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The European Physical Journal. C, Particles and Fields
 
Eur Phys J C Part Fields. 2017; 77(7): 497.
Published online 2017 July 26. doi:  10.1140/epjc/s10052-017-5019-y
PMCID: PMC5586975

An analytic analysis of the pion decay constant in three-flavoured chiral perturbation theory

Abstract

A representation of the two-loop contribution to the pion decay constant in SU(3) chiral perturbation theory is presented. The result is analytic up to the contribution of the three (different) mass sunset integrals, for which an expansion in their external momentum has been taken. We also give an analytic expression for the two-loop contribution to the pion mass based on a renormalized representation and in terms of the physical eta mass. We find an expansion of Fπ and Mπ2 in the strange-quark mass in the isospin limit, and we perform the matching of the chiral SU(2) and SU(3) low-energy constants. A numerical analysis demonstrates the high accuracy of our representation, and the strong dependence of the pion decay constant upon the values of the low-energy constants, especially in the chiral limit. Finally, we present a simplified representation that is particularly suitable for fitting with available lattice data.

Introduction

The mass and decay constants of the pions, kaons and the eta have been worked out to two-loop accuracy in three-flavoured chiral perturbation theory (ChPT) in [1] some time ago. The expressions for these at this order bring about a class of diagrams known as the sunsets. For the decay constants, in addition to the sunset integral, derivatives of the sunsets with respect to the square of the external momentum (also known as ‘butterfly’ diagrams), evaluated at a value equal to the square of the mass of the particle in question, are needed. The sunset diagrams themselves have been studied in field theory literature for many years now, and for particular mass configurations analytic expressions exist in Laurent series expansions in ϵ = (4 - d)/2. In general, however, the sunsets and their derivatives have to be evaluated numerically and publicly available software [2] does this with user driven inputs.

There is, however, a need for an analytic study of the observables in ChPT since one would like to have an intuitive sense for the results appearing therein. More importantly, with recent advances allowing lattice simulations to tune the quark masses to near physical values, a combining of lattice and ChPT results has become possible. However, at next to next to leading order (NNLO), three-flavoured ChPT amplitudes are available only numerically or take a complicated form, and thus have not been used much by the lattice community. With this in mind [3, 4] has advocated a large Nc motivated approach to replace the two-loop integrals by effective one-loop integrals, and find it fruitful for the study of the ratio FK/Fπ as well as Fπ. The analytic studies of SU(3) amplitudes in the strange-quark mass expansion of [57] are also steps in that direction, but, as the results presented there are in the chiral limit, mumd = 0, the need for more general expressions is left unfulfilled.

Some years ago, Kaiser [8] studied the problem of the pion mass in the analytic framework, and was able to employ well known properties of sunset integrals to reduce a large number of expressions to analytic ones. One exception was the sunset integral with kaons and an eta propagating in the loops with the external momentum at s=mπ2, for which an expansion around mπ2 was used. Kaiser [8] also replaced the mη in his work by the leading-order Gell-Mann–Okubo (GMO) formula. In principle, therefore, one can get an expansion in mπ2 to arbitrary accuracy, proving thereby the accessibility of an analytical approach to the full two-loop result. For practical purposes, we have used the expansion up to and including mπ4 terms. These are more than sufficient for the numerical accuracy wanted.

The reason why it is possible to attain the objectives above is that for many purposes, the sunset integrals are accessible analytically for kinematic configurations known as threshold and pseudo-threshold configurations [9], as well as for the case when the square of the external momentum vanishes [10]. Indeed, this is the case for most of the sunset integrals appearing in the expressions for the mass and decay constants. These properties also allow one to isolate the divergent parts in closed form, while the finite part remains calculable in analytic form only for special cases. On the other hand, there is always an integral representation for the finite part which can be evaluated numerically. Furthermore, for the most general case, all sunsets can be reduced to a set of master integrals. All other vector and tensor integrals, as well their derivatives with respect to the square of the external momentum, can also be reduced to master integrals. The work of [11] in developing this work is noteworthy, as is the automation of these relations with the publicly available Mathematica package Tarcer [12]. Application of these methods and tools to sunset diagrams in chiral perturbation theory is elucidated in [13].

Inspired by the developments above, we now seek to extend the work of [8] for the case of the pion decay constant in an expansion around s = 0, which also brings about the butterfly diagrams. In contrast to the approach of [8], we will retain the mass of the eta without recourse to the GMO. This is the main objective of the present work. As a side result, we also give the expression for the two-loop pion mass with the full eta mass dependence.

In principle, this may also be extended to the mass and decay constant of the kaon and the eta, but the expansion about s = 0 for these particles when particles of unequal mass are running around in the loops is bound to converge poorly, and one would have to go to very high orders in the expansion, thereby losing the appeal of such a result. Thus we confine ourselves to the pion in this work. We present expressions for the kaon and eta masses and decay constants in a future publication [14].

As an application of the expressions given here, we give their expansion in the strange-quark mass in the isospin limit and perform the ‘matching’ of the three-flavoured low-energy constants F0 and B0 with their two-flavoured counterparts F and B, respectively. We compare our results with those given in [15] and the chiral limit results of [5]. The results given in this work, however, go beyond the chiral limit matching done in the aforementioned papers. Indeed, the full expressions presented here allow for an expansion up to an arbitrary order in the quark masses.

The scheme of this paper is as follows. In Sect. 2 we briefly review sunset diagrams and their evaluation. In Sect. 3 we give the expressions for the analytical results up to O(mπ4) for the pion decay constant at two loops. We repeat the analysis for the two-loop pion mass contribution in Sect. 4. In Sect. 5, we give the s-quark expansion for both the pion decay constant and the pion mass, and we perform the matching of the two- and three-flavour low-energy constants (SU(2) and SU(3) LECs). We present a numerical analysis of our results in Sect. 6, and in Sect. 7 we discuss the fitting of lattice data with the expressions given in this paper, and present them in a form that allows one to perform these fits relatively easily. In Sect. 7, we discuss several possible ways of expressing the results of this paper, and present a simplified representation that is particularly suitable for performing fittings with available lattice data. We conclude in Sect. 8 with a discussion of possible future work in this area.

Sunset diagrams and their derivatives

The sunset diagram, shown in Fig. 1, represents the two-loop Feynman integral,

H{α,β,γ}d(m1,m2,m3;s)=1i2ddq(2π)dddr(2π)d×1[q2-m12]α[r2-m22]β[(q+r-p)2-m32]γ.
1

Aside from the basic scalar integral, there exist tensor varieties of the sunset integral with loop momenta in the numerator. The two tensor integrals that are of relevance to this work are Hμ and Hμν, in which the momenta qμ and qμqν, respectively, appear in the numerator. These may be decomposed into linear combinations of scalar integrals via the Passarino–Veltman decomposition as

Hμd=pμH1,Hμνd=pμpνH21+gμνH22.
2

The representation of the pion decay constants in [1] involves the scalar integrals H1 and H21. Taking the scalar product of Hμd with pμ allows us to express the integral H1 in terms of the sunset integral with the scalar numerator q.p. Similarly, we may express H21 in terms of sunset integrals with numerators (q.p)2 and q2:

H1=q.pp2,H21=(q.p)2d-q2p2p4(d-1),
3

where ⟨⟨X⟩⟩ represents a sunset integral with numerator X.

Fig. 1
The two-loop self-energy “sunset” diagram

Another class of integrals that appear in the representation of [1] is the derivative of the sunset integrals and the H1 and H21 with respect to the external momentum. In some places in the literature, these are sometimes known as ‘butterfly’ diagrams. These butterfly integrals may be expressed as sunset integrals of higher dimension by means of the following expression, which can be derived from the Feynman parameter representation of the sunset integrals, and a more general version of which is given in [8]:

snH{α,β,γ}d=(-1)n(4π)2nΓ(α+n)Γ(β+n)Γ(γ+n)Γ(α)Γ(β)Γ(γ)×H{α+n,β+n,γ+n}d+2n.
4

Tarasov [11] has shown that by means of integration by parts relations, all sunset integrals may be expressed as linear combinations of four master integrals, namely H{1,1,1}d, H{2,1,1}d, H{1,2,1}d and H{1,1,2}d, and the one-loop tadpole integral:

Ad(m)=1iddq(2π)d1q2-m2=-Γ1-d/2(4π)d/2md-2.
5

This includes sunset integrals of dimensions greater than d, permitting us to express the butterfly integrals in terms of the four master integrals and tadpoles. Scalar sunset integrals with non-unit numerators, such as those appearing in Eq. (2) may also be expressed in terms of the four master integrals and tadpoles. The Tarcer package [12], written in Mathematica, automates the application of Tarasov’s relations, and we have made extensive use of it in this work. We have also made use of the package Ambre [16, 17], which allows for a direct evaluation of many scalar and tensor Feynman integrals using a Mellin–Barnes approach, to numerically check our breakdown of the sunset and butterfly diagrams into master integrals. The theory of analytic (rather than numeric) evaluation of multi-fold Mellin–Barnes integrals is described with examples in [18, 19].

As is the usual practice in chiral perturbation theory, we use a modified version of the MS¯ scheme to handle the divergences arising from the evaluation of the sunset diagrams. The subtraction procedure to two-loop order in ChPT is equivalent to multiplying Eq. (1) by (μχ2)4-d, where

μχ2μ2eγE-14π,
6

and taking into consideration only the 𝒪(ϵ0) part of the result in a Laurent expansion about ϵ = 0. We denote such renormalized sunset integrals by use of the subscript χ instead of d, i.e.

H{a,b,c}χ(μχ2)4-dH{a,b,c}d.
7

The inclusion of factor μ raised to a power of the dimension d introduces terms involving chiral logarithms, i.e.

lPr12(4π)2logmP2μ2P=π,K,η.
8

In the results presented in this paper, we group together all terms containing chiral logarithms, whether or not they arise from the renormalized sunset integrals. We therefore use the notation

H{a,b,c}χH¯{a,b,c}χ+H{a,b,c}χ,log
9

where Hχ,log are the terms of the sunset integral containing chiral logarithms, and H¯χ is the aggregation of the remainder. All results given hereafter have been renormalized using this subtraction scheme, and they are presented using the notation above.

Analytic expressions for the master integrals themselves have been studied thoroughly, and several results exist in the literature [9, 10, 2023]. For sunset integrals with only one mass scale, there is a further reduction in the number of master integrals, and all sunsets can be expressed in terms of the tadpole integral, Aχ=μχ4-dAd, and H{1,1,1}χ, which is given in [9, 20], amongst others, as

H{1,1,1}χ=-(μ2eγE-1)2ϵ(m2)1-2ϵ(4π)4Γ2(1+ϵ)(1-ϵ)(1-2ϵ)×-32ϵ2+14ϵ+198+O(ϵ).
10

Analytic expressions for the two mass scale integrals can be found by means of the pseudo-threshold results of [9].

Expressions for the three mass sunset integrals are given in [23] in terms of elliptic dilogarithmic functions. However, as one of the principal reasons for the lack of use of ChPT results by the lattice community is the complicated form of many of the results, we wish to keep the expression derived here as simple and accessible as possible. To this end, and to stay true to the spirit of the method of [8], instead of using the results of [23] we take an expansion in the external momentum s up to order 𝒪(s2):

H{α,β,γ}χ=K{α,β,γ}+sK{α,β,γ}+s22!K{α,β,γ}+O(s3)
11

where K{α,β,γ}H{α,β,γ}χ|s=0. In this special case of s = 0, as in the case of the single mass scale sunsets, all sunset integrals may be expressed solely in terms of K{1,1,1} and tadpole integrals [11].

The pion mass and decay constant at two loops both involve a sunset integral with the following three mass scale configuration:

H{α,β,γ}χ(mK,mK,mη;s=mπ2).

This may be expanded in s by making use of the result [1, 8, 10]

24π4M2H{1,1,1}χ{M,M,m;0}=2+m2M21ϵ2+m2M21-2logm2μ2+21-2logM2μ21ϵ-2(μ2)2ϵ(m2M2logm2μ21-logm2μ2+2logM2μ21-logM2μ2)-m2M2log2m2M2+m2M2-4Fm2M2+2+m2M2π26+3+O(ϵ)
12

where

F[x]=1σ[4Li2(σ-1σ+1)+log2(1-σ1+σ)+π23],σ=1-4x.
13

The pion decay constant to two loops

The pion decay constant is given in [1] as

Fπ=F0(1+F¯π(4)+F¯π(6))+O(p8)
14

where the 𝒪(p6) contribution can be broken up into a piece that results from the model-dependent counterterms (F¯π(6))CT, and one that results from the chiral loop (F¯π(6))loop. For the pion, the explicit form of these terms are given by

Fπ2F¯π(4)=4mπ2(L4r+L5r)+8L4rmK2-lKrmK2-2lπrmπ2,
15

Fπ4(F¯π)CT(6)=8mπ4(C14r+C15r+3C16r+C17r)+16mK2mπ2(C15r-2C16r)+32C16rmK4,
16

where mP with PπKη are the physical meson masses, and lPr are the chiral logarithms defined in Eq. (8). Note that the Ci used in this paper are dimensionless.

The loop contributions can be subdivided as follows:

Fπ4(F¯π)loop(6)=d¯sunsetπ+dlog×logπ+dlogπ+dlog×Liπ+dLiπ+dLi×Ljπ.
17

The terms containing the LECs Li but no chiral logarithms are given by

(16π2)dLiπ=89L2r+L3r3mK2mπ2-2L1r+379L2r+2827L3rmπ4-529L2r+4327L3rmK4,
18

and the terms bilinear in the LECs are contained in

dLi×Ljπ=32mK2mπ27(L4r)2+5L4rL5r-8L4rL6r-4L5rL6r+32mK4L4r(7L4r+2L5r-8L6r-4L8r)+8mπ4(L4r+L5r)(7L4r+7L5r-8L6r-8L8r).
19

The remaining three terms of Eq. (17) give the terms containing the chiral logs. Explicitly, the following gives the terms linear in chiral logarithms:

(16π2)dlogπ=mK423lηr+238lKr+98lπr+mK2mπ213972lπr-172lηr-12lKr+mπ41381288lπr-11288lηr
20

while the terms bilinear in the lPr are contained in

dlog×logπ=mK4772(lηr)2-5536lηrlKr+536(lKr)2-34lKrlπr+38(lπr)2+mπ4418(lπr)2-124(lηr)2+mK2mπ219(lηr)2+49lηrlKr+19(lKr)2+253lKrlπr-76(lπr)2+12mK6mπ2lηr-lKr2.
21

The contributions from terms involving products of chiral logarithms and the LECs are collected in

dlog×Liπ=4mπ4lπr(14L1r+8L2r+7L3r-13L4r-10L5r)+49(4mK2-mπ2)2lηr(4L1r+L2r+L3r-3L4r)+4mK4lKr(16L1r+4L2r+5L3r-14L4r)-mK2mπ2(4lKr(3L4r+5L5r)+48lπrL4r).
22

Finally, the contributions from the sunset diagrams are given by

dsunsetπ=1(16π2)2(35288mπ4π2+41128mπ4+1144mπ2mK2π2-532mπ2mK2+1172mK4π2+1532mK4)+512mπ4H¯πππχ-12mπ2H¯πππχ-516mπ4H¯πKKχ+116mπ2H¯πKKχ+136mπ4H¯πηηχ+12mπ2mK2H¯KπKχ-12mK2H¯KπKχ-512mπ4HKKηχ-116mπ4H¯ηKKχ+14mπ2mK2H¯ηKKχ+116mπ2H¯ηKKχ-14mK2H¯ηKKχ+12mπ4H¯1χπKK+mπ4H¯1χKKη+32mπ4H¯21χπππ-316mπ4H¯21πKKχ+32mπ4H¯21KπKχ+916mπ4H¯21ηKKχ
23

where we use the notation

H¯aPbQcRχ=H¯{a,b,c}χ{mP,mQ,mR;s=mπ2}
24

with H¯{a,b,c}χ as defined in Eq. (9). abc will be suppressed if equal to 1. The terms resulting from the sunset integrals involving chiral logarithms have been included in dlogπ or dlog×logπ as appropriate.

Evaluating the sunset integrals as described in Sect. (2), dsunsetπ can be re-expressed as

dsunsetπ=1(16π2)2[34451728+107π2864mK4+125864+17π2324mK2mπ2-32-π212mK6mπ2-356912+13π22592mπ4]+dπKKπ+dπηηπ+dKKηπ
25

where

dπKKπ=-916mK4mπ2+34mK2+148mπ2H¯πKKχ+34mK4+16mK2mπ2+mπ412H¯2πKKχ,
26

dπηηπ=-136mπ2H¯πηηχ+136mπ4H¯2πηηχ,
27

dKKηπ=1516mK4mπ2-1336mK2+13144mπ2H¯KKηχ+12mK4-2mK6mπ2-16mK2mπ2H¯2KKηχ+91108mK4-mK6mπ2-527mK2mπ2+1108mπ4H¯KK2ηχ.
28

Closed form expressions, at 𝒪(ϵ0), for the master integrals H¯χ appearing in dπKK and dπηη are given in Appendix B. The master integrals appearing in dKKη are of three mass scales, for which there exist no simple closed form expressions. For these, therefore, we take an expansion around s=mπ2=0. Up to order Omπ4, we have

(16π2)2dKKη=dKKη(-1)(mπ2)-1+dKKη(0)+dKKη(1)(mπ2)+dKKη(2)(mπ2)2,
29

where

dKKη(-1)=5116+π296mK6-3548mK4mπ2+112-π296mK2mπ4-196mπ6-18mK6+332mK4mπ2-132mK2mπ4log243,
30

dKKη(0)=-42353456+25π21728mK4+4851728-π2864mK2mπ2-1936912mπ4-1532mK4-116mK2mπ2+164mπ4log[ρ]+116mK4-164mK2mπ2log43+572mK4-5288mK2mπ2log243+13mK4+124mK2mπ2F43,
31

dKKη(1)=11152+5π2288mK2-314608+π2576mπ2-512mπ4mK2+17144mK2-7288mπ2log[ρ]+2274608mπ2-512mπ4mK2-471152mK2log43+196mπ2-124mK2log243-748mK2+7384mπ2F43,
32

(4mK2-mπ2)2dKKη(2)=-1λ2161162mK8-295324mK6mπ2+712mK4mπ4+4955,296mπ10mK2-126510,368mK2mπ6+3541,472mπ8+1λ3(5093243mK10-1981162mK8mπ2+38331296mK6mπ4+182,944mπ14mK4-34317776mK4mπ6+2962,208mπ12mK2+172592mK2mπ8+10320,736mπ10)×log43-(4mK2-mπ2)2192log[ρ]-1λ3(50536mK10-6316mK8mπ2+512mK6mπ4-13144mK4mπ6+112,288mπ12mK2+3256mK2mπ8+1512mπ10)F43.
33

In the above expressions, τmη2/mK2, ρmπ2/mK2, λ-(8mK2+mπ2)/3, and F[x] is defined in Eq. (13). Note that in this expansion, divergences appear in the mπ → 0 limit. The divergences from the dKKη(-1) term cancel against the divergences in Eq. (25) and in Eq. (104), while those arising from the log[ρ] and log2[ρ] in dKKη(0) cancel against divergences in Eqs. (104), (21) and (26). Therefore the overall F¯π(6) remains non-divergent in the mπ20 limit.

The pion mass to two loops

We repeat the steps of the previous section for the pion mass. A representation for this is given in [1] as

Mπ2=mπ02+(mπ2)(4)+(mπ2)CT(6)+(mπ2)loop(6)+O(p8)
34

where mπ02=2B0m^ is the bare pion mass squared, and mP are the physical meson masses.

Fπ2mπ2(mπ2)(4)=-8mπ2(L4r+L5r-2L6r-2L8r)-16mK2(L4r-2L6r)+mπ2lπr+19lηr-49mK2lηr,
35

-Fπ416mπ2(mπ2)CT(6)=2mK2mπ2(2C13r+C15r-2C16r-6C21r-2C32r)+4mK4(C16r-C20r-3C21r)+mπ4(2C12r+2C13r+C14r+C15r+3C16r+C17r-3C19r-5C20r-3C21r-2C31r-2C32r).
36

The (mπ2)loop(6) term can be subdivided into the following components:

Fπ4(mπ2)loop(6)=csunsetπ+clog×logπ+clogπ+clog×Liπ+cLiπ+cLi×Ljπ
37

where

16π2mπ2cLiπ=29mπ418L1r+37L2r+283L3r+83L5r-32L7r-16L8r+19mK4104L2r+863L3r+163L5r-64L7r-32L8r-169mK2mπ2L2r+13L3r+23L5r-8L7r-4L8r,
38

-cLi×Ljπ128mπ2=(L4r-2L6r)(mK4(4L4r+L5r-8L6r-2L8r)+mK2mπ2(4L4r+3L5r-8L6r-6L8r))+mπ4(L4r+L5r-2L6r-2L8r)2,
39

16π2mπ2clogπ=116lηr-1199144lπrmπ4-2027lηr+27736lKr+34lπrmK4-7108lηr+13lKr+4736lπrmK2mπ2,
40

clog×logπmπ2=739324(lηr)2-4318lηrlKr+8318(lKr)2+12lKrlπr-14(lπr)2mK4+32(lπr)2-67162(lηr)2+13lηrlKr+209lηrlπr+29(lKr)2-3lKrlπrmK2mπ2+12136(lπr)2-11324(lηr)2-13lηrlπrmπ4-13mK6mπ2(lηr-lKr)2,
41

clog×Liπmπ2=16mK2mπ2(19lηr(16L1r+4L2r+4L3r-21L4r-8L5r+26L6r-24L7r+4L8r)+lKr(L4r+L5r-2L6r-2L8r)+5lπr(L4r-2L6r))-8mK4(49lηr(16L1r+4L2r+4L3r-18L4r-3L5r+20L6r-12L7r+2L8r)+lKr(16L1r+4L2r+5L3r-20L4r-4L5r+24L6r+8L8r))-8mπ4(19lηr(4L1r+L2r+L3r-6L4r-4L5r+8L6r+6L8r)+lπr(14L1r+8L2r+7L3r-18L4r-12L5r+32L6r+22L8r)).
42

The contribution from the sunset integrals is given by

csunsetπ=1(16π2)2[1-π218mK6-2435864+97π2432×mK4mπ2+235144-23π2648mK2mπ4+47573456-41π21296mπ6]+cπKKπ+cπηηπ+cKKηπ
43

where

cπηηπ=mπ418H¯πηηχ,
44

cπKKπ=38mK4+34mπ2mK2-18mπ4H¯πKKχ+12mπ6-12mπ2mK4H¯2πKKχ,
45

cKKηπ=4336mK2mπ2-58mK4-1772mπ4H¯KKηχ+43mK6-53mK4mπ2+13mK2mπ4H¯2KKηχ+23mK6-6554mK4mπ2+1727mK2mπ4-554mπ6H¯KK2ηχ.
46

With ρmπ2/mK2 and τmη2/mK2, expanding cKKηπ about s=mπ2=0 gives

(16π2)2cKKηπ=cKKη(0)+cKKη(1)(mπ2)+cKKη(2)(mπ2)2+O((mπ2)3)
47

where

cKKη(0)=-178+π2144mK6+3572mK4mπ2-118-π2144mK2mπ4+1144mπ6+112mK6+116mK4mπ2-148mK2mπ4log243,
48

cKKη(1)=79451728+95π2864mK4-751864+7π2432×mK2mπ2+1553456mπ4+196mK2mπ2-124mK4log43+13144mK2mπ2-1336mK4log243+516mK4-124mK2mπ2+196mπ4log[ρ]-23mK4+112mK2mπ2F43,
49

(4mK2-mπ2)cKKη(2)=π2864-1092304mπ4-289144+π227mK4+205288+π2216mK2mπ2-1768mπ6mK2-1λ6154mK6-2348mK4mπ2-12304mπ8mK2-548mK2mπ4+2776912mπ6log43-4mK2-mπ22144log243-209mK4-79mK2mπ2+118mπ4log[ρ]+1λ1324mK4mπ2+11144mK2mπ4+51728mπ6-13727mK6F43.
50

The expressions of this section agree fully with those given in [8] when the eta masses here are expressed in terms of the pion and kaon masses by means of the GMO formula. As with the expansion of the pion decay constant in mπ2, here too divergences appear in the mπ20 limit. These are offset by the divergences appearing in Eqs. (95), (96), (98) and (45) in the same limit. In a similar way, the terms that do not vanish as mπ20 cancel.

Expansion in the strange-quark mass in the isospin limit

As an application of the expressions presented in the preceding sections, we present their expansion in the strange-quark mass, ms. More specifically, for the pion decay constant, we keep the physical kaon mass constant and expand in the small quark ratio Rqm^/ms where m^(mu+md)/2. Our choice of such an expansion, rather than one in which we keep ms fixed and vary m^, is to facilitate comparison with the results given in [5]. For the pion mass we expand in ms to compare with [15].

The isospin limit expansion of Fπ is

FπF0=1+d1MK2(4πF0)2+d2MK2(4πF0)22+O(ms3)
51

where

d1=8(4π)2L4r-12logmK2μ2+8(4π)2(L4r+L5r)-2logmK2μ2-2log[2Rq]Rq+{2-8(4π)2(L4r+L5r)+2log[mK2μ2]+2log[2Rq]}Rq2+O(Rq3),
52

d2=d2tree+d2loop,
53

and

d2tree32(4π)4=C16r+L4r(3L4r+2L5r-8L6r-4L8r)+C15r-2C16r+6(L4r)2+4L4rL5r-16L4rL6r-4L4rL8r+2(L5r)2-8L5rL6r-4L5rL8rRq+C14r+5C16r+C17r-3(L4r)2-2L4rL5r+8L4rL6r+4L4rL8r-3(L5r)2+4L5rL8rRq2+O(Rq3),
54

d2loop=-1112log2MK2μ2+(329D1(0)+73-13log43)×logMK2μ2-7332+13log43-169(D2(0)-2log43D3(0))+13F43+{54log2MK2μ2+-169D1(1)+3512+53log43+13log2RqlogMK2μ2+15748+76log43-89D2(1)+2D3(1)log43-524F43+43log43+16(4π)2(L4r-L5r+2L8r)log2Rq}Rq+{-416log2MK2μ2+29D1(2)+10136-2912log43-434log2RqlogMK2μ2-84551536-6144518432log43+89D2(2)+D3(2)log43+787324576F43-5log22Rq+8D4(2)+294-2log43log2Rq}Rq2+O(Rq3),
55

and

D1(0)=(4π)213L1r+134L2r+6116L3r-518L4r,D2(0)=(4π)2134L2r+4348L3r,D3(0)=(4π)2(4L1r+L2r+L3r-3L4r),
56

D1(1)=(4π)28L1r+2L2r+2L3r-574L4r+574L5r-18L8r,D2(1)=(4π)28L1r+43L3r-6L4r+18L5r-36L8r,D3(1)=(4π)2(8L1r+2L2r+2L3r-3L4r+3L5r),
57

D1(2)=(4π)2(584L1r+308L2r+272L3r-258L4r+234L5r-432L8r),D2(2)=(4π)25L1r-17L2r-116L3r-512L4r+75L5r-144L8r,D3(2)=(4π)2(20L1r+5L2r+5L3r-6L4r+9L5r),D4(2)=(4π)2(14L1r+8L2r+7L3r-6L4r+5L5r-12L8r).
58

We can then connect the chiral SU(2) constant F in terms of the chiral SU(3) LECs as follows:

FF0=limmu,md0FπF0=1+d1MK2(4πF0)2+d2MK2(4πF0)22+O(ms3)
59

where d1 and d2 are understood to be in the limit mumd = 0. In this limit Eq. (51) agrees perfectly with the one-loop matching done in [5].

A similar expansion for the pion mass representation given in this paper is given below. In this case, we express the expansion in terms of the parameter B0ms rather than MK2 so as to facilitate comparison with the results of [15]. We have

Mπ2(mu+md)B0=1+c1msB0(4πF0)2+c2msB0(4πF0)22+O(ms3)
60

where

c1=-16(4π)2(L4r-2L6r)-29log4B0ms3μ2-16(4π)2(2L4r+L5r-4L6r-2L8r)+19+log43-89log4B0ms3μ2-log2RqRq-136Rq2+O(Rq3),
61

c2=c2tree+c2loop,
62

and

c2tree64(4π)4=-C16r+C20r+3C21r+4L4r(L4r-2L6r)-2C13r+C15r-2C20r-12C21r-2C32r-8L4r(2L4r+L5r-4L6r-L8r)-L5rL6rQ-{2C12r+4C13r+C14r+2C15r+2C16r+C17r-3C19r-6C20r-12C21r-2C31r-4C32r-42L4r+L5r2L4r+L5r-4L6r-2L8r}Rq2+O(Rq3),
63

c2loop=1112log2B0msμ2-329C1(0)+38081-29log43×logB0msμ2-3881log43+29log243+169C2(0)-2log43C3(0)+7316-23F43+{9754log2B0msμ2-169C1(1)+1549162+527log43×logB0msμ2-407324log43+827log243-89C2(1)+2log43C3(1)+1075648-79144F43-16C4(1)+49log43-59logB0msμ2log[2Rq]}Rq+{1165108log2B0msμ2-89C1(2)+6347324-754log43logB0msμ2-116636912-7111782944log43-154log243+49C2(2)-4log43C3(2)-137336864F43-89C4(2)+272-13log43-1196logB0msμ2×log[2Rq]+172log2[2Rq]}Rq2+O(Rq3),
64

and

C1(0)=(4π)226L1r+132L2r+618L3r-29L4r-132L5r+30L6r-6L7r+11L8r,C2(0)=(4π)2132L2r+4324L3r+2L4r+43L5r-4(L6r+L7r+L8r),C3(0)=(4π)2(8L1r+2(L2r+L3r)-11L4r-2L5r+12L6r-6L7r+2L8r),
65

C1(1)=(4π)2(88L1r+22L2r+532L3r-76L4r-26L5r+72L6r+52L8r),C2(1)=(4π)2(88L1r+623L3r-86L4r-743L5r+80L6r-28L7r+40L8r),C3(1)=(4π)2(16L1r+4(L2r+L3r)-31L4r-8L5r+36L6r+16L8r),C4(1)=(4π)23L4r-4L6r,
66

C1(2)=(4π)2(332L1r+164L2r+3012L3r-200L4r-78L5r+312L6r+24L7r+164L8r),C2(2)=(4π)2-204L1r+32L2r-1513L3r+203L4r+1003L5r-148L6r-22L7r-74L8r,C3(2)=(4π)2(4L1r+L2r+L3r-10L4r-3L5r+12L6r+12L7r+10L8r),C4(2)=(4π)2(252L1r+144L2r+126L3r-108L4r-54L5r+216L6r+108L8r).
67

From Eq. (60) we obtain the matching for B, which agrees completely with [15] in the chiral limit:

BB0=1+c1msB0(4πF0)2+c2msB0(4πF0)22+O(ms3).
68

Numerical analysis

We present in this section a numerical analysis of the expressions given in the preceding sections, and discuss some of their implications.

Fπ

We begin by giving a breakdown of the relative numerical contributions of the different terms constituting the 𝒪(p6) term of Fπ. As the expressions used in Sects. 3 and 4 of [1] correspond to those expressed in physical meson masses, we use the physical values of the masses. The caption of Table Table22 gives the numerical input values we used. Our expressions are exact except for the approximation used for dKKηπ. The value calculated using the approximate expression Eq. (29) agrees with using precise numerical expressions for the sunset integrals in Eq. (28) to 8 significant digits. The parts that do not depend on the LECs are given in Table 1. The large cancellations are due to the terms that diverge for mπ → 0.

Table 1
Numerical contributions (in units of 10-6 GeV4) of different terms to (F¯π)loop(6), the parts not depending on LECs. The inputs to these were FπFπ phys = 0.0922 GeV, mπmπ0 = 0.1350 GeV, mK=mKavg=0.4955 GeV, ...
Table 2
Numerical contributions (in units of 10-6 GeV4) of different terms to the (F¯π)loop(6) of Appendix A.2, the part depending on the LECs. The inputs are the same as in Table 1

The most recent fit of LECs with a number of different assumptions are given in Ref. [24]. Their main fit is called BE14 and can be found in Table Table33 [24]. We show results both for the exact fit results (BE14exact) and with the two digit precision given in the reference (BE14paper). The free fit in Table 2 in [24] was done with L4r free and a slightly different choice of p6 LECs, this fit we call free fit and finally we take the fit with the p6 LECs estimated with a chiral quark model of Table 2 in [24], labelled CQMfit. The results for the three Lir-dependent contribution, their sum and the sum including the contributions from Table 1 are given in Table 2.

Table 3
Numerical contributions (in units of 10-6 GeV4) of different terms to the GMO simplified (F¯π)loop(6) of Sect. 3, the part depending on the LECs. The inputs are the same as in Table 1

We examine the contributions calculated using the BE14exact LECs. The largest contribution arises from the dlog term, followed by the dlog×Li term. The sign of these two terms being opposite, however, reduces the overall contribution of the explicitly μ-scale dependent terms to the decay constant. In absolute value terms, the bilinear chiral log terms dlog×log provide the next largest contribution. The bilinear Li terms are of an order of magnitude smaller. The sunsets have a relatively small contribution in absolute value terms, but due to cancellations of the other contributions, the value of dsunset is little over a third of the total contribution to the sum.

The sum of the contributions calculated using BE14exact (free-fit) LECs yields

FπF0=1+F¯π(4)+(F¯π(6))loop+(F¯π(6))CT=1+0.2085(0.3143)+0.0126(0.1081)+0.0755(0.0193)=1+0.2085(0.3143)+0.0881(0.1274)=1.2966(1.4414).
69

Using the expressions simplified using the GMO relation, we obtain

FπF0=1+0.2085(0.3143)+0.0873(0.1263).
70

The value given in [24] is

FπF0=1+0.208(0.313)+0.088(0.127),
71

which agrees excellently with the physical representation and decently with the GMO simplified representation. Note that the last term has been calculated with exact p6 LECs as used in [24].

The numerical values calculated using the free-fit LECs demonstrate the sensitivity of the two-loop contribution to Fπ to the values of the LECs. In particular, it is to be noted that L4r and L6r are difficult to determine precisely, and the free-fit values for these two low-energy constants have relatively large uncertainties. The variation of (F¯π(6))loop with L4r and L6r over their possible range in the free fit is shown in Figs. Figs.22 and and3.3. The trend is of a progressively smaller value of (F¯π(6))loop for increasing L6r and decreasing L4r. A more thorough fit and detailed analysis of the LECs with the Fπ representation is planned for the future after a similar representation for the kaon and eta have been obtained.

Fig. 2
L4r dependence of (F¯π(6))loop. The full line is the value for L6r=0.49×10-3, while the shaded area indicates the range of possible values corresponding to the ±0.25 uncertainty of L6r in the free fit
Fig. 3
L6r dependence of (F¯π(6))loop. The dashed line is the value for L4r=0.76×10-3, while the shaded area indicates the range of possible values corresponding to the ±0.18 uncertainty of L4r in the free fit

The dependence of Fπ/F0 on MK2 given in Eq. (59), with MK = 0.4955 GeV and F0 on the r.h.s. replaced by the physical Fπ phys, has the following numerical form in the chiral limit:

FF0=1+0.1499(0.2562)+0.0157(-0.0516)+.
72

The first set of numbers correspond to the use of the BE14exact LECs, while the numbers in parentheses are calculated using the free fit. Figure Figure44 shows the MK dependence of F/F0 using these inputs, keeping F0Fπ fixed on the. A significant divergence in the two sets of values is observed as MK2 increases.

Fig. 4
MK2 dependence of F/F0 in the chiral limit

The largest contribution to F/F0 at O(ms2) comes from the d2tree term, followed by the term proportional to log(B0ms/μ2). In absolute terms, the pure number contribution to d2 is greater than that of the ( - 11/12)log(B0ms/μ2) term, but its sign being negative, the pure number serves to decrease the numerical size of d2, as do all the remaining terms as well. Ignoring the terms proportional to the Li in d2loop, one gets a value of - 1.4244 for d2, in contrast to 0.4698 when the Li proportional terms are retained. The Li therefore contribute significantly to the O(MK2) contribution to Fπ.

The effect of the higher-order terms in Rq can be seen by comparing Eq. (72) with Eq. (76) below, which gives numerical values for Fπ/F0. We use a value of Rq=m^/ms=1/24.4 obtained from [25], the numerical value of d1, Eq. (52), with corrections up to O(Rq2), is

d1=0.8198(1.4009)+0.3454(0.3425)-0.0108(-0.0107)=1.1544(1.7327).
73

Similarly,

d2tree=2.5022(-0.0863)-0.3229(-0.2641)+0.0170(0.0129)=2.1963(-0.3375),
74

d2loop=-2.0324(-1.4574)-0.0180(-0.1834)-0.0729(-0.0718)=-2.1233(-1.7126).
75

Note that the 𝒪(Rq) contribution of d2loop evaluated using the BE14exact LECs is numerically smaller than the O(Rq2). Note too that the 𝒪(Rq) value calculated using the free-fit value differs from the one calculated using BE14exact by an order of magnitude. Putting it all together we obtain up to O(Rq2,s2) the following expansion:

FπF0=1+0.2111(0.3169)+0.0024(-0.0686)+,
76

which gives a more accurate numerical representation of the effect on Fπ of integrating the strange-quark mass out. The effect of the correction due to m^ to the chiral limit is particular pronounced at O(Rq2), with the value of the chiral limit number at this order given in Eq. (72) calculated using the BE14 fit differs from its analogous value in Eq. (76) by one order of magnitude, due to cancelations between the different parts.

mπ2

An analysis of the expression for the pion mass produces the numerical results given in Tables Tables44 and and5.5. The large cancellations in the sunset contributions follow from the fact that the separate parts do not vanish in the limit mπ → 0 but their sum does. Except for CQMfit, which was not a good fit in [24], the largest contribution comes from the pure logarithmic terms, the contribution of which, however, is cancelled to a large degree by the log × Li term of similar magnitude but opposite sign. The bulk of the net contribution to (Mπ(6))loop therefore comes from the sunsets diagrams and the bilinears in the chiral logs. The cLi and cLi×Lj contribute very little. Using the BE14exact (free-fit) LECs, we get (Table (Table66):

Mπ2mπ2=1.057(0.940)+(mπ2)(4)+(mπ2)loop(6)+(mπ2)CT(6)=1.057(0.940)-0.0051(0.1044)+0.1254(0.1292)-0.1769(-0.1732)=1.057(0.940)-0.0051(0.1044)-0.0515(-0.0440).
77

Using the expressions simplified using the GMO relation, we get

Mπ2mπ2=1.057(0.940)-0.0060(0.1035)-0.0476(-0.0407).
78

The lowest-order term is determined by having the right-hand side sum to 1. This agrees well with the numerical values given in [24].

Table 4
Numerical contributions (in units of 10-7 GeV6) of different terms to mπ2loop(6) of Appendix A.1, the parts not depending on LECs. The inputs are the same as in Table 1
Table 5
Numerical contributions (in units of 10-7 GeV6) of different terms to mπ2loop(6), the part depending on the LECs. The inputs are the same as in Table 1
Table 6
Numerical contributions (in units of 10-7 GeV6) of different terms to the GMO simplified (mπ2)loop(6) of Sect. 4, the part depending on the LECs. The inputs are the same as in Table 1

Numerically, with msB0=0.484 GeV, F0 = 0.0922 GeV and BE14exact (free-fit) LECs, we have for the expansion given in Eq. (68) in the chiral limit

BB0=1+0.0197(0.1219)-0.0586(-0.1027)+.
79

Figure Figure55 shows the ms dependence of B/B0 for two sets of LECs, BE14exact and free fit. The two sets of LECs produce the same general behaviour, but they are different numerically.

Fig. 5
ms dependence of Mπ2/mπ2 in the chiral limit

Fitting lattice data

In the equal mass case the formulae have a simple form in terms of the physical mass and decay constant. For the two-flavour case these can be found in the FLAG report [27], and for the three-flavour case in [28]. Here, the only non-analytic dependences that occur are logarithms, allowing for a compact expression. Even here there are a number of different ways to express the result. In terms of the physical mass mπ2, the physical decay constant Fπ, the lowest-order mass M2, and the chiral limit decay constant F, the first option is

mπ2=M21+x12logM2μ2+lMr+x2178log2M2μ2+c1MrlogM2μ2+c2Mr+O(x3),Fπ=F1+x-logM2μ2+lFr+x2-54log2M2μ2+c1FrlogM2μ2+c2Fr+O(x3).
80

Here the left-hand side is the physical observable, and the right-hand-side is expressed purely in terms of lowest-order quantities. The expansion parameter here is xM2/(16π2F2).

An alternative is to write the lowest order on the left-hand side and the physical quantities on the right-hand side:

M2=mπ21+ξ-12logmπ2μ2+l~Mr+ξ2-58log2mπ2μ2+c~1Mrlogmπ2μ2+c~2Mr+O(x3),F=Fπ1+ξlogmπ2μ2+l~Fr+ξ2-14log2mπ2μ2+c~1Frlogmπ2μ2+c~2Fr+O(ξ3).
81

Here the expansion is in terms of ξ=mπ2/(16π2Fπ2).

A third alternative is to have the physical quantities on the left hand side but do the expansion on the right-hand side in terms of physical masses.

mπ2=M2+mπ2ξ12logmπ2μ2+l^Mr+mπ2ξ258log2mπ2μ2+c^1Mrlogmπ2μ2+c^2Mr+O(ξ3),Fπ=F1+ξ-logmπ2μ2+l^Fr+ξ254log2mπ2μ2+c^1Frlogmπ2μ2+c^2Fr+O(ξ3).
82

There are obviously even more possibilities but these are the three that we know have been used to fit data. The coefficients in the three options are clearly related by recursively using the expansions. The three options differ by higher orders (NNNLO).

We use a generic notation for all of the coefficients below with a  ·  over the letter and IMF. The coefficients l˙Ir,c˙1Ir depend on the NLO LECs while the c2Ir in addition depend on the NNLO LECs. The expressions (8082) are exactly μ-independent when the μ-dependence of the coefficients l˙Ir,c˙iIr, is taken into account. The FLAG report uses a slightly different form where lIr is traded for the scale of NLO leading logarithm Λ3,4 and c1I for the scale of the log2 terms ΛI and a similar notation for the ξ-expansion.

A side comment is that the leading logarithms are known to higher orders [2931].

When different masses come into play there are clearly more ways of writing some masses as lowest-order and others as physical ones, as well as the complication that the lowest-order masses satisfy the GMO relation allowing for having different choices for which physical masses to use. The final complication is that the non-analytic dependence from the sunset diagram is considerably more involved than just logarithms, and in fact a large aim of this program is to find faster numerical ways to handle exactly this.

In the three-flavour fitting of LECs to data [24, 32, 33] traditionally forms corresponding to the third option, Eq. (82), have been used, called “expansion in physical masses and Fπ.” The equivalent to the x-expansion of Eq. (80) is usually called expansion in lowest-order quantities. Both cases were calculated in [1] and can be downloaded from [34]; they are included in CHIRON [2].

In lattice calculations one has easy access to the physical masses for the charged pion and kaon while the eta mass is more difficult. On the other hand one would still like to have the expansion in terms of physical quantities since part of the higher corrections are precisely changing lowest-order masses in the loop diagrams to physical masses. For fitting lattice data we thus choose an option where one uses the physical pion decay constant and the physical charged pion and kaon masses. The eta mass in the loops is then replaced by the value obtained by using the GMO relation with the physical pion and koan mass as input. These are the formulae quoted in the main text.

We can now check how many parameters are needed for the expressions for the pion mass and decay constant to NNLO. We use here the notation mπ2 and mK2 for the physical pion and kaon masses while mη2=(4/3)mK2-(1/3)mπ2.

The GMO expressions can be written as

mπ2=M2+mπ2{12ξπλπ-29ξK-118ξKλη+ξKL^1Mr+ξπL^2Mr}+mπ2(K^1Mrλπ2+K^2MrλπλK+K^3Mrλπλη+K^4MrλK2+K^5MrλKλη+K^6Mrλη2+ξK2FMmπ2mK2+C^1Mλπ+C^2MλK+C^3Mλη+C^4M),FπF=1+-ξπλπ-12ξKλK+ξKL^1Fr+ξπL^2Fr+(K^1Frλπ2+K^2FrλπλK+K^3Frλπλη+K^4FrλK2+K^5FrλKλη+K^6Frλη2+ξK2FFmπ2mK2+C^1Fλπ+C^2FλK+C^3Fλη+C^4F)
83

where we defined the quantities ξπ=mπ2/(16π2Fπ2), ξK=mK2/(16π2Fπ2) and λi=log(mi2/μ2). The coefficients L^iIr are a function of the NLO LECs Lir. Each of the K^iIr,C^iIr has three terms proportional to ξπ2,ξπξK,ξK2, respectively. The K^iI and FI are fully determined, the C^iIr,i=1,2,3 depend linearly on the NLO LECs and C^4F depends up to quadratically on the NLO LECS and linearly on the NNLO LECs. There is some ambiguity in dividing the terms not depending on LECs between the various terms since log(mi2/mK2)=λi-λK for iπη.

The FI can be subdivided as

FI[ρ]=116π2a1I+(a2I+a3Ilog[ρ]+a4Ilog2[ρ])ρ+(a5I+a6Ilog[ρ]+a7Ilog2[ρ])ρ2+a8Ilogmη2μ2+O(ρ3).
84

Explicitly, the coefficients for the pion mass are given by

L^1Mr=-16(4π)2(L4r-2L6r),L^2Mr=-128π2(L4r+L5r-2L6r-2L8r),
85

K^1Mr=38ξπξK+121144ξπ2,K^2Mr=-34ξπξK,K^3Mr=59ξπξK-112ξπ2,K^4Mr=175144ξK2+118ξπξK,K^5Mr=112ξπξK-4372ξK2,K^6Mr=7391296ξK2-67648ξπξK-111296ξπ2,
86

C^1Mr=-4(4π)2(14L1r+8L2r+7L3r-18L4r-12L5r+32L6r+22L8r)+1199288ξπ2+40(4π)2(L4r-2L6r)-4772ξπξK,

C^2Mr=-4(4π)2(16L1r+4L2r+5L3r-20L4r-4L5r+24L6r+8L8r)+389ξK2+8(4π)2(L4r+L5r-2L6r-2L8r)-16ξπξK,

C^3Mr=-169(4π)2(16L1r+4L2r+4L3r-18L4r-3L5r+20L6r-12L7r+2L8r)+1027ξK2+89(4π)2(16L1r+4L2r+4L3r-21L4r-8L5r+26L6r-24L7r+4L8r)-7216ξπξK+132-49(4π)2(4L1r+L2r+L3r-6L4r-4L5r+8L6r+6L8r)ξπ2,

C^4Mr=227(4π)2{(54L1r+111L2r+28L3r+8L5r-96L7r-48L8r)ξπ2+(156L2r+43L3r+8L5r-96L7r-48L8r)ξK2-8(3L2r+L3r+2L5r-24L7r-12L8r)ξπξK-8(8π)4{(L4r-2L6r)(4L4r+L5r-8L6r-2L8r)ξK2-(L4r-2L6r)(4L4r+3L5r-8L6r-6L8r))ξπξK+(L4r+L5r-2L6r-2L8r)2ξπ2},
87

a1M=-23F43+7316-43144log243-124log43a2M=113288F43-1291864+124log243+35288log43a3M=4772a4M=-38a5M=5576F43+84896912-148log243-2632304log43a6M=1972+118log43a7M=-31144a8M=124.
88

It may be noted that the aiMi = 1, .., 8 have an elegant structure. Similarly, for the pion decay constant, we have

L^1Fr=8(4π)2L4r,L^2Fr=4(4π)2(L4r+L5r),
89

K^1Fr=4132ξπ2-724ξπξK,K^2Fr=2512ξπξK,K^3Fr=0,K^4Fr=136ξπξK-17288ξK2,K^5Fr=19ξπξK-55144ξK2,K^6Fr=7288ξK2+136ξπξK-196ξπ2,
90

C^1Fr=139144-24(4π)2L4rξπξK+ξπ2(2(4π)2(14L1r+8L2r+7L3r-13L4r-10L5r)+1381576),

C^2Fr=2(4π)2(16L1r+4L2r+5L3r-14L4r)+2×ξK2-2(4π)2(3L4r+5L5r)+14ξπξK,

C^3Fr=329(4π)2(4L1r+L2r+L3r-3L4r)+13ξK2+29(4π)2(4L1r+L2r+L3r-3L4r)-11576ξπ2-169(4π)2(4L1r+L2r+L3r-3L4r)+1144ξπξK,

C^4Fr=-127(4π)2{(156L2r+43L3r)ξK2-8(3L2r+L3r)ξπξK+(54L1r+111L2r+28L3r)ξπ2}+(8π)4{27(L4r)2+5L4rL5r-8L4rL6r-4L5rL6rξπξK+2L4r(7L4r+2L5r-8L6r-4L8r)ξK2+12(L4r+L5r)(7L4r+7L5r-8L6r-8L8r)ξπ2},
91

a1F=13F43-7332-7288log243+116log43,a2F=-548F43+10964-136log243-37576log43,a3F=-139144,a4F=724,a5F=275398304F43-23756144+196log243-53373,728log43,a6F=47576,a7F=-132,a8F=-116.
92

For the equal mass case we had one free parameter at NLO for the mass and decay constant and two each at NNLO. For the three-flavour case in the isospin limit there is a significantly larger number, two each at NLO but, three each at NNLO not involving logarithms and 9 each for the terms involving logarithms. The latter are clearly not independent since they at most depend on the eight NLO LECs L1r,,L7r.

We defer a full study to future work when kaon and eta quantities will be included.

Conclusions

In this work, we have used the explicit representations of the two-loop contribution to the pion decay constant and mass in three-flavour chiral perturbation theory [1] to derive (semi-)analytic expressions for them. That it is semi-analytic and not fully analytic stems from the fact that we treated the three mass configuration sunset integrals appearing in them as an expansion in the square of the external momentum and have retained only the first few terms. This semi-analytic representation is nonetheless very accurate and numerically reproduces the full result to a high degree [1, 2].

We have used these expressions to expand Fπ and Mπ in the strange-quark mass to O(ms2) and to perform the matching of two-flavour low-energy constants B and F with their three-flavour counterparts in the chiral limit. The results obtained fully agree with those previously derived in [5, 15, 26].

Aside from an investigation of the numerical implications of the strange-quark expansion of both Fπ and B0, we have also done a preliminary study of the dependence of Fπ on the low-energy constants L4r and L6r. These show trends that are possibly in contradiction with the large Nc analysis of these LECs, and a more detailed study needs to be done. The breakdown of the relative numerical contributions to the pion decay constant at two loops shows that the contribution from the terms involving the Lir and Cir, although not large, is not insignificant. Their contribution is amplified partially due to the cancellation of other terms that have a larger absolute value. Furthermore, in the chiral limit ms expansion, the terms proportional to the low-energy constants contribute greatly to the O(ms2) term. All these point to the need for a thorough study into the dependence of such quantities on the LECs for a better understanding of the chiral perturbation series.

We also present a discussion of the various ways in which NNLO results for the pion mass and decay constant may be represented, and their relative merits. We then rewrite the expressions given in this paper in a manner which allows for east fitting with data from lattice simulations.

In forthcoming work, we will present similar semi-analytic expressions for the three-flavour two-loop contributions to the kaon and eta mass and decay constants, and use those results and the ones presented in this work to do a preliminary fit of lattice data to obtain new values for some low-energy constants. That exercise, along with the results and analyses presented in this work, is indicative of the usefulness of such analytic representations of ChPT amplitudes and other quantities, and it will hopefully encourage and facilitate the lattice community in making use of full NNLO results from ChPT.

Acknowledgements

JB is partially supported by the Swedish Research Council grants contract numbers 621-2013-4287 and 2015-04089, and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 668679). SG thanks the authors of [57] for clarifying the precise relation between their and our results, and G. Ecker, H. Leutwyler, S. Friot and M. Misiak for correspondence and discussion. BA is partly supported by the MSIL Chair of the Division of Physical and Mathematical Sciences, Indian Institute of Science.

Appendix A: Expressions without the use of GMO

A.1 Pion mass

We have

Fπ2mπ2(mπ2)(4)=-8mπ2(L4r+L5r-2L6r-2L8r)-16mK2(L4r-2L6r)-13mη2lηr+mπ2lπr,
93

16π2mπ2cLiπ=mπ44L1r+749L2r+5627L3r+19mK4104L2r+863L3r-169mK2mπ2L2r+13L3r,
94

(16π2)clogπ=-316mη4mπ2+14mη2mK2mπ2+13mη2mπ4-34mK4mπ2-116mK2mπ4-29936mπ6lπr+-294mK4mπ2-13mK2mπ4lKr+316mη4mπ2-54mη2mK2mπ2-172mη2mπ4lηr,
95

clog×logπ=12136mπ6+32mπ4mK2-14mπ2mK4(lπr)2+12mπ2mK4-3mπ4mK2lπrlKr+53mπ4mη2lπrlηr+52mK4mη2-32mK2mη4-32mπ2mK2mη2lKrlηr+16mπ4mK2+194mπ2mK4+112mπ2mK2mη2-54mK4mη2+34mK2mη4(lKr)2+118mπ4mη2+2512mπ2mK2mη2-54mK4mη2-2936mπ2mη4+34mK2mη4(lηr)2,
96

clog×Liπmπ2=89mη2mπ2lηr(12L1r+3L2r+3L3r-18L4r-8L5r+24L6r-48L7r-6L8r)-169mη2mK2lηr(24L1r+6L2r+6L3r-27L4r-4L5r+30L6r-24L7r)-8mK4lKr(16L1r+4L2r+5L3r-20L4r-4L5r+24L6r+8L8r)-8mπ4lπr(14L1r+8L2r+7L3r-18L4r-12L5r+32L6r+22L8r)+16mK2mπ2(lKr(L4r+L5r-2L6r-2L8r)+5lπr(L4r-2L6r)).
97

The contribution from the sunset integrals is given by

csunsetπ=1(16π2)2[316mη6-14+π216mη4mK2-155384mη4mπ2+98+π26mη2mK4-2532+π2144mη2mK2mπ2+25192mη2mπ4-12+π26mK6-5596+31π2144mK4mπ2+677864-5π2162mK2mπ4+25431728-41π21296mπ6]+cπKKπ+cπηηπ+cKKηπ
98

where cπηηπ is given by Eq. (44), cπKKπ is given by Eq. (45), and

cKKηπ=-548mπ4+23mπ2mK2+13mπ2mη2-58mK4+14mK2mη2-316mη4H¯KKηχ+124mπ4mK2-1924mπ2mK4-58mπ2mK2mη2+52mK6-158mK4mη2+34mK2mη4H¯2KKηχ+748mπ4mη2-18mπ2mK2mη2-13mπ2mη4+18mK2mη4+316mη6H¯KK2ηχ.
99

With ρmπ2/mK2 and τmη2/mK2, expanding cKKηπ about s=mπ2=0 gives

(16π2)2cKKηπ=cKKη(0)+cKKη(1)(mπ2)+cKKη(2)(mπ2)2+O((mπ2)3)
100

where

cKKη(0)=-316mη6+14+π216mη4mK2-98+π26mη2mK4-58-5π248mK6+516mη2mK4-316mη4mK2log2[τ],
101

cKKη(1)=155384mη4+353192+13π2288mK4+4932+7π2144×mη2mK2+14mη2mK2-mK4F[τ]+18mη2mK2-332mη4log[τ]-1348mη2mK2log2[τ]+332mη4-18mη2mK2+516mK4log[ρ],
102

cKKη(2)=-1796-π2288mη2-1348+π272mK2+1λmη448-mK62mη2-mη2mK224-13mK424F[τ]+1λmη46-mη2mK224-mK42log[τ]-148mη2log2[τ]-mη26+mK23log[ρ].
103

A.2 Pion decay constant

We have

(16π2)dlogπ=932mη4-38mη2mK2-748mη2mπ2+98mK4+94mK2mπ2+679144mπ4lπr+238mK4-12mK2mπ2lKr+-932mη4+78mη2mK2+148mη2mπ2lηr,
104

dlog×logπ=158mK4mη2mπ2-98mK2mη4mπ2+14mπ2mη2-1724mK2mη2+38mη4lηr2+253mπ2mK2-34mK4lπrlKr+418mπ4-76mπ2mK2+38mK4lπr2+-154mK4mη2mπ2+94mK2mη4mπ2-712mK2mη2lKrlηr+158mK4mη2mπ2-98mK2mη4mπ2+13mπ2mK2-58mK4+724mK2mη2lKr2,
105

dlog×Liπ=4mπ214mπ2L1r+8mπ2L2r+7mπ2L3r-13mπ2L4r-12mK2L4r-10mπ2L5rlπr+4mK216mK2L1r+4mK2L2r+5mK2L3r-3mπ2L4r-14mK2L4r-5mπ2L5rlKr-43mη2mπ2-4mK24L1r+L2r+L3r-3L4rlηr.
106

The term involving the sunset integrals dsunsetπ is given by

dsunsetπ=1(16π2)2[-932mη6mπ2+38+3π232mη4mK2mπ2+193768mη4-2716+π24mη2mK4mπ2-1364+7π2288mη2mK2+49384+π2216mη2mπ2+34+π24mK6mπ2+209192+5π232mK4+41192+π236mK2mπ2-11152+π2288mπ4]+dπKKπ+dπηηπ+dKKηπ
107

where dπKKπ is given by Eq. (26), dπηηπ is given by Eq. (27), and

dKKηπ=196mπ2-124mK2+1516mK4mπ2-748mη2-38mK2mη2mπ2+932mη4mπ2H¯KKηχ+548mπ2mK2-748mK4-154mK6mπ2+716mK2mη2+4516mK4mη2mπ2-98mK2mη4mπ2H¯2KKηχ+596mπ2mη2-116mK2mη2+748mη4-316mK2mη4mπ2-932mη6mπ2H¯KK2ηχ.
108

This can be expressed as an expansion in s=mπ2 as

(16π2)2dKKη=dKKη(-1)(mπ2)-1+dKKη(0)+dKKη(1)(mπ2)+dKKη(2)(mπ2)2+O((mπ2)3)
109

where

dKKη(-1)=932mη6-38+3π232mη4mK2+2716+π24mη2mK4+1516-5π232mK6+932mη4mK2-1532mη2mK4log2[τ],
110

dKKη(0)=-193768mη4-1164-π2288mη2mK2-211384+11π2576mK4+12mK4-18mη2mK2F[τ]+596mη2mK2log2[τ]+964mη4-316mη2mK2log[τ]+-964mη4+316mη2mK2-1532mK4log[ρ],
111

dKKη(1)=19384+π2192mη2-11192-π296mK2+F[τ]mη232-mK28+796mη2+148mK2log[ρ]-132mη2log2[τ]-796mη2log[τ],
112

dKKη(2)=1λ223576mη4-14mK6mη2-235576mη2mK2+139288mK4+1λ3-12mK10mη4+1748mK8mη2-748mη2mK4-13mK6F[τ]+1λ31192mη6-132mη4mK2-12mK8mη2+8396mη2mK4+1348mK6log[τ]-1192log[ρ].
113

In the above expressions, τmη2/mK2, ρmπ2/mK2, λmη2-4mK2, and F[x] is defined in Eq. (13).

Appendix B: Two mass sunset master integrals

The finite parts of the master integrals appearing in the expressions for dπKK and dπηη are presented here. The chiral logarithms arising from these integrals do not appear in the expressions below, having been removed and included in the clog, clog×log, dlog or dlog×log term as appropriate. We have

H¯πKKχ=mK2(16π2)2(2+π26+mπ2mK2π212-18-mπ22mK2log2mπ2mK2+logmπ2mK2+mK2mπ2+mπ2mK2-2Li2mπ2mK2+log1-mπ2mK2logmπ2mK2),
114

H¯2πKKχ=116π22(π212-12-12log2mπ2mK2+1-mK2mπ2Li2mπ2mK2+logmπ2mK2log1-mπ2mK2).
115

The expressions for H¯πηηχ and H¯2πηηχ can be obtained from the above by making the replacement mK → mη.

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