A refined determination of the η-η′ mixing angle and η → 2γ decay

A refined determination of the η-η′ mixing angle and η → 2γ decay

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  Volume 58B, number 3 PHYSICS LETTERS 15 September 1975 A REFINED DETERMINATION OF THE tyrf MIXING ANGLE AND q 27 DECAY G.,.CICOGNA Institute di Fisica dell’lmiversith, Pisa and Institute di Matematiche Applicate dell’Universit4 Pisa, taly and F. STROCCHI Instituto Nazionale di Fisica Nucleare, Sezione di Pisa and Scuola Normale Superiore, F’isa, taly Retiived 7 June 1975 A rigorous determination of the ~11’ mixing angle 0 is obtained within the SU (3) X SU (3) scheme of Gell-Mann, Oakes and Renner, by carefully taking into account possible contributions due to unequal renormalization constants and possible deviations from the pole dominance of the propagators at zero momentum. The result is tan 0 = 0.045 f 0.025. It leads to a determination of ~~~2~ in agreement with the recent data and to a value for rr)‘+2y = 10 keV in agreement with other models. The Cell-Mann-Okubo (GMO) formula for pseudo- scalar mesons was derived under the simplifying assump- tion that the physical states transform as the octet re- presentation of SU(3) and that the vacuum is SU(3) invariant. By Coleman’s theorem [ 1 ] this may at most be considered as a first order approximation and one must therefore expectZcorrections arising from the SU(3) non-invariance of the vacuum and from the non simple transformation properties of the states under SU(3). As a matter of fact, according to the SU(3) X SU(3) scheme [2], the SU(3) non-invariance of the vacuum is rather large [3] (x&O E (Olu810)/(01uO10) z 20%) and the expected corrections may therefore be quite significant. Their determination has been dis- cussed in the literature by making suitable and often questionable approximations. A rigorous way of treating this problem is provided by the Ward identities h la Glashow and Weinberg. A detailed analysis was pro- vided in a previous paper [3], under very general assump- tions, where a generalized GM0 formula and an inde- pendent sum rule yielding the q-71’ mixing angle were derived. Here, a derivation is presented by taking into account the possible contributions due to unequal re- normalization constants and by carefully discussing the deviations from the pole dominance of the propa- gatprs at zero momentum transfer. This refinement of the argument is cruciuZ n order to get a precise and ri- gorous determination of the 7-v ’ mixing angle a param- eter which plays an important role in the q+ 2y decay. The evaluation of the 9~’ mixing angle 0 is in fact strongly dependent on the approximations usually made in the literature [e.g. 41 and reliable conclusions can be drawn only if the effect of the approximations is under control. The main result of this note is to pro- vide a proof that 8 is very small, in agreement with the recent experimental dat on the r) + 27 decay rate. In the framework speciRed by the Cell-Mann, Oakes and Renner Hamiltonian [2] 91= 81, + E&-J + E&3 , (1) one gets the following Ward identities A1i’(0)(gLYX)i=(pe)i (i,i=O, l,..., 8)) (2) where A,j(O) = lim +o il exp (- iqx) T SiCxMj 0)) ) is the “propagator’ at zero momentum transfer and all notations are the same as in [3 1. It is easy to see that At:’ can be diagonalized by an orthogonal trans- formatton A, and one may introduce “physical” fields qh = A, #k and the corresponding asymptotic states Ii phys): <OIQphliphys)=6. Z1’2. m Irn i Putting (3) 313  Volume 58B, number 3 PHYSICS LETTERS 15 September 1975 one obtains (Ola”.$liphys) = Zr:“‘&$( 1 Cj)-l(ApX)j = M;F; , (5) where Mj is the mass of the ith particle and F,? the cor- responding PCAC constant. In terms of the above quan- tities the Ward identities (2) yield the following two in- dependent relations 4Zz’/2 FK K _ Z-l/2 s Flrn = 3(Zm1j2F r) rl COST? Z-‘12F ,~‘sin8) , rl’ 9 (6) 2di (Zz112FK K- Z,1’2F,rr) =-3(Z-1’2F rl rl nsin8 tZ1’12F ,q’cosB) 71 1) , (7) (where, as usual, K means Mi, etc. and the v-17’ mixing angle 0 is defined by $:h = u8 cos B t u. sin e), which generalize the GM0 formula and the relation found by Khuri [5,3]. It is important to stress that the above equations (6,7) are exact consequences of (2), with no approximation involved about the renormalization constants and the pole dominance of the propagators at zero momentum. Similarly, eq. (5) yields the fol- lowing relations between the PCAC constants 4Zg2(1 + C,)F, - Z;‘2(1 + C,)F,, (8) = 3 (Zi”( 1 t C,)F, cos 0 Zi12( 1 Cql)FV1 in 0) , 2fi(Z$‘(l + C,)F,- Z;‘2(1+C,)F,) (9) = -3(Z, ,‘2(ltc,,)F,, sine tz~ 2(1tC,r~)F,~~~~8). By combining eqs. (6,7) and using eqs. (8,9), one ob- tains 1 r-4 tan0 =- - 2d’-1’ where F7l r=- FK (10) (10’) The above eqs. (10, 10’) involve only the propagators of ?r, K and 17 7’ has been eliminated). The renormali- zation constants Zj and the deviations Ci from pole do- minance can be estimated by using chiral perturbation theory [6] : one gets*’ Z$ 2/Zi’2 - 1 = 6 (FK/F,, - 1) , 4(ZI$Zi’2 - 1) = 3(Zi’2/Zi’2- 1) . (11) One should stress that: i) the deviations of ZK/Z,, Z,/Z, from one are of an order.of magnitude smaller than the deviation of FK/F, from one, in agreement with the content of Glashow’s theorem [7]. This jus- tifies our previous treatment in which SU(3) non-in- varance of the vacuum (FK/F= f 1) was carefully taken into account, but equal renormalization con- stants, within the rr, K, 77 octet, were used; ii) Since Zj = 1 Zij~idS, the above eqs. (11) yields interesting bounds on the contributions of the conti- nuum, e.g. K ” ’ (MK t 2M,)2 s 1-Z, &(S)dS = K -- (& + m,)2 Z, and similarly for II and r) *2 This shows that even for deviations of Z, from 1 of the order of 15%, C, can at most be of the order of 1% and can be neglected. Moreover, from eqs. (8,9) and from chiral perturbation theory [6], one finds 4FgCg * 3F,,C, , (12) so that Cg and C’n are rather close. Everything is known in eq. (10, lo’), once a value of CK is given. For values of CK ranging from 5% to 20%, the value of tan 8 does not change significantly. One gets *3 tan0 = 0.045 + 0.025 , (13) *’ We do not enter here in a detailed discussion of the approxi- mations involved in the derivation of eqs. (11) and (12) be- low. Corrections to eqs. (11) and (12) are expected to be at most of the order of a further SU(3) breaking with respect to the leading SU (3) breaking term. *2 The Ci can in fact be neglected under smoothness assump- tions about the propagators, as shown by Glashow and Wein; berg [2] (see also [ 71. Most of the papers on chiral symmetry followed Glashow and Weinberg in neglecting the Cr: see e.g. [8] s *3 We have used FKIF, = 1.25 0.03. 314  Volume 58B, number 3 PHYSICS LETTERS 15 September 1975 in agreement with our previous determination [3]. It is clear from the above discussion that no questionable assumption has been made in deriving the result [ 131 from the Ward identities (2) and that all the approxima- tions are completely under control*‘. Besides eq. (IO), one obtains from eqs. (6-9) an in- dependent equation which can be used to evaluate Z;l(l+Cn~)-ln’/Z;l(l+C,)-l. One gets z, u+q z, fl-+) 9’ = (0.83 f 0.1) GeV2 . r) r) (14) This shows that the corrections due to unequal renor- malization constants and to the deviations from the pole dominance definitely favour the identification of 17’ ith Xo(95~8)*~ and thus improve the agreement with the experimental data. With the above value (13) of 19 one may compute the 7) + 2 y decay rate in terms of the ratio n’+ 2y/a + 27. One has r7?‘2Y 1 Mv 3 H _-- r n-+27 3 co& M, 1- tane (@)_,. (15) Since 6 is small, the ratio R = (q'l27)/(77 27) needs not to be known at high precision. An estimation of R can be obtained from the divergence of Ag with a proce- dure *6 similar to that used for computing the II + 2y decay [8-lo] : R=( 2,) ’2 (G--c)cose+ficsin8 -&c0set(fi-c)sine +(a (2,’ (5J2 (16) & tc e8 cc0se -(fi--c)sin8] ' c=-. '0 By putting Z, = Z,, , as suggested by the above analysis (eq. (14)), one gets R = .32 and r,r,_,2y = 9.3 keV if *’ The error in eq. (13) takes into account also the effect of possible approximations. Addition of a “tadpole” term e3u3 in the Hamlltonian (1) would change eqs. (6,7) only by terms of the order 10e5 - 10e6. *’ With the identification r)’ = X0(958), Z,~(l+C’,,~) turns out rather close to one, as for the members of the octet. For ZR = Zn = Z,, Ci = 0 one would get Mn’ IJ 1050 MeV. *6 We are aware that significant corrections may occur. We ex- pect however that the sign of R and its order of magnitude should be correct. the recent experimental value [ 1 l] of q + 2y is used. It is interesting to stress that with the above value of 8, also the quark model [ 121 yields essentially the same prediction: r rI,+201/rtl_+2r = 33, or Pll r n’+2r ~10.5 keV . Actually, a deeper connection between the SU(3) X SU(3) scheme and the quark model emerges since the first implies (through the determination’ of 0 and the di- vergences of the axial currents) (vnl2-v) ==2.64, bgw to be compared with the value 24 of the quark model W A rate of the n’+ 2 y decay of the same order as above is also predicted by the vector dominance model 1131 For values of r 17~-+27&-*2~ = 33 + 10, one gets from eq. (15) r q 2y (29 * 5) m+27 , (17) (where also the error in the knowledge (13) of 0 has been taken into account), quite consistently*’ with the recent experimental data [ 111. It should be stressed that a careful determination of t9 is crucial for the above agreement: a zero mixing angle*s would lead to a r ,,_,2,r much too small. We acknowledge Dr. R. Vergara Caffarelli for useful remarks. *’ The agreement becomes excellent if the value of n + 27 ob- tained the Primakoff effect [ 141 is used. A more sensible check of the validity of the above scheme is obtained by comparing the experimental and theoretical values of the matrix elements (TJ 27): one has (n I z eor = 0.18 GeV-r , or 0.21 GeV-r if the va ue [ 141 of P,r_,2y is used, whereas [ll] (s12y)exP = 0.216GeV-r. *a We do not agree on a recent claim [ 15,161 that 0 must be zero. This conclusion is in fact reached by neglecting in the generalized GM0 formula (eq. (59) of ref. [ 31) the term 3 (qq’)fi sin 0 cos 0 = 24 (F, - F&Mao. The neglection. of this term is responsible for the inconsistency of eqs. (471 and (48b) of ref. 1151, or of eqs. (15) and (17) of ref. 1161. 315  Volume 58B, number 3 PHYSICS LETTERS 15 September 1975 eferences [l] S. Coleman, Journ. Math. Phys. 7 (1966) 787. (21 M. Cell-Mann, R.J. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195; S.L. Glashow and S. Weinberg, Phys. Rev. Letters 20 (1968) 224. [ 31 G. Cicogna, F. Strocchi and R. Vergara Caffarelli, Phys. Rev. D6 (1972) 301. [4] S.J. Han, Phys. Letters 47B (1973) 169. [5] N.N. Khuri, Phys. Rev. Letters 16 (1966) 75. [6] P. Langacker and H. Pagels, Phys. Rev. D8 (1973) 4595. (7) S.L. Glashow, in Particles currents symmetries, ed. P. Urban (Springer, 1968). [8] S.L. Glashow, R. Jackiw and S.S. Shei, Phys. Rev. 187 (1969) 1916. (91 S. Adler, Phys. Rev. 177 (1969) 2426. [lo] F. Strocchi and R. Vergara CaffareBi; Phys. Letters 35B (1971) 595. [ 111 A. Browman et al., Phyr. Rev. Letters 32 (1974) 1067. [ 121 G. Morpurgo, in Properties of the fundamental interac- tions, vol. 9B, ed. A. Zichichi, Bologna 1973; A. Bramon and M. Greco, Phys. Letters 48B (1974) 137. [ 131 G.J. Gounaris, University of Ioannina preprint, June 1974. [ 141 G. Bellettini et al., Nuovo Cimento 66A (1970) 243. [.lS] B.G. Kenny, Phys. Rev. D8 (1973) 2172. [ 161 B.G. Kenny, Phys. Rev. D9 (1974) 2677.
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