Pseudoscalar Decays into Two Photons in Chiral Perturbation Theory

The Langrangian of QCD is known to have a U(3)L SU(3)tt chiral symmetry when the three lightest quark masses are set to zero. This symmetry is believed to be spontaneously broken to its vector U(3)L+R subgroup via nonvanishing quark vacuum expectation values (qq)0. The nonet of (pseudoscalar) Goldstone bosons associated with the spontaneous breaking of the symmetry can be conveniently parametrized in terms of

The nonet of (pseudoscalar) Goldstone bosons associated with the spontaneous breaking of the symmetry can be conveniently parametrized in terms of Z =exp(2iM/f), with M =tr'k'/E2+ rit I/J3 and transforming under U(3) L U(3) tt as Z Z' =LZR .The corresponding Lagrangian, to lowest order in derivatives and mass terms (dimension two), is given by tr(D"ZD"Z~)+ v tr(mZ+mZ )v'ri j, (2) 8 3f 2 where the first term provides the kinetic energy for the mesons and interaction terms consistent with current algebra and the second one describes the explicit breaking of U(3)LU(3)tt by the quark masses.The effect of the breaking of U(1)A through quantum loops (in-  stanton effects) is included only via an extra mass term for rit.The covariant derivative is D"X=&"X+ie[Q, Z)A", where A" is the photon field, and the quark mass matrix m and quark charge matrix (in units of e), Q, are given by m =diag(m", md, m, ) and Q =diag( -, ', --, ', --, ' ).The Lagrangian X2, Eq. ( 2), describes the lowest-order strong and electromagnetic interactions of the pseudoscalar-meson nonet.
Chiral perturbation theory' starts with the Lagrang- ian X2 and proceeds to a higher-order expansion in terms of momenta and quark masses.The resulting efkctive low-energy theory is completely known to next- to-leading order, once the g~h as been integrated out.It contains two distinct types of terms.The first ones are originated by pseudoscalar loops with vertices deducible from L2 giving rise to analytic and nonanalytic contribu- tions.The latter are particularly interesting, since they are considered to be the dominant ones and, on the other hand, involve a dependence on the renormalization scale p.As a result of this, the second type of (counter)terms, eliminating that p dependence and the associated diver- gences, is required.The set of all the counterterms ap- pearing in the next-order Lagrangian, L4, has been identified and extensively analyzed by Gasser and Leutwyler.
The problem of treating anomalies in an eA'ective low-energy theory was solved long ago by Wess and Zu- mino' and has been elegantly reformulated by Witten.

The preceding nonanomalous
Lagrangians have to be implemented by anomalous terms.To lowest order these extra terms in the effective Lagrangian are +anom t(e2/8tt2)eu~P(tl A a)A &tr(g'atXZt+Q2Zte&Z+ -, ' QZQZtatZZt -, QZtgxa&rtZ-) i(e/16~2)e""'-t'A" x tr [g(e,Z Z') (e. xz') (e,Z Z') + g (Z' e,Z) (Z' e.Z) (Z' e, Z) ]+ (3) where the dots refer to nonphotonic terms.As in the previously discussed nonanomalous part, three types of contribu- tions are expected to appear at next order in the chiral perturbation expansion: (i) loop diagrams involving one vertex photon decays of the neutral pseudoscalar mesons.
The decay widths P yy and P yy* strongly de- pend on the value of the decay constant fp.A brief dis- cussion of the first-order corrections to fp is therefore unavoidable.Writing the result in terms of (4) f. =f 1 2p.-pK-+ 4 [(2mu, d+ m, ) L4+m"dLs], 64v from X4"' and the other vertices from Xq, Eq. ( 2); (ii) tree diagrams involving one vertex from X4"', one from X4 and any number of vertices from X2 (these diagrams, however, do not contribute to the processes we study in this Letter); and (iii) tree diagrams from a dimension-six anomalous Lagrangian L6"' .The coefficients in L6"' can be used to absorb possible divergences appearing in m m mp mp the calculation from contributions (i) and (ii).Some ex- amples of terms of X6"' are given in Ref. 5. In this pa- 16z f p per we want to study their contribution to the two- which appears in the loop calculations, one has where the simplicity of the last line (concerning the loop part) is due to the SU(3) singlet nature of rl~a nd the dots represent unknown, higher-dimension terms contributing only to g~.L'4 and L5 are the renormalized coupling constants of the two relevant terms of the Lagrang- ian X4.There, the first three equations of (5) can also be found and the value of L4 has been argued to be con- sistent with zero.The experimental value fg/f, =1.22 +0.01 allows then to estimate the constant L5.At p =m"=0.55 GeV, the contribution of the chiral loga- rithms to fx/f, is negligible and one obtains L5 =(2.2 +. 0.5) x10 3. 2 Using these values for L4 and L5 and neglecting the extra counterterms we get f", /f =1.1.
Alternatively, if we choose p -1.5 GeV, where f~/f is described by the chiral logarithms (Ls =0), we obtain f", /f, -0. 9, still neglecting the extra counterterms.In any case, the absolute correction in f, is around 10%%uo.
We agree with Ref. 7 on the globally vanishing z and E loop corrections, although we disagree on the independent con- tributions.Introducing gg' mixing in the usual way, i.e. , with q = gscosO -g~s inO, g' = pssinO+ g] cosO, (9) one can construct the following reduced ratios': where the numerical values are the experimental results deduced from Eqs. ( 6).Taking 8 = -19.5' (sin8 = -3 ), as follows from the theoretical predictions and phenomenological analyses, ' ' one can deduce f", /f, from the experimental value of p".The obtained result is f", /f =1. 1, thus leading to the prediction p"=2.6, in good agreement with the experimental result (10).This numerical discussion indicates that Xs"' coun- terterms do not play a relevant roll in P yy decays.
On the one hand, divergent (p dependent) counterterms FIG. 1. One-loop diagrams contributing to P yy*.cannot be present in Xs"' and, on the other hand, finite counterterms should have low-energy constants compatible with zero because of the just obtained reasonable agreement between the theoretical prediction and experimental data for P yy decays.
A drastic change in the situation occurs when turning P yy to P yy* decays.This case is the relevant one in P y/+l (k )0) and in two-photon formation processes yy* P (k &0).Data on the k dependence in n ye+e, g yp+p, or in yy n, ri and ri' are (or are expected to be) available even if their quality at the moment is rather poor.The k dependence, up to first-order corrections, is exclusively generated by the n+ and E+ loops in Fig. 1(b).Therefore the expression for A(P~yy*) is given by with Ap 2 for P m and res and Ap=1 for P=ri& and F(mp, k ) is defined by where x =k /mp.Here one clearly observes the cancellation of chiral logarithms as k =0 (as before) and the noncan- cellation in the terms proportional to k .This latter fact implies unambiguously the existence of counterterms propor- tional to k: =is""~F, pF.
+Bt tr(g )tr[Z 8"Zl+B2tr[QZQZ ]tr[Zt8"Z]l+ (14) where the counterterms containing 2; or B, give contri- butions to x, g8, and qi, proportional to the lowest-order amplitude coming from Eq. (3) times k .The terms containing 8, ' or 8 contribute only to gi yy*.This can be rather easily seen since (suppressing derivatives) there are only two types of terms contributing to P yy*.Namely, tr(Q'M), trQ'trM.(1 S) The consequence of this is that the siopes at small k are the same for n and g8, while we cannot say anything about the slope for the gi.
Experimentally, nothing is known about the slope for the g' and the results for the n are contradictory.'' A single experiment for the q gives a measured slope in the decay g yp+p of 1.9+ 0.4 GeV .The predictions for the slopes in quark models and vector-meson domi- nance have been analyzed in Ref. 12.They predict values slightly dependent on the s-quark content of the meson involved, namely, 1.7 GeV for the z and, with 0= -19.5', 1.9 GeV for the g and 1.4 GeV for the ri'.The loop contributions in Eq. ( 12) amount to about one-third of the observed value for the t) (1.9+ 0.4 GeV ).The rest should come from the counterterms (14).In any case, since the physical t) is mainly t)s the slopes for n and g should be very similar.
The branching ratio for the n decay, r(zo -ye+e ) = 1.2%, r(~oyy) is clearly reproduced in our context but it is not a specific test of the k behavior.Just pure QED eff'ects in the photon propagator and the ye+e vertex explain the ra- tio above.
In conclusion, the eff'ects of the next-to-leading-order contributions to the two-photon decays of the pseudosca- lar mesons are rather diA'erent when the two photons are on mass shell than when one of the photons is allowed to be ofI' mass shell.In the first case the loop contributions cancel in such a way that their only eff'ect is the U(3) breaking in the decay constants.Using the gg' mixing angle 0= -19.5, the obtained theoretical values for the decay widths are in good agreement with the experimental results.Therefore, the contribution of the terms from the dimension-six anomalous Lagrangian is expected to be very small.In the second case, i.e. , the decays P yl + l and y y* P transitions, the terms propor- tional to the invariant mass of the lepton pair, k, are divergent.
These divergences can be absorbed in the coefficients of the terms of Xs"', such as the ones shown in Eq. ( 14).The dimension-six terms proportion- al to k can be classified in two groups: terms contribut- ing with the same weight to z and g8 decays, relative to the lowest order, and terms that only contribute to gl de- cays.Since the slope at k 0 obtained from the loop contributions is the same as for z and g8, one can con- clude that this result still holds when the complete first- order corrections are taken into account.