Published for SISSA by Springer
Received: March 14, 2015 Revised: May 20, 2015
Accepted: June 2, 2015 Published: June 25, 2015
Xu-Kun Guo,a Zhi-Hui Guo,a,b,1 Jos Antonio Ollerc and Juan Jos Sanz-Cillerod
aDepartment of Physics, Hebei Normal University, Shijiazhuang 050024, Peoples Republic of China
bState Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing 100190, Peoples Republic of China
cDepartamento de Fsica, Universidad de Murcia,
E-30071 Murcia, Spain
dDepartamento de Fsica Terica and Instituto de Fsica Terica, IFT-UAM/CSIC, Universidad Autnoma de Madrid,Cantoblanco, 28049 Madrid, Spain
E-mail: mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , [email protected], mailto:[email protected]
Web End [email protected]
Abstract: We study the - mixing up to next-to-next-to-leading-order in U(3) chiral perturbation theory in the light of recent lattice simulations and phenomenological inputs. A general treatment for the - mixing at higher orders, with the higher-derivative, kinematic and mass mixing terms, is addressed. The connections between the four mixing parameters in the two-mixing-angle scheme and the low energy constants in the U(3) chiral e ective theory are provided both for the singlet-octet and the quark-avor bases. The axial-vector decay constants of pion and kaon are studied in the same order and confronted with the lattice simulation data as well. The quark-mass dependences of m, m and mK are found to be well described at next-to-leading order. Nonetheless, in order to simultaneously describe the lattice data and phenomenological determinations for the properties of light pseudoscalars , K, and , the next-to-next-to-leading order study is essential.
Furthermore, the lattice and phenomenological inputs are well reproduced for reasonable values of low the energy constants, compatible with previous bibliography.
Keywords: E ective eld theories, Chiral Lagrangians
ArXiv ePrint: 1503.02248
1Corresponding author.
Open Access, c
The Authors.
Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP06(2015)175
Web End =10.1007/JHEP06(2015)175
Scrutinizing the - mixing, masses and pseudoscalar decay constants in the framework of U(3) chiral e ective eld theory
JHEP06(2015)175
Contents
1 Introduction 1
2 Theoretical framework 32.1 Relevant chiral Lagrangian 32.2 The - mixing at NNLO in expansion 52.3 Insights into previous studies of the - mixing 92.4 Masses and decay constants of pion and kaon up to NNLO in expansion 10
3 Phenomenological discussions 153.1 Leading-order analyses 163.2 Next-to-leading order analyses 173.3 NLO ts focusing on the masses 193.4 Next-to-next-to-leading order analyses 21
4 Conclusions 27
A Higher order corrections to the and bilinear terms 30
1 Introduction
The phenomenology of light avor pseudoscalar mesons and provides a valuable window on many important nonperturbative features of Quantum Chromodynamics (QCD). It includes such important aspects as:
The spontaneous breaking of chiral symmetry, which gives rise to the appearance of
the multiplet of light pseudoscalar mesons.
The U(1)A anomaly of strong interactions, which gives mass to the singlet 0 in
NC = 3 QCD, even in the chiral limit.
The explicit SU(3)-avor symmetry breaking, due to the splitting ms 6= m between
the strange and up/down quark masses (the isospin limit, where mu = md = m and
the electromagnetic corrections are neglected, will be assumed all through the article).
The 1/NC expansion of QCD in the limit of large NC, with NC the number of colors
in QCD.
The interaction between the pseudo-Nambu-Goldstone bosons (pNGBs) (, K, 8) from the spontaneous chiral symmetry breaking can be systematically described through a low-energy e ective eld theory (EFT) based on SU(3)L SU(3)R chiral symmetry, namely
1
JHEP06(2015)175
Chiral Perturbation Theory (~PT) [1, 2]. Following largeNC arguments [35], this approach was later extended, incorporating the singlet 0 into a U(3) ~PT Lagrangian [614].
This combination of ~PT and the 1/NC expansion provides a consistent framework which addresses all the previous issues.
More precisely, in this article we show that this largeNC ~PT framework yields an excellent description of the and masses from lattice simulations at di erent light-quark masses [1519]. Constraints from phenomenological studies of , !, , J/ decays [20, 21] and kaon mass lattice simulations [22, 23] are compatible and easily accommodated in a joint t. The problems arise when one tries to also describe lattice simulations for F, FK
and FK/F [2224]. Nevertheless, the issue of these observables in ~PT is known and has been widely discussed in previous bibliography [2529]. It constitutes a problem in its own and it is not the central goal of this article. It is discussed for sake of completeness and to show its impact in a global t.
The and mesons not only attract much attention from the chiral community but they have been also intensively scrutinized in lattice QCD simulations, where enormous progresses have been recently made by di erent groups [1519]. Varying the light-quark masses m and ms, both their masses and mixing angles have been extracted in the range 200 MeV < m <700 MeV. We will focus on the simulation points with m < 500 MeV in the present work. By observing the dependence of these observables with the light-quark masses we will determine the ~PT low energy constants (LECs) and further constrain the theoretical models. At the practical level we have recast all m dependencies in terms of m and study the observables as functions of m. The and lattice simulations have not been thoroughly analyzed in the chiral framework yet and it is the central goal of the present work. However, the numerical uncertainties resulting from our analyses in this work must be taken with a grain of salt as correlations between the di erent lattice data points and other systematic errors are not considered here.
In addition to lattice QCD, there are also phenomenological studies of the and mixing, which has been extensively investigated in radiative decays of light-avor vector resonances , !, and J/ V P, P processes [20, 21, 3036]. In these works, the modern
two-mixing-angle scheme for the and mesons, which was rst advocated in refs. [10, 11], was employed to t various experimental data. The common methodology in these works is that the two-mixing-angle pattern for the and is simply adopted to perform the phenomenological discussion and the mixing parameters are then directly determined from data. This is a bottom-up approach to address the - mixing problem and it is quite useful for the phenomenological analysis. Contrary to the bottom-up method, it is also very interesting to study the - mixing from a top-down approach in which one rst constructs the relevant ~PT Lagrangian and then calculates the - mixing pattern and parameters in terms of the LECs. In this case, one can predict the - mixing parameters once the values of the unknown LECs are given. The present work belongs to the latter category of top-down approaches.
Though the singlet 0 meson, which is the main component of the physical state, is not a pNGB due to the strong U(1)A anomaly, it can be formally introduced into ~PT from the large-NC point of view. The argument is that the quark loop induced U(1)A
2
JHEP06(2015)175
anomaly, which is responsible for the large mass of the singlet 0, is 1/NC suppressed and hence the 0 becomes the ninth pNGB in the large NC limit [3739]. Based on this argument, the leading-order (LO) e ective Lagrangian for U(3) ~PT, which simultaneously includes the pNGB octet , K, 8 and the singlet 0 as dynamical elds, was formulated in refs. [69]. Later on, a full O(p4) U(3) chiral Lagrangian was constructed in ref. [13]
and the discussion on the O(p6) unitary group chiral Lagrangian has been very recently
completed in ref. [40]. Subtle problems about the choice of suitable variables for the higher order U(3) ~PT Lagrangian in the large NC framework were analyzed in ref. [12].
The standard power counting employed in SU(2) and SU(3) ~PT in powers of the external momenta and quark masses [1, 2], is not valid any more in U(3) ~PT, due to the appearance of the large 0 mass. However, since the singlet 0 mass squared behaves like 1/NC in large NC limit, the 0 mass can be harmonized with the other two expansion parameters if one assigns the same counting to 1/NC, the squared momenta p2 and the light quark masses mq. As a result of this, in order to have a systematic power counting, the combined expansions on momentum, light quark masses and 1/NC are mandatory in U(3) ~PT [12, 13]. We will work in this combined expansion in our study and denote it as expansion throughout the paper, where O( ) O(p2) O(mq) O(1/NC). This counting rule is di erent from the one proposed in ref. [41], where the 0 mass is counted as O(1) and the infrared regularization method is employed to handle the chiral loops.
Some recent works in refs. [14, 4245] have addressed the - mixing in the chiral framework up to next-to-leading order (NLO). As an improvement, we will perform the systematic study of the - mixing in the -expansion scheme up to next-to-next-to-leading order (NNLO) and take into account the very recent lattice simulation data, which are not considered in the previous works [14, 4245]. In addition, we also simultaneously analyze the m dependences of other physical quantities from lattice simulations, such as the axial , K decay constants and the mass ratio of the strange and up/down quarks, in order to further constrain the ~PT LECs.
This article is organized as follows. In section 2, we introduce the theoretical framework and calculate the relevant physical quantities. In section 3, the phenomenological discussions will be presented. Conclusions will be given in section 4. Further details about the calculations up to NNLO are relegated to appendix A.
2 Theoretical framework
2.1 Relevant chiral Lagrangian
At leading order in the expansion, i.e. O( 0), the U(3) ~PT Lagrangian consists of three
operators
L(0) =
2
F , ~ = 2B(s + ip) , ~ = u~u u~u , X = log (det U) , u = iuDUu , DU = @U i(v + a)U + iU(v a) , (2.2)
3
JHEP06(2015)175
F 2
4 huui +
F 2
4 h~+i +
F 2
12 M20X2 , (2.1)
where the chiral building blocks are dened as [1, 2, 1214]
U = u2 = ei
with the pNGB octet+singlet matrix
=
12 0 +
16 8 +
13 0 + K+ 1
2 0 +
16 8 +
13 0 K0
K K0 2
6 8 +
1 3 0
, (2.3)
and s, p, v, a being the external scalar, pseudoscalar, vector and axial-vector sources, respectively. The coupling F appearing in eqs.(2.1) and (2.2) corresponds to the pNGB axial decay constant in the large NC and chiral limits. The light quark masses are introduced by setting (s + ip) =diag{ m, m, ms}, being m the averaged up and down quark masses and
ms that of the strange quark.
Notice the structure of the LO Lagrangian in eq. (2.1): the rst operator is of O(NC, p2) type, the second one corresponds to the type of O(NC, mq) and the last one stems from the
QCD U(1)A anomaly and is of O(N0C, p0) type, where U is counted as O(1),F 2 O(NC)
and M20 O(N1C) in the classication O(NjC, pk, mq) of the EFT Lagrangian operators
in eq. (2.1). In the following, we will denote the chiral expansions in powers of squared momenta p2 and quark masses mq simply as a generic expansion in p2.
The NLO U(3) chiral Lagrangian, i.e., O( ), contains O(NC, p4) and O(N0C, p2) oper
ators. The relevant ones in our work read [12]
L() = L5huu~+i +
L8
2 h~+~+ + ~~i +
F 2 1
12 DXDX
F 2 2
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12 Xh~i , (2.4) with the dimensionless LECs scaling like L5, L8 O(NC) and 1, 2 O(N1C).
At NNLO, i.e. O( 2), there are three types of operators: O(N1C, p2), O(N0C, p4) and O(NC, p6). Their explicit forms read [13, 46]
L(2) =
F 2 v(2)2
4 X2h~+i+L4huuih~+i + L6h~+ih~+i + L7h~ih~i + L18huihu~+i + L25Xh~+~i
+C12hhh~+i + C14huu~+~+i + C17hu~+u~+i+C19h~+~+~+i + C31h~~~+i , (2.5)
where the rst line corresponds to the O(N1C, p2) type, the second line is of the O(N0C, p4)
type and the last two lines are of the O(NC, p6) type. The LECs carry the scalings
v(2)2 O(N2C), L4, L6, L7, L18, L25 O(N0C) and C12, C14, C17, C19, C31 O(NC). Notice that we have only shown the operators at di erent orders in eqs. (2.1), (2.4) and (2.5)
that are pertinent to our present study, not aiming at giving the complete sets of operators. The conventions to label the LO, NLO and NNLO operators in eqs. (2.1), (2.4) and (2.5) follow closely the notations in refs. [12, 13, 46]. Unless it is explicitly stated, the LECs will correspond to U(3) ~PT and must not be confused with those in SU(3) ~PT. The matching between these two EFTs can be found in ref. [12]. The terms Lj are denoted as j in refs. [13, 14].
Comparing the U(3) and SU(3) theories one can observe that some terms have been reshu ed in the expansion of the U(3) Lagrangian. For example, the Li=4,5,6,7,8 terms
are NLO in SU(3) ~PT, but they are now split into NLO and NNLO in the expansion
4
(see eqs. (2.4) and (2.5)). We have several additional new operators, namely the last one in eq. (2.1), the i=1,2 in eq. (2.4) and the v(2)2, L18, L25 terms in eq. (2.5), that are absent in the SU(3) ~PT case. Finally, the chiral loops start contributing at NNLO in the
expansion, while they appear at NLO in the conventional SU(3) case.
2.2 The - mixing at NNLO in expansionNext we calculate the - mixing order by order in the expansion. In literature, there are two bases to address the - mixing, namely the singlet-octet basis with 0 and 8, and the quark-avor basis with q and s. The relations between elds in these two bases are
8 0
!
=
q13
q2 3
q2 3
q s
!
. (2.6)
In the largeNC limit where the U(1)A anomaly is absent, q and s are the mass eigenstates and they are generated by the axial-vector currents with the quark avors qq = (u + d d)/2 and ss, respectively. The two bases are related to each other through an orthogonal transformation and provide an equivalent description for the - mixing.
As noticed in refs. [45, 47], when doing the loop calculations with and , it is rather cumbersome to work with the 0 and 8 states. The reason is that at leading order the Lagrangian in eq. (2.1) gives the mixing between 0 and 8, and the mixing strength is proportional to m2K m2, which in the expansion is formally counted as the same order
as the diagonal terms in the mass matrix for 0 and 8. As a result, the insertion of the 0- 8 mixing in the chiral loops will not increase the order of the loop diagrams. This makes the loop calculation technically much more complicated, as one needs to consider the arbitrary insertions of the 0- 8 mixing in the chiral loop diagrams. Nevertheless, refs. [45, 47] provide a simple recipe to handle this problem by expressing the Lagrangian in terms of the and states which result from the diagonalization of 0 and 8 at leading order in . The main di erence is that the mixing between and is now at least a NLO e ect in , while the 0- 8 mixing was appearing at LO. The relation between the LO mass eigenstates
and
and the singlet-octet basis is given by the mixing angle :
!
JHEP06(2015)175
q1 3
= c s s c
!
8 0
!
, (2.7)
with c = cos and s = sin . The LO mixing angle and masses of and are given by the leading order Lagrangian L(0) in eq. (2.1) (see e.g. ref. [45]):
m2 = M20
2 + m2K q
M40 4M
20 2
3 + 4 42 , (2.8)
m2 =
M20
2 + m2K + q
M40 4M
20 2
3 + 4 42 , (2.9)
sin =
1
s1 + 3M20 2 2 +
p9M40 12M20 2 + 36 4
2
32 4
, (2.10)
with 2 = m2K m2. Here mK and m denote the LO kaon and pion masses, respectively.
5
When higher order corrections are taken into account, the LO diagonalized and will get mixed again. Up to the NNLO, a general parametrization of the bilinear terms involving the and states can be written as
L =
1
2 @@ @@ +
2
2 @@ @@ + 3 @@ @@
+1 +
2 @ @ +
1 +
2 @ @ + k @ @
m2 + m2
2
m2 + m2
2 m
2 , (2.11)
where the is contain the NLO and NNLO corrections. Here these operators must be understood as the terms of the e ective action that provide the pseudoscalar meson self-energies. The higher-derivative terms j=1,2,3 in the rst line of eq. (2.11) are exclusively contributed by the O(p6) operator C12 in eq. (2.5), which belongs to the NNLO Lagrangian.
The remaining is receive contributions from the NLO operators in eq. (2.4), the NNLO ones in eq. (2.5) and the one-loop diagrams, which contribute at NNLO. Their explicit expressions can be found in appendix A.
At leading order, there is only the mass mixing term from eq. (2.1) whereas at NLO and NNLO one has to deal in addition with the kinematic mixing terms in eq. (2.11), apart from the mass mixing. The physical states of and can be obtained from the perturbative-expansion ( -expansion) in three steps: as a rst step, we eliminate the higher-derivative terms through the eld redenitions of and ; then we transform and rescale the elds resulting from the rst step in order to write the kinematic terms in the canonical form; after the preceding two steps, there is only the mass mixing term left, which is straightforward to handle.
In the rst step, we make the following eld redenitions for the and states
+ 1
+ 2 , + 2
JHEP06(2015)175
+ 3 , (2.12)
with the dAlembert operator @@. After some algebra manipulations, it is straight
forward to obtain
1 =
1
2 , 2 =
3
2 , 3 =
2
2 , (2.13) so that the three higher-derivative terms in eq. (2.11) will be eliminated. Notice that the 1,2,3 are NNLO, i.e., O( 2). Substituting the eld redenitions from eq. (2.12) into the
general mixing structure in eq. (2.11) and keeping the terms up to NNLO, the resulting bilinear Lagrangian reads
L =
1 + + m2 1
2 @ @ +
1 + + m2 2
2 @ @ +
k + 3
2 (m2 + m2 )
@ @
m2 + m2
2
m2 + m2
2 m
2 . (2.14)
In the second step, we need to eliminate the kinematic mixing term in eq. (2.14), and then to rescale the elds to have them in the canonical forms. This can be done
6
perturbatively. In the nal step, we take care of the mass mixing term. The last two steps can be achieved through the following eld transformations
!
= cos sin
sin cos
!
1 + A B
B 1 + C
!
!
, (2.15)
with , the physical states and
A = 2 +
m2 1
JHEP06(2015)175
2
2,NLO
8
2k,NLO
8 ,
B = k
2 +
3
4 (m2 + m2 )
,NLO k,NLO
8
,NLO k,NLO 8 ,
8 , (2.16)
where ,NLO, ,NLO, k,NLO stand for the NLO parts of the three quantities respectively. We point out that , , k receive both NLO and NNLO contributions, while 1, 2, 3 are only contributed by the NNLO e ect, which is the C12 operator in eq. (2.5). Comparing with the NLO results in eq. (15) from our previous paper [45], we have generalized the expression to the NNLO case in the present eq. (2.15). Another way to treat the mixing of pseudoscalar mesons in ~PT was also previously studied in ref. [48] and applied to the 0- case up to the two-loop level.
In the practical calculation, it is more often to use the inverse of the relations in eq. (2.15), where the perturbative expansion leads to
!
= 1 + A B B 1 + C
C =
2 +
m2 2
2
2,NLO
8
2k,NLO
!
cos sin
sin cos
!
!
, (2.17)
with
A =
2
m2 1
2 +
3 2,NLO
8 +
3 2k,NLO
8 ,
3
4 (m2 + m2 ) +
B =
k
2
3 ,NLO k,NLO
8 +
3 ,NLO k,NLO
8 ,
C =
2
m2 2
2 +
3 2,NLO
8 +
3 2k,NLO
8 . (2.18)
The appearing in eqs. (2.15) and (2.17) is determined through
tan =
b m2 m2
bm2 , (2.19)
7
with
b m2 = m2
1
2
k + 32 (m2 + m2 ) m2 + m2
+ 1
8 k,NLO ,NLO
5m2 + 3m2
1
2 k,NLO
m2 ,NLO + m2 ,NLO + 18 k,NLO ,NLO
3m2 + 5m2
1
2 m
2,NLO ,NLO + ,NLO
,
bm2 = m2 + m2
m2 + m2 1
+ m2 2,NLO + 34m2 2k,NLO +
14m2 2k,NLO
JHEP06(2015)175
k,NLO m2,NLO ,NLO m2 ,NLO ,
bm2 = m2 + m2 m2
+ m2 2
+ m2 2,NLO +
1 4m2 2k,NLO +
3
4m2 2k,NLO
k,NLO m2,NLO ,NLO m2 ,NLO ,
2m2 =
bm2 +
bm2 r
bm2
bm2
2+ 4
b 2m2 ,
b 2m2 , (2.20)
where i,NLO stand for the NLO parts of i.
In the phenomenological discussions, the popular two-mixing-angle parametrization in the singlet-octet basis [10, 11] takes the form
!
= 1
F
2m2 =
bm2 +
bm2 +
r
bm2
bm2
2+ 4
F8 cos 8 F0 sin 0
F8 sin 8 F0 cos 0
!
8 0
!
. (2.21)
Combining eqs. (2.7) and (2.15), it is straightforward to derive the relations between the four parameters in the two-mixing-angle scheme in eq. (2.21) and the ~PT LECs:
F 28 = F 2
cos( + ) + B sin( ) + A cos cos C sin sin
2
+ sin( + ) + B cos( ) + A cos sin + C sin cos
2
,
F 20 = F 2
sin( + ) + B cos( ) A sin cos C cos sin
2
+ cos( + ) B sin( ) A sin sin + C cos cos
2
,
tan 8 = sin( + ) + B cos( ) + A cos sin + C sin cos
cos( + ) + B sin( ) + A cos cos C sin sin
,
tan 0 =
sin( + ) + B cos( ) A sin cos C cos sin
cos( + ) B sin( ) A sin sin + C cos cos
, (2.22)
where the ~PT LECs are implicitly included in , , A, B and C. Since , A, B, C O( ) or O( 2), at LO one has F8 = F0 = F and one mixing-angle 8 = 0 = .
8
The relations between the physical , states and the quark-avor basis is commonly parametrized as
!
= 1
F
Fq cos q Fs sin s
Fq sin q Fs cos s
!
q s
!
. (2.23)
Combining eqs. (2.6), (2.7) and (2.15), it is straightforward to obtain the parameters in eq. (2.23):
F 2q = 2F 20 + F 28 2
2F0F8 sin( 0 8)
3 ,
2F0F8 sin( 0 8)
3 ,
tan q = 2F8 cos 8 + F0 sin 0
2F0 sin 0 F8 cos 8
,
JHEP06(2015)175
F 2s = F 20 + 2F 28 + 2
tan s = 2F0 cos 0 + F8 sin 8
2F8 sin 8 F0 cos 0
, (2.24)
where at LO in the -expansion one has Fq = Fs = F and q = s = id , with the ideal
mixing id = arcsin
p2/3.
2.3 Insights into previous studies of the - mixing
In the previous subsection we have performed the full computation of the mixing up to NNLO in the expansion. It is interesting to make a brief summary of the assumptions made in previous works, where plenty of mixing formalisms have been proposed to address the - system [14, 30, 4245, 49, 50]. In ref. [49], only the lowest order in the quark masses and 1/NC, i.e. the LO contributions in the expansion, were taken into account. Even though it provided a reasonable rst approximation, it failed to give an accurate description of the experimentally observed mass ratio m2/m2. The O(p2) contributions
were studied up to NLO in 1/NC in ref. [50] (including the terms in eq. (2.1) and 1 and 2 in eq. (2.4)), perfectly explaining the experimental value of m2/m2. However, it turned
out to be inadequate to give a proper value for the - mixing angle. On the other hand, the authors in refs. [4244] went up to NLO in the p2 expansion but keeping just the LO in 1/NC (including the terms in eq. (2.1) and L5 and L8 in eq. (2.4)). Both the - mixing angle and the ratio FK/F were qualitatively reproduced in this case. The full set of NLO contributions in the -expansion (i.e., the e ects up to NLO both in 1/NC and p2) was analyzed in ref. [14], together with the mixing angle and the , K, and axial-vector decay constants. In ref. [45], the contributions from the tree-level resonance exchanges and partial NNLO e ects, e.g. the loop diagrams, were considered for the masses of and .
In this work, we generalize the discussions up to the full NNLO study in the -expansion and confront our theoretical expressions with the very recent lattice simulation data and the phenomenological inputs from the two-mixing-angle scheme.
Reference [30] introduced a quark-model inspired approach to the - mixing, which is commonly referred as the FKS formalism and used in many phenomenological analyses [51].
9
The essence of the FKS formalism is the assumption that the axial decay constants in the quark-avor basis takes the same mixing pattern as the states
F q F s
F q F s
!
= cos sin
sin cos
!
Fq 0
0 Fs
!
, (2.25)
where the decay constants are dened as the matrix elements of the axial currents
h0|Aa(0)|P (k)i = i2F
aP k , (a = q, s; P = , ) ,
Aq = 1
2 5u +
d 5d
, As = s 5s . (2.26)
From another point of view, the pattern of eq. (2.25) employed in the FKS formalism relies on the assumption that there is no mixing between the decay constants of the avor states q and s. In the ~PT framework, the physical masses and decay constants can be obtained from the bilinear parts of nonet elds in the e ective action with the correlation function of two axial currents. Since the correlation function is the second derivative with respect to the axial-vector external source a, and a always appears in the Lagrangian together with the partial derivative @ as shown in eq. (2.2), the absence of the mixing for the q and s decay constants in eq. (2.25) implies that there are no kinematic mixing terms for the quark-avor states q and s in the FKS formalism. In fact, the assumption in ref. [42] is in accord with the FKS formalism. This can be simply demonstrated by expanding the chiral operators considered in refs. [42, 43], i.e. those in eq. (2.1) and L5, L8
in eq. (2.4), up to quadratic terms in q and s.1 No kinematic mixing terms for the q and s elds result from these chiral operators. This also conrms the nding in ref. [44] that only when the NLO of 1/NC operator is excluded the FKS formalism is recovered with their chiral Lagrangian calculations.
Since general terms up to NNLO in expansion are kept in our discussion, unlike in the previous works [14, 4245, 49, 50] where di erent assumptions, such as the preference of the higher order p2 and 1/NC e ects, are made, it is important and interesting for us to justify these assumptions in later discussions.
2.4 Masses and decay constants of pion and kaon up to NNLO in expansion
The NLO expression of the pion decay constant in the expansion reads
F = F
1 + 4L5 m2
F 2
, (2.27)
or, up to the precision considered, one can also use the physical F in the expression inside brackets,
F = F
1 + 4L5 m2
F 2
JHEP06(2015)175
. (2.28)
1Our L5 and L8 operators correspond to the 2 and 1 terms in refs. [4244], respectively. The term in the previous references corresponds to our 2 operator in eq. (2.4). The term, though introduced from the beginning in these references, is dropped in their later discussions, since it is 1/NC suppressed.
10
The di erences between eqs. (2.27) and (2.28) are NNLO e ects. We mention that at a given order there is always ambiguity in choosing the renormalized quantities in the higher order expressions. In contrast, there is formally no ambiguity in the expressions in terms of the quantity F , which is the pNGB axial decay constant in the chiral and large NC limits. For example, if we limit our analysis up to NLO, formally, it is equally good to use F or FK in the denominators of the NLO part in eq. (2.28), since the di erence is beyond the NLO precision. A typical solution in the chiral study is to express the quantities, such as m, F, mK, FK, in terms of the renormalized F in the higher order corrections, as done in the two-loop calculations in SU(3) ~PT [52]. We follow this rule throughout the current work to estimate the uncertainty due to the truncation of the expansion when one works at a given order in perturbation theory. We mention that the notation of m2 in the above equations stands for the renormalized pion mass squared and the leading order mass squared is denoted by m2. Notice the LO pion mass squared m2 is the one that is linear in the quark masses. The expressions relating m2 and m2 will be discussed below.
Similarly up to NNLO, we can either use F or F in the NLO and NNLO expressions for other quantities such as FK and the is in eq. (2.11). In the NNLO expressions, the di erence between using F or F in the denominators is a next-to-next-to-next-to-leading order e ect (N3LO). Since in this work we study lattice simulation data up to pion mass of 500 MeV, the convergence of the chiral series is expected to be much slower than that in the physical case with m = 135 MeV. Therefore it is a priori not trivial to judge whether the two approachesusing 1/F 2 and 1/F 2 are numerically equivalent or the lattice data prefer one of them. Indeed in ref. [53], it is already noticed that to use F or F could cause some noticeable e ects. We will use the di erence between both approaches as an estimate of the truncation error at a given order in .
We take the pion decay constant as an example to illustrate the di erences of using F and F in the higher order expressions. Using F in the higher order corrections, its expression reads
F = F
1 + 4L5 m2
F 2 + 4L4m2 + 2m2KF 2 + (24L25 64L5L8)m4
F 4 + (8C14 + 8C17) m4F 2
+A0(m2)162F 2 +
A0(m2K) 322F 2
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. (2.29)
The one-point loop function A0(m2) is calculated in dimensional regularization within the MS 1 scheme [1, 2] and it reads
A0(m2) = m2 ln
m2
2 , (2.30)
with the renormalization scale xed at 770 MeV throughout. Using eq. (2.27) to replace F by F in the NLO and NNLO corrections, the resulting form is
F = F
1 + 4L5 m2
F 2 + 4L4m2 + 2m2KF 2 + (56L25 64L5L8)m4
F 4 + (8C14 + 8C17) m4F 2
+A0(m2)162F 2 +
A0(m2K) 322F 2
. (2.31)
11
In the expansion, the expressions for a physical quantity with F or F in the higher order chiral corrections di er only for the L5Lj=5,8 and L5 j=1,2 terms, since the di erences by replacing F by F are originated from the NLO expressions of F in eq. (2.28) and we only retain terms up to NNLO in this work. It is clear that the di erence between eqs. (2.29) and (2.31) is the L25 term. Notice that in the expansion scheme, the terms like L5L4 are
N3LO and will be dropped throughout the article.
The corresponding expression for the kaon decay constant when one uses F to express the NLO and NNLO corrections reads
FK = F 1+4L5 m2KF 2 + 4L4m2+ 2m2KF 2 + (24L2564L5L8)m4KF 4 + 8C142m4K 2m2Km2+m4F 2
+8C17 m2(2m2K m2)F 2 +
3A0(m2) 1282F 2 +
3A0(m2K)
642F 2 +
3c2A0(m2)
1282F 2 +
3s2A0(m2) 1282F 2
. (2.32)
On the other hand, expressing the NLO and NNLO contributions in terms of F yields
FK = F 1 + 4L5 m2KF 2 + 4L4m2 + 2m2KF 2 + 8L253m4K + 4m2Km2
F 4 64L5L8 m4K F 4
+8C14 2m4K 2m2Km2 + m4F 2 + 8C17
m2(2m2K m2)
F 2
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+3A0(m2)1282F 2 +
3A0(m2K)
642F 2 +
3c2A0(m2)
1282F 2 +
3s2A0(m2) 1282F 2
. (2.33)
The expanded expression for the ratio of FK/F in terms of F up to NNLO in expansion, takes the form
FK
F = 1 + 4L5
m2K m2F 2 + 8L25
3m4K 2m2Km2 m4F 4 + 64L5L8
m4 m4K F 4
+16C14 m4K m2Km2
F 2 + 16C17
m2Km2 m4 F 2
3s2A0(m2)
1282F 2 . (2.34)
When expressing the previous result in terms of F, it reads
FK
F = 1 + 4L5
m2K m2
F 2 + 8L25
5A0(m2) 1282F 2 +
A0(m2K)
642F 2 +
3c2A0(m2)
1282F 2 +
3m4K + 2m2Km2 5m4
F 4 + 64L5L8
m4 m4K F 4
+16C14 m4K m2Km2
F 2 + 16C17
m2Km2 m4 F 2
5A0(m2) 1282F 2 +
A0(m2K)
642F 2 +
3c2A0(m2)
1282F 2 +
3s2A0(m2)
1282F 2 , (2.35)
which di ers from eq. (2.34) in the L25 term.
The pion squared mass up to NNLO is given by
m2 = m2 + m2,NLO + m2,NNLO , (2.36)
12
with
m2 = 2B
bm , (2.37) m2,NLO = 8(2L8 L5)m4
F 2 , (2.38)
m2,NNLO = 8(2L6 L4)m2(2m2K + m2)
F 2
64(L25 6L5L8 + 8L28)m6 F 4
16(2C12 + C14 + C17 3C19 2C31)m6F 2 +
m2(c2 22cs + 2s2)A0(m2) 962F 2
+
m2(2c2 + 22cs + s2)A0(m2 )962F 2
m2A0(m2)322F 2 . (2.39)
When expressing the renormalized m in terms of F, the only di erences are the L5L8 and L25 terms in eq. (2.39) and the other parts are the same as in eq. (2.36) with the explicit replacement of F by F in eq. (2.27). Therefore we only give the di erent parts for simplicity when expressing in terms of F and they read
m2,(F),L5L8 ,L
25
= 128(4L5L8 L25)m6
F 4 . (2.40)
The mass squared for kaon up to NNLO is provided by
m2K = m2K + m2,NLOK + m2,NNLOK (2.41)
with
m2K = B(
bm + ms) , (2.42) m2,NLOK =8(2L8 L5)m4K
F 2 , (2.43)
m2,NLOK =
8(2L6 L4)m2K(2m2K + m2)
F 2
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64(L25 6L5L8 + 8L28)m6K F 4
32C12m6K
F 4 +
32C31m6K
F 2 +
16C17m2Km2(2m2K + m2) F 2
16C14m2K(2m4K 2m2Km2 + m4)
F 2 +
48C19m2K(2m4K 2m2Km2 + m4) F 2
[c2(3m2 + m2) + 22cs(2m2K + m2) 4m2Ks2]A0(m2)
1922F 2
[4c2m2K + 22cs(2m2K m2) + (3m2 + m2)s2]A0(m2 )
1922F 2 . (2.44)
When expressing eq. (2.41) in terms of F, the di erences are the L5L8 and L25 terms in eq. (2.44) and the new expressions are
m2,(F),L5L8 ,L
128L5L8m4K(3m2K + m2)
F 4 . (2.45)
Notice that the masses of pion and kaon appearing in NLO and NNLO parts in the above equations correspond to the renormalized quantities, instead of their LO expressions. In addition, this gives the quark mass ratio relation ms/ m = 2m2K/m2 1.
13
K =
64L25m4K(m2K + m2)
F 4 +
25
When performing the chiral extrapolation of the lattice data, instead of the renormalized m2K as in the previous equations, it is convenient to use the LO kaon mass squared in the higher order corrections. In this way, we do not need to iteratively solve eq. (2.41) in order to give the value of mK for a given m. The result in terms of mK in the NLO and
NNLO expressions becomes
m2,LatK = m2K + m2,LatNLOK + m2,LatNNLOK , (2.46)
with
m2,LatNLOK =
8(2L8 L5)m4KF 2 , (2.47)
m2,LatNNLOK =
8(2L6L4)m2K(2m2K +m2)
F 2 +
64(L252L5L8)m6KF 4
32C12m6K
F 4 +
32C31m6K F 2
+16C17m2Km2(2m2K + m2)F 2
16C14m2K(2m4K 2m2Km2 + m4) F 2
+48C19m2K(2m4K 2m2Km2 + m4)F 2
[c2(3m2 + m2) + 22cs(2m2K + m2) 4m2Ks2]A0(m2)
1922F 2
[4c2m2K + 22cs(2m2K m2) + (3m2 + m2)s2]A0(m2 )
1922F 2 . (2.48)
When expressing eq. (2.46) in terms of F, the di erences are the L5L8 and L25 terms in eq. (2.48) and the new expressions are
m2,Lat,(F),L5L8 ,L
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64L5(L5 2L8)m4K(m2K m2)
F 4 . (2.49)
When confronting with the lattice data, we only consider the simulated points with the physical strange-quark mass, i.e. the lattice ensembles that when extrapolating to the physical pion masses lead simultaneously to physical kaon masses. In this case, we can express the LO kaon mass squared as
m2K =B(mPhys +
bm) = B(mPhys+ bmPhy)B
bmPhy+B
K =
25
bm = m2,PhyK
m2,Phy
2 +
m2
2 , (2.50)
where m2,PhyK and m2,Phy can be obtained through eqs. (2.36) and (2.41) by substituting the physical masses of , K, , in the NLO and NNLO expressions. For m2, which varies in the lattice simulation, we can extract its value by using eq. (2.36). In this case, m in
eq. (2.36) takes the value from lattice simulation, and mK, m, m , which only appear in the NNLO part, can be approximated by their LO expressions.
In the above discussions, we have distinguished the situations of using 1/F 2 and 1/F 2 in the higher order corrections for various observables. Similarly, we can also generalize the discussions by replacing the renormalized masses (m and mK) with the LO ones (m
and mK) in the higher order corrections. We take the observables F and FK as examples to illustrate the di erences. The renormalized m and mK have been used in eqs. (2.29)
14
and (2.32) for F and FK with 1/F 2 in the higher order terms, respectively. After replacing m and mK in eqs. (2.29) and (2.32) with their expressions in terms of the LO masses m and mK through eqs. (2.36) and (2.41) respectively, the corresponding expressions are found to be
F = F
1+4L5 m2
F 2 +4L4m2+2m2KF 2 8L25m4
F 4 +(8C14+8C17)m4
F 2 +A0(m2)162F 2 +A0(m2K) 322F 2
FK = F
. (2.52)
As in the discussion of 1/F 2 versus 1/F 2 up to the NNLO precision, the expressions for a specic observable by using the renormalized masses m, mK and the LO m, mK only di er in the terms like LiLj, being Li and Lj the NLO LECs in eq. (2.4). This can be clearly seen when comparing eqs. (2.29) and (2.51). E.g. the di erences caused by using the renormalized masses and the LO ones are the L25 and L5L8 terms, apart from the explicit replacement of m and mK by m and mK respectively. Similar rules are also applied to eqs. (2.32) and (2.52).
To replace m, mK by m and mK in the NLO and NNLO corrections in eq. (2.36), the only changes happen for the LiLj terms and the corresponding new expressions read
25 ,L28
,
(2.51)
1 + 4L5 m2KF 2 + 4L4m2 + 2m2KF 2 8L25m4KF 4 + 8C142m4K 2m2Km2 + m4 F 2
+8C17 m2(2m2K m2)F 2 +
3A0(m2) 1282F 2 +
3A0(m2K)
642F 2 +
3c2A0(m2)
1282F 2 +
3s2A0(m2) 1282F 2
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= 64L5(L5 2L8)m6
F 4 . (2.53)
In principle, we should also present the results expressed with the LO masses m, mK
and the renormalized decay constants F, FK, which can be straightforwardly obtained by substituting the relations in eqs. (2.36) and (2.41) into the corresponding observables. We consider the expressions given in terms of the renormalized masses and 1/F 2 as our preferred ones in this work. The reason to choose the renormalized masses is for practical purpose, since in lattice simulations the di erent observables are typically given as functions of the renormalized m2. Also most of the chiral studies choose to express the quantities with the renormalized masses in the higher order corrections, such as in refs. [52, 53].
Following this rule we consider the results with the renormalized masses and 1/F 2 as an estimate of systematic errors due to the truncation of the expansion when one works at a given order in perturbation theory. While for the case with the LO masses, we shall also comment the results in the following numerical discussions.
3 Phenomenological discussions
The big challenge in the present general discussions on the - mixing is the determination of the unknown LECs in eqs. (2.1), (2.4) and (2.5). The recent lattice simulations on the light pseudoscalar mesons provide us valuable sources to constrain these free parameters. The considered lattice simulations include the m dependences of the masses of , [1519]
15
m2,(m,mK,F),L5L8 ,L
and kaon [22, 23], and the , K decay constants [22, 23] and their ratios [24]. Moreover, relevant phenomenological results and experimental data will be also included to constrain the LECs.
Since we do not consider the isospin violating e ects, we will take the values for the physical pion and kaon masses in the isospin limit from ref. [54], where the corrections from the electromagnetic contributions are removed,
m = 135.0 MeV , mK = 494.2 MeV . (3.1)
These values will be used in later chiral extrapolations, while for the physical masses of and and the decay constants of pion and kaon, we will take their world-average values from ref. [55].
In order to show the results step by step, we present the discussions in the following sections split in three parts: we consider ts performed at leading order, next-to-leading order and next-to-next-to-leading order.
3.1 Leading-order analyses
At leading order, the - mixing is described by one free parameter, namely the singlet 0 mass M0 in eq. (2.1) and the explicit expressions for the masses and mixing angle are given in eqs. (2.8), (2.9) and (2.10). At this order, the , K decay constants are degenerate and given by their chiral and large NC limits, i.e. F = FK = F . Therefore we shall not take the lattice simulations of the decay constants into account for the LO discussion as they clearly show the need of higher order corrections for a suitable description. Also at leading order, F will not enter the masses and mixing angle, as shown in eqs. (2.8), (2.9) and (2.10). As a result of this, we do not need to distinguish the two situations with F or F discussed previously in the expressions of di erent observables. Apart from the lattice simulation data, we also t the physical values of the and masses. Nonetheless, tting the physical masses with the experimental precision at the level of several hundred-thousandth is too ambitious. Since the ultimate goal of the present work is the NNLO study, the ballpark estimate of our theoretical uncertainty, starting from the N3LO part, should be around 3%. This value is obtained from the general rule that each higher order correction in expansion, either the SU(3)-avor breaking or the 1/NC e ect, is around 30%. In fact, the estimated three-percent uncertainty is also similar to the typical error bars reported in many lattice simulations, in the range from 3 10% [1519]. Consistently, we assign a
1% uncertainty to the physical values of m and m in the ts.The value of the singlet mass M0 from the LO t is
M0 = (835.7 7.5) MeV . (3.2) The physical masses for the , and their LO mixing angle from the t are found to be
m = (496.4 1.3) MeV , m = (969.8 5.8) MeV , = 18.9 0.3 . (3.3) The resulting plots can be seen in gure 1. We verify that if the physical masses are excluded in the t, M0 = 813 11 MeV results. If we only include the physical masses
16
JHEP06(2015)175
JHEP06(2015)175
Figure 1. The masses of and from the LO t. The left most two points correspond to the physical masses. The remaining lattice simulation data are taken from refs. [18, 19] (ETMC), [17] (UKQCD), [16] (RBC/UKQCD), [15] (HSC), where we only take into account the simulation points with m < 500 MeV. The shade area surrounding each curve stands for the statistical uncertainty from the t.
and exclude the lattice simulation data in the t, M0 = 859 11 MeV is obtained. These
determinations of M0 lie within the broad range summarized in ref. [51] and are quite close with the commonly used values of M0 = 850 MeV [51]. Taking into account the large uncertainties of the lattice simulation data, specially for m , and the concise formalism of the LO mixing, it is impressive that the lattice simulation data can already be qualitatively described with the LO analysis, as shown in gure 1. This also indicates that the higher order mixing e ects can only give moderate corrections to the masses of and .
Nevertheless, in order to describe the lattice data more accurately, specially the masses, the chiral corrections beyond the leading order are needed. For the physical masses, it has also been shown that the LO description fails to explain the mass ratio of and accurately enough [49]. Therefore it is essential to generalize the discussions to
NLO and NNLO in order to achieve a precise description both for lattice simulations and physical data.
3.2 Next-to-leading order analyses
At next-to-leading order, in addition to the parameter M0 at leading order, there are ve additional free parameters: the decay constant F at chiral and large NC limits, and the four NLO LECs L5, L8, 1 and 2 in eq. (2.4). At this order, as well as at next-to-
17
next-to-leading order, one can rewrite the chiral expansion of the observables in various equivalent ways up to the perturbative order in under consideration. In the following discussion, we will perform two types of ts: one using F in the theoretical NLO and NNLO expressions and the other employing F, as discussed in section 2.4. Since the di erences of the theoretical expressions used in the two types of ts are beyond the considered precision, the variances of the outputs from the two ts can be considered as systematic errors from the theoretical models by neglecting higher order contributions. In the following, we will explicitly present the t results by using F in the theoretical expressions, which is the most straightforward option, as discussed in section 2.4. The outputs of the ts with the theoretical formulas expressed in terms of F will be used to estimate the systematic errors: the di erence between the central values of the two types of ts will be used to estimate the truncation uncertainty due to working just up to a given order in the -expansion, providing the second error for each quantity in the following tables.
In refs. [4244], it is argued that at each chiral order, the leading NC e ects are dominant, or in other words that the 1 and 2 terms are assumed to be much less irrelevant than the L5 and L8 terms in the NLO expansion. This assumption has been more or less conrmed when focusing on the masses of , and the LO mixing angle at the physical points [4244]. In ref. [45], the local higher order LECs were estimated by the tree-level resonance exchanges and it was found that with those LECs 2 seems to be more important than 1 when focusing on the physical masses for and . It is interesting to check how these assumptions work when including the lattice simulations and the phenomenological results of the two-mixing-angle parameters, which are not considered in refs. [4245]. Di erent sets of ts to the lattice data and phenomenological inputs from the two-mixing-angle scheme are performed either by xing i=1,2 to zero or releasing their values, in order to reexamine the assumptions. Interestingly we do not nd qualitative changes between the ts with xed i=1,2 = 0 and the ones with free values for these parameters. This tells us that indeed the 1 and 2 terms do not signicantly improve the t results, even after taking into account the lattice simulations. Nevertheless, we nd that these two terms are quite important to reproduce the phenomenological mixing angles 0 and 8 in the ts where M0 is xed at its LO value. If M0 is released in the ts we nd that including 1 and 2 improves the descriptions of m from lattice simulations. Therefore, we will not further discuss ts with 1 and 2 set to zero in the following. Instead, we focus on the results given in table 1 with all the four NLO LECs in the ts, namely L5,
L8, 1 and 2 in eq. (2.4).
For the parameter M0, we take two strategies to estimate its value in NLO analysis. In one of them we x M0 = 835.7 MeV from its LO determination (NLOFit-A) and in the other case we free its value for the NLO t (NLOFit-B). These two NLO ts are given in table 1. The rst error bar for each tted parameter corresponds to the statistical one from the ts and the second error bar is estimated from the variation of the ts between those using F and F in the NLO (and later also NNLO) theoretical expressions. From the two ts shown in table 1, one can see that releasing M0 in the ts barely changes the t quality with respect to the cases when its value is xed, although there are slight variations in the determinations of M0 and 2.
18
JHEP06(2015)175
Concerning the results of the LECs in table 1, the resulting values for F from the two ts are quite compatible and close to the physical pion decay constant. For 1 and 2, their values are poorly known in literature and it is helpful to compare our values with the following estimate for their ranges: we take the LO determination M0 = 835.7 MeV, and we then separately include the 1 and 2 terms in the - mixing and vary their values to obtain new results for m and m with the physical m. Since the 1 and 2 terms are
NLO 1/NC e ects, it is reasonable to assume that their corrections to m2 or m2 should be
at most around 30% of the LO results. In this way we can set up conservative and rough estimates for the ranges of 1 and 2, which are found to be
| 1| < 0.4 , | 2| < 0.7 . (3.4)
The resulting magnitudes of 1 in our ts are tiny and consistent with zero, as shown in table 1. For the parameter 2, our determinations lie within the ranges estimated in eq. (3.4). Its value, specially the one from NLOFit-A, is close to the one used in ref. [56], where the mixing was discussed at next-to-leading order. However the determinations for 2 in table 1 become much more precise than those given in refs. [45, 47], where the lattice simulations for m and m are not included, indicating the usefulness of incorporating the lattice data in the U(3) ~PT study. Our determinations of L5 and L8 are in good agreement with the leading NC predictions from resonance chiral theory [57], the SU(3) one-loop results in refs. [1, 2] and the one-loop resonance chiral theory determination for L8 [58, 59]. But the values here are clearly larger than those from the recent two-loop determinations [26, 27], the results from K scattering in the scalar channels [60], and the one-loop resonance chiral theory estimates for L5 [29]. The discrepancies of L5 and L8, comparing with the recent two-loop determinations [26, 27], can be eliminated once the
O(p6) LECs are taken into account, as we will show in the NNLO discussion.
The values of the parameters in the two-mixing-angle scheme and the mass ratio of strange and up/down quarks resulting from the ts are given in table 2. Similarly, the rst error bar for each quantity is the statistical error and the second one corresponds to the systematic error, which is obtained in the same way as the one in table 1. Notice that these inputs have already been satisfactorily reproduced in NLO analyses.
The other quantities in the ts are presented in gures 2, 3, 4 and 5, together with the lattice simulation data and the experimental inputs. We nd that the nal outputs from NLOFit-A and NLOFit-B are quite similar, so only the plots from NLOFit-B are given explicitly. The shaded area surrounding each curve corresponds to the statistical error band for each quantity. In gure 2, we show the resulting gures from NLOFit-B for the masses of and . In gures 3, 4 and 5, we show the corresponding plots for m2K, F,K and FK/F as functions of m2, respectively.
3.3 NLO ts focusing on the masses
In this section, we present another kind of NLO ts by focusing on the masses of , , K
and excluding the decay constants F, FK and their ratio. This kind of discussion is well motivated, since it is known that the NNLO corrections in counting, such as the LEC
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JHEP06(2015)175
NLOFit-A NLOFit-B ~2/(d.o.f) 481.2/(76-5) 477.7/(76-6) M0 (MeV) 835.7* 767.331.532.3
F (MeV) 92.10.20.6 92.10.20.6
103 L5 1.450.020.30 1.470.020.29
103 L8 1.000.070.10 1.080.050.04
1 0.020.050.06 -0.090.080.02
2 0.250.060.02 0.140.070.03Table 1. Parameters from the NLO ts. The meaning of di erent notations to label di erent ts are explained in detail in the text. In the row of M0, the columns with 835.7* denote the t results by xing the value of M0 from its LO determination. The rst error bar for each parameter is the statistical one given by the ts and the second one corresponds to the systematic error. The way to estimate the systematic error is explained in detail in the text.
Parameters Inputs NLOFit-A NLOFit-B F0 (MeV) 118.0 16.5 104.92.90.3 99.73.61.6
F8 (MeV) 133.7 11.1 113.20.34.4 113.50.34.2
0 (Degree) -11.0 3.0 -7.22.11.3 -10.62.40.1
8 (Degree) -26.7 5.4 -21.52.23.9 -25.42.62.3
ms/
bm 27.5 3.0 22.60.80.6 21.90.61.2 Fq (MeV) 106.0 11.1* 94.11.91.7 90.62.40.4
Fs (MeV) 143.8 16.5* 122.31.25.1 120.91.25.5
q (Degree) 34.5 5.4* 40.43.13.6 35.03.71.6
s (Degree 36.0 4.2* 39.91.72.2 37.21.81.1Table 2. The outputs from NLO ts. Notice that Fq, Fs, q and s are not the phenomenological inputs in the ts, since they are related to F0, F8, 0 and 8 through eq. (2.24). The phenomenological values for the mixing parameters are taken from ref. [21] and we triple the error bands here in order to make a conservative estimate. The input of ms/
JHEP06(2015)175
bm is taken from the FLAG working group in ref. [54] and we assign the 10% error bar as done in ref. [26]. For the error bars of each quantity, the rst one corresponds to the statistic error and the second one is for the systematic error, which are explained in detail in the text.
L4, are important to simultaneously describe F and FK [2628]. But this LEC is absent in NLO study. We have also provided another independent conrmation on this nding in gure 4, where one can see that the decay constants of pion and kaon are poorly reproduced at next-to-leading order in expansion. When only focusing on the , and K masses and the ratio ms/ m at next-to-leading order the parameter F can not be resolved, because it always appears in the form L5/F 2 or L8/F 2. We will x its value to F = 90 MeV, close to the values given in table 1. For the mixing parameters we consider the mixing angles of 0 and 8, but exclude the constants F0 and F8. This is because F0 and F8 are dependent on the parameter F and should be determined together with F and FK. For simplicity in later discussion, we call the ts performed in this section as the mass-focusing type throughout.
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Figure 2. The masses of and from the NLO and NNLO ts. The left most two points correspond to the physical masses. The remaining lattice simulation data are taken from refs. [18, 19] (ETMC), [17] (UKQCD), [16] (RBC-UKQCD), [15] (HSC), where we only take into account the points with m < 500 MeV. The shaded areas around the black solid and red dashed lines stand for the statistical error bands from the NLOFit-B and NNLOFit-B ts, respectively. The meaning of notations for di erent lines are explained in detail in the text.
As in the previous section, we present the ts with F in the denominators of the theoretical expressions (e.g. eq. (2.27)) and use the ts with F to estimate the systematic errors. For each case, we perform the ts either by xing M0 at its LO determination (NLOFit-C) or by freeing its value (NLOFit-D). The tted parameters are given in table 3 and the ms/ m ratio and mixing angles are given in table 4. The resulting gures from
NLOFit-C and NLOFit-D are quite similar and we explicitly show one set of them, e.g. NLOFit-D in gures 2 and 3 for the () and kaon masses, respectively.
A signicant di erence between the results in table 1 and the mass-focusing ts in table 3 is that much larger statistical error bars are obtained in the latter case, especially for the LECs L5 and L8, as they are constrained by fewer data. Likewise, there are large systematic errors for the values of L5 and L8 in table 3, indicating a larger truncation uncertainty due to higher orders. We do not see a signicant improvement when freeing the value of M0 in the ts.
3.4 Next-to-next-to-leading order analyses
From the NLO discussions in the previous two sections, we observe that the phenomenological results and the lattice simulations on and states can be reasonably reproduced.
21
JHEP06(2015)175
Figure 3. Kaon mass from the NLO and NNLO ts. The lattice simulation data are taken from RBC and UKQCD [22, 23]. Only the unitary points simulated with the physical strange quark mass are included. The shaded areas around the black solid and red dashed lines stand for the statistical error bands from the NLOFit-B and NNLOFit-B ts, respectively. The meaning of notations for di erent lines are explained in detail in the text.
NLOFit-C NLOFit-D ~2/(d.o.f) 168.8/(44-4) 168.7/(44-5) M0 (MeV) 835.7* 821.543.5
103 L5 1.400.580.75 1.510.680.91
103 L8 0.880.290.35 0.940.340.44
1 -0.060.040.02 -0.090.110.09
2 0.170.190.25 0.180.190.25
Table 3. Parameters from the mass-focusing NLO ts. The meaning of di erent notations to label di erent ts are explained in detail in the text. F is xed at 90 MeV in these ts. The rst error for each parameter corresponds to the statistical one and the second error denotes the systematic uncertainty. See the text for details.
22
JHEP06(2015)175
Figure 4. Pion and kaon decay constants from the NLO and NNLO ts. The left-most points for F and FK correspond to the physical experimental inputs. The remaining lattice simulation data are taken from RBC and UKQCD [22, 23], where we have only included the unitary points simulated with the physical strange quark mass. The shaded area around each curve stands for the statistical error band from the ts. The meaning of notations for di erent lines are explained in detail in the text.
Parameters Inputs NLOFit-C NLOFit-D 0 (Degree) -11.0 3.0 -10.62.43.3 -11.03.42.1
8 (Degree) -26.7 5.4 -25.32.44.4 -26.74.57.1
ms/
bm 27.5 3.0 23.70.30.3 23.60.50.1 q (Degree) 34.5 5.4* 35.61.41.1 34.14.34.1
s (Degree) 36.0 4.2* 37.00.90.7 36.32.22.1
Table 4. The outputs from the mass-focusing NLO ts. See table 2 for the phenomenological inputs. The rst error for each quantity corresponds to the statistical one and the second error denotes the systematic uncertainty. See the text for details.
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JHEP06(2015)175
Figure 5. Ratio FK/F from the NLO and NNLO ts. The left most point corresponds to the experimental input. The remaining lattice simulation data are taken from ref. [24] (BMW). The shaded area around each curve stands for the statistical error band from the ts. The meaning of notations for di erent lines are explained in detail in the text.
This is an important improvement comparing with the LO study, since at this order we only have the conventional one-mixing-angle formalism. The two-mixing-angle formalism only shows up beyond LO. However, observing mK, F, FK and their ratio in gures 3, 4 and 5, it is clear that the NLO analysis is still inadequate. We need to include higher order contributions beyond NLO in order to further improve the descriptions. Moreover, the chiral logarithms predicted by ~PT at one loop start at NNLO in the expansion. Due to their importance in other observables, we consider it is relevant to discuss the impact of these chiral logs.
As in the NLO case, we perform two types of ts, using the NLO and NNLO theoretical expressions given in terms of F and F for various observables. We explicitly present the t results with F in the theoretical expressions and use the alternative ts expressed in terms of F to estimate the systematic errors, due to working up to NNLO in and neglecting higher orders. According to the Lagrangian in eq. (2.5), eleven additional unknown LECs appear at NNLO and there will be seventeen parameters in total for the NNLO study. At the present precision of the lattice simulations and phenomenological inputs, it is impossible to obtain sensible and stable ts if we free all of the seventeen parameters. Therefore, we need to take other independent determinations for some of the LECs in order to proceed the NNLO study.
24
We mention that the state-of-art determinations of the O(p4) LECs in SU(3) ~PT
su er uncertainties from the many poorly known O(p6) LECs [26, 27]. Because of the
large number of barely known O(p6) LECs, it is rather di cult to get conclusive results in
the present two-loop SU(3) ~PT studies [26, 27]. In the present work, there are ve O(p6) LECs, i.e. C12, C14, C17, C19, C31, in eq. (2.5) and we cannot make precise determinations
of these Ci parameters here. Maybe when taking into account the scattering data, one can make more stringent constraints on the Ci LECs in U(3) ~PT. But this is beyond the scope of current work. Instead we take the Ci values from the Dyson-Schwinger-like approach given in ref. [61], where all of the O(p6) Ci at leading NC are predicted. In order
to show the dependences of the nal results on the Ci values, we also perform other ts by using their updated determinations [62]. Like in ref. [26], we multiply the O(p6) Ci from refs. [61, 62] by a global factor and consider as a free parameter in the ts. In this way, we partially compensate the large uncertainties of the Ci parameters.
For the operators proportional to v(2)2, L18 and L25 in eq. (2.5), they are not present in SU(3) ~PT and purely contribute to the - mixing, being irrelevant to the pion and kaon observables. Since the - mixing parameters have already been satisfactorily described in the NLO ts, we do not further include v(2)2, L18 and L25 at NNLO study.2 Their inclusion in the present analysis tend to make the t unstable. Clearly studying more () related observables it would be possible to extract these parameters but this is out of the reach of the present analysis. A global t is too unconstrained, being unstable and producing values of the latter couplings compatible with zero within uncertainties. Then we are left with three O(N0C, p4) operators: L4, L6 and L7, which have corresponding parts
in SU(3) ~PT. Since U(3) and SU(3) ~PT contain di erent dynamical degrees of freedom, the corresponding LECs from the two theories can be di erent. A typical example is the L7 parameter in SU(3) ~PT, which is demonstrated to be dominated by the singlet 0 state [1, 2]. Since in U(3) ~PT the singlet 0 has been explicitly introduced, the value of L7 in this theory can be totally di erent from LSU(3)7 in SU(3) case. While for other O(p4) LEcs, such as Li=4,5,6,8, the di erences between U(3) and SU(3) ~PT are not expected to be as large as the L7 case, since they do not receive the tree-level contributions from the 0 state.
Another subtlety to take into account is that m and m appear in the chiral loops and, at the same time, the nal expressions of m and m depend on the these loops as well. In order to avoid making the complicated iterative procedure to obtain the - mixing parameters, we use the LO formulas for m and m in the chiral loops. The di erences caused by this simple treatment and the strict iterative procedure are beyond the NNLO precision in expansion, since the chiral loops themselves are already NNLO. Our simple solution is also justied by the fact that the LO description of m and m is in qualitative agreement with the lattice simulation data, as shown in section 3.1. Since the
2Indeed, in this work M0 and v(2)2 only enter in the mass Lagrangian in eq. (2.14) explicitly. They always appear combined in the e ective form M20, eff = M20 + 6v(2)2(2m2K + m2), which is the parameter
we are actually extracting. The contribution v(2)2 could be singled out through the study of the 0 0 scattering. However we point out that the anti-correlation between M0 and v(2)2 in general can not be recovered in the present numerical ts, due to the presence of far too many parameters in the problem and the large uncertainties of the lattice simulation data, specially for the determinations of m .
25
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qualitative agreement between the LO formulas and the lattice simulation data requires the value of M0 to be around 835.7 MeV, as given in eq. (3.2), we x M0 = 835.7 MeV in the following discussions. This also helps to stabilize the NNLO ts, with its many free parameters. Other useful criteria to discriminate reasonable ts are the a priori ranges estimated in eq. (3.4), since the ts with large magnitudes of 1 and 2 imply unphysically large corrections to the - mixing parameters and the breakdown of the expansion. In the following we only present the t results that are consistent with eq. (3.4).
With all of the above setups, the values of parameters from the NNLO ts are summarized in table 5. The ts labeled by NNLOFit-A and NNLOFit-B correspond to using di erent values of the O(p6) LECs. For NNLOFit-A, the Ci values are taken from ref. [61]:
C12 = 0.34 , C14 = 0.83 , C17 = 0.01 , C19 = 0.48 , C31 = 0.63 , (3.5)
which are given in units of 103GeV2. For NNLOFit-B, we take their updated O(p6) Ci
values from ref. [62]:
C12 = 0.34 , C14 = 0.87 , C17 = 0.17 , C19 = 0.27 , C31 = 0.46 , (3.6)
in the same units as before.
It is clear that the parameters resulting from ts with di erent Ci inputs slightly di er from one another. We remind that the rst error bar for each parameter in table 5 corresponds to the statistical one directly from the ts and the second error bar stands for the systematic one, which is estimated, as usual, from the variation of the parameter from the NNLO ts with the theoretical expressions in terms of F and those expressed as functions of F.
At NNLO, one has the contributions from the chiral loops and the O(p6) LECs, which
make our determinations in table 5 closer to the recent two-loop results of the SU(3) ~PT LECs, comparing with the NLO determinations in table 1. Some typical trends of the values of parameters from the NLO study in table 1 to the NNLO one in table 5 are summarized now. The axial-vector decay constant F at leading NC and chiral limit is reduced at
NNLO, which is mainly due to the inclusion of L4. Our conclusion is based on the fact that strong correlations between F and L4 always appear, which has been conrmed in previous study [28, 29]. For L5 and L8, we nd that their values are obviously reduced compared to the NLO determination and become closer to the two-loop results in ref. [27]. As mentioned in the former reference, the discussions in the two-loop SU(3) ~PT are sensitive to the value of the 1/NC suppressed LEC L4. The present study provides an independent determination for this parameter and for the 1/NC suppressed LEC L6 as well. We mention that our determinations of L4 have opposite signs with respect to that in ref. [27], which may be the source of the smaller F obtained in that reference. Notice that the present values of L4, L5, L6, L8 are rather compatible with the combinations of 2L8 L5 and 2L6 L4 given
in ref. [63]. Fit solutions with larger 1 and 2 than those in eq. (3.4) (out of the a priori range (3.4)) are discarded: they are not considered as reasonable physical solutions and will not be discussed any further. According to the values of in the two ts, it seems that our study somewhat prefers smaller magnitudes of the O(p6) LECs than those from
26
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the Dyson-Schwinger approach given in refs. [61, 62] and also prefers a global change of sign with respect to eqs. (3.5) and (3.6). We have investigated the impact of tting but releasing one of the O(p6) LECs as an independent parameter (e.g., C14), but no denitive
conclusion could be extracted. These puzzles cannot be resolved here and it is denitely interesting and necessary to further investigate the values of the O(p6) LECs in the future.
The various plots from the NNLO ts are shown in gure 2 for m and m , gure 3 for mK, gure 4 for F and FK, and gure 5 for the ratio FK/F, together with the
NLO results and the lattice simulation data and experimental inputs. The shaded area surrounding each curve represents the statistical error band. The gures from NNLOFit-B are compatible with those from NNLOFit-A within the uncertainties, so we only show the results for the former in gures 2, 3, 4 and 5.
In addition, to demonstrate the e ects by using the LO masses in the higher order corrections, instead of the renormalized ones, we explicitly show the results for m and m
expressed in terms of the LO masses m, mK and 1/F 2 in gure 2, with the lines labeled as NNLOFit-B-m,K. The values of the LECs when plotting these lines are exactly the same as those from the NNLOFit-B column in table 5. In this way, one can directly see the di erences due to the N3LO truncation uncertainty caused by using the renormalized masses and the LO ones at the NNLO level. According to gure 2, we conclude that the di erences for m and m caused by using di erent types of masses in the higher order corrections are rather within the statistical uncertainties from the ts and therefore the di erences should be perfectly compatible within the total uncertainties after taking into account the systematic ones in table 5. We verify that similar conclusions are obtained for other cases. In order not to overload the plots in other gures, we shall not explicitly show the results given in terms of m and mK.
From gures 2, 3, 4 and 5, we observe, when compared with the curves of the NLO study, slight improvements in the reproduction of the masses for , and signicant ones for mK, F, FK and the ratios of FK/F. Moreover the ~2 for the NNLO ts are greatly reduced compared with ~2 for the NLO ones, indicating that the NNLO corrections are important at the present level of precision and essential to simultaneously describe the lattice simulation data and experimental inputs of the light pseudoscalar mesons , K, and .
4 Conclusions
In this article we have performed a thorough study on the - mixing, and axial-vector decay constants for the pion and kaon, up to next-to-next-to-leading order in expansion within U(3) chiral perturbation theory. We have carried on a detailed scrutiny and discussions of our results, which have been carefully compared to other works in literature for the - mixing. A general mixing formalism, including the higher-derivative terms and kinematic mixing cases, has been addressed in detail. The connections between the mixing parameters from the popular two-mixing-angle scheme and the low energy constants from chiral perturbation theory have been established, both for the singlet-octet basis and the quark-avor basis.
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NNLOFit-A NNLOFit-B ~2/(d.o.f) 212.4/(76-9) 231.9/(76-9) F (MeV) 81.71.55.3 80.81.66.1
103 L5 0.600.110.52 0.450.120.78
103 L8 0.250.070.31 0.300.060.30
1 -0.0030.0600.093 -0.040.060.13
2 0.080.110.20 0.140.100.40
103 L4 -0.120.060.19 -0.090.060.23
103 L6 -0.050.040.02 0.030.030.02
103 L7 0.260.050.06 0.360.050.12
-0.590.090.18 -0.760.080.44
Table 5. Parameters from the NNLO ts. In all of these ts, M0 is xed at 835.7 MeV from its LO determination. The meaning of di erent notations to label di erent ts are explained in detail in the text. The rst error bar for each parameter corresponds to the statistical one and the second error denotes the systematic uncertainty. See the text for details.
The considered quantities, including the masses of , and K, the quark mass ratio of ms/
JHEP06(2015)175
bm, the parameters in the two-mixing-angle scheme and the , K decay constants have been confronted with recent lattice simulations and phenomenological inputs. We nd that the next-to-leading-order ts yield satisfactory descriptions for the masses of the three pseudoscalar mesons as functions of m2 and the four mixing parameters (F0, F8, 0, 8), producing in addition reasonable values of low energy constants. Nonetheless, when the and K decay constants are included together with the masses and mixing parameters in the ts, the next-to-leading-order analyses are inadequate and it is necessary to step into the next-to-next-to-leading-order study. Using the O(p6) LECs determinations from a
Dyson-Schwinger-like approach [61, 62] multiplied by a global factor, we are able to achieve a reasonable description for all of the physical quantities considered above and the resulting values for the leading NC O(p4) low energy constants L5 and L8 turn to be compatible with
the very recent two-loop determinations in ref. [27]. Therefore we conclude that the large NC U(3) chiral perturbation theory o ers a concise theoretical framework that is able to simultaneously reproduce accurately the general - mixing and to provide sophisticated enough expressions to describe the chiral extrapolations of the and K decay constants and masses.
Our results are also useful for future phenomenological studies of di erent processes involving and . Combining eq. (2.21) or eq. (2.23) with table 6, one can directly nd the relations between the physical states , and the octet-singlet bases 8, 0 or the quark-avor bases q, s. These relations are consistent with the requirements from the recent lattice simulations and phenomenology.
Finally, it is worthy to remark that some of the parameters in our best analysis (NNLOFit-B) in table 5 have been determined with relatively small errors. For instance,
28
Parameters Inputs NNLOFit-A NNLOFit-B F0 (MeV) 118.0 16.5 108.01.53.6 109.11.35.9
F8 (MeV) 133.7 11.1 124.71.28.7 126.51.211.8
0 (Degree) -11.0 3.0 -6.81.12.6 -6.80.93.7
8 (Degree) -26.7 5.4 -26.81.10.2 -27.91.01.4
ms/
bm 27.5 3.0 27.00.60.4 29.40.40.6 Fq (MeV) 106.0 11.1* 92.81.11.2 92.71.01.0
Fs (MeV) 143.8 16.5* 136.41.510.0 139.01.414.9
q (Degree) 34.5 5.4* 36.41.40.2 35.81.20.3
s (Degree) 36.0 4.2* 37.80.91.5 37.10.81.1
Table 6. The outputs from NNLO ts. See table 2 for the explanation of the phenomenological inputs. The rst error for each quantity corresponds to the statistical one and the second error denotes the systematic one. See the text for details.
the NLO parameters 1,2, which are tted up to O(N2C) in the NNLO analysis, become
1 = 0.04 0.06 0.13 , 2 = 0.14 0.10 0.40 . (4.1) The NNLO t also determines some U(3) NNLO couplings with relatively high precision.
NNLOFit-B yields
103 L4 = 0.09 0.06 0.23 , 103 L6 = 0.03 0.03 0.02 ,
103 L7 = 0.36 0.05 0.12 . (4.2) Even though the error estimates in the present article must be considered with some caution, as some lattice systematic uncertainties escape our control, this hints the potentiality of this U(3) ~PT framework. We hope these results may encourage future lattice analyses along this line.
Acknowledgments
We thank Shao-Zhou Jiang for communication on the updated values of the O(p6) LECs.
This work is supported in part by the National Natural Science Foundation of China (NSFC) under Grant No. 11105038, the Natural Science Foundation of Hebei Province with contract No. A2015205205, the grants from the Education Department of Hebei Province under contract No. YQ2014034, the grants from the Department of Human Resources and Social Security of Hebei Province with contract No. C201400323, and the Doctor Foundation of Hebei Normal University under Contract No. L2010B04, the Spanish Government (MINECO) and the European Commission (ERDF) [FPA2010-17747, FPA2013-44773-P, FPA2013-40483-P, SEV-2012-0249 (Severo Ochoa Program), CSD2007-00042 (Consolider Project CPAN)], the grants with contract No. FIS2014-57026-REDT from MINECO (Spain), and EPOS network of the European Community Research Infrastructure Integrating Activity Study of Strongly Interacting Matter (HadronPhysics3, Grant No. 283286). J.J. Sanz-Cillero wants to thank the Center for Future High Energy Physics and the Institute of High Energy Physics in Beijing for their hospitality.
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A Higher order corrections to the and bilinear terms
In the following we provide the explicit expressions of the is in eq. (2.11). When expressing the results in terms of F , they take the form
1 = 32C12
3F 2 c2(4m2K m2) + 42cs(m2K m2) + s2(2m2K + m2)
, (A.1)
2 = 32C12
3F 2
c2(2m2K + m2) 42cs(m2K m2) + s2(4m2K m2)
, (A.2)
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3 =
64C12
3F 2 (m2K m2) 2c2 cs 2s2
, (A.3)
= 8L5[c2(4m2K m2) + 42c
(m2K m2)s + (2m2K + m2)s2]
3F 2 + s2 1
8L18s[22c(m2K m2) + (2m2K + m2)s]
F 2
+64L5(L5 2L8)[c2(4m4K m4) + 42c(m4K m4)s + (2m4K + m4)s2]
3F 4
+16(C14 + C17)
3F 2 [c2(8m4K 8m2Km2 + 3m4) + 82cm2K(m2K m2)s+(4m4K 4m2Km2 + 3m4)s2] , (A.4)
=
+c2A0(m2K)162F 2 +
8L4(2m2K + m2)F 2 +
8L5[c2(2m2K + m2) + 42c(m2K + m2)s + (4m2K m2)s2]
3F 2 + c2 1
+s2A0(m2K)162F 2 +
8L4(2m2K + m2)
F 2 +
8L18c[c(2m2K + m2) + 22(m2K + m2)s]
F 2
+64L5(L5 2L8)[c2(2m4K + m4) + 42c(m4K + m4)s + (4m4K m4)s2]
3F 4
+16(C14 + C17)
3F 2 [c2(4m4K 4m2Km2 + 3m4) + 82cm2K(m2K + m2)s+(8m4K 8m2Km2 + 3m4)s2] , (A.5)
k =
16L5(m2K m2)(2c2 cs 2s2)
3F 2 cs 1
+csA0(m2K)
162F 2
8L18[2c2
(m2K m2) + c(2m2K + m2)s + 2(m2K + m2)s2]
F 2
128L5(L5 2L8)(m4K m4)(2c2 cs 2s2)
3F 4
64(C14 + C17)m2K(m2K m2)(2c2 cs 2s2)
3F 2 , (A.6)
m2
3F 2 [c2(8m4K 8m2Km2 + 3m4) + 82cm2K(m2K m2)s +(4m4K 4m2Km2 + 3m4)s2]
+2
3s[22c(m2K m2) + (2m2K + m2)s] 2
30
=
16L8
+ 1
162
hc4(16m2K 7m2) + 42c3(8m2K 5m2)s + 12c2(4m2K m2)s2
+162c(m2K m2)s3 + 2(2m2K + m2)s4i
1 18F 2
A0(m2)
+(4m2K m2)(2c4 22c3s 3c2s2 + 22cs3 + 2s4)
18F 2 A0(m2 )
[c2m2 + 22c(2m2K + m2)s 4m2Ks2]3F 2 A0(m2K)
+m2(c2 22cs + 2s2)2F 2 A0(m2)
16L25s[42cm2K(m2K m2)+(4m4K 4m2Km2+3m4)s]
F 2 +6(2m2K +m2)s2v(2)2
+16L6(2m2K + m2)[c2(4m2K m2) + 42c(m2K m2)s + (2m2K +m2)s2]
3F 2
+16L7[8c2(m2K m2)2 + 42c(2m4K m2Km2 m4)s + (2m2K + m2)2s2]
3F 2
+256(L5 2L8)L83F 4
hc2(8m6K 4m4Km2 4m2Km4 + 3m6) +42cm2K(2m4K m2Km2 m4)s + (4m6K 2m4Km2 2m2Km4 + 3m6)s2i +16(L5 2L8) 2s[22c(m4K m4) + (2m4K + m4)s]
3F 2
+16(3C19 + 2C31)3F 2
hc2(16m6K 24m4Km2 + 12m2Km4 m6) +42c(4m6K 6m4Km2 + 3m2Km4 m6)s
+(8m6K 12m4Km2 + 6m2Km4 + m6)s2i
, (A.7)
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m2
=
2
3c[c(2m2K + m2) + 22(m2K + m2)s] 2
+16L8
3F 2 [c2(4m4K 4m2Km2 + 3m4) + 82cm2K(m2K + m2)s +(8m4K 8m2Km2 + 3m4)s2]
+ 1
162
(4m2K m2)(2c4 22c3s 3c2s2 + 22cs3 + 2s4)18F 2 A0(m2)
+ 1
18F 2
h2c4(2m2K + m2) 162c3(m2K m2)s + 12c2(4m2K m2)s2
42c(8m2K 5m2)s3 + (16m2K 7m2)s4i A0(m2 )
[4c2m2K + 22c(2m2K m2)s + m2s2]
3F 2 A0(m2K)
+m2(2c2 + 22cs + s2)
2F 2 A0(m2)
16L25c[c(4m4K 4m2Km2 + 3m4) + 42m2K(m2K + m2)s] F 2
31
+6c2(2m2K + m2)v(2)2
+16L7[c2(2m2K + m2)2 + 42c(2m4K + m2Km2 + m4)s + 8(m2K m2)2s2]
3F 2
+16L6(2m2K + m2)[c2(2m2K + m2) + 42c(m2K + m2)s + (4m2K m2)s2]
3F 2
+256(L5 2L8)L83F 4 [c2(4m6K 2m4Km2 2m2Km4 + 3m6) +
42cm2K(2m4K + m2Km2 + m4)s + (8m6K 4m4Km2 4m2Km4 + 3m6)s2]
+16(L5 2L8) 2c[22s(m4K + m4) + (2m4K + m4)c]
3F 2
+16(3C19 + 2C31)3F 2
hc2(8m6K 12m4Km2 + 6m2Km4 + m6)
42c(4m6K 6m4Km2 + 3m2Km4 m6)s +(16m6K 24m4Km2 + 12m2Km4 m6)s2i
, (A.8)
2
3[2c2(m2K m2) + c(2m2K + m2)s + 2(m2K + m2)s2] 2
1
2882F 2
h42c4(m2K m2) + 32c2(4m2K m2)s2 + c(8m2K + m2)s3
+2(8m2K + 5m2)s4 + 4c3s(5m2K + 2m2)i
+6[2c2
(2m2K m2) + c(4m2K + m2)s + 2(2m2K + m2)s2]A0(m2K) +9m2(2c2 + cs + 2s2)A0(m2)
6c(2m2K + m2)sv(2)2
32L6(2m4K m2Km2 m4)(2c2 cs 2s2)
3F 2
h22c2(2m4K m2Km2 m4) + c(4m4K + 20m2Km2 7m4)s +
22(2m4K + m2Km2 + m4)s2i
512(L5 2L8)L8m2K(2m4K m2Km2 m4)(2c2 cs 2s2)
3F 4
16(L5 2L8) 2[2c2(m4K m4) + c(2m4K + m4)s + 2(m4K + m4)s2]
3F 2
32(3C19 + 2C31)(4m6K 6m4Km2 + 3m2Km4 m6)(2c2 cs 2s2)
3F 2 . (A.9)
32
m2 =
64L8m2K(m2K m2)(2c2 cs 2s2)
3F 2
+
h
JHEP06(2015)175
2c4(8m2K 5m2) + c3(8m2K + m2)s + 32c2(4m2K + m2)s2
+4c(5m2K + 2m2)s3 + 42(m2K + m2)s4i
A0(m2)
A0(m2 )
h22c2m2K(m2K m2) + c(4m4K 4m2Km2 + 3m4)s
+22m2K(m2K + m2)s2i
+16L25
F 2
16L7 3F 2
When expressing the above results in terms of F from eq. (2.27), the terms with L25 and L5L8 can be di erent from the expressions in terms of F and the other parts remain the same, apart from the obvious replacement of F by F. Therefore, for the expressions of i expressed in F, we only give the parts that are di erent from those in terms of F
(F),L
hc2(2m4K + 2m2Km2 m4) + 22cs(m4K + m2Km2 2m4)
+s2(m4K + m2Km2 + m4)
i
, (A.10)
(F),L
= 128L25
3F 4
25
hc2(m4K + m2Km2 + m4) 22cs(m4K + m2Km2 2m4)
+s2(2m4K + 2m2Km2 m4)i
, (A.11)
(F),L
25
=
128L25 3F 4
JHEP06(2015)175
k =
128L25(m4K + m2Km2 2m4)(2c2 cs 2s2)3F 4 , (A.12)
(F),L5L8m2
=
25
hc2(16m6K 16m2Km4 + 9m6) + 162cm2K(m4K m4)s
+(8m6K 8m2Km4 + 9m6)s2i
, (A.13)
(F),L5L8m2
=
128L5L8
3F 4
hc2(8m6K 8m2Km4 + 9m6) 162cm2K(m4K m4)s
+(16m6K 16m2Km4 + 9m6)s2i
, (A.14)
(F),L5L8m2 =
128L5L8
3F 4
1024L5L8m2K(m4K m4)(2c2 cs 2s2)
3F 4 . (A.15)
In order to obtain the full expressions for the is given in terms of F one has to make use of eq. (2.27) up to the precision required. Taking k for example, its nal expression in terms of F is
k =
16L5(m2K m2)(2c2 cs 2s2)
3F 2 cs 1
+csA0(m2K)
162F 2
(m2K m2) + c(2m2K + m2)s + 2(m2K + m2)s2]
F 2
+256L5L8(m4K m4)(2c2 cs 2s2)
3F 4
8L18[2c2
128L25(m4K + m2Km2 2m4)(2c2 cs 2s2) 3F 4
64(C14 + C17)m2K(m2K m2)(2c2 cs 2s2)
3F 2 , (A.16)
which di ers from eq. (A.6) in the L25 term. For 1, 2 and 3, their expressions are the same regardless of whether F or F is chosen up to next-to-next-to-leading order.
For completeness, we also give the results in terms of the LO masses m and mK and 1/F 2. Only the terms with LiLj, being Li and Lj the NLO LECs in eq. (2.4), will be di erent, comparing with the expressions in terms of m and mK and the other parts
33
remain the same, apart from the obvious replacement of the renormalized masses by the LO ones. Therefore, we only give the parts that are di erent from those in terms of m, mK
and 1/F 2 and it turns out that in this case all of the LiLj terms for , , k, m2 , m2
, m
vanish.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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JHEP06(2015)175
37
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SISSA, Trieste, Italy 2015
Abstract
Abstract
We study the [eta]-[eta]' mixing up to next-to-next-to-leading-order in U(3) chiral perturbation theory in the light of recent lattice simulations and phenomenological inputs. A general treatment for the [eta]-[eta]' mixing at higher orders, with the higher-derivative, kinematic and mass mixing terms, is addressed. The connections between the four mixing parameters in the two-mixing-angle scheme and the low energy constants in the U(3) chiral effective theory are provided both for the singlet-octet and the quark-flavor bases. The axial-vector decay constants of pion and kaon are studied in the same order and confronted with the lattice simulation data as well. The quark-mass dependences of m ^sub [eta]^, m ^sub [eta]'^ and m ^sub K^ are found to be well described at next-to-leading order. Nonetheless, in order to simultaneously describe the lattice data and phenomenological determinations for the properties of light pseudoscalars [pi], K, [eta] and [eta]', the next-to-next-to-leading order study is essential. Furthermore, the lattice and phenomenological inputs are well reproduced for reasonable values of low the energy constants, compatible with previous bibliography.
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