Approximating chiral \({SU}(3)\) amplitudes
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Abstract
We construct large\(N_c\) motivated approximate chiral \({SU}(3)\) amplitudes of nexttonexttoleading order. The amplitudes are independent of the renormalisation scale. Fitting lattice data with those amplitudes allows for the extraction of chiral coupling constants with the correct scale dependence. The differences between approximate and full amplitudes are required to be at most of the order of N\(^3\)LO contributions numerically. Applying the approximate expressions to recent lattice data for meson decay constants, we determine several chiral couplings with good precision. In particular, we obtain a value for \(F_0\), the meson decay constant in the chiral \({SU}(3)\) limit, that is more precise than all presently available determinations.
1 Introduction
Hadronic processes at low energies cannot be treated with perturbative QCD. The main protagonists in this field, lattice QCD and chiral perturbation theory (CHPT), have mutually benefited from a cooperation started several years ago. The emphasis of this cooperation has shifted in recent years. Although extrapolation to the physical quark (and hadron) masses and finitevolume corrections, both accessible in CHPT, are still useful for lattice simulations, improved computing facilities and lattice algorithms allow for simulations with ever smaller quark masses and larger volumes. On the other hand, the input of lattice QCD for CHPT has become more important over the years to determine the coupling constants of chiral Lagrangians, the socalled lowenergy constants (LECs). This input is especially welcome for LECs modulating quark mass terms: unlike in phenomenological analyses, quark (and hadron) masses can be tuned on the lattice.
While this program has been very successful for chiral \({SU}(2)\), the situation is less satisfactory for \({SU}(3)\) [1]. In the latter case, the natural expansion parameter (in the meson sector) is \(M_K^2/16 \pi ^2 F_\pi ^2 \simeq 0.2\). To match the precision that lattice studies can attain nowadays, it is therefore mandatory to include NNLO contributions in CHPT. Although NNLO amplitudes are available for most quantities of interest for lattice simulations [2], there has been a certain reluctance in the lattice community to make full use of those amplitudes for two reasons mainly: for chiral \({SU}(3)\), NNLO amplitudes are usually quite involved and they are mostly available in numerical form only.

We set up numerical criteria for the amplitudes to qualify as acceptable approximations. Those criteria can be checked by comparing with available numerical results making use of the full NNLO amplitudes for some given sets of meson masses.

The proposed approximation includes all terms leading and nexttoleading order in large \(N_c\). In addition, it contains all chiral logs, independently of the large\(N_c\) counting. In order to meet the numerical criteria just mentioned, it may sometimes be useful to go beyond the strict large\(N_c\) counting by including also products of oneloop functions occurring in twoloop diagrams.

In addition to the ratio \(F_K/F_\pi \) of meson decay constants investigated in Ref. [3], here we also study the pion decay constant \(F_\pi \) itself. By confronting our approximation with lattice data, we demonstrate the possibilities to extract information on both NLO and NNLO LECs. While the NNLO LECs have the expected large uncertainties, the NLO LECs can be determined quite well. Our numerical fits of lattice data are not intended to compete with actual lattice results for obvious reasons. Instead, we hope to encourage lattice groups to use NNLO amplitudes that are much simpler than the full amplitudes and yet offer considerably more insight than, e.g., polynomial fits. These amplitudes can also be considered as relatively simple tools to study convergence issues of chiral \({SU}(3)\) with lattice data.
2 Analytic approximations of NNLO amplitudes
The procedure how to actually calculate an amplitude corresponding to Eq. (2.1) was described in Ref. [3]. In many cases, the relevant amplitudes can be extracted from available calculations of \(O(p^6)\) [2].

All chiral logs are included.

The functional \(Z_6^{I}\) is independent of the renormalisation scale \(\mu \). Unlike the doublelog approximation [8], it therefore allows for the extraction of LECs with the correct scale dependence.

In addition to single and double logs, the residual dependence on the scale \(M\) is the only other vestige of the twoloop part.
 In dropping the genuine twoloop contributions, Approximation I respects the large\(N_c\) hierarchy of \(O(p^6)\) contributions:$$\begin{aligned} C_a, L_i L_j ~\longrightarrow ~L_i \times \,\mathrm{loop} ~\longrightarrow ~ 2\text {}\mathrm{loop}\,\mathrm{part}. \end{aligned}$$

Only tree and oneloop amplitudes need to be calculated.
Approximation I is motivated by large \(N_c\), but in some cases the accuracy may be improved by including in the approximate functional (2.1) also products of oneloop amplitudes (from diagrams a,c in Fig. 1, subleading in \(1/N_c\)), which also have a simple analytic form. We call this extension Approximation II. In contrast to Approximation I, this extension is not uniquely defined^{1} because it depends on the representation of the matrix field \(U\). In the standard representation used, e.g., also in Refs. [4, 9], it amounts to omitting (except for chiral logs) the sunset diagram b from the full functional \(Z_6\) in Eq. (7.21).
3 \({\mathbf {F_K/F}}_{\varvec{\pi }}\) and the lowenergy constant \({\mathbf {L_5}}\)
The explicit expression for Approximation II of \(F_K/F_\pi \) is given in Appendix B where all masses are lowestorder masses of \(O(p^2)\). Since we work to \(O(p^6)\) the masses in \(R_4\) [Eq. 8.2] must be expressed in terms of the lattice masses to \(O(p^4)\) [6]. The chiral limit value \(F_0\) is expressed in terms of the experimental value \(F_\pi =92.2\) MeV and physical meson masses, using again the \(O(p^4)\) relation. In \(R_6\) and \(R_6^\mathrm{ext}\) [Eqs. 8.3, 8.4], \(F_0\) and the meson masses can be replaced by \(F_\pi \) and lattice masses, respectively.
Fit results for \(F_K/F_\pi \) and LECs for Approximations I (statistical errors only) and II
\(F_K/F_\pi \)  \(10^3 L_5^r\)  \(10^3 (C_{14}^r + C_{15}^r)\)  \(10^3 (C_{15}^r + 2 C_{17}^r)\)  

App. I (\(M=M_K\))  \(1.198(5)\)  \(0.76(8)\)  \(0.37(7)\)  \(1.29(14)\) 
App. II  \(1.200(5)\)  \(0.75(8)\)  \(0.20(8)\)  \(0.71(22)\) 
BMW [11]  \(1.192(7)_\mathrm{stat}(6)_\mathrm{syst}\) 
The fitted values of \(F_K/F_\pi \) agree with the detailed analysis of Ref. [11]. For both \(F_K/F_\pi \) and \(L_5\), there is practically no difference between the two approximations but the LECs of \(O(p^6)\) show a bigger spread. For Approximation II, we have added the uncertainty due to varying \(M\) in the range \(0.9 \le M/M_K \le 1.1\) in quadrature to the statistical lattice errors. The effect of this variation is small, for \(F_K/F_\pi \) and \(L_5\) in fact negligible. Since \(C_{15}\) is subleading in \(1/N_c\) the fit determines essentially \(C_{14}\) and \(C_{17}\) [10]. Although the values depend of course on the input for the \(L_i\), the results in Table 2 suggest that both \(C_{14}^r\) and \(C_{17}^r\) are positive and smaller than \(10^{3} ~\mathrm{GeV}^{2}\), always taken at \(\mu =0.77\) GeV. We will use these fit results with Approximation II for \(L_5^r\), \(C_{14}^r\) and \(C_{17}^r\) in the analysis of \(F_\pi /F_0\) in the following section.
The fit also demonstrates very clearly that NNLO terms are essential. While the NNLO fit (Approximation II) is well behaved (\(\chi ^2\)/dof = 1.2, statistical errors only), the NLO fit with the single parameter \(L_5\) is unacceptable (\(\chi ^2\)/dof = 4). Analysing presentday lattice data with NLO chiral \({SU}(3)\) expressions does not make sense.
4 \({\mathbf {F}}_{\varvec{\pi }}\) and the lowenergy constants \({\mathbf {F_0,L_4}}\)
The lowenergy expansion in chiral \({SU}(3)\) is characterised by the ratio \(p^2/(4\pi F_0)^2\) where \(p\) stands for a generic meson momentum or mass. \(F_0\) thus sets the scale for the chiral expansion. In practice, \(F_0\) is usually traded for \(F_\pi \) at successive orders of the chiral expansion. Nevertheless, \(F_0\) affects the ‘convergence’ of the chiral expansion: a smaller \(F_0\) tends to produce bigger fluctuations at higher orders.
Why has it been so difficult both for lattice and phenomenological studies to determine \(F_0\)? One clue is the apparent anticorrelation with the NLO LEC \(L_4\) in the fits of Ref. [13]: the bigger \(F_0\), the smaller \(L_4^r(M_\rho )\), and vice versa. The large\(N_c\) suppression of \(L_4\) is not manifest in the fits with small \(F_0\).
\({SU}(3)\) lattice data for \(F_\pi \) seem well suited for a determination of \(F_0\) and \(L_4\) although the emphasis in most lattice studies has been to determine \(F_\pi \) itself. As for \(F_K/F_\pi \), the use of CHPT to NNLO, \(O(p^6)\) [9], is essential for a quantitative analysis.
In the following, we are going to apply Approximation I for the analysis of \(F_\pi \). It turns out that, unlike for \(F_K/F_\pi \), Approximation I agrees better with the numerical results of Ref. [10] than Approximation II. The explicit representation for \(F_\pi \) is given in Appendix C. The lowestorder masses appearing in the terms of \(O(p^4)\) must again be expressed in terms of lattice masses. Unlike in the previous section, we leave \(F_0\) in Eq. (9.1) untouched.
We confront the expression (9.1) for \(F_\pi \) with lattice data from the RBC/UKQCD Collaboration [15, 16]. In our main fit we only consider (five) unitary lattice points with \(M_\pi < 350\) MeV. In this case, \(F_\pi \) for physical meson masses emerges as a fit result but the fitted value is lower than the experimental value. Another alternative is therefore to use in addition to the lattice points also the experimental value \(F_\pi = (92.2 \pm 0.3)\) MeV as input where we have doubled the error assigned by the Particle Data Group [17].
In addition, we added the theoretical uncertainties related to \(M\), \(L_5\) and the \(C_a\) in quadrature. Lattice and theoretical errors are of similar size. For instance, keeping only the lattice errors, the error of \(F_0\) moves from \(4.1\) down to \(2.8\) MeV. The \(\chi ^2/\)dof is 0.5 (statistical errors only), suggesting once more that we have at least not underestimated the errors.
The two ellipses are roughly compatible with each other. The green ellipse is lower because from the RBC/ UKQCD data alone the fitted value of \(F_\pi \) is smaller than the experimental value. The value for \(L_4\) is consistent with large \(N_c\) and with available lattice results [1]. The result for \(F_0\) is more precise than existing phenomenological and lattice determinations. It is somewhat bigger than expected [18], roughly of the same size as the \({SU}(2)\) LEC \(F\) in Eq. (4.1).
\(F_0\) and \(L_4\) in Eq. (4.6) are compatible with the comparison between \({SU}(2)\) and \({SU}(3)\) to \(O(p^6)\) [14], as indicated by the orange bands in Fig. 4. \(C_{16}\) is the only NNLO LEC appearing in the relation between \(F_0\) and \(F\). As always in this paper, we have used fit 10 [12] for the NLO LECs. However, unlike for our fit results (4.6), the orange bands in Fig. 4 are rather sensitive to the precise values of the \(L^r_i\). Therefore, although the consistency between the ellipses and the lower orange band is manifest, it can hardly be used as a determination of \(C_{16}\).
Raising the range in pion masses to \(M_\pi < 425\) MeV, two more lattice points [15] can be added. Repeating the fit with the bigger sample moves the ellipses down, because with the original data set of RBC/UKQCD the fitted value of \(F_\pi \) comes out too low [15].
The strong anticorrelation between \(F_0\) and \(L_4\) persists because the kaon masses in the RBC/UKQCD data are all close to the physical kaon mass. Simulations with smaller kaon masses would not only be welcome from the point of view of convergence of the chiral series [19], but they could also provide a better lever arm for reducing the anticorrelation and the fit errors of \(F_0\) and \(L_4\). This expectation is supported by the fact that the quantity \(F(M_\rho )\) defined in Eq. (4.3) can be determined much better than \(F_0\).
5 Remarks on \({\mathbf {f^{K\pi }_+(0)}}\)
The kaon semileptonic vector form factor at \(t=0\) is a crucial quantity for a precision determination of the CKM matrix element \(V_{us}\). Both approximations discussed here do not appear very promising in this case.
First of all, unlike for \(F_\pi \) and \(F_K/F_\pi \), the chiral expansion of \(f^{K\pi }_+(0)\) shows a rather atypical behaviour. Due to the Ademollo–Gatto theorem [20], the \(O(p^4)\) contribution of \( 0.0227\) [21] is very small. On the basis of recent lattice studies, which find \(f^{K\pi }_+(0)=0.967\) with errors of \(<\)1 % [22, 23], all higherorder contributions in CHPT would have to sum up to about \( 1\) %. On the other hand, the genuine twoloop contributions at the usual scale \(\mu =770\) MeV are positive and slightly bigger than 1 % [10, 24, 25], suggesting that the remainder is about \( 2\) % to match the lattice value. In other words, the remainder would have to be as big as the NLO contribution, certainly not the typical behaviour for a chiral expansion.
In principle, Approximation I fulfills our criterion in Sect. 2 in differing from the full twoloop result [10, 24, 25] by \(<\)2 %. However, especially in view of the accuracy of recent lattice studies claiming a precision of better than \(1\) % for \(f^{K\pi }_+(0)\), the accuracy of Approximation I is simply not sufficient in this case. Approximation II does not improve the situation.
To sum up, lattice determinations of \(f^{K\pi }_+(0)\) seem to be able to do without CHPT. Moreover, only the full NNLO expression may allow for a meaningful extraction of LECs if at all [10].
6 Conclusions
 1.
Lattice QCD has become a major source of information for the lowenergy constants of CHPT. We have argued that the meson decay constants \(F_\pi \), \(F_K\) are especially suited for extracting chiral \({SU}(3)\) LECs of different chiral orders. The ratio \(F_K/F_\pi \) allows for a precise and stable determination of the NLO LEC \(L_5\). In addition, it gives access to some NNLO LECs although the accuracy is of course more limited in that case. Phenomenological analyses have had difficulties in determining the LEC \(F_0\), the meson decay constant in the chiral \({SU}(3)\) limit. We have shown that lattice data for \(F_\pi \) allow for the extraction of \(F_0\) together with the NLO LEC \(L_4\). The strong anticorrelation between \(F_0\) and \(L_4\) observed in phenomenological analyses can in principle be lifted by varying the lattice masses. From a fit to the RBC/UKQCD data for \(F_\pi \), we have obtained a value for \(F_0\) that is more precise than other presently available determinations.
 2.
Confronting presentday lattice data with chiral \({SU}(3)\) requires chiral amplitudes to NNLO in most cases. Chiral \({SU}(3)\) amplitudes are often rather unwieldy and mostly available in numerical form only. We have therefore proposed large\(N_c\) motivated approximate NNLO amplitudes that contain only oneloop functions. Unlike simpler approximations as the doublelog approximation, our amplitudes are independent of the renormalisation scale and can therefore be used to extract LECs with the correct scale dependence. However, approximations of NNLO amplitudes can only be successful if the differences to the full amplitudes are at most of the order of N\(^3\)LO contributions. We have checked that this criterion can be fulfilled with our approximate amplitudes both for \(F_\pi \) and \(F_K/F_\pi \). Therefore, we expect our results for the different LECs to be as reliable as CHPT to NNLO, \(O(p^6)\), permits. Although our general criterion is also satisfied for the kaon semileptonic form factor at \(t=0\), the approximate expression for \(f^{K\pi }_+(0)\) is not precise enough compared to recent lattice data.
Footnotes
 1.
Hans Bijnens, private communication.
Notes
Acknowledgments
We are grateful to Hans Bijnens for making the full results of Ref. [9] accessible to us. We also thank Véronique Bernard, Claude Bernard, Hans Bijnens, Gilberto Colangelo, Elvira Gámiz, Laurent Lellouch, Heiri Leutwyler, Emilie Passemar and Lothar Tiator for helpful comments and suggestions. Special thanks are due to Elvira Gámiz for helping us to understand lattice data. Finally, we thank Hans Bijnens for suggesting several improvements for the original manuscript. P.M. acknowledges support from the Deutsche Forschungsgemeinschaft DFG through the Collaborative Research Center ‘The LowEnergy Frontier of the Standard Model’ (SFB 1044).
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