Two-Point Functions and S-Parameter in QCD-like Theories
aa r X i v : . [ h e p - ph ] N ov LU TP 11-40November 2011
Two-Point Functions and S-Parameterin QCD-like Theories
Johan Bijnens and Jie Lu
Department of Astronomy and Theoretical Physics, Lund University,S¨olvegatan 14A, SE 223-62 Lund, Sweden
Abstract
We calculated the vector, axial-vector, scalar and pseudo-scalar two-pointfunctions up to two-loop level in the low-energy effective field theory for threedifferent QCD-like theories. In addition we also calculated the pseudo-scalardecay constant G M . The QCD-like theories we used are those with fermions ina complex, real or pseudo-real representation with in general n flavours. Thesecase correspond to global symmetry breaking pattern of SU ( n ) L × SU ( n ) R → SU ( n ) V , SU (2 n ) → SO (2 n ) or SU (2 n ) → Sp (2 n ). We also estimated the Sparameter for those different theories. ontents Q a case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4.2 Singlet case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 The Pseudo-Scalar Two-Point Functions . . . . . . . . . . . . . . . . 163.5.1 The meson pseudo-scalar decay constant G M . . . . . . . . . . 163.5.2 X a case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5.3 Singlet case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Large n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A.1 One-loop integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A.2 Sunset integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
The different global symmetry breaking patterns of QCD-like theories with a vector-like gauge group have been summarized in [1, 2, 3] around 30 years ago. The globalsymmetry and its spontaneous breaking depend on whether the fermions live in acomplex, real and pseudo-real representation of the gauge group. For n identicalfermions this corresponds to the symmetry breaking pattern SU ( n ) L × SU ( n ) R → SU ( n ) V , SU (2 n ) → SO (2 n ) and SU (2 n ) → Sp (2 n ) respectively. These theories canbe used to characterize some of technicolor models with vector-like gauge bosons.QCD-like theories are also important in the theory of finite baryon density. Here thereal and pseudo-real case allow to investigate the mechanism of diquark condensateand finite density without the sign problem. A main nonperturbative tool in study-ing strongly interacting theories is lattice gauge theory. Numerical calculations areperformed at finite fermion mass and need in general to be extrapolated to the zeromass limit. In the case of QCD Chiral Perturbation Theory (ChPT) is used to helpwith this extrapolation. Our work has the intention of providing similar formulas1or the QCD-like theories using the effective field theory (EFT) appropriate for thealternative global symmetry patterns.These EFT have been used at lowest order (LO) [4] with earlier work to be foundin [5, 6, 7] and some studies at next-to-leading order (NLO) have also appeared[8, 9, 10]. The former two are the usual QCD case with n flavours. In our earlierpapers [11, 12] we have systematically studied the effective field theory of these threedifferent QCD-like theories to next-to-next-to-leading order (NNLO). We managed towrite the EFT of these cases in an extremely similar form. We calculated the quark-antiquark condensates, the mass and decay constant of the pseudo-Goldstone bosons[11], and meson-meson scattering [12]. In this paper we extend the analysis to two-point correlation functions. We obtain expressions for the vector, axial-vector, scalarand pseudo-scalar two-point functions as well as the pion pseudo-scalar coupling G M to NNLO or order p .In our earlier work [11, 12], we called the three different cases QCD or complex,adjoint or real and two-colour or pseudo-real. In this paper we use only the latter,more general, terminology.One motivation for this set of work was the study of strongly interacting Higgssectors, reviews are [13, 14]. For any model beyond the Standard Model, passing thetest of oblique corrections, or precision LEP observables, is crucial [15, 16]. Over theyears, the impact of the oblique corrections in those models have been studied quiteintensively but in strongly interacting cases mainly an analogy with QCD has beeninvoked. Lattice gauge theory methods allow to study strongly interacting modelsfrom first principles. The contributions from these theories to the S -parameter can becalculated using the two-point functions studied here and our formulas are useful toperform the extrapolation to the massless case. This was in fact the major motivationfor the present work but we included the other two-point functions for completeness.The paper is organized as follows. In Section 2 we give a brief introduction toEFT for the three different cases. Section 3 is the main part of the paper. We definethe fermion currents and the two-point functions in Section 3.1. In Sections 3.2to 3.5, we present the calculation of vector, axial-vector, scalar, pseudo-scalar two-point functions. In Section 4, we discuss the oblique corrections and the S-parameter.Section 5 summarizes our main results and we present the definition. In this section we briefly review the EFT of QCD-like theories, the details can befound in the earlier paper [11]. The basic methods are those of Chiral PerturbationTheory [17, 18]. The counting of orders is in all cases the same as in ChPT, we countmomenta as order p and the fermion mass m as order p . The case of n fermions in a complex representation is essentially like QCD. TheLagrangian with external left and right vector, scalar and pseudos-calar external We use LO, NLO and NNLO as synomyms for order p , order p and order p calculations evenif the order p vanishes. l µ , r µ , s and p , is L = q Li iγ µ D µ q Li + q Ri iγ µ D µ q Ri + q Li γ µ l µij q Lj + q Ri γ µ r µij q Rj − q Ri M ij q Lj − q Li M † ij q Rj i, j = 1 , , ..., n . (1)The covariant derivative is given by D µ q = ∂ µ q − iG µ q , and the mass matrix M = s − ip . The sums shown are over the flavour index. The sums over gauge groupindices are implicit.The Lagrangian (1) has a symmetry SU ( n ) L × SU ( n ) R which is made local bythe external sources [8, 18]. The quark-anti-quark condensate h ¯ qq i breaks SU ( n ) L × SU ( n ) R spontaneously to the diagonal subgroup SU ( n ) V . According to the Nambu-Goldstone theorem, n − m explicitly by setting s = m + s . This mass term explicitly breaksthe symmetry SU ( n ) L × SU ( n ) R down to SU ( n ) V as well and gives the Goldstonebosons a small mass.The Goldstone boson manifold SU ( n ) L × SU ( n ) R /SU ( n ) V can be parametrizedby u = exp i √ F π a T a ! a = 1 , , ..., n − . (2)The T a are the generators of SU ( n ) normalized to h T a T b i = δ ab . The notation h A i stands for the trace over flavour indices. u transforms under g L × g R ∈ SU ( n ) L × SU ( n ) R as u → g R uh † = hug † L where h is the “compensator” and is a function of u , g L and g R . The methods are those of [19], but we use the notation of [20, 21]. We canconstruct quantities which transform under the unbroken group H as : O → hOh † u µ = i [ u † ( ∂ µ − ir µ ) u − u ( ∂ µ − l µ ) u † ] , ∇ µ O = ∂ µ O + Γ µ O − O Γ µ ,χ ± = u † χu † ± uχ † u ,f ± µν = ul µν u † ± u † r µν u . (3)The field strengths l µν and r µν are l µν = ∂ µ l ν − ∂ ν l µ − i [ l µ , l ν ] ,r µν = ∂ µ r ν − ∂ ν r µ − i [ r µ , r ν ] . (4)The covariant derivative ∇ µ containsΓ µ = 12 [ u † ( ∂ µ − ir µ ) u + u ( ∂ µ − l µ ) u † ] . (5) χ contains the matrix M , which is the combination of scalar and pseudo-scalarsources χ = 2 B M = 2 B ( s − ip ) . (6)Using the quantities in (3), we can find the leading order, p , Lagrangian which isinvariant under Lorentz and chiral symmetry: L = F h u µ u µ + χ + i . (7)The subscript “2” stands for the order of p . The p and p Lagrangian will beexplained in Section 2.3. 3 .2 Real and Pseudo-Real representation
The case of n fermions in a real or pseudo-real representation of the gauge groupwe can treat in a similar way as the complex case. In the real case, the globalsymmetry breaking pattern is SU (2 n ) → SO (2 n ), and the number of generatedGoldstone bosons is 2 n + n −
1. In the pseudo-real case, the symmetry breaking is SU (2 n ) → Sp (2 n ), and the number of generated Goldstone is 2 n − n −
1. In bothcases anti-fermions are in the same representation of the gauge group and can be puttogether in a 2 n vector ˆ q , see [11] for more details.The condensate can now be a diquark condensate as well as a quark-antiquarkcondensate. Our choice of vacuum corresponds to a quark-anti-quark condensate. Interms of the 2 n fermion vector ˆ q they can be written asReal : h ˆ q T CJ S ˆ q i + h . c . J S = ! , (8)Pseudo − Real : h ˆ q α ǫ αβ CJ A ˆ q β i + h . c . J A = − II 0 ! . (9)Here C is the charge conjugation operator. J S and J A are symmetric and anti-symmetric 2 n × n matrices, I is the n × n unit matrix. Since J S and J A often appearin the same way in the expressions, we use J for both cases unless a distinction isneeded.The generators, T a , of the global symmetry group SU (2 n ) can be separatedinto belonging to the broken, X a , or unbroken part, Q a . They satisfy the followingrelations with J : J Q a = − Q aT J , J X a = X aT J , (10)The Goldstone boson manifold can be parametrized with U = uJ u T → gU g T , with u = exp i √ F π a X a ! . (11)where J = J S and a runs from 1 to 2 n + n − J = J A and a runs from 1 to 2 n − n − u µ = i [ u † ( ∂ µ − iV µ ) u − u ( ∂ µ + iJ V Tµ J ) u † ] , Γ µ = 12 [ u † ( ∂ µ − iV µ ) u + u ( ∂ µ + iJ V Tµ J ) u † ] .f ± µν = J uV µν u † J ± uV µν u † ,χ ± = u † χJ u † ± uJ χ † u . (12)The 2 n × n matrix V µ includes the left and right-handed external sources V µ = r µ − l Tµ ! (13)and V µν is the field strength V µν = ∂ µ V ν − ∂ ν V µ − i ( V µ V ν − V ν V µ ) . (14)4 include the matrix ˆ M via χ = 2 B ˆ M [11]. Those quantities behave similarly asthose (3) if we take − J V Tµ J → l µ , V µ → r µ . (15)With this correspondence, the Lagrangian of the real and pseudo-real case has thesame form as the complex one. However one has to remember there are differencesin the generators, external sources, coupling constants,. . . . Anyway, now we can usethe techniques of ChPT to perform the calculations. Using Lorentz and chiral invariance, we can write down the p EFT lagrangian [18]for all three cases using the quantities listed in (3) and (12): L = L h u µ u ν u µ u ν i + L h u µ u µ ih u ν u ν i + L h u µ u ν ih u µ u ν i + L h u µ u µ u ν u ν i + L h u µ u µ ih χ + i + L h u µ u µ χ + i + L h χ + i + L h χ − i + 12 L h χ + χ − i− iL h f + µν u µ u ν i + 14 L h f − f − i + H h l µν l µν + r µν r µν i + H h χχ † i . (16)To do the renormalization, we use the ChPT MS scheme with dimensional regular-ization [18, 8, 21]. The bare coupling constants L i are defined as L i = ( cµ ) d − [Γ i Λ + L ri ( µ )] , (17)where the dimension d = 4 − ǫ , andΛ = 116 π ( d − , (18)ln c = −
12 [ln 4 π + Γ ′ (1) + 1] . (19)The coefficients Γ i for the complex case have been obtained in [8], for the real andpseudo-real case we have calculated them earlier in [11]. However, there are mistakesin the coefficients of L , L and H in the Table 1 of [11]. These mistakes had noeffects on our previous calculations. We therefore list all the coefficients here againin Table 1.The p Lagrangian for the complex case and general n has been obtained in [20],it contains 112+3 terms. The divergence structure of the bare coupling constants K i in the p can be written as K i = ( cµ ) d − (cid:20) K ri − Γ (2) i Λ − (cid:18) π Γ (1) i + Γ ( L ) i (cid:19) Λ (cid:21) . (20)The coefficients Γ (2) i , Γ (1) i and Γ ( L ) i for the complex case have been obtained in [21].For the real and pseudo-real case, the p Lagrangian has the same form as in thecomplex case but some terms might be redundant. The divergence structure as givenin (20) still holds but the coefficients are not known. One check on our results thatremains is that all the non-local divergences cancel. The K i have been made dimensionless by including a factor of 1 /F explicitly in the order p Lagrangian. n/
48 ( n + 4) /
48 ( n − /
481 1 /
16 1 /
32 1 /
322 1 / /
16 1 / n/
24 ( n − /
24 ( n + 2) /
244 1 / /
16 1 / n/ n/ n/
86 ( n + 2) / (16 n ) ( n + 1) / (32 n ) ( n + 1) / (32 n )7 0 0 08 ( n − / (16 n ) ( n + n − / (16 n ) ( n − n − / (16 n )9 n/
12 ( n + 1) /
12 ( n − / − n/ − ( n + 1) / − ( n − / − n/ − ( n + 1) / − ( n − / n − / (8 n ) ( n + n − / (8 n ) ( n − n − / (8 n )Table 1: The coefficients Γ i for the three cases that are needed to absorb the diver-gences at NLO. The last two lines correspond to the terms with H and H . This isthe same as Table 1 in [11] but with the error for L , L and H corrected. The effective action of the fermion level theory with external sources isexp { iZ ( l µ , r µ , s, p ) } = Z D q D ¯ q D G µ exp (cid:26) i Z d x L QCD ( q, ¯ q, G µ , l µ , r µ , s, p ) (cid:27) (21)At low energies, i.e. below 1 GeV in QCD, the effective action can be obtained alsofrom the low-energy effective theoryexp { iZ ( l µ , r µ , s, p ) } = Z D U exp (cid:26) i Z d x L eff ( U, l µ , r µ , s, p ) (cid:27) . (22)With this effective action, the n-point Green functions can be easily derived by takingthe functional derivative w.r.t. the external sources of Z ( J ) G ( n ) ( x , . . . , x n ) = δ n i n δj ( x ) . . . δj ( x n ) Z [ J ] (cid:12)(cid:12)(cid:12)(cid:12) J =0 . (23)Here j stands for any of the external sources l µ , r µ , s, p and J for the whole set ofthem.The vector current v µ and axial-vector current a µ are included via l µ = v µ − a µ , r µ = v µ + a µ . (24)In this paper we will calculate the two-point functions of vector, axial-vector,scalar and pseudo-scalar currents. The fermion currents in the complex case are6efined as V aµ ( x ) = q i T aij γ µ q j , (25) A aµ ( x ) = q i T aij γ µ γ q j , (26) S a ( x ) = − q i T aij q j , (27) P a ( x ) = iq i T aij γ q j . (28) T a is an SU ( n ) generator or in addition for the singlet scalar and pseudo-scalarcurrent the unit matrix which we label by T . These currents also exist the forreal and pseudo-real case. In this case also currents with two fermions or two anti-fermions exist. These can be combined with those above. The generators can thenbecome SU (2 n ) generators. All conserved generators are like the vector or scalarcase while the broken generators are like the axial-vecor or pseudo-scalar case. Allthose cases are related to the ones with the currents of (25)-(28) via transformationsunder the unbroken part of the symmetry group.The definitions of the two-point functions areΠ V aµν ( q ) ≡ i Z d x e iq · x h | T ( V aµ ( x ) V aν (0)) † | i , Π Aaµν ( q ) ≡ i Z d x e iq · x h | T ( A aµ ( x ) A aν (0)) † | i , Π SMaµ ( q ) ≡ i Z d x e iq · x h | T ( V aµ ( x ) S a (0)) † | i , Π P Maµ ( q ) ≡ i Z d x e iq · x h | T ( A aµ ( x ) P a (0)) † | i , Π Sa ( q ) ≡ i Z d x e iq · x h | T ( S a ( x ) S a (0)) † | i , Π P a ( q ) ≡ i Z d x e iq · x h | T ( P a ( x ) P a (0)) † | i . (29)Using Lorentz invariance the two-point functions with vectors and axial-vectors canbe decomposed in scalar functionsΠ V aµν = ( q µ q ν − q g µν )Π (1) V a ( q ) + q µ q ν Π (0) V a ( q ) . (30)where Π (1) V a ( q ) is the transverse part and Π (0) V a ( q ) is the longitudinal part or alter-natively the spin one and spin 0 part. The same definition holds for the axial-vectortwo-point functions. The mixed functions can be decomposed asΠ SMaµ = q µ Π SMa , Π P Maµ = iq µ Π P Ma . (31)Using the divergence of fermion currents and equal time commutation relations, wefind that some two-point functions are related to each other by Ward identities. Inthe equal mass case considered here, they areΠ (0) V a = Π
SMa = 0 ,q Π (0) Aa = 2 m Π P Ma ,q Π (0) Aa = 4 m Π P a + 4 m h ¯ qq i . (32) We have defined here the current with fermion-anti-fermion operators, hence the SU ( n ) for n fermions. For the real and pseudo-real case, the unbroken symmetry relates them also to difermionot dianti-fermmion operators.
1) (2) (3)(4) (5) (6) (7)(8) (9) (10) (11)(12) (13) (14) (15)
Figure 1: The diagrams for the vector two-point function. A filled circle is a vertexfrom L , a filled square a vertex from L , and an open square a vertex from L . Thetop line is order p . The remaining ones are order p .The vacuum expectation value is the single quark-anti-quark one. We will use thelast relation to double check our results of axial-vector and pseudo-scalar two-pointfunctions.The mixed two-point functions, Π SMa and Π
P Ma we do not discuss further sincethey are fully given by the Ward identities.
The vector two-point function is defined in (29). The longitudinal part vanishes forall three cases because of the Ward identities.The Feynman diagrams for the vector two-point function are listed in Fig. 1.There is no diagram at lowest order. The diagrams at NLO are (1–3) in Fig. 1. TheNNLO diagrams are (4–15). The 3-flavour QCD case is known to NNLO [22, 23].We have rewritten the results in terms of the physical mass and decay constant.For these we use the notation M M and F M rather than the M phys and F phys used in[11, 12]. Their expression in terms of the lowest order quantities F and M = 2 B m can be found in [11]. We also use the quantities L = 116 π log M M µ and π = 116 π . (33)The loop integral B is defined in Appendix A.1.8he results up to NNLO for three different cases are listed below, where the firstline in each case is the NLO contributions, the rest are NNLO contributions. Complex Π (1) V V = − nq h B ( M M , M M , q ) + 2 LM M i − L r − H r + 1 F M ( M M q Ln − L r n ! B ( M M , M M , q ) + 4 n q [ B ( M M , M M , q )] + M M q L n − q K r + 8 M M ( LL r n − K r − K r n ) ) , (34) Real Π (1) V V = − q ( n + 1) h B ( M M , M M , q ) + 2 M M L i − L r − H r + 1 F M ( " M M q L ( n + 1) − n + 1) L r B ( M M , M M , q )+ 4 q ( n + 1) [ B ( M M , M M , q )] + M M q L ( n + 1) − q K r + 8 M M [ LL r ( n + 1) − K r − K r n ] ) , (35) Pseudo − Real Π (1) V V = − q ( n − h B ( M M , M M , q ) + 2 M M L i − L r − H r + 1 F M ( " M M q L ( n − − L r ( n − B ( M M , M M , q )+ 4 q ( n − [ B ( M M , M M , q )] + M M q L ( n − − q K r + 8 M M [ LL r ( n − − K r − K r n ] . (36)The complex result with n = 3 agrees with [22] when the masses there are set equal. The axial-vector two-point function is defined in (29). Similar to the vector two-pointfunction, it also can be decomposed in a transverse and longitudinal part.Π µνAA = ( q µ q ν − q g µν )Π (1) AA ( q ) + q µ q ν Π (0) AA ( q ) . (37)The diagrams contributing at LO are shown in (1–2) in Fig. 2. The LO resultsare the same for all three cases. The result isΠ µνAA ( q ) = 2 F g µν − q µ q ν q − M ! . (38) F and M are the LO decay constant and mass respectively. Note that in the masslesslimit this has only a transverse part as follows from the Ward identities.The diagrams at NLO are (3–10) in Fig. 2 and the NNLO diagrams are (11–48)in Fig. 3. 9
1) (2)(3) (4) (5) (6)(7) (8) (9) (10)
Figure 2: The axial-vector two-point function at LO and NLO. The filled circle is avertex from L , The filled square is a vertex from L , and the open square is a vertexfrom L .. Many of the diagrams are one-particle-reducible and at first sight have doubleand triple poles at q = M . From general properties of field theory these should beresummable in into a single pole at the physical mass, q = M M and a nonsingularpart that only has cuts. The residue at the pole is the decay constant squared. Wemust thus find contributions that allow for the last term in (38) the lowest order F , M to be replaced by F M , M M . It turns out to be advantageous to also do thisin the first term. Most of the corrections are already included in this way.At NLO the remaining part is only from the tree level diagram (3) in Fig. 2 andis Π (1) AA = 4 L r − H r , Π (0) AA = 0 . (39)So we can express our result up to NNLO asΠ µνAA ( q ) = 2 F M g µν − q µ q ν q − M M ! + ( q µ q ν − q g µν )(4 L r − H r )+ 1 F M " ( q µ q ν − q g µν ) ˆΠ (1) AA ( q ) + q µ q ν ˆΠ (0) AA ( q ) . (40)The ˆΠ (0) AA ( q ) and ˆΠ (1) AA ( q ) are the remainders at NNLO and have no singularity at q = M M .The transverse part can be obtained from the part containing g µν as an overallfactor. So the transverse part cannot come from the one-particle reducible diagramsand only gets contributions from diagrams (11–16) at NNLO. The sunset integrals H F and H F appearing in the results are defined in Appendix A.2.10
11) (12) (13) (14) (15)(16) (17) (18) (19) (20)(21) (22) (23) (24) (25)(26) (27) (28) (29) (30)(31) (32) (33) (34) (35)(36) (37) (38) (39) (40)(41) (42) (43) (44) (45)(46) (47) (48)
Figure 3: The axial-vector two-point function at NNLO. The filled circle is a vertexfrom L , the filled square is a vertex from L , and the open square is a vertex from L H M and H M defined in Appendix A.2. Complex ˆΠ (1) AA ( q ) = n " M M q H F ( M M , M M , M M , q ) − H F ( M M , M M , M M , q ) + M M " L n − nLL r −
32 ( K r + K r n + K r + K r n ) − q (16 K r + 8 K r ) − M M q (cid:18) K r + 32 n L (cid:19) + π L " M M q n n − ! + M M n + π " M M q n − n π − n − ! + M M n π
36 + 172 ! + n q , (41)ˆΠ (0) AA ( q ) = " M M (cid:18) − n (cid:19) − M M q n H M ( M M , M M , M M , q ) − M M n H M ( M M , M M , M M , q ) + 8 M M q K r + M M q " π L − n − n ! + π − n − n ! , (42) Real ˆΠ (1) AA ( q ) = n n + 1) " M M q H F ( M M , M M , M M , q ) − H F ( M M , M M , M M , q ) + M M " L n ( n + 1) − nLL r − K r + 2 nK r + K r + 2 nK r ) − q (16 K r + 8 K r ) − M M q (cid:20) K r + 32 n ( n + 1) L (cid:21) + π L " M M q n n + 43 n − n − ! + M M n ( n + 1) + π " M M q − n π − nπ − n
64 + 78 n + 5 n − n − ! + M M n ( n + 1) π
36 + 172 ! + n
96 ( n + 1) q , (43)ˆΠ (0) AA ( q ) = " − M M q n n + 1) + M M ( n − (cid:18) n − (cid:19) H M ( M M , M M , M M , q ) − M M n ( n + 1) H M ( M M , M M , M M , q ) + 8 M M q K r π M M q L − n − n − n
24 + 12 n + 16 ! + π M M q − n − n − n
192 + 78 n + 724 ! , (44) Pseudo − Real ˆΠ (1) AA ( q ) = n n − " M M q H F ( M M , M M , M M , q ) − H F ( M M , M M , M M , q ) − M M ( K r + 2 nK r + K r + 2 nK r ) − q (2 K r + K r ) − M M q K r − M M q n n − L + M M n ( n − L − M M nLL r + π LM M q n n − n
24 + 12 n − ! + π LM M n − n ! + π " M M q − n π − n
64 + 78 n + nπ − n
192 + 78 n − ! + M M n ( n − π
36 + 172 ! + q n
96 ( n − , (45)ˆΠ (0) AA ( q ) = " − M M q n n − − M M ( n + 1) (cid:18) n − (cid:19) H M ( M M , M M , M M , q ) − M M n ( n − H M ( M M , M M , M M , q ) + 8 M M q K r + π M M q L − n − n + 7 n − n + 16 ! + π M M q − n − n + 101 n − n + 724 ! . (46)The axial two-point function is known in 3-flavour ChPT [22, 24]. We havechecked that our result agrees with the one in [22] in the limit of equal masses. The scalar two-point function is defined in (29), which contains the unbroken gener-ator case ( T a = Q a ) and the singlet case ( a = 0).The Feynman diagrams for both cases are the same as those for the vector two-point function shown in Figure 1 except that diagrams (2) and (5–7) are absent.Diagrams (1) and (3) are at NLO, and the diagrams (4) and (8–11) are at NNLO. Q a case The scalar two-point functions are similar to the vector two-point functions, the LOresults are zero for all the three cases since the vertex at LO is absent. We haverewritten again everything in terms of the physical mass and decay constant, M M and F M . The results for the three cases are given below. The first line is the NLOcontribution and the remainder is the NNLO contribution.13 omplex Π SS = B (cid:26) H r + 16 L r + 1 n (cid:16) n − (cid:17) B ( m , q ) (cid:27) + B F M ( q (8 K r + 32 K r ) + M M K r + 64 K r n ! + M M L " (cid:18) n − n (cid:19) L r − L r + (cid:16) n − (cid:17) n L r + B ( m , q ) (cid:16) n − (cid:17) " q n L r + M M n L − L r − n L r − L r + 64 n L r ! + B ( m , q ) (cid:16) n − (cid:17) q − M M n ! ) , (47) Real Π SS = B (cid:26) H r + 16 L r + 1 n ( n −
1) ( n + 2) B ( m , q ) (cid:27) + B F M ( q (8 K r + 32 K r ) + M M K r + 128 nK r ! + M M L " (cid:18) n − − n (cid:19) L r − L r + ( n −
1) ( n + 2) 16 n L r + B ( m , q ) ( n −
1) ( n + 2) " q n L r + M M (cid:18) − n + 1 n (cid:19) L − L r − n L r + 64 L r + 64 n L r ! + B ( m , q ) ( n −
1) ( n + 2) " q M M (cid:18) n − n (cid:19) , (48) Pseudo − Real Π SS = B (cid:26) H r + 16 L r + 1 n ( n + 1) ( n − B ( m , q ) (cid:27) + B F M ( q (8 K r + 32 K r ) + M M K r + 128 nK r ! + M M L " (cid:18) n + 32 − n (cid:19) L r − L r + ( n + 1) ( n −
2) 16 n L r + B ( m , q ) ( n + 1) ( n − " q n L r + M M (cid:18) n + 1 n (cid:19) L − L r − n L r + 64 L r + 64 n L r ! + B ( m , q ) ( n + 1) ( n − " q M M (cid:18) − n − n (cid:19) . (49)The definition of the one-loop function B ( m , q ) can be found in Appendix A.1.14 .4.2 Singlet case We have also calculated the singlet case. This is the quark-antiquark combinationthat shows up in the mass term.We write the expression up to NNLO as:
Complex Π SS = B ( nH r + 32 n L r + 16 nL r + 2( n − B ( m , q ) ) + B F M ( q (cid:16) nK r + 4 nK r + 4 n K r (cid:17) + 192 M M (cid:16) nK r + n K r + n K r (cid:17) + M M L (cid:16) n − (cid:17) (32 nL r + 32 L r − nL r − L r )+ B ( m , q ) (cid:16) n − (cid:17) " q ( nL r + L r )+ M M (cid:18) n L + 64 (2 L r + 2 nL r − L r − nL r ) (cid:19) + B ( m , q ) (cid:16) n − (cid:17) nq − M M n ! ) , (50) Real Π SS = B ( nH r + 128 n L r + 32 nL r + 2(2 n + n − B ( m , q ) ) + B F M ( q (cid:16) nK r + 4 nK r + 8 n K r (cid:17) + 384 M M (cid:16) nK r + 2 n K r + 4 n K r (cid:17) + M M L (cid:16) n + n − (cid:17) (64 nL r + 32 L r − nL r − L r )+ B ( m , q ) (cid:16) n + n − (cid:17) " q (2 nL r + L r )+ M M (cid:18)(cid:18) − n (cid:19) L + 64(2 L r + 4 nL r − L r − nL r ) (cid:19) + B ( m , q ) (cid:16) n + n − (cid:17) (cid:20) nq + M M (cid:18) − n (cid:19)(cid:21) ) , (51) Pseudo − Real Π SS = B ( nH r + 128 n L r + 32 nL r + 2(2 n − n − B ( m , q ) ) + B F M ( q (cid:16) nK r + 4 nK r + 8 n K r (cid:17) + 384 M M (cid:16) nK r + 2 n K r + 4 n K r (cid:17) + M M L (cid:16) n − n − (cid:17) (64 nL r + 32 L r − nL r − L r )+ B ( m , q ) (cid:16) n − n − (cid:17) " q (2 nL r + L r )+ M M (cid:18)(cid:18) n (cid:19) L + 64(2 L r + 4 nL r − L r − nL r ) (cid:19) B ( m , q ) (cid:16) n − n − (cid:17) (cid:20) nq + M M (cid:18) − − n (cid:19)(cid:21) ) . (52)We also written the result in term of physical M M and F M . Notice that all loopdiagrams are proportional to the number of Goldstone bosons in each case, i.e. n − n + n −
1, 2 n − n − The pseudo-scalar two-point function is defined in (29). Just as in the case of theaxial-vector two-point function there are one-particle-reducible diagrams. The dia-grams are the same as those for the axial-vector two-point function with the axial-vector current replaced by a pseudo-scalar current. These are shown in Figure 2 and3. There is also no vertex with two pseudo-scalar currents at LO so the equivalent ofdiagrams (1) and (7) in Figure 2 and (13–15) in Figure 3 vanish immediately. Justas in the scalar case, one should distinguish here between two cases: The adjointcase for the complex representation case which generalizes to the broken generatorsfor the real and pseudo-real case, and the singlet operator with T a in (28) the unitoperator.In Section 3.3 we could simplify the final expressions very much by writing thefinal expression with the single pole at the meson mass in terms of the decay constant.The same happens here if we instead rewrite the result in terms of the meson pseudo-scalar decay constant G M . So we first need to obtain that quantity to NNLO. G M The decay constant of the pseudoscalar density to the mesons, G M is defined simi-larly to F M : h | ¯ qiγ T a q | π b i = 1 √ δ ab G M (53)The calculation of G M is very similar to F M , the diagrams are exactly those shownin Figure 2 in [11] with one of the legs replaced by the pseudo-scalar current. Thereis here also a contribution from wave-function renormalization. In [11] we reportedall the quantities M M , F M and h ¯ qq i as an expansion in the bare or lowest orderquantities F and M = 2 B m . We therefore do the same here. We therefore use thequantity L = 116 π log M F (54)instead of L as in the other sections of this paper.This quantity has been calculated to NLO in two-flavour ChPT in [18] and wascalled G π there. We have checked that our NLO result agrees with theirs.At leading order, all the three case have same expression: G M = G = 2 B F . (55) The √
16e express the full results up to NNLO in terms of the LO meson mass M anddecay constant F as G M = 2 B F M F a G + M F b G ! (56)At NLO and NNLO, the coefficients a G and b G are Complex a G = (cid:18) n − n (cid:19) L + 4( − nL r − L r + 4 nL r + 4 L r ) b G = − L r + nL r )( L r + nL r ) + 24( L r + nL r ) − n K r + 48 n K r − K r − K r − K r + 48 K r + 32 K r − nK r − nK r − nK r + 48 nK r + 32 nK r + L " − (32 − n ) (cid:18) L r + 1 n L r (cid:19) + (4 − n )( L r + 4 L r )+ (cid:18) n − n (cid:19) L r + (cid:18) n − n (cid:19) L r − (cid:16) n (cid:17) L r + (cid:18) n − n (cid:19) L r + π " (2 − n ) (cid:18) n L r + 8 L r − n L r − L r − n L r (cid:19) + n L r + 2 L r + 2 (cid:18) n − n (cid:19) L r + π n − n ! − π L n − n ! + L n −
32 + 92 n ! , (57) Real a G = − (cid:18) n − n (cid:19) L + ( − nL r − L r + 32 nL r + 16 L r ) b G = − L r + 2 nL r )( L r + 2 nL r ) + 24( L r + 2 nL r ) − K r n + 192 K r n − K r − K r − K r + 48 K r + 32 K r − K r n − K r n − K r n + 96 K r n + 64 K r n + L " ( −
16 + 16 n + 22 n ) (cid:18) L r + 1 n L r (cid:19) + (4 − n − n ) L r +16(1 − n − n ) L r − (cid:18) − n + 48 n (cid:19) L r − (cid:18) − n + 10 n (cid:19) L r − (cid:16) n + 4 n (cid:17) L r − (cid:18) − n + 4 n (cid:19) L r + π " (1 − n − n ) (cid:18) n L r + 16 L r − n L r − L r − n L r (cid:19) +( n + 2 n ) L r + 2 L r + (cid:18) − n + 2 n (cid:19) L r + π n
256 + 443 n − − n + 1332 n ! π L n
96 + 67 n − − n + 78 n ! + L n
16 + 13 n − − n + 98 n ! , (58) Pseudo − Real a G = − L (cid:18) n − − n (cid:19) + ( − nL r − L r + 32 nL r + 16 L r ) b G = − L r + 2 nL r )( L r + 2 nL r ) + 24( L r + 2 nL r ) − K r n + 192 K r n − K r − K r − K r + 48 K r +32 K r − K r n − K r n − K r n + 96 K r n + 64 K r n + L " ( − − n + 22 n ) (cid:18) L r + 1 n L r (cid:19) + (4 + 8 n − n ) L r +16(1 + 3 n − n ) L r + (cid:18)
40 + 40 n − n (cid:19) L r + (cid:18) n − n (cid:19) L r − (cid:16) − n + 4 n (cid:17) L r + (cid:18) n − n (cid:19) L r + π " (1 + n − n ) (cid:18) n L r + 16 L r − n L r − L r − n L r (cid:19) +( − n + 2 n ) L r + 2 L r + (cid:18) − − n + 2 n (cid:19) L r + π n − n − n + 1332 n ! − π L n − n −
38 + 58 n + 78 n ! + L n − n −
18 + 32 n + 98 n ! . (59) X a case The pseudo-scale two point functions are similar to the axial-vector ones in thediagrams as described above. The LO result is the same for all the three cases:Π aP P = − G q − M . (60)The superscript “ a ” indicates the case with T a in (28) an SU ( n ) generator. For thereal and pseudo-real case this is related by the conserved part of the symmetry groupalso to a number of diquark currents.Subtracting the pole contribution in terms of the physical mass and decay con-stants, M M , F M and G M , absorbs the major part of the higher order corrections.The final results are thus much simpler when written in this way. The remainingpart at NLO is Π aP P = B (8 H r − L r ) . (61)18hus we can define the full NNLO results asΠ aP P = − G M q − M M + B (8 H r − L r ) + B F M ˆΠ aP P , (62)where the ˆΠ P P is the remainder at NNLO. Its expression for the three different casesis:
Complex ˆΠ aP P = − n q H M ( M M , M M , M M , q )+ "(cid:18) − n (cid:19) q − n M M q H M ( M M , M M , M M , q )+8 q K r + 64 M M ( K r + nK r − K r − nK r )+ L M M − n − n + 2 ! + LM M (cid:18) L r + 32 nL r − L r n (cid:19) + π L " M M − − n
12 + 8 n ! + − n − n ! q + π " M M − n − n ! + q − n − n ! , (63) Real ˆΠ aP P = − q n ( n + 1) H M ( M M , M M , M M , q )+ (cid:20)(cid:18) − n − n n + 13 (cid:19) q − M M q n ( n + 1) (cid:21) H M ( M M , M M , M M , q )+8 q K r + 64 M M ( K r + 2 nK r − K r − nK r )+ M M L − n − n − n + 32 n + 12 ! + 32 M M L (cid:18) L r + L r n − L r n + L r (cid:19) + π L " M M − n
12 + 2 n + 7 n − n − ! + q − n − n − n
24 + 12 n + 16 ! + π " M M − n − n − n
96 + 54 n + 512 ! + q − n − n − n
192 + 78 n + 724 ! , (64) Pseudo − Real ˆΠ aP P = 32 q n (1 − n ) H M ( M M , M M , M M , q )+ (cid:20)(cid:18) − n + n − n + 13 (cid:19) q + 12 M M q n (1 − n ) (cid:21) H M ( M M , M M , M M , q )+8 q K r + 64 M M ( K r + 2 nK r − K r − nK r )+ M M L − n − n + n − n + 12 ! + M M L (cid:20) L r + 32 (cid:18) n − n − (cid:19) L r (cid:21) + π L " M M − n
12 + 2 n − n
12 + 2 n − ! + q − n − n + 7 n − n + 16 ! π " M M − n − n + 125 n − n + 512 ! + − n − n + 101 n − n + 724 ! q . (65)The loop integrals H M and H M are defined in Appendix A.2. In the singlet case with a = 0, there is no contribution with poles. Only the one-particle-irreducible diagrams contribute. As a consequence, there is no order p contribution and at order p there is only a tree level contribution from the equivalentof diagram (3) in Figure 2. At order p or NNLO only the one-particle-irreduciblediagrams contribute and since there is no order p vertex with two pseudo-scalarcurrents only diagram (11–12) and (16) in Figure 3 contribute.Since there is no single pole contribution, there is also no need here to expand inthe integrals around the meson mass. The integral H F is defined in Appendix A.2.The singlet pseudo-scalar two-point function we write asΠ P P = B Π P P + B F M ˆΠ P P . (66)The results for the three cases are Complex :Π P P = 8 nH r − nL r − n L r , ˆΠ P P = − n (cid:16) n − (cid:17) (cid:16) n − (cid:17) H F ( M M , M M , M M , q )+ q (cid:16) K r n − K r n (cid:17) − M M (cid:16) K r n + K r n + K r n + K r n (cid:17) + L M M n (cid:16) n − (cid:17) (cid:16) n − (cid:17) + 64 LM M ( n − nL r + L r )+ M M π n (cid:16) n − (cid:17) (cid:16) n − (cid:17) π ! , (67) Real :Π P P = 16 nH r − n L r − nL r , ˆΠ P P = − n (cid:16) n + n − (cid:17) (cid:16) n + n − (cid:17) H F ( M M , M M , M M , q )+ q (cid:16) K r n − K r n (cid:17) − M M (cid:16) nK r + 2 n K r + 2 n K r + 4 n K r (cid:17) + L M M n (cid:16) n + n − (cid:17) (cid:16) n + n − (cid:17) + 64 M M L (2 n + n − nL r + L r )+ M M π n (cid:16) n + n − (cid:17) (cid:16) n + n − (cid:17) π ! , (68) Pseudo − Real :Π P P = 16 nH r − n L r − nL r , ˆΠ P P = − n (cid:16) n − n − (cid:17) (cid:16) n − n − (cid:17) H F ( M M , M M , M M , q )20igure 4: The one-loop oblique correction to LEP process e + + e − → q + ¯ q .+ q (cid:16) K r n − K r n (cid:17) − M M (cid:16) nK r + 2 n K r + 2 n K r + 4 n K r (cid:17) + L M M n (cid:16) n − n − (cid:17) (cid:16) n − n − (cid:17) + 64 M M L (2 n − n − nL r + L r )+ M M π n (cid:16) n − n − (cid:17) (cid:16) n − n − (cid:17) π ! . (69)Notice that just as for the scalar singlet two-point function, all loop contributionsare proportional to the number of Goldstone bosons. n As one can see from all the explicit formulas, many of the expressions become equalfor the different cases in the large n limit . The physical process at the CERN LEP collider is e + + e − → q + ¯ q . There arethree types of one loop correction to this process: vacuum polarization corrections,vertex corrections, and box corrections. The vacuum polarization contribution isindependent of the external fermions and it dominates the contributions from physicsbeyond SM. For the light fermions, the other two corrections are suppressed byfactor of m f /m Z . That’s why the vacuum polarization corrections are called “obliquecorrections,”, and the vertex and box corrections are called “nonoblique corrections.”The oblique polarization only affect the gauge bosons propagators and their mix-ing. The vacuum polarization amplitude can be defined as g µν Π XY + ( q µ q ν terms) = i Z d x e iq · x h | T ( J µX ( x ) J νY (0)) | i . (70)The influence of new physics to the oblique corrections can be summarized to threeparameters: S , T and U . One can find their definition in Ref. [15]. These parametersare chosen to be zero at a reference point in the SM. In the past 20 years, they havebeen studied intensively in many models beyond the Standard Model physics.For a beyond the Standard Model with strong dynamics at the TeV scale, therewill in general be many resonances and other nonperturbative effects. At low mo-menta one can use the EFT as described above for these cases. In this paper, we willestimate the S parameter contribution from pseudo-Goldstone Boson sector within21he EFT. The parameter T and U vanish because of the exact flavor symmetry, i.e.we work in the equal mass case.The S parameter can be written as [15] S = − π h Π ′ V V (0) − Π ′ AA (0) i = 2 π ddq (cid:16) q Π (1) V V − q Π (1) AA (cid:17) q =0 . (71)Π ′ V V (0) and Π ′ AA (0) are the derivatives of the vector and axial-vector two-point func-tions at q = 0. One should keep in mind that S is defined to be zero at a particularplace in the standard model, as discussed at the end of section V in [15]. Our formulasare the equivalent of (5.12) in that reference.The result can be written as S = S + πM M F M ˆ S , (72)with
Complex : S = − πL r − nπ L + π ) , ˆ S = 64 ( K r − K r + K r + nK r − nf K r + nK r ) + n L +16 n ( L r + 2 L r ) L − π n L + π n (cid:18) −
527 ˜ ψ (cid:19) (73) Real : S = − πL r − n + 1) π L + π ) , ˆ S = 64 ( K r − K r + K r + 2 nK r − nf K r + 2 nK r ) + n ( n + 1)3 L +16 [( n + 1) L r + (2 n + 1) L r ] L − π n ( n + 1)9 L + π n ( n + 1) (cid:18) −
527 ˜ ψ (cid:19) , (74) Pseudo − real : S = − πL r − n − π L + π ) , ˆ S = 64 ( K r − K r + K r + 2 nK r − nf K r + 2 nK r ) + n ( n − L +16 [( n − L r + (2 n − L r ] L − π n ( n − L + π n ( n − (cid:18) −
527 ˜ ψ (cid:19) . (75)The quantity ˜ ψ is ˜ ψ = 6 √ (cid:18) π (cid:19) = 7 . . (76) Our two point functions are normalized differently from those in [15]. S M M [GeV]n=2 complexp p +p p L r10 only (a) S M M [GeV]n=4 complexp p +p p L r10 only (b)Figure 5: The S -parameter for the values of L r and L r given in the text for thecomplex case. (a) n = 2 (b) n = 4. S M M [GeV]n=2 real p p +p p L r10 only (a) S M M [GeV]n=4 real p p +p p L r10 only (b)Figure 6: The S -parameter for the values of L r and L r given in the text for the realcase. (a) n = 2 (b) n = 4.The real purpose of (73)-(75) is to be able to study the S -parameter in more gen-eral theories than just scaling up from QCD. However to provide some feeling aboutnumerical results we choose parameters as if they are scaled up from QCD/ChPT.We change F π = 0 . F M = 243 GeV and the subtraction scale from0 .
77 GeV to 2 TeV. We set the K ri = 0 and keep L r = 0 . L r = − . n = 2 and n = 4. Shown are the full p and p contributionsas well as the p part proportional to L r only. The latter is what is the usualcontribution to S corrected for the pieces that go into the reference point at p . Wecannot do the same for the full result since that depends on how one treats the extrapseudo-Goldstone bosons that occur in the other models.23 S M M [GeV]n=2 pseudo-realp p +p p L r10 only (a) S M M [GeV]n=4 pseudo-realp p +p p L r10 only (b)Figure 7: The S -parameter for the values of L r and L r given in the text for thepseudo-real case. (a) n = 2 (b) n = 4. In this paper, we have calculated the two-point correlation functions of vector, axial-vector, scalar and pseudo-scalar currents for QCD-like theories.In the beginning of the paper, we gave a very brief overview of the QCD-liketheories and their EFT treatment as developed earlier.We then gave the analytic results of those two-point functions up to NNLO. Theresults are significantly shortened by using the physical meson mass M M and decayconstants F M and G M when rewriting the pole contributions.The main use of these formulas is expected to be in extrapolations to zero fermionmass of technicolour related lattice calculations. We have therefore also includedprecisely the combination needed for the S -parameter. Acknowledgments
This work is supported in part by the European Community-Research InfrastructureIntegrating Activity “Study of Strongly Interacting Matter” (HadronPhysics2, GrantAgreement n. 227431) and the Swedish Research Council grants 621-2008-4074 and621-2010-3326. This work used FORM [27].
A Loop integrals
We use dimensional regularization and
M S scheme to evaluate the loop integrals, d = 4 − ǫ . 24 .1 One-loop integrals The loop integral with one propagator is A ( m ) = 1 i Z d d q (2 π ) d q − m = m π ( λ − ln( m ) + ǫ " C π
12 + 12 ln ( m ) − C ln( m ) + O ( ǫ ) . (77)Here C = ln(4 π ) + 1 − γ λ = 1 ǫ + C The extra +1 in C is the ChPT version of M S .The loop integrals with two propagators are B ( m , m , p ) = 1 i Z d d q (2 π ) d q − m )(( q − p ) − m ) ,B µ ( m , m , p ) = 1 i Z d d q (2 π ) d q µ ( q − m )(( q − p ) − m ) (78)= p µ B ( m , m , p ) ,B µν ( m , m , p ) = 1 i Z d d q (2 π ) d q µ q ν ( q − m )(( q − p ) − m )= p µ p ν B ( m , m , p ) + g µν B ( m , m , p ) . The two last integrals can be reduced to simpler integrals A and B via B ( m , m , p ) = 12 B ( m , m , p ) ,B ( m , m , p ) = 12( d − h A ( m ) + (cid:18) m − p (cid:19) B ( m , m , p ) i ,B ( m , m , p ) = 1 p h A ( m ) + m B ( m , m , p ) − dB ( m , m , p ) i . (79)We quote here only the equal mass case results relevant for this paper. The explicitexpression for B is B ( m , m , p ) = 116 π λ + B ( m , p ) + O ( ǫ ) ,B ( m , p ) = 116 π − − m log m µ ! + ¯ J ( m , p ) , ¯ J ( m , p ) = − π Z dx ln m − x (1 − x ) p m ! , (80)The function ¯ J ( m , p ) is¯ J ( m , p ) = σ ln (cid:16) σ − σ +1 (cid:17) , p < , − q x − · arccot (cid:16)q x − (cid:17) , ≤ p < m , σ ln (cid:16) − σ σ (cid:17) + iπσ, p > m , ( x ) = s − x , x = m p / ∈ [0 , . (81)Taking derivatives w.r.t. p at p = 0 is most easily done in the form with theFeynman parameter integration explicit. A.2 Sunset integrals
The sunset integrals are done with the methods of [22, 28]. They are defined as hh X ii = 1 i Z d d q (2 π ) d d d r (2 π ) d X ( q − m ) ( r − m ) [( q + r − p ) − m ] , (82)The various sunset integrals with Lorenz indices are H ( m , m , m ; p ) = hh ii ,H µ ( m , m , m ; p ) = hh q µ ii = p µ H ( m , m , m ; p ) , (83) H µν ( m , m , m ; p ) = hh q µ q ν ii = p µ p ν H ( m , m , m ; p ) + g µν H ( m , m , m ; p ) . and hh r µ ii = p µ H ( m , m , m ; p ) , hh r µ r ν ii = p µ p ν H ( m , m , m ; p ) + g µν H ( m , m , m ; p ) , hh q µ r ν ii = hh r µ q ν ii , hh q µ r ν ii = p µ p ν H ( m , m , m ; p ) + g µν H ( m , m , m ; p ) , (84)The function H is fully symmetric in m , m and m , while H , H and H aresymmetric under the interchange of m and m . The relation between the above 3functions p H ( m , m , m ; p ) + dH ( m , m , m ; p ) = m H ( m , m , m ; p ) + A ( m ) A ( m ) , (85)allows to express H in terms of H .Similar to the integral B and B , there is also a relation between H and H whichin the equal mass case becomes H ( m , m , m ; p ) = 13 H ( m , m , m ; p ) . (86)The other functions, H and H , can be written in term of H , H and H by usingrelations derived from redefining the momenta and masses in its definition [22].The full sunset integral expressions and the definition for finite part H Fi = { H F , H F , H F } can be found in the appendix of [22]. In our case we take m = m = m = m . 26n order to eliminate the extra poles in the expressions, sometimes we need toexpand the H Fi ( m , m , m ; q ) around the pseudoscalar mass m , and we define H Mi ( m , m , m ; q ) = 1( q − m ) " H Fi ( m , m , m ; q ) − H Fi ( m , m , m ; m ) − ( q − m ) H F ′ i ( m , m , m ; m ) , (87)where H F ′ i ( m , m , m ; m ) = ∂H Fi ( m , m , m ; q ) ∂q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = m . (88) References [1] S. Dimopoulos, Nucl. Phys. B (1980) 69.[2] M. E. Peskin, Nucl. Phys. B (1980) 197.[3] J. Preskill, Nucl. Phys. B , 21 (1981).[4] J. B. Kogut, M. A. Stephanov, D. Toublan, J. J. M. Verbaarschot and A. Zhit-nitsky, Nucl. Phys. B (2000) 477 [arXiv:hep-ph/0001171].[5] Y. I. Kogan, M. A. Shifman and M. I. Vysotsky, Sov. J. Nucl. Phys. (1985)318 [Yad. Fiz. (1985) 504].[6] H. Leutwyler and A. V. Smilga, Phys. Rev. D (1992) 5607.[7] A. V. Smilga and J. J. M. Verbaarschot, Phys. Rev. D (1995) 829 [arXiv:hep-th/9404031].[8] J. Gasser and H. Leutwyler, Nucl. Phys. B (1985) 465.[9] J. Gasser and H. Leutwyler, Phys. Lett. B (1987) 83.[10] K. Splittorff, D. Toublan and J. J. M. Verbaarschot, Nucl. Phys. B (2002)290 [arXiv:hep-ph/0108040].[11] J. Bijnens and J. Lu, JHEP (2009) 116 [arXiv:0910.5424 [hep-ph]].[12] J. Bijnens and J. Lu, JHEP (2011) 028 [arXiv:1102.0172[hep-ph]].[13] J. R. Andersen, O. Antipin, G. Azuelos, L. Del Debbio, E. Del Nobile, S. DiChiara, T. Hapola, M. Jarvinen et al. , Eur. Phys. J. Plus (2011) 81.[arXiv:1104.1255 [hep-ph]].[14] C. T. Hill and E. H. Simmons, Phys. Rept. (2003) 235 [Erratum-ibid. (2004) 553] [arXiv:hep-ph/0203079].[15] M. E. Peskin, T. Takeuchi, Phys. Rev. D46 (1992) 381-409.2716] G. Altarelli, R. Barbieri, Phys. Lett.
B253 (1991) 161-167.[17] S. Weinberg, Physica A (1979) 327.[18] J. Gasser and H. Leutwyler, Annals Phys. (1984) 142.[19] S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. (1969) 2239;C. G. . Callan, S. R. Coleman, J. Wess and B. Zumino, Phys. Rev. (1969)2247.[20] J. Bijnens, G. Colangelo and G. Ecker, JHEP (1999) 020 [arXiv:hep-ph/9902437].[21] J. Bijnens, G. Colangelo and G. Ecker, Annals Phys. (2000) 100 [arXiv:hep-ph/9907333].[22] G. Amoros, J. Bijnens and P. Talavera, Nucl. Phys.
B 568 (2000) 319 [hep-ph/9907264].[23] E. Golowich, J. Kambor, Nucl. Phys.
B447 (1995) 373-404. [hep-ph/9501318].[24] E. Golowich and J. Kambor,
Phys. Rev.
D 58, 036004 (1998) [hep-ph/9710214].[25] J. Bijnens, P. Talavera, JHEP (2002) 046. [hep-ph/0203049].[26] M. Gonzalez-Alonso, A. Pich, J. Prades, Phys. Rev.
D78 (2008) 116012.[arXiv:0810.0760 [hep-ph]].[27] J. A. M. Vermaseren, arXiv:math-ph/0010025.[28] J. Gasser and M. E. Sainio, Eur. Phys. J. C6