The Pion Mass and Decay Constant at Three Loops in Two-Flavour Chiral Perturbation Theory

A calculation of the pion mass and decay constant at NNNLO in two-flavour chiral perturbation theory is presented. The results are cross-checked by using both the exponential and square root parameterizations of the Goldstone matrix field, as well as by comparing to the known leading log coefficients of the two quantities. A small numerical study of the quark mass dependence is performed, and for a physical quark mass there is good agreement with lower order results.

φ is a vector of N real fields φ a and f is the external current. We have indicated here the higher order terms with c i , l i and d i as well and have used the equations of motion (or field redefinitions) to discard the higher order terms in the coupling to the external current f . Up to two-loop order the counterterms are given by 1 l 1 = (cµ) −2ε (l r 1 ) l 2 = (cµ) −2ε l r 2 + N + 2 16π 2 ε 1 These are slightly different from [5] since we have chosen not to put l r 1 = 0.
We use dimensional regularization with d = 4 − 2ε and modified minimal subtraction (M S). The choice of c determines which version of M S is used. In this manuscript we use the usual ChPT [2] version with The mass is defined as the pole of the two point function, see e.g. the discussion in [6], as where Σ is the sum of one-particle irreducible diagrams. The relevant diagrams are shown in Fig. 1 when neglecting those involving vertices with more than four legs. p 2 corresponds to terms with 1, p 4 with l i , p 6 with c i and p 8 with d i in (1). The physical mass M φ is given by the solution of We write the mass as and the self-energy as Where we used the subscript 4,6,8 for NLO, NNLO and NNNLO respectively. With this expansion we can solve for the mass perturbatively. We evaluate diagrams at p 2 = M 2 φ as a perturbative expansion away from M 2 . This is why derivatives of the self-energy show up. Taking into account that here ∂ 2 Σ 4 /(∂p 2 ) 2 = 0 we get All nonlocal divergences cancel as they should and we get a finite result by setting This result can be checked in a number of ways. The nonlocal divergences cancelled as they should. The terms leading in N can be derived using a gap equation similar to what was done for the nonlinear sigma model in [11,12]. The renormalization group equations are known to five loop order [13], these can be used to check the 1/ε terms in (9). All checks are satisfied.

Chiral perturbation theory
The effective Lagrangian in ChPT is expanded in powers of p 2 as The relevant degrees of freedom are the Goldstone Bosons from the spontaneous breakdown of SU (N f ) L × SU (N f ) to SU (N f ) V . These can be described by a special unitary N f × N f matrix u. For two flavours the lowest order Lagrangian is The fields l µ , r µ , s and p are the usual N f × N F external fields of ChPT. F and B are the two low-energy-constants (LECs) at leading order for the two-flavour case. The nextorder Lagrangian was classified in [2]. The NNLO Lagrangian can be found in [14]. The NNNLO Lagrangian L 8 is at present not known, but there will be one combination of p 8 LECs contributing to the mass and another to the decay constant, we will call these combinations r M 8 and r F 8 , respectively. The NLO and NNLO low-energy-constants (LECs) are conventionally denoted as l i and c i , respectively. The divergent parts needed to one- [2] and two-loop order [15] are known in general and the equivalent formulas to (2) can be found there. For later convenience we introduce the lowest order order pion mass where 2m = m u + m d . In the remainder we will work in the isospin limit with m u = m d .

The calculation and checks
The diagrams contributing are shown in Fig. 1. The Feynman diagrams are programmed in FORM [16]. The derivatives w.r.t. p 2 needed are obtained by taking the derivative diagram by diagram at this stage. Then the expressions are rewritten in integrals. These are reduced to a set of master integrals. This is done using integration-by-parts and Lorentz invariance identities through a Laporta algorithm. We have used the program Reduze [17] for this. The resulting master integrals are all known to the order in ε required and we quote them in App. A. A three-loop calculation needs a large number of checks. We have checked that the nonlocal divergences cancel, that we reproduce the known two-loop and leading logarithm results. As a final check we use two different parametrizations for u in terms of a traceless Hermitian 2 × 2 matrix Φ, namely the exponential, u = exp i √ 2F Φ, and a square root, The diagrams are quite different in these two parametrizations but the final result must of course be the same. All checks are satisfied by our results. Since essentially the same programs were used for φ 4 the checks discussed in Sect. 2 are another partial check on our main results.

The pion mass and decay constant
The physical mass is defined as the pole of the two-point function (4) with Σ(p 2 ) the self-energy. The physical pion mass M 2 π is then found as the solution of The decay constant is defined through the relation and is calculated using diagrams of the same topology as those in Σ (the only difference is that one of the external legs corresponds to the axial current). For the decay constant one also needs to calculate the wave function renormalization factor Z defined as the residue of the propagator in (4), i.e.
at p 2 = M 2 π . The physical pion mass and decay constant can be written in expanded form
In order to obtain a finite result we need the subtraction − π 16 ε (416c r 18 + 208c r 17 + 32c r 16 − 96c r 14 − 8c r 13 + 48c r 12 + 384c r 11 + 192c r 10 − 80c r 9 − 160c r 8 − 80c r 7 − 96c r 6 + 8c r 5 + 56c r 4 − 16c r 3 − 32c r 2 + 96c r 1 − 14l r 3 l r 4 + 16l r r M 8 is the combination of p 8 LECs that contributes to the mass. This is a single scale problem and only logarithms of the mass scale show up, the expression is thus fairly compact. We use the abbreviations The results can be written in the form The coefficients are where ζ k is the Riemann-Zeta function. The leading log coefficient a M 33 agrees with [11,12]. The LECs l r i are well known but the c r i less well. In [5,15] combinations of the p 6 LECs appearing at p 6 in ππ-scattering, the mass and decay constant were defined, r 1 , . . . , r 6 , r M , r F , and numerical estimates using resonance saturation were done in [5]. The expressions in terms of the c r i are given in App. B. We can check whether the c r i dependence can be rewritten in terms of those. This can be done for a M 20 by definition and also for a M 31 . However not completely for a M 30 which in any case contains the free p 8 LEC combination r r M 8 . Defining r q i = 16π 2 r r i for i = 1, . . . , 6, M, F , we obtain

Decay constant
For the decay constant everything is analogous except that we need to evaluate the diagrams with one leg replaced by the axial current and take into account the wave function renormalization factor Z. Denoting the sum of one-particle-irreducible diagrams of the axial current as A(p 2 = M 2 π ) = A 4 (p 2 ) + A 6 (p 2 ) + A 8 (p 2 ) the expression for the decay constant is (normalized to 1 at lowest order) Putting in the expanded expressions for Σ and A and using ∂ 2 Σ 4 /(∂p 2 ) 2 = ∂A 4 /∂p 2 = 0, we obtain with all right hand sides evaluated at p 2 = M 2 .

Numerical study: mass dependence
Now that the analytic forms of the mass and decay constant have been obtained, we can do a first numerical analysis of the mass dependence. We present only results for one choice of input parameters to give an impression of the size of the NNNLO correction.
The expansions given in the previous section correspond to an expansion expressed in terms of the lowest order mass M and decay constant F in the form There are many ways to rewrite this expansion but the second most standard version is the inverse, namely where ξ = M 2 π /(16π 2 F 2 π ) and L π = log(M 2 π /µ 2 ). The analytic expressions of the a F.M ij are in the previous section and the b F,M ij can be found in App. C. These are often referred to as the x-and ξ-expansion, see e.g. [18].
As input we use µ = 0.77 GeV,l 1 = −0.4,l 2 = 4.3 from [19],l 3 = 3.41,l 4 = 4.51 from the N f = 2 estimates of [18]. The numerical values for the r r i are taken from [5] 10 4 r r 1 = − 0.6 10 4 r r 2 = 1.  Table 1. Note that the numerical values of a F 30 and b F 30 are rather large. This is due to the very large numerical coefficient 383293667/1555200 ≈ 246.5 appearing there, the remaining coefficients are of more natural size.
The quantities in (31)-(32) are plotted in Fig. 2(a-d), with the same inputs as above. For the ξ-expansion we kept F π = 92.2 MeV constant while varying M π and for the xexpansion we kept F = 92.2/1.073 MeV constant while varying M . The convergence around the physical value M 2 π ≈ 0.02 GeV 2 is excellent. For the mass, the ξ-expansion converges much better, for the decay constant it is somewhat better. The effect of the very large constants a F 30 and b F 30 is clearly visible in the results for the decay constant.

Conclusions
In this paper we presented the calculation of the NNNLO contributions to the pion mass and decay constant in the isospin limit of two-flavour ChPT. We also calculated the mass in the O(N ) φ 4 case to show the principle. This required the evaluation of 2 × 34 diagrams and their derivatives w.r.t. the momentum. The master integrals needed for the calculation were known. The tree level contributions from the NNNLO Lagrangian are unknown, but were here parameterized as free renormalized parameters. We reproduced the known NNLO results and the known leading logarithms. A small numerical study of the two quantities was performed, and there was continuing good convergence at the physical pion mass. The coefficients for the decay constant are b F 10 = − l q