Revisiting the pion's scalar form factor in chiral perturbation theory
aa r X i v : . [ h e p - l a t ] O c t CERN-PH-TH/2011-254
Revisiting the pion’s scalar form factor in chiralperturbation theory
Andreas J¨uttnerCERN, Physics Department, TH Unit, CH-1211 Geneva 23, Switzerland
Abstract:
The quark-connected and the quark-disconnected Wick contractions con-tributing to the pion’s scalar form factor are computed in the two and in the three flavourchiral effective theory at next-to-leading order. While the quark-disconnected contribu-tion to the form factor itself turns out to be power-counting suppressed its contributionto the scalar radius is of the same order of magnitude as the one of the quark-connectedcontribution. This result underlines that neglecting quark-disconnected contributionsin simulations of lattice QCD can cause significant systematic effects. The techniqueused to derive these predictions can be applied to a large class of observables relevantfor QCD-phenomenology.
Key words:
Chiral Perturbation Theory; Lattice QCD
PACS: ontents χ PT for the scalar pion form factor 44 Results for the form factor 5
A.1 Traces of SU ( N | M ) generators . . . . . . . . . . . . . . . . . . . . . . . 10A.2 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11A.2.1 2 π − s -vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11A.2.2 4 π and 4 π − s vertices . . . . . . . . . . . . . . . . . . . . . . . . 12A.2.3 Counter terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 B Results for the octet and singlet form factors 13C Kinematical functions 14
At first sight the scalar form factor of the pion appears to be a purely academic quantitysince despite being a well-defined object in QCD, nature does not seem to provide alow-energy scalar probe. The experimental information on the scalar I = 0 ππ phaseshift however serves as an input parameter in the dispersive representation of the formfactor. Indirectly a comparison with experiment is therefore in principle possible.Lattice QCD is the tool of choice for determining low energy properties of hadrons [1].However, while computing the scalar form factor is in principle a straight forward pro-cedure it turns out to be numerically much more complicated than the computation ofthe closely related pion vector form factor in the iso-spin limit [2–5]. The reason liesin the contribution of quark-disconnected Wick contractions to its defining correlationfunctions in the scalar case. There is no conceptual difficulty in treating these [5] but thecomputational effort is immense compared to quark-connected contributions. The dis-connected part is therefore often neglected, not always providing supporting argumentsthat the systematic effect introduced in this way is under control.1his paper provides the expressions for the contributions of the quark connectedand the quark disconnected Wick contractions to the scalar form factor of the pion inchiral perturbation theory at next-to-leading order (NLO) [6–8]. As one might haveexpected and as first numerical studies indicate [5], the relative contribution of thequark-disconnected part to the form factor in the space-like region turns out to besmall for small values of the momentum transfer. However, its contribution to the formfactor above threshold and to the scalar radius which is defined as the form factor slopeat vanishing momentum transfer turn out to be of about the same magnitude as theone of the quark-connected contribution.While underlining the potential shortcomings of neglecting quark-disconnected con-tributions in lattice computations of QCD observables the results derived in this workallow to predict the disconnected contribution at NLO in the effective theory. Moreover,the arguments presented here justify the use of (partially) twisted boundary conditionsin a lattice computation [9–14] allowing to compute the form factor on the finite latticevolume for arbitrary values of the momentum transfer.The technique used to derive these predictions has recently been introduced in [15]where it was applied to the two-point correlator of vector currents in QCD. After a briefsummary of the idea the important steps in the computation for the scalar form factorwithin this frame work are presented for the theory with both N f = 2 and N f = 2 + 1flavours, followed by an illustration and discussion of the results. The technique will be briefly discussed for the example of the 2-flavour theory. Thegeneralisation to the 2+1-flavour theory is straight forward and will become clear in thefollowing sections. The scalar form factor of the pion is defined through h π i ( p ′ ) | ¯ uu + ¯ dd | π j ( p ) i = δ ij F S, ( t ) . (2.1)where t = ( p ′ − p ) is the squared momentum transfer and where the sub-script 2 indi-cates N f = 2 flavours. The matrix element on the l.h.s. is the ground state contributionto the Fourier transform of the QCD quark three point function D O i ( z ) S ( y ) O i † ( x ) E , (2.2)constructed of the interpolating operators O i ( x ) = ¯ ψ τ i γ ψ ( x ) and S ( x ) = ¯ ψ ( x ) ψ ( x ).The ψ T = ( u, d ) are SU (2) flavour vectors of u - and d -quarks and the matrices τ i = σ i / v , which is degener-ate with the light dynamical flavours, each Wick contraction can be rewritten in termsof a single fermionic correlation function defined in an un-physical theory. The phys-ical result is recovered by summing over the correlation functions in the un-physical,2 O S O O S Figure 1 : Wick contractions contributing to the scalar form factor.partially quenched, theory [16–18]. In the case we consider here, e.g., h ¯ uγ d ¯ dd ¯ dγ u i = h ¯ uγ v ¯ vd ¯ dγ u i + h ¯ uγ d ¯ vv ¯ dγ u i . (2.3)An expression for the l.h.s. in the effective theory has been found many years ago atNLO [6, 8] and at NNLO in [19, 20] for both the N f = 2 and N f = 2 + 1 theory. Thispaper provides expressions for the ground state matrix elements of the terms on the r.h.s.of eq. (2.3) in partially quenched chiral perturbation theory (PQ χ PT) [16–18], which isan extension of chiral perturbation theory [6, 7] that provides an asymptotic low energydescription of partially quenched QCD (PQQCD). Following [15], the flavour group ispromoted to the graded SU (3 |
1) with the flavour vector ψ T | = ( u, d, v, g ) extended byan additional valence and ghost quark. Both are assumed to be mass degenerate tothe up- and down quarks. The generators of the thus enlarged flavour group are T a , a = 1 , . . . ,
15. We use the conventions also employed in [15, 21], T a = ( T a ) † , Str { T a } = 0 , Str n T a T b o = 12 g ab , a = 1 , . . . , , (2.4)with the super-trace Str { A } = A + A + A − A . The matrix g ab reads g = − σ − σ − σ − − − (cid:9)
15 (2.5)where σ is the second Pauli matrix. T , . . . , T are the generators of the SU (3) sub-group that acts on the sea and valence components, T , . . . , T mix the quark withthe ghost components, T is the diagonal matrix diag(1 , , , / (2 √ T with Str { T } = 1 / √ D π (cid:12)(cid:12)(cid:12) S + S − S √ (cid:12)(cid:12)(cid:12) π E = D π (cid:12)(cid:12)(cid:12) S (cid:12)(cid:12)(cid:12) π E + D π (cid:12)(cid:12)(cid:12) S − S + S √ (cid:12)(cid:12)(cid:12) π E , (2.6)where the scalar source now is of the form S i = ¯ ψ | T i ψ | .3 πs (a) tree level π πs (b) bubble π πs (c) tadpole π πs (d) counter Figure 2 : Contributing diagrams in the effective theory. Solid lines represent pions builtof fermionic or bosonic quarks and the dashed line represents the external scalar source.All previous steps can also be applied to the 2+1-flavour theory. There, the SU (3)flavour symmetry is promoted to a graded SU (4 |
1) and the flavour content of the theoryis represented by the flavour vector ψ T | = ( u, d, s, v, g ) with s being the strange quarkfield. The graded group SU (4 |
1) has 24 generators and the super-trace as well as themetric g ab and T introduced above for SU (3 |
1) need to be modified correspondingly.In the next section the chiral effective theory corresponding to the SU (3 |
1) and SU (4 |
1) flavour symmetry groups is set up. χ PT for the scalar pion form factor
The degrees of freedom of the partially quenched theory are parameterised in terms of U = exp (cid:16) i φ a T a F (cid:17) , where the φ a are the Goldstone-boson/fermion fields ( a = 1 , . . . , SU (3 |
1) and a = 1 , . . . ,
24 for SU (4 | L (2) = F Str n ∂ µ U ∂ µ U † o + F Str n χU † + U χ † o , (3.1)where χ = 2 B ( s + M ). The mass matrix has the form SU (3 | SU (4 | M = diag ( m q , m q , m q , m q ) , M = diag ( m q , m q , m s , m q , m q ) . (3.2)and we define the external scalar source as s = 2 T a s a . The relevant counter termsillustrated in figure 5(d) can be derived from the Lagrangian , L (4) = L Str n ∂ µ U ( ∂ µ U ) † o Str n χU † + U χ † o + L Str n(cid:0) ∂ µ U ( ∂ µ U ) † (cid:1)(cid:0) χU † + U χ † (cid:1)o + L Str n χU † + U χ † o + L Str n U χ † U χ † + χU † χU † o . (3.3)The partially quenched propagators are constructed in analogy to the procedure pre-sented in detail in [15, 21]. Note however that in the current paper we use Minkowski-signature. 4 Results for the form factor
For the 2-flavour theory the expressions for the form factor are F F S, ( t ) = 2 B n F (cid:16) − ¯ A ( m π ) + Λ F2 + (2 t − m π ) ¯ B ( m π , t ) (cid:17)o ,F C S, ( t ) = 2 B n F (cid:16) − ¯ A ( m π ) + Λ C2 + ( t − m π ) ¯ B ( m π , t ) (cid:17)o ,F D S, ( t ) = 2 B n F (cid:16) Λ D2 + ( t + m π ) ¯ B ( m π , t ) (cid:17)o , (4.1)where, Λ F2 = 4 n m π ( − L r − L r +16 ˜ L r +8 ˜ L r )+ t (2 ˜ L r + ˜ L r ) o , Λ C2 = 4 n m π ( − L r − L r +8 ˜ L r +8 ˜ L r )+ t ˜ L r o , Λ D2 = 4 n m π ( − L r +8 ˜ L r ) + t L r o . (4.2)and where t = ( p ′ − p ) is the squared momentum transfer between the two pions. Thekinematical functions ¯ A ( m ) and ¯ B ( m , t ) are defined in appendix C. The superscriptsF, C and D indicate the Full form factor and the Connected and Disconnected contri-butions, respectively and the superscript r indicates the subtraction of divergences inthe M S -scheme at subtraction scale µ = 0 . A = ¯ A ( m η ) − ¯ A ( m π ) , (4.3)the results for the N f = 2 + 1 theory are F F S, ( t ) = 2 B n F (cid:16) A +Λ F3 + m π ¯ B ( m η , t ) + (2 t − m π ) ¯ B ( m π , t )+ t ¯ B ( m K , t ) (cid:17)o ,F C S, ( t ) = 2 B n F (cid:16) A +Λ C3 + ( t − m π ) ¯ B ( m π , t )+ t ¯ B ( m K , t )+ m π ¯ B ( m η , m π , t ) (cid:17)o ,F D S, ( t ) = 2 B n F (cid:16) +Λ D3 + m π ¯ B ( m η , t ) + ( t + m π ) ¯ B ( m π , t ) − m π ¯ B ( m η , m π , t ) (cid:17)o , (4.4)whereΛ F3 = 4 n m π ( − L r − L r +12 L r +8 L r )+ m K ( − L r + 8 L r )+ t (2 L r + L r ) o , Λ C3 = 4 n m π ( − L r − L r +4 L r +8 L r )+ m K ( − L r + 8 L r )+ t L r o , Λ D3 = 4 n m π ( − L r +8 L r ) + t L r o . (4.5)5he results for the octet and singlet scalar form factors h π i | ¯ uu + ¯ dd − ss | π k i = δ ik F S ( t ) , h π i | ¯ uu + ¯ dd + ¯ ss | π k i = δ ik F S ( t ) , (4.6)are given in appendix B. An interesting observation is the absence of low-energy con-stants in the expression for the disconnected contribution to the octet form factor dueto the vanishing super-trace of the octet-current. This allows for its parameter-freeprediction at this order of the chiral expansion. The chiral Lagrangian in the 2-flavour theory in [6] was presented in a different para-meterisation which is related to the one used here through the relations l = 4( − L − ˜ L + 4 ˜ L + 2 ˜ L ) and l = 4(2 ˜ L + ˜ L ), hence,Λ F2 = t l r +4 m π l r , Λ C2 = t ( l r − L r )+4 m π ( l r +4 ˜ L r − L r ) , Λ D2 = t ( +8 ˜ L r )+4 m π ( − L r +8 ˜ L r ) . (4.7)The results for F F S, and F F S, agree with the ones found in [6, 8]. The low-energy pa-rameters ˜ L r and ˜ L r entering F C S, and F D S, are in principle unknown from the 2-flavourtheory - a corollary of the unphysical nature of the connected and the disconnectedcontributions considered on their own. The corresponding terms are naively present inthe SU (2) chiral Lagrangian but trace-identities and the use of the equations of motionallow to reduce the relevant set of terms, hence the above linear combinations defining l and l in terms of the ˜ L i .In both the 2- and the 3-flavour theory the disconnected contribution is suppressedby power counting relative to the connected contribution and the full form factor, re-spectively. Power counting also suggests that the contribution of the disconnected con-tractions to the scalar radius h r i defined through F S ( t ) = F S (0) (cid:0) h r i t + O ( t ) (cid:1) , (4.8)(and similarly for the octet and singlet charge radius) can be of about the same mag-nitude. Since the tree-level contribution is purely real, the connected and the discon-nected contractions also contribute democratically to the imaginary part of the formfactor above the two-pion threshold in the time-like region.In order to analyse the two contributions to the form factor more quantitatively weuse the lattice estimates for the 3-flavour low-energy constants L r , L r , L r and L r by theRBC/UKQCD lattice collaboration [22] as given in table 1. Further input parameters See [1] for a summary of determinations of low-energy parameters in lattice QCD. r [1, 22] 0 . × − L r [1, 22] 0 . × − L r [1, 22] 0 . × − L r [1, 22] 0 . × − Table 1 : Values for the 3-flavour low energy constants used in the illustration of theresults. The subtraction scale is µ = 0 . F = 93MeV, m π = 0 . m K = 495MeV and m η = 548MeV. With thisexternal input it is straight forward to analyse the 3-flavour case. Although ˜ L r and ˜ L r are not known they can be related to the low-energy constants L r and L r by matchingthe two theories. In particular, we use l r = 4( − L r − L r + 16 L r + 8 L r ) − ˜ L ( m η ) ,l r = 8 L r + 4 L r − ˜ L ( m K ) , L r − ˜8 L r = 4 L r − L r −
54 118 ˜ L ( m η ) −
112 1 N , ˜ L r = L r , (4.9)where ˜ L ( m ) = π (cid:0) m /µ ) (cid:1) .The dependence on the momentum of the full, the connected and the disconnectedform factor are illustrated in the plots in figure 3. As anticipated F D S (dashed) issuppressed relative to F C S (dotted), predominantly due to the absence of a tree-levelcontribution in the former. Above threshold the contribution from the imaginary partof the form factor sets in and the quark disconnected contribution is larger than 50%over the whole range plotted. Figure 4 shows the scalar radius and as expected thedisconnected contribution is not suppressed. In fact, both the connected and the dis-connected contractions contribute to roughly equal parts.The observations made here for the form factor F S apply qualitatively in the sameway for the form factors F S and F S . Clearly this study has been motivated by the technical problems encountered in the com-putation in lattice QCD of observables receiving contributions from quark-disconnecteddiagrams. Here we highlight three ways in which the results of this paper might be ofuse: • When computing the scalar form factor in lattice QCD with the aim of determininglow-energy parameters it is possible to gain valuable information by comparingthe dependence of the lattice data for only the connected part F C S on the quark7 f = 2 N f = 2 + 1 - - PSfrag replacements t [GeV ] R e (cid:16) F F , C , D S , ( t ) / ( B ) (cid:17) ∆ Re - - PSfrag replacements t [GeV ]Re (cid:16) F F , C , D S, ( t ) / (2 B ) (cid:17) ∆ Re R e (cid:16) F F , C , D S , ( t ) / ( B ) (cid:17) PSfrag replacements t [GeV ]Re (cid:16) F F , C , D S, ( t ) / (2 B ) (cid:17) ∆ Re Re (cid:16) F F , C , D S, ( t ) / (2 B ) (cid:17) I m (cid:16) F F , C , D S , ( t ) / ( B ) (cid:17) ∆ Im PSfrag replacements t [GeV ]Re (cid:16) F F , C , D S, ( t ) / (2 B ) (cid:17) ∆ Re Re (cid:16) F F , C , D S, ( t ) / (2 B ) (cid:17) Im (cid:16) F F , C , D S, ( t ) / (2 B ) (cid:17) ∆ Im I m (cid:16) F F , C , D S , ( t ) / ( B ) (cid:17) Figure 3 : The plots show the momentum dependence of the real (top) and imaginary(bottom) parts of the scalar form factor for N f = 2 (left) and N f = 2 + 1 (right):Full form factor (solid red), connected contribution (dotted blue) and disconnectedcontribution (dashed green). 8 f = 2 N f = 2 + 1
0. 0.02 0.04 0.06 0.08 0.100.20.40.60.81.
PSfrag replacements h r i [ f m ] m π [GeV]
0. 0.02 0.04 0.06 0.08 0.100.20.40.60.81.
PSfrag replacements h r i [ f m ] m π [GeV] Figure 4 : The plots show the scalar radius for N f = 2 (left) and N f = 2 + 1 (right) as afunction of the squared pion mass ( m K is fixed to the physical value and m η = m K − m π ): full radius (solid red), connected contribution (dotted blue) and disconnectedcontribution (dashed green).mass and the momentum transfer, to the corresponding expressions in the effectivetheory for N f = 2 (eq. (4.4)) and N f = 2 + 1 (eq. (4.7)). A special case in thiscontext is the octet scalar form factor (cf. appendix B) since the expression forthe disconnected contribution is entirely parameter-free. • If one is interested in the result for the full form factor F F S , the only approachwhich will eventually provide a reliable control of systematic effects is a completenumerical evaluation in lattice QCD. An approximate picture for the form fac-tor can be obtained by combining an evaluation of the connected contribution inlattice QCD with predictions for the disconnected contribution by chiral pertur-bation theory. With the exception of the octet form factor the latter of courserelies on external input in terms of estimates of the low-energy constants. • In [23] an argument was given that justifies the use of (partially) twisted boundaryconditions for the computation of the pion’s iso-spin limit vector form factor.From the flavour structure of the correlation functions on the r.h.s. of eq. (2.3) itfollows that this technique can also be applied to the quark-connected correlatorthat contributes to the scalar form factor.9
Conclusions and outlook
This paper provides the expressions for quark-connected and quark-disconnected Wickcontractions, respectively, contributing to the pion’s scalar form factor and its radius innext-to-leading order (partially quenched) chiral perturbation theory. While the quark-disconnected part in the scalar form factor is sub-dominant, it turns out to contributewith about the same magnitude to the scalar radius as the quark-connected one.This result has implications for lattice QCD simulations where quark-disconnectedcontributions are often neglected ad hoc because of their typically bad signal-to-noiseratio. Computing quark-disconnected contributions in the effective theory can providepower-counting arguments for or against neglecting them for QCD-observables wherechiral perturbation theory is expected to be a reliable effective description of the low-energy properties of the underlying fundamental theory. The example of the octetscalar form factor shows that for a suitable choice of the observable a prediction of thedisconnected contribution in the effective theory is possible without loss of predictivity.Quark-disconnected contractions contribute to a large class of matrix elements rel-evant for the precision phenomenology of the Standard Model. Examples are the yetto be fully understood process K → ππ or iso-spin breaking effects. The technique em-ployed here and previously in [15] can potentially be applied to guide the computationof quark-disconnected contributions in these cases. Acknowledgments:
The author would like to thank Martin L¨uscher for comments onthe manuscript and Michele Della Morte for fruitful discussions.
Appendix:A Vertices in the partially quenched theory
The Feynman rules in partially quenched chiral perturbation theory differ from the onesin the purely bosonic theory since the commutators of the involved fields depend on their grading η . In the following we first provide expressions for the traces of products of SU ( N | M ) generators and then give explicit expression for the relevant vertices thatderive from the Lagrangians (3.1) and (3.3). A.1 Traces of SU ( N | M ) generators Assuming the normalisation of the SU ( N | M ) generators as in section 2 their commu-tators and anti-commutators are[ T a , T b ] ≡ T a T b − ( − ) η a η b T b T a = iC cab T c { T a , T b } ≡ T a T b + ( − ) η a η b T b T a = N g ab + D cab T c (A.1)10here η a is the grading of the generator a , i.e. η a = 0(1) if T a generates a bo-son(fermion). The explicit expressions for the structure constants C and D can beobtained by projection, C cab = − i n [ T a , T b ] T k o g kc ,D cab = + 2Str n { T a , T b } T k o g kc . (A.2)With these definitions the following results for the super-traces of products of SU ( N | M )-generators are obtained:Str (cid:8) T a T b (cid:9) = g ab , Str (cid:8) T a T b T c (cid:9) = ( iC oab + D oab ) g oc , Str (cid:8) T a T b T c T d (cid:9) = Str (cid:8)(cid:0) N g ab + (cid:0) iC kab + D kab (cid:1) T k (cid:1) × (cid:0) N g cd + (cid:0) iC lcd + D lcd (cid:1) T l (cid:1)(cid:9) =
14 1 N g ab g cd + (cid:0) iC kab + D kab (cid:1) g kl (cid:0) iC lcd + D lcd (cid:1) , Str (cid:8) T a T b T c T d T e (cid:9) = Str (cid:8)(cid:0) N g ab + (cid:0) iC kab + D kab (cid:1) T k (cid:1) × (cid:0) N g cd + (cid:0) iC lcd + D lcd (cid:1) T l (cid:1) T e (cid:9) = N g ab (cid:0) iC lcd + D lcd (cid:1) g le + N g cd (cid:0) iC kab + D kab (cid:1) g ke + (cid:0) iC kab + D kab (cid:1) (cid:0) iC lcd + D lcd (cid:1) ( iC okl + D okl ) g oe . (A.3) A.2 Vertices
A.2.1 π − s -vertex The vertex illustrated in figure 5(a) is V γ,α,β πs = − i B n ( − η α ( η β + η γ ) (cid:16) Str (cid:8) T α T γ T β (cid:9) + ( − η β η γ Str (cid:8) T α T β T γ (cid:9) +Str (cid:8) T γ T β T α (cid:9) + ( − η β η γ Str (cid:8) T β T γ T α (cid:9) (cid:17)o (A.4)11 γ βp βµ p γµ (a) V γ,α,β π ǫ αβγδ (b) V α,β,γ,δ,ǫ π − s αβγδ p αµ p δµ p βµ p γµ (c) V α,β,γ,δ π, kinetic , V α,β,γ,δ π, mass α βγ s αµ p βµ p γµ LEC (d) V α,β,γl , Figure 5 : Vertices entering the computation of the scalar form factor.
A.2.2 π and π − s vertices The 4 π − s vertex in figure 5(b) is V γ,δ,ǫ,α,β πs = i BF P P S ( σ ( α ) , σ ( β ) , σ ( γ ) , σ ( δ )) × n ( − η ǫ ( η α + η β + η γ + η δ ) Str (cid:8) T ǫ T σ ( α ) T σ ( β ) T σ ( γ ) T σ ( δ ) (cid:9) +Str (cid:8) T σ ( α ) T σ ( β ) T σ ( γ ) T σ ( δ ) T ǫ (cid:9) o , (A.5)and the vertex deriving from the mass term as in figure 5(c) is V γ,δ,α,β π M = i BF P P S ( σ ( α ) , σ ( β ) , σ ( γ ) , σ ( δ ))Str (cid:8) M T σ ( α ) T σ ( β ) T σ ( γ ) T σ ( δ ) (cid:9) . (A.6)Here P indicates that the sum is over all permutations σ of the indices. Starting from( α, β, γ, δ ) a given permutation ( σ ( α ) , σ ( β ) , σ ( γ ) , σ ( δ )) may be reached by a series ofexchanges of pairs of neighbouring indices. S ( α, β, γ, δ ) is the product of signs that arepicked up in each such exchange of two indices depending on its grading. For example,for ( σ ( α ) , σ ( β ) , σ ( γ ) , σ ( δ )) = ( γ, β, α, δ ), S ( γ, β, α, δ ) = ( − η γ ( η β + η α )+ η β η α .The four point vertex deriving from the kinetic term is V γ,δ,α,β π, kin = i F P P S ( σ ( α ) , σ ( β ) , σ ( γ ) , σ ( δ )) P ( σ ( α ) , σ ( β ) , σ ( γ ) , σ ( δ )) × Str (cid:8) T σ ( α ) T σ ( β ) T σ ( γ ) T σ ( δ ) (cid:9) , (A.7)where P ( a, b, c, d ) = p a · p b − p a · p c + p a · p d − p b · p d − p b · p c + p c · p d . In thecase where all external legs are bosons the known vertices of SU ( N ) chiral perturbationtheory are recovered. 12 .2.3 Counter terms The counter terms needed derive from L (4) as given in eq. (3.3) and the correspondingvertices with the indices associated to the external legs as illustrated in figure 5(d) are(we assume here that the external legs fulfil η β = η γ = 0), V γ,α,βL = − i L BF p · p ′ Str { T α } Str (cid:8) T β T γ + T γ T β (cid:9) ,V γ,α,βL = − i L BF p · p ′ Str (cid:8) T α (cid:0) T β T γ + T γ T β (cid:1)(cid:9) ,V γ,α,βL = + i L BF n Str { M } Str (cid:8) T α (cid:0) T β T γ + T γ T β (cid:1)(cid:9) +Str { T α } Str (cid:8) M (cid:0) T β T γ + T γ T β (cid:1)(cid:9) o ,V γ,α,βL = − i L BF n Str (cid:8) T α M T β T γ (cid:9) + Str (cid:8) T α M T γ T β (cid:9) + (cid:0) Str (cid:8)
M T α T β T γ (cid:9) + Str (cid:8) M T α T γ T β (cid:9)(cid:1) +2 (cid:0) Str (cid:8)
M T β T α T γ (cid:9) + Str (cid:8) M T γ T α T β (cid:9)(cid:1) o . (A.8) B Results for the octet and singlet form factors
Here we provide the results for the octet and the singlet scalar form factor as definedin eq. (4.6), F F , S, ( t )= 2 B n F (cid:16) A +Λ F , − m π ¯ B ( m η , t )+ (2 t − m π ) ¯ B ( m π , t ) − t ¯ B ( m K , t ) (cid:17)o ,F C , S, ( t )= 2 B n F (cid:16) A +Λ C , + ( t − m π ) ¯ B ( m π , t )+ t ¯ B ( m K , t )+ m π ¯ B ( m η , m π , t ) o ,F D , S, ( t )= 2 B n F (cid:16) +Λ D , − m π ¯ B ( m η , t )+ ( t + m π ) ¯ B ( m π , t ) − t ¯ B ( m K , t ) − m π ¯ B ( m η , m π , t ) o , (B.1)13nd F F , S, ( t ) = 2 B n F (cid:16) A +Λ F , + m π ¯ B ( m η , t )+ (2 t − m π ) ¯ B ( m π , t )+ t ¯ B ( m K , t ) o ,F C , S, ( t ) = 2 B n F (cid:16) A +Λ C , + ( t − m π ) ¯ B ( m π , t )+ t ¯ B ( m K , t )+ m π ¯ B ( m η , m π , t ) o ,F D , S, ( t ) = 2 B n F (cid:16) +Λ D , + m π ¯ B ( m η , t )+ ( t + m π ) ¯ B ( m π , t ) + t ¯ B ( m K , t ) − m π ¯ B ( m η , m π , t ) o . (B.2)These form factors have the following dependence on low-energy constants:Λ F , = 4 n m π ( − L r − L r + 4 L r + 8 L r ) + m K ( − L r + 8 L r ) + tL r o , Λ C , = 4 n m π ( − L r − L r + 4 L r + 8 L r ) + m K ( − L r + 8 L r ) + tL r o , Λ D , = 0 , (B.3)andΛ F , = 4 n m π ( − L r − L r +16 L r + 8 L r )+ m K ( − L r + 8 L r )+ t (3 L r + L r ) o , Λ C , = 4 n m π ( − L r − L r + 4 L r + 8 L r )+ m K ( − L r + 8 L r )+ t L r o , Λ D , = 4 n m π ( − L r +12 L r ) + t L r o . (B.4)Note that F D , S is an entirely parameter-independent prediction for the disconnectedcontribution. C Kinematical functions
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