The π −→π−γ decay rate and vector-meson dominance

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  • LiETTER:E AL NUOVO CIM]BNTO VOL. 15, N. 12 20 Marzo 1976

    The p -~ 7:- T Decay Rate and Vector-Meson Dominance (').

    D. H. BOAL and R. TORGERSON

    Theoretical Physics Institute, Department o] Physics University o] Alberta - Edmonton, Alta., Canada

    (ricevuto il 12 Gennaio 1976)

    The recent measurement of the p--~=-~, decay width (1) has caused nmeh concern in that it is approximately three times smaller than anticipated using quark or higher- symmetry models (~). We note that this rate is in direct disagreement with simple vector-meson dominance (VMD) and the well-known 7:~165 rate, without the use of higher symmetries or singlet-oetet mixing (3). Using two methods, we estimate the corrections to simple VMD due to higher-mass vector mesons and find that they are insufficient to remove the disagreement.

    The coupling constants g~'vv and gp~v for the processes ~o~u165 and p-+~:T are re- lated through VDM by

    2eg~.~ ( 1 ) g~,.~.~ --

    go

    where e is the electric charge and go is related to the direct p-photon coupling vertex Gpy via

    (2) G~y = e m~. g0

    Although gp can be measured directly through the decays p-+~, the value so obtained has a rather large error associated with it. Fortunately, simple VMD pro- vides the relation

    (3) gp = g~=,

    (*) Work suppor ted in par t by the Nat iona l Research Counci l of Canada. (1) B. GOBBI, J . L. ROSEN, H. A. SCOTT, S. L. SHAPIRO, L. STRAWCZYNSKI and C. M. i~ELTZER: Phys. Rev. Left., 33, 1450 (1974). (=) B. T. FELD: Mo~els o/ Elementary Particles (New York , N .Y . , 1969) reviews the quark model . See for example ref. (~3.~) for h igher symmetry models . (a) A. KOTLEWSKI, W. LEE, M. SUZUKI and J . THALER: Phys. Rev. D, 8, 348 (1973).

    417

  • ~l~ D.H. BOAL and R. TORG]~RS01~

    where gp~ is the p-+ =r~ coupling constant. Since this decay rate is now known (4) with greater accuracy, it is the one we shall choose for determining g~. We wish to point out that eq. (1) follows from VDM and the isospin invarianee of the strong inter- actions alone and does not depend on the imposition of a higher symmetry on meson- photon coupling constants. Neither does it depend on the spectrum of the isosealar vector mesons that dominate the isosealar component of the electromagnetic current.

    From eq. (1) one computes

    (4) 2 2 2 3 /'(P- -->" w-Y) gp==/4z~(mp--m=- I

    /,(=o._,.yy) 6~ \ m=omp / "

    The coupling constant gpn~ is determined by using the p-meson mass and width found by Roos (4), who was able to achieve consistent results from ~r~ phase shift and pion form factor data. We find

    (5) g~ ~ -- 2.934-0.02. 4z~

    For the r:o~Ty decay width we adopt (5)

    (6) /,(=o_+yy) = (7.894-0.38) eV.

    The resulting value for the p-~ r~y decay width

    (7) F(p---~ r~-y) = (91 4-4) keV

    is more than five standard deviations away from the experimental result (1)

    (8) FexD(p--+~:ry) = (354-10) keV.

    One would expect that the generalized vector-meson dominance model (e) (GVMD) would yield corrections to eq. (1). The higher-mass resonances (predicted, for example, by the Veneziano model C)) change eq. (1) into a series

    (9) g~%y=2e~g: '~v , i :/p#

    where the sum is over all the I = 1, jPo = 1-- mesons Pi. The quantity gp, is the generalization of gp (el. eq. (2)) for these mesons. Here, one no longer has the equality

    (4) M. Roos: Hels ink i Un ivers i ty Report No. ISBN 951-45-06-3-0 (1975), finds mp= (776.3 and /~(p --> rr~) = (154.5 1.0) MeV. (5) Th is i s the weighted average of the recent determinat ion of A. BROWMAI~, J. I)EWIRE, B. GITTEL,MAN, K. M. HANSO~r D. L~SOI% E. LOH and R. LEWIS: Phys. Rcv. Left., 33, 1400 (1974) and the tabulat ion g iven by the PARTICLE DATA GROUP: Phys. Left., 50 B, 1 (1974). (') J . J. SAKURAI: Proceedings o! the Canadian Institute o! Particle Physics Summer School (Montrea], 1972), p. 437. (7) See J. L. PETERSON: Phys. Lelt., 2C, 155 (1971), for a review.

  • THE p - - ->~-~ DECAY RATE AND VECTOR-M~SON DOMINANCE 419

    gp, = g~,~= but ra ther the sum rule

    (10) 1 ~ gP 'uu . i gp,

    Again, eq. (9) is independent of the spectrum of I - 0, j Pc= 1-- mesons. Let us first examine eq. (9) f rom a theoret ica l s tandpo int . Tr iangle anomal ies in

    the part ia l ly conserved axial vector cur rent can be used to eva luate possible correct ions to eq. (9). In such an approach (s) the coupl ing constant g~,~v factor izes:

    ( 11 ) gp,gv = gp~r: C= v ,

    where C~ is a constant independent of ~ . This result is obta ined in fermion loop models (9) as well, prov ided that F un iversa l i ty (10) is mainta ined. This faetor izat ion is also independent of h igher -symmetry arguments .

    Combin ing eqs. (9)-(I1), we obta in

    (12) g~'vv = ze'~,~v 2,

    a resul t ident ical to eq. (1). This ra ther model -dependent conclusion shows that GVMD does not change the VMD result.

    We may est imate the correct ion in other ways. For example, the results of a dua l i ty analysis of the pion form factor data 01) ind icate that the sum of the higher- order terms of eq. (I0) is - - 11% of the first te rm gar:rJgp. Thus GVMD rate predic- t ions could change VMD predict ions by 22%, lowering our ~- -~-u result to 70 keV or rais ing i t to 110 keV. This is sti l l at least three s tandard deviat ions from the exper i menta l value.

    The analogous decay ~.0_+ ~oy has also been measured recent ly (1~). Again, the w idth found is approx imate ly three t imes smal ler than expected from the quark model. I f we assume that the ~*o_+~ou decay w idth is re lated to that of p -~y through S U 3, we find that

    (13) F(]~*~ K(;y) = (206 4-9) keV,

    again using =o_>yu as input . This is more than three s tandard deviat ions away from the exper imenta l value

    (14) F~p(I~o*__> ]~oy) = (75 ~35) keV .

    (*) P. G. O. FREUND an4 S. NANDI: Phys. Rev. Left., 32, 181 (1974); I~. TORGERSON: Phys. Rev. D, 10, 2951 (1974). (*) References to very early work may bc found in R. TORGERSON: Phys. Rev. D, 10, 2951 (1974). For the baryon loop model, see, for example, R. ROCK,lORE: Phys. Rev. D, 11, 620 (1974) and references contained therein; see also B. EDWARDS and A. N. KAMAL: Phys. Rev. Left. (to be published), for its application to PVy interactions without the imposition of F universality. (1o) j. j . SAKURAI: Ann. o] Phys., 11, 1 (1960); M. GELL-MANN: Calteeh Report No. CTSL-20 (1961) (published in M. GELL-MANN and Y. NE'EMANN: The Eight/old Way (New York, N. Y., 1964)). (11) p. L. BRUNINI, F. ]{IMONDI and G. VENTURI: Left. Nuovo Cimento, 10, 693 (1974). We quote results derived from their model labelled 2a siuce it is in closest agreement with known meson decay data. (lB) W. C. CARITHERS, 1). MOHLEMA~, D. UNDERWOOD a/ld D. G. RYAN: Phys. Rev. Lett.,3$, 349 (1975)o

  • 420 D.H. BOAL and R. TORC-F, RSON

    I t is of interest to compare the value for the p->=y decay width found here to that found in other calculations. In four recent approaches, nonet symmetry (13), SU 3 with broken nonet symmetry (1~), honer symmetry with broken SU 3 (15) and strong PCAC anomalies (le), a p~uu decay width close to 80 keV is found, in good agreement with what has been found here. It should also be noted that the broadest inter- pretation of the p--~ 7:-u experiment implies a range for F (p -~ ~:-y) from (30 10) keV to (804-10) keV. This upper limit is not in disagreement with VMD. The desirability of a new measurement of p-->~$ is clearly indicated since it tests VMD alone. In addition, if VMD passes this test, more precise information on K*->Ku rates will enable us to properly assess the assumption of SU 3 symmetry for pseudoscalar-vcctor- vector interactions.

    The authors wish to thank B. EDWARDS, Drs. A. N. KAMAL and J. W. MOFFAT for stimulating discussions. One of us (D. H. B.) is indebted to the National Research Council of Canada for financial support in the form of an N.R.C. Post-doctoral Fellowship.

    (an) A. BRAMON and M. GRECO: Phys. Left., 48 B, 137 (1974). (~4) D. H . BOAL, n . H . GRAHAI~I and J . W. :MOFFAT: Un ivers i ty of Toronto Repor t ( Ju ly 1975). (,5) B. EDWARDS and A. N. KAMAL: Phys. Rev. Left. (to be publ ished) . (~6) R. TORGERSON: Phys. Rev. D, 10, 2951 (1974).