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BUTP-96/3, hep-ph/9601285 ¯ ¯ The Chiral Coupling Constants l and l from ππ Phase 1 2 Shifts 6 9 9 1 n a J 7 1 B. Ananthanarayan 1 P. Bu¨ttiker v 5 8 Institut fu¨r theoretische Physik 2 1 Universität Bern, CH–3012 Bern, Switzerland 0 6 9 / h p - p e h Abstract : v i X A Roy equation analysis of the available ππ phase shift data is r a performed with the I = 0 S- wave scattering length a0 in the range 0 predictedbytheone-loop standardchiral perturbationtheory. Asuit- able dispersive framework is developed to extract the chiral coupling constants¯l ,¯l andyields¯l = 1.70 0.15 and¯l 5.0. Weremarkon 1 2 1 2 − ± ≈ the implications of this determination to (combinations of) threshold parameter predictions of the three lowest partial waves. 1 1 Introduction Chiralperturbationtheory [1,2,3]provides the lowenergyeffective theoryof the standard model that describes interactions involving hadronic degrees of freedom and exploits the near masslessness of the u, d (and s) quarks and the observation that the pions, kaons and the η could be viewed as the Goldstone bosons of the spontaneously broken axial symmetry of massless QCD. It is a non-renormalizable theory and involves additional coupling constants that havetobeintroducedateachorderinthederivativeormomentumexpansion. [From here on we confine our attention to the SU(2) flavor subgroup.] At leading order, O(p2), we have the pion decay constant, F 93 MeV in π ≃ addition to the mass of the pion, m = 139.6 MeV, henceforth set equal to 1. π As a result, one has forwhat is arguably the simplest purely hadronic process ofππ scatteringapredictionforakeythresholdparameter,theI = 0,S-wave scattering length a0 = 7/(32πF2) 0.16 [4]. There are ten more coupling 0 π ≃ constants at the next to leading order, O(p4); four of them, ¯l , ¯l , ¯l and ¯l 1 2 3 4 enter the ππ scattering amplitude [1]. As a result, at this order a0 (and a2) 0 0 are predicted in terms of these as well. One of the cornerstones of standard chiral perturbation theory at O(p4) is a relatively definite prediction of a0 in 0 the range 0.19-0.21. ¯ ¯ The coupling constants l and l have been fixed in the past from ex- 1 2 perimental values available in the literature [5] for the D- wave scattering 2 lengths: a0 = 17 3 10−4, a2 = 1.3 3 10−4. 2 ± · 2 ± · The one-loop expressions for these are [2]: 1 53 1 103 a0 = (¯l +4¯l ), a2 = (¯l +¯l ) (1.1) 2 1440π3F4 1 2 − 8 2 1440π3F4 1 2 − 40 π π and yield ¯ ¯ l = 2.3 3.7, l = 6.0 1.3. 1 2 − ± ± These have also been determined from analysis of K decays [6] which yield l4 ¯ ¯ l = 0.7 0.9, l = 6.3 0.5 1 2 − ± ± and more recently by estimating higher order corrections to these decays [7] ¯ ¯ l = 1.7 1.0, l = 6.1 0.5. 1 2 − ± ± ¯ ¯ l andl arecoupling constantsconsistent withthepresence ofresonances. In 1 2 particular the ρresonance may make a significant contribution [see Appendix C in Ref. [2]] and has also been discussed more recently [8]. Furthermore, generalized vector meson dominance [9] leads to numerical values for these consistent with the numbers above. Tensor resonances also have been found to make non-trivial contributions [10]. ¯ The constants l has been estimated from the analysis of SU(3) mass 3 relations which yields [2]: ¯ l = 2.9 2.4. 3 ± 3 The variation of a0 is essentially equivalent to the variation of ¯l . While 0 3 here ¯l would have to be 70 in order to accommodate a0 = 0.26, there 3 − 0 is an extended framework which re-orders the chiral power counting in order to accommodate such large values of a0 modifying even the tree-level 0 prediction [11]. Here we confine ourselves to the more predictive standard chiral perturbation theory whose stringent predictions will come under ex- perimental scrutiny [12]. Thus we note that in our final analysis we cannot ¯ claim an independent determination of l via the Roy equation analysis of 3 this work since a0 is varied anyway in the range predicted by standard chiral 0 perturbation theory. ¯ The constant l enters the one-loop expression of the relatively accurately 4 determined value of the “universal curve” combination 2a0 5a2 [13] and is 0 − 0 also related to the independent estimates of the scalar charge radius of the pion. An SU(3) analysis that has been performed for the ratio of the kaon and pion decay constants F /F also provides a measure of this constant K π ¯l 4.6 [14]. In the present analysis we treat a0 as the only free parameter 4 ≈ 0 to the fit to the data and a2 is produced as an output corresponding to the 0 optimal solution of our data fitting. In particular, the values we find remain consistent with the universal-curve band. Thus we have a determination of the constant ¯l . However, we also perform constrained fits with a2 computed 4 0 ¯ from certain universal curve relations that fix l a priori. The influence on 4 the actual numerical fits is found to be minimal reflecting the weakness of 4 ¯ ¯ the I = 2 channel and influences the determination of l and l minimally 1 2 due to reasons we discuss in subsequent sections. On the other hand ππ scattering has been studied in great detail in ax- iomaticfieldtheory[15]. (Fixed-t)dispersionrelationswithtwosubtractions, a number dictated by the Froissart bound, have been rigorously established in the axiomatic framework. The properties of crossing and analyticity have been exploited in order to establish the Roy equations [16, 17], a system of integral equations for the partial waves. The Roy equations have been the basis of analysis of phase shift data [18] and a knowledge of the threshold parameters involved in ππ scattering has been obtained. Best fits to Roy equation analysis of data are obtained with a0 = 0.26 0.05 [19]. Note that 0 ± the D- wave scattering lengths cited earlier have also been extracted from Roy equation analysis. The properties of analyticity, unitarity and crossing andpositivityofabsorptivepartshavealsobeenshowntoproducenon-trivial ¯ ¯ constraints on the magnitudes of a certain combination of l and l [20]. 1 2 Here we report on a direct determination of the chiral coupling constants from the existing phase shift data itself by performing a Roy equation fit to it when a0 is confined to the range predicted by chiral perturbation theory. 0 The chiral amplitude is rewritten in terms of a dispersive representation with a certain number of effective subtractions consistent with O(p4) accuracy, where the subtraction constants are expressed in terms of the chiral coupling constants. The fixed-t dispersion relations of axiomatic field theory are also 5 rewritten in a form whereby a direct comparison can be made with the chiral dispersive representation, while the effective subtraction constants are now computed in terms of physical partial waves produced by the Roy equation fit, the input value of a0 and the resulting value of a2 generated by the fit. In 0 0 most of our treatment we limit ourselves to a certain approximation where we account for the absorptive parts of l 2 states only through the “driving ≥ terms” of the Roy equations for the S- and P- waves. Furthermore, we also perform an analysis of certain threshold parameters computed from the Roy equation fits which reveals the magnitude of O(p6) corrections their one-loop predictions are expected to suffer from. The dispersive framework can be extended to meet the needs of two-loop chiral perturbation theory and used to pin down the associated coupling constants [21, 22]. The work reported here summarizes the first stage of our computations and is presently being extended to meet the needs of the two- loop computation of [22]. 6 2 ππ Scattering to O(p4) in chiral perturbation theory and the Roy equation solutions The notation and formalism that we adopt in this discussion follows that of Ref. [18]. Consider ππ scattering: πa(p )+πb(p ) πc(p )+πd(p ), a b c d → where all the pions have the same mass. The Mandelstam variables s, t and u are defined as s = (p +p )2, t = (p p )2, t = (p p )2, s+t+u = 4. (2.1) a b a c a d − − The scattering amplitude F(a,b c,d) (our normalization of the amplitude → differs fromthat of Ref. [18] by 32π and is consistent with that of Ref.[2, 23]) can then be written as F(a,b c,d) = δ δ A(s,t,u)+δ δ A(t,s,u)+δ δ A(u,t,s). ab cd ac bd ad bc → From A(s,t,u) we construct the three s-channel isospin amplitudes: T0(s,t,u) = 3A(s,t,u)+A(t,s,u)+A(u,t,s), T1(s,t,u) = A(t,s,u) A(u,t,s), (2.2) − T2(s,t,u) = A(t,s,u)+A(u,t,s). We introduce the partial wave expansion: ∞ t u TI(s,t,u) = 32π (2l+1)P ( − )fI(s), (2.3) l s 4 l Xl=0 − 7 f0(s) = f2(s) = 0,l odd, f1(s) = 0,l even. l l l The unitarity condition for the partial wave amplitudes fI(s) is: l 1 (ηI(s))2 ImfI(s) = ρ(s) fI(s) 2 + − l , (2.4) l | l | 4ρ(s) where ρ(s) = (s 4)/s and ηI(s) is the elasticity at a given energy √s. − l q We also introduce the threshold expansion: s 4 l s 4 RefI(s) = − aI +bI − +... , (2.5) l 4 l l 4 (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:19) where the aI are the scattering lengths and the bI are the effective ranges, l l namely the leading threshold parameters. Chiral perturbation theory at next to leading order gives an explicit representation for the function A(s,t,u) at O(p4) [2]: A(s,t,u) = A(2)(s,t,u)+A(4)(s,t,u)+O(p6), (2.6) with s 1 A(2)(s,t,u) = − , F2 π 1 A(4)(s,t,u) = 3(s2 1)J¯(s) 6F4 − π (cid:16) + [t(t u) 2t+4u 2]J¯(t)+(t u) − − − ↔ 1 (cid:17) + 2(¯l 4/3)(s 2)2 +(¯l 5/6)[s2 +(t u)2] 1 2 96π2F4{ − − − − π ¯ ¯ ¯ +12s(l 1) 3(l +4l 5) 4 3 4 − − − } 1 1 ρ(s) and J¯(z) = dxln[1 x(1 x)z], ImJ¯(z) = Θ(z 4). −16π2 − − 16π − Z0 8 NotealsothatatO(p4)theimaginarypartsofthepartialwaves abovethresh- old computed (s > 4) from the amplitude above is: ρ(s) Imf0(s) = (2s 1)2 0 1024π2F 4 − π ρ(s) Imf1(s) = (s 4)2 1 9216π2F 4 − π ρ(s) Imf2(s) = (s 2)2 (2.7) 0 1024π2F 4 − π ImfI(s) = 0, l 2. l ≥ [Notethatthechiralpowercountingenforcesthepropertythattheabsorptive parts of the D- and higher waves arise only at O(p8).] Furthermore these verify the property of perturbative unitarity, viz., when the O(p2) predictions for the threshold parameters a0 = 7/(32πF 2), a2 = 1/(16πF 2), b0 = 0 π 0 − π 0 1/(4πF 2), b2 = 1/(8πF 2) and a1 = 1/(24πF 2) are inserted into the π 0 − π 1 π pertinent form of the perturbative unitarity relations: ImfI(s) = ρ(s)(aI +bI(s 4)/4)2, I = 0,2 0 0 0 − Imf1(s) = ρ(s)(a1(s 4)/4)2. 1 1 − In order to carry out the comparison between the chiral expansion and the physical scattering data, we first recall that up to O(p6), it is possible to decompose A(s,t,u) into a sum of three functions of single variables as follows [24]: 9 1 3 3 A(s,t,u) = 32π W (s)+ (s u)W (t)+ (s t)W (u) 0 1 1 3 2 − 2 − (cid:20) 1 2 + W (t)+W (u) W (s) . (2.8) 2 2 2 2 − 3 (cid:18) (cid:19)(cid:21) One convenient decomposition of the chiral one-loop amplitude is: 3 s 1 2 W (s) = − + (s 1/2)2J¯(s) 0 32π " Fπ2 3Fπ4 − 1 + (2(¯l 4/3)(s 2)2 +4/3(¯l 5/6)(s 2)2 96π2F4 1 − − 2 − − π ¯ ¯ ¯ +12s(l 1) 3(l +4l 5)) , (2.9) 4 3 4 − − − # 1 W (s) = (s 4)J¯(s), (2.10) 1 576πF4 − π 1 1 1 W (s) = (s 2)2J¯(s)+ (¯l 5/6)(s 2)2 ,(2.11) 2 16π "4Fπ4 − 48π2Fπ4 2 − − # where we note that this decomposition is not unique, with ambiguities in the real part only. We observe that the imaginary parts of these functions verify the relation: ImW (x) = ImfI(x), I = 0,2 I 0 ImW (x) = Imf1(x)/(x 4), 1 1 − which may be used to demonstrate the following dispersion relations: 10