Nucleon axial charge and pion decay constant from two-flavor lattice QCD

The axial charge of the nucleon $g_A$ and the pion decay constant $f_\pi$ are computed in two-flavor lattice QCD. The simulations are carried out on lattices of various volumes and lattice spacings. Results are reported for pion masses as low as $m_\pi=130\,\mbox{MeV}$. Both quantities, $g_A$ and $f_\pi$, suffer from large finite size effects, which to leading order ChEFT and ChPT turn out to be identical. By considering the naturally renormalized ratio $g_A/f_\pi$, we observe a universal behavior as a function of decreasing quark mass. From extrapolating the ratio to the physical point, we find $g_A^R=1.29(5)(3)$, using the physical value of $f_\pi$ as input and $r_0=0.50(1)$ to set the scale. In a subsequent calculation we attempt to extrapolate $g_A$ and $f_\pi$ separately to the infinite volume. Both volume and quark mass dependencies of $g_A$ and $f_\pi$ are found to be well decribed by ChEFT and ChPT. We find at the physical point $g_A^R=1.24(4)$ and $f_\pi^R=89.6(1.1)(1.8)\,\mbox{MeV}$. Both sets of results are in good agreement with experiment. As a by-product we obtain the low-energy constant $\bar{l}_4=4.2(1)$.


I. INTRODUCTION
The axial charge g A of the nucleon is a fundamental measure of nucleon structure. While g A has been known accurately for many years from neutron β decays, a calculation of g A from first principles still presents a significant challenge. Present lattice calculations [1][2][3][4][5], except perhaps [4], underestimate the experimental value by a large amount. The resolution of this problem is of great importance to any further calculation of hadron structure.
Lattice calculations of g A are in many ways connected to calculations of the pion decay constant f π . Both quantities involve the axial vector current, which is not conserved and thus needs to be renormalized. Though it is standard practice nowadays to compute the renormalization constant nonperturbatively (see, for example, [6,7]), some scope of uncertainty remains [8]. Another common feature is that g A and f π seem to be affected by large finite size corrections, in particular at small pion masses, which to leading order ChEFT and ChPT [9][10][11] appear to be the same in both cases. This led us to suggest to determine g A from the ratio g A /f π . Preliminary results [12] were encouraging, and we present here the final analysis of this investigation.
The calculations are done with two flavors of nonperturbatively O(a) improved Wilson fermions and Wilson plaquette action [13], including simulations at virtually physical pion mass and on a variety of lattice volumes. This allows for a separate extrapolation of both g A and f π to the infinite volume and the physical point.

II. LATTICE SIMULATION
Our lattice ensembles are listed in Table I. The pion masses and the chirally extrapolated values of r 0 /a are taken from our preceding paper [13] on the nucleon mass and sigma term.
The Sommer parameter was found to be r 0 = 0.50(1) fm, which we will use to set the scale throughout this paper. The ensembles cover three β values, β = 5.25, 5.29 and 5.40, with lattice spacings a = 0.076, 0.071 and 0.060 fm.
We employ the improved axial vector current where c A is taken from [14]. The improvement term does not contribute to forward matrix elements, but it will contribute to f π .     Table 3 of [17]). In Fig. 1 we plot the effective mass m N of the nucleon at our smallest, nearly physical pion mass, which indicates that excited states have died out at times t 5. To determine the nucleon mass on this ensemble, we chose the conservative fit range t = 8 − 16. In [4] it has been argued that contributions from excited states might be the reason for lattice calculations to underestimate g A , when compared to its experimental value. To investigate this scenario (beyond tuning the smearing parameters, see Fig. 1), we have performed additional simulations on the 24 3 × 48 lattice at β = 5.29, κ = 0.13590 with a large range of different source-sink separations, t = 11, · · · , 19 (0.79, · · · , 1.36 fm), albeit with somewhat lower statistics than our reference point at t = 13 (0.93 fm) on this ensemble.
In Fig. 2 we show the ratio R for various time separations t between source and sink. If our g A determinations were affected by excited state contaminations, then we should find a larger value at separations t > 13. However, we do not see any systematic deviation of R from our result at t = 13 within the error bars, not even for t = 11. This provides us with confidence that our choices of t are sufficient with our choice of source and sink smearing.
Similar conclusions were found in [3].
Our smearing parameters are tuned to give a rms radius of ≈ 0.5 fm, which is about half the radius of the nucleon. For this level of smearing no further improvement of the extracted result for g A was found by employing variational techniques [17], which systematically separate excited states out from the ground state at source and sink.
The calculation of f π follows [18]. We use the notation employed in ChPT, with the experimental value f π + = 92.2 MeV. Our final results for the bare quantities, g A and af π , on all our ensembles, are given in Table I.

III. RESULTS
Except for the very lowest pion mass, m π L 4 (L being the spatial extent of the lattice) on our larger lattices at any other κ value, which is state of the art for pion masses of O(200) MeV. But even on lattices of this size g A , f π and m π are found to suffer from finite size effects, which we have to deal with in one way or another.
Finite size corrections to g A , f π and m π have been studied extensively in the literature.
In the Appendix we show, based on predictions of ChEFT and ChPT adapted to the finite volume, that the leading corrections to g A and f π are identical and cancel in the ratio g A /f π .
This makes g A /f π the preferred quantity for computing g A .
Denoting the physical, renormalized axial charge and pion decay constant in the infinite volume by g R A and f R π , respectively, and making use of the fact that the renormalization constant Z A of the axial vector current cancels in the ratio g A /f π , we then have To test this relation, we plot g A /af π for three different lattice volumes at our second lowest pion mass in Fig. 3. The ratio is found to be independent of the volume, within the errors, which demonstates that finite size corrections cancel indeed in g A /f π . The curve shows a fit of eq. (5) to the data.
Let us now turn to the calculation of g A . In Fig. 4 we plot the ratio g A (L)/f π (L) for our (raw) data points listed in Table I, restricting ourselves to pion masses m π ≤ 750 MeV, and taking r 0 = 0.50(1) fm to set the scale. If we have more than one volume at a given κ value, we show the result of the largest volume. The lowest pion mass in Fig. 4 is 157 MeV. The data points of all three β values lie nicely on a universal curve. At m 2 π ≈ 0.06, 0.23 and 0.44 GeV 2 , for example, where we have results for more than one lattice spacing, the data points coincide with each other, indicating that discretization effects are negligible.
With finite size corrections being practically absent, the leading order chiral expansion of g A /f π can be cast in the form [10,19,20] We have fitted eq. (5) to the data points in Fig. 4. The result is shown by the solid curve.
At the physical point this gives The second error is due to the error on r 0 . Multiplying the ratio (6) by the physical value of f π , f R π = 92.2 MeV, we then obtain g R A = 1.29 ± 0.05 ± 0.03 .
Alternatively, we could have set the scale by the physical value of f π , using the results of Sec. III B. That would give the value g R A = 1.27 (5). Now that we have presented the main result of the paper, i.e. g R A from the ratio g A (L)/f π (L), we proceed to study g R A and f R π separately and present results in the context of established expressions from finite volume ChEFT and ChPT.
B. The axial coupling g A and pion decay constant f π in the infinite volume We now consider explicitly the finite size formulae as given in the Appendix. In the following fits we take f 0 = 86 MeV [21]. There is some freedom in which pion mass to   Table II we list our final pion masses. Our lowest mass turns out to be m π = 130(5) MeV.
Let us now turn to the axial charge and the pion decay constant. In Fig. 6 we show the fits of eqs. (A.3) and (A.4) to g A and af π , respectively, for our three lowest pion masses.
In this case the fits involve one free parameter each, g A (∞) and af π (∞), only. The leading order expressions are able to describe the data at β = 5.29, κ = 0.13632 on all three volumes, which include data with m π L < 3 as well as m π L > 4. This gives us confidence that the fits provide a reasonable infinite volume extrapolation at the lighter mass point as well, where we do not have access to data with larger m π L. 2 All fits gave χ 2 /d.o.f < 1.4.
To obtain continuum numbers, we need to renormalize the axial vector current. The latter reads The coefficient b A is required to maintain O(a) improvement for nonvanishing quark masses m q as well. The renormalization constant Z A has been computed nonperturbatively in [7], employing the Rome-Southampton method [6], with the result  The coefficient b A is only known perturbatively [23], In Table II we give g R A in the infinite volume. For pion masses m π ≤ 300 MeV we demand that we have at least two lattice volumes to ensure a controlled extrapolation. For this reason we excluded the point at β = 5.25, κ = 0.13620 from the analysis.
Our results for g R A are plotted in Fig. 7. Since after finite volume corrections the lightest pion mass is 130 MeV, no extrapolation is required. At the lightest pion mass we find in good agreement with the previous determination and the experimental value. It turns out that g R A hovers around ≈ 1.1 for m π 250 MeV, a feature it shares with most other lattice calculations [5]. Only within the last 100 MeV from the physical point does g R A rise to its final value. This phenomenon is not totally unexpected, from general arguments [24] and from ChEFT [10,20,25]. Near the chiral limit ChEFT predicts, following [10], To this order, both sets of chiral expansions, [10,20] and [25], are equivalent with B 9 = d 16 and B 20 = d 28 . In (12) we have chosen λ = m π phys (λ being the scale parameter of the dimensional regularization). The coupling B r 9 cannot be observed independent of B r 20 . Taking B r 20 (m π phys ) ≡ 0, the preferred value is [26] B r 9 (m π phys ) = (−1.4 ± 1.2) GeV −2 . A fit of the leading order chiral formula (12) to the data points in Fig. 7 is shown by the shaded area. The fit gives g 0 A = 1.26(7) and B r 9 (m π phys ) = (−2.1 ± 1.0) GeV −2 . Our results for f R π are plotted in Fig. 8. Again, no extrapolation to the physical point is needed. At the lightest pion mass we find using r 0 = 0.50(1) fm. The second error in eq. (13) is due to the error on r 0 .
Instead of taking f R π at the lowest pion mass, eq. (13), it might be a better idea to include the adjacent data points in the analysis as well and fit the data by a chiral ansatz [19], The result of the fit is shown in Fig. 8. At the physical point we obtain in full agreement with the result (13). The main effect is that the statistical error has reduced by 30%. In the chiral limit we obtain f 0 = 86(1) MeV, which agrees with the assumption made in Sec. III B. A fit of the chiral ansatz (14) to the lowest four data points with A = 0 gives the low-energy constantl The main result of this paper is a determination of g R A from the ratio g A /f π , which is free of finite size effects and renormalization errors. We found that this ratio has a smooth behavior as a function of quark mass, and can be essentially described by a polynomial in m 2 π , leading to g R A = 1.29 (5)(3) at the physical point, using r 0 = 0.50(1) [13] to set the scale, in excellent agreement with experiment.
We attempted a direct calculation of g R A and f R π , taking account of finite size corrections and renormalization. To our knowledge, this is the first time finite size corrections have been applied to g A at physical pion masses. Both approaches give consistent results, suggesting that finite size corrections to both g A and f π are well described by ChEFT and ChPT.

Appendix: Finite size corrections
Let us first consider g A . Utilizing the (nonrelativistic) small scale expansion (SSE) of the ChEFT, including pion, nucleon (N) and ∆(1232) degrees of freedom, we obtain to O(ǫ 3 ) [10] where λ = m π L and λ(y) = f (m π , y)L with f (m π , y) = m 2 π + y 2 + 2y∆ 0 , ∆ 0 being the ∆ − N mass difference. K 0 and K 1 denote the modified Bessel functions, and c A and g 1 are the leading axial ∆N and ∆∆ couplings. The parameter c A should not be confused with the improvement coefficient c A in eq. (1).
The second term in eq. (A.1), ∆(L), receives contributions from chiral loops, which renormalize the axial charge and act on intermediate ∆ baryons [10]. It turns out that the various contributions to ∆(L) effectively cancel each other over a wide range of λ values. This has been noticed by the authors of [27] as well. To state an example, let us consider the 48 3 × 64 lattice at β = 5.29, κ = 0.13640. This lattice has the lowest pion mass and is especially important for our final conclusions. Taking c A = 1.5 from [28] and g 1 = 2.16 from SU(6), we find −0.044 for the total contribution, but only +0.001 for ∆(L). We thus may The finite size corrections to f π have been computed in [11] within the context of ChPT.
To NLO (∝ m 2 π ) the outcome is The NNLO corrections are found to be very small on our configurations and, thus, can safely be neglected.
The investigations above show that the leading finite size corrections to g A and f π , The NLO correction to the pion mass reads [11] m π (L) − m π (∞) m π (∞) = m 2 π 16π 2 f 2 0 n |n| =0 At smaller values of m π L, m π L 3, this expression alone cannot describe the observed finite size effects [13]. That is not surprising, since in a finite spatial box chiral symmetry does not break down spontaneously. This is because giving the system enough time it will rotate through all vacua. This results in a mass gap at vanishing quark masses [29][30][31], m π res = 3 2f where Λ i are the intrinsic scale parameters of the low-energy constantsl i = ln Λ 2 i /m 2 π phys [19], with m π phys being the physical pion mass. In [22] we found that the pion mass extrapolates indeed to a finite value in the chiral limit, in good agreement with the expected result (A.6). This also has an effect on m π in the region of small, but nonvanishing, quark masses [22]. We thus expect the finite size correction to be effectively given with the parameter c(m π ) rapidly dropping to zero at larger pion masses.