta I=2\) P-odd amplitudes [32, 33], there is so-far only one very preliminary study of \(h_\pi ^1\) by Wasem in Ref. [34] with no follow-ups. In that work, a three-point correlation function is computed to obtain the matrix element \(\left\langle n\pi ^+\right| {\mathcal {O}}_{\mathrm {PV}}^{\Delta I=1}\left| p\right\rangle \) with \(L=2.5~\hbox {fm}\), \(a=0.123~\hbox {fm}\) and \(m_\pi =389~\hbox {MeV}\), and the reported result is \(h_\pi ^1=\left( 1.099\pm 0.505^{+0.058}_{-0.064}\right) \times 10^{-7}\). Despite being consistent with the NPDGamma result, this number should not be taken seriously due to the existence of several unquantified assumptions as pointed out in Ref. [33]: (1) the three-quark representation of the \(N\pi \) interpolator; (2) the negligence of the so-called “quark loop diagrams”; (3) the calculation was done with only a single choice of volume, lattice spacing and pion mass; and (4) the lattice renormalization was not performed. We find the current situation not totally satisfactory because although the lattice calculation in the \(\Delta I=2\) channel is technically simpler, there is no existing HPV experiment to our knowledge that depends only on the \(\Delta I=2\) couplings (see, e.g. Ref. [4] for a summary) so that its comparison with experiments will not be straightforward. In contrast, a successful calculation of \(\Delta I=1\hbox { HPV}\) can be directly confronted to the NPDGamma result. Therefore, despite all the technical difficulties, we believe a renewed lattice study of \(h_\pi ^1\) is extremely worthwhile, and in this work we discuss how the proper application of a chiral EFT in the continuum space may help in alleviating part, if not all, of such difficulties.

The contents of this paper are as follows. We first introduce the theoretical basis of the \(\Delta I=1\hbox { HPV}\), including the underlying four-quark operators, their Wilson coefficients and the rigorous definition of the coupling \(h_\pi ^1\) as a soft-pion matrix element. Next, we review the soft-pion theorem derived in our previous work [35] and present some of the technical details not included in that Letter. Then, we begin the analysis of contraction diagrams by rigorously defining them in terms of three-point correlation functions. With the aid of the partially-quenched chiral perturbation theory (PQChPT), we derive the theoretical expression for each contraction diagram that contributes to \(h_\pi ^1\) as a function of the pion mass; such expressions are useful in performing chiral extrapolations from unphysical light quark masses to the physical ones. We point out that there are only a small number of LECs needed to fix the matrix elements, and provide approximate relations between different LECs that may facilitate their global fit. Finally, we briefly discuss the four-quark operators with strange quark fields and draw our conclusions.

## 2 Theoretical basis

*W*or

*Z*boson leads to a P-odd interaction between a pair of quarks. At the energy scale \(E\ll m_W,m_Z\), the

*W*or

*Z*propagator shrinks to a point, so we obtain effective four-quark interactions involving the product of two weak currents.

In this work we focus on the \(\Delta I=1\) P-violation in nucleon-nucleon interactions, and one may deduce from Eqs. (2) and (3) that they are dominated by neutral current interactions. An easy way to understand this is to realize that in the \(\theta _C\rightarrow 0\) limit, the first and the second generations of quarks completely decouple in the current level, and the charged weak current involving light quarks then reads \(J_W^\mu ={\bar{u}}\gamma ^\mu (1-\gamma _5)d\) which is purely an isovector. Therefore, the symmetric combination \(J_W^{\mu \dagger }J_{W,\mu }\) can only form \(\Delta I=0,2\) objects but not \(\Delta I=1\). In reality, the Cabibbo angle is not zero but the charged weak current contribution is suppressed by \(\sin ^2\theta _C\approx 0.05\) so the neutral current contribution is still dominant. This is an important observation as it identifies the \(\Delta I=1\) HWI as one of the very few direct experimental probes of the quark-quark neutral current effects at low energy.

^{1}

*a*,

*b*are the color indices. The running of the Wilson coefficients \(\{C_i^{(1)},S_i^{(1)}\}\) has been calculated to leading order (LO) in Refs. [7, 37] and to next-to-leading order (NLO) in Ref. [36]. We quote the results of the latter at the scale \(\Lambda _\chi \approx 1\hbox { GeV}\):

## 3 From P-odd to P-even matrix element

This section mainly serves as a review of the results in our previous work [35] with some more technical details added.

### 3.1 PCAC relation

The matrix element in Eq. (8) involves a soft pion in the final state that greatly complicates its analysis. We shall illustrate this point by considering a possible lattice QCD calculation of such a matrix element. First, one needs to choose a form of the interpolator for the \(n\pi ^+\) state. The most natural choice with the largest overlap with the physical state is obviously a five-quark interpolator, for example, \(\varepsilon ^{abc}d^a(u^{bT}C\gamma _5d^c){\bar{d}}^e\gamma _5 u^e\). Such a choice will however lead to many contraction diagrams, some of which involving up quark propagators between *n* and \(\pi ^+\) are noisy and expensive. Another possible choice is a three-quark interpolator with negative parity, as adopted in Ref. [34], \(\varepsilon ^{abc}\gamma _5 u^a(d^{bT}C\gamma _5 u^c)\). This choice avoids the calculations of the \(n\pi ^+\) contraction diagrams, but cannot avoid a large overlap with single-nucleon excited states, e.g., the *N*(1535). Thus, the use of a three-quark interpolator would be unjustified without properly taking into account the excited-state contaminations. Next, the rescattering effect between the final-state \(n\pi ^+\) modifies the finite-volume correction on lattice from an exponentially-suppressed effect to a power-suppressed effect. Finally, while we want the final-state pion to have a vanishing momentum squared, lattice QCD only computes matrix elements of on-shell states. As a result, the lattice calculation returns not just \(h_\pi ^1\) but its linear combination with the LECs of total-derivative operators that must be introduced to compensate the energy difference between the initial *p* and the final \(n\pi ^+\) [39]. Although the leading effect can be canceled by considering the difference between \({p\rightarrow n\pi ^+}\) and \({n\rightarrow p\pi ^-}\), but the \(m_q\)-suppressed terms still retain, and they are in principle indistinguishable from the \(m_q\)-dependent terms of \(h_\pi ^1\) and lead to a sizable systematic error.

*a*,

*b*are hadrons. This applies exactly to our case: Instead of computing \(\left\langle n\pi ^+\right| {\mathcal {L}}_\mathrm {PV}^w\left| p\right\rangle \), one may compute \(\left\langle n\right| [{\mathcal {L}}_\mathrm {PV}^w,{\hat{Q}}_A^-]\left| p\right\rangle \) which is much simpler. To that end, it is beneficial to introduce the following four-quark operators,

We would like to point out that the idea above is not at all new. To our knowledge, the first application of PCAC in the study of the \(\Delta S=0\) weak pion-baryon vertex appeared in Ref. [40] in the late 60s; it was also adopted in the DDH paper [6] as well as Ref. [22] as a starting point of their model-based estimation of \(h_\pi ^1\). The originality of Ref. [35] is really not in its application of PCAC, but rather in its quantitative analysis of the higher-order corrections which determines the degree of accuracy of the PCAC result, as we shall describe later.

### 3.2 Chiral perturbation theory analysis

The PCAC relation in Eq. (9) holds rigorously only in the exact chiral limit, i.e., when \(m_\pi =0\). For example, it predicts that the matrix element at the left hand side should vanish if \({\hat{O}}\) is chirally-invariant, which is obviously incorrect. Here we shall provide an immediate counter-example in a closely-related problem, namely the study of the P, T-odd pion-nucleon coupling \({\bar{g}}_\pi ^i\) induced by higher-dimensional operators. Eq. (9) suggests that chirally-invariant operators such as the Weinberg three-gluon operator \(f^{ABC}{\tilde{G}}^A_{\mu \nu }G^{B\nu }_{\rho }G^{C\rho \mu }\) would not contribute to \({\bar{g}}_\pi ^i\); however, we know in reality that the contribution of such an operator is non-zero, but just suppressed by powers of \(m_\pi \) [41]. Therefore, a truly practical application of the PCAC relation will need to take into account all the \(m_\pi \)-related corrections to the level of desired precision.

The above-mentioned task is made possible by recasting the PCAC statement in the language of chiral perturbation theory (ChPT), where Eq. (9) then becomes a simple consequence of two observables sharing the same LEC at the tree level. Higher-order corrections such as loop diagrams and counterterms to the left and the right sides of the equation can be computed order-by-order; any mismatch will then signifies a quantifiable violation of the tree-level matching. This idea was born of the in-depth studies of the P, T-odd pion-nucleon coupling \({\bar{g}}_\pi ^i\) [42, 43, 44, 45, 46, 47], and Ref. [35] constitutes its first implementation in HPV. Below we shall describe the method in detail.

*U*defined as

*N*appears as a matter field and can be chosen to transform as \(N\rightarrow KN\) under the chiral rotation, where

*K*is a spacetime-dependent matrix defined through the transformation property of \(u=\sqrt{U}\),

*N*is effectively replaced by its “light” component \(N_v\) which appears as a massless excitation, and the Dirac structures are effectively reduced as \(\gamma ^\mu \rightarrow v^\mu \) and \(\gamma ^\mu \gamma _5\rightarrow 2S^\mu \) where \(v^\mu \) is a constant four-velocity vector and \(S^\mu =i\gamma _5\sigma ^{\mu \nu }v_\nu /2\) is the nucleon spin-matrix satisfying \(S\cdot v=0\). With this, the LO nucleon Lagrangian becomes

^{2}Finally, one also needs to include the standard pion [56] and nucleon [49, 50] wavefunction renormalization as well as the higher-order correction to the pion decay constant \(F_\pi \),

^{3}

## 4 Baryon interpolators and the contraction diagram analysis

The objective of this paper is to investigate the possibility of performing a high-precision calculation of the coupling \(h_\pi ^1\) on the lattice. To that end, it is instructive to go through the existing limitations of the original Wasem calculation (as pointed out in Ref. [33]) and ask ourselves how many of them can be properly taken into account from a theoretical point of view. Clearly, our strategy in Ref. [35] avoids the unjustified application of the \(N\pi \) interpolator and alleviates the effect of the finite-volume correction, but does not resolve the other issues and therefore is not the end of the story. We shall devote the rest of this paper to the discussion of the remaining problems that could be at least addressed partially in a continuous field theory. In particular, we shall discuss the properties of different Wick contraction diagrams that occur in the the calculation of \((\delta m_N)_{4q}\) on lattice and the chiral extrapolation of the lattice result. We shall assume isospin symmetry throughout the rest of the paper.

*aabb*but the discussion of the operators with color contraction of the type

*abba*proceeds exactly the same way. We are interested in the matrix element of \({\hat{O}}\) with respect to the static proton state. Following the spirit of lattice QCD, we define a two-point and a three-point correlation function (with \(\tau>\tau _0>\tau '\)):

*L*the lattice size. Here, the summation also runs over all possible quantized momentum modes.

^{4}When \(\tau -\tau _0\rightarrow +\infty \) and \(\tau _0-\tau '\rightarrow +\infty \), only the static proton contribution that scales as \(\exp \{-m_N(\tau -\tau ')\}\) survives. Therefore one obtains

*C*is the charge conjugation matrix. It satisfies the following symmetry relations under the exchange of quark flavors,

*X*. With the exchange symmetry of \(\chi \) as shown in Eq. (38), it is straightforward to demonstrate that there are altogether eight types of independent contraction diagrams as depicted in Fig. 2. In each diagram, the ellipse at the left represents \({\bar{\chi }}(q_1,q_2,q_3;\vec {x}',\tau ')\), that at the right represents \(\chi (q_1,q_2,q_3;\vec {x},\tau )\), whereas a solid line represents the contraction between a pair of quark fields and the short dashed line denotes the four-quark vertex. We shall name the first two diagrams as “connected”, the next four as “quark loop” and the last two as “vacuum” diagrams, respectively. One may read off the explicit form of each contraction function \(X(\tau ,\tau _0,\tau ')\) from Fig. 2. For instance,where we have suppressed the spacetime coordinates: \((\vec {x},\tau )\) for the first three quark fields, \((\vec {x}_0,\tau _0)\) for the next four inside the square brackets and \((\vec {x}',\tau ')\) for the last three. Notice also that in such a definition the contraction function \(X(\tau ,\tau _0,\tau ')\) includes all possible minus signs due to the switching of positions between quark fields. For the vacuum contractions \(V_1(\tau ,\tau _0,\tau ')\) and \(V_2(\tau ,\tau _0,\tau ')\), we choose to normalize them according to the two-point function \(F_2(\tau ,\tau ')\) which is independent of the choice of baryon in the flavor degenerate limit. With these, we may now define the quantity \(M_X\) as

In practice, the lattice computations of different types of \(X(\tau ,\tau _0,\tau ')\) contractions will involve very different techniques. While the \(C_i(\tau ,\tau _0,\tau ')\) can usually be computed quite economically with high precision, the quark loop contractions \(D_{ai}(\tau ,\tau _0,\tau ')\) and \(D_{bi}(\tau ,\tau _0,\tau ')\) contain a quark propagator that starts and ends at the same spacetime point and is extremely noisy. For such propagators, one needs to average its value over all lattice points in order to improve its signal, but then it requires the use of all-to-all propagators that are computationally expensive [62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76]. As a consequence, with a given computational power, the precision level for the lattice outcomes of \(C_i\) and \(\{D_{ai},D_{bi}\}\) are very different. Of course, the physical results require summing all of them in the way given in Eq. (42) for the proton. However, we may employ the ability of calculating contractions separately on lattice as a handle to improve the precision of the final results. For that we can carry out the chiral extrapolations for different types of contractions separately, and this requires the analytic expression of each \(M_X\) as a function of the pion mass.

In the physical world it is usually not possible to separate the connected contractions from the quark loops in a given matrix element. Such separation is however possible in a QCD with an extended flavor sector. Let us consider a strong interaction theory with four fermionic quarks \(\{u,d,j,k\}\) and two bosonic “ghost” quarks \(\{{\tilde{j}},{\tilde{k}}\}\) with degenerate masses, which can be written collectively as \(q'=(u\,\,d\,\,j\,\,k\,|\,{\tilde{j}}\,\,{\tilde{k}})^T\). All internal dynamics of such a theory will be identical to the ordinary two-flavor QCD because all the loop effects brought up by the two extra fermionic quarks \(\{j,k\}\) (known as “valence quarks”) are exactly canceled by their corresponding bosonic partners \(\{{\tilde{j}},{\tilde{k}}\}\), keeping the sea DOFs unchanged. The net effect of this extension is that one introduces quark flavors that can only appear in external states but not in loops, and the strong interaction theory of such system is known as the partially-quenched QCD (PQQCD) [77, 78]. Within this framework, any contraction diagram of interest can be constructed by appropriately choosing the quark contents in either the external states or the operators [79]; such an idea has been previously applied in studies of the hadronic vacuum polarization [80], the pion scalar form factor [81] and the \(\pi \pi \) scattering amplitudes [82]. In our case, one could easily demonstrate that each quantity \(M_X\) can be written as a linear combination of four-quark matrix elements in PQQCD; an explicit example is given in Appendix B. With that we also show that each contraction function \(X(\tau ,\tau _0,\tau ')\) has the correct asymptotic exponential behavior of \(\exp \{-m_N(\tau -\tau ')\}\) that guarantees the existence of the limits in Eq. (41).

## 5 PQChPT analysis

In the previous section we have successfully separated each contraction diagram into well-defined matrix elements in PQQCD, and here we shall proceed to study the low-energy behavior of each individual contraction that contributes to the four-quark matrix element \(\left\langle p\right| {\hat{O}}\left| p\right\rangle \) of our interest. This involves the application of the low-energy EFT of PQQCD as follows.

### 5.1 Tree-level and one-loop results

*p*, \({\tilde{\Sigma }}^0\) and \({\tilde{\Lambda }}\). One may then compute the latter to one loop using HB PQChPT to obtain the former. Below we summarize the main results. First, at tree-level, we obtain

Let us try to understand the results above. The first important observation is that the two LECs \(\{\alpha _{C_1},\alpha _{C_2}\}\), which contribute only to connected diagrams at tree level, enter the quark loop diagrams in the form of chiral logarithms; this feature can be understood diagrammatically as depicted in Fig. 3. On the other hand, there is no way that the LECs \(\{\alpha _{D_i}\}\) can induce connected diagrams through loop corrections, and therefore we observe that both the tree-level and chiral logarithms of \(M_{C_1}\) depend only on \(\{\alpha _{C_i}\}\). Since connected diagrams can be readily computed on lattice, they may be computed with several values of \(m_\pi \) which then, using our derived formula, allow for a determination of the LECs \(\{\alpha _{C_i}\}\). By doing so, we do not just acquire the full information of the connected diagrams, but also fix a part of the chiral logarithms in the quark loop diagrams. Even though the leading terms and the remaining chiral logarithms of the latter can only be fixed by direct lattice calculations of such diagrams, now they depend on a smaller amount of unknown LECs (i.e., just \(\alpha _{D_1}\) and \(\alpha _{D_2}\)), making their chiral extrapolation much easier.

## 6 Spin-flavor symmetry

*N*fermionic quark flavors, such a symmetry means that the quarks \(\{q_i\}\) of a definite flavor

*q*and spin

*i*form the fundamental representation of a \(\mathrm {SU}(2N)\) group and all hadrons can be grouped into irreducible representations of that symmetry group. For instance, the baryon octet and decuplet collectively form an irreducible 56-plet of the spin-flavor \(\mathrm {SU}(6)\). It is well-known that the spin-flavor symmetry is a direct consequence of the large-\(N_c\) limit (with \(N_c\) the number of colors), and for most of the practical purposes it is simply equivalent to the non-relativistic quark model [87]. Therefore, for the discussion here let us consider the spin-flavor wave function of a spin-up \({\tilde{\Sigma }}^0\) state in the quark-model (QM) representation,

We end this section by commenting on the spin-flavor symmetry at one loop. We observe that the one-loop corrections of type 1PI(e) preserve the spin-flavor symmetry while those of type 1PI(f) do not. The reason is that under the spin-flavor symmetry the spin-1/2 and 3/2 baryons belong to the same multiplet and thus have to be taken simultaneously as dynamical DOFs. Our treatment of Fig. 1f, however, includes only spin-1/2 baryons whereas the effects of the rest get buried in the counterterms. This results in the explicit breaking of the symmetry in this particular diagram.

## 7 Operators with strange quarks

## 8 Conclusions

HPV has been studied for many years. Among others, the \(\Delta I=1\hbox { HPV}\) holds a special role as a unique probe of hadronic neutral weak current as well as one of the main contributors of of long-range nuclear PV. Moreover, with the release of the NPDGamma result, the \(\Delta I=1\) P-odd pion-nucleon coupling \(h_\pi ^1\) is now the only DDH coupling with a definite isospin that has been numerically measured through a single experiment. Therefore, the first-principle calculations of \(h_\pi ^1\) are highly desirable as they are directly comparable to experimental results.

Despite the above, currently we observe a lack of progress in the lattice study of the \(\Delta I=1\hbox { HPV}\) comparing to its \(\Delta I=2\) counterpart. The latter involves a direct computation of nucleon-nucleon scattering amplitudes with the insertion of \(\Delta I=2\) four-quark operators. It does not require computations of noisy disconnected diagrams but the total amount of contractions is tremendous even in the exact isospin limit. On the other hand, although disconnected diagrams are unavoidable in the \(\Delta I=1\) channel, the total amount of contractions is much less. Furthermore, given its more straightforward relation to experimental data, we believe that the study of the \(\Delta I=1\hbox { HPV}\) on lattice should receive the same amount, if not more, of attention as the \(\Delta I=2\) one. In this paper we investigate in some detail how a continuous EFT may help in the future lattice calculation of \(h_\pi ^1\).

In Ref. [35] we show that \(h_\pi ^1\) can be recast as a neutron-proton mass splitting induced by a set of \(\Delta I=1\) P-even four-quark operators. Improving from the limitations of PCAC, we show by considering the long- and short-range higher-order corrections that such a relation holds with a precision better than \(10\%\) even with a conservative estimation. This observation turns the lattice study of \(h_\pi ^1\) into computations of P-even three-point correlation functions involving only five sets (three independent combinations) of contractions. Two combinations among them are quark loop contractions which are in principle noisy, but in this work we show that one can obtain partial information of the chiral logarithms in the quark loop diagrams by studying the much easier connected diagrams. We further demonstrate that one only needs to perform a global fit with four independent LECs, two of which can be obtained easily from connected diagrams, in order to completely determine the LO and chiral logarithmic terms in \(h_\pi ^1\) induced by non-strange operators. Approximate relations among LECs based on the spin-flavor symmetry are also derived to facilitate the global fit. For operators with strange quark fields, fitting of one single LEC from the computation of one quark loop contraction is needed to determine the LO and chiral logarithmic terms. We hope that the analysis above will provide extra motivations for the lattice community to perform an up-to-date first-principle computation of \(h_\pi ^1\) which will constitute a new breakthrough in our understanding of HWI.

## Footnotes

- 1.
- 2.
Notice that no isospin symmetry is assumed here.

- 3.
We take this opportunity to correct a typo in Eq. (13) of Ref. [35]: The denominator in the first term of \(\delta (F_\pi )\) should be \(F_\pi \) instead of \(F_\pi ^2\).

- 4.
The three-momentum is quantized as \(2\pi \vec {m}/L\), with \(\vec m\in {\mathbb {Z}}^3\) a three-dimensional vector of integers, in a finite volume with periodic boundary conditions.

- 5.
The expression of \(F_{lmn,ijk}^{1/2}\) in Ref. [90] contains a couple of typos which are corrected here.

## Notes

### Acknowledgements

The authors thank Jordy de Vries, Xu Feng and Liuming Liu for many inspiring discussions. We thank Ulf-G. Meißner for a careful reading of this manuscript and for his useful comments. This work is supported in part by the National Natural Science Foundation of China (NSFC) under Grant nos. 11575110, 11655002, 11735010 and 11747601, by NSFC and Deutsche Forschungsgemeinschaft (DFG) through funds provided to the Sino–German Collaborative Research Center “Symmetries and the Emergence of Structure in QCD” (NSFC Grant no. 11621131001), by the Natural Science Foundation of Shanghai under Grant nos. 15DZ2272100 and 15ZR1423100, by Shanghai Key Laboratory for Particle Physics and Cosmology, by the Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education, by the CAS Key Research Program of Frontier Sciences (Grant no. QYZDB-SSW-SYS013), by the CAS Key Research Program (Grant no. XDPB09), and by the CAS Center for Excellence in Particle Physics (CCEPP). We also appreciate the supports through the Recruitment Program of Foreign Young Talents from the State Administration of Foreign Expert Affairs, China, and the Thousand Talents Plan for Young Professionals.

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