Finite volume for three-flavour Partially Quenched Chiral Perturbation Theory through NNLO in the meson sector

Abstract

Abstract

We present a calculation of the finite volume corrections to meson masses and decay constants in three flavour Partially Quenched Chiral Perturbation Theory (PQChPT) through two-loop order in the chiral expansion for the flavour-charged (or off-diagonal) pseudoscalar mesons. The analytical results are obtained for three sea quark flavours with one, two or three different masses. We reproduce the known infinite volume results and the finite volume results in the unquenched case. The calculation has been performed using the supersymmetric formulation of PQChPT as well as with a quark flow technique.

Partial analytical results can be found in the appendices. Some examples of cases relevant to lattice QCD are studied numerically. Numerical programs for all results are available as part of the CHIRON package.

Published for SISSA by Springer

Received: September 1, 2015

Accepted: October 19, 2015 Published: November 16, 2015

Finite volume for three- avour Partially Quenched Chiral Perturbation Theory through NNLO in the meson sector

Johan Bijnens and Thomas RosslerDepartment of Astronomy and Theoretical Physics, Lund University, Solvegatan 14A, SE 223-62 Lund, Sweden

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We present a calculation of the nite volume corrections to meson masses and decay constants in three avour Partially Quenched Chiral Perturbation Theory (PQChPT) through two-loop order in the chiral expansion for the avour-charged (or o -diagonal) pseudoscalar mesons. The analytical results are obtained for three sea quark avours with one, two or three di erent masses. We reproduce the known in nite volume results and the nite volume results in the unquenched case. The calculation has been performed using the supersymmetric formulation of PQChPT as well as with a quark ow technique.

Partial analytical results can be found in the appendices. Some examples of cases relevant to lattice QCD are studied numerically. Numerical programs for all results are available as part of the CHIRON package.

Keywords: Lattice QCD, Quark Masses and SM Parameters, E ective eld theories, Chiral Lagrangians

ArXiv ePrint: 1508.07238

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2015)097

Web End =10.1007/JHEP11(2015)097

JHEP11(2015)097

Contents

1 Introduction 1

2 Partially Quenched Chiral Perturbation Theory 32.1 The Lagrangian 32.2 The propagator and notation for masses and residues 52.3 The quark ow case 7

3 The nite volume integrals 8

4 Analytical results 10

5 Numerical examples 125.1 dval = dsea = 1=1 125.2 The pion mass 125.3 The pion decay constant 145.4 The kaon mass and decay constant 15

6 Conclusions 17

A Expressions for the mass 18

B Expressions for the decay constant 23

1 Introduction

Quantum chromodynamics (QCD) is nowadays accepted to be the theory describing the strong force. The smallness of the coupling constant at high energies makes it possible to test and con rm the theory in highly energetic scattering. It also provides | at least in principle | a way to obtain various low-energy hadronic observables, such as masses and decay constants, but it has hitherto been impossible to derive such quantities of interest in terms of analytical expressions by means of ab initio calculations. A numerical approach that can circumvent the problem is lattice QCD. A review of the applications to avour and low-energy hadron physics is [1]. To calculate observables, one uses a numerical evaluation of the QCD path integral in a Monte Carlo approach. A number of restrictions follow from the nature of the calculation. Since it is carried out on a space-time lattice in a nite volume, it is of high interest to have the e ect of the nite volume under good control. Furthermore, although lattice computations in meson physics are now feasible when using physical parameters for the light quark masses, a lot of calculations still use unphysically high masses for the quarks. It is also useful to vary quark masses to study a number of

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JHEP11(2015)097

phenomena. A common solution to study quark mass dependence with lower computational needs is given by partial quenching. In partially quenched QCD (PQQCD), one associates di erent masses (usually larger ones) to the sea quarks and the valence quarks. Valence quarks are those connected to the external operators while sea quarks are those in the fermion determinant or equivalently in closed loops. Sea quarks are only connected to external states via gluons.

The preferable way to correct for unphysical quark masses is by means of Chiral Perturbation Theory (ChPT) [2{4]. Finite volume e ects for ChPT have been introduced in [5{7]. The corresponding e ective theory for PQQCD is given by Partially Quenched Chiral Perturbation Theory (PQChPT) [8]. The arguments underlying this are elaborated in [9].

The proper matching of calculations in PQChPT to results from Partially Quenched Lattice QCD allows a whole new landscape of possibilities, such as improved validation and extrapolation of lattice results, or a more accurate determination of the chiral low-energy constants (LECs), see e.g. [10]. It should be stressed that, as opposed to fully quenched calculations, partially quenched calculations are connected to their corresponding unquenched scenarios by a continuous change in variables, making it possible to immediately extract physical results from otherwise unphysical simulations.

In this paper, we address the nite volume corrections through two-loop order in the PQChPT framework, speci cally for the avour-charged or o -diagonal mesons. The in nite volume (IV) results in PQChPT to this order are known for three [11{13] and two [14] sea quark avours. The nite volume (FV) corrections in (unquenched) ChPT at two-loop order have been addressed in our earlier study [15]. The needed integrals have been worked out in [16]. Our expressions are valid in the frame with vanishing spatial momentum, ~p = 0, often called the center-of-mass frame. In the so-called moving frames or with twisted boundary conditions there will be additional terms. We have chosen to present our result in terms of lowest order masses given the ambiguity in expressing the results in terms of the large number of possible di erent physical masses.

Earlier work on nite volume corrections at NNLO are besides our own work [15], the pion mass in two- avour ChPT [17] and the vacuum expectation value in three avour ChPT [18]. Extensions of the latter work to partially quenched are in [19]. We did not nd published results for the nite volume corrections at one-loop order in the partially quenched case. They are however implicit in the expressions given for the staggered partially quenched case in [20, 21].

We give a short list of references for ChPT and discuss some small points in section 2. The de nitions of the integrals we use and how they relate to the results in [16] is given in section 3. The next section describes our major result which is the full nite volume correction to the pion mass and decay constant to two-loop order in ChPT. Section 4 contains the results for the three- avour case for pion, kaon and eta for both the mass and decay constant. The large two-loop order formulas are collected for one case in the appendices and all of them can be downloaded from [22]. A numerical discussion of our results is in section 5.

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2 Partially Quenched Chiral Perturbation Theory

This section is very similar to the description of PQChPT given in [13] since our work is the extension to nite volume of that paper.

An introduction to ChPT can be found in [23, 24] and in the two-loop review [25]. The lowest order and p4-Lagrangian can be found in [4]. The order p6 Lagrangian is given in [26]. We use the standard renormalization scheme in ChPT. An extensive discussion of the renormalization scheme can be found in [27] and [28]. Important for our work is that the LECs do not depend on the volume [7]. An introduction with applications to lattice QCD is [29]. References to more introductory literature can be found in [22].

The expansion in ChPT is in momenta p and quark-masses. We count the latter as two powers of p. This counting is referred to as p-counting. We prefer to designate orders by the p-counting order at which the diagram appears. Thus we refer to lowest order (LO) as order p2, next-to-leading order (NLO) as order p4 or one-loop order and next-to-next-to-leading order (NNLO) as order p6 or two-loop order and include in the terminology oneor two-loop order also the diagrams with fewer loops but the same order in p-counting.

2.1 The Lagrangian

Three massless quark avours QCD has a chiral symmetry

G = SU(nf)L [notdef] SU(nf)R ; (2.1) which is spontaneously broken to the diagonal subgroup SU(3)V . The Goldstone bosons following from this spontaneous breakdown are described by the meson octet matrix

(x) =

0 B

@

: (2.2)

The avour-singlet component has been integrated out since it is heavy due to the U(1)A anomaly. The spontaneous symmetry breaking is the basis of ChPT.

In partially quenched QCD one distinguishes between valence and sea quarks. Valence quarks are connected to the external states (or operators) while the sea-quarks are those contributing in closed loops only connected via gluons to external states. These can be given di erent masses in lattice QCD calculations. The ChPT for this partial quenching can be done by studying the quark ow generalizing the quenched case studied in [30]. One can then treat the sea and valence lines di erently. Alternatively, one can make use of the supersymmetric formulation of PQChPT [8]. In the latter, three corresponding sets of quarks are introduced instead of only two: in addition to the valence and sea sector, a set of so-called ghost quarks is added. These are \bosonic" in the sense that they are treated as commuting variables. With their masses xed to the same numerical values as present in the valence sector, they will cancel exactly the contribution coming from closed valence quark loops. Most of the remainder of this section will be concerned with the supersymmetric formulation. The changes needed to use a quark ow technique are discussed at the end.

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1 C

A

1p2 0 +

1p6 + K+ 1p2 0 +

1p6 K0

K

K0 1p3

The chiral symmetry group is formally extended to the graded1

G = SU(nval + nsea[notdef]nval)L [notdef] SU(nval + nsea[notdef]nval)R ; (2.3)

for the case of nval valence and nsea quarks. The chiral group G is spontaneously broken to the diagonal subgroup SU(nval + nsea[notdef]nval)V . We will work in the avour basis rather than in the meson basis. We will thus use elds ab corresponding to the avour content of qa

qb. The mixing of the neutral eigenstates and the integrating out of the singlet degree of freedom is taken care of by using a more complicated propagator.2

The corresponding Goldstone degrees of freedom are put in a matrix with the generic structure

=

0 B

B

B

B

B

B

B

@

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h

qV

qV

i h

qV

qS

i h

qV

qB

i

h

: (2.4)

V denotes valence, S denotes sea and B denotes the bosonic ghost quarks. Note that the meson elds containing one single ghost quark only will themselves obey fermionic, i.e. anticommuting, statistics.

The structure of the Lagrangian is similar to standard ChPT for a generic number of avours. The lowest order Lagrangian is

L2 = F 204 [angbracketleft]u u + ~+[angbracketright] : (2.5)

At one-loop, it is given by

L4 = ^

L0 [angbracketleft]u u u u [angbracketright] +

^

L1 [angbracketleft]u u [angbracketright]2 +

qS

qV

i h

qS

qS

i h

qS

qB

i

h

qB

qV

i h

qB

qS

i h

qB

qB

i

1 C

C

C

C

C

C

C

A

^

L2 [angbracketleft]u u [angbracketright][angbracketleft]u u [angbracketright] +

^

L3 [angbracketleft](u u )2[angbracketright] (2.6)

+ ^

L4 [angbracketleft]u u [angbracketright][angbracketleft]~+[angbracketright] +

^

L5 [angbracketleft]u u ~+[angbracketright] +

^

L6 [angbracketleft]~+[angbracketright]2 +

^

L7 [angbracketleft]~[angbracketright]2 +

^ 2 [angbracketleft]~2+ + ~2[angbracketright] + : : : :

We show only the terms relevant for our work.

The generalized Goldstone manifold is parametrized by

u exp

i =(p2 ^

F )

(2.7)

similar to the exponential representation in standard ChPT. It is a 9 [notdef] 9 matrix with

fermionic parts. We have furthermore introduced

u = i

nu(@ ir ) u u (@ il ) uo;

~[notdef] = u~ u [notdef] u ~ u : (2.8)

1The precise structure of the symmetry group is somewhat di erent, but the one given here is su cient for both the present discussion as well as for practical calculations. The \approximate" symmetry group reproduces the right Ward identities [10, 31].

2This is described in detail in [31]. It is possible to use the same method also in standard ChPT.

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The matrix ~ is for this work restricted to

~ = 2B0 diag(m1; : : : ; m9) (2.9)

with mi the quark mass of quark i and B0 a LEC. We have here m1 = m7; m2 = m8; m3 = m9 as the valence masses and m4; m5; m6 as the sea quark masses. Ordinary traces have been replaced by supertraces, denoted by [angbracketleft] [angbracketright], de ned in terms of the ordinary ones by

Str A B

C D

!

= Tr A Tr D : (2.10)

B and C denote the fermionic blocks in the matrix. The supersinglet 0, generalizing the [prime], is integrated out to account for the axial anomaly as in standard ChPT, implying the additional condition

h [angbracketright] = Str ( ) = 0 : (2.11)

However, as mentioned above, we will work in the avour basis enforcing the constraint (2.11) via the propagator.

A calculation in PQChPT has to be performed using a larger set of operators since no further reduction by means of Cayley-Hamilton relations can be performed. The three- avour PQChPT Lagrangian (equation (2.6)) thus has 11 LECs for PQChPT.

The LECs for standard three avour ChPT are related to those of three avour PQChPT via

Lr1 = ^

Lr1 + ^

Lr0=2; Lr2 = ^

Lr2 + ^

Lr0; Lr3 = ^

Lr3 2

^

Lr0; (2.12)

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and Lri = ^

Lri for the others. Note that a numerical value for ^

L0 cannot be obtained by experiment, but can be determined only via PQQCD lattice simulations or modelling.

An additional comment is that the divergences for PQChPT are directly related to those for nsea- avour ChPT [28] when all traces are replaced by supertraces. This can be argued using the formal equivalence of the equations of motion used or via the replica trick [32].

2.2 The propagator and notation for masses and residues

The variant of PQChPT, considered in this paper, comes with three valence quarks, with masses m1; m2; m3 and three sea quarks with masses m4; m5; m6. The additional ghost quarks emerging only in the supersymmetric formulation have masses m7; m8; m9. They do not appear explicitly since they are xed to the ones in the valence sector, i.e. m7 = m1, m8 = m2, m9 = m3.

We use the numbers dval and dsea to denote the number of non-degenerate quark masses in each sector. In the case of two non-degenerate mass scales for one sector, it is the two masses with the lowest indices that we set degenerate, which will in turn both be represented by the mass scale with the lowest index, e.g. in the case dsea = 2 we have m4 = m5 [negationslash]= m6 and expressions will be explicitly dependent on m4 and m6 only.

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In fact we will always absorb a factor 2B0 in the notation and we use

~i 2B0mi ; ~ij

1

2 (~i + ~j) : (2.13)

The lowest order masses for o -diagonal mesons with avour content qi

qj are given by

~ij and we will use ~i rather then ~ii for equal masses. Dealing with masses for the diagonal valence mesons in PQChPT is not trivial. This is discussed in detail in [31] and extended to NNLO in [33]. The diagonal sea quark sector has two masses associated with it, corresponding to the neutral pion and eta masses. These we denote by ~ and ~ . They are de ned as the solutions to the equations

~ + ~ = 23 (~4 + ~5 + ~6) ;

~~ = 13 (~4~5 + ~5~6 + ~4~6) : (2.14)

They are non-polynomial in the sea masses ~j for three non-degenerate quark masses, i.e. dsea = 3. For dsea = 2 one has instead ~ = ~4 and ~ = (1=3)(~4 + 2~6).

The avour-charged propagator, connecting ij with ji, is given by [8, 10, 31]

i Gcij(k) =

jk2 ~ij + i"

(i [negationslash]= j) ; (2.15) with ~ij (~i + ~j)=2, the lowest order meson mass, and the signature j is de ned as +1

for the avor indices of the nval + nsea fermionic quarks, and as 1 for the avor indices of

the nval bosonic ghost quarks. In the present calculation, with the number of valence and sea quarks as given above, j thus takes the values

j =

(+1 for j = 1; : : : ; 6

1 for j = 7; 8; 9 : (2.16)

The avour-neutral propagator, connecting a avour eld ii to jj, on the other hand su ers from additional contributions emerging from the elimination of the 0 and the partial quenching [8, 10, 31]. We write it as

Gnij(k) = Gcij(k) ij Gqij(k)=nsea: (2.17)

The additional terms are either

i Gqij(k) =

JHEP11(2015)097

Rij k2 ~i + i"

+ Rji k2 ~j + i"

+ R ijk2 ~ + i"

+ R ijk2 ~ + i"

; (2.18)

for the case with i [negationslash]= j and ~i [negationslash]= ~j, or

i Gqij(k) =

Rdi(k2 ~i + i")2

+ Rcik2 ~i + i"

+ R iik2 ~ + i"

+ R iik2 ~ + i"

; (2.19)

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for the case with ~i = ~j which clearly includes i = j. In the second case, the sum of single poles is supplemented with an unphysical double pole. Since double poles emerge due to the partial quenching in the valence sector, they disappear by taking the appropriate unquenched limit.

Using the ratios of products of di erences of masses

Rzab = ~a ~b;

Rzabc = ~a ~b

~a ~c

Rzabcd = (~a ~b)(~a ~c)

~a ~d

; (2.20)

the residues R of the neutral meson propagator in equations (2.18) and (2.19) are (for dsea = 3)

Rijkl = Rzi456jkl; Rdi = Rzi456 ; Rci = Ri4 + Ri5 + Ri6 Ri Ri : (2.21)

Note that many of these quantities vanish when i takes the value of a sea quark index. The sea-quark propagators thus do not contribute any double poles as expected since these originate from the quenching in the valence sector.

For dsea = 2 or ~ = ~5 = ~4. The needed residues simplify to

Rijk = Rzi46jk; Rdi = Rzi46 ; Rci = Ri4 + Ri6 Ri : (2.22)

The corresponding propagator can be obtained by removing all pion indices as well as the pion mass pole from equations (2.18) and (2.19).

The physically less interesting case dsea = 1 immediately yields ~ =~ =~6 =~5 =~4. All residues from the sea quark sector are reduced to numbers, only

Rij = Rzi4j; Rdi = Rzi4 ; (2.23)

appear.

2.3 The quark ow case

We have performed the calculation using the supersymmetric method described above but also with the quark ow method [30]. We use the same Lagrangians as in (2.5) and (2.6) but with normal traces everywhere. The matrix is now written in terms of generic elds ij and all indices are kept symbolic implying summations.

Connecting propagators of a eld ij to kl should be done by using

Gijkl(k) = Gcij(k) il jk ij klGqik(k)=nsea : (2.24)

The propagators Gcij(k); Gqij(k) remain the same but we can now disregard the factors j

since with this method there are no bosonic ghost quarks.

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;

;

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Rzabcdefg = (~a ~b)(~a ~c)(~a ~d)

(~a ~e)(~a ~f)(~a ~g)

After constructing the Feynman diagrams using the above, the quark ow is visible following the symbolic avour indices. Next, one replaces the index lines that connect to external elds or operators by their appropriate valence value. The remaining index lines are now sea indices and are summed over with the sea quark indices.

The results obtained with the quark ow method agreed in all cases with those of the supersymmetric method.

3 The nite volume integrals

The loop integrals at nite volume at one-loop are well known. There is a sum over discrete momenta in every direction with a nite size rather than a continuous integral. The Poisson summation formula allows to identify the in nite volume part and the nite volume corrections. The remainder can be done with two di erent methods. For one-loop tadpole integrals the rst method was introduced by [5{7] and a sum over Bessel functions, that for large ML converges fast, remains to be done. With the other method one remains instead with an integral over a Jacobi theta function, this method can be used for small and medium ML as well. It can be found in [34]. The extensions to other one-loop integrals is done in both cases by combining propagators with Feynman parameters. The rst method was extended to the equal mass two-loop sunset integral [17] and later to the more general mass case in [16]. The latter extended the Jacobi theta function method as well to the sunset case. Details and further references can be found in [16]. In this paper we use Minkowski notation for the integrals.

For the one-loop integrals needed here, we use a notation that does a rst classi- cation according to the sum of the powers of the propagators with di erent masses, m1; m2; : : : ; mmax. We label the integrals A; B; C; D for a total power of propagators of n = 1; 2; 3; 4 respectively, since total powers of up to 4 can appear in the calculation as follows from the discussion of double poles in section 2.2. The di erent mass scales are given as consecutive arguments of the integral. Alternatively, if only one mass scale in total is present, we omit its repetition as a shorthand notation. For the present calculation at most two di erent scales can appear.

Both scalar and tensor integrals will occur, e. g. in the simplest case of one single propagator raised to single power

A(m2); A (m2) = 1 i

ZVddr(2)d [notdef]1; r r [notdef](r2 m2): (3.1)

We used the subscript V to indicate it is a nite volume sum and integral.

More Lorentz structures are possible than in the in nite volume case. We de ne the tensor t as the spatial part of the Minkowski metric g , to express these. For the center-of-mass (cms) case this is su cient. The needed functions for the above example are

A (m2) = g A22(m2) + t A23(m2) : (3.2)

We then use Passarino-Veltman identities in order to further simplify the result. In in nite volume the relation obtained by considering g A (m2) can be used to remove A22. In

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n1 n2 n3

n = 1 1 1 1 n = 2 2 1 1 n = 3 1 2 1 (n = 4) 1 1 2 n = 5 2 2 1 (n = 6) 2 1 2 n = 7 1 2 2 n = 8 2 2 2

Table 1. Overview of the notation for the possible con gurations of powers of propagators in the H functions in PQChPT. Redundant con gurations are given in parentheses.

nite volume, we again remove the A22-type integrals from the extended relation

dA22(m2) + 3A23(m2) = m2A(m2) : (3.3)

Each integral is split into an in nite volume contribution and a nite volume correction by means of the Poisson summation formula, while simultaneously being expanded in up to the necessary order.

A(m2) = 0 m2162 + A(m2) + AV (m2) + A (m2) + AV (m2)

+ log(4) + 1 . The same split is done for all one-loop integrals. The

expressions can be obtained by using the relations

B(m2) = @@m2 A(m2);

C(m2) = 12

D(m2) = 13

The sunset integrals, de ned as

H; H ; Hs ; H ; Hrs ; Hss (n; m21; m22; m23; p) =

ZVddr(2)ddds(2)d [notdef]1; r ; s ; r r ; r s ; s s [notdef]

r2 m21

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+ [notdef] [notdef] [notdef] : (3.4)

Here, 0 = 1

@@m2 B(m2);

@@m2 C(m2);

B(m21; m22) = A(m21) A(m22)

m21 m22

: (3.5)

1 i2

n3 ; (3.6)

now come with eight di erent pole con gurations. We label these by the index n according to table 1 analoguous to the in nite volume de nitions of [11{14].

The interchange (r; m21; n1) $ (s; m22; n2) allows to show that Hs ; Hss are related

directly to Hr ; Hrr . Hrs can also be related to H using the trick shown in [35] and

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s2 m22

(r + s p)2 m23

n1

n2

also used in [16], now taking the pole con gurations into account properly. The resulting H and H can then be reduced to six pole con gurations only, cf. table 1, the bracketed ones can be eliminated via the interchange above. In the scalar case H, only four pole con gurations are needed.

For the partially quenched calculation we thus generalized the sunset integrals used in our earlier work via

H(~i; ~j; ~k; p2) ! H(n; ~i; ~j; ~k; p2); (3.7)

introducing the new index n for the pole con gurations as the rst argument. Note on the side that all new pole con gurations are related to the simplest one by di erentiation with respect to the mass scales.

In the cms frame, we reduce the tensor structure of the sunsets as

H = p H1 (3.8)

H = p p H21 + g H22 + t H27 :

As in [15], we renormalize the FV sunsets by not only subtracting the in nite part but also an additional nite part containing O( ) contributions of one-loop integrals. In this way,

the latter integrals will cancel out of the nal result, and thus do not need to be computed. The splitting for n = 1

V = 0162 AV (m21) + AV (m22) + AV (m23)

+ 1

162 AV (m21) + AV (m22) + AV (m23)

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+ HV ;

V1 = 0

162

1

2 AV (m22) + AV (m23)

+ 1

1

2 AV (m22) + AV (m23)

+ HV1 ;

162

V21 = 0

13 AV (m22) + AV (m23)

+ 1

13 AV (m22) + AV (m23)

+ HV21 ;

162

AV23(m21) + 13A23(m22) +1 3AV23(m23)

+ 1

162

V27 = 0

162

AV 23(m21) + 13AV 23(m22) +13AV 23(m23) + HV27 ; (3.9)

has to be generalized for the other pole con gurations by taking the appropriate derivatives w.r.t. the masses.

4 Analytical results

The calculation of the masses proceeds in the usual way from the Feynman diagrams for the self-energy shown in gure 1. We have performed the calculation for the o -diagonal mesons, i.e. consistening of a valence quark and a di erent valence anti-quark, and for the case of three avours of sea quarks. The calculation has been done for all mass cases, equal and di erent valence quark-masses, dval = 1; 2, and sea quark masses all equal, dsea = 1, two equal and the third di erent, dsea = 2 and all three di erent, dsea = 3.

A large number of checks have been done on the calculations. They have been performed both in the supersymmetric formalism and using quark ow techniques. The in nite

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162

(a) (b) (c) (d) (e)

(f) (g) (h) (i)

Figure 1. Diagrammatic contributions to the pseudoscalar self-energy, up to O(p6). Circular

vertices are of O(p2), the lled boxes are of O(p4), the open box is of O(p6). The tree level

diagrams (a,b,i) do not contribute to nite volume corrections.

volume results are also in full agreement with [11{13]. The nite volume parts agree with our earlier results [15] when these are expressed in terms of lowest order masses and when the sea masses are put equal to the valence masses.

The formulas especially for the case of three di erent sea quark masses are very long. In appendix A we list the case of equal valence masses and two sea quark masses. This corresponds to the charged pion mass in the isospin limit. The other cases can be downloaded from [22].

The masses are given as

m2ij = ~ij + m2(4)ij + V m2(4)ij + m2(6)ij + V m2(6)ij : (4.1)

In addition a superscript indicating dvaldsea is added. The in nite volume and the one-loop nite volume corrections were known before. The new parts are the two-loop nite volume corrections. These we split in addition in an Lri dependent part and a pure two-loop

contribution

V m2(6)ij = V m2(6L)ij + V m2(6R)ij : (4.2)

The subscript ij is set to 12 for dval = 1 and to 13 for dval = 2 similar to the in nite volume work.

The decay constant is de ned in the usual way as

h0[notdef] qj 5qi[notdef]Mij(p)[angbracketright] = ip2Fijp ; (4.3)

for the pseudoscalar meson Mij with quark content i [negationslash]= j and momentum p. The calculation

needs the the diagrams of gure 1 for the wave function renormalization and the same ones with one external meson leg replaced by an insertion of the axial current.

We split the result as

Fij = F0 + F (4)ij + V F (4)ij + F (6)ij + V F (6)ij : (4.4)

The NNLO part is split again in

V F (6)ij = V F (6L)ij + V F 2(6R)ij : (4.5)

{ 11 {

JHEP11(2015)097

The calculations have been done using the supersymmetric and the quark ow methods. The in nite volume and NLO results agree with the known expressions and the result reduces in the correct limit to the unquenched results of our earlier work [15]. The formulas are rather long, the case for equal valence masses and two di erent sea masses corresponding to the charged pion decay constant in the isospin limit is given in appendix B. The expressions for the other cases can be downloaded from [22].

5 Numerical examples

The intention is that various lattice QCD collaborations can use our formulas. All cases discussed have been included in the package CHIRON [36] available from [37]. The numerical results shown in this section have been obtained with that implementation. The programs have been cross-checked with an independent version. It has been checked that the results reduce in the appropriate limits to those of our earlier work [15]. For this purpose the expressions obtained in [15], but rewritten in terms of lowest order masses and decay constants, have been implemented and included in CHIRON [37]. In addition, a check has been done that the di erent mass cases reduce to each other numerically.

For input values we have chosen the recent global t for the Lri [38]. We have set the

extra LEC Lr0 = 0. We always use a scale of = 0:77 GeV. For the size of the lattice we

present results for a length L such that ML = 2 for M = 0:13 GeV. The lowest order pion decay constant we have chosen throughout as F0 = 87:7 MeV.

The numerical results are presented via

VM =

VF =

JHEP11(2015)097

m2Vij m21ij ~ij

F Vij F 1ij

F0 : (5.1)

We thus plot the size of the nite volume corrections relative to the lowest order value of the quantity under consideration. Note that the results are for charged or o -diagonal mesons. They consist of a quark and a di erent anti-quark which might have equal mass.

5.1 dval = dsea = 1=1

Here we set all valence and all sea masses equal, dval = dsea = 1. The size of the nite volume corrections as a function of ~1 and ~4 is shown in gure 2. The corrections in this case are reasonable, at most a few %, except for very low masses and become very large for low valence and high sea quark mass.

5.2 The pion mass

In this subsection we look at the case where the lowest order mass is around the pion mass. We plot VM with p~12 = 0:13 GeV. The strange sea quark mass we have always

chosen such that the average lowest order kaon mass is 0:45 GeV. This corresponds to p~6 =

p2(0:45)2 (0:13)2 GeV 0:623 GeV. The other input parameters are chosen as

{ 12 {

DVM

DVF

0.25

0.2

0.25

0

0

0.2

-0.02

-0.2

0.2

-0.04

-0.4

2 ]

0.15

-0.6

2 ]

c 4[GeV

0.15

-0.06

0.1

-0.8

c 4[GeV

-0.08

0.1

-1

-0.1

0.05

-1.2

-0.12

0.05

JHEP11(2015)097

-1.4

-0.14

0

-1.6

0

-0.16

0 0.05 0.1 0.15 0.2 0.25

c1 [GeV2]

0 0.05 0.1 0.15 0.2 0.25

c1 [GeV2]

Figure 2. The nite volume corrections relative to the lowest order value as de ned in (5.1) for the case with all valence masses equal and all sea masses equal. Left: VM the correction to the mass-squared, contour lines are drawn at 0:03; 0:01; 0:003; 0; 0:03; 0:01 starting from the bottom

left and going counterclockwise. Right: the correction to the decay constant, contour lines are drawn at 0:001; 0:002; 0:005; 0:01; 0:02; 0:05 going from top-right to bottom-left.

given in the introduction of this section. We have restricted the sea up and down quark masses corresponding to a lowest order sea quark pion mass of 100 to 300 MeV.

The rst case we look at is dval = 1; dsea = 2. This corresponds to taking the up and down quark masses equal in both the valence and sea quark sector and a di erent strange quark mass. This is the isospin limit. The result is shown in gure 3(a). There is a rather large cancellation between the p4 and p6 correction while the p6 contribution coming from the Lri is fairly small.

We now include isospin breaking in the valence sector. We thus look at the case with dval = 2; dsea = 2. We x the valence quark masses such that ~1 + ~2 = 2~12 and ~1=~2 = 1=2. There is a sizable isospin breaking visible in the nite volume corrections, as shown in gure 3(b).

The opposite case, isospin breaking in the sea sector, but not in the valence sector, leads to numerically similar but opposite sign corrections. Here we used ~1 = ~2, ~4 = ~5=2 and ~4 + ~5 = 2~av. The results are shown in gure 3(c).

Finally, we introduce isospin breaking in both the valence and sea quark sector with ~1=~2 = 1=2, ~4 = ~5=2 and ~4 + ~5 = 2~av. The results are shown in gure 3(d). The total isospin corrections are rather small.

The numerical cancellation between the isospin breaking in the valence and sea quark case is accidental. The corrections due to valence and sea quark masses are all second order in isopin breaking. The same argument as in the unquenched case goes through both for the valence and sea quark masses. We have compared four scenarios in gure 4. We show the p4 and the full p4 + p6 result rst with no isospin breaking, then only in the valence sector or only in the sea sector and nally in both sectors. The curves are those shown

{ 13 {

0.15

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.15

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.1

p4 p6 Li p6 R

p4+p6

0.1

p4 p6 Li p6 R

p4+p6

0.05

M, d val=1 d sea=2

M, d val=2 d sea=2

0.05

0

0

DV

-0.05

DV

-0.05

-0.1

-0.1

JHEP11(2015)097

-0.15

-0.15

cav [GeV2]

cav [GeV2]

(a)

(b)

0.15

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.15

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.1

p4 p6 Li p6 R

p4+p6

0.1

p4 p6 Li p6 R

p4+p6

M, d val=1 d sea=3

0.05

M, d val=2 d sea=3

0.05

0

0

DV

-0.05

DV

-0.05

-0.1

-0.1

-0.15

-0.15

cav [GeV2]

cav [GeV2]

(c)

(d)

Figure 3. The corrections for the pion mass relative to the lowest order mass as a function of the average up and down sea quark mass via ~av. (a) The isospin limit, ~1 = ~2, ~4 = ~5 = ~av. (b)

Isospin breaking in the valence sector, ~1 = ~3=2 and ~4 = ~5 = ~av. (c) Isospin breaking in the sea sector, ~1 = ~2 and ~4 = ~5=2. (d) Isospin breaking in both sectors, ~1 = ~3=2 and ~4 = ~5=2.

in gure 3(a{d). We have checked numerically by using a di erent ratio for the isospin breaking that the corrections are indeed second order in isospin breaking.

5.3 The pion decay constantIn this subsection we look at the same cases as before. The lowest order mass is around the pion mass. We plot VF with p~12 = 0:13 GeV, and as before p~6 =

p2(0:45)2 (0:13)2 GeV 0:623 GeV. The other input parameters are again chosen as given in the introduction of this section. We have restricted the sea up and down quark masses corresponding to a lowest order sea quark pion mass of 100 to 300 MeV.

{ 14 {

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

-0.02

-0.04

-0.06

Figure 4. Comparing the nite volume correction for the meson masses for the cases with no isospin breaking (none), only in the valence sector (val), only in the sea sector (sea) and in both (full) for the meson mass squared. The upper curves are the p4, the bottom the p4 + p6 results.

The rst case we look at is dval = 1; dsea = 2. This corresponds to taking the up and down quark masses equal in both the valence and sea quark sector and a di erent strange quark mass, i.e. the isospin limit. The result is shown in gure 5(a). The total p6 correction is fairly small.

We now include isospin breaking in the valence sector. We thus look at the case with dval = 2; dsea = 2. We x the valence quark masses such that ~1 + ~2 = 2~12 and ~1=~2 = 1=2. There is a sizable isospin breaking visible in the nite volume corrections, as shown in gure 3(b).

The opposite case, isospin breaking in the sea sector but not in the valence sector leads to numerically much smaller e ects. Here we used ~1 = ~2, ~4 = ~5=2 and ~4 +~5 = 2~av.

The results are shown in gure 3(c).

Finally, we introduce isospin breaking in both the valence and sea quark sector with ~1=~2 = 1=2, ~4 = ~5=2 and ~4 + ~5 = 2~av. The results are shown in gure 3(d). The total isospin corrections are failry small.

The corrections due to valence and sea quark masses are all second order in isospin breaking. The same argument as in the unquenched case goes through both for the valence and sea quark masses. We compare the same four scenarios as for the pion mass, no isospin breaking, only in the valence sector, only in the sea sector and in both sectors. The curves are those shown in gure 5(a{d). In gure 6 we compare the di erent isospin breaking cases for p4 and p4 + p6.

5.4 The kaon mass and decay constant

We now look only at the dval = dsea = 2 case but choose the valence masses such that we have a lowest order pion mass of 130 MeV and a lowest order kaon mass of 450 MeV. This corresponds to p~1 = 130 MeV and p~3 623 MeV. We plot the nite volume corrections

{ 15 {

none val sea full

DV

M

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0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

cav [GeV2]

0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0

0

-0.01

-0.01

F, d val=1 d sea=2

F, d val=2 d sea=2

-0.02

-0.02

-0.03

-0.03

p4 p6 Li p6 R

p4+p6

p4 p6 Li p6 R

p4+p6

DV

-0.04

DV

-0.04

JHEP11(2015)097

-0.05

-0.05

-0.06

-0.06

cav [GeV2]

cav [GeV2]

(a)

(b)

0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0

0

-0.01

-0.01

F, d val=1 d sea=3

F, d val=2 d sea=3

-0.02

-0.02

-0.03

-0.03

p4 p6 Li p6 R

p4+p6

p4 p6 Li p6 R

p4+p6

DV

-0.04

DV

-0.04

-0.05

-0.05

-0.06

-0.06

cav [GeV2]

cav [GeV2]

(c)

(d)

Figure 5. The corrections for the pion decay constant relative to its lowest order value as a function of the average up and down sea quark mass via ~av. (a) The isospin limit, ~1 = ~2, ~4 = ~5 = ~av.

(b) Isospin breaking in the valence sector, ~1 = ~3=2 and ~4 = ~5 = ~av. (c) Isospin breaking in the sea sector, ~1 = ~2 and ~4 = ~5=2. (d) Isospin breaking in both sectors, ~1 = ~3=2 and ~4 = ~5=2.

relative to the lowest order value of the quantity in gure 7 as a function of ~4 = ~5. For the sea quark strange mass we use ~6 = 1:02~3. The LECs are again the ones from [38] and L such that ML = 2 for M = 130 MeV.

For the kaon we see that we reproduce the results of [15] that near the physical case the p4 corrections are very small. The total nite volume corrections to the mass remain fairly small. The kaon decay constant has larger corrections but they remain in the few % region for the parameters considered.

{ 16 {

0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0

0

-0.01

-0.01

-0.02

-0.02

DV

F

DV

F

-0.03

-0.03

p4 none p4 val p4 sea p4 full

p4+p6 none p4+p6 val p4+p6 sea p4+p6 full

-0.04

-0.04

-0.05

-0.05

JHEP11(2015)097

-0.06

-0.06

cav [GeV2]

cav [GeV2]

(a)

(b)

Figure 6. Comparing the nite volume correction for the meson decay constant and masses for the cases with no isospin breaking (none), only in the valence sector (val), only in the sea sector (sea) and in both (full) for the meson mass squared. (a) p4 (b) p4 + p6.

-0.006

0.008

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.03

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.006

0.02

0.004

0.01

0.002

DV

M

DV

F

0

0

p4 p6 Lir p4 R

p4+p6

p4 p6 Lir p4 R

p4+p6

-0.01

-0.002

-0.004

-0.02

-0.03

c4 [GeV2]

c4 [GeV2]

(a)

(b)

Figure 7. The nite volume corrections for a valence mass close to the kaon mass relative to the lowest order value. (a) the kaon mass squared. (b) the kaon decay constant.

6 Conclusions

We have computed the NNLO expressions for the masses and decay constants in three- avour partially quenched ChPT for all possible mass cases. The calculation has been performed using two di erent formalisms, quark ow and the supersymmetric method. The known in nite volume expressions have been reproduced. We quoted the expressions for the case of equal valence and two di erent sea quark masses in the appendices. The other cases can be obtained from [22].

{ 17 {

The numerical work shows nite volume corrections of a similar size as those in the unquenched case [15]. We have presented some representative numerics. The numerical work has been done using C++. The programs are available together with the in nite volume results in [37]. The analytical work relied heavily on FORM [39].

Acknowledgments

This work is supported in part by the Swedish Research Council grants 621-2011-5080 and 621-2013-4287. JB thanks the Centro de Ciencias de Benasque Pedro Pascual, where part of this paper was written, for hospitality.

A Expressions for the mass

F 20 Vm2(4)1212 = +AV (~1) 1=3 ~1Rc146 +AV (~ )

JHEP11(2015)097

2=3 ~1Rz 612

+BV (~1)

1=3 ~1Rz146 (A.1)

F 40 Vm2(6L)1212 = +AV (~1)

64=3 ^

Lr8~21 Rc146 32=3

^

Lr7~1 ~6 Rz14 64=3

Lr7~1 ~4 Rz16

^

+ 32 ^

Lr7~21 + 32 ^

Lr6~21 + 8=3 ^

Lr5~1 Rz146 8=9

^

Lr5~1 ~6 Rz14 2 16=9

Lr5~1 ~4 Rz16 2

^

+ 40=3 ^

Lr5~21 Rc146 8

^

Lr3~1 Rz146 8

^

Lr3~21 Rc146 + 20 ^

Lr2~21 + 8 ^

Lr1~21 8

Lr0~1 Rz146

^

8

^

Lr0~21 Rc146 16=3 (~6 + 2 ~4)

^

Lr6~1 Rc146 16=9 (~6 + 2 ~4)

Lr4~1 Rz16 2

^

8=9 (~6 + 2 ~4)

^

Lr4~1 Rz14 2 + 8

~6 + 2 ~4) ^

Lr4~1 Rc146 + AV (~14)

32 (~4 + ~1) ^ Lr8~1

16(~4 + ~1)

^

Lr5~1 + 20(~4 + ~1)^

Lr3~1 + 8(~4 + ~1)^

Lr0~1

+ AV (~16)

16(~6 + ~1)^ Lr8~1

8 (~6 + ~1)

^

Lr5~1 + 10 (~6 + ~1) ^

Lr3~1 + 4 (~6 + ~1) ^

Lr0~1

+ AV (~4)

48 ^

Lr6~1 ~4

48

^

Lr4~1~4 + 12^

Lr2~1~4 + 48^

Lr1~1~4

+ AV (~46)

32(~6 + ~4)^

Lr6~1 32(~6 + ~4)^Lr4~1

+ 8 (~6 + ~4) ^

Lr2~1 + 32 (~6 + ~4) ^

Lr1~1

+ AV (~ )

128=3 ^

Lr8~21 Rz 612

+ 32=9 ^

Lr5~1 ~6 Rz14 Rz 61 32=9

^

Lr5~1 ~4 Rz16 Rz 61 + 16 ^

Lr3~1 ~ Rz 612 + 4 ^

Lr2~1 ~

+ 16 ^

Lr1~1 ~ + 16 ^

Lr0~1 ~ Rz 612 8=3 (3 ~ + 2 ~6 + ~4)

^

Lr4~1

16=9 (3 ~ + 2 ~6 + ~4 + 12 ~1)

^

Lr5~1 Rz 612 + 64=3 (~6 ~4)

Lr7~1 Rz 61

^

+ 32=3 (~6 + 2 ~4) ^

Lr6~1 Rz 612 64=3 (~6 + 2 ~4)

Lr4~1 Rz 612

^

32=9(~6 + 2~4)

^

Lr4~1Rz16 Rz 61 + 32=9(~6 + 2~4)^

Lr4~1Rz14 Rz 61 + 16=3(2~6 + ~4)^

Lr6~1

+ BV (~1)

16=9^

Lr8~1~26Rz14 2 + 32=9^

Lr8~1~24Rz16 2 64=3^

Lr8~21Rz146 32=3^Lr8~31Rc146

+ 16=9^

Lr7~1~26Rz14 2 + 64=9^

Lr7~1~4~6Rz14 Rz16 + 64=9^

Lr7~1~24Rz16 2 32=3

Lr7~21~6Rz14

^

64=3

^

Lr7~21~4Rz16 + 16^

Lr7~31 + 40=3^

Lr5~21Rz146 8=9

^

Lr5~21~6Rz14 2 16=9

Lr5~21~4Rz16 2

^

+ 16=3 ^

Lr5~31 Rc146 8

^

Lr3~21 Rz146 8

^

Lr0~21 Rz146 16=3 (~6 + 2 ~4)

Lr6~1 Rz146

^

+ 16=9 (~6 + 2 ~4) ^

Lr6~1 ~6 Rz14 2 + 32=9 (~6 + 2 ~4) ^

Lr6~1 ~4 Rz16 2

32=3 (~6 + 2 ~4)

^

Lr6~21 Rc146 + 8 (~6 + 2 ~4) ^

Lr4~1 Rz146 16=9 (~6 + 2 ~4)

Lr4~21 Rz16 2

^

{ 18 {

8=9 (~6 + 2 ~4)

^

Lr4~21 Rz14 2 + 16=3 (~6 + 2 ~4) ^

Lr4~21 Rc146

+ BV (~1; ~ )

64=9 ^

Lr8~1 ~26 Rz14 Rz 61 + 64=9 ^

Lr8~1 ~24 Rz16 Rz 61 + 64=3 ^

Lr8~31 Rz 612

+ 32=9 ^

Lr5~21 ~6 Rz14 Rz 61 32=9

^

Lr5~21 ~4 Rz16 Rz 61 32=3

Lr5~31 Rz 612

^

64=9 (~6 ~4)

^

Lr7~1 ~6 Rz14 Rz 61 128=9 (~6 ~4)

Lr7 ~1 ~4 Rz16 Rz 61

^

+ 64=3 (~6 ~4)

^

Lr7~21 Rz 61 64=9 (~6 + 2 ~4)

Lr6~1 ~6 Rz14 Rz 61

^

+ 64=9 (~6 + 2 ~4) ^

Lr6~1 ~4 Rz16 Rz 61 + 64=3 (~6 + 2 ~4) ^

Lr6~21 Rz 612

32=3 (~6 + 2 ~4)

^

Lr4~21 Rz 612 32=9 (~6 + 2 ~4)

Lr4~21 Rz16 Rz 61

^

JHEP11(2015)097

+ 32=9 (~6 + 2 ~4) ^

Lr4~21 Rz14 Rz 61

+ BV (~ )

64=9 (~6 ~4)2^

Lr7~1 Rz 612

16=3 (~6 + 2 ~4)

^

Lr4~1 ~ Rz 612 + 32=9 (~6 + 2 ~4) (2 ~6 + ~4) ^

Lr6~1 Rz 612

16=9 (2 ~6 + ~4)

^

Lr5~1 ~ Rz 612 + 32=9 (2 ~26 + ~24) ^

Lr8~1 Rz 612

+ CV (~1)

32=3 ^

Lr8~31 Rz146 + 16=3 ^

Lr5~31 Rz146 32=3 (~6 + 2 ~4)

Lr6~21 Rz146

^

+ 16=3(~6+2~4)^

Lr4~21Rz146

+AV23(~1)

8^

Lr3~1Rc146 12^

Lr2~124^

Lr1~1+8^ Lr0~1Rc146

+ AV23(~14)

24 ^

Lr3~1 48

^

Lr0~1

+ AV23(~16)

12 ^

Lr3~1 24

^

Lr0~1

+ AV23(~4)

36 ^

Lr2~1

+ AV23(~46)

48 ^

Lr2~1

+ AV23(~ )

16 ^

Lr3~1 Rz 612 12

^

Lr2~1

16

^

Lr0~1 Rz 612

+ BV23(~1)

8 ^

Lr3~1 Rz146 + 8 ^

Lr0~1 Rz146

(A.2)

F 40 Vm2(6R)1212 = +A(~1)AV (~1)

~1 + 1=9~1Rc146 2

+ A(~1)AV (~14)

4=45Rz146 + 8=9~1Rc146 + 4=45(~4 11~1)Rz16

+ A(~1)AV (~16)

2=45Rz146 + 4=9~1Rc146 + 2=45(~6 11~1)Rz14

+ A(~1)AV (~46)

2=27~1Rz16 2 + 4=27~1Rz14 Rz16 2=27~1Rz14 2

+ A(~1)AV (~ )

2=9~1Rc146 Rz 612

+ A(~1)BV (~1)

1=9~1Rz146 Rc146

+ ~21 + 2=9~21Rc146 2

+ A(~1)BV (~1; ~ )

8=9~21Rc146 Rz 612

+ A(~1)CV (~1)

2=9~21Rz146 Rc146 + A(~14)AV (~1)

4=45Rz146 + 8=9~1Rc146

+ 4=45(~4 11~1)Rz16

+ A(~14)AV (~14)

4=27Rz146 + 10=27~1Rz16 + 1=27(2~ 2~4 47~1) + 2=27(4~ + 4~4 + 9~1)Rz 612 + 2=27(4~ + ~1)Rz 61

1=27(4~4 + 13~1)Rc146

+ A(~14)AV (~16)

~1

+ A(~14)AV (~ )

2=45(~ ~4)

8=45(~ + 9~1)Rz 612 4=45(2~ ~4 + 9~1)Rz 61

+ A(~14)BV (~1)

8=9~1Rz146

4=9(~4 + 2~1)~1Rz16

+ A(~14)BV (~1; ~ )

4=9(~4 + 2~1)~1Rz 61

+ A(~16)AV (~1)

2=45Rz146 + 4=9~1Rc146 + 2=45(~6 11~1)Rz14

{ 19 {

+ A(~16)AV (~14)

~1

+ A(~16)AV (~16)

2=27Rz146 + 5=27~1Rz14 + 2=27(2~ 2~6 5~1) + 1=27(4~ + 4~6 + 9~1)Rz 612 2=27(4~ + ~1)Rz 61

1=54(4~6 + 13~1

Rc146 + A(~16)AV (~ ) 4=45(~ ~6)

4=45(~ + 9~1)Rz 612 + 4=45(2~ ~6 + 9~1)Rz 61

+ A(~16)BV (~1)

4=9~1Rz146

2=9(~6 + 2~1)~1Rz14

+ A(~16)BV (~1; ~ )

4=9(~6 + 2~1)~1Rz 61

+ A(~4)BV (~1)

1=3~1~4Rz16 2 + A(~4)BV (~1; ~ )

2=3~1~4Rz16 Rz 61

1=3~1~4Rz 612 + A(~46)AV (~1) 2=27~1Rz16 2 + 4=27~1Rz14 Rz16

2=27~1Rz14 2

+ A(~46)AV (~ )

+ A(~4)BV (~ )

JHEP11(2015)097

2=3~1Rz 612 4=9~1Rz16 Rz 61 + 4=9~1Rz14 Rz 61

+ A(~46)BV (~1)

4=27(~6 + ~4 + ~1)~1Rz14 Rz16 + 2=27(~6 ~1)~1Rz14 2 + 2=27(~4 ~1)~1Rz16 2

+ A(~46)BV (~1; ~ )

4=27(~6 ~4 3~1)~1Rz14 Rz 61

4=27(2~6 + ~4 + 3~1)~1Rz16 Rz 61

+ A(~46)BV (~ )

2=9(3~ + ~4)~1Rz 612

+ A(~ )AV (~1)

2=9~1Rc146 Rz 612

+ A(~ )AV (~14)

2=45(~ ~4)

8=45(~ + 9~1)Rz 612 4=45(2~ ~4 + 9~1)Rz 61

+ A(~ )AV (~16)

4=45(~ ~6)

4=45(~ + 9~1)Rz 612 + 4=45(2~ ~6 + 9~1)Rz 61

+ A(~ )AV (~46)

2=3~1Rz 612

4=9~1Rz16 Rz 61 + 4=9~1Rz14 Rz 61

+ A(~ )AV (~ )

4=9~1Rz 614

2=9~1Rz146 Rz 612 + 2=27~1~6Rz14 2 + 1=27~1~4Rz16 2

4=9~21Rc146 Rz 612

+ A(~ )BV (~1; ~ )

+ A(~ )BV (~1)

8=27~1~6Rz14 Rz 61 + 2=27~1~4Rz16 Rz 61

+ 8=9~21Rz 614 + 4=9~21Rc146 Rz 612

+ A(~ )BV (~ )

1=27(8~6 + ~4)~1Rz 612

+ A(~ )CV (~1)

4=9~21Rz146 Rz 612

+ B(~1)AV (~1)

1=9~1Rz146 Rc146 + ~21

+ 2=9~21Rc146 2

+ B(~1)AV (~14)

8=9~1Rz146 4=9(~4 + 2~1)~1Rz16

+ B(~1)AV (~16)

4=9~1Rz146 2=9(~6 + 2~1)~1Rz14 + B(~1)AV (~4)

1=3~1~4Rz16 2

+ B(~1)AV (~46)

4=27(~6 + ~4 + ~1)~1Rz14 Rz16 + 2=27(~6 ~1)~1Rz14 2 + 2=27(~4 ~1)~1Rz16 2

+ B(~1)AV (~ )

2=9~1Rz146 Rz 612 + 2=27~1~6Rz14 2

+ 1=27~1~4Rz16 2 4=9~21Rc146 Rz 612

+ B(~1)BV (~1)

1=9~1Rz146 2 + 4=9~21Rz146 Rc146

+ B(~1)BV (~1; ~ )

4=9~21Rz146 Rz 612

+ B(~1)CV (~1)

2=9~21Rz146 2

+ B(~1; ~ )AV (~14)

4=9(~4 + 2~1)~1Rz 61

+ B(~1; ~ )AV (~16)

4=9(~6

+ B(~1; ~ )AV (~4)

2=3~1~4Rz16 Rz 61

{ 20 {

+ 2~1)~1Rz 61

+ B(~1; ~ )AV (~46)

4=27(~6 ~4 3~1)~1Rz14 Rz 61 4=27(2~6 + ~4

+ 3~1)~1Rz16 Rz 61

+ B(~1; ~ )AV (~ )

8=27~1~6Rz14 Rz 61 + 2=27~1~4Rz16 Rz 61

+ 8=9~21Rz 614 4=9~21Rc146 Rz 612

+ B(~1; ~ )BV (~1)

4=9~21Rz146 Rz 612

1=3~1~4Rz 612 + B(~ )AV (~46) 2=9(3~ + ~4)~1Rz 612

+ B(~ )AV (~ )

1=27(8~6 + ~4)~1Rz 612 + C(~1)AV (~1) 2=9~21Rz146 Rc146

+ C(~1)AV (~ )

+ B(~ )AV (~4)

4=9~21Rz146 Rz 612

+ C(~1)BV (~1)

2=9~21Rz146 2

+ AV (~1) 1

JHEP11(2015)097

162

1=4~21 + 1=6~21Rz16 + 1=12~21Rz14

+ AV (~1)2

1=2~1 + 1=18~1Rc146 2

+ AV (~1)AV (~14)

4=45Rz146 + 8=9~1Rc146 + 4=45(~4 11~1)Rz16

+ AV (~1)AV (~16)

2=45Rz146 + 4=9~1Rc146 + 2=45(~6 11~1)Rz14

+ AV (~1)AV (~46)

2=27~1Rz16 2 + 4=27~1Rz14 Rz16 2=27~1Rz14 2

+ AV (~1)AV (~ )

2=9~1Rc146 Rz 612

+ AV (~1)BV (~1)

1=9~1Rz146 Rc146 + ~21

+ 2=9~21Rc146 2

+ AV (~1)BV (~1; ~ )

4=9~21Rc146 Rz 612

2=9~21Rz146 Rc146 + AV (~14) 1162 1=6~1Rz146 + 1=36(3~ + 18~6

+ 42~4 ~1)~1 + 1=18

+ AV (~1)CV (~1)

6~ + 3~4 + ~1)~1Rz 61 + 1=18(6~ + 3~4 + ~1)~1Rz 612

+ 1=18(3~4 + 7~1)~1Rz16 1=36(3~4 + 7~1)~1Rc146

+ AV (~14)2

2=27Rz146

+ 5=27~1Rz16 + 1=54(2~ 2~4 47~1) + 1=27

4~ + 4~4 + 9~1)Rz 612

+ 1=27(4~ + ~1)Rz 61 1=54(4~4 + 13~1)Rc146

+ AV (~14)AV (~16)

~1

+ AV (~14)AV (~ )

2=45(~ ~4) 8=45(~ + 9~1)Rz 612 4=45(2~ ~4 + 9~1)Rz 61

+ AV (~14)BV (~1)

8=9~1Rz146 4=9(~4 + 2~1)~1Rz16

+ AV (~14)BV (~1; ~ )

4=9(~4 + 2~1)~1Rz 61

+ AV (~16) 1

1=12~1Rz146

1=18(6~ + 3~6 + ~1)~1Rz 61 + 1=36(6~ + 3~6 + ~1)~1Rz 612 + 1=72(12~ + 15~6 + 36~4 ~1)~1 + 1=36(3~6 + 7~1)~1Rz14 1=72(3~6 + 7~1)~1Rc146

+ AV (~16)2

162

1=27Rz146 + 5=54~1Rz14 + 1=27(2~ 2~6 5~1) + 1=54(4~ + 4~6 + 9~1)Rz 612 1=27(4~ + ~1)Rz 61 1=108(4~6 + 13~1)Rc146

+ AV (~16)AV (~ )

4=45(~ ~6) 4=45(~ + 9~1)Rz 612 + 4=45(2~ ~6 + 9~1)Rz 61

+ AV (~16)BV (~1)

4=9~1Rz146 2=9(~6 + 2~1)~1Rz14 + AV (~16)BV (~1; ~ )

4=9(~6

1=3~1~4Rz16 2

{ 21 {

+ 2~1)~1Rz 61

+ AV (~4)BV (~1)

+ AV (~4)BV (~1; ~ )

2=3~1~4Rz16 Rz 61 + AV (~4)BV (~ )

1=3~1~4Rz 612

+ AV (~46)AV (~ )

2=3~1Rz 612 4=9~1Rz16 Rz 61 + 4=9~1Rz14 Rz 61

+ AV (~46)BV (~1)

4=27(~6 + ~4 + ~1)~1Rz14 Rz16 + 2=27(~6 ~1)~1Rz14 2 + 2=27(~4 ~1)~1Rz16 2

+ AV (~46)BV (~1; ~ )

4=27(~6 ~4 3~1)~1Rz14 Rz 61

4=27(2~6 + ~4 + 3~1)~1Rz16 Rz 61

+ AV (~46)BV (~ )

2=9(3~ + ~4)~1Rz 612

2=9~1Rz 614 + AV (~ )BV (~1) 2=9~1Rz146 Rz 612 + 2=27~1~6Rz14 2

+ 1=27~1~4Rz16 2 4=9~21Rc146 Rz 612

+ AV (~ )2

+ AV (~ )BV (~1; ~ )

8=27~1~6Rz14 Rz 61

JHEP11(2015)097

+ 2=27~1~4Rz16 Rz 61 + 8=9~21Rz 614

+ AV (~ )BV (~ )

1=27(8~6 + ~4)~1Rz 612

+ AV (~ )CV (~1)

4=9~21Rz146 Rz 612

+ BV (~1)2

1=18~1Rz146 2 + 2=9~21Rz146 Rc146

+ BV (~1)BV (~1; ~ )

4=9~21Rz146 Rz 612

+ BV (~1)CV (~1)

2=9~21Rz146 2

+ AV23(~1) 1

162

3=4~1 1=2~1Rz16 1=4~1Rz14 + AV23(~14) 1 162

1=6~1 1=3~1Rz 61

1=3~1Rz 612 1=3~1Rz16 + 1=6~1Rc146

+ AV23(~16) 1

1=12~1 + 1=3~1Rz 61

162

1=6~1Rz 6121=6~1Rz14 +1=12~1Rc146

+ HV (1; ~1; ~1; ~1; ~1)

1=3~21 + 2=9~21Rc146 2

+ HV (1; ~1; ~1; ~ ; ~1)

8=9~21Rc146 Rz 612

+ HV (1; ~1; ~14; ~14; ~1)

11=18~1Rz146

5=6~21Rc146 1=9(4~4 7~1)~1Rz16

+ HV (1; ~1; ~16; ~16; ~1)

11=36~1Rz146

5=12~21Rc146 1=18(4~6 7~1

~1Rz14

+ HV (1; ~1; ~ ; ~ ; ~1)

8=9~21Rz 614

+ HV (1; ~14; ~14; ~ ; ~1)

1=27(~ ~4)(~ ~4 6~1) + 4=27(~ + 2~1)(~ 4~1)Rz 612 + 4=27(~2 ~4~ 4~1~ + ~1~4 6~21)Rz 61

+ HV (1; ~16; ~16; ~ ; ~1)

2=27(~

~6)(~ ~6 6~1) + 2=27(~ + 2~1)(~ 4~1)Rz 612 4=27(~2 ~6~ 4~1~ + ~1~6 6~21)Rz 61

+ HV (1; ~4; ~14; ~14; ~1)

3=4~1~4 + HV (1; ~46; ~14; ~16; ~1)

1=2(~6

+ ~4)~1

+ HV (1; ~ ; ~14; ~14; ~1)

1=12~1~ + 1=3~1~ Rz 61 + 1=3~1~ Rz 612

+ HV (1; ~ ; ~16; ~16; ~1)

1=6~1~ 1=3~1~ Rz 61 + 1=6~1~ Rz 612

+ HV (2; ~1; ~1; ~1; ~1)

4=9~21Rz146 Rc146 + HV (2; ~1; ~1; ~ ; ~1) 8=9~21Rz146 Rz 612

+ HV (2; ~1; ~14; ~14; ~1)

5=6~21Rz146

+ HV (2; ~1; ~16; ~16; ~1)

5=12~21Rz146

+ HV (5; ~1; ~1; ~1; ~1)

2=9~21Rz146 2 + HV1 (1; ~1; ~14; ~14; ~1)

4=9~1Rz146 + 4=3~21Rc146

+ 4=9(~4 4~1)~1Rz16

+ HV1 (1; ~1; ~16; ~16; ~1)

2=9~1Rz146 + 2=3~21Rc146

+ 2=9(~6 4~1)~1Rz14

+ HV1 (1; ~14; ~14; ~ ; ~1)

4=9(~ ~4)~1 + 16=9(~

{ 22 {

+ 2~1)~1Rz 612 + 8=9(2~ ~4 + 2~1)~1Rz 61

+ HV1 (1; ~16; ~16; ~ ; ~1)

8=9(~ ~6)~1

+ 8=9(~ + 2~1)~1Rz 612 8=9

2~ ~6 + 2~1)~1Rz 61

+ HV1 (2; ~1; ~14; ~14; ~1)

4=3~21Rz146 + HV1 (2; ~1; ~16; ~16; ~1)

2=3~21Rz146

+ HV21(1; ~1; ~14; ~14; ~1)

~21Rz16 1=2~21Rc146 + HV21(1; ~1; ~16; ~16; ~1)

1=2~21Rz14

1=4~21Rc146

+ HV21(1; ~4; ~14; ~14; ~1)

9=4~21 + HV21(1; ~46; ~14; ~16; ~1)

3~21

+ HV21(1; ~ ; ~14; ~14; ~1)

1=4~21 + ~21Rz 61 + ~21Rz 612 + HV21(1; ~ ; ~16; ~16; ~1)

1=2~21

~21Rz 61 + 1=2~21Rz 612

+ HV21(2; ~1; ~14; ~14; ~1)

1=2~21Rz146

JHEP11(2015)097

+ HV21(2; ~1; ~16; ~16; ~1)

1=4~21Rz146

+ HV27(1; ~1; ~14; ~14; ~1)

~1Rz16

+ 1=2~1Rc146

+ HV27(1; ~1; ~16; ~16; ~1)

1=2~1Rz14 + 1=4~1Rc146

+ HV27(1; ~4; ~14; ~14; ~1)

9=4~1

+ HV27(1; ~46; ~14; ~16; ~1)

3~1

+ HV27(1; ~ ; ~14; ~14; ~1)

1=4~1 ~1Rz 61 ~1Rz 612

+ HV27(1; ~ ; ~16; ~16; ~1)

1=2~1+~1Rz 611=2~1Rz 612

+ HV27(2; ~1; ~14; ~14; ~1)

1=2~1Rz146 + HV27(2; ~1; ~16; ~16; ~1)

1=4~1Rz146 (A.3)

B Expressions for the decay constant

F0 VF 2(4)1212 = +AV (~14)

1 + AV (~16)

1=2

(B.1)

F 30 VF 2(6L)1212 = +AV (~1)

4=3^

Lr5~1Rc146 +4^

Lr3Rz146 +4^

Lr3~1Rc146 10^

Lr2~14^Lr1~1

+ 4 ^

Lr0 Rz146 + 4 ^

Lr0 ~1 Rc146

+ AV (~14)

4 ^

Lr5 ~1 4 (~6 + 2 ~4)^

Lr4 10 (~4 + ~1)^Lr3

4(~4 + ~1)

^ + AV (~16)

2^

Lr5~1 2(~6 + 2~4)^

Lr4 5(~6 + ~1)^

Lr3 2(~6 + ~1)^Lr0

+ AV (~4)

12 ^

Lr4 ~4 6^

Lr2 ~4 24^

Lr1 ~4 + AV (~46)

8 (~6 + ~4) ^

Lr4 4 (~6 + ~4)^

Lr2

16 (~6 + ~4)

^ + AV (~ )

8=3 ^

Lr5 ~1 Rz 612 8^

Lr3 ~ Rz 612 2^

Lr2 ~ 8^

Lr1 ~

8

^

Lr0 ~ Rz 612 + 4=3 (2 ~6 + ~4) ^

Lr4

+ BV (~1)

4=3 ^

Lr5 ~1 Rz146 + 4 ^

Lr3 ~1 Rz146

+ 4 ^

Lr0 ~1 Rz146

+ BV (~14)

8 (~4 + ~1) (~6 + 2 ~4) ^

Lr6 4 (~4 + ~1) (~6 + 2 ~4)^Lr4

+ 4 (~4 + ~1)2 ^

Lr8 2 (~4 + ~1)2

^ + BV (~16)

4 (~6 + ~1) (~6 + 2 ~4) ^ Lr6

2 (~6 + ~1) (~6 + 2 ~4)

^

Lr4 + 2 (~6 + ~1)2 ^

Lr8

~6 + ~1)2 ^

Lr5

+ AV23(~1)

4 ^

Lr3 Rc146

+ 6 ^

Lr2 + 12 ^

Lr1 4

Lr0 Rc146

^

+ AV23(~14)

12 ^

Lr3 + 24 ^

Lr0 + AV23(~16)

6 ^

Lr3 + 12 ^ Lr0

+ AV23(~4)

18 ^

Lr2 + AV23(~46)

24 ^

Lr2 + AV23(~ )

8 ^

Lr3 Rz 612 + 6 ^

Lr2 + 8 ^

Lr0 Rz 612

{ 23 {

+ BV23(~1)

4 ^

Lr3 Rz146 4

Lr0 Rz146

^

(B.2)

F 30 VF 2(6R)1212 = +A(~1)BV (~14)

1=18Rz146 1=9(~4 + 2~1)Rz16

+ A(~1)BV (~16)

1=36Rz146 1=18(~6 + 2~1)Rz14

+ A(~14)AV (~14)

5=54 5=27Rz 61 5=27Rz 612 5=27Rz16 + 5=54Rc146

+ A(~16)AV (~16)

5=108 + 5=27Rz 61 5=54Rz 612 5=54Rz14 + 5=108Rc146

+ A(~ )BV (~14)

1=36(~ ~4) 1=9(~ + ~4 + ~1)Rz 61 1=9(~ ~1)Rz 612

+ A(~ )BV (~16)

1=18(~ ~6) + 1=9(~ + ~6 + ~1)Rz 61 1=18(~ ~1)Rz 612

+ B(~14)AV (~1)

1=18Rz146 1=9(~4 + 2~1)Rz16 + B(~14)AV (~ ) 1=36(~ ~4)

1=9(~ + ~4 + ~1)Rz 61 1=9(~ ~1)Rz 612

+ B(~16)AV (~1)

JHEP11(2015)097

1=36Rz146

1=18(~6 + 2~1)Rz14

+ B(~16)AV (~ )

1=18(~ ~6) + 1=9(~ + ~6 + ~1)Rz 61

1=18(~ ~1)Rz 612

+ AV (~1) 1

162

1=8~1 1=12~1Rz16 1=24~1Rz14

+ AV (~1)BV (~14)

1=18Rz146 1=9(~4 + 2~1)Rz16 + AV (~1)BV (~16)

1=36Rz146

1=18(~6 + 2~1)Rz14

+ AV (~14) 1

1=12Rz146 1=24(~ + 6~6 + 14~4 + 5~1)

1=12(2~ + ~4 + ~1)Rz 61 1=12(2~ + ~4 + ~1)Rz 612 1=12(~4 + 3~1)Rz16 + 1=24(~4 + 3~1)Rc146

+ AV (~14)2

162

5=108 5=54Rz 61 5=54Rz 612 5=54Rz16

1=24Rz146 + 1=12(2~ + ~6 + ~1)Rz 61

1=24(2~ + ~6 + ~1)Rz 612 1=48

+ 5=108Rc146

+ AV (~16) 1

162

4~ + 5~6 + 12~4 + 5~1) 1=24(~6 + 3~1)Rz14

+ 1=48(~6 + 3~1)Rc146

+ AV (~16)2

5=216 + 5=54Rz 61 5=108Rz 612 5=108Rz14

+ 5=216Rc146

+ AV (~ )BV (~14)

1=36(~ ~4) 1=9(~ + ~4 + ~1)Rz 61

1=9(~ ~1)Rz 612

+ AV (~ )BV (~16)

1=18(~ ~6) + 1=9(~ + ~6 + ~1)Rz 61

1=18(~ ~1)Rz 612

+ AV23(~1) 1

162

3=8 + 1=4Rz16 + 1=8Rz14

1=12 + 1=6Rz 61 + 1=6Rz 612 + 1=6Rz16 1=12Rc146 + AV23(~16) 1

162

1=24 1=6Rz 61 + 1=12Rz 612 + 1=12Rz14 1=24Rc146

+ HV (1; ~1; ~14; ~14; ~1)

1=12Rz146 1=6~1Rz16 + 1=12~1Rc146

+ HV (1; ~1; ~16; ~16; ~1)

1=24Rz146 1=12~1Rz14 + 1=24~1Rc146

+ HV (1; ~4; ~14; ~14; ~1)

+ AV23(~14) 1

162

3=8~4

+ HV (1; ~46; ~14; ~16; ~1)

1=4(~6 + ~4)

{ 24 {

+ HV (1; ~ ; ~14; ~14; ~1)

1=24~ 1=6~ Rz 61 1=6~ Rz 612

+ HV (1; ~ ; ~16; ~16; ~1)

1=12~ + 1=6~ Rz 61 1=12~ Rz 612

+ HV (2; ~1; ~14; ~14; ~1)

5=54~1Rz146 + HV (2; ~1; ~16; ~16; ~1)

5=108~1Rz146

+ HV1 (2; ~1; ~14; ~14; ~1)

1=108~1Rz146

+ HV1 (2; ~1; ~16; ~16; ~1)

1=216~1Rz146

+ HV1 (3; ~14; ~1; ~14; ~1)

1=54~1Rz146

+ HV1 (3; ~16; ~1; ~16; ~1)

1=108~1Rz146

+ HV27(1; ~1; ~14; ~14; ~1)

1=2Rz16 1=4Rc146 +HV27(1; ~1; ~16; ~16; ~1)

1=4Rz14 1=8Rc146

JHEP11(2015)097

+ HV27(1; ~4; ~14; ~14; ~1)

9=8 + HV27(1; ~46; ~14; ~16; ~1)

3=2

+ HV27(1; ~ ; ~14; ~14; ~1)

1=8 + 1=2Rz 61 + 1=2Rz 612

+ HV27(1; ~ ; ~16; ~16; ~1)

1=4 1=2Rz 61 + 1=4Rz 612

+ HV27(2; ~1; ~14; ~14; ~1)

1=4Rz146

+ HV27(2; ~1; ~16; ~16; ~1)

1=8Rz146

+ H[prime]V (1; ~1; ~1; ~1; ~1)

1=6~21+1=9~21Rc146 2

+H[prime]V (1; ~1; ~1; ~ ; ~1)

4=9~21Rc146 Rz 612

+ H[prime]V (1; ~1; ~14; ~14; ~1)

11=36~1Rz146 5=12~21Rc146 1=18(4~4 7~1)~1Rz16

+ H[prime]V (1; ~1; ~16; ~16; ~1)

11=72~1Rz146 5=24~21Rc146 1=36(4~6 7~1)~1Rz14

+ H[prime]V (1; ~1; ~ ; ~ ; ~1)

4=9~21Rz 614 + H[prime]V (1; ~14; ~14; ~ ; ~1)

1=54(~ ~4)(~ ~4

6~1) + 2=27(~ + 2~1)(~ 4~1)Rz 612 + 2=27(~2 ~4~ 4~1~ + ~1~4 6~21)Rz 61

+ H[prime]V (1; ~16; ~16; ~ ; ~1) 1=27(~ ~6)(~ ~6 6~1) + 1=27(~ + 2~1)(~ 4~1)

Rz 612 2=27(~2 ~6~ 4~1~ + ~1~6 6~21)Rz 61

+ H[prime]V (1; ~4; ~14; ~14; ~1)

3=8~1~4

+ H[prime]V (1; ~46; ~14; ~16; ~1)

1=4(~6 + ~4)~1 + H[prime]V (1; ~ ; ~14; ~14; ~1)

1=24~1~

+ 1=6~1~ Rz 61 + 1=6~1~ Rz 612

+ H[prime]V (1; ~ ; ~16; ~16; ~1)

1=12~1~ 1=6~1~ Rz 61

+ 1=12~1~ Rz 612

+ H[prime]V (2; ~1; ~1; ~1; ~1)

2=9~21Rz146 Rc146

+ H[prime]V (2; ~1; ~1; ~ ; ~1)

4=9~21Rz146 Rz 612

+ H[prime]V (2; ~1; ~14; ~14; ~1)

5=12~21Rz146

+ H[prime]V (2; ~1; ~16; ~16; ~1)

5=24~21Rz146

+ H[prime]V (5; ~1; ~1; ~1; ~1)

1=9~21Rz146 2

+ H[prime]V1 (1; ~1; ~14; ~14; ~1)

2=9~1Rz146 + 2=3~21Rc146 + 2=9(~4 4~1)~1Rz16

+ H[prime]V1 (1; ~1; ~16; ~16; ~1)

1=9~1Rz146 + 1=3~21Rc146 + 1=9(~6 4~1)~1Rz14

+ H[prime]V1 (1; ~14; ~14; ~ ; ~1)

2=9(~ ~4)~1 + 8=9(~ + 2~1)~1Rz 612 + 4=9(2~ ~4

+ 2~1)~1Rz 61

+ H[prime]V1 (1; ~16; ~16; ~ ; ~1)

4=9(~ ~6)~1 + 4=9(~ + 2~1)~1Rz 612

4=9(2~ ~6 + 2~1)~1Rz 61

+ H[prime]V1 (2; ~1; ~14; ~14; ~1)

2=3~21Rz146

{ 25 {

+ H[prime]V1 (2; ~1; ~16; ~16; ~1)

1=3~21Rz146

+H[prime]V21 (1; ~1; ~14; ~14; ~1)

1=2~21Rz16 1=4~21Rc146

+ H[prime]V21 (1; ~1; ~16; ~16; ~1)

1=4~21Rz14 1=8~21Rc146 + H[prime]V21 (1; ~4; ~14; ~14; ~1)

9=8~21

+ H[prime]V21 (1; ~46; ~14; ~16; ~1)

3=2~21 + H[prime]V21 (1; ~ ; ~14; ~14; ~1)

1=8~21 + 1=2~21Rz 61

+ 1=2~21Rz 612

+ H[prime]V21 (1; ~ ; ~16; ~16; ~1)

1=4~21 1=2~21Rz 61 + 1=4~21Rz 612

+ H[prime]V21 (2; ~1; ~14; ~14; ~1)

1=4~21Rz146

+ H[prime]V21 (2; ~1; ~16; ~16; ~1)

1=8~21Rz146

+ H[prime]V27 (1; ~1; ~14; ~14; ~1)

1=2~1Rz16 + 1=4~1Rc146

+ H[prime]V27 (1; ~1; ~16; ~16; ~1)

1=4~1Rz14 +1=8~1Rc146 + H[prime]V27 (1; ~4; ~14; ~14; ~1)

9=8~1

JHEP11(2015)097

+ H[prime]V27 (1; ~46; ~14; ~16; ~1)

3=2~1

+ H[prime]V27 (1; ~ ; ~14; ~14; ~1)

1=8~1 1=2~1Rz 61

1=2~1Rz 612

+ H[prime]V27 (1; ~ ; ~16; ~16; ~1)

1=4~1 + 1=2~1Rz 61 1=4~1Rz 612

+ H[prime]V27 (2; ~1; ~14; ~14; ~1)

1=8~1Rz146 (B.3)

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

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