Publication: Precision of chiral-dispersive calculations of π π scattering
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2003-10-01
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Ynduráin, F. J.
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Amer Physical Soc
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We calculate the combination 2a⁰_(0)-5a₀(^2) (the Olsson sum rule) and the scattering lengths and effective ranges a₁,a₂^(2) and b₁,b₂^(I) dispersively (with the Froissart-Gribov representation) using, at low energy, the phase shifts for π π scattering obtained by Colangelo, Gasser, and Leutwyler (CGL) from the Roy equations and chiral perturbation theory, plus experiment and Regge behavior at high energy, or directly, using the CGL parameters for a's and b's. We find mismatch, both among the CGL phases themselves and with the results obtained from the pion form factor. This reaches the level of several (2 to 5) standard deviations, and is essentially independent of the detai᾿ls of the intermediate energy region (0.82 ≤ E ≤ 1.42 GeV) and, in some cases, of the high energy behavior assumed. We discuss possible reasons for this mismatch, in particular in connection with an alternate set of phase shifts.
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©2003 The American Physical Society.
One of us (F.J.Y.) is grateful to G. Colangelo, J. Gasser, and H. Leutwyler for very interesting discussions that triggered his interest in this subject. Both of us thank again Professor Leutwyler for a very useful critical reading of a preliminary draft, which, in particular, allowed us to correct a few inconsistencies. J.R.P. acknowledges support from the Spanish CICYT projects PB98-0782 and BFM2000 1326, as well as from the Marie Curie Fellowship, grant No. MCFI- 2001-01155.
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[4] Actually, b₁ is not the effective range, though it is related to it. We use the definitions of ACGL and CGL for the a and b, except that we take the dimensions of a to be M_π ^(2l+1). Here M_π is the charged pion mass, M_=≈139.57 MeV.
[5] The method of the Froissart-Gribov representation to calculate scattering lengths and effective ranges was introduced in [6,7]. It is also discussed in some detail in [8].
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[8] F. J. Ynduráin, ‘‘Low energy pion interactions,’’ FTUAM02- 28, hep-ph/0212282.
[9] We are actually simplifying in that Eqs. (2) and (3) should take into account the different isospin structure of s and u channels, which the reader may find in, e.g. the text of Martin, Morgan, and Shaw
[10].B. R. Martín, D. Morgan, and G. Shaw, Pion-Pion Interactions in Particle Physics (Academic, New York, 1976).
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[12] For analyticity properties of pp scattering see for example [10].
[13] The (slight) differences with some of the parameters in [8] occur because now we are using M_π =m_π^+ = 139.57 MeV instead of the average pion mass, 138 MeV, and we are also essentially eliminating from the fit the data for energies above 0.96 MeV. The change in the X^2/DOF of the I=0 S wave corrects an error there. We send to this reference for details on the fitting procedure.
[14] For the more recent determination, see A. Aloisio et al., Phys. Lett. B 538, 21 (2002); the older one is from P. Pascual and F. J. Ynduráin, Nucl. Phys. B83, 362 (1974).
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[16] The fact that the errors for a₁ ,b₁ are smaller when using Eq. (15) than when using Eq. (13), which at first sight appears counterintuitive, can be understood as follows. The errors in the parameters B₀ ,M_π in Eq. (15) are larger than those in Eq (13)—as suggested by intuition. The error in B₁ , however, is smaller, and it is this quantity that influences most b1 (the error in a1 stays essentially constant). Including systematic errors makes the determinations of the pion from factor in the timelike and spacelike regions more compatible one with another, and this allows a more precise determination of low energy parameters.
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[19] Equation (21) corrects a mistake in the corresponding wave in [8].
[20] M. J. Losty et al., Nucl. Phys. B69, 185 (1974); W. Hoogland et al., ibid. B126, 109 (1977).
[21] M. Gell-Mann, Phys. Rev. Lett. 8, 263 (1962); V. N. Gribov and I. Ya. Pomeranchuk, ibid. 8, 343 (1962); for more references in general Regge theory, see V. D. Barger and D. B. Cline, Phenomenological Theories of High Energy Scattering (Benjamin, New York, 1969); for references to the QCD analysis, F. J. Ynduráin, The Theory of Quark and Gluon Interactions (Springer, Berlin, 1999).
[22] In potential theory the proof can be made mathematically rigorous; in relativistic theory, it follows from extended unitarity or, in QCD, in the DGLAP formalism, as is intuitively obvious from Fig. 3.
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[24] Fits to deep inelastic scattering processes, and references to previous literature, may be found in the book by Ynduráin in [21].
[25] Consistency requires a more complicated form for the residue functions f_i^(I_t) (t); see [6,23]. For the small values of t in which we are interested, our expressions are sufficiently accurate.
[26] In the case of the rho trajectory, exact linearity would imply α᾿_ρ(0)=1/2(M_ρ–^2_π)=≈.0.87 GeV^-2, not far from the value 1.01 GeV^-2 that the actual fits give, and which we have used here.
[27] This unreliability is reflected, for example, in the Particle Data Tables (e.g., the edition of [28])!, where no number is given for the branching ratios of pp resonances with masses at or above 1.2 GeV [with the exception of the ρ₃(1690)] and even the S0 phase around the f₀(980) has a dubious status: this last due to the ambiguity caused by the inelastic K̅K channel. In fact, the resonances that appear in ππ production are not the same that one finds in e⁺e⁻, τ decay or J/ѱ decay, and the inelasticities in both cases are also quite different.
[28] Particle Data Group, D. E. Groom et al., Eur. Phys. J. C 15, 1 (2000).
[29] D. Atkinson, G. Mahoux, and F. J. Ynduráin, Nucl. Phys. B54, 263 (19739; B98, 521 (1975). [30] M. R. Pennington, Ann. Phys. (N.Y.) 92, 164 (1975).
[31] This should not be taken as a criticism of the work of Pennington; at that time the data, very poor, did indeed suggest possible deviations of Regge behavior for ππ. On the other hand, the fact that QCD also implies factorization was of course unknown.
[32] This fit is actually a refinement of that of Eq. (7.6.2) in [8]; more details about this will be presented in a separate publication.
[33] J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.9 158, 142 (1984).
[34] S. Descotes, N. H. Fuchs, L. Girlanda, and J. Stern, Eur. Phys. J. C 24, 469 (2002).
[35] These errors are particularly large, and uncertain, above 1.3 GeV, where inelasticity begins to be important. For example, the error on the D0 wave contribution to J₁ₑ due to a 50% change in the inelasticity of the f₂ (1270) resonance is as large as the nominal error due to only the errors in the parameters in Eq. (20). [36] I. Caprini, G. Colangelo, J. Gasser, and H. Leutwyler, following paper, Phys. Rev. D 68, 074006 (2003).