CPT-98/P.3716 LU TP 98/23 ZU-TH 21/98 February 1, 2008 arXiv:hep-ph/9904452v1 26 Apr 1999 Low–energy photon–photon fusion into three pions in Generalized Chiral Perturbation Theory ∗ Ll. Ametllera, J. Kamborb, M. Knechtc and P. Talaverad a b c Dept. de Fı́sica i Enginyeria Nuclear, UPC, E-08034 Barcelona, Spain. Institut für Theoretische Physik, Universität Zürich, CH-8057 Zürich, Switzerland. Centre de physique Théorique, CNRS–Luminy, Case 907F–13288 Marseille Cedex 9, France. d Dept. of Theoretical Physics, University of Lund, Sölvegatan 14A, S-22362 Lund, Sweden. PACS: 11.30.Rd, 12.39.Fe, 13.75.-n, 14.70.Bh, 13.60.Le Keywords: Chiral Symmetry, Chiral Perturbation Theory, Scattering at low energy, Photon-photon Physics, Meson production. Abstract The processes γγ → π 0 π 0 π 0 and γγ → π + π − π 0 are considered in Generalized Chiral Perturbation theory, in view of their potential sensitivity to the mechanism of spontaneous breaking of chiral symmetry and to various counterterms. The amplitudes are computed up to order O(p6 ). The event production rates are estimated for the Daphne φ–Factory and for a future τ –Charm Factory. ∗ Work supported in part by TMR, EC–Contract No. ERBFMRX–CT980169 1 Introduction In the limit where the masses of the lightest quark flavours u, d and s are set to zero, the QCD lagrangian becomes invariant under a chiral SU(3)L × SU(3)R global symmetry. This symmetry is not reproduced by the hadronic spectrum, and must therefore be spontaneously broken towards the diagonal SU(3)V subgroup. Actually, this spontaneous breaking of chiral symmetry can be shown to follow from very general properties of QCD [1, 2, 3]. However, besides the existence of eight massless pseudoscalar states coupling to the eight conserved axial currents, nothing is known from “first principles” about the actual mechanism of spontaneous breaking of chiral symmetry. The widespread belief in that matter is that it proceeds through the formation of a strong quark–antiquark condensate, < q̄q >∼ −(250 MeV )3 , where < q̄q > denotes the single flavour quark–antiquark condensate in the SU(2) chiral limit, ¯ > |mu =m =0 . < q̄q > = < ūu > |mu =md =0 = < dd d (1.1) In particular, this means that once the light quark masses mu , md and ms are turned on, the mass of the pion is assumed to be dominated by the contribution linear in quark masses [4], − c < q̄q > mu + md 2m c = ∼1, m , 2 2 F Mπ 2 (1.2) where F stands for the pion decay constant Fπ (the normalization we use corresponds to the numerical value Fπ = 92.4 MeV [5]) in the same two–flavour chiral limit, F = Fπ |mu =md =0 . (1.3) However, our present theoretical knowledge of non–perturbative aspects in QCD does not exclude a picture where the condensate would be much smaller, say < q̄q >∼ −(100 MeV )3 , or even vanishing. How the latter possibility may arise in QCD has been discussed recently [6, 7] in terms of spectral properties of the Dirac operator. On the other hand, it has also been suggested [8] that a strictly vanishing condensate in the chiral limit could be excluded by an inequality [9] between the correlator < Aµ (x)Aµ (0) > of two axial currents and the two–point function < P (x)P (0) > of the pseudoscalar quark bilinear density P (x) ≡ (q̄iγ5 q)(x). This inequality, however, has only been established so far for bare quantities, i.e. in the presence of an ultraviolet cut–off ΛUV . As the cut–off is removed, the pseudoscalar density P (x) needs to be renormalized, whereas the (partially) conserved axial current Aµ (x) remains unaffected. Stricktly speaking, the claim of Ref. [8] is that < q̄q > (ΛUV ) 6= 0 for finite ΛUV , which does not, a priori, exclude the possibility of a vanishing condensate in the limit ΛUV → ∞ [6]. Finally, Ref! . [10] provides an example of a different approach which, within a variational framework, leads to a non–vanishing, but nevertheless small, value of < q̄q >. 1 Clearly, in order to settle the question of the mechanism of spontaneous chiral symmetry breaking on a purely theoretical level, a major breakthrough in our understanding of non–perturbative aspects of confining gauge theories is required. Instead, one may try a phenomenological approach, and look for experimental observables which could provide the relevant information. In this respect, low–energy π–π scattering in the S–wave has been put forward as a process particularly sensitive to the size of the condensate [11, 12] (for recent discussions, see e.g. [13, 7]). This can be most conveniently seen in the framework of the effective lagrangian [14, 15, 16]. In order to incorporate the possibility of a small condensate, the usual counting has however to be modified. As explained in Refs. [11, 17], a consistent expansion scheme, usually refered to as Generalized Chiral Perturbation Theory (GChPT), is obtained by taking (from now on, we restrict ourselves to the case of two massless flavours only), mu , md ∼ O(p) , B ∼ O(p) , (1.4) with p being a generic momentum, much smaller than the typical hadronic scale ΛH ∼ 1 GeV , and < q̄q > . (1.5) B≡− F2 With this counting, the effective lagrangian at lowest order, which consists of all chiral invariant terms of order O(p2 ), reads [11] L̃(2) = 1 2n F hDµ U + D µ Ui + 2BhU + χ + χ+ Ui 4 +Ah(U + χ)2 + (χ+ U)2 i + Z P hU + χ − χ+ Ui2 + h0 hχ+ χi + h1 (detχ + detχ+ ) o (1.6) . The matrix U collects the pion fields (throughout, we adopt the Condon and Shortley phase convention),  √ +  0 π − 2π  U = eiφ/F , φ =  √ − . (1.7) 2π −π 0 The notation is as in Refs. [15, 16], except that χ, the quantity that contains the scalar and pseudoscalar sources, is defined without the usual factor 2B, χ = s + ip = M + · · · , M = diag(mu , md ) . (1.8) The covariant derivative contains the external vector and axial sources, Dµ U = ∂µ U − i[vµ , U] − i{aµ , U} . 2 (1.9) This is to be contrasted with Standard Chiral Perturbation Theory (SChPT) [15, 16], which assumes a large condensate, and hence the counting rule mu , md ∼ O(p2 ), B ∼ ΛH , so that only the first two terms on the right–hand side of Eq. (1.6) are taken into account at leading order. It easily follows from (1.6) that the lowest order expression of the pion mass is now given by 1 c B + 4m c2 A + · · · , Mπ2 = 2m (1.10) where the ellipsis stands for higher order corrections. At the same level of approximation in the chiral expansion, the π–π scattering amplitude can be written as [11] (s − 43 Mπ2 ) Mπ2 + α + ··· , A(s|t, u) = β F2 3F 2 (1.11) c), while at this order the parameter α is directly related to the < q̄q > with β = 1 + O(m condensate through   cB 2m c) . + O(m (1.12) α= 4−3 Mπ2 The case (1.2) of a strong condensate corresponds to α = 1, whereas the extreme limit where the condensate would vanish yields α = 4. Notice that the above expression for A(s|t, u) is not affected by the additional terms in (1.6) and reproduces the result first obtained by cB)/F 2 + · · ·. The difference between the standard case Weinberg [18], A(s|t, u) = (s − 2m and deviations from it lies here only in the leading–order expression (1.10) of the pion mass and its relation to the condensate. Higher orders in the chiral expansion will modify the simple expression (1.11), but the correlation between low–energy π–π scattering and the value cB/Mπ2 subsists, and can be studied in a controled way within the generalized of the ratio 2m chiral expansion [12]. Available data on low–energy π–π phases, which are dominated by the Geneva–Saclay Kℓ4 experiment [19], do however not possess the required accuracy in order to distinguish between the different alternatives at present. Forthcoming experiments, such as new Kℓ4 experiments, conducted by the BNL865 collaboration [?] or planed at the Daphne φ–factory [21], and the DIRAC experiment at CERN [22], represent promissing prospects in this direction. In order to create a (hopefully convergent) set of evidences pro or contra a specific picture of spontaneous breaking of chiral symmetry, it remains however important to explore other possibilities, and to find other processes which, at the theoretical level, can be shown to exhibit a reasonably strong dependence on the value of the condensate. The present work was motivated by the above consideration and the following observation. A straightforward re–analysis in GChPT of the existing SChPT calculations [23, 24, 25] at lowest order shows that the two 1 We neglect the mass difference mu − md and set mu = md = m. b 3 amplitudes for the production of three pions in low–energy photon–photon collisions involve the parameter α already at tree level. In the case of γγ → π 0 π 0 π 0 , the amplitude reads AN = while for γγ → π + π − π 0 we find A C e2 µναβ αMπ2 ′ ′ ǫ k ǫ k ǫ + ··· , µ ν α β 4π 2 Fπ3 s − Mπ2  (p+ + p− )2 − Mπ2 + 31 (α − 1)Mπ2 e2 µ ν ′α ′β ǫµναβ k ǫ k ǫ 1 − = 4π 2 Fπ3 s − Mπ2  e2 ǫ · p− β ǫ · p+ β  + 2 3 ǫµναβ k ′µ ǫ′ν pα0 −ǫβ + p+ + p 4π Fπ k · p− k · p+ −  ǫ′ · p− β ǫ′ · p+ β  e2 p+ + ′ p− + · · · . + 2 3 ǫµναβ k µ ǫν pα0 −ǫ′β + ′ 4π Fπ k · p− k · p+ (1.13)  (1.14) In these expressions, k and k ′ ( ǫ and ǫ′ ) denote the momenta (polarizations) of the two photons, while p+ , p− and p0 are the pion momenta. The ellipses stand for higher order corrections which will be considered later. Diagrammatically, the origin of the dependence on α lies in the presence of one–pion reducible contributions to the two amplitudes, see Fig. 1 below. Figure 1 : The lowest order contributions to the γγ → πππ amplitudes. The α–dependence in AN and AC comes from the vertex for (virtual) π–π scattering in the first graph, which is the only one to contribute in the case of the neutral amplitude. Therefore, the leading–order neutral amplitude AN increases as the condensate decreases. For a strictly vanishing condensate (α = 4), the cross section at low energies is thus enhanced by a factor 16 as compared to the standard case of a strong condensate (α = 1) ! In the charged case, the situation is less favourable. At threshold, the average over the photon polarizations of the modulus squared of the amplitude is still proportional to the square of α,  e2 1 1X C 2 |A |thr = 4 pol 4 4π 2 Fπ3 2 9Mπ4 2 α , 128 (1.15) but the sensitivity on α of the corresponding total cross section σ C (s, α) rapidly decreases with increasing energy. For instance, the ratio σ C (s, α = 3)/σ C (s, α = 1) is equal to 4.48 at 4 √ √ s = 450 MeV, i.e. just above threshold, but drops to 1.68 at s = 500 MeV and becomes √ less than 1.10 at s > the corresponding ratio is equal to 2.21 ∼ 600 MeV. For the case α = 2, √ √ at s = 450 MeV, but the effect is less than 25% at s = 500 MeV, while it barely reaches √ a few percent at s > ∼ 600. Thus, the interference between the two kinematical structures contributing to the amplitude AC , which is responsible, at low energies, for the suppression of the cross section in the charged channel as compared to the cross section in the neutral channel [25], also washes out the dependence on the value of the < q̄q > condensate as soon as one leaves the threshold region. Considering definite polarization configurations for the two photons does not improve the situation : For parallel polarizations of the two photons, the part of AC which is sensitive to α does not contribute to the cross section, whereas for polarizations taken along orthogonal axes, the same interference effect is again fully at work. Furthermore, both cross sections rapidly rise above threshold, so that the effect of higher orders also needs to be investigated. The purpose of this paper is precisely to investigate these higher order effects at the one–loop level in GChPT. In fact, at next–to–leading order new tensorial structures appear both in the neut! ral and in the charged amplitude [25], whereas the pion loops provide additional sources of dependence with respect to α, so that their behaviour with respect to changes in the value of the condensate could be modified to some extent. Accordingly, the rest of the paper is organized as follows : The general kinematical structure of the two amplitudes AN and AC , as well as their properties under isospin symmetry, are the subject of Section 2. The construction and the renormalization of the effective lagrangian of GChPT at order one–loop in the two–flavour case are treated in Section 3. Section 4 is devoted to the actual calculation of the two amplitudes to one–loop precision. The counterterms which are involved in these expressions and several numerical results are discussed in Section 5. A summary and conclusions are presented in Section 6. Details on the evaluation of some counterterms have been gathered in an Appendix. 2 Kinematics and Isospin Symmetry The amplitudes AN and AC for the processes γ(k) γ(k ′ ) γ(k) γ(k ′ ) → π 0 (p1 ) π 0 (p2 ) π 0 (p3 ) , → π + (p+ ) π − (p− ) π 0 (p0 ) , (2.1) are obtained from the matrix elements < π 0 (p1 )π 0 (p2 )π 0 (p3 ) out | γ(k, ǫ)γ(k ′, ǫ′ ) in > = i(2π)4 δ 4 (Pf − Pi )AN , < π + (p+ )π − (p− )π 0 (p0 ) out | γ(k, ǫ)γ(k ′, ǫ′ ) in > = i(2π)4 δ 4 (Pf − Pi )AC , 5 (2.2) respectively, with AN (k, ǫ; k ′ , ǫ′ ; p1 , p2 , p3 ) = ie2 ǫµ (k)ǫ′ν (k ′ ) Z d4 xe−ik·x < π 0 (p1 )π 0 (p2 )π 0 (p3 ) out | T {j µ(x)j ν (0)} | Ω > , Z d4 xe−ik·x < π + (p+ )π − (p− )π 0 (p0 ) out | T {j µ (x)j ν (0)} | Ω > .(2.3) AC (k, ǫ; k ′ , ǫ′ ; p0 , p+ , p− ) = 2 ie ǫµ (k)ǫ′ν (k ′ ) In the above expressions, jµ (x) denotes the electromagnetic current, with its usual decomposition into an isotriplet and an isosinglet component, jµ = jµ3 + jµ0 , while | Ω > stands for the QCD vacuum with massive light quarks, but in the absence of electromagnetism (radiative corrections to the two processes (2.1) are not considered here). Since we also neglect isospin breaking effects due to mu 6= md , Bose symmetry and G–parity constrain the three final pions to be in an I = 1 total isospin state. Thus the matrix elements in Eq. (2.3) only involve the I = 1 component of the product of the two electromagnetic currents, T {jµ (x)jν (0)}I=1 = T {jµ3 (x)jν0 (0) + jµ0 (x)jν3 (0)} . (2.4) Therefore, isospin invariance relates the two amplitudes AC and AN in a simple way. With the Condon and Shortley phase convention adopted in Eq. (1.7), this relation reads − AN (k, ǫ; k ′ , ǫ′ ; p1 , p2 , p3 ) = AC (k, ǫ; k ′ , ǫ′ ; p1 , p2 , p3 ) + cyclic (p1 , p2 , p3 ) , (2.5) where “cyclic (p1 , p2 , p3 )” indicates that the contributions arising from cyclic permutations over the pion momenta p1 , p2 and p3 have to be added. Up to permutations of the momenta and polarizations of the photons, and/or permutations of the momenta of the charged pions, the amplitude AC may be decomposed into six independent Lorentz invariant amplitudes ′ ′ ′ AC (k, ǫ; k ′ , ǫ′ ; p0 , p+ , p− ) = AC 1 (k, k ; p0 , p+ , p− ) t1 (k, ǫ; k , ǫ ) ! ! X  k′ k C ′ ′ ′ ↔ ′ Ai (k, k ; p0 , p+ , p− ) ti (k, ǫ; k , ǫ ; p0 , p+ , p− ) + + ǫ ǫ i=2,3 + X i=4,5,6  ′ ′ ′ AC i (k, k ; p0 , p+ , p− ) ti (k, ǫ; k , ǫ ; p0 , p+ , p− ) + ! k′ k ↔ ′ ǫ ǫ (2.6) !  + p+ ↔ p− with (pij ≡ pi + pj , i, j = +, −, 0) t1 (k, ǫ; k ′ , ǫ′ ) = ǫµναβ k µ ǫν k ′α ǫ′β ,  t2 (k, ǫ; k ′ , ǫ′ ; p0 , p+ , p− ) = ǫµναβ k ′µ ǫ′ν pα0 −ǫβ +  t3 (k, ǫ; k ′ , ǫ′ ; p0 , p+ , p− ) = ǫµναβ −ǫµ + 6 ǫ · p+ β  ǫ · p− β p+ + p , k · p− k · p+ − ǫ · p+− µ  ′ν α ′β k k p0 ǫ , k · p+−  , ǫ · p+0 µ  ′ν α ′β k k p− ǫ , k · p+0 ǫ·p ǫ · p+  +0 t5 (k, ǫ; k ′ , ǫ′ ; p0 , p+ , p− ) = ǫµναβ k ′µ pν+ pα− ǫ′β , − k · p+0 k · p+  ǫ · p+ µ  ′ν α ′β t6 (k, ǫ; k ′ , ǫ′ ; p0 , p+ , p− ) = ǫµναβ −ǫµ + k k p− ǫ . k · p+  t4 (k, ǫ; k ′ , ǫ′ ; p0 , p+ , p− ) = ǫµναβ −ǫµ + (2.7) In the sequel, we shall compute the amplitudes AC and AN within the framework of generalized chiral perturbation theory up to order one–loop. At this level of accuracy of the chiral expansion, the amplitude AN does not yet receive its full structure as implied by Eqs. (2.6) and (2.5). Rather, it takes the simpler form (pij ≡ pi + pj , i, j = 1, 2, 3) ′ ′ ′ AN (k, ǫ; k ′ ǫ′ ; p1 , p2 , p3 ) = AN 1 (k, k ; p1 , p2 , p3 ) t1 (k, ǫ; k , ǫ )    ǫ′ · p12 ′µ  ν α β ′ ′µ p ǫ k + cyclic (p , p , p ) k + AN (k, k ; p , p , p )ǫ ǫ − 1 2 3 1 2 3 µναβ 12 2 k ′ · p12 ! !   k′ k ↔ ′ + ··· , + ǫ ǫ (2.8) where the ellipsis stands for higher order terms in the chiral expansion. In order that (2.8) follows from (2.5) and (2.6), it is sufficient that the one–loop charged amplitudes AC i satisfy the following conditions : ′ C ′ i) AC 2 (k, k ; p0 , p+ , p− ) − A5 (k, k ; p+ , p− , p0 ) is entirely symmetric under permutations of the pion momenta p+ , p− , p0 ; ′ C ′ ii) k · p0 [AC 5 (k, k ; p0 , p+ , p− ) + A6 (k, k ; p0 , p+ , p− )] = ′ C ′ k · p+ [AC 5 (k, k ; p+ , p0 , p− ) + A6 (k, k ; p+ , p0 , p− )]; ′ C ′ iii) AC 5 (k, k ; p0 , p+ , p− ) = A5 (k, k ; p+ , p0 , p− ). Furthermore, the two neutral amplitudes have then the following expressions in terms of the charged ones : ′ C ′ AN 1 (k, k ; p1 , p2 , p3 ) = −A1 (k, k ; p1 , p2 , p3 )  1 C ′ A (k, k ′ ; p1 , p2 , p3 ) + AC + 3 (k, k ; p3 , p1 , p2 ) 3 2 ′ C ′ +AC 4 (k, k ; p1 , p2 , p3 ) + A4 (k, k ; p2 , p1 , p3 ) ′ ′ C +AC 6 (k, k ; p1 , p2 , p3 ) + A6 (k, k ; p2 , p1 , p3 ) !   ! 2 C k′ k ′ + A5 (k, k ; p2 , p3 , p1 ) + ↔ ′ ǫ 3 ǫ + cyclic (p1 , p2 , p3 ) , (2.9) and ′ C ′ C ′ − AN 2 (k, k ; p1 , p2 , p3 ) = [A4 (k, k ; p1 , p2 , p3 ) + A6 (k, k ; p1 , p2 , p3 )] + [ p1 ↔ p2 ] ′ C ′ +AC 3 (k, k ; p1 , p2 , p3 ) + A5 (k, k ; p2 , p1 , p3 ) . 7 (2.10) 3 The Effective Lagrangian in GChPT Before proceeding with the calculation of the amplitudes AC and AN in the next section, we first discuss the structure of the low–energy generating functional in the case of two light flavours that we need for our subsequent calculation. Since most of the results of this section are not available from the existing literature, we discuss them in some detail. The structure of the effective lagrangian Leff is independent of the underlying mechanism of spontaneous chiral symmetry breaking. It consists of an infinite tower of chiral invariant contributions Leff = X (k,l) L(k,l) , (3.1) where L(k,l) contains k powers of covariant derivatives and l powers of the scalar or pseudoscalar sources. In the chiral limit, these terms behave as L(k,l)  p ∼ ΛH k  mquark ΛH l , (3.2) with mquark = mu , md , and p stands for a typical external momentum. The standard approach not only assumes mquark ≪ ΛH , but also mquark ≪ B/2A, such as to enforce the dominance of the term linear in mquark in the expression of the pion mass (1.10). This allows to reorganize the double expansion (3.1) as [15, 16] Leff = L(2) + L(4) + L(6) + · · · , (3.3) P where L(d) = L(k,l) with k + 2l = d. The generalized framework considers the possibility that the condensate could be much smaller than usually believed, so that for the actual values of the quark masses one could have mquark ∼ B/2A and still mquark ≪ ΛH . This leads to a different reorganization of the double expansion (3.1), namely Leff = L̃(2) + L̃(3) + L̃(4) + L̃(5) + L̃(6) + · · · , P (3.4) where now, according to (1.4), L̃(d) = B n L(k,l) with k + l + n = d [11, 17]. The leading order of the generalized expansion is described by L̃(2) , which in the two flavour case was given in Section 1, Eq. (1.6). For our purposes, we need only to consider the situation without axial source, and with the vector source restricted to the (classical) photon field Aµ , aµ = 0 , vµ = eAµ Q , (3.5) 8 where Q stands for the charge matrix of the two light quark flavours u and d,  2 3  0  Q =  . 0 − 13 (3.6) In GChPT, the next–to–leading–order corrections are of order O(p3 ), and still occur before the loop corrections. They are embodied in L̃(3) = L(2,1) + L(0,3) , which reads 2 L̃(3) = 1 2 n (2) F ξ hDµ U + D µ U(χ+ U + U + χ)i 4 (2) (2) +ρ1 h(χ+ U)3 + (U + χ)3 i + ρ2 h(χ+ U + U + χ)χ+ χi (2) +ρ3 hχ+ U − U + χih(χ+ U)2 − (U + χ)2 i (3.7) (2) +ρ4 h(χ+ U)2 + (U + χ)2 ihχ+ U + U + χi (2) +ρ5 hχ+ χihχ+ U + U + χi o . ¿From order O(p4 ) onward, the contributions to Leff come with either even or odd intrinsic (d) (d) − parity, L̃(d) = L̃+ +L̃− for d ≥ 4, or L(k,l) = L+ (k,l) +L(k,l) for k ≥ 4. The tree–level contributions at order O(p4 ) in the even intrinsic parity sector are contained in (4) 2 ′ ′ ′ L̃+ = L+ (4,0) + L(2,2) + L(0,4) + B L(0,2) + BL(2,1) + BL(0,3) . (3.8) The part without explicit chiral symmetry breaking, L+ (4,0) , is described by the same low–energy constants l1 , l2 , l5 , l6 and h2 as in the standard case [15], L+ (4,0) = l1 l2 h D µ U + Dµ U i 2 + h D µ U + D ν U i h Dµ U + Dν U i 4 4 il6 L R µ L µ + ν + l5 h Fµν UF L µν U + i + h Fµν d Udν U + + Fµν d U d Ui 2 1 R R µν L F + Fµν F L µν i . − (2h2 + l5 ) h Fµν 2 (3.9) The part with two powers of momenta and two powers of quark masses is given by L(2,2)  1 2 F a1 hDµ U + D µ U(χ+ χ + U + χχ+ U)i = 4 +a2 hDµ U + Uχ+ D µ UU + χi +a3 hDµ U + U(χ+ D µ χ − D µ χ+ χ) + Dµ UU + (χD µ χ+ − D µ χχ+ )i +b1 hDµ U + D µ U(χ+ Uχ+ U + U + χU + χ)i +b2 hDµ U + χD µ U + χ + χ+ Dµ Uχ+ D µ Ui 2 (2) The superscript (2) is meant to distinguish the low energy constants ξ (2) , ρi occur in the expression of L̃(3) in the three flavour case [11, 17]. 9 from the similar ones that +b3 hU + Dµ χU + D µ χ + Dµ χ+ UD µ χ+ Ui +c1 hDµ U + χ + χ+ Dµ UihD µ U + χ + χ+ D µ Ui +c2 hDµ χ+ U + U + Dµ χihD µ U + χ + χ+ D µ Ui +c3 hDµ χ+ U + U + Dµ χihD µ χ+ U + U + D µ χi (3.10) +c4 hDµ U + χ − χ+ Dµ UihD µ U + χ − χ+ D µ Ui +c5 hDµ χ+ U − U + Dµ χihD µ χ+ U − U + D µ χi + µ +  +h3 hDµ χ D χ i . Finally, the tree–level contributions which behave as O(m4quark ) in the chiral limit are contained in L(0,4) , which reads L(0,4)  1 2 F e1 h(χ+ U)4 + (U + χ)4 i = 4 +e2 hχ+ χ(χ+ Uχ+ U + U + χU + χ)i +e3 hχ+ χU + χχ+ Ui +f1 h(χ+ U)2 + (U + χ)2 i2 +f2 h(χ+ U)3 + (U + χ)3 ihχ+ U + U + χi +f3 hχ+ χ(χ+ U + U + χ)ihχ+ U + U + χi +f4 h(χ+ U)2 + (U + χ)2 ihχ+ U + U + χi2 +f5 h(χ+ U)3 − (U + χ)3 ihχ+ U − U + χi +h4 hχ+ χχ+ χi +h5 hχ+ χi(detχ + detχ+ ) +h6 (detχ + detχ+ )2 + 2 +h7 (detχ − detχ )  . Notice that in the standard framework the contributions from L(2,2) and from L(0,4) would count as order O(p6 ) and order O(p8 ), respectively 3 . Next, we turn to the odd intrinsic parity sector. There, the first contribution starts at order O(p4 ), and since it is entirely fixed by the short distance properties of QCD, there is no (4) (4) difference between the standard and the generalized case, L− = L̃− = L− (4,0) . In the two flavour − case, L(4,0) vanishes in the absence of external sources. In the presence of an electromagnetic field, it reads e µναβ ǫ Aµ hQ(∂ν U∂α U + ∂β UU + − ∂ν U + ∂α U∂β U + U)i (3.11) L− (4,0) = 16π 2 ie2 1 1 − 2 ǫµναβ ∂µ Aν Aα hQ2 ∂β UU + + Q2 U + ∂β U − QUQ∂β U + + QU + Q∂β Ui . 8π 2 2 3 The standard O(p6 ) effective lagrangian L(6) = L(6,0) + L(4,1) + L(2,2) + L(0,3) has been worked out in Ref. [27] for the case of three light flavours, and very recently, for both two and three light flavours, in Ref. [28]. 10 The computation of the amplitudes AN and AC to one loop also involves the counterterms (5) (6) − − − − from L̃− = L− (4,1) and from L̃− = L(6,0) + L(4,2) . In the standard case, both L(4,1) and L(6,0) count as order O(p6 ), and have been discussed before in the literature 4 [30, 31, 27]. Borrowing from the last and most recent of these references, we obtain L− (4,1)  1 ǫµναβ iA4 h[χ]− [Gµν ]+ [Gαβ ]+ i + iA6 h[χ]− ih[Gµν ]+ [Gαβ ]+ i = 2 4π +A12 h[D µ U]− [D ν U]− ([χ]− [Gαβ ]+ + [Gαβ ]+ [χ]− )i µ ν αβ +A13 h[D U]− [χ]− [D U]− [G ]+ i · · ·  , (3.12) and L− (6,0)  1 = ǫµναβ iA2 h[D µ U]− ([D ν Gγα ]+ [Gγ β ]+ − [Gγα ]+ [D ν Gγ β ]+ )i 4π 2 +iA3 h[D µ U]− ([Dγ Gγν ]+ [Gαβ ]+ − [Gγν ]+ [Dγ Gαβ ]+ −[Dγ Gαβ ]+ [Gγν ]+ + [Gαβ ]+ [Dγ Gγν ]+ )i +A7 h[D α D γ U]− ([Dγ U]− [D β U]− [Gµν ]+ − [Gµν ]+ [D β U]− [Dγ U]− )i +A8 h[D α D γ U]− ([D β U]− [Dγ U]− [Gµν ]+ − [Gµν ]+ [Dγ U]− [D β U]− )i + ···  . (3.13) Here, we have only listed those terms that will actually contribute to the processes under study, when the mass–shell conditions for the momenta and polarizations of the photons are taken into account. We have however kept the numbering of the low-energy constants introduced in [27], but we have, for convenience, changed their normalization by an overall factor 1/4π 2 . The notation is otherwise as in [27], except for the fact that the source χ does not contain the factor 2B, see Eq. (1.8). The last piece we need for a full one–loop computation of the amplitudes (2.3) is L− (4,2) . 8 It counts as order O(p ) in the standard case, and is not available from the existing literature. c2 ) corrections to L− These contributions, which are order O(m (4,0) , are expected to be small in the two–flavour chiral expansion, and will be parametrized appropriately in the one–loop expressions of the amplitudes AC and AN given in the next section. The determination of the combinations of low–energy constants that enter these amplitudes will be discussed in Section 5 below. When studying a given process one also needs to take into account contributions from pion loops, which produce divergences that are eliminated by a renormalization of the low– energy constants of the effective lagrangian. We have computed this divergent part of the 4 For a review, see [29]. 11 one–loop generating functional in the even intrinsic parity sector using standard heat–kernel techniques, and we have then performed the corresponding renormalization of the low–energy constants in the same dimensional renormalization scheme as described in [15, 16]. Thus the low–energy constants display a logarithmic scale dependence (X(µ) denotes generically any of these renormalized low-energy constants) X(µ) = X(µ′ ) + ΓX · ln(µ′ /µ) . (4π)2 (3.14) The full list of the resulting β–function coefficients ΓX is given in Table 1 below.5 F 2 · ΓX 3B 2 − 23 B 2 0 6B 2 X A ZP h0 h1 X ξ (2) (2) ρ1 (2) ρ2 (2) ρ3 (2) ρ4 (2) ρ5 F 2 · ΓX 4B −4B(A + Z P ) −4B(A − 3Z P ) 2B(A + 3Z P ) 2B(3A + Z P ) 4B(A − 2Z P ) X l1 l2 l5 l6 h2 ΓX 1 3 2 3 − 61 − 31 1 12 Table 1a: Scale dependence of the low energy constants of L̃(2) , L̃(3) and of L+ (4,0) . X a1 a2 a3 b1 b2 b3 c1 c2 c3 c4 c5 h3 F 2 · ΓX −2Z P −12Z P 0 6(A + Z P ) −2(A + Z P ) 0 2(A + 2Z P ) 0 0 2A 0 0 X e1 e2 e3 f1 f2 f3 f4 f5 h4 h5 h6 h7 F 2 · ΓX −4A2 − 22(Z P )2 − 20AZ P −4A2 − 16(Z P )2 − 12AZ P −12A2 − 64(Z P )2 − 48AZ P 3A2 + 15(Z P )2 + 12AZ P 2A2 + 8(Z P )2 + 10AZ P 4A2 + 24(Z P )2 + 20AZ P −6(Z P + A)Z P 2A2 + 8(Z P )2 + 10AZ P 4A2 + 8(Z P )2 + 8AZ P −4A2 − 32(Z P )2 − 28AZ P 4A2 + 6(Z P )2 + 8AZ P −14(Z P )2 Table 1b: Scale dependence of the low energy constants of L(2,2) and of L(0,4) . Notice that at order O(p4 ), the low-energy constants of L̃(2) and L̃(3) also need to be renormalized. The corresponding counterterms, however, are of order O(B 2 ) and O(B), respectively, 5 These results have also been established independently by L. Girlanda, private communication to M.K. and [32]. The renormalization of the L+ (4,0) counterterms has, of course, already been obtained before in [15]. 12 and they are gathered in the three last terms of Eq. (3.8) : in GChPT, renormalisation proceeds order by order in the expansion in powers of B. Alternatively, one may think of Eqs. (1.6) and (3.7) as standing for the combinations L̃(2) + B 2 L′(0,2) and L̃(3) + BL′(2,1) + BL′(0,3) , respectively, with the corresponding low-energy constants representing the renormalized, scale dependent, quantities. We shall adopt the latter point of view in the sequel. We have not worked out the general structure of the divergent part of the one–loop generating functional in the odd intrinsic parity sector. For SChPT, this has been done in Refs. [33, 30, 34]. 4 The One–Loop Amplitudes Having constructed the effective lagrangian in the preceding section, the computation of the amplitudes AN and AC at next–to–leading order is a straightforward exercise. We begin with the one–loop expression of the neutral amplitude AN , whose structure at that level of the chiral expansion is given by (2.8), with ′ AN 1 (k, k ; p1 , p2 , p3 ) Aπ 0 →γγ A00;00 (p212 , p213 , p223 ) = s − Mπ2  e2 1 1 ct(β − 3) − αMπ2 t′ − 4m c2 t′′ − γ00 − 2 3 (β − 1) + m 4π Fπ 2 2 1 + 2 (λ1 + 2λ2 )(s − 3Mπ2 )) Fπ 1 1 ¯ 2 2 2 2 + 2 J(p 12 )[p12 − Mπ + (α − 1)Mπ ] Fπ 3 1 ¯ 2 1 2 2 2 + 2 J(p 13 )[p13 − Mπ + (α − 1)Mπ ] Fπ 3 1 ¯ 2 1 2 2 2 + 2 J(p 23 )[p23 − Mπ + (α − 1)Mπ ] Fπ 3 1 2 + 2 [R̄(p212 , k · p12 ) + R̄(p212 , k ′ · p12 )][p212 − Mπ2 + (α − 1)Mπ2 ] Fπ 3 1 2 + 2 [R̄(p213 , k · p13 ) + R̄(p213 , k ′ · p13 )][p213 − Mπ2 + (α − 1)Mπ2 ] Fπ 3  2 1 2 2 ′ 2 2 2 + 2 [R̄(p23 , k · p23 ) + R̄(p23 , k · p23 )][p23 − Mπ + (α − 1)Mπ ] , Fπ 3 (4.1) and ′ AN 2 (k, k ; p1 , p2 , p3 ) = 1 e2 R̄(p212 , k · p12 )[p212 − Mπ2 + (α − 1)Mπ2 ] . 2 5 2π Fπ 3 13 (4.2) This second amplitude, which was absent at tree level, is entirely generated by the pion loops. The numerator of the contribution which develops a pole at s = Mπ2 in the expression (4.1) of π 0 →γγ AN , which is related to the on-shell amplitude of the π 0 → γγ 1 is given by the product of A 0 decay through A(π 0 → γγ) = −t1 (k, ǫ, k ′ , ǫ′ )Aπ →γγ , times the amplitude A00;00 (p212 , p213 , p223 ) of virtual π 0 − π 0 scattering (p212 + p213 + p223 = s + 3Mπ2 ). The expressions, at order O(p6 ) and order O(p4 ), respectively, of these amplitudes read Aπ and A00;00 (p212 , p213 , p223 ) = 0 →γγ = e2 ct + Mπ2 t′ + m c2 t′′ + 2m c2 a3 ] , [1 + m 4π 2 Fπ (4.3) αMπ2 1 + 4 (λ1 + 2λ2 )[(p212 − 2Mπ2 )2 + (p213 − 2Mπ2 )2 + (p223 − 2Mπ2 )2 ] 2 Fπ Fπ   4 2 α 2 2 α2 4 1 ¯ 2 2 + 4 J(p12 ) (p12 − Mπ + Mπ ) + Mπ Fπ 3 3 2   1 ¯ 2 4 2 α 2 2 α2 4 2 + 4 J(p13 ) (p13 − Mπ + Mπ ) + Mπ Fπ 3 3 2   4 2 α 2 2 α2 4 1 ¯ 2 2 (p − ) . (4.4) M + M ) + M + 4 J(p 23 23 Fπ 3 π 3 π 2 π The various parameters α, β, λ1,2 and γ00 involved in the expressions (4.1) and (4.4) are given in terms of combinations of the low–energy constants of the effective lagrangian Leff and of chiral logarithms due to the pion loops. They read Fπ2 2 cB + 16m c2 A M α = 2m F2 π (2) (2) (2) (2) c3 (81ρ1 + ρ2 + 164ρ4 + 2ρ5 ) − 4Mπ2 m cξ (2) +m c4 (16e1 + e2 + 32f1 + 34f2 + 2f3 + 72f4 + 6a3 A) +16m c2 (2b1 − 2b2 − a3 − 4c1 ) −8Mπ2 m M2 1 4 c2 Mπ2 A + 264m c4 A2 ] ln π [2M + 102 m − π 16π 2 Fπ2 µ2 Mπ4 1 c2 Mπ2 A + 264m c4 A2 ] , [ + 44m − 16π 2 Fπ2 2 (4.5) cξ (2) − 4m c2 (ξ (2) )2 + 2m c2 (3a2 + 2a3 + 4b1 + 2b2 + 4c1 ) β = 1 + 2m   Mπ2 Mπ2 − ln 2 + 1 [6 + 5(α − 1)] , 48π 2 Fπ2 µ (4.6) 14 λ1 = λ2 = 4 1 ¯ (l1 − ) , 2 48π 3 (4.7) 5 1 ¯ (l2 − ) , 2 48π 6 c2 (3a2 + 2a3 + 6b2 + 12c1 ) γ00 = m   Mπ2 Mπ2 ln 2 + 1 (α − 1) . − 32π 2 Fπ2 µ (4.8) − − The parameters t, t′ and t′′ contain the contributions from L− (4,1) , L(6,0) and L(4,2) , respectively. In particular, from the formulae (3.12) and (3.13) we derive 32 A4 , 3 8 = − (A2 − 2A3 ) . 3 t = t′ (4.9) ¯ Finally, J(s) denotes the Chew–Mandelstam function [35], the usual scalar two–point loop integral subtracted at s = 0 (for its expression, see Ref. [15]), whereas R̄(p2 , k · p), which is related to the three–point scalar loop function, is given, for k 2 = 0, by R̄(p2 , k · p) = C̄(p2 , k · p) − with (σ = q 1 − 4Mπ2 /p2 , σ ′ = q (k − p)2 ¯ ¯ 2 )] + 1 , [J((k − p)2 ) − J(p 4(k · p) 32π 2 (4.10) 1 − 4Mπ2 /(k − p)2 ) 16π 2 C̄(p2 , k · p) = Mπ2 σ−1 σ′ − 1 [ ln2 ( ) − ln2 ( ′ )] . 4(k · p) σ+1 σ +1 (4.11) For the charged amplitude AC , the general structure is more involved, see Eq. (2.6), and at one loop we obtain ′ AC 1 (k, k ; p0 , p+ , p− ) = Aπ 0 →γγ A00;+− (p2+− , p2+0 , p2−0 ) s − Mπ2  1 1 1 e2 ct(β − 1) − (α − 1)Mπ2 t′ + 2 3 (β + 1) + m 4π Fπ 2 2 3 2 ′ +γ+− + γ+− 3 1 1 λ1 2 )(p2 + p2−0 − 4Mπ2 ) + 2 (p+− − 2Mπ2 ) + 2 (λ2 − Fπ Fπ 288π 2 +0 15 1 ¯ 2 J(p+− )[3p2+− + 4(α − 1)Mπ2 ] 6Fπ2 1 ¯ 2 + J (p+0 )[5p2+0 − 14Mπ2 − 2(α − 1)Mπ2 ] 12Fπ2  1 ¯ 2 2 2 2 J (p )[5p − 14M − 2(α − 1)M ] , + −0 −0 π π 12Fπ2 +  e2 1 ′ 6 + 6γ − [(k ′ − p0 )2 + p2+0 + p2−0 ] +− 24π 2 Fπ3 48π 2 Fπ2 1 ¯ 2 1 ¯ 2 2 2 − 2 J(p J(p−0 )[4Mπ2 − p2−0 ] +0 )[4Mπ − p+0 ] − Fπ Fπ2  1 ¯ ′ 2 2 ′ 2 − 2 J((k − p0 ) )[4Mπ − (k − p0 ) ] , Fπ ′ AC 2 (k, k ; p0 , p+ , p− ) = ′ AC 3 (k, k ; p0 , p+ , p− ) = − ′ AC 4 (k, k ; p0 , p+ , p− ) ′ AC 5 (k, k ; p0 , p+ , p− ) (4.12) 4 e2 R̄(p2+− ; k · p+− )[p2+− + (α − 1)Mπ2 ] , 2 5 4π Fπ 3 (4.13) (4.14)  k · p+ 1¯ ′ e2 − 1) − p− )2 )Mπ2 (4 = − 2 5 J((k 4π Fπ 3 k · p+0  1 ¯ ′ 1 p2+0 + [J((k − p− )2 ) − J¯(p2+0 )] p2+0 + 2Mπ2 + (4Mπ2 − p2+0 ) 6 2 k · p+0  k · p+ k · p+ p2+0 + (4Mπ2 − 7p2+0 + 6k · p+0 ) + 2 (p2+0 − 4Mπ2 ) k · p+0 k · p+0 k · p+0 2 p 1 +R̄(p2+0 , k · p+0 )[ − Mπ2 + 2k · p+ ( +0 − 1) − (α − 1)Mπ2 ] k · p+0 3  2 p 1 1 2 + k · p+ (1 − +0 ) + p , (4.15) 2 24π k · p+0 96π 2 +0   1 e2 J¯((k ′ −p− )2 )[4Mπ2 −(k ′ −p− )2 ]−J¯(p2+0 )(4Mπ2 −p2+0 )− k·p+0 , =− 2 5 24π Fπ 24π 2 (4.16) ′ AC 6 (k, k ; p0 , p+ , p− )   e2 1 ¯ ′ ¯ 2 )] p2 − 4M 2 − 6k · p+ = − 2 5 [J((k − p− )2 ) − J(p +0 +0 π 4π Fπ 6  k · p+ 2 k · p+ 2 2 2 (p+0 − Mπ2 ) + p (4M − p ) +4 π +0 k · p+0 (k · p+0 )2 +0 k · p+ 1¯ ′ − p− )2 )[k · p0 + 2Mπ2 ] − J((k 3 k · p+0 16 p2+0 ] − p+0 ) )(k · p+ )[2 − k · p+0   1 1 k · p+ ′ 4 2 2 k · p+ (k − p ) + k · p − p + − + +0 32π 2 k · p+0 96π 2 3 k · p+0  1 + k · p+0 . 144π 2 +R̄(p2+0 , (k 2 (4.17) 2 The amplitude AC 1 again contains a contribution with a pole at s = Mπ , which is given 0 by the product of the O(p6 ) π 0 → γγ amplitude Aπ →γγ times the (off–shell) π 0 π 0 → π + π − amplitude A00;+− , with − A00;+− (p2+− , p2+0, p2−0 ) β 2 4 1 (p+− − Mπ2 ) + αMπ2 2 Fπ 3 3Fπ2 λ2 λ1 + 4 (p2+− − 2Mπ2 )2 + 4 [(p2+0 − 2Mπ2 )2 + (p20− − 2Mπ2 )2 ] Fπ Fπ   1 ¯ 2 4 2 5 4 2 2 2 2 2 2 2 2 + 4 J(p+− ) 4(p+− − Mπ + αMπ ) − (p+− − Mπ − αMπ ) 6Fπ 3 6 3 3   4 2 1 ¯ 2 2 2 2 2 2 2 2 2 J(p ) 3(p − M − αM ) + (p − p )(p − 4M ) + +0 +0 π +− −0 +0 π 12Fπ4 3 π 3   4 2 2 1 ¯ 2 2 2 2 2 2 2 2 J(p−0 ) 3(p−0 − Mπ − αMπ ) + (p+− − p+0 )(p−0 − 4Mπ ) , + 12Fπ4 3 3 (4.18) = where p2+− + p2+0 + p2−0 = s + 3Mπ2 . ′ C The remaining parameters γ+− and γ+− which appear in the amplitudes AC 1 , ... A6 contain the contributions from the low–energy constants and chiral logarithms : c2 (a2 + 2b2 + 4c1 ) γ+− = −m Mπ2 Mπ2 7 Mπ2 c2 δγ+− , + ln (α − ) + m (α − 1) + 2 2 2 2 2 96π Fπ µ 96π Fπ 3 (4.19) ′ c(2A12 − A13 ) + 8Mπ2 (A7 − A8 ) + 6m c2 a3 γ+− = 8m Mπ2 Mπ2 ′ c2 δγ+− − ln +m . 2 2 2 32π Fπ µ (4.20) With the expressions given above, it is straightforward to check that the three conditions listed before Eq. (2.9) as well as Eq. (2.10) are satisfied, which provides a non–trivial check of our calculation. The isospin relation (2.9) implies that the condition 2 + 3γ+− + γ00 + Mπ2 c2 (2a3 − 3t′′ ) =m 24π 2 Fπ2 17 (4.21) must hold. This requires that the contributions of the L− (4,2) counterterms to γ+− , which we 2 c δγ+− in the expression (4.19), have to satisfy have denoted as m 3t′′ + 3δγ+− = 0 . (4.22) ′ c2 ′ The contributions of the L− (4,2) counterterms to γ+− , m δγ+− in the expression (4.20), are not constrained by isospin symmetry. Upon using the information provided by Table 1, it is straightforward to check that α, β and γ00 are scale independent by themselves (the parameters λ1 and λ2 were directly expressed in terms of the scale independent quantities ¯l1 and ¯l2 defined in Ref. [15]). In order that the 0 amplitudes Aπ →γγ , AN and AC be independent of the subtraction scale µ, the parameters t, ′ ′ t′ , t′′ , γ+− and γ+− must be separately scale independent. This requires that δγ+− and δγ+− themselves are scale independent. Since we have not worked out the structure of the one–loop divergences of the generalized generating functional in the odd intrinsic parity sector, we could not perform these checks explicitly. ¿From the above formulae, one may infer the expressions of the amplitudes in the standard case [25, 26]. Since only the result for the neutral amplitude AN was displayed explicitly in Ref. [25], and the expressions of the amplitudes AC i in the charged channel are only available from the unpublished work [26], we describe in some detail the necessary steps to obtain them. Their general structure is of course unchanged, the differences occur only in the expressions of the various combinations of low–energy constants that are involved. In particular, the contributions from L(0,3) , L(2,2) , L(0,4) , and L− (4,2) are relegated to higher orders. For the remaining constants, the correspondance with the usual SChPT notation is given as follows (2) m̂ξst = αst = βst = γ00,st = γ+−,st = ′ γ+−,st = 2 1 ¯l4 + ln Mπ ) , ( 16π 2 Fπ2 µ2 2 Mπ 1+ (4¯l4 − 3¯l3 − 1) , 32π 2 Fπ2 M2 1 + 2 π 2 (¯l4 − 1) , 8π Fπ 0, Mπ2 , − 72π 2 Fπ2 2A12 − A13 Mπ2 Mπ2 8Mπ2 (A7 − A8 + ln )− . 2B 32π 2Fπ2 µ2 (4.23) We have checked that upon substituting these expressions into the one–loop amplitudes AN and AC given above, we recover the results of the standard case, up to the contributions from the counterterms contained in t and t′ , which were not included in Refs. [25, 26]. 18 5 Counterterm Estimates and Numerical Results In order to make numerical estimates for the cross sections based on the O(p6 ) amplitudes, we first need to fix or estimate the values of the various counterterms involved. i) λ1 , λ2 : At order O(p4 ), these parameters are related to ¯l1 and ¯l2 through Eq. (4.7). The values of these low–energy constants in the standard case have been the subject of numerous studies in the literature [15, 36, 37, 38, 39]. A first determination at order O(p4 ) was given in Ref. [15], using information from the D–wave π–π scattering lengths. The corresponding values, taken from a recent numerical re–analysis [40], are ¯l1,GL = −2.15 ± 4.30 , ¯l2,GL = 5.84 ± 1.72 , (5.1) λ1,GL = (−7.35 ± 9.06) × 10−3 , λ2,GL = (10.57 ± 3.63) × 10−3 . (5.2) leading to The parameters λ1 and λ2 can also be determined directly, via a set of rapidly convergent sum– rules [41], from the knowledge, at order two loops, of the π–π scattering amplitude A(s|t, u) in GChPT [12], and from medium energy data on π–π phase shifts. The values obtained this way correspond to a determination at order O(p6 ). They depend only very weakly on the values of the parameters α and β when the latter are varied within the ranges specified below, and read λ1 = (−6.1 ± 2.2) × 10−3 , λ2 = (9.6 ± 0.5) × 10−3 . (5.3) These values are compatible with those given in eq. (5.2), but are affected by much smaller error bars. The analysis may even be refined in the standard case, using the information on the SChPT two–loop π–π amplitude obtained in Ref. [38], leading to the following values [39], λ1,st = (−5.7 ± 2.2) × 10−3 , λ2,st = (9.3 ± 0.5) × 10−3 . (5.4) ii) α, β : At leading order, α is directly correlated to the size of the condensate, see Eq. (1.12). As such, the value of α is not predicted by GChPT, but remains a free parameter, that can a 4 < priori be varied in the range 1 < ∼ α ∼ 4. At order O(p ), the relationship between α and the cB/Mπ2 becomes more complicated, as shown in Eq. (4.5). The corresponding next– ratio 2m to–leading and next–to–next–to–leading corrections have been estimated in Ref. [12] (see in particular the figures 8 and 10 in that reference). Notice that the analysis of Ref. [12] was done within the framework of SU(3)L × SU(3)R chiral perturbation theory. Working with only two light flavours as in the present paper might further reduce the uncertainties in the correspondance between the value of α and the size of the condensate, due to the lower number of unknown counterterms involved, and due to the absence of large contributions from the chiral 19 logarithms induced by the kaon loops [32]. However, the results of Ref. [12] are sufficient for our present purposes, and we shall not pursue that matter further. Once α is given, the parameter β is also constrained by low–energy π–π data. The correlation between α and β, which results from the Morgan–Shaw universal curve [42], has also been studied beyond leading order in [12], and is summarized in Fig. 6 of that reference. For the subsequent numerical analyses, we shall take the values given in Table 2 below. The values given in the second line of this table correspond to the one–loop values αst and βst of the standard case, which follow from the expressions given in Eq. (4.23), and from the values ¯l3 = 2.9 ± 2.4 [15], ¯l4 = 4.4 ± 0.3 [43]. Two points are worth being remembered. The first is that, independently of the value of α, β stays close to unity. The second point is that the standard case allows to make a very precise prediction for the value of α at order O(p4 ), viz. αst = 1.06 ± 0.06. Furthermore, this value is barely affected by the corrections at order O(p6 ) : The analysis of Ref. [39], based on the results of [38], gives αst = 1.07 ± 0.01 and βst = 1.105 ± 0.015 at next–to–next–to–leading order. Therefore, any significant deviation of the value of α from unity would provide evidence for a departure from the standard scenario of chiral symmetry breaking with a strong condensate. iii) t, t′ , t′′ : The constants t, t′ and t′′ appear in the expression of the π 0 → γγ amplitude (4.3). The uncertainty on the experimental value of the decay rate [5], Γ(π 0 → γγ) = 7.74±0.56 0 ct + eV, only yields a very weak constraint on the combination that appears in Aπ →γγ , viz. m 2 ′ 2 ′′ −2 c t = (0.0 ± 3.6) × 10 . This is comparable to the estimate one would obtain through Mπ t + m naive dimensional analysis [44]. In addition, as shown in Ref. [45], isospin breaking effects can 0 be sizeable in Aπ →γγ . Further information may be obtained by making use of the sum–rules considered in Ref. [45]. The corrections due to t′ were however not taken into account there, but the analysis of [45] is easily extended to the more general situation. In the generalized case, a similar set of sum–rules can be established, but they do not yield complete information on the three constants t, t′ and t′′ . We have summarized this analysis in the Appendix for the interested reader. Here, we only quote the values that we shall use in the sequel (the estimate for t′′ follows from naive dimensional analysis), ct = (6 ± 12) × 10−3 , Mπ2 t′ = (−3 ± 3) × 10−3 , m c2 t′′ ∼ ±1 × 10−3 . m (5.5) ′ iv) γ00 , γ+− , γ+− : The main difficulty in obtaining numerical estimates for these parameters comes from the lack of knowledge, in the generalized case, on the contributions from the low– energy constants of L+ (2,2) . In the spirit of Vector Meson Dominance (VMD), we have estimated the contributions of vector mesons to the two amplitudes AN and AC , as done in Ref. [25] for the standard case. This way, we find that t, t′ and t′′ receive no contribution, whereas the ′ contribution to γ00 , γ+− and γ+− read ′ γ00 |VMD = γ+− |VMD = 0 , γ+− |VMD = − 20 3 Mπ2 ∼ −0.024 . 4 MV2 (5.6) We take (5.6) as the values of these constants at the scale µ ∼ MV =770 MeV, and estimate the error associated to the VMD approximation and to the presence of the low-energy constants from L+ (2,2) in the expressions of γ00 and of γ+− by varying the scale µ of the corresponding chiral logarithms between 500 MeV and 1 GeV. The resulting values for γ00 and γ+− are shown ′ in Table 2. In the case of γ+− , we obtain a constant value which, to a very good precision, is compatible with zero. α β γ00 × 103 1.06 ± 0.06 1.103 ± 0.008 0 1.5 1.06 ± 0.06 8±3 2 1.07 ± 0.06 16 ± 5 2.5 1.08 ± 0.06 24 ± 8 3 1.11 ± 0.06 32 ± 10 γ+− × 103 −3 −5.7 ± 0.8 −8.4 ± 1.6 −11.0 ± 2.4 −13.7 ± 3.2 Table 2:Values of β, γ00 and γ+− for different values of α. With the above inputs at hand, we may now consider a few numerical applications. In Fig. 2, we have plotted the tree–level and one–loop cross sections for the charged channel, √ C σtree (s, α) and σ C (s, α), in the threshold region, 3Mπ ≤ s ≤ 0.5 GeV, where we expect the one–loop expression of the amplitude to be reliable, and for different values of α. We have used Mπ± = Mπ0 = 135 MeV in the amplitude and the experimental values, as quoted in [5], in the phase space integrals. Let us first concentrate on the standard case (α ∼ 1) which has been discussed before in the literature. In the second half of the energy region that we have shown, the correction as compared to the tree level cross section amounts approximatively to a factor of two, if we consider the central values. This agrees with the unpublished result [26], but is much less than previously found in [25]. On the other hand, the error bars induced by the uncertainties attached to the various conterterm contributions that enter the O(p6 ) amplitude are important. A closer analysis (see also below) reveals that the main contribution comes from the uncertainty on the value of λ1 given in (5.4). This shows also that the sensitivity of the cross section on the O(p6 ) counterterms becomes already sizeable even at such low energies. Coming now to the dependence with respect to α, we see that here also the higher order corrections have a deep influence and upset the situation that prevailed at tree level. For α ≥ 2 the one–loop cross sections are suppressed as compared to their tree–level values, and become even smaller than σ C (s, 1) as the energy increases. Unfortunately, the present theoretical error bars make it difficult to disentangle the different situations in practice from the knowledge of the total cross 21 section alone. 0.01 0.001 σ(s,1) σ(s,2) σ(s,3) σtree(s,1) σtree(s,2) σtree(s,3) σ (nb) 0.0001 1e-05 1e-06 1e-07 1e-08 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 √ s (GeV) Figure 2 : The cross section σ(s, α) (in logarithmic scale) for γγ → π + π − π 0 as a function of the center of mass total energy, for three values of α. Also shown are the corresponding curves for the tree level cross section σtree (s, α). N The cross sections σtree (s, α) and σ N (s, α) in the neutral channel have been plotted in N Fig. 3. Whereas σtree (s, α) ∼ α2 , the corrections are seen to have an even more drastic influence on the behaviour of the cross section as a function of energy than in the charged channel. Unfortunately, as far as the dependence on α is concerned, the picture is again totally blurred by the uncertainties, which, for the sake of clarity, we have not shown, but which are even more important than in the charged case. In both channels, the origin of the large error bars is a consequence of the highly destructive interferences between the various amplitudes. In the charged case, this interference was already present at tree level, and is even accentuated by the loop effects. In the neutral case, the strong α dependence of the single amplitude that contributes at lowest order is, in a similar way, washed out by the interferences between the two one–loop amplitudes AN 1 and 22 AN 2 . In order to illustrate this point, we show, in Fig. 4, how the neutral cross section looks like if only the amplitude AN 1 is considered. The importance of the interference effects with the second amplitude AN 2 appears clearly upon comparing Figs. 3 and 4 (for the standard case, a similar observation was already made by the authors of Ref. [25]). Unfortunately, we have not found a simple way to extract the contribution from AN 1 only : for photons with perpendicular polarizations, the two amplitudes contribute, and the destructive interference between them is again at work, while for photons with parallel polarizations, only the very small contribution from AN 2 , which vanishes at tree level, is singled out. We have tried to investigate whether looking at more refined observables allows to reach better perspectives from this point of view. 0.01 σ(s,1) σ(s,2) σ(s,3) σtree(s,1) σtree(s,2) σtree(s,3) 0.001 σ (nb) 0.0001 1e-05 1e-06 1e-07 0.4 0.41 0.42 0.43 0.44 0.45 0.46 √ s (GeV) 0.47 0.48 0.49 0.5 0.51 Figure 3 : The cross section σ(s, α) (in logarithmic scale) for γγ → π 0 π 0 π 0 as a function of the center of mass total energy, for three values of α. Also shown are the corresponding curves for the tree–level cross section σtree (s, α). We have, for instance, looked at the invariant mass distribution of the two charged pions in the γγ → π + π − π 0 channel. The result is shown in Fig. 5. As one may observe, the error bars are much less important than for the total cross section, and the different values of α can be 23 distinguished over a substantial portion of the energy range that has been considered. Actually, analyses of this type usually require sufficiently high statistics, which also represents a problem in the present case. 0.01 0.001 σAN1 (s,1) σAN1 (s,2) σAN1 (s,3) σtree (s,1) σtree (s,2) σtree (s,3) σ (nb) 0.0001 1e-05 1e-06 1e-07 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 √ s (GeV) Figure 4 : The cross section σAN1 (s, α) for γγ → π 0 π 0 π 0 obtained by taking into account the contribution from the amplitude AN 1 alone, for different values of α. Indeed, in both channels, the cross sections are very small at low energies, orders of magnitude below, for instance, the corresponding cross sections for the γγ → ππ processes. We have therefore also estimated the numbers of events that could be expected at an e+ − e− √ collider for the two–photon total invariant mass s below a maximal energy Emax . As typical examples, we have considered two instances of symmetric e+ − e− colliders. The first case corresponds to the Daphne φ–factory [21], with a total beam energy of Ebeam = 510 MeV, and a nominal integrated luminosity of 5 × 106 nb−1 per year. The second case concerns a τ –Charm Factory configuration [46], with a beam energy four times as large as for Daphne, and a design integrated luminosity of 107 nb−1 per year. The numbers of events are obtained upon convoluting the above cross sections with the corresponding photon luminosities quoted in 24 [47] (since we are not interested in resonant η production, whenever necessary we avoided the η peak by applying a cut on mγγ such that only the events with mγγ < Mη − ∆ or mγγ > Mη + ∆, with ∆ = 20 MeV are accepted), 4 α Ebeam 2 dLγγ = ( ln ) [ − (2 + z 2 )2 ln z − (1 − z 2 )(3 + z 2 )], dmγγ mγγ π me (5.7) √ where mγγ = s is the photon–photon center of mass energy, z = mγγ /2Ebeam , and me is the electron mass. For the luminosity quoted above, the expected total number of γγ → π + π − π 0 √ events per year at energies s < ∼ 0.6 GeV is around 5 ± 1 for Daphne (independently of α) and even less in the neutral case. In the case of a τ –Charm Factory, the total number of events becomes sizeable, and the results are shown, for the two modes, in the fourth column of Table 3, 18 1 loop, α=1 1 loop, α=2 1 loop, α=3 Tree, α=1 Tree, α=2 Tree, α=3 mπ+π- event distribution (Arbitrary Units) 16 14 12 10 8 6 4 2 0 0.28 0.3 0.32 0.34 0.36 0.38 mπ+π- (GeV) 0.4 0.42 0.44 0.46 Figure 5 : The histogram of the distribution of the invariant mass mπ+ π− of the two final √ charged pions in the process γγ → π + π − π 0 for different values of α and s ≤ E max = 0.6 GeV. Also shown are the curves correponding to the same distribution, but from the lowest order amplitude alone. 25 Mode Emax (GeV ) 0.50 π+π−π0 0.55 0.60 0.50 π0π0π0 0.55 0.60 α 1 2 3 1 2 3 1 2 3 #events 16 11 12 56 40 38 327 273 246 ∆(#events) 4 1 2 11 5 3 56 39 26 ∆λ1 3.3 1 1 9.7 4.5 0.2 50 35 24 ∆λ2 1.3 0.2 0.6 3.6 1.3 0.6 18 11 5.7 ∆β 0.3 0.2 1.8 0.6 0.1 3.1 1.8 2.7 4.4 ct ∆m 1.3 0.4 0.5 3.5 1.8 0.1 14 10 7.3 1 2 3 1 2 3 1 2 3 14 1 5 42 8 6 211 91 51 7 2 4 18 9 3 81 59 34 5.9 1.2 2.6 15.8 5.9 1.7 71 45 26 2.7 0.6 1.2 7.2 2.7 0.8 32 20 12 0.7 1 2.6 1.8 4.8 2.0 6.3 28 15 2.3 0.5 1.3 5.6 2.1 1.0 19 12 6.6 Table 3: Number of events for γγ → π + π − π 0 and for γγ → π 0 π 0 π 0 at a τ –Charm Factory as a function of the maximal energy Emax (GeV ), and the principal sources of error. for different choices of Emax (second column) and for different values of α (third column). Also shown are the corresponding uncertainties ∆(#events) (fifth column), while the remaining entries show various sources of contributions to the total error ∆(#events). The latter was obtained upon adding the individual contributions in quadrature. As mentioned above, the main source of error comes from the uncertainty on the value of λ1 . A sizeable but not necessarily drastic reduction of the latter would already allow to distinguish the standard case of a strong condensate (α ∼ 1) from situations where spontaneous breakdown of chiral symmetry would be triggered by a much weaker condensate (α > ∼ 2). 6 Summary and Conclusions In the present paper, the amplitudes of the processes γγ → π 0 π 0 π 0 and γγ → π + π − π 0 have been computed in the framework of SU(2)L × SU(2)R Generalized Chiral Perturbation Theory 26 to O(p6 ) precision. The corresponding generating functional has been constructed explicitly in Section 3, and the structure of its divergences in the sector of even intrinsic parity has been analysed. The resulting amplitudes, worked out in Section 4, satisfy the isospin relations that we have established in Section 2 (to the best of our knowledge, these relations have not been discussed previously in the literature). When restricted to the standard case, specified by the choice of parameters as indicated in Eq. (4.23), we recover the results obtained by previous authors [25, 26], both in the neutral and in the charged case. Finally, we have estimated the counterterms that enter the one-loop amplitudes and we have performed some numerical analyses in Section 5. We have, in particular, shown that the error bars associated to the counterterm estimates become important, especially in the neutral channel, as a consequence of highly destructive interference effects between the various amplitudes that build up the total cross sections. We have also considered the possible detection of these processes at Daphne and at a τ –Charm Factory. Unfortunately, and precisely because of these large interference effects, the expected number of events is rather discouraging in the first case. Depending on the actual values of the counterterms and on α, it is hard to expect more than ≈ 5 events per year with total invariant mass lower than 500 MeV. The number of events increases substantially when allowing larger invariant masses, but at the expense of working in an energy region where the O(p6 ) expressions are probably less reliable, since higher order terms can become important. The computation at O(p8 ) would allow a better control of the cross sections at larger momenta, thus allowing a substantial increase of the number of events already at Daphne. The required amount of work seems excessive, though, and would make sense only if conducted in parallel with a better determination of ¯l1 (for instance, from a two–loop analysis of Kℓ4 decays), by far the main source of theoretical uncertainties at present. We rather expect interesting and realistic prospects in this field to come from future machines, like the τ –Charm Factory, which run at higher energy and higher luminosity. Acknowledgments We are pleased to thank A. Bramon for discussions and for his interest and collaboration at the early stages of this work, and J. Bijnens for discussions. One of us (M.K.) wishes to acknowledge clarifying correspondance and/or discussions with I. Kogan and J. Stern on the content of Ref. [8], and L. Girlanda for discussions and for sharing information on unpublished work. He also thanks the Universitat Politècnica de Catalunya and the Universitat Autònoma de Barcelona for their hospitality. Ll. A. thanks the Institut de Physique Nucléaire in Orsay for its hospitality. P. T. thanks FEN department, where most of his contribution to the present work was done. This work has been partially supported by the TMR Program, EC–Contract N. 27 CT98–0169. Ll. A. and P. T. received partial support from the CICYT research project AEN95– 0815, while the work of J. K. and of P. T. was supported by the Schweizerischer Nationalfonds and by the Swedish Research Council (NFR), respectively. Appendix In this Appendix, we give a brief description of our analysis of the anomalous counterterms A2 , A3 , A4 and A6 which is based on the approach of Ref. [45]. The starting point are the invariant amplitudes ΠV V P (p2 , q 2 , r 2 ) and Π0V V P (p2 , q 2 , r 2 ) of the vector–vector–pseudoscalar three–point correlation functions in the three flavour chiral limit (we take the definition given by Eqs. (3) and (4) of [45]). At p2 = q 2 = 0, the loop contribution vanish, and one has (notice the absence of the pion pole in the second equality) 1 2B0 Nc + [8A4 − 16B0 (A2 − 2A3 )] , 2 2 16π r 4π 2 1 [8A4 + 24A6 ] . Π0V V P (0, 0, r 2) = 4π 2 ΠV V P (0, 0, r 2) = (A.1) In the chiral limit, the counterterms from L− (4,2) do not contribute, so that the above result holds both in SChPT and in GChPT. Although the contributions from the low–energy constants A2 and A3 from L− 6,0 were omitted in [45], one may follow the same steps as described there (we have also kept the same notation), and end up with the following set of sum–rules :  Nc 1 F02 B0 + [8A − 16B (A − 2A )] = − 4 0 2 3 2 2 2 4π 2MV MV 4π 2  MV MP 2 tanΘ A(π ′ → γγ) A(π → γγ)  , (A.2) and 1 3 B0 Gη′ √ 6A(η ′ → γγ) . [24A + 16B (A − 2A )] = − 6 0 2 3 4π 2 8 Mη2′ (A.3) In addition, one needs to know the expression of the amplitude of the two–photon decay of the 0 η, A(η → γγ) = −t1 (k, ǫ, k ′ , ǫ′ )Aη→γγ , which, in analogy to the π 0 → γγ amplitude Aπ →γγ of Eq. (4.3), may be written as Aη→γγ = 128 2 2 Fπ 1 e2 c(5 − 2r)t + c(1 − r)A6 + Mη2 t′ + m c2 te′′ (r) + m c (1 + 2r 2 )a3 ] . m [ + m 2 3 3 4 3π Fπ Fη 3 (A.4) √ In this last expression, t and t′ are related to A2 , A3 and A4 according to Eq. (4.9), while r c, and m c2 te′′ (r) denote the corrections coming from L− stands for the quark mass ratio ms /m (4,2) , 2 2 c ), O(m cms ), and O(ms ). which can be of order O(m In the standard case, the two last terms in Eq. (A.4) would appear only at higher orders, and one may thus proceed as described in Ref. [45]. At the order we are working, the quark 28 mass ratio is then given as rst = r2 ≡ 2MK2 /Mπ2 − 1 ∼ 25.9 [16]. Using the numerical values given in [45], one obtains, from Eqs. (A.2) and (A.3), respectively, cA4,st − Mπ2 (A2 − 2A3 )st = (−5.9 ± 1.8) × 10−4 m (A.5) cA6,st + Mπ2 (A2 − 2A3 )st = (−2.1 ± 0.4) × 10−3 . 3m (A.6) Mπ2 (A2 − 2A3 )st = (1 ± 1) × 10−3 . (A.7) and Upon using the experimental rate for η → γγ [5], we then determine the combination Adding errors in quadrature, the previous results give ctst + Mπ2 t′st = (1.7 ± 8.2) × 10−3 , m (A.8) ctst = (4.4 ± 10.8) × 10−3 , Mπ2 t′st = (−2.7 ± 2.7) × 10−3 . m (A.9) which is four times more accurate than the value obtained directly from the experimental rate of π 0 → γγ. For the separate pieces, we obtain In the generalized case, the analysis may, unfortunately, not be pursued quite that far. The main drawback are the corrections from L− (4,2) , which in particular produce potentially 2 large O(ms ) corrections to the η → γγ decay amplitude, and on which the sum–rules (A.2) and (A.3) give no information. If one restricts the analysis to the order O(p5 ) precision, then the contributions from L− (6,0) are also absent, and the situation becomes even simpler than in the standard case, since (cf. Eq. (1.4)) the left–hand sides of the sum–rules (A.2) and (A.3) now only involve A4 and A6 , respectively. Keeping in mind that r is now a free parameter (α and r are however related, see [17]), and taking the necessary inputs from [45], the O(p5 ) determination of A4 and A6 reads Nc cA4 = − m 32 |λ(r)MS2 − (1 − λ(r))2 Mπ2 | MP2 − MS2 3π 2 Mπ cA6 = − m 32 Mη′ !1 ∆GMO λ(r) − (r − 1)2 2 Mπ A(π ′ → γγ) + ··· , MP A(π → γγ) !1 (A.10) 2 Fπ A(η ′ → γγ) + · · · , (A.11) where the ellipses stand for higher order corrections, λ(r) = 2(r2 − r)/(r 2 − 1) and ∆GMO ≡ (3Mη2 − 4MK2 + Mπ2 )/Mπ2 ∼ −3.6. For r = r2 , the above expressions give values compatible with the previous SChPT analysis. Furthermore, as r decreases (i.e as α increases), the variation 29 ct stays within the bounds given in Eq. (A.9). On the other hand, in the extreme case of of m a vanishing condensate, the L− (6,0) contributions to the two sum–rules also disappear, and the expressions (A.10) and (A.11) become exact at order O(p6 ). 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