Comparison of Chiral Perturbation Theory and QCD Sum Rule Results for Pseudoscalar Isoscalar-Isovector Mixing ADP-95-2/T169 hep-ph/9504252

The forms of the neutral, non-strange pseudoscalar propagator matrix and mixed axial current correlator, 〈0|T (AμAν)|0〉, are discussed at next-toleading (one-loop) order in chiral perturbation theory, and the results compared to those obtained using QCD sum rules. This comparison provides a check of the truncations employed in the sum rule treatment of the current correlator. Values for the slope of the correlator with q2 in the two approaches are found to differ by more than an order of magnitude and the source of this discrepancy is shown to be the incorrect chiral behavior of the sum rule result. 12.39.Fe, 24.85.+p, 21.30.+y, 21.45.+v Typeset using REVTEX 1 In the traditional treatment of charge symmetry breaking (CSB) in the meson-exchange framework, contributions to CSB observables arising from isoscalar-isovector meson mixing are obtained under the assumption that the strength of meson mixing is q-independent. A number of recent papers [1–10] , however, have demonstrated that this assumption is suspect. Although the quantity in question (the off-diagonal element of the matrix propagator) is the element of an off-shell Green function and, as such, not in general invariant under allowable field redefinitions (in the sense of Haag’s theorem), nonetheless, the existence of qdependence for any particular choice of interpolating field throws the standard treatment of CSB into question. Of special interest, among the papers mentioned above, is the treatment of ρ−ω and π−η mixing using QCD sum rules [7,8] since, without having made any apparent explicit choice for the meson interpolating fields, the authors claim to extract the leading q-dependence of the off-diagonal propagator element. If true, this result would be extremely interesting, suggesting that at least the leading q-dependence, was field reparametrization independent, and hence to be incorporated in all treatments of CSB. In this paper we critically investigate this claim in the pseudoscalar sector (cf. Ref. [8] ) and come to two main conclusions. The first is that it is not possible to extract the qdependence of the off-diagonal propagator matrix element from that of the off-diagonal element of the axial vector current correlator matrix. That this is true in general is a consequence of the well-known behavior of quantum field theories under allowed field redefinitions [11] : S-matrix elements are unchanged, but off-shell Green functions are not. The off-diagonal element of the propagator is not a physical observable; the axial vector current correlator is. They cannot, therefore, in general, be related. The second, and more useful, conclusion is that chiral perturbation theory (ChPT) provides formal constraints useful for investigating the reliability of the trunctations employed in the sum rule treatment of the axial vector current correlator. The reason this is true is that ChPT, being constructed solely via symmetry arguments, provides the most general possible representation of the physics of the pseudoscalar Goldstone bosons realizing the exact symmetries of QCD and breaking the approximate chiral symmetries in exactly the way they are broken in QCD. The 2 off-diagonal element of the axial current correlator, therefore, has a low-energy expansion in terms of current quark masses, momenta and the low-energy constants of the effective chiral Lagrangian, the terms of which can be calculated in a reliable and systematic manner. The only approximations enter when one truncates this expression to a given order in the chiral expansion. However, even if the expansion converges slowly for a given observable (typically, expansion to one-loop order for SU(3)L × SU(3)R is sufficient, but a small number of observables are known for which this is not the case – for a general discussion see the recent review by Ecker [12] ) the formal dependence on the current quark masses and momenta obtained to a given order is a rigorous consequence of QCD. For the case at hand, namely the off-diagonal element of the axial current correlator, we will see below that, while the leading chiral behavior of the q-independent part is correctly reproduced by the truncations employed in Ref. [8] , the leading chiral behavior of the q-dependent piece is not. The sum rule result for the slope with respect to q of this correlator, which differs numerically from the one-loop ChPT expression by more than an order of magnitude, thus cannot be correct. The remainder of the paper is organized as follows. We first revisit the attempt to relate the off-diagonal propagator and axial current correlator matrix elements, fixing our notation in the process. A general expression for the meson pole contributions to the axial current correlator is then given in terms of the meson decay constants and two isospin breaking parameters describing the couplings of the axial currents Aμ and A 8 μ to the physical η and π, respectively. The calculation of these parameters is then reviewed, the development providing, in addition, an explicit realization of the source of error in the treatment of the relation of the propagator and correlator matrix elements in Ref. [8] . Finally, we compare the one-loop ChPT result for the off-diagonal element of the axial correlator with that obtained via the sum rule analysis. Using the known dependences of the physical meson masses on the current quark masses we show that the truncations employed in the sum rule treatment of Ref. [8] remove the leading chiral behavior of the slope with respect to q, and hence are unsuitable for use in treating this feature of the correlator. We begin with some notation. Let π, η represent the physical mixed-isospin π and η 3 fields, and π3, π8 the pure I = 1, 0 flavor octet neutral fields. Then one has, in general, two mixing angles, θπ and θη, such that, to O(θπ,η), π = π3 + θππ8 π3 = π − θπη η = −θηπ3 + π8 π8 = θηπ + η . (1) (There is, in general, q-dependent mixing so that θπ 6= θη.) To O(θπ,η), i.e., to first order in isospin breaking, one defines the isospin-breaking parameter, θ(q), by Π38(q ) = i ∫ dx e〈0|T (π3(x)π8(0))|0〉 ≡ θ(q) (q2 −mπ)(q −mη) (2) where, from Eqns. (1), θ(q) = q(θη − θπ) + (mηθπ −mπθη) . The axial current correlator, Π μν , is similarly defined via Π μν = i ∫ dx e〈0|T (Aμ(x)Aν(0))|0〉 ≡ Π 1 (q)qμqν +Π 2 (q)gμν (3) where Aμ, A 8 ν are the 3, 8 members of the axial current octet A a μ = q̄γμγ5 λ 2 q. In Ref. [8] the authors evaluate this correlator using QCD sum rules, and attempt to determine θ(q) by considering the pseudoscalar pole contributions to the form factor Π 1 . Invoking the PCAC relations 〈0|Aμ|π3(q)〉 = ifπqμ and 〈0|Aμ|π8(q)〉 = ifηqμ and, evaluating Π 1 in the pole approximation, they write