Parity-violating πNN coupling constant in the chiral quark–soliton model

Abstract We investigate the parity-violating πNN Yukawa coupling constant h π NN 1 within the framework of the SU(2) chiral quark–soliton model, based on the Δ S = 0 effective weak Lagrangian derived within the same framework. We find that the parity-violating πNN coupling constant is about 1 × 10 − 8 at the scale of 1 GeV. The results of h π NN 1 turn out to be sensitive to the Wilson coefficient. We discuss how the gluonic renormalization suppresses the parity-violating πNN coupling constant.


I. INTRODUCTION
The parity-violating (PV) hadronic processes in low-energy regions have been one of the most fundamental issues in nuclear and hadronic physics for long time (see a recent review [1] for some historical and phenomenological background). However, the weak interactions of hadrons are yet poorly understood because of the strong interaction, compared to lepton-lepton or lepton-hadron weak processes. For example, the long-standing puzzle of the ∆I = 1/2 rule in strangeness-changing weak interactions indicates that the effect of the strong interaction in weak processes raises a non-trivial problem [2][3][4]. It is even more difficult to study parity-violating nuclear processes because of experimental feasibility and theoretical complication caused by the nonperturbative strong interaction of quarks and gluons. The standard model (SM) asserts that charged weak boson exchange induces flavor-changing weak interactions whereas the neutral current conserves the flavor. The basic ingredient to describe low-energy hadronic weak processes is the quark current-current interaction with W and Z bosons. However, in order to describe low-energy phenomena below 1 GeV, one has to scale down this interaction from the mass scale of the W and Z. In the course of this scaling, the quark-gluon interactions are encoded in the Wilson coefficients by the renormalization group equation [5][6][7][8], which, however, explains only a perturbative part of the strong interaction.
Desplanques, Donoghue, and Holstein (DDH) [8] suggested that hadronic and nuclear PV processes can be described by one-boson exchange such as π-, ρ-, and ω-exchanges [8][9][10]à la the strong nucleon-nucleon (N N ) potential. The main factors of the PV N N potential are the seven weak meson-N N coupling constants, i.e. h 1 πN N , h 0 ρN N , h 1 ρN N , h 2 ρN N , h 0 ωN N , h 1 ωN N , and h ′1 ρN N , where superscripts denote the isospin difference ∆I. Among these coupling constants, it is of utmost importance to understand the PV πN N coupling constant, because it governs the long-range part of the PV N N interaction, so that it plays the most signicant role in explaining the PV nuclear processes. The PV πN N coupling constant can in principle be extracted from various PV reactions np → dγ [11][12][13], and 18 F * → 18 F [ [14][15][16] but is fraught with large undertainties. We refer to a recent review [17] for the present status of hadronic PV experiments. It has been also calculated in various theoretical frameworks: the SU(6) W quark model [8,18] with the effective weak Hamiltonian, the Skyrme model [19][20][21], and QCD sum rules [22], and so on. Even though a great amount of efforts was made on understanding h 1 πN N experimentally as well as theoretically, its quantitative value is still elusive.
In the present work, we investigate the PV πN N coupling constant, h 1 πN N , within the framework of the SU(2) chiral quark-soliton model (χQSM) which is an effective chiral model for QCD in the low-energy region with constituent quarks and the pseudoscalar mesons as the relevant degrees of freedom. The model respects the spontaneous breakdown of chiral symmetry and describes baryons fully relativistically. Moreover, it is deeply related to the QCD vacuum based on instantons [23] and contains only a few free parameters. These parameters can mostly be fixed to the meson masses and meson decay constants in the mesonic sector. The only remaining free parameter is the constituent quark mass or dynamical quark mass that is also fixed by reproducing the electric properties of the proton. The χQSM was successful in describing lowest-lying baryon properties [24]. Furthermore, the renormalization scale for the χQSM is naturally given by the cut-off parameter for the regularization which is about 0.36 GeV 2 . Note that it is implicitly related to the inverse of the size of instantons (ρ ≈ 0.35 fm) [25,26]. This renormalization scale is very important in general, because the essential feature of the PV hadronic interactions comes from the effective weak Hamiltonian that has a specific scale dependence, as mentioned previously. Thus, the matching of this scale consists of an essential part in investigatng any nonleptonic decays and PV hadronic processes.
While the χQSM provides a plausible framework to study the PV πN N coupling constant, there are at least two theoretical difficulties. Firstly, the effective weak Hamiltonian has two-body operators and one has to treat the four-point correlation functions in order to compute the PV πN N coupling constant. Secondly, since the momentumdependent dynamical quark mass is known to play a significant role in describing K → ππ nonleptonic decays [27,28], one can expect that it would also contribute to h 1 πN N substantially. This is in particular important, because a certain amount of non-perturbative effects is reflected in the momentum-dependent quark mass, which arises from the zero mode of instantons. However, it is very difficult to handle these problems in the self-consistent χQSM. In order to circumvent all technical difficulties in the self-consistent approach, we will use the gradient expansion to calculate h 1 πN N , taking the limit of a large soliton size, so that valence quarks in a nucleon plunge into the Dirac sea and the soliton emerges as a topological one [24], which is quite similar to a skyrmion. Equivalently, we can start directly from the ∆S = 0 effective weak chiral Lagrangian derived in Ref. [29] and quantize the chiral soliton collectively. Then, we introduce a physical pion through quantum fluctuations around the soliton field. This procedure will lead to the results for h 1 πN N without fitting any parameter. In the present work, we will restrict ourselves the SU(2) case for simplicity and will concentrate on how the low-energy constants (LECs) found in Ref. [29] feature the PV πN N coupling constant.
This paper is organized as follows: In Section II, we describe briefly a general formalism for the derivation of the PV weak πN N coupling constant. In Section III we present the numerical results for h 1 πN N and discuss the role of the LECs of the ∆S = 0 effective weak chiral Lagrangian. The last section is devoted to the summary and outlook of this work.

II. GENERAL FORMALISM
In this Section, we will show how to incorporate the ∆S = 0 effective weak Hamiltonian into the effective chiral action. We employ the ∆S = 0 effective weak Hamiltonian derived in Ref. [8]. The Hamiltonian reads Euclidean space, and t A denotes the generator of the color SU (3) group, normalized as tr t A t B = 2δ AB . The definitions of the matrices A i and B i , and the coefficients α, β, γ and ρ can be found in [8]. These coefficients are the functions of the scale-dependent Wilson coefficient K(µ) defined as where g(µ 2 ) denotes the strong running coupling constant, µ stands for the renormalization point that specifies the energy scale, b = 11 − 2N f /3, and M W is the mass of the W boson. The coefficient K encodes the effect of the strong interaction from perturbative gluon exchanges. The four-quark operators are expressed generically by where i(= 1, · · · , 12) labels each four-quark operator in the effective weak Hamiltonian and Γ i 1 (2) consist of the Dirac gamma and flavor matrices. Thus, the effective weak Hamiltonian can be rewritten as follows: where C i denotes α, β, γ and ρ according to Eq. (1). In order to derive h 1 πN N in the χQSM, we have to solve the following matrix element: The nucleon state is defined in terms of the Ioffe-type current in Euclidean space (x 0 = −ix 4 ): The nucleon current J † N (J N ) plays a role of creating (annihilating) nucleons. The N * (N ) represensts the normalizing factor depending on the initial (final) momentum. The J † N (J N ) consists of N c quarks: where s 1 · · · s Nc and c 1 · · · c Nc denote respectively spin-isospin and color indices. The Γ {s} (T T3Y )(JJ3YR) are matrices with the quantum numbers (T T 3 Y )(JJ 3 Y R ). For the nucleon, T = 1/2, Y = 1 and J = 1/2. The right hypercharge will be constrained by the baryon number. The creation baryon current is written as The partial conservation of the axial-vector current (PCAC) being considered, the matrix elements in Eq.(5) can be related to the following four-point correlation function where A a µ stands for the axial-vector current. In the χQSM, the correlation function K can be expressed as a functional integral in the chiral limit, where M (−∂ 2 ) denotes the momentum-dependent dynamical quark mass and U γ5 represents the chiral field defined as with the Goldstone boson field U = exp(iλ a π a /f π ). It is, however, extremely complicated to solve Eq. (10) numerically, since the PV πN N coupling constant involves the two-body quark operators Q i and the axial-vector one, which will lead to laborious triple sums in quark levels already at the leading order. Moreover, the momentum-dependent quark mass, which is known to be of great significance in describing nonleptonic processes [27], introduces in addition technical difficulties [30]. One way to avoid these complexities is to use a gradient expansion taking (/ ∂U/M ) ≪ 1 [31] or equivalently is to start from the effective weak chiral Lagrangian already derived in Ref. [29]. Note that though we did not carry out the derivative expansion to order p 4 , it is not difficult to estimate how large the corresponding LECs could be. In Ref. [28], the ∆S = 1 effective weak chiral Lagrangian to order p 4 was investigated in the case of the local chiral quark model. As one can see, all of the LECs are order-of-magnitude smaller than the O(p 2 ) LECs. In this sense, even though we go further beyond the leading order, the contribution from higher derivative terms will not enhance or suppress h πN N much. It will be at most below (5 − 10) %. Thus, we will use the ∆S = 0 effective weak chiral Lagrangian derived in Ref. [29] as our starting point, instead of dealing with Eq. (10). Nevertheless, the present approach goes beyond the previous analyses in the Skyrme model [19][20][21], because the present scheme incorporates properly the effects of the perturbative quark-gluon strong interaction in the derivation of the PV πN N coupling constant.
The leading-order (LO) term of the ∆S = 0 effective weak chiral Lagrangian in the large N c can be expressed in terms of the vector and axial-vector currents where the vector and axial-vector currents are defined as The parameter f π stands for the pion decay constant f π = 93 MeV. The explicit expressions for the coefficientsã ij (a = α, β, γ, ρ) can be found in Ref. [29].
The classical soliton field U 0 is assumed to have a structure of the trivial embedding of the SU(2) hedgehog field as with the profile function of the soliton P (r). This classical soliton field can be fluctuated in such a way that the pion field can be coupled to a weak two-body operator Similarly, the vector and the axial-vector currents transform as where the indices a, b = 1, · · · , 8 and i = 1, 2, 3. The current with a tilde indicates that arising from the background soliton field. Since the PV πN N interaction Lagrangian is expressed as which is linear in the pion field and defined in the SU(2) flavor space (proportional to (τ × π) 3 ), one can easily see that the term 5 i=4 V i µ A i µ does not contribute to the PV πN N Lagrangian. Moreover, since f 8bi = f 0bi = 0, the pion fields for the PV πN N Lagrangian can survive in the vector and axial-vector currents only when a = i. Writing them explicitly, we have Considering the terms contributing to the PV πN N vertex, we obtain for the LO Lagrangian Extracting the terms linear in the pion field from Eq. (19) and rearranging them, we finally derive the LO PV πN N Lagrangian: For simplicity, we have omitted the tildes in the currents. In a similar manner, the next-to-leading order (NLO) effective weak chiral Lagrangian in the large N c expansion derived in [29] yields the Lagrangian for the PV πN N vertex as where and λ 0 is defined as the unit matrix in SU(3) divided by 3. The integrals I i in Eq. (21) were already evaluated in Ref. [29] and are expressed as where ψ ψ M denotes the quark condensate in Minkowski space andM ′ = (dM (k)/dk)/2k. The next step is to carry out the zero-mode collective quantization of the soliton where R(t) stands for the unitary time-dependent SU(3) orientation matrix of the soliton R(t) = exp(iΩ a (t)λ a /2) with its angular velocity Ω a (t) that is of order O(1/N c ). Each current is transformed as where Italic (Greek) indices run over 1, 2, 3 (4, · · · , 7), respectively, and dot (prime) means the derivative with respect to time (radius), respectively. The Wigner D functions and the angular velocity are defined as Before we proceed the calculation of h 1 πN N , we want to emphasize that we will investigate h 1 πN N first in the SU(2) case in this work. Of course, the strange quarks may still play a certain role in describing h 1 πN N . In fact, Ref. [35] showed that the strange quark operator (qλ 3 γ µ q)(sγ µ γ 5 s) induced by Z 0 exchange could contribute significantly to the N N coupling constant. The main argument of Ref. [35] lies in the fact that the ∆I = 1 operator proportional to h 1 πN N can be related to the ∆S = 1 operator by an SU(3) rotation followed by an isospin rotation. Then, it was found that the linear combination of the strange operators made a large contribution to h 1 πN N , which indicates that it has large matrix elements in the nucleon state. The SU(3) Skyrme model came to the similar conclusion that the four quark operators with the strange quark contributed to h 1 πN N significantly [21] because of the induced kaon field. Note, however, that Ref. [21] has not used the renormalized effective weak Hamiltonian but started from the bare Hamiltonian. On the other hand, in a recent lattice study [38], the strange quark operators can only contribute to the quark-loop diagrams for which the signal-to-noise ratio remains far too small to bring out any reasonable signal, so that they were neglected. Moreover, recent findings have it that the content of strange quarks in the nucleon in the vector channel is negligible small [36] and that the strangeness in the scalar and axial-vector channels is still hampered by uncertainties [37]. Thus, it is still too early to reach a conclusion on the contribution of strange quark operators to h 1 πN N . In the present work, we will concentrate on the case of SU (2), since it does not vanish even in SU (2). As we will discuss later in detail, this finite result is distinguished from that of the SU(2) Skyrme model [34] in which h 1 πN N turns out to be equal to zero. The extension of the investigation to SU(3) will be found elsewhere. Since we will calculate h 1 πN N in the process of nπ + → p, we can rewrite the LO and NLO Lagrangians in SU(2) as follows: where we have used the identity with the definitions O ± = 1 2 (O 1 ± iO 2 ) and π ± = 1 √ 2 (π 1 ± iπ 2 ). The eighth component of the Gell-Mann matrices becomes the unity matrix with factor 1/ √ 3 in going from SU(3) to SU(2). The PV πN N coupling constant, h 1 πN N , can be directly read from the matrix element where Let us first compute h 1 πN N with the LO Lagrangian. Note that the iso-scalar current vanishes identically in the present model. By using the results in the previous Section, we can see that the temporal component can contribute to h 1 πN N because of the orthogonality of D ab . This produces the following expression: Because of the zero-mode quantization, the angular velocity is expressed in terms of the spin operator S i Ω i = S i /I, where I is the moment of inertia of the soliton. The spin operator and Wigner D function satisfy the commutation relation [S i , D aj ] = iǫ ijk D ak . Then, the matrix elements of Eq.(38) are written as Since the LO Lagrangian is symmetric under the exchange of the indices 3 and +, it turns out that This null result of the LO h 1 πN N was also obtained in the minimal Skyrme model [34]. The NLO Lagrangian has a rather complicated structure, so that it is convenient to analyze first Λ i 0,3 . Introducing r i µ and l i µ as we rewrite the expressions for Λ i j as Here, index a runs over a = 1, 2, 3. Then, r i µ and l i µ become where the transverse Kronecker delta is expressed as δ ab T = δ ab −r arb . Putting these results together, we arrive at the expressions for Λ + 3 and Λ + 0 : Since one can easily see that only Λ + 0 contributes to h 1 πN N . As a result, h 1 πN N from the NLO Lagrangian turns out to be dr r 2 sin 2 P (r)(sin 2 P (r) − 3 cos 2 P (r)) , where the LECs N 9 and N 10 are given as [29] N 9 = 4N c 4I 1 I 3 (γ 12 + +γ 21 + 2ρ 12 + 2ρ 21 ) + I 2 2 (γ 12 −γ 21 + 2ρ 12 − 2ρ 21 ) = 4 ψψ M I 3 (γ 12 +γ 21 + 2ρ 12 + 2ρ 21 As we will discuss later, the LECs N 9 and N 10 are essential to describe the PV πN N coupling constant.

III. RESULTS AND DISCUSSION
We are now in a position to calculate Eq.(50) numerically. In doing so, we make use of the momentume dependent quark mass derived from the instanton vacuum [26] and the corresponding results of the LECs obtained in Ref. [29]. The value of M 0 = M (k = 0) is taken to be 350 MeV as in Ref. [29], which was fixed by the saddle-point equation from the instanton vacuum [26]. Moreover, we employ three different types of the solitonic profile function to examine the dependence of h 1 πN N on them. The first one is the arctangent profile function P (r) [32] P (r) = 2 arctan where r 0 is given by r 0 = 3gA 16πf 2 π . Employing g A = 1.26 and f π = 93 MeV, we obtain r 0 = 0.582 fm. We use a physical profile function as a second one, which associates with the proper pion tail of the nucleon P (r) = 2 arctan r0 where m π denotes the pion mass and A = 2r 2 0 . r x is determined by the intersection of the arctangent function (r ≤ r x ) and pion tail (r > r x ). If one takes the limit m π → 0 for the pion tail, the physical profile function becomes identical with the arctangent one at large r. With the physical pion mass considered, we have r x = 0.749 fm. The final one is the linear profile function initially proposed by Skyrme [33] where u ≡ 2ef π r with e = 4.84 and λ = 3.342. Using these three profile functions, we can immediately compute the PV πN N coupling constant h 1 πN N . Figure     profile function, whereas the dashed and short-dashed ones correspond to those with the arctangent and linear profile functions respectively. One can regard the difference between the results with the physical profile function and those with the arctangent one as effects of the finite pion mass, which contribute to h 1 πN N approximately by 10 %. Moreover, the type of the profile function does not change much the general features of h 1 πN N , though we preferably take the results with the physical one as our final values.
We find out from Fig. 1 that h 1 πN N is rather sensitive to the Wilson coefficient K and it decreases monotonically, as K increases. We notice that its sign is even changed around K = 6. This can be easily understood. The LECs N 9 and N 10 in Eq. (50) play essential roles in determining the K dependence of h 1 πN N . Figure 2 draws the results of the LECs N 9 and N 10 as functions of K. While N 9 depends rather strongly on K, N 10 does mildly on K. Moreover, N 9 is dominant over N 10 , so that the PV πN N coupling constant is mainly governed by N 9 . Since N 9 is the main contribution to h 1 πN N , we want to examine it in detail. We can easily see that the first term of Eq.(51) containing the quark condensate is much larger than the second one. Moreover, sinceγ 12 andρ 21 are much smaller than the other two coefficientsγ 21 andρ 12 , we can neglect them in N 9 . Then, N 9 can be expressed as As shown in Eq.(1), the coefficientγ 21 comes from the original effective weak Hamiltonian at the mass scale of the W boson µ = M W = 80.4 GeV corresponding to K = 1. In this case, onlyγ 21 survives in N 9 . However, when we start to scale the Hamiltonian down to µ ≈ 1 GeV that corresponds to K ≈ 4, the gluonic renormalization arising from gluon exchange parallel to Z-boson exchange is turned on. As a result, the 2ρ 12 term becomes as large as a half of theγ 21 one [8,29] at this scale. If one goes further down to the scale at which K ≈ 6 1 , the correction ofρ 12 cancels out the contribution ofγ 21 , so that h 1 πN N almost vanishes, as already shown in Figs. 1 and 2. This cancellation implies that the effects of the gluon renormalization leads to the suppression of the PV πN N coupling constant in the present approach of the SU(2) χQSM.
Since Ref. [29] derived the effective weak chiral Lagrangian based on the ∆S = 0 effective weak Hamiltonian, it is plausible to take the value K = 4 for h 1 πN N , which corresponds to the renormalization scale of the charm quark mass µ ≈ 1 GeV as done for the ∆S = 1 case [7]. Thus, we obtain h 1 πN N ≈ 1 × 10 −8 for K = 4. However, if one neglects all the renormalization effects, i.e. if one takes K = 1, we have h 1 πN N ≈ 4 × 10 −8 , which is similar to that of the SU(2) Skyrme model with vector mesons (h 1 πN N = (2 − 3) × 10 −8 ) [19] in which the effective weak Hamiltonian at µ = M W or with K = 1 was used. Note that the present result is almost 40 times smaller than the "best" value of  Very recently, Ref. [38] has reported the first result of lattice QCD: h 1 πN N = (1.099 ± 0.505 +0.058 −0.064 ) × 10 −7 with the pion mass m π = 389 MeV. Thus, it is interesting to compare the present result with the lattice one. In order to do that, we need to examine the dependence of h 1 πN N on the pion mass m π . Figure 3 depicts the PV πN N coupling constant as a function of m π . We employ here the physical profile function with K = 4. Interestingly, h 1 πN N starts to increase, as m π does. As a result, h 1 πN N turns out to be around 1.8 × 10 −8 for m π = 400 MeV. Though it is still around five times less than that of the lattice calculation, we can infer from Fig. 3 that lattice results with the physical pion mass might be quite smaller than that of Ref. [38]. Moreover, if one takes K = 1, h 1 πN N with m π = 389 MeV would become h 1 πN N ≈ 6.77 × 10 −8 that is comparable to the lattice one, though the value K = 1 does not seem tenable for h 1 πN N as discussed previously. However, one has to keep in mind that Ref. [38] has not performed the calculation of the full matrix element, since the quark-loop diagrams were omitted because of technical difficulties. A quantitative comparison with full lattice calculations is still being awaited.

IV. SUMMARY AND OUTLOOK
We have investigated the parity-violating pion-nucleon coupling constant h 1 πN N within the framework of the chiral quark-soliton model with the gradient expansion used. Starting from the ∆S = 0 effective weak chiral Lagrangian derived in the same framework, we have calculated the parity-violating πN N coupling constant. It was found that it vanished at the leading order in the large N c , i.e. Employing three different profile functions, that is, the arctangent, physical, and linear ones, we calculated the parity-violating πN N coupling constant to the next to the leading oder. It turns out that the values of h 1 πN N depend sensitively on the values of the Wilson coefficient K and vanishes around K ≈ 6. The reason can be found in the fact that the contribution of the gluonic renormalization constantρ 12 cancels out that of theγ 21 , which is the leading one. It indicates that the perturbative gluonic contribution supresses the parity-violating πN N coupling constant. Taking the scale of the charm quark mass, i.e. µ ≈ 1 GeV, we found h 1 πN N ≈ 1 × 10 −8 , which is almost 40 times smaller than the "best value" of Ref. [8]. If the µ = M W is selected, the value of h 1 πN N turns out to be similar to that from the Skyrme model with vector mesons [19]. We also compared the present result with that of the lattice calculation. Thus, we examined the dependence of the parity-violating πN N coupling constant on the pion mass and found that h 1 πN N increased as m π did. If one uses m π = 400 MeV, the result turns out to be almost two times larger than that with the physical value m π = 140 MeV but is still about five times smaller than the lattice one. However, we want to emphasize that neither the present result nor the lattice one is the final one.
In order to understand the parity-violating πN N coupling constant h 1 πN N more completely and quantitatively, we have to consider the following important physics: Since we have considered the SU(2) case in the present work, the effects of strangeness were left out. As already mentioned in Section II , however, the strange quark operators may play a certain role in describing the parity-violating πN N coupling constant. Extending from SU(2) to SU(3) is lengthy but straightforward in the present framework. Starting from Eqs. (20,21), we employ the quantization with the embedding (14). In particular, the singlet current is distinguishable from the octet one in SU (3), so that this would make difference in predicting the parity-violating πN N coupling constant. Moreover, the fourth and fifth flavor components of the vector currents enter the next-to-leading order Lagrangian, this would also contribute to h 1 πN N . As was seen in Section III, the gluon renormalization plays a role of suppressing the parity-violating πN N coupling constant. However,the effective weak Hamiltonian at two-loop order was derived very recently in Ref. [39], where the QCD penguin diagrams were also considered. This Hamiltonian is more complete than that from Ref. [8]. Thus, it is of great significance to investigate the ∆S = 0 effective weak chiral Lagrangian based on this effective weak Hamiltonian. It is also of great interest to see whether these penguin diagrams enhance the h 1 πN N or not, since they give part of answers of explaining the ∆I = 1/2 rule in nonleptonic decays [6,7]. The corresponding investigations are under way.