QCD anomaly mediation of ψ′ → ψπ0 and ψ′ → ψη

QCD anomaly mediation of ψ′ → ψπ0 and ψ′ → ψη

Volume 100B, number 4 PHYSICS LETTERS 9 April 1981 QCD ANOMALY MEDIATION OF ~ ' - + ~ n 0 AND ~ ' - ~ t II Kimball A. MILTON 1, William F. PALMER a...

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Volume 100B, number 4

PHYSICS LETTERS

9 April 1981

QCD ANOMALY MEDIATION OF ~ ' - + ~ n 0 AND ~ ' - ~ t II Kimball A. MILTON 1, William F. PALMER and Stephen S. PINSKY

Department of Physics, The Ohio State University, Columbus, 0H43210, USA Received 17 November 1980 Revised manuscript received 17 January 1981

We comment on various decay processes mediated dominantly by pseudoscalar couplings to the QCD axial-vector anomaly. Using coupling strengths of our earlier calculation, we compute ~/' ~ n°~A0 ' ---,~0 and obtain good agreement with experiment.

Several successful calculations [ I - 5 ] of inhibited processes involving pseudoscalar mesons (P) have appeared in the past year. They all involve the use of the QCD axial-vector anomaly and PCAC in one way or another; they all require knowledge of pseudoscalar decay constants, including SU(3) mixing effects. Recently we have carried through a program [5] for determining these decay constants by a combined chiral and 1/N expansion of the normal and anomalous Ward identities. Saturating with pseudoscalar mesons, as well as with bound glue states and a contact term contribution, we combine these constraints with data from ~ ~ r/'7 and ~ -* r/7 and (,?, r/', rrO) ~ 27. The solution to this set of equations is rather insensitive to the unknown gluonic and contact term contribution and yields nearly unique values for the decay constants (010ttAtz lip~) = m2 F/i , "

(1)

(0l a~A~ 0 - (2/3)l/2(3as/4rr) tr F~Ip/) = m:Tjo. We then used these decay constants in a successful calculation of the rl -+ 37r rate [4,5]. In this note we will use them to calculate ~ ' --> lr0 ~ / ~ ' ~ r/~. This ratio has been recently calculated by G6rard et al. [6] and by Ioffe and Shifman [7]. However, we feel there are significant defects in their calculations, which our I On leave from Department of Physics, University of California, Los Angeles, CA 90024, USA. 336

methods avoid. Before proceeding to the result, let us make some general remarks concerning the kinds of processes we can calculate using the QCD axial-vector anomaly. Some are amenable to direct current algebra methods. For example, ~ ~ 3~r can be calculated by reducing in the 7r0 and relating the amplitude, in the soft rr0 limit, to a matrix element of tr FF. The processes ~k(~') -~ P7 and ~k' ~ ~P, however, are p-wave, with amplitudes that vanish in the soft pseudoscalar limit, and current algebra tells us nothing about the rate * 1 To proceed, we assume that in these processes the pseudoscalar is radiated from the heavy quark state via two gluons [1,2], A ~ B + GG where A and B are ~c states. Then the GG system decays into the pseudoscalar, GG P. If the amplitude A ~ B + GG does not vary significantly with GG invariant mass, and the two gluons make the pseudoscalar at a single point ,2, M(A ~ BP 1)

M(A ~ BGG)M(GG -> P1 )

M ( A ' + BP2)

M(A -+ BGG)M(GG -+ P2)

(2)

(01tr F F I P 1) (01tr F F IP2)" Care must be taken when calculating this process in that there may be other mechanisms than that ofeq. (2) ,1 For this reason, the erroneous use of PCAC, the first calculation ofref. [7] is wrong. ,2 For a justification see ref. [8].

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PHYSICS LETTERS

which may dominate the amplitude. Consider for example the process ¢ ~ PT, P --- 0 r0, 77, r/'). By the mechanism of eq. (2) these amplitudes are of order Otsm~r2 2 (m u _ md)/(m u + rod), ot2m 2 s n,-s"'r/tv2rn2', respectively, where a2 is from the two-gluon exchange, and all decay constants have been put equal. The factor of quark-mass ratios for the lr0 is there because (01 tr Fffl rrO) violates isospin in the "large" sense [9] * 3 The competing mechanism for this process is GGG and GGG ~ P + % Here all amplitudes go like a3sM2 since the photon has both I = 0 or 1 = 1. [In all cases M = O(1 GeV)]. For the ~7and 7' channels it is clear that the first process is the dominant one; however, in the rr0 channel the 3-gluon mechanism dominates and eq. (2) should not be used for this decay ,4 (Indeed it gives bad results.) For the OZI inhibited 4' ~ ~bP process, however, only the two-gluon exchange process is relevant and we have r--

F(~',~ ~klrO) (P~rl 3 (01tr FFITr0) 2 P(~ -~brT) =\p-£n/ (0[trFfflr/)

(3)

9 April 1'981

The agreement is within experimental accuracy. Let us compare our results with those of Ioffe and Shifman [7] and Gdrard et al. [6]. Ioffe and Shifman fred eq. (7) but without the factor of -~ [F~r/(Fno - fin0)] 2 which amounts to a numerical factor of two as determined in our decay parameter solution in ref. [5]. They have two derivations of their result, the first of which we believe to be in error * 1 as discussed above. We agree in principle with their second derivation based on ref. [1 ]. However, we have calculated in ref. [5] the terms which they drop, namely, light quark mass terms compared to the strange quark mass and the term we denote Fn0. Inclusion of these effects results in a 50% reduction of their estimate of the decay rate of 4' ~ ~r/. G6rard et al., [6], with numerical results similar to ours, determine their decay parameters from a pole model mixing formalism quite different in spirit from the 1/Nand chiral perturbation expansion of ref. [5]. Indeed, the approximations of Gdrard et al. are certainly inconsistent. They obtain the result

Are = A Xro + An, X~rn,

The ~r0 matrix element is calculated in ref. [9],

(9)

+ (3/2) 1/2 ½ [(m d - mu)/(m d + mu)] Fm27r,

I<01(3as/41r) tr FFI rr0)l = ~Fm2r(md - mu)/(m d + mu).

(4)

where in their notation Ap = (Pl(3/2)l/2(as/4Zr) tr Fffl0),

(10)

l(0[(3as/47r ) tr FF[r/)[ = (3/2)l/2m2n[F 0 - Fn01, (5)

[~r0) = Irr3) + Xrnln) + X~rn,lrl'),

(11)

was evaluated to be

with 7r3 a pure triplet. But clearly, since the anomaly is a pure singlet,

The r/matrix element, defined by us earlier as [5]

(6)

- 'Z,o)/F = 0.53.

Arp =Aria + X~rnAn + Xnn,An, = X~rnAn + X~rn,An,,

Thus we have the theoretical ratio F2~r (P~r/3 3 [md-mu]2(mTr] 4 rth =2~mdd +mul \ m n l IF o - f r o 12~p~/ (7) = 3.6

X

10 -2,

(12) which is inconsistent with eq. (9) unless m u = md. The source of the inconsistency is their assumption that the divergences auA/=3'8_ transform like unmixed i = 3 and 8 members of an octet. In essence the effect of their inconsistent approximation is to double count

to be compared with the experimental values = ( 4 - + 2 ) X 1 0 -2

(ref.[ll]),

rexp = (6 + 4) X 10 -2

(ref. [12]).

(8)

, a Note there is an error of a factor of 2 in their tr F/~ term. ,4 Yet another mechanism for ~0~ ~r ° has been discussed by Lipkin and Rubinstein [10].

This work was supported in part by the Department of Energy. We are grateful for the helpful comments of K. Lane.

References [1 ] V.A. Novikov et al., Nuel. Phys. B165 (1980) 55.

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[2] W. Bardeen and V. Zakharov, Phys. Lett. 91B (1980) 111. [3] H. Goldberg, Phys. Rev. Lett. 44 (1980) 363. [4] K. Milton, W. Palmer and S. Pinsky, Phys. Rev. D22 (1980) 1124. [5] K. Milton, W. Palmer and S. Pinsky, Phys. Rev. D22 (1980) 1647. [6] J. G6rard, J. Pestieau and J. Weyers, Phys. Rev. Lett. 94B (1980) 227.

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[7] B.L. Ioffe and M.A. Shifman, Phys. Lett. 95B (1980) 99. [8] M. Voloshin and V. Zakharov, Phys. Rev. Lett. 45 (1980) 688. [9] D. Gross, S. Treiman and F. Wilczek, Phys. Rev. D19 (1979) 2188. [10] H. Lipkin and H. Rubinstein, Phys. Lett. 76B (1978) 324. [11] M. Oreglia et al., Phys. Rev. Lett. 45 (1980) 959. [12] T. Himel et al., Phys. Rev. Lett. 44 (1980) 920.