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Corrections to VMD in the photoproduction of vector mesons I: Mass dependence of amplitudes
Volume 60B, number 2
PttYSICS LETTERS
CORRECTIONS OF VECTOR
5 January 1976
TO VMD IN THE PHOTOPRODUCTION
MESONS
I: MASS DEPENDENCE
OF AMPLITUDES
*
T. B A U E R and D.R. Y E N N I E
Laboratory of Nuclear Studies, Cornell University, Ithaca, N. Y. 14853, USA Received 15 August 1975 We study the dependence of the vector meson scattering amplitudes on the external masses, assuming that the most important effects arise from the coupling of the vector mesons to low mass intermediate states which mediate the scattering. The consequence is that the amplitudes for p0 and q photoproduction are modified by approximately + 10% and 15% respectively. This result is equivalent to naively using the coupling constants from the e+e - decay widths and ignoring the mass variation effects. The analysis also calls attention to residual uncertainties at the several percent level.
Several authors [1,2] have n o t e d that the vectorm e s o n propagator o f f o u r - m o m e n t u m k is affected by (hadronic) vacuum polarization, which alters its strength over the range k 2 = 0 to k 2= m 2 in a way w h i c h affects the evaluation o f coupling constants. We write this propagator D v ( k 2) = l / ( k 2 - m 2 +
nv(k2)),
(1)
II v is calculated f r o m bubble diagrams indicated schematically in fig. la. Since vector mesons are unstable, H v has an imaginary part w h i c h is related to the w i d t h by Im I1v = m v I'v(k2).
(2)
In turn, the real part o f II v is related to Im II v by subtracted dispersion relations. The subtraction procedure m a y be defined in various ways. Here we adopt the c o n v e n t i o n that D v should have a pole o f unit strength 2 Hence near mv. Re I l v ( m 2) = Re dllv . . .(k2) . . k. 2. . 0. (3) dk 2 =m 2 v When evaluated at k 2 = 0, the propagator differs f r o m what w o u l d be o b t a i n e d f r o m a siml~le pole w i t h o u t vacuum polarization (namely - 1 / m ~ ) b y the factor N v 1= [ 1 - 1-1v (0)/m 2 ] -1
(4)
In VMD, the coupling o f the p h o t o n to the vector m e s o n is assumed to be i n d e p e n d e n t o f k 2 . However, within this restriction we m a y admit k 2 m o d i f i c a t i o n s * Supported in part by the National Science Foundation.
,v
v vAv (a)
(b)
N
N
(c) Fig. 1. Mass variation correction in photoproduction, a) The vacuum polarization bubble, llv, that renormalizes the vector meson propagator, b) Vector meson photoproduction through elastically scattering the material in law c) Usual VMD mechanism for vector-meson photoproduction through elastically scattering a real vector meson, including the on-mass-shell (or renormalization) contribution from (b). o f the propagator a n d / o r matrix elements. It is the purpose o f this n o t e to estimate these effects in diffractive p h o t o p r o d u c t i o n o f vector mesons. A m o n g various possible conventions, we define the photonvector coupling to be - e m 2 / f v , corresponding to
rwe+e- = 7~U~/ mY"
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Volume 60B, number 2
PHYSICS LETTERS
On the other hand, coupling of the vector mesons to hadrons usually employs a different coupling constant than fv" For example, the decay p -+ rr+rr- yields a co~ pling fp given by )'~ rp = 7~87 mp
(1
4 -'21-~2'3/2 '".l"o~
(6) •
In case tile pTrlr vertex is point-like and other decay channels are negligible, the two couplings are related by fl
/2 .q 4n-
4rr NO"
(7)
This is simply another fornmlation of the well known Gounaris-Sakurai [I] finite width correction. With other conventions for the couplings, (5), (6) and (7) would be modified by factors o f N v , but the physical content of the analysis would be unchanged. There are differe'~t approaches in applying VMD to real photon processes leading to results differing by factors o f N v. In a field-theoretic formulation, it seems natural to incorporate the change in propagator yielding a VMD coefficient e/(fvNv) = e / fv. In a somewhat different view, the photon state is assumed to contain a superposition of hadrons which can be calculated from perturbation theory [3]. In this treatment, the vector meson is replaced by a resonant state o f strength elf v. One then assumes that this resonant superposition acts like a single incident particle. If one neglects any further off-mass-shell effects, cross sections calculated in these two ways then differ by a factor of N~. Using simple models, one estimates that No ~ 1.09 from rr+rr- intermediate states [1], N o 0.85 from K-pair intermediate states [2], and No: 1.00 [2]. Model uncertainties could modify any o f these by a few percent. Obviously, a factor Nv2 corresponds to a significant ambiguity in the phenomenological treatment of photoproduction data. In the field-theoretic approach, it is clearly inconsistent to incorporate the effect of vacuum polarization while ignoring the related contribution suggested by fig. lb in which the same low mass states scatter diffractively from the target. We shall now argue that the principal effect o f these contributions is to make *1 From the condition that the pion form factor is unity at k 2 = 0. This coupling cannot be quite right, however, since it gives a spurious pole for very large space-like k 2. 166
5 January 1976
e2/fv2 the correct VMD coefficient for this process. We make a gross simplifying assumption that the content of the bubble scatters from the target with a fixed amplitude J~Cv(t), depending perhaps only on the incident energy and not the internal configuration. Then it is not very difficult to show that the contribution of fig. lb to the photoproduction amplitude is
nv(O) - nv(m ~) f'~v (t) =fv
-rn 2
fvCv(t).
(8)
v
Eq. (8) is in no sense a rigorous result. (For example, our derivation applies only at t = 0 [4] and at high energies.) We use it only to estimate the size and sign of the correction. As the simplest example to motivate (8), suppose the content of the vacuum polarization bubble of fig. la is a single particle C o f m a s s M c. Then the unrenormalized II function is
II u = --a2/(k 2 M 2 + ie),
(9)
where a represents the coupling of V to C. In a scattering where V changes from k 2 = m 2 to k 2 = rn '2 through the mediation of C, we have
fCv(t, m '2 , m 2) = a 2
1 1 fcc(t) m'2_ M 2 rn 2_ M 2 C
C
IlvU(m2) - Ilu (rn '2)
=
rn2--rn '2
fcc(t).
(10)
Next, suppose that the intermediate state in fig. la is a collection of particles, one of which (particle B) scatters from the nucleon in fig. lb. Suppose the incident (final) V has four momentum, k(k') and the scattered particle from the set has momentum q (q'). Then q ' - q = k ' - k , and the relevant factors involving this particle (propagators and scattering amplitude) are 1
- -
1
4
JBB (t)
4
Iql OB g(t). Now
where in the diffractive limit fBB(t) = I
q2 and
1
ia2Bq'2_la2B
_
1
q'2_q2
I
1
q2-tj2B
121(11)
q'2~-Ia
q2 _ q ' 2 = (q + q')u (k-k')U"
For forward scattering, we have energy conservation (aside from the small recoil energy of the target) and
(q+q')u (k k ' ) U ~ - q z ( m
2
rn'2)/v.
Volume 60B, number 2
PHYSICS LETTERS
Now we make tile plausible assumption that the m o m e n t u m of the particle B is primarily along the direction of k, i.e., in the high m o m e n t u m limit, the transverse momentum is much less than the longitudinal. Then qz ~ Iql and the two terms o f ( 1 1 ) may be separated and combined with the factors from the other particles to obtain a result of the same form as (10). Now use IlU(m 2) = Hv(m 2) + Re IlU(m~) +(m 2
and note that ReIlvu cancels and Re dllU/dm 2 gives a "renormalization' contribution which can be incorporated in fig. lc. Thus we may drop the " u " in (10). For m 2 = 0 , m ' 2 = m 2 we recover (8). For m 2 = rn '2 , we find din2
,
2
fcc(O) = llv(m )fcc(0).(12)
This describes a vector meson contribution to Compton scattering (with m 2 = 0). Using (2) and (3), the contributions from fig. lb,c may be written e
e [Iv(O)
fvv by at most 50% (+ or - ). The modification to Compton scattering is somewhat different. In the diagonal approximation fVMD = v ~ e2 , c ~t'~ ~ [fvv + IIv(0)fvvl"
(14)
v
Note that here fv is not obviously singled out as the ' natural" coupling coustant. In our physical discussion below, we shall argue that in Compton scattering the effective value of fvCv should differ from £ v ' For example, a model estimate by Yennie [3] yields Joo = 1.7 )'oo" Estimates for the other vector mesons have not been made. In the case of the p 0, we accept the factor 1.7 as reasonable and allow an uncertainty of +0.5 in this coefficient. In a forthcoming review [6], the status o f coupling constants and cross sections will be discussed more comprehensively. For present purposes, we illustrate the size of the effects for the 00 meson by taking some typical numbers from experiments. The Orsay measurement of e+e- ~ ~+ rr- yields.f2/4rr = 2.26 -+0.25. Using N = 1.09, this implies f2/4~r =2-".69 -+0.30, corresponding to I"o = 140 MeV. Experimentally the p0 width is rather uncertain, but it is probably compatible with this value. The p0 width also enters the determination o f the photoproduction cross sections [7]. These cross sections are uncertain at the 1 0 - 2 0 % levels because of difficulties stemming from ambiguities in interpreting broad resonances. Using r o = 140 MeV, the experiments on nuclei [8] yield a forward cross section of 110/Jb/GeV 2 on a single nucleon. We may compare this with VMD which yields *2 'C
dW'l m 2 ) R e ---v]
fCv( t , m 2, m 2) - dIlv(m2)
5 January 1976
Pv
c
v Ivy Iv m2 (r:v-lvv)+i v< VV The last term on the R.H.S. o f (13) has a special meaning. It has been described as anomalous photoproduction, and in the case of the p0 it is significant and leads to the well-known "double-counting" correction to the Drell amplitude [5]. Note the factor e l f v in place o f effv in the first term of (13). It was obtained by combining a piece o f fig. l b with scattering amplitude fvv with the direct contribution from fig. lc. If this first term were the only contribution, the finite width correction in the propagator would have been completely undone by the mass variation o f the amplitude and the difference between the two approaches would be reconciled. The middle term of [13] represents the residual effects of non-resonant photon constituents scattering into the resonant state. F o r a number of reasons a realistic evaluation of this term is probably impossible. We defer until later the physical interpretation o f this term. Simply to indicate what is involved numerically, we shall estimate its uncertainty by assuming fv c differs from
d°~'° = & ( f ~ t -1 (l +o~2) 2 di t=0 1-6~-\ 4 ~ ! %N = (116 +-9 +- 13) ~b/GeV 2,
(15)
assuming up = 0.2, oo = 26 mb. Had we followed the usual formulation, the result would have been 98 ob/ GeV 2. It is clear that the off-mass-shell effects lead to a significant change in the predictions o f the VMD model +a. ,2 The first error represents the uncertainty in the model, the second in the coupling; uncertainties from op are not included These numbers may change slightly in the final analysis [6]. ,3 Before claiming that the new model improves agreement between the VMD model and experiment, a more comprehensive discussion is necessary [6]. In particular there are disagree ments between counter and bubble chamber experiments which ought to be resolved. 167
Volume 60B, number 2
PHYSICS LETTERS
To estimate the dipion contribution to the total cross section, we use the model calculation of Yennie [3], which allows for absorption of non-resonant pion pairs, and allow a model uncertainty of 0.5 l-l'p(0), yielding +2 °Yl°
e2 f 7 op [1 + (1.7 +0.5) II'o(0)]
= (84 -+4 +-9)/Jb,
(16)
where we have take ll'o(0 ) = 0.11. For comparison, the "old" VMD estimate with two options for the coupling constant yields e2 e2 J`2 o0 = ( 7 1 + - 8 ) o r 7 2 o o ¢u,
do
(84-+9)•b.
Again, the naive use of the coupling from (5) yields a result in agreement with the more sophisticated treatment. We next seek to justify our estimates of the middle term o f ( 1 3 ) and the last term of (14) for the p 0 meson, using the physical picture developed in ref. [3]. In that picture, the dipion consituent of the photon is envisaged as made up of/90 resonance (strength e/.~) and a low mass non-resonant pion pair structure *4. The p°-resonance has a normal hadronic size, but the nonresonant structure is more spatially extended and corresponds to two separate pions. In the diffractive photoproduction of pion pairs, the resonant part contributes with an amplitude c~ oo ~ 26 mb while the non-resonant amplitude oc ( % . + %_) ~ 52 mb. This corresponds to the Drell-S6ding model of photoproduction [8]. The result obtained is in good agreement with the observed mass distribution of pion pairs. Incidentally, the total dipion probability in the free photon turns out to be (e2/.f2)_ [1 + 7r'o(0)], which should be compared with (14). *s Our intuitive justification of the estimates for ff# is the following: (1) In the case of diffractive photo-
5 January 1976
production, we are considering the resonant part of the dipion structure in the final state. This had a good "overlap" only with the material in the bubble which has the resonant distribution and a similar spatial size. Its amplitude should be ~ f#o" The non-resonant distribution in the bubble is assumed to have a small over lap with the final resonance. (2) In Compton scattering, all masses must be considered, and we must take into account the larger total cross section of the lowmass non-resonant pion pairs. The high-mass constituents give a small contribution to the probability, and we ignore their effect on the Compton amplitude for small t. We expect that ffo/fp # lies between 1 and 2; the model calculation result of focp = 1.7 Joo seems to be a reasonable estimate. In summary, the effects of the mass dependence of the propagator and the scattering amplitude are separately quite significant, but they tend to cancel so that naive use of the coupling from rv_+e+e- yields the most reasonable estimate of diffractive photoproduction and the total cross section. At the same time, it must be recognized that residual uncertainties at the several percent level remain. It seems unlikely that these estimates can be improved without a considerably more sophisticated analysis. In the following article, we use these results and add the feature of co-~ mixing to revise the VMD estimate of co and q~ photoproduction and the VMD contribution to the total cross section.
References
[ 1] G.J. Gounaris and J.J. Sakurai, Phys. Rev. Letters 21 (1968) 244. M.T. Vaughn and K.C. Wali, Phys. Rev. Letters 21 (1968) 938. [2] F.M. Renard, Nucl. Phys. B15 (1970) 267. 13] D.R. Yennie, Rev. Mod. Phys. 47 (1975) 311. [4] T.H. Bauer, Nucl. Phys. B57 (1973) 109. [5] T.tt. Bauer, Phys. Rev. D3 (1971) 2671. ,4 This separation is not precise and probably has a meaning [6] T.H. Bauer, F. Pipkin, R. Spital and D.R. Yennie, to be only for constituent masses less than ~ 1100-1200 MeV. submitted to Rev. Mod. Phys. ,5 If one formally calculates the resonant or non-resonantprob[7] R. Spital and D.R. Yennie, Phys. Rev. D9 (1974) 138. abilities separately, they turn out to be infinite because of the [8] H. Alvensleben et al., Nucl. Phys. B18 (1970) 333; P-wave phase space. However, higher masseshave a destructive G. McClellanet al., Phys. Rev. D4 (1971) 2683; interference which makes the net probability finite. This isone R. Spital and D.R. Yennie, Phys. Rev. D9 (1974) 138. one reason we cannot take the separation seriously for high masses. [91 P. S6ding, Phys. Letters 19 (1966) 702.
168
PttYSICS LETTERS
CORRECTIONS OF VECTOR
5 January 1976
TO VMD IN THE PHOTOPRODUCTION
MESONS
I: MASS DEPENDENCE
OF AMPLITUDES
*
T. B A U E R and D.R. Y E N N I E
Laboratory of Nuclear Studies, Cornell University, Ithaca, N. Y. 14853, USA Received 15 August 1975 We study the dependence of the vector meson scattering amplitudes on the external masses, assuming that the most important effects arise from the coupling of the vector mesons to low mass intermediate states which mediate the scattering. The consequence is that the amplitudes for p0 and q photoproduction are modified by approximately + 10% and 15% respectively. This result is equivalent to naively using the coupling constants from the e+e - decay widths and ignoring the mass variation effects. The analysis also calls attention to residual uncertainties at the several percent level.
Several authors [1,2] have n o t e d that the vectorm e s o n propagator o f f o u r - m o m e n t u m k is affected by (hadronic) vacuum polarization, which alters its strength over the range k 2 = 0 to k 2= m 2 in a way w h i c h affects the evaluation o f coupling constants. We write this propagator D v ( k 2) = l / ( k 2 - m 2 +
nv(k2)),
(1)
II v is calculated f r o m bubble diagrams indicated schematically in fig. la. Since vector mesons are unstable, H v has an imaginary part w h i c h is related to the w i d t h by Im I1v = m v I'v(k2).
(2)
In turn, the real part o f II v is related to Im II v by subtracted dispersion relations. The subtraction procedure m a y be defined in various ways. Here we adopt the c o n v e n t i o n that D v should have a pole o f unit strength 2 Hence near mv. Re I l v ( m 2) = Re dllv . . .(k2) . . k. 2. . 0. (3) dk 2 =m 2 v When evaluated at k 2 = 0, the propagator differs f r o m what w o u l d be o b t a i n e d f r o m a siml~le pole w i t h o u t vacuum polarization (namely - 1 / m ~ ) b y the factor N v 1= [ 1 - 1-1v (0)/m 2 ] -1
(4)
In VMD, the coupling o f the p h o t o n to the vector m e s o n is assumed to be i n d e p e n d e n t o f k 2 . However, within this restriction we m a y admit k 2 m o d i f i c a t i o n s * Supported in part by the National Science Foundation.
,v
v vAv (a)
(b)
N
N
(c) Fig. 1. Mass variation correction in photoproduction, a) The vacuum polarization bubble, llv, that renormalizes the vector meson propagator, b) Vector meson photoproduction through elastically scattering the material in law c) Usual VMD mechanism for vector-meson photoproduction through elastically scattering a real vector meson, including the on-mass-shell (or renormalization) contribution from (b). o f the propagator a n d / o r matrix elements. It is the purpose o f this n o t e to estimate these effects in diffractive p h o t o p r o d u c t i o n o f vector mesons. A m o n g various possible conventions, we define the photonvector coupling to be - e m 2 / f v , corresponding to
rwe+e- = 7~U~/ mY"
(S) 165
Volume 60B, number 2
PHYSICS LETTERS
On the other hand, coupling of the vector mesons to hadrons usually employs a different coupling constant than fv" For example, the decay p -+ rr+rr- yields a co~ pling fp given by )'~ rp = 7~87 mp
(1
4 -'21-~2'3/2 '".l"o~
(6) •
In case tile pTrlr vertex is point-like and other decay channels are negligible, the two couplings are related by fl
/2 .q 4n-
4rr NO"
(7)
This is simply another fornmlation of the well known Gounaris-Sakurai [I] finite width correction. With other conventions for the couplings, (5), (6) and (7) would be modified by factors o f N v , but the physical content of the analysis would be unchanged. There are differe'~t approaches in applying VMD to real photon processes leading to results differing by factors o f N v. In a field-theoretic formulation, it seems natural to incorporate the change in propagator yielding a VMD coefficient e/(fvNv) = e / fv. In a somewhat different view, the photon state is assumed to contain a superposition of hadrons which can be calculated from perturbation theory [3]. In this treatment, the vector meson is replaced by a resonant state o f strength elf v. One then assumes that this resonant superposition acts like a single incident particle. If one neglects any further off-mass-shell effects, cross sections calculated in these two ways then differ by a factor of N~. Using simple models, one estimates that No ~ 1.09 from rr+rr- intermediate states [1], N o 0.85 from K-pair intermediate states [2], and No: 1.00 [2]. Model uncertainties could modify any o f these by a few percent. Obviously, a factor Nv2 corresponds to a significant ambiguity in the phenomenological treatment of photoproduction data. In the field-theoretic approach, it is clearly inconsistent to incorporate the effect of vacuum polarization while ignoring the related contribution suggested by fig. lb in which the same low mass states scatter diffractively from the target. We shall now argue that the principal effect o f these contributions is to make *1 From the condition that the pion form factor is unity at k 2 = 0. This coupling cannot be quite right, however, since it gives a spurious pole for very large space-like k 2. 166
5 January 1976
e2/fv2 the correct VMD coefficient for this process. We make a gross simplifying assumption that the content of the bubble scatters from the target with a fixed amplitude J~Cv(t), depending perhaps only on the incident energy and not the internal configuration. Then it is not very difficult to show that the contribution of fig. lb to the photoproduction amplitude is
nv(O) - nv(m ~) f'~v (t) =fv
-rn 2
fvCv(t).
(8)
v
Eq. (8) is in no sense a rigorous result. (For example, our derivation applies only at t = 0 [4] and at high energies.) We use it only to estimate the size and sign of the correction. As the simplest example to motivate (8), suppose the content of the vacuum polarization bubble of fig. la is a single particle C o f m a s s M c. Then the unrenormalized II function is
II u = --a2/(k 2 M 2 + ie),
(9)
where a represents the coupling of V to C. In a scattering where V changes from k 2 = m 2 to k 2 = rn '2 through the mediation of C, we have
fCv(t, m '2 , m 2) = a 2
1 1 fcc(t) m'2_ M 2 rn 2_ M 2 C
C
IlvU(m2) - Ilu (rn '2)
=
rn2--rn '2
fcc(t).
(10)
Next, suppose that the intermediate state in fig. la is a collection of particles, one of which (particle B) scatters from the nucleon in fig. lb. Suppose the incident (final) V has four momentum, k(k') and the scattered particle from the set has momentum q (q'). Then q ' - q = k ' - k , and the relevant factors involving this particle (propagators and scattering amplitude) are 1
- -
1
4
JBB (t)
4
Iql OB g(t). Now
where in the diffractive limit fBB(t) = I
q2 and
1
ia2Bq'2_la2B
_
1
q'2_q2
I
1
q2-tj2B
121(11)
q'2~-Ia
q2 _ q ' 2 = (q + q')u (k-k')U"
For forward scattering, we have energy conservation (aside from the small recoil energy of the target) and
(q+q')u (k k ' ) U ~ - q z ( m
2
rn'2)/v.
Volume 60B, number 2
PHYSICS LETTERS
Now we make tile plausible assumption that the m o m e n t u m of the particle B is primarily along the direction of k, i.e., in the high m o m e n t u m limit, the transverse momentum is much less than the longitudinal. Then qz ~ Iql and the two terms o f ( 1 1 ) may be separated and combined with the factors from the other particles to obtain a result of the same form as (10). Now use IlU(m 2) = Hv(m 2) + Re IlU(m~) +(m 2
and note that ReIlvu cancels and Re dllU/dm 2 gives a "renormalization' contribution which can be incorporated in fig. lc. Thus we may drop the " u " in (10). For m 2 = 0 , m ' 2 = m 2 we recover (8). For m 2 = rn '2 , we find din2
,
2
fcc(O) = llv(m )fcc(0).(12)
This describes a vector meson contribution to Compton scattering (with m 2 = 0). Using (2) and (3), the contributions from fig. lb,c may be written e
e [Iv(O)
fvv by at most 50% (+ or - ). The modification to Compton scattering is somewhat different. In the diagonal approximation fVMD = v ~ e2 , c ~t'~ ~ [fvv + IIv(0)fvvl"
(14)
v
Note that here fv is not obviously singled out as the ' natural" coupling coustant. In our physical discussion below, we shall argue that in Compton scattering the effective value of fvCv should differ from £ v ' For example, a model estimate by Yennie [3] yields Joo = 1.7 )'oo" Estimates for the other vector mesons have not been made. In the case of the p 0, we accept the factor 1.7 as reasonable and allow an uncertainty of +0.5 in this coefficient. In a forthcoming review [6], the status o f coupling constants and cross sections will be discussed more comprehensively. For present purposes, we illustrate the size of the effects for the 00 meson by taking some typical numbers from experiments. The Orsay measurement of e+e- ~ ~+ rr- yields.f2/4rr = 2.26 -+0.25. Using N = 1.09, this implies f2/4~r =2-".69 -+0.30, corresponding to I"o = 140 MeV. Experimentally the p0 width is rather uncertain, but it is probably compatible with this value. The p0 width also enters the determination o f the photoproduction cross sections [7]. These cross sections are uncertain at the 1 0 - 2 0 % levels because of difficulties stemming from ambiguities in interpreting broad resonances. Using r o = 140 MeV, the experiments on nuclei [8] yield a forward cross section of 110/Jb/GeV 2 on a single nucleon. We may compare this with VMD which yields *2 'C
dW'l m 2 ) R e ---v]
fCv( t , m 2, m 2) - dIlv(m2)
5 January 1976
Pv
c
v Ivy Iv m2 (r:v-lvv)+i v< VV The last term on the R.H.S. o f (13) has a special meaning. It has been described as anomalous photoproduction, and in the case of the p0 it is significant and leads to the well-known "double-counting" correction to the Drell amplitude [5]. Note the factor e l f v in place o f effv in the first term of (13). It was obtained by combining a piece o f fig. l b with scattering amplitude fvv with the direct contribution from fig. lc. If this first term were the only contribution, the finite width correction in the propagator would have been completely undone by the mass variation o f the amplitude and the difference between the two approaches would be reconciled. The middle term of [13] represents the residual effects of non-resonant photon constituents scattering into the resonant state. F o r a number of reasons a realistic evaluation of this term is probably impossible. We defer until later the physical interpretation o f this term. Simply to indicate what is involved numerically, we shall estimate its uncertainty by assuming fv c differs from
d°~'° = & ( f ~ t -1 (l +o~2) 2 di t=0 1-6~-\ 4 ~ ! %N = (116 +-9 +- 13) ~b/GeV 2,
(15)
assuming up = 0.2, oo = 26 mb. Had we followed the usual formulation, the result would have been 98 ob/ GeV 2. It is clear that the off-mass-shell effects lead to a significant change in the predictions o f the VMD model +a. ,2 The first error represents the uncertainty in the model, the second in the coupling; uncertainties from op are not included These numbers may change slightly in the final analysis [6]. ,3 Before claiming that the new model improves agreement between the VMD model and experiment, a more comprehensive discussion is necessary [6]. In particular there are disagree ments between counter and bubble chamber experiments which ought to be resolved. 167
Volume 60B, number 2
PHYSICS LETTERS
To estimate the dipion contribution to the total cross section, we use the model calculation of Yennie [3], which allows for absorption of non-resonant pion pairs, and allow a model uncertainty of 0.5 l-l'p(0), yielding +2 °Yl°
e2 f 7 op [1 + (1.7 +0.5) II'o(0)]
= (84 -+4 +-9)/Jb,
(16)
where we have take ll'o(0 ) = 0.11. For comparison, the "old" VMD estimate with two options for the coupling constant yields e2 e2 J`2 o0 = ( 7 1 + - 8 ) o r 7 2 o o ¢u,
do
(84-+9)•b.
Again, the naive use of the coupling from (5) yields a result in agreement with the more sophisticated treatment. We next seek to justify our estimates of the middle term o f ( 1 3 ) and the last term of (14) for the p 0 meson, using the physical picture developed in ref. [3]. In that picture, the dipion consituent of the photon is envisaged as made up of/90 resonance (strength e/.~) and a low mass non-resonant pion pair structure *4. The p°-resonance has a normal hadronic size, but the nonresonant structure is more spatially extended and corresponds to two separate pions. In the diffractive photoproduction of pion pairs, the resonant part contributes with an amplitude c~ oo ~ 26 mb while the non-resonant amplitude oc ( % . + %_) ~ 52 mb. This corresponds to the Drell-S6ding model of photoproduction [8]. The result obtained is in good agreement with the observed mass distribution of pion pairs. Incidentally, the total dipion probability in the free photon turns out to be (e2/.f2)_ [1 + 7r'o(0)], which should be compared with (14). *s Our intuitive justification of the estimates for ff# is the following: (1) In the case of diffractive photo-
5 January 1976
production, we are considering the resonant part of the dipion structure in the final state. This had a good "overlap" only with the material in the bubble which has the resonant distribution and a similar spatial size. Its amplitude should be ~ f#o" The non-resonant distribution in the bubble is assumed to have a small over lap with the final resonance. (2) In Compton scattering, all masses must be considered, and we must take into account the larger total cross section of the lowmass non-resonant pion pairs. The high-mass constituents give a small contribution to the probability, and we ignore their effect on the Compton amplitude for small t. We expect that ffo/fp # lies between 1 and 2; the model calculation result of focp = 1.7 Joo seems to be a reasonable estimate. In summary, the effects of the mass dependence of the propagator and the scattering amplitude are separately quite significant, but they tend to cancel so that naive use of the coupling from rv_+e+e- yields the most reasonable estimate of diffractive photoproduction and the total cross section. At the same time, it must be recognized that residual uncertainties at the several percent level remain. It seems unlikely that these estimates can be improved without a considerably more sophisticated analysis. In the following article, we use these results and add the feature of co-~ mixing to revise the VMD estimate of co and q~ photoproduction and the VMD contribution to the total cross section.
References
[ 1] G.J. Gounaris and J.J. Sakurai, Phys. Rev. Letters 21 (1968) 244. M.T. Vaughn and K.C. Wali, Phys. Rev. Letters 21 (1968) 938. [2] F.M. Renard, Nucl. Phys. B15 (1970) 267. 13] D.R. Yennie, Rev. Mod. Phys. 47 (1975) 311. [4] T.H. Bauer, Nucl. Phys. B57 (1973) 109. [5] T.tt. Bauer, Phys. Rev. D3 (1971) 2671. ,4 This separation is not precise and probably has a meaning [6] T.H. Bauer, F. Pipkin, R. Spital and D.R. Yennie, to be only for constituent masses less than ~ 1100-1200 MeV. submitted to Rev. Mod. Phys. ,5 If one formally calculates the resonant or non-resonantprob[7] R. Spital and D.R. Yennie, Phys. Rev. D9 (1974) 138. abilities separately, they turn out to be infinite because of the [8] H. Alvensleben et al., Nucl. Phys. B18 (1970) 333; P-wave phase space. However, higher masseshave a destructive G. McClellanet al., Phys. Rev. D4 (1971) 2683; interference which makes the net probability finite. This isone R. Spital and D.R. Yennie, Phys. Rev. D9 (1974) 138. one reason we cannot take the separation seriously for high masses. [91 P. S6ding, Phys. Letters 19 (1966) 702.
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