PLB32655S03702693(17)30152110.1016/j.physletb.2017.02.039The Author(s)PhenomenologyTable 1The Wilson coefficients ci taken from Ref. [31].Table 1LONLO (Z)NLO (Z+W)
c10.264−0.054−0.055
c20.9810.8030.810
c3−0.592−0.629−0.627
c4000

c55.974.855.09
c6−2.30−2.14−2.55
c75.124.274.51
c8−3.29−2.94−3.36
Table 2The results for the lowenergy constants in units of 10−4 MeV2. Notations are the same as in Table 1.Table 2WLECLONLO (Z)NLO (Z+W)
N10.3070.2240.235
N2−0.381−0.328−0.344
N30.1860.1420.154
N4−0.382−0.327−0.343
N51.1060.9010.910
N6−0.0050.0020
N72.7162.2142.236
N8−0.0120.0040
N90.4690.3820.386
N10−0.0020.0010
N11−0.171−0.164−0.203
N120.4390.3930.449

M10.3200.2160.215
M20.001−0.001−0.001
M30.7860.5310.529
M40.003−0.002−0.004
M50.1360.0920.091
M60.0004−0.0004−0.001
M70.1640.1220.130
M8−0.254−0.218−0.229
Table 3The values of βi in units of 10−4 MeV2. Notations are the same as in Table 1.Table 3WLECLONLO (Z)NLO (Z+W)
β10.031−0.006−0.006
β2−0.069−0.073−0.073
β3−1.334−1.092−1.102
β4−0.434−0.309−0.308
Table 4hπNN1 in units of 10−8. Notations are the same as in Table 1.Table 4LONLO (Z)NLO (Z+W)
hπNN110.968.698.74
Table 5hπNN1 in units of 10−8 with various theoretical works.Table 5DDH [9]DZ [20]KS [19]QCD sum rules [23]Skyrme Model [18]Lattice QCD [24]Present work
45.611.46028.0−1310.99±5.05−0.64+0.588.74
Parityviolating πNN coupling constant from the flavorconserving effective weak chiral LagrangianChang HoHyunahch@daegu.ac.krHyunChulKimbchchkim@inha.ac.krHeeJungLeed⁎hjl@chungbuk.ac.kraDepartment of Physics Education, Daegu University, Gyeongsan 38453, Republic of KoreaDepartment of Physics EducationDaegu UniversityGyeongsan38453Republic of KoreabDepartment of Physics, Inha University, Incheon 22212, Republic of KoreaDepartment of PhysicsInha UniversityIncheon22212Republic of KoreacSchool of Physics, Korea Institute for Advanced Study (KIAS), Seoul 02455, Republic of KoreaSchool of PhysicsKorea Institute for Advanced Study (KIAS)Seoul02455Republic of KoreadDepartment of Physics Education, Chungbuk National University, Cheongju 28644, Republic of KoreaDepartment of Physics EducationChungbuk National UniversityCheongju28644Republic of Korea⁎Corresponding author.Editor: W. HaxtonAbstractWe investigate the parityviolating pion–nucleon–nucleon coupling constant hπNN1, based on the chiral quarksoliton model. We employ an effective weak Hamiltonian that takes into account the nexttoleading order corrections from QCD to the weak interactions at the quark level. Using the gradient expansion, we derive the leadingorder effective weak chiral Lagrangian with the lowenergy constants determined. The effective weak chiral Lagrangian is incorporated in the chiral quarksoliton model to calculate the parityviolating πNN constant hπNN1. We obtain a value of about 10−7 at the leading order. The corrections from the nexttoleading order reduce the leading order result by about 20%.1IntroductionThe electroweak interactions have been tested and confirmed mainly by parityviolating lepton scattering, decays of hadrons, and β decays of nuclei. Recently, Parityviolating (PV) hadronic processes play yet another important role of a touchstone to examine the standard model (SM) and physics beyond the standard model (BSM) (see for example recent reviews [1–6]). There are mainly two different ways of describing PV hadronic reactions: One is to consider oneboson exchanges such as π, ρ, and ω mesons à la the strong nucleon–nucleon (NN) potential [7–9]. The other is to employ effective field theory [6,10]. In both methods, the PV pion–nucleon coupling constant is the most essential quantity, since it governs the PV hadronic processes in long range. Desplanques, Donoghue and Holstein (DDH) [9] estimated the value of the PV πNN coupling constant, also known as the socalled “DDH best value”: hπNN1=4.6×10−7. A great deal of experimental and theoretical efforts have been devoted to extracting the precise value of the πNN coupling constant (for recent reviews, see [11,12]), since the contribution of πexchange is dominant over those of ρ and ωmeson exchanges in the PV asymmetry in n→p→dγ [13–15], and n→d→tγ [16]. The PV πNN coupling constant has been studied in various different theoretical approaches such as the Skyrme models [17–19], quark models [20], the chiralquark soliton model [21,22], QCD sum rule [23], and so on. However, all these values of hπNN1 are far from consensus and are given in the wide range between 10−8 [17] and ∼5×10−7 [19]. A recent analysis of lattice QCD yields hπNN1,con=(1.099±0.505−0.064+0.058)×10−7 for which only the contribution of the connected diagrams to hπNN1 has been considered [24]. On the experimental side, though the accuracy of the measurements has been much improved, an upper bound on the value of hπNN1 [25] is only known. Thus, more systematic and quantitative studies are required in order to obtain the value of the PV πNN coupling constant.The main dynamical origin of hadronic parity violation comes from the flavorconserving effective weak Hamiltonian, which was already investigated [8,9,26–30]. In particular, the PV πNN coupling constant can be obtained from the isovector (ΔI=1) effective weak Hamiltonian, which was first derived in Ref. [30] at the oneloop level with the effects of heavy quarks taken into account. Very recently, Tiburzi [31] investigated systematically the ΔS=0 effective weak Hamiltonian with QCD corrections at nexttoleading order (NLO). The effects of the NLO corrections have changed the Wilson coefficients about (10–20)% at the typical scale of light hadrons (μ=1 GeV). Considering the fact that the PV πNN coupling constant is very tiny, we expect that the corrections from QCD at NLO may come into play. Thus, it is of great interest to examine the NLO corrections to the PV πNN coupling constant.In the present work, we investigate the PV πNN coupling constant, hπNN1, within the framework of the chiral quarksoliton model (χQSM) together with the effective weak Hamiltonian at NLO [31]. The χQSM is constructed based on the effective chiral action that explains both the pseudoscalar mesons and lowlying baryons on the equal footing [32] (see also a review [33]). The presence of the Nc valence quarks makes the vacuum polarized. As a result, a baryon is described as a soliton of the classical pion mean field. Since the χQSM is constructed in a relativistic fieldtheoretic way, it has been very successful in explaining not only usual properties of baryons such as the vector form factors [34–36], axialvector form factors and semileptonic decay constants [37,38], but also internal structures of the nucleon like the parton distributions [39–41], generalized parton distributions [42,43], transversities [44–48], etc. Moreover, the χQSM provides a more realistic picture than the Skyrme model, because the effective chiral action on which the model is based contains all orders of the effective chiral Lagrangian including the Gasser–Leutwyler terms [49,50], Wess–Zumino term [51], and so on. The gradient expansion of the effective chiral action yields the corresponding lowenergy constants in good agreement with the empirical data [52,53]. Recently, the present authors computed the PV πNN coupling constant [22] in the same framework, employing the effective weak Hamiltonian from Ref. [9]. We first derived the effective weak chiral Lagrangian, based on the nonlocal chiralquark model (NχQM) from the instanton vacuum associating with the effective weak Hamiltonian [21]. If one performs the gradient expansion for the effective chiral action of the χQSM with the effective weak Hamiltonian, we would obtain exactly the same expressions starting directly from the effective weak chiral Lagrangian. Using this gradient expansion, we were able to obtain the PV πNN coupling constant to be about 1×10−8 at μ=1 GeV. We also found that the hπNN1 is rather sensitive to the Wilson coefficients. In this respect, it is of great importance to reexamine the PV πNN coupling constant, the ΔI=1 effective weak Hamiltonian being employed with the NLO QCD effects. As we will show in this work, the value of hπNN1 indeed turns out to be different from the previous result. Moreover, the effects from the NLO corrections reduce the leadingorder result by about 20%.The paper is organized in the following order: In Section 2, we present briefly the general procedure to obtain the PV πNN coupling constants within the χQSM. We derive the flavorconserving effective weak chiral Lagrangian, with the nonlocal chiral quark model from the instanton vacuum. In Section 3 we compute the correlation function corresponding to the PV πNN coupling constant, and discuss the result. In Section 4, we conclude the work.2ΔI=1 effective weak chiral LagrangianWe start with the ΔI=1 flavorconserving effective weak Hamiltonian including the NLO corrections [31], which is expressed as(1)HWΔI=1=GF2sin2θW3∑i=18ci(μ)Oi(μ), where GF and θW denote the Fermi constant and the Weinberg angle, respectively. The eight different operators Oi are defined generically as twobody operators: Oi=(ψ¯γμγ5Miψ)(ψ¯γμNiψ). The ci stands for the Wilson coefficient corresponding to the Oi, which depends on the renormalization scale μ. Applying the following Fiertz identity to O2,O4,O6, and O8,(2)δbcδad=12(tA)ab(tA)cd where tA denote the GellMann matrices in color space, we are able to express the effective weak Hamiltonian in the following form(3)HWΔI=1=GF6sin2θW3{(ψ¯γμγ5λ3ψ)(ψ¯γμ[λ0(2c1+c52)+λ8(c1−c5)]ψ)+12(ψ¯γμγ5λ3tAψ)(ψ¯γμ[λ0(2c2+c62)+λ8(c2−c6)]tAψ)+(ψ¯γμλ3ψ)(ψ¯γμγ5[λ0(2c3+c72)+λ8(c3−c7)]ψ)+12(ψ¯γμλ3tAψ)(ψ¯γμγ5[λ0(2c4+c82)+λ8(c4−c8)]tAψ)}. We rewrite the Hamiltonian in terms of the effective fourquark operators that contain already the Wilson coefficients Qi(z;μ)(4)HWΔI=1=GF6sin2θW3(Q1+Q2+Q3+Q4), where the fourquark operators Qi(z;μ) are defined as(5)Qi(z;μ)=αi(ψ¯Γ1(i)ψ)(ψ¯Γ2(i)ψ), where αi=1 for i=1,3 and αi=1/2 for i=2,4. The Γj(i) are defined as(6)Γ1(1)=γμγ5λ3,Γ2(1)=γμΛ(1),Γ1(2)=γμγ5λ3tA,Γ2(2)=γμΛ(2)tA,Γ1(3)=γμλ3,Γ2(3)=γμγ5Λ(3),Γ1(4)=γμλ3tA,Γ2(4)=γμγ5Λ(4)tA with flavor matrices defined as(7)Λ(1)=λ0(2c1+c52)+λ8(c1−c5),Λ(2)=λ0(2c2+c62)+λ8(c2−c6),Λ(3)=λ0(2c3+c72)+λ8(c3−c7),Λ(4)=λ0(2c4+c82)+λ8(c4−c8).In order to compute the ΔI=1 flavorconserving effective weak chiral Lagrangian, we employ the NχQM from the instanton vacuum. The effective weak chiral Lagrangian is defined as a vacuum expectation value (VEV) of the effective weak Hamiltonian [54,55](8)LWΔI=1=∫DψDψ†HWΔI=1exp[∫d4zψ†(z)Dψ(z)], where D represents the nonlocal covariant Dirac operator defined as(9)D(−i∂)≡iγμ∂μ+iM(−i∂)Uγ5(x)M(−i∂), and Uγ5 represents the chiral field defined as(10)Uγ5=1+γ52U+1−γ52U† with the Goldstone boson field U=exp(iλaπa/fπ). Then, the flavorconserving effective weak chiral Lagrangian can be expressed in terms of the VEV of the fourquark operator(11)Leff=GF6sin2θW3∑i=14〈Qi〉. We refer to Refs. [21,54,55] for details of how to compute the VEV of Qi.The flavorconserving effective weak chiral Lagrangian in the ΔI=1 channel is obtained in terms of the lowenergy constants Ni and Mi(12)LeffΔI=1=N1〈(Rμ−Lμ)λ3〉〈(Rμ+Lμ)λ0〉+N2〈(Rμ−Lμ)λ3〉〈(Rμ+Lμ)λ8〉+N3〈(Rμ−Lμ)λ0〉〈(Rμ+Lμ)λ3〉+N4〈(Rμ−Lμ)λ8〉〈(Rμ+Lμ)λ3〉+N5〈λ3Uλ0U†−λ3U†λ0U〉+N6〈λ3Uλ8U†−λ3U†λ8U〉+N7〈Lμλ3LμU†λ0U−Rμλ3RμUλ0U†〉+N8〈Lμλ3LμU†λ8U−Rμλ3RμUλ8U†〉+N9〈(λ3RμRμ+RμRμλ3)Uλ0U†−(λ3LμLμ+LμLμλ3)U†λ0U〉+N10〈(λ3RμRμ+RμRμλ3)Uλ8U†−(λ3LμLμ+LμLμλ3)U†λ8U〉+N11〈(Rμλ3Rμ−Lμλ3Lμ)λ0〉+N12〈(Rμλ3Rμ−Lμλ3Lμ)λ8〉,+M1〈λ3Uλ0U†−λ3U†λ0U〉+M2〈λ3Uλ8U†−λ3U†λ8U〉+M3〈Lμλ3LμU†λ0U−Rμλ3RμUλ0U†〉+M4〈Lμλ3LμU†λ8U−Rμλ3RμUλ8U†〉+M5〈(λ3RμRμ+RμRμλ3)Uλ0U†−(λ3LμLμ+LμLμλ3)U†λ0U〉+M6〈(λ3RμRμ+RμRμλ3)Uλ8U†−(λ3LμLμ+LμLμλ3)U†λ8U〉+M7〈(Rμλ3Rμ−Lμλ3Lμ)λ0〉+M8〈(Rμλ3Rμ−Lμλ3Lμ)λ8〉 where 〈⋯〉 means the trace over the flavor. The right and left currents Rμ and Lμ are defined respectively as(13)Rμ=iU∂μU†,Lμ=iU†∂μU. The weak lowenergy constants (WLECs) Ni are the leading order in the large Nc limit whereas Mi are of the subleading order. They are expressed as(14)N1=4Nc2J22C(2c1+c52),N2=4Nc2J22C(c1−c5),N3=4Nc2J22C(2c3+c72),N4=4Nc2J22C(c3−c7),N5=8Nc2J12C(2c2−2c4+c62−c82),N6=8Nc2J12C(c2−c4−c6+c8),N7=16Nc2J1J3C(2c2−2c4+c62−c82),N8=16Nc2J1J3C(c2−c4−c6+c8),N9=8Nc2J1J4C(2c2−2c4+c62−c82),N10=8Nc2J1J4C(c2−c4−c6+c8),N11=4Nc2J22C(2c2+2c4+c62+c82),N12=4Nc2J22C(c2+c4−c6−c8),M1=8NcJ12C(2c1−2c3+c52−c72),M2=8NcJ12C(c1−c3−c5+c7),M3=16NcJ1J3C(2c1−2c3+c52−c72),M4=16NcJ1J3C(c1−c3−c5+c7),M5=8NcJ1J4C(2c1−2c3+c52−c72),M6=8NcJ1J4C(c1−c3−c5+c7),M7=4NcJ22C(2c1+2c3+c52+c72),M8=4NcJ22C(c1+c3−c5−c7). Definitions of the integrals J1, J2, J3, and J4 are given in Ref. [21], and the constant C contains the Fermi constant and the Weinberg angle(15)C=GF6sin2θW3.To compute the WLECs in Eq. (14), we use the momentumdependent quark mass derived from the instanton vacuum [56](16)M(k)=M0F2(kρ) with(17)F(kρ)=2z(I0(z)K1(z)−I1(z)K0(z)−1zI1(z)K1(z)), where Ii and Ki are the modified Bessel functions, and z=k/2Λ. The value of the dynamical quark mass at the zero virtuality of the quark is also obtained from the instanton vacuum, i.e. M0=350 MeV, given the average size of the instanton and the interdistance between instantons R≈1 fm [56]. The parameter Λ is determined to reproduce the physical value of fπ.As discussed already in Ref. [55], the vector and axialvector currents are not conserved in the presence of the nonlocal interaction arising from the momentumdependent quark mass, that is, the corresponding gauge symmetries are broken. In order to keep the currents conserved, we need to make the effective chiral action gaugeinvariant. In Ref. [57,58], the gauged effective chiral action was derived, based on the instanton vacuum. Had we naively computed J2=fπ2/4Nc without the current conservation being considered, then we would have ended up with the Pagels–Stokar formula for fπ2 [59], which does not satisfy the gauge invariance. The numerical results for the integrals are obtained as(18)J1=(−112.31)3 MeV3,J2=(26.673)2 MeV2,J3=−1.7403 MeV,J4=−0.601 MeV. Note that the value of J1 is related to that of the quark condensate 〈ψ‾ψ〉M=(−257.13 MeV)3 and that of J2 corresponds to the value of fπ=92.4 MeV.In Table 1, the values of the Wilson coefficients taken from Ref. [31] are listed. The first column lists the results for the Wilson coefficients in the LO, and the second and third ones correspond to those from the NLO contributions together with the LO terms. The Z and Z+W in the second and third columns stand respectively for the considerations of Z and Z+W boson exchanges. As already discussed in Ref. [31], there are certain effects from the NLO contributions.Based on these values of the Wilson coefficients, we list in Table 2 the results for the WLECs given in Eq. (14). Note that the WLECs N6, N8, and N10 are null. This is due to the fact that they correspond to the operators(19)O2′=−O2+O4+O6−O8, for which the corresponding Wilson coefficient vanishes because O2′ is not generated by QCD radiative corrections [31].Though there are arguments that sizable contributions in ΔI=1 channel come from the operators with strangeness [19], we will restrict ourselves to the case of SU(2). The calculation in SU(2) has several merits in particular in the present work. Firstly, the chiral solitonic approach in SU(2) is much simpler and physically clearer than that in SU(3). Secondly, the SU(2) approach allows one to understand better the PV πNN constant based on the effective weak Hamiltonian. A more quantitative work within SU(3) will appear elsewhere. In the case of SU(2), we reduce λ0, λ3, and λ8 to(20)λ0 → 231,λ3 → τ3,λ8 → 131. As a result, the ΔI=1 effective weak Lagrangian is simplified as(21)LeffSU(2)=β1〈(Rμ−Lμ)τ3〉〈Rμ+Lμ〉+β2〈Rμ−Lμ〉〈(Rμ+Lμ)τ3〉+β3〈(RμRμ−LμLμ)τ3〉,+β4〈(RμRμ−LμLμ)τ3〉, where βi are defined in terms of the WLECs(22)β1=13(2N1+N2),β2=13(2N3+N4),β3=13[2N11+N12+2(2N9−N7)+2N10−N8],β4=13[2M7+M8+2(2M5−M3)+2M6−M4]. As will be shown soon, β1 and β2 do not contribute at all to the PV πNN coupling constant. On the other hand, β3 and β4 do come into play, so that we need to examine them in detail. We can explicitly express β3 and β4 in terms of the Wilson coefficients such that we can see which terms contribute dominantly to the PV πNN coupling constant. β3 and β4 are rewritten as(23)β3=GFsin2θW122[(c2+c4)fπ4−16Nc(c2−c4)〈ψ‾ψ〉M(J3−J4)],β4=GFsin2θW122Nc[(c1+c3)fπ4−16Nc(c1−c3)〈ψ‾ψ〉M(J3−J4)], which clearly shows that β4 is the subleading order in the large Nc limit with respect to β3. Note that the structure of the β4 is the same as that of β3 except for the Wilson coefficients and the 1/Nc factor. The magnitudes of the second terms in Eq. (23) are much larger than those of the first ones, so we can ignore approximately the first terms. That is, β3 and β4 can be expressed as(24)β3≈4GFsin2θWNc32c2〈ψ‾ψ〉M(J4−J3),β4≈−c3c2Ncβ3, which indicates that β3 is larger than β4 approximately by (70–75)%.In Table 3, we list the results for the βi. Note that β3 and β4 have the same sign because c2 and c3 have different relative signs as shown in Table 1. The magnitude of β3 indeed turns out to be about 75% larger than the β4, as expected from Eq. (24).3Parityviolating πNN coupling constantWe are now in a position to determine the PV πNN coupling constant. Starting from Eq. (21), we are able to derive the PV πNN coupling constant. We already have shown explicitly how one can obtain the PV πNN coupling constant, based on the χQSM [22]. Thus, we want to briefly explain the procedure of computing the hπNN1 within the model. The PV πNN coupling constant can be derived by solving the following matrix element:(25)〈NHWΔI=1πaN〉=GF6sin2θW3∑i=14〈NQi(z;μ)πaN〉=GF6sin2θW3∑i=14∫d4ξ(k2+mπ2)×eik⋅ξ〈NT[Qi(z;μ)πa(ξ)]N〉, where the nucleon states can be constructed by using the Ioffetype current in Euclidean space (x0=−ix4) [32,33]:(26)N(p1)〉=limy4→−∞ep4y4N⁎(p1)∫d3yeip1⋅yJN†(y)0〉,〈N(p2)=limx4→+∞e−p0x4N(p2)∫d3xe−ip2⋅x〈0JN(x). The JN† (JN) constitutes Nc quarks(27)JN(x)=1Nc!ϵc1c2⋯cNcΓ(TT3Y)(JJ3YR)s1s2⋯sNcψs1c1(x)⋯ψsNccNc(x), where s1⋯sNc and c1⋯cNc stand for spin–isospin and color indices, respectively. The Γ(TT3)(JJ3){s} provides the quantum numbers (TT3)(JJ3) for the nucleon: T=1/2, Y=1 and J=1/2. The nucleon creation operator JN† can be obtained by taking the Hermitian conjugate of JN. The matrix element in Eq. (25) is just the fourpoint correlation function given as(28)limy0→−∞x0→+∞∑i=14〈0T[JN(x)Qi(z;μ)∂μAμa(ξ)JN†(y)]0〉=limy0→−∞x0→+∞K, where Aμa stands for the axialvector current. Note that we have used the partial conservation of the axialvector current (PCAC). In principle, the fourpoint correlation function K can be computed by solving the following functional integral(29)K=1Z∫DψDψ†DUJN(x)Qi(z;μ)∂μAμa(ξ)JN†(y)×exp[∫d4xψ†(i/∂+iM(−∂2)Uγ5M(−∂)2)ψ].As was already mentioned in the previous work [22], it is extremely complicated to deal with Eq. (29) technically, since the PV πNN coupling constant arises from both the twobody quark operators Qi and the axialvector one, which causes laborious triple sums over quark levels already at the leading order in the large Nc expansion. Thus, we employ the gradient expansion method as in Ref. [22]. In the gradient expansion, (/∂U/M)≪1 is used as an expansion method [32] to expand the quark propagator in the pion background field, with the pion momentum assumed to be small. This approximation of the gradient expansion is known to describe static properties of the nucleon qualitatively well [32]. Equivalently, we can directly start from the effective weak chiral Lagrangian in Eqs. (12), (21) already derived in the previous Section.The classical soliton is assumed to have a hedgehog symmetry, so that it can be parametrized in terms of the soliton profile function P(r)(30)U0=exp(iτ⋅rˆP(r)). In principle, P(r) can be found by solving the equations of motion selfconsistently [33]. However, we will employ a parametrized form of P(r) which is very close to the selfconsistent one. The classical soliton field can be fluctuated such that the pion field can be coupled to a ΔI=1 twobody quark operator(31)U=exp(iτ⋅π2fπ)U0exp(iτ⋅π2fπ). Since the traces of the left and right currents over flavor space vanish, i.e.(32)〈Rμ+Lμ〉=0,〈Rμ−Lμ〉=0, the terms with N1, N2, N3, and N4 do not contribute to hπNN1 as shown in our previous analysis [22] with the DDH effective Hamiltonian [9]. Considering the fact that N1 and N2 contain the Wilson coefficient c5, which is the most dominant one, and N3 and N4 have c7 that is the second largest one, one can explain a part of the reason why hπNN1 turns out to be rather small in the present approach.When it comes to all other terms, we can approximately rewrite Lμ2 and Rμ2 as(33)LμLμ≃i2fπ(Lμ0Lμ0τ⋅π−τ⋅πLμ0Lμ0),RμRμ≃i2fπ(τ⋅πRμ0Rμ0−Rμ0Rμ0τ⋅π) with Lμ0=iU0†∂μU0 and Rμ0=iU0∂μU0†, so that we get(34)〈(RμRμ−LμLμ)τ3〉=i2fπ〈(Rμ0Rμ0+Lμ0Lμ0)(τ−π+−τ+π−)〉, where τ± and π± are defined in the spherical basis as(35)τ±=12(τ1±iτ2),π±=12(π1±iπ2). The relevant effective Lagrangian is then expressed as(36)LeffSU(2)=(β3+β4)i2fπ〈(Rμ0Rμ0+Lμ0Lμ0)(τ+π−−τ−π+)〉, where β3 and β4 are defined already in Eq. (22).Since we have already explained how the quantization of the soliton is performed in the context of the PV πNN coupling constant in Ref. [22], we proceed to compute the hπNN1 within this framework. For simplicity, let us consider the PV process n+π+→p. Then, we need to compute the following trace(37)〈(Rμ0Rμ0+Lμ0Lμ0)τ+〉. Defining isovector fields rμi and lμi as(38)Rμ0=−rμiτi,Lμ0=−lμiτi and using an identity 〈τiτjτk〉=2iϵijk, we obtain(39)〈Rμ0Rμ0τ+〉=(rμ1+irμ2)rμ3−rμ3(rμ1+irμ2),〈Lμ0Lμ0τ+〉=(lμ1+ilμ2)lμ3−lμ3(lμ1+ilμ2). Thus, we arrive at the final form of the effective Lagrangian(40)LeffSU(2)=(β3+β4)i2fπ×[(rμ1+irμ2)rμ3−rμ3(rμ1+irμ2)+(r→l)]π−,from which we can derive the PV πNN coupling constant. Using the collective quantization discussed in Ref. [22], we get(41)∫d3x〈p↑rμ3(rμ1+irμ2)n↑〉=−∫d3x〈p↑(rμ1+irμ2)rμ3n↑〉=2π3I2∫drr2sin2P(r)[sin2P(r)−3cos2P(r)],where I denotes the moment of inertia [32] expressed as(42)I=Nc12∫−∞∞dω2πTr(τi1ω+iHτi1ω+iH)≈83πfπ2∫0∞drr2sin2P(r). Here, ω is the energy frequencies of the quark levels and Tr stands for the functional trace over coordinate space, isospin and Dirac spin space. The second term was derived approximately by the gradient expansion. Similarly, we obtain the same result for lμ. Having carried out the calculation of the matrix element for the collective operators, we finally derive the PV πNN coupling constant hπNN1 as(43)hπNN1=i〈p↑LeffSU(2)n↑,π+〉=82π3fπI2(β3+β4)×∫drr2sin2P(r)[sin2P(r)−3cos2P(r)]. It is interesting to see that Eq. (43) is exactly the same as the expression obtained in Ref. [22] except for the coefficient β3+β4.In order to compute the PV πNN coupling constant, we employ the following numerical values of the constants involved in the present work: the Fermi constant GF=1.16637×10−5 GeV−2, the Weinberg angle sin2θW=0.23116, and the pion decay constant fπ=0.0924 GeV. Concerning the profile function, we have already examined the dependence of hπNN1 on types of the profile functions [22]. In the present work, we employ the physical profile function expressed as(44)P(r)={2arctan(r0r)2,(r≤rx),P0e−mπr(1+mπr)/r2,(r>rx), where r0 is defined as r0=3gA16πfπ2 with the axialvector constant gA=1.26. P0 and rx are given as P0=2r02, and rx=0.752 fm, respectively. The profile function in Eq. (44) satisfies a correct behavior of the Yukawa tail. Then, the moment of inertia is obtained to be I=3.32 GeV−1.Numerical results for hπNN1 are summarized in Table 4. In Ref. [31], it was shown that NLO contributions alter the values of the Wilson coefficients at μ=1 GeV by about (10–20)%, which actually lessens the value of the hπNN1 by about 25% as shown in Table 4. As already examined in Eqs. (23), (24), β3 plays a dominant role in determining hπNN1. Thus, the most important operator in the ΔI=1 effective weak Hamiltonian is O2, which contains the Wilson coefficient c2. As clearly shown in Table 4, the NLO QCD radiative corrections suppress the PV πNN coupling constant. In fact, we have shown already in the previous work [22], the QCD radiative corrections strongly diminish the value of hπNN1. This behavior contrasts with the case of K nonleptonic decays, where the penguin diagrams enhance the contribution to the ΔI=1/2 channel. Concerning the dependence on the profile functions, the linear and arctangent functions give the NLO (Z+W) results 6.44×10−8 and 7.53×10−8, respectively. Therefore dependence on the profile function does not change the essential conclusion of this work.In Table 5, we compare the present result with those of various theoretical works. The present result turns out to be about 5 times smaller than the DDH “best value”. We find that the result from the QCD sum rules predicts the smallest value of hπNN1 whereas Ref. [19] yields the largest result, in which the importance of the strangeness contribution was emphasized. Compared to the value of hπNN1 from lattice QCD with connected diagrams considered only, the present result is in good agreement with it. On the experiment side, forbidden γray transition of F18 reported an upper limit hπNN1<1.3×10−6 [60,61]. Recent analyses of the PV analyzing power in the pp scattering give wide ranges, hπNN1=(1.1±2)×10−6 [62], and −1×10−6<hπNN1<2×10−6 [63]. Our result is within the range of these experiment analyses, but the large magnitude of the boundary values stimulates extension of our investigation to the SU(3) flavor space.4Summary and outlookIn the present work, we investigated the parityviolating pion–nucleon coupling constant. Starting from the ΔI=1 effective weak Hamiltonian [31] that considered the nexttoleading order QCD radiative corrections, we derived the effective weak chiral Lagrangian with the weak lowenergy constants determined in the ΔI=1 and ΔS=0 channel. In order to calculate the parityviolating pion–nucleon coupling constant hπNN1, we employed the chiral quarksoliton model. Using the gradient expansion, which is equivalent to using the effective weak chiral Lagrangian directly, we were able to compute the values of hπNN1. We found that the first four terms of the Lagrangian did not contribute at all to hπNN1, which partially explains why the value of hπNN1 should be small. It was also found that the main contribution to hπNN1 arose from the operator O2 in the effective weak Hamiltonian. We also noted that the nexttoleading order QCD radiative corrections further suppress the value of hπNN1 and as a result we obtained hπNN1=8.74×10−8. We compared this result with those from various theoretical models including the recent result from lattice QCD. The present result was shown to be in agreement with that from lattice QCD.The present work can be extended to the SU(3) case in which the strange quark comes into play. Another merit of the chiral quarksoliton model is that the explicit breaking of flavor SU(3) symmetry can be treated systematically, the strange quark mass being considered as a perturbation. Thus, it is interesting to examine the contribution of the strange quark and its current quark mass to the parityviolating pion–nucleon coupling constant. Other coupling constants such as hρNN and hωNN can be studied within the same framework. The related works are under way.AcknowledgementsThe work of H.Ch.K. was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant Number: NRF2015R1D1A1A01060707). The work of H.J.L. was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant Number: NRF2013R1A1A2009695). The work of C.H.H. was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF2013S1A5B6043901).References[1]M.J.RamseyMusolfS.A.PageAnnu. Rev. Nucl. Part. Sci.5620061[2]B.R.HolsteinJ. Phys. G362009104003[3]V.CiriglianoS.GardnerB.HolsteinProg. Part. Nucl. Phys.71201393[4]V.CiriglianoM.J.RamseyMusolfProg. Part. Nucl. Phys.7120132[5]W.C.HaxtonB.R.HolsteinProg. Part. Nucl. Phys.712013185[6]M.R.SchindlerR.P.SpringerProg. Part. Nucl. Phys.7220131[7]D.TadicPhys. Rev.17419681694[8]J.F.DonoghuePhys. Rev. D1319762064[9]B.DesplanquesJ.F.DonoghueB.R.HolsteinAnn. Phys.1241980449[10]S.L.ZhuC.M.MaekawaB.R.HolsteinM.J.RamseyMusolfU.van KolckNucl. Phys. A7482005435[11]W.C.HaxtonB.R.HolsteinProg. Part. Nucl. Phys.712013185[12]M.R.SchindlerR.P.SpringerProg. Part. Nucl. Phys.7220131[13]C.H.HyunT.S.ParkD.P.MinPhys. Lett. B5162001321[14]C.H.HyunS.J.LeeJ.HaidenbauerS.W.HongEur. Phys. J. A242005129[15]C.H.HyunS.AndoB.DesplanquesPhys. Lett. B6512007257[16]B.DesplanquesJ.J.BenayounNucl. Phys. A4581986689[17]N.KaiserU.G.MeissnerNucl. Phys. A4891988671[18]U.G.MeissnerH.WeigelPhys. Lett. B44719991[19]D.B.KaplanM.J.SavageNucl. Phys. A5561993653[20]V.M.DubovikS.V.ZenkinAnn. Phys.1721986100[21]H.J.LeeC.H.HyunC.H.LeeH.Ch.KimEur. Phys. J. C452006451[22]H.J.LeeC.H.HyunH.Ch.KimPhys. Lett. B7132012439[23]E.M.HenleyW.Y.P.HwangL.S.KisslingerPhys. Lett. B367199621[24]J.WasemPhys. Rev. C852012022501[25]M.T.GerickePhys. Rev. C832011015505[26]J.G.KornerG.KramerJ.WillrodtPhys. Lett. B811979365[27]B.GuberinaD.TadicJ.TrampeticNucl. Phys. B1521979429[28]T.KarinoK.OhyaT.OkaProg. Theor. Phys.651981693[29]T.KarinoK.OhyaT.OkaProg. Theor. Phys.6619811389[30]J.DaiM.J.SavageJ.LiuR.P.SpringerPhys. Lett. B2711991403[31]B.C.TiburziPhys. Rev. D852012054020[32]D.DiakonovV.Y.PetrovP.V.PobylitsaNucl. Phys. B3061988809[33]C.V.ChristovA.BlotzH.Ch.KimP.PobylitsaT.WatabeT.MeissnerE.Ruiz ArriolaK.GoekeProg. Part. Nucl. Phys.37199691[34]H.Ch.KimA.BlotzM.V.PolyakovK.GoekePhys. Rev. D5319964013[35]A.SilvaH.Ch.KimK.GoekePhys. Rev. D652002014016Erratum:Phys. Rev. D662002039902[36]A.SilvaD.UrbanoH.Ch.KimarXiv:1305.6373 [hepph][37]A.SilvaH.Ch.KimD.UrbanoK.GoekePhys. Rev. D722005094011[38]T.LedwigA.SilvaH.Ch.KimK.GoekeJ. High Energy Phys.08072008132[39]D.DiakonovV.PetrovP.PobylitsaM.V.PolyakovC.WeissNucl. Phys. B4801996341[40]D.DiakonovV.Y.PetrovP.V.PobylitsaM.V.PolyakovC.WeissPhys. Rev. D5619974069[41]M.WakamatsuT.KubotaPhys. Rev. D601999034020[42]K.GoekeM.V.PolyakovM.VanderhaeghenProg. Part. Nucl. Phys.472001401[43]J.OssmannM.V.PolyakovP.SchweitzerD.UrbanoK.GoekePhys. Rev. D712005034011[44]H.Ch.KimM.V.PolyakovK.GoekePhys. Lett. B3871996577[45]H.Ch.KimM.V.PolyakovK.GoekePhys. Rev. D5319964715[46]P.SchweitzerD.UrbanoM.V.PolyakovC.WeissP.V.PobylitsaK.GoekePhys. Rev. D642001034013[47]T.LedwigA.SilvaH.Ch.KimPhys. Rev. D822010034022[48]T.LedwigA.SilvaH.Ch.KimPhys. Rev. D822010054014[49]J.GasserH.LeutwylerAnn. Phys.1581984142[50]J.GasserH.LeutwylerNucl. Phys. B2501985465[51]J.WessB.ZuminoPhys. Lett. B37197195[52]D.DiakonovM.I.EidesJETP Lett.381983433Pis'ma Zh. Eksp. Teor. Fiz.381983358[53]H.A.ChoiH.Ch.KimPhys. Rev. D692004054004[54]M.FranzH.Ch.KimK.GoekeNucl. Phys. B5621999213[55]M.FranzH.Ch.KimK.GoekeNucl. Phys. A6992002541[56]D.DiakonovV.Y.PetrovNucl. Phys. B2721986457[57]M.M.MusakhanovH.Ch.KimPhys. Lett. B5722003181[58]H.Ch.KimM.MusakhanovM.SiddikovPhys. Lett. B608200595[59]H.PagelsS.StokarPhys. Rev. D2019792947[60]E.G.AdelbergerM.M.HindiC.D.HoyleH.E.SwansonR.D.Von LintingW.C.HaxtonPhys. Rev. C2719832833[61]S.A.PageH.C.EvansG.T.EwanS.P.KwankJ.R.LeslieJ.D.MacarthurW.MclatchieP.SkensvedPhys. Rev. C3519871119[62]J.de VriesUlfG.MeißnerE.EpelbaumN.KaiserEur. Phys. J. A492013149[63]M.VivianiA.BaroniL.GirlandaA.KievskyL.E.MarcucciR.SchiavillaPhys. Rev. C892014064004