HISKP-TH-03/18 IPNO/DR-03-08/ LPT-ORSAY/03-76 February 1, 2008 A new analysis of πK scattering from Roy and Steiner type equations 1 4 0 0 2 P. Bu¨ttikera, S. Descotes-Genonb and B. Moussallamc n a a Helmholtz-Institut fu¨r Strahlen- und Kernphysik, J Universita¨t Bonn, D-53115 Bonn, Germany 5 b Laboratoire de Physique Th´eorique2 3 Universit´e Paris-Sud, F-91406 Orsay, France v 3 8 c Institut de Physique Nucl´eaire3 2 Universit´e Paris-Sud, F-91406 Orsay, France 0 1 3 0 / h p - p e h Abstract : v With the aim of generating new constraints on the OZI suppressed couplings of chiral i X perturbation theory a set of six equations of the Roy and Steiner type for the S- and P- r wavesof the πK scatteringamplitudes is derived. The rangeof validity andthe multiplicity a of the solutions are discussed. Precise numerical solutions are obtained in the range E < 1 GeV which make use as input, for the first time, of the most accurate experimental d∼ata available at E >1 GeV for both πK πK and ππ KK amplitudes. Our main result is the determinat∼ion of a narrowallowed→regionfor the→two S-wavescattering lengths. Present experimentaldatabelow1GeVarefoundtobeingenerallypooragreementwithourresults. Asetofthresholdexpansionparameters,aswellas sub-thresholdparametersarecomputed. For the latter, a matching with the SU(3) chiral expansion at NLO is performed. 1Work supported in part by the EU RTN contract HPRN-CT-2002-00311 (EURIDICE) and by IFCPAR contract 2504-1. 2LPT is an unit´emixte derecherche duCNRS et del’Universit´e Paris-Sud (UMR 8627). 3IPN is an unit´emixtede recherchedu CNRS et del’Universit´e Paris-Sud (UMR 8608). 1 Introduction Scattering amplitudes of pseudo-Goldstone bosons at low energies probe with a unique sensitiv- ity the scalar-source sector of Chiral Perturbation Theory (ChPT) [1, 2]. For instance, recent progress in the domain of ππ scattering has provided valuable information on the SU(2) chiral limitwherethemasses oftheu,dquarksaresettozero. For thispurpose,theππ Roy equations, which have been extensively studied in the past [3, 4, 5], were re-analyzed [6] (in particular, a formulation as a boundary value problem was developped) and solved numerically [6, 7] (see also [8]). These equations constrain the low-energy ππ-scattering amplitude by exploiting si- multaneously theoretical requirements and data at higher energies. New data on Kl decays 4 from the E865 experiment [9] could thus be studied with the help of the solutions to the Roy equations and a bound on the coupling constant ℓ¯ of the SU(2) chiral Lagrangian [10] was 3 derived for the first time. Constraints on the SU(2) quark condensate were also obtained along similar lines [11]. In a parallel way, scattering amplitudes involving both pions and kaons at very low energy should allow one to unveil features of the SU(3) chiral vacuum, i.e. that in the limit where m , m and m vanish. The structure of the SU(3) chiral vacuum is worth studying for its u d s own sake, since SU(3) ChPT provides relations between many low-energy processes involving π-, K- and η-mesons. In addition, it is interesting to compare SU(2) and SU(3) chiral limits, especially in the scalar sector. A sizable difference between the two limits would indicate that sea-quark effects areparticularly significant in the case of the strange quark[12, 13]. Inprevious works[14,15]itwasshownthattheratio ofthepion’sdecay constant inSU(2)andSU(3)chiral limits could be determined from a sum rule based on πK scattering amplitudes. The deviation of this ratio from 1 would indicate a violation of the large-N approximation. Let us emphasize c that the latter is often relied upon to attribute values to some O(p4) couplings arising in the scalar sector of thechiral Lagrangian [2, 16]. Ourwork is motivated by thedesireof determining from πK-scattering experimental data as many chiral couplings as possible (in principle, five out of the ten independent O(p4) couplings of the SU(3) chiral Lagrangian [17, 18]), without relying on the large-N approximation. In this paper, we provide the first step of this analysis, c by deriving the analogue of Roy equations for the πK system and solving them numerically. A simple matching with the SU(3) expansion is performed while more detailed comparisons with SU(2) and SU(3) expansions are left for future work. Further motivation for the study of πK scattering can be found in refs. [19, 20]. Inthecaseofππscattering,Royobserved[21]thatgeneralpropertiesofanalyticity, unitarity, combined with crossing symmetry, lead to a set of non-linear integral equations that the S- and P-partialwavesmustsatisfy. AsimilarprogramwascarriedoutbySteinerforπN scattering[22]. Given experimental input at high energies (typically E > 1 GeV), Roy-Steiner (RS) equations constrain the low-energy behaviour of partial-wave ampl∼itudes. In the present paper, we derive and perform a detailed analysis of a system of RS equations for πK scattering. In this case, s t crossing relates the πK πK and the ππ KK amplitudes, leading to six coupled − → → 1/2 1/2 3/2 3/2 equations that involve the four πK S and P partial-wave amplitudes f , f , f , f 0 1 0 1 and the two ππ KK amplitudes g0, g1. Equations of a similar kind have been considered → 0 1 earlier [23, 24, 25]. However, some approximations were invoked in the treatment of these 1 equations and,moreover, noaccurate experimentalinputdatawereavailable atthattime. Since then, high-statistics production experiments have been performed for both πK πK [26, 27] → and ππ KK amplitudes [28, 29]. These experiments provide the necessary input data for the → RS equations with a level of accuracy comparable to the case of ππ scattering. Experimental data at lower energies should also be available in the near future: the FOCUS experiment [30] has demonstrated the feasibility of determining the πK S-wave phase shifts at energies lower that 1 GeV from the weak decays of D mesons [31], P-wave phase shifts should be measured soon in τ decays [32] and finally, direct determinations of combinations of S-wave scattering lengths are expected from planned experiments on kaonic atoms [33]. The plan of the paper is as follows. After reviewing the notation, we derive the set of RS equations that we intend to solve. The setting is similar to a previous work [25] but we differ in the number of subtractions used in the dispersive representations. We aim here at an optimal use of the energy region where accurate experimental data are available, while avoiding to rely on slowly convergent sum rules. After discussing the domains of validity of such equations, we explain our treatment of the available experimental input and of the asymptotic regions. Next, we start solving the equations. One first eliminates g0 and g1 and the remaining four equations 0 1 then have a similar structure to the ππ Roy equations such that recent results concerning the multiplicity of the solutions [34, 35] can be exploited. Finally, we turn to the numerical resolution and discuss the resulting constraints on the S-wave scattering lengths. Finally, the πK amplitudes near and below threshold are constructed and estimates for the O(p4) chiral coupling constants obtained from matching with the SU(3) expansion are given. 2 Derivation of the equations 2.1 Notation Let us recall briefly some standard notation [36]. Firstly, we define from the pion and kaon masses m = m m , Σ = m2 +m2, ∆ = m2 m2 . (1) ± K ± π K π K − π In this paper, exact isopin symmetry will always be assumed. In the isospin limit, there are two independent πK amplitudes FI(s,t), with isospin I = 1 and I = 3. Making use of s u 2 2 − crossing, the I = 1 amplitude can be expressed in terms of the I = 3 one, 2 2 1 3 1 3 3 F2(s,t,u) = F2(s,t,u)+ F2(u,t,s) . (2) −2 2 It is convenient to introduce the amplitudes F+ and F which are, respectively, even and odd − under s u crossing. In terms of isospin amplitudes, they are defined as − 1 2 F+(s,t,u) = F21(s,t,u)+ F32(s,t,u) 3 3 1 1 1 3 F−(s,t,u) = F2(s,t,u) F2(s,t,u) . (3) 3 − 3 2 The partial-wave expansion of the πK isospin amplitudes is defined as FI(s,t) = 16π (2l+1)P (z )fI(s) . (4) l s l l X where P (z) are the standard Legendre polynomials and z is the cosine of the s-channel scat- l s tering angle 2st z = 1+ with λ =(s m2)(s m2) . (5) s λs s − + − − In a similar way we can expand F+ and F , and the corresponding partial-wave projections are − denoted by f+(s) and f (s). The amplitudes can be projected over the partial waves through l l− s 0 fI(s) = dt P (z )FI(s,t) . (6) l 16πλ l s s Z−λs/s ThevaluesoftheamplitudesatthresholddefinetheS-wavescatteringlengths,withthefollowing conventional normalization 2 aI = fI(m2) (7) 0 m 0 + + (and similarly for a in terms of f (m2)). ±0 0± + Under s t crossing, one generates the I = 0 and I = 1 ππ KK amplitudes, − → G0(t,s,u) = √6F+(s,t,u) G1(t,s,u) = 2F (s,t,u) . (8) − The partial-wave expansion of the ππ KK amplitudes is conventionally defined as → GI(t,s) = 16π√2 (2l+1)[q (t)q (t)]lP (z )gI(t) , (9) π K l t l l X wherethesummationrunsover even (odd)values of l forI = 0(I = 1) duetoBose symmetryin theππ channel. Inthisexpressionthemomentaq , q andthecosineofthet-channelscattering π K angle z are given by t 1 s u q (t) = t 4m2, z = − . (10) P 2 − P t 4q (t)q (t) π K q The relations between these partial-wave amplitudes and the S-matrix elements are easily worked out √λ SI(s) = 1+2i sθ(s m2)fI(s) l πK πK s − + l h i → (q (t)q (t))l+1/2 SI(t) = 4i π K θ(t 4m2 )gI(t) . (11) l ππ KK √t − K l h i → 3 2.2 Fixed-t based dispersive representation To derive RS equations, we assume the validity of the Mandelstam double-spectral representa- tion [37] from which one can derive a variety of dispersion relations (DR’s) for one variable 4. According to the Froissart bound [40], two subtractions are needed at most for F+ and one sub- traction for F (because s u can befactored out in thelatter case). More detailed information − − about asymptotic behaviour is provided by Regge phenomenology [41], according to which two subtractions are indeed necessary for F+ while an unsubtracted DR is expected to converge for F . However, convergence is rather slow in the latter case since the integrand behaves like − (s) 3/2 asymptotically. Therefore, we choose to make use of a once-subtracted DR for F in ′ − − order to improve the convergence and reduce the sensitivity to the high-energy domain. Fixed-t DR’s for F+ and F , with the number of subtractions as discussed above can be − written in the following form 1 1 1 2s 2Σ t F+(s,t)= c+(t)+ ∞ds′ + ′− − ImF+(s′,t) . π Zm2+ (cid:20)s′−s s′−u − λs′ (cid:21) F (s,t) 1 1 1 − = c (t)+ ∞ds ImF (s,t) . (12) − ′ − ′ s−u π Zm2+ (cid:20)(s′−s)(s′−u) − λs′(cid:21) These expressions involve two unknown functions of t: c+(t) and c (t). The basic idea for − determining these functions is to invoke crossing [21, 22], which can be implemented in various ways: for instance, one can use fixed-s or fixed-(s u) DR’s. After trying several possibilities, − we found that DR’s at fixed us provide the largest domain of applicability (these relations, sometimes called hyperbolic DR’s, were exploited in ref. [25]). We start with a special set of hyperbolic DR’s (more general hyperbolic DR’s will be considered later) in which us = ∆2 . (13) The condition above fixes s and u to be functions of t 1 s s (t) = 2Σ t+ (t 4m2)(t 4m2 ) ≡ ∆ 2 − − π − K (cid:18) q (cid:19) 1 u u (t) = 2Σ t (t 4m2)(t 4m2 ) . (14) ≡ ∆ 2 − − − π − K (cid:18) q (cid:19) According to Regge theory, the function F+(s ,t) satisfies a once-subtracted DR which is ∆ slowly converging. Like in the case of the fixed-t DR for F , we choose to improve the conver- − gence by using a twice-subtracted representation. On the other hand, the function F (s ,t) is − ∆ expected to satisfy an unsubtracted DR which is well converging. Making use of the fact that s (0) = m2, these DR’s can be written in the following way ∆ + 1 2s 2Σ+t 2s 2Σ t F+(s∆,t) = 8πm+a+0 +b+t+ π Zm∞2+ds′ (cid:20) λ′s−′ +s′t − ′−λs′ − (cid:21)ImF+(s′,t′∆) 4For the πK amplitude, the existence of fixed-t DR can be established on more general grounds in a finite domain of t [38, 39]. 4 t2 dt + ∞ ′ ImG0(t,s ) √6π Z4m2π (t′)2(t′−t) ′ ′∆ F (s ,t) 8πm a 1 1 1 − ∆ = + −0 + ∞ ds ImF (s,t ) s∆−u∆ m2+−m2− π Zm2+ ′ (cid:20)λs′ +s′t − λs′(cid:21) − ′ ′∆ t dt G1(t,s ) + ∞ ′ Im ′ ′∆ . (15) 2π Z4m2π t′(t′−t) (t′−4m2π)(t′−4m2K) q In these equations, we have used the following notation ∆2 s = s (t), t = 2Σ s , (16) ′∆ ∆ ′ ′∆ − ′− s ′ together with the relation (s s (t))(s u (t)) = λ +st . ′ ∆ ′ ∆ s′ ′ − − These representations involve three subtraction constants: the two scattering lengths a+, a 0 −0 and an additional parameter denoted b+. Let us now show that the latter can be computed through a rapidly convergent sum rule. We notice first that a and b+ satisfy slowly convergent −0 sum rules, 8πm a 1 ds 1 dt G1(t,s ) + −0 = ∞ ′ImF (s,t )+ ∞ ′Im ′ ′∆ . m2+−m2− π Zm2+ λs′ − ′ ′∆ 2π Z4m2π t′ (t′−4m2π)(t′−4m2K) 1 ds 1 dt q b+ = −π Zm∞2+ λs′′ImF+(s′,t′∆)+ √6π Z4m∞2π (t′)′2ImG0(t′,s′∆) . (17) By combining these two sum rules, we can express the parameter b+ as a sum rule which has better convergence property: 8πm a 1 ds b+ = + −0 ∞ ′Im F+(s,t )+F (s,t ) m2+−m2− − π Zm2+ λs′ (cid:2) ′ ′∆ − ′ ′∆ (cid:3) 1 dt G0(t,s ) G1(t,s ) + ∞ ′Im ′ ′∆ ′ ′∆ . (18) π Z4m2π t′  √6t′ − 2 (t′−4m2π)(t′−4m2K)  q  Why does this sum rule converge more quickly ? In the first integral, the combination F++F − appears, which is the amplitude for the process π+K π+K . The asymptotic region of − − → the integrand corresponds to s , u 0. The amplitude in this region is controlled by the → ∞ → Regge trajectories in the u channel which is exotic, leading to a fast decrease of the integrand. − In the second integral, the high-energy tail involves the combination 1 G0(t,s) 1G1(t,s) √6 ′ ′ − 2 ′ ′ for t and s 0. The leading Regge contributions are generated by the K and K ′ ′ ∗∗ ∗ → ∞ → trajectories 1 1 lim Im G0(t,s) G1(t,s) = βK∗∗(s)tαK∗∗(s) βK∗(s)tαK∗(s) . (19) t , s 0 √6 − 2 − →∞ → (cid:20) (cid:21) This difference would vanish if Regge trajectories satisfied exactly the property of exchange de- generacy. Innature,thispropertyisnotexactbutithaslongbeenobservedtobeapproximately 5 fulfilled 5 (see e.g. [41] ), which should lead to a significant suppression of the integrand at high energies. Therefore, the two integrals involved in eq. (18) are expected to converge quickly, providing a determination of b+ with only a mild sensitivity to high energies. Combining the two dispersive representations eqs. (12) and (15) for the amplitudes F+ and F , the subtraction functions in eqs. (12) get determined in terms of the two S-wave scattering − lengths and we obtain the following representation for the two amplitudes 1 1 1 2s 2Σ+t F+(s,t) = 8πm+a+0 +b+t+ π Zm∞2+ds′ (cid:20)s′−s + s′−u − λ′s−′ +s′t (cid:21)ImF+(s′,t) 1 2s 2Σ+t 2s 2Σ t + π Zm∞2+ds′ (cid:20) λ′s−′ +s′t − ′−λs′ − (cid:21)ImF+(s′,t′∆) t2 dt + ∞ ′ ImG0(t,s ) , √6π Z4m2π (t′)2(t′−t) ′ ′∆ 8πm a 1 1 1 s u F−(s,t) = m2+−+m−02−(s−u)+ π Zm∞2+ds′ (cid:20)s′−s − s′−u − λs′−+s′t(cid:21)ImF−(s′,t) 1 1 1 +(s u) ∞ds ImF (s,t ) − (π Zm2+ ′ (cid:20)λs′ +s′t − λs′(cid:21) − ′ ′∆ t dt G1(t,s ) + ∞ ′ Im ′ ′∆ (20) 2π Z4m2π t′(t′−t) (t′−4m2π)(t′−4m2K)) q where the parameter b+ is to be expressed in the terms of the sum rule eq. (18). The domain of applicability of this representation is limited by the domain of validity of the fixed t DR’s, − eq. (12). In sec. 3, we will show that the fixed-t DR’s hold for t < 4m2, which enables us to π performtheprojectionofeq.(20)onπK πK partialwaves. Wewillalsoneedarepresentation → which is valid for t 4m2 in order to obtain equations for the ππ KK¯ partial waves. For ≥ π → this purpose, we now consider a family of hyperbolic DR’s. 2.3 Fixed us dispersive representation Let us consider a general family of hyperbolic DR’s for which us =b (21) is fixed. b is a parameter with (a priori) arbitrary values and should not be confused with the subtraction constant b+ introduced in the previous section. We write down a twice-subtracted representation for F+(s ,t) and a once-subtracted one for F (s ,t), b − b 1 2s 2Σ+t 2s 2Σ t F+(sb,t) = f+(b)+th+(b)+ π Zm∞2+ds′" λ′bs−′ +s′t − ′−λbs′ − #ImF+(s′,t′b) 5The underlying reason for this property is not understood but could be related to the possibility that the large-N limit of QCD is described bya string theory [42, 43]. c 6 t2 dt + ∞ ′ ImG0(t,s ) , √6π Z4m2π t′2(t′−t) ′ ′b F (s ,t) 1 1 1 − b = f (b)+ ∞ ds ImF (s,t ) sb−ub − π Zm2+ ′"λbs′ +s′t − λbs′# − ′ ′b t dt G1(t,s ) + ∞ ′ Im ′ ′b (22) 2π Z4m2π t′(t′−t) s′b−u′b with the notation 1 s = 2Σ t + (2Σ t)2 4b ′b 2 − ′ − ′ − (cid:18) q (cid:19) b t = 2Σ s (23) ′b − ′− s ′ λb = (s)2 2Σs +b . s′ ′ − ′ Therepresentationseqs.(22)areageneralizationoftheDR’seqs.(15)derivedforus = ∆2. They involve three unknown functions of b: f+(b), f (b) and h+(b) (which generalize the subtraction − constants of eqs. (15) ) Thetwo functions f+(b), f (b) can bedetermined by matching eqs. (22) − with the representations eqs. (20) at the point t = 0 (which lies inside their domain of validity). Next, thefunctionh+(b)canbeexpressedasarapidlyconvergent sumruleanalogoustoeq.(18). Putting things together, one finally obtains the following representations involving the two S- wave scattering lengths a+, a as the only arbitrary constants, 0 −0 a F+(s ,t) = 8πm a++t −0 b + 0 m2+−m2−! 1 2s 2Σ+t 2s 2Σ + π Zm∞2+ds′( λ′bs−′ +s′t ImF+(s′,t′b)− ′λ−bs′ Im[F+(s′,t′b)−F+(s′,0)] t Im[F (s,t ) F (s,0)] − λb − ′ ′b − − ′ s′ 2s 2Σ t ′− ImF+(s′,0) ImF−(s′,0) − λs′ − λs′ ) t dt ImG0(t,s ) G1(t,s ) + ∞ ′ ′ ′b Im ′ ′b . π Z4m2π t′ " √6(t′−t) − 2(s′b−u′b)# F (s ,t) 8πm a 1 1 1 − b = + −0 + ∞ds ImF (s,t ) ImF (s,0) sb−ub m2+−m2− π Zm2+ ′(λbs′ +s′t − ′ ′b − λs′ − ′ 1 Im[F (s,t ) F (s,0)] − λbs′ − ′ ′b − − ′ ) t dt G1(t,s ) + ∞ ′ Im ′ ′b (24) 2π Z4m2π t′(t′−t) s′b−u′b These representations will allow us to perform projections on the t-channel partial waves for t 4m2. ≥ π 7 2.4 RS equations for fI(s) l RS equations can now be obtained by performing the partial-wave projections of the dispersive representations obtained above. Projecting eqs. (20) on the l = 0,1 πK πK amplitude we → get the first four equations, 1 1 Ref2(s) = k2(s) l l +π1Z−m∞2+ds′ l′X=0,1((cid:18)δll′(s′−λss)λs′ − 31Klαl′(s,s′)(cid:19)Imfl21′(s′)+ 43Klαl′(s,s′)Imfl32′(s′)) 1 1 + ∞ dt K0(s,t)Img0(t)+2K1(s,t)Img1(t) +d2(s) π Z4m2π ′n l0 ′ 0 ′ l1 ′ 1 ′ o l 3 3 Ref2(s) = k2(s) l l 1 λ 1 3 2 1 +πZ−m∞2+ds′ l′X=0,1((cid:18)δll′(s′−ss)λs′ + 3Klαl′(s,s′)(cid:19)Imfl2′(s′)+ 3Klαl′(s,s′)Imfl2′(s′)) 1 3 + ∞ dt K0(s,t)Img0(t) K1(s,t)Img1(t) +d2(s) . (25) π Z4m2π ′n l0 ′ 0 ′ − l1 ′ 1 ′ o l The domain of validity in s of these equations is given by eq. (53) below. In these equations, the terms kI(s) contain the contributions associated with the subtraction constants, l 1 λ 7 8πm a 3s+m2 k0I(s)= 2m+aI0+ 32πss −b++(−3I + 2)m2+−+m−02− s−m2−−! λ 7 8πm a kI(s)= s b+ +( 3I + ) + −0 . (26) 1 96πs − 2 m2+−m2−! TheequationsinvolvethreekindsofkernelsKα(s,s),KI (s,t),andKσ(s,s)(whichappear ll′ ′ ll′ ′ ll′ ′ only in the driving terms dI). The kernels Kα read, for l, l = 0, 1, l ll′ ′ λ +2s(s s) K0α0(s,s′)= − s 2sλs′′− +L(s,s′) 3(λ +2s(s +s)) 3(λ +2ss 2∆2) K0α1(s,s′)= s 2sλs′′ − s′ λs′′− L(s,s′) λ2+12s2λ (λ +2ss 2∆2) K1α0(s,s′)= s6sλsλs′ s′ − s λs′− L(s,s′) K1α1(s,s′)= −12s2(λs′ +22ssλss′λ−s′2∆2)+λ2s 3(λs +2ss′ 2∆2)(λs′ +2ss′ 2∆2) + − − L(s,s′) (27) λsλs′ with s ∆2 L(s,s) = log(s +s 2Σ) log s (28) ′ ′ ′ λs " − − − s !# 8 Next, the kernels K0 , K0 (with l even) read, 0l′ 1l′ ′ 2l +1 s λ λ λ K0 (s,t)= ′ (q q )l′ log 1+ s s 1 s 0l′ ′ √3 π′ K′ λs (cid:26) (cid:18) st′(cid:19)− st′ (cid:18) − 2st′(cid:19)(cid:27) 2l +1 s 2st λ 1 λ 2 K0 (s,t)= ′ (q q )l′ 1+ ′ log 1+ s 2 s . (29) 1l′ ′ √3 π′ K′ λs ((cid:18) λs (cid:19) (cid:18) st′(cid:19)− − 6(cid:18)st′(cid:19) ) Finally, the kernels K1 , K1 (with l odd) read 0l′ 1l′ ′ √2(2l +1) s(2s 2Σ+t) λ λ λ K01l′(s,t′) = 8′ (qπ′qK′ )l′−1 −λ ′ log 1+ sts − sts + 2sst (cid:26) s (cid:20) (cid:18) ′(cid:19) ′(cid:21) ′(cid:27) √2(2l +1) K1 (s,t) = ′ (q q )l′ 1 1l′ ′ 8 π′ K′ − × s(2s 2Σ+t) 2st λ λ ′ ′ s s − 1+ log 1+ 2 . (30) λ λ st − − 6st (cid:26) s (cid:20)(cid:18) s (cid:19) (cid:18) ′(cid:19) (cid:21) ′(cid:27) The analyticity properties of the partial-wave amplitudes fI(s) were established in ref. [44]. l They can be recovered by considering the various kernels. In particular, the circular cut is generated by the kernels KI (s,t). ll′ ′ The terms dI(s) are the so-called driving terms in which the contributions from the partial l waves with l 2 are collected ′ ≥ 1 2 1 dI(s) = ∞ ds Kσ(s,s)+ (I 1)Kα(s,s) Imf2(s) l π Zm2+ ′ lX′≥2((cid:18) ll′ ′ 3 − ll′ ′ (cid:19) l′ ′ 1 3 + ( 2I +5)Kα(s,s)Imf2(s) 3 − ll′ ′ l′ ′ ) 1 7 + ∞ dt K0 (s,t)Img0 (t)+( 3I + )K1 (s,t)Img1 (t) . (31) π Z4m2π ′lX′≥1n l2l′ ′ 2l′ ′ − 2 l2l′+1 ′ 2l′+1 ′ o The kernels Kσ(s,s) appear in the driving terms only; the first few which are non-vanishing ll′ ′ read 5λ Kσ (s,s) = s 02 ′ s(λ )2 s′ 35(λ )2s(ss ∆2) K0σ3(s,s′) = − 3ss2(′λs′)′3− 7λ (ss ∆2)((s+s)(ss +∆2) 4ssΣ) K1σ3(s,s′) = s ′− 3s2(λ′ )3′ − ′ . (32) s′ 2.5 RS equations for g0(t), g1(t) 0 1 In order to obtain a closed system of equations we now need two equations yielding the real parts of g0(t) and g1(t) valid for positive values of t. They can be obtained from the family of 0 1 9