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On the Connection Between the Charged and Neutral Pion-Nucleon Coupling Constants in the Yukawa Model

In the Yukawa model for nuclear forces, a simple relation between the charged and neutral pion-nucleon coupling constants is derived. The relation implies that the charged pion-nucleon coupling constant is larger than the neutral one since the np interaction is stronger than the pp interaction. The derived value of the charged pion-nucleon constant shows a very good agreement with one of the recent experimental values. The relative splitting between the charged and neutral pion-nucleon coupling constants is predicted to be practically the same as that between the charged and neutral pion masses. The charge dependence of the NN scattering length arising from the mass difference between the charged and neutral pions is also analyzed.

Exact knowledge of their values is instrumental for quntitatively describing and qualitatively understanding a broad variety of hadron-and nuclear-physics phenomena [3][4][5][6][7][8]. For this reason, the pion-nucleon coupling constants have been investigated and their values have been refined throughout the whole period of nuclear-physics studies. Historically, the evolution of these investigations is detailed in [5,7,9].
In this paper we investigate the pion-nucleon coupling constant relying on the standard classical Yukawa model [1][2][3]31] for the nucleon-nucleon interaction and invoking contemporary experimental data on low-energy parameters of nucleon-nucleon scattering. It is well known [1][2][3] that, unfortunately, no accurate quantitative description of the nucleonnucleon system can be obtained with the Yukawa potential as soon as its parameters are defined and selected using fundamental quantities of the field theory: the pion masses and coupling constants. Therefore, in selecting the parameter values of the Yukawa potential, we rather rely on measured low-energy parameters of nucleon-nucleon scattering in the effective range theory.
Analyzing the pion-nucleon coupling constants in the Yukawa-model framework is justified, since the one-pion exchange which corresponds to the Yukawa potential is the dominant mechanism of the nucleon-nucleon interaction at the lowest collision energies, which imply long-range interactions. The two-pion exchange and the heavier ρ-and ω-meson exchanges are dominant at medium and small distances, and quark-gluon degrees of freedom gain an important role at the smallest interaction ranges. Quite understandably, in determining the pion-nucleon coupling constant as a characteristic of the pion-nucleon interaction, one should rely on the data for long-range (or peripheral) nucleon-nucleon interaction, which is dominated by one-pion exchange.
In quantum field theory, the pion-nucleon interaction may be described by either a pseudoscalar (P S) or a pseudovector (P V ) Lagrangian formulated as [3,9,20,32,33] where in the latter formula the scaling mass m s is introduced so as to render the pseudovector pion-nucleon coupling constant f π dimensionless. In Eqs. (1) and (2), the nucleon and pion fields are denoted as ψ (x) and φ (x), respectively. As was demonstrated in [9,34], the Lagrangian forms (1) and (2) are interrelated by a gauge transformation and therefore are equivalent.
Also note that our Lagrangian definitions (1) and (2) feature explicit √ 4π factors so that the 4π factor no longer enters the coupling constants themselves. In other words, we follow the convention adopted in the review paper [9] and write simply g 2 π instead of the often-used denotation g 2 π /4π. According to the convention adopted in [9,32], the scaling mass m s is usually assumed to be equal to the charged-pion mass: m s = m π ± . Then, the pseudoscalar pion-nucleon coupling constant g π and the pseudovector coupling constant f π are interrelated by the equivalence equation [9,34] implied by the equivalence of Lagrangian forms (1) and (2), where M 1 and M 2 are the masses of interacting nucleons. Therefore, the π 0 and π ± pseudoscalar coupling constants, g π 0 and g π ± , and the corresponding pseudovector constants f π 0 and f π ± may be interrelated through [5,9,14] where M p and M n denote the proton and neutron masses, respectively.

YUKAWA MODEL
According to the meson-field theory, strong interaction between two nucleons at low energies is dominated by the exchange of virtual pions, which determine the form of the long-range nucleon-nucleon interaction. The classical one-pion-exchange potential of the nucleon-nucleon interaction in the meson-field theory, referred to as the Yukawa potential, for the pure singlet 1 S 0 state has a simple and well-known form of [1-3, 31, 33] V Y (r) = −V 0 e −µr µr .
Here, r is the distance between the nucleons and µ is expressed through the pion mass m π as µ = m π c , where c and are the speed of light and the reduced Planck constant, respectively. According to (7), the nuclear-force range R ≡ 1/µ is inversely proportional to the pion mass m π and has a small value of R ∼ 1.4 fm. For this case, the depth of the Yukawa potential, V 0 , is expressed through the pseudovector pion-nucleon coupling constant f π via a simple relation [1-3, 5, 33,

35]
which, like the Yukawa-potential form (6), is a direct consequence of the quantum meson-field theory represented by the Lagrangians (1) and (2).
For the interaction between two charged protons mediated by the exchange of a neutral π 0 meson, the µ pp and V pp 0 parameters of the Yukawa potential (6) are expressed through the π 0 mass m π 0 and the coupling constant f π 0 according to equations (7) and (8). The neutron-proton interaction involves the exchanges of both the neutral π 0 mesons and charged π ± mesons. In estimating the µ np and V np 0 parameters of the potential (6) for the latter case, one should [8,36] substitute the averaged pion mass and the averaged pion-nucleon coupling constant We know [1][2][3] that, unfortunately, no accurate quantitative description of the nucleonnucleon system can be obtained with the Yukawa potential as soon as its parameters are defined and selected using fundamental quantities of the field theory the pion masses and coupling constants. Therefore, in this paper the parameters of the Yukawa potential are assigned values consistent with the measured parameters of low-energy nucleon-nucleon scattering in the effective range theory [1,4,[37][38][39][40][41][42][43][44].
Indeed, one may estimate the "effective" mass m Y π 0 and pion-nucleon coupling constant f Y π 0 2 for the neutral π 0 meson from the measured purely nuclear proton-proton scattering length and effective range, assuming the Yukawa form for the proton-proton potential. Thus estimated m Y π 0 and f Y π 0 2 values prove to significantly exceed the directly measured values [26,27], so that we have where the factors C 1 and C 2 can be computed numerically [26,27]. Their exact values are not needed for our purposes. It is quite natural to assume that equations analogous to (11) also hold for the masses and pion-nucleon coupling constants of charged π ± mesons, and therefore also for the averaged pion mass (9) and averaged pion-nucleon coupling constant (10).
Under the latter assumption and Eqs. similar in form to (11), from (8)-(10) we obtain the following relations between the parameters of the neutron-proton Yukawa potential, µ np and V np 0 , and the analogous parameters of the proton-proton potential, µ pp and V pp 0 : Equations (10) and (13) directly imply a relation between the pseudovector pion-nucleon coupling constants for the charged and neutral pions, where the factor C 2 f is expressed as From Eqs. (4), (5), and (14) we obtain that the pseudoscalar charged and neutral pionnucleon coupling constants are interrelated as where Equation (15) features directly measured pion masses, so in the considered model the proportion between the charged and neutral pion-nucleon coupling constants is fully determined by that between the depths of the neutron-proton and proton-proton Yukawa potentials, Further we demonstrate that the neutron-proton potential is appreciably deeper than the proton-proton one: As a consequence, the factor in parentheses in Eq. (15) is in excess of two. Therefore, in the considered formalism, the charged pion-nucleon coupling constant proves to be larger than the neutral one: That the charged pion-nucleon constant is greater than the neutral one is, within the considered scheme, a consequence of stronger np than pp interaction in the spin-singlet 1 S 0 state, which is a reliably established phenomenon. One of its manifestations is that the absolute singlet length of np scattering is larger than the purely nuclear pp scattering length: |a np | > |a pp |.
a np = −23.715(8) fm (20) and applying the variable phase approach [49], for the Yukawa-potential parameters of pp and np interactions we obtain Note that the neutron-proton interaction parameters have been obtained using Eq. (12) and substituting the neutron-proton scattering length as (20).
The Yukawa potential with parameters (22) results in an np-scattering effective range of which fully agrees with the measured value [41,47,50,51] r np = 2.70(9) fm .
As expected, we have so that the aforementioned condition leading to inequalities (18) is satisfied.
Substituting the derived values (21) and (22) of the depths of pp and np potentials and the measured pion and nucleon masses [52] in (15) and (17), for the factors relating the charged and neutral pion-nucleon coupling constants we obtain The C 2 f and C 2 g values are seen to be very similar, as expected from the proximity of the proton and neutron masses.
The g 2 π ± value (28) derived by us in the Yukawa-model framework practically coincides with one of the most recent experimental values: The measurement (31), reported in [6] by the Uppsala Neutron Research Group, is close to measurements of the same group, g 2 π ± = 14.62 (35) [23] and g 2 π ± = 14.74(33) [11], and to the value g 2 π ± = 14.28 (18) earlier obtained in [20][21][22]. On the other hand, the charged pion-nucleon coupling constant was extracted by the Nijmegen group as g 2 π ± = 13.54(5) [9,10], which practically coincides with the neutral π 0 constant g 2 π 0 . Other recent experimental evaluations [16][17][18][19] have yielded values of g 2 π ± ∼ 13.7 ÷ 13.8 which are close to the neutralpion constant g 2 π 0 . Thus, possible charge dependence of the pion-nucleon coupling constant, or possible difference between those for charged and neutral pions, is still an open problem which is of paramount and fundamental importance. The proposed model involves an explicit violation of charge independence of the pion-nucleon constant; see Eqs. (27) and (28).
A measure of the charge independence breaking (CIB) of pion-nucleon couplings is the difference between the charged and neutral pion-nucleon coupling constants: In the considered model, Eqs. (14) and (16) imply the following explicit expressions for these quantities: Substituting the numerical value (26) for the factor C 2 g , which relates the pseudoscalar charged and neutral pion-nucleon coupling constants, as well as the reliably measured value (27) for the neutral constant g 2 π 0 , for the absolute breaking of charge independence of pion-nucleon coupling constants in the Yukawa model we obtain In relative units, the charge independence breaking in pion-nucleon coupling constants is formulated as Thus, the relative violation of charge independence of pion-nucleon coupling constants is as high as 7.4 % in the considered scheme.
Equations (36) and (37) suggest that the relative charge independence breaking is larger by 0.15 % for the pseudoscalar pion-nucleon coupling constant g 2 π than for the pseudovector constant f 2 π . According to Eqs. (15) and (17), this effect arises from the neutron-proton mass difference (M n > M p ). Therefore, even as soon as the pseudovector coupling constant is strictly charge-independent (f 2 π ± = f 2 π 0 ), charge invariance should be violated for the pseudoscalar coupling constant g 2 π [9]. It should be noted that the values (36) and (37) derived for relative charge independence breaking of coupling constants are actually independent of particular values of the charged and neutral pion-nucleon constants, but are rather determined by the pion and nucleon masses according to (15) and (17) and by the experimental input parameters of the model quoted in (19) and (20).

CONNECTION BETWEEN CHARGE SPLITTINGS OF THE PION-NUCLEON COUPLING CONSTANT AND OF THE PION MASS
Equations (14) and (26) imply that the charged and neutral pseudovector pion-nucleon coupling constants are in a ratio of which closely agrees with the ratio between measured masses of the charged and neutral pions [52] m π ± m π 0 = 1.0340 .
Therefore, to a high precision we have Thus, charge splitting of the pion-nucleon coupling constant practically coincides with that of the pion mass.
Equation (40) has a simple physical justification. Since the pion-nucleon coupling constant f π is a measure of the strength of pion-field action on the nucleon, this action is stronger the greater the pion mass m π is. Since we have m π ± > m π 0 , the nucleon is more strongly affected by the ambient field of charged π ± mesons than by that of neutral π 0 mesons. Equation (40) directly implies that, to a high precision, the ratio f π /m π is a charge independent quantity, in contrast with the coupling constant f π by itself.
As soon as the equality in (40) is exact, for the charged pseudovector coupling constant one obtains Under the same assumption, for the pseudoscalar coupling constant, Eqs. (5) and (41) imply the value which practically coincides with the value (28) computed with formula (16). Then, from (27) and (42), the charge independence breaking in the pion-nucleon coupling constant arising from the mass difference between the π ± and π 0 mesons (∆m π = 4.59 MeV) is estimated as which amounts to ∼ 7 % in relative units.
Pseudoscalar coupling constants obey the approximate equation which is analogous to Eq. (40) for pseudovector coupling constants. Formula (44) first attracted attention in [5,14], but in these analyses it was treated as an accidental coincidence which has no physical meaning. In our analysis, Eqs. (40) and (44) result from implementing the traditional classical Yukawa model and relying on measured values of low-energy pp-and npscattering parameters quoted in (19) and (20).
Equation (40) may be rewritten as where the radius R π 0 of the meson-exchange potential for the π 0 -exchange is larger than the radius R π ± corresponding to the π ± -exchange: Thus, the pion-nucleon coupling constant f π and the radius of the meson-exchange potential R π prove to be charge-dependent quantities due to the mass splitting between the π ± and π 0 mesons. That π ± mesons are heavier than the π 0 meson effectively increases the charged-pion constant f π ± with respect to the neutral-pion constant f π 0 and effectively reduces the π ±exchange radius R π ± with respect to the π 0 -exchange radius R π 0 . As a result, the product of the pion-nucleon constant f π and the π-exchange radius R π is a charge-invariant quantity: Substituting the well-measured value of the neutral pion-nucleon constant f π 0 = 0.2735 (49) and the value (46) for the π 0 -exchange radius, the constant B is numerically estimated as Thus, the pion-nucleon coupling constant f π and the π-exchange radius R π are correlated through the relation which is valid to high precision.

CHARGE DEPENDENCE OF THE NUCLEON-NUCLEON SCATTERING LENGTH
Since the 1 S 0 state of the two-nucleon system features a virtual level with a nearly zero energy, scattering length is the parameter which is most sensitive to small variations of the nucleon-nucleon potential. For this reason, the charge independence breaking of nuclear forces is often quantitatively estimated using the difference between the proton-proton and neutronproton scattering lengths, According to (19) and (20), the experimental value of this difference is which amounts to ∼ 30 % in relative units. That this difference is nonzero significantly beyond the experimental uncertainty indicates that the hypothesis of charge independence of nuclear forces is violated at low energies [35,[53][54][55][56]. The charge dependence of nuclear forces is often attributed to the mass difference between charged and neutral pions [14,35,53,[57][58][59][60][61]. However, only a half of the difference ∆a еxpt CIB has been shown to be due to the mass difference between the π ± and π 0 mesons [14,53,60,61].
The value of the singlet np-scattering length computed by us assuming that equality (40) is exact, differs from the pp-scattering length of a pp = −17.3(4) fm. The difference between the computed pp-and np-scattering lengths, is consistent with experimental value (53).
Thus, within the discussed model framework, the violation of charge independence of nuclear forces is fully explained by the mass difference between charged and neutral pions. The predicted difference between the pp-and np-scattering lengths, ∆a π CIB , amounts to ∼ 90 % of the corresponding experimental value ∆a еxpt CIB . In contrast with this, the ∆a π CIB values derived in previous analyses reached only ∼ 50 % of ∆a еxpt CIB [14,60,61].

MAJOR CONCLUSIONS AND SUMMARY
In this paper, which is based on the Yukawa meson-field model, we develop a physically consistent formalism of the nucleon-nucleon interaction in which the parameters of the np and pp systems in the spin-singlet state 1 S 0 are related to major characteristics of the pion-nucleon interaction: the pion masses m π and the pion-nucleon coupling constants f 2 π . Within this model framework, a simple relation between the charged and neutral pion-nucleon coupling constants is formulated by Eqs. (14)- (17).
The results of our analysis suggest that the charged pion-nucleon coupling constant f 2 π ± is larger than the corresponding neutral constant f 2 π 0 , so that charge independence of nuclear forces is violated for the pseudovector and pseudoscalar pion-nucleon coupling constants, f 2 π and g 2 π . Using the derived formulae, for the pseudoscalar charged constant we obtain the value g 2 π ± = 14.53 (25), which virtually coincides with the recent experimental value reported by the Uppsala Neutron Research Group [6]: g 2 π ± = 14.52 (26). The absolute value of the difference between the charged and neutral pseudoscalar pionnucleon coupling constants, ∆f CIB ≡ f π ± − f π 0 , is derived by us as ∆f CIB = 0.0093. In relative units, the ratio ∆f CIB /f π 0 = 3.58 % proves to be very close to ∆m π /m π 0 = 3.40 %. Therefore, in relative units the charge splitting of the pion-nucleon coupling constant is practically the same as that of the pion mass.
Our analysis demonstrates that, while both the pion-nucleon coupling constant f π and the π-meson-exchange radius R π are charge dependent, their product f π R π is, to high precision, a charge independent quantity. In relative units, the difference between such products for charged pions and neutral pions does not exceed 0.2 %.
In our model approach, 90 % of the difference between experimental values of the pp-and np-scattering lengths is accounted for by the mass difference between the π ± and π 0 mesons, ∆m π = 4.59 MeV.