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Vector-meson electroproduction from generalized vector dominance
11 March 1999
Physics Letters B 449 Ž1999. 328–338
Vector-meson electroproduction from generalized vector dominance Dieter Schildknecht a
a,b,1
, Gerhard A. Schuler
a,2
, Bernd Surrow
c
Theoretical Physics DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland Fakultat ¨ fur ¨ Physik, UniÕersitat ¨ Bielefeld, D-33501 Bielefeld, Germany Experimental Physics DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland
b c
Received 20 October 1998; revised 5 January 1999 Editor: R. Gatto
Abstract Including destructively interfering off-diagonal transitions of diffraction-dissociation type, we arrive at a formulation of GVD for exclusive vector-meson production in terms of a continuous spectral representation of dipole form. The transverse cross-section, s T,g ) p ™ V p , behaves asymptotically as 1rQ 4 , while R V ' s L ,g ) p ™ V prs T,g ) p ™ V p becomes asymptotically constant. Contributions violating s-channel helicity conservation stay at the level established in low-energy photoproduction and diffractive hadron–hadron interactions. The data on r 0-meson production from the Fermilab E665 Collaboration and on f- and r 0-meson production from HERA are found to be in agreement with these predictions. q 1999 Elsevier Science B.V. All rights reserved.
The key role played by the vector mesons in the dynamics of hadron photoproduction on nucleons, at energies sufficiently above the vector-meson production thresholds, became clear in the late sixties and early seventies. Indeed, the total photoproduction cross-section on protons, sg p ŽW 2 ., was found to be
related to forward vector-meson photoproduction, d s 0rdt < g p ™ V p ŽW 2 ., extrapolated to t s 0 w1x 3 ,
sg p Ž W 2 . s
Vs r 0 , v , f ,Jr c
=
1
Supported by the BMBF, Bonn, Germany, Contract 05 7BI92P and the EC-network contract CHRX-CT94-0579. 2 Heisenberg Fellow; supported in part by the EU Fourth Framework Programme ‘‘Training and Mobility of Researchers’’, Network ‘‘Quantum Chromodynamics and the Deep Structure of Elementary Particles’’, contract FMRX-CT98-0194 ŽDG 12MIHT..
'16p
Ý
ž
ds 0 dt
(
ap g V2 1r2
/
,
Ž 1.
and to the total cross-sections for the scattering of transversely polarized vector mesons on protons,
3
A precision evaluation of Ž1. requires a correction for the Žsmall. ratio of real to imaginary forward scattering amplitudes to be inserted in the right-hand side of Ž1..
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 0 5 2 - 0
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
s V p , obtained w2x by applying the additive quark model for hadron–hadron interactions ap
sg p Ž W 2 . s
Ý
g V2
0
Vs r , v , f ,Jr c
sV p Ž W 2 . .
Ž 2.
The factor aprg V2 in Ž1. and Ž2. denotes the strength of the coupling of the Žvirtual. photon to the vector meson V, as measured in eqey annihilation by the integral over the vector-meson peak:
ap g V2
1 s
4p 2a
Ý Hse
q y
e ™V™ F
Ž s. d s ,
Ž 3.
F
or by the partial width of the vector meson:
G V ™ eq eys
a 2 mV 12 Ž g V2r4p .
.
Ž 4.
The sum rules Ž1. and Ž2. are based on Ži. the direct couplings of the vector mesons to the photon and on Žii. subsequent strong-interaction diffractive scattering of the vector mesons on the proton. Relations Ž1. and Ž2. accordingly provide the theoretical basis for applying concepts of strong-interaction physics, such as Regge-pole phenomenology, to the interaction of the photon with nucleons. Compare w3x for a recent analysis of the experimental data for the total photoproduction cross-section in terms of Regge phenomenology. The sum rule Ž1. is an approximate one. The fractional contributions of the different vector mesons to the total cross-section, sg p , were found to be w4x 4 rr s 0.65 ,
rv s 0.08 ,
rf s 0.05 ,
Ž 5.
adding up to approximately 78% of the total crosssection. An additional contribution of rJr c , 1–2% has to be added for the Jrc vector meson. To saturate the sum rule Ž1., the contributions of the leading vector mesons have to be supplemented by more massive contributions also coupled to the pho-
4
Compare also the review w5x.
329
ton, as observed in eqey annihilation. From the point of view of generalized vector dominance ŽGVD. w4x, the sum rules Ž1. and Ž2. appear as an approximation that is reasonable for the Q 2 s 0 case of photoproduction, while breaking down with increasing space-like Q 2 , the role of r 0 , v and f being taken over by more massive states. Relations Ž1. and Ž2. implicitly contain the propagators of the different vector mesons. Being evaluated for real photons at Q 2 s 0, no explicit propagator factors appear in Ž1. and Ž2., and the photon vector-meson transition with subsequent vector-meson propagation is reduced to a multiplication of the various cross-sections by coupling constants characteristic of the vector-meson photon junctions. It was pointed out a long time ago w6x 5 that an experimental study of vector-meson electroproduction would provide an additional and particularly significant test of the underlying photoproduction dynamics. The presence of the vector-meson propagators in the respective production amplitudes for the various vector mesons would be explicitly tested in vectormeson electroproduction. In addition, vector-meson production by virtual photons, at values of Q 2 4 m2V , would allow to test the expected dominance of the production by longitudinal photons over the production by transverse ones. Moreover, the Žapproximate. hypothesis of helicity conservation with respect to the centre-of-mass frame of the reaction g ) p ™ V p, the hypothesis of ‘s-channel helicity conservation’ ŽSCHC., introduced in Ref. w6x by generalizing experimental results from photoproduction w8x to electroproduction, would become subject to experimental tests. More recently, it was conjectured w9–14x that vector-meson electroproduction at large values of Q 2 was calculable in perturbative QCD ŽpQCD. and would provide experimental tests of it. We will comment on the results from the pQCD approach below. Expressing the cross-section for forward Ž t , 0. production of vector mesons on nucleons by transversely polarized virtual photons in terms of the
5
See also Ref. w7x.
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
330
respective real-photon cross-sections, we have the vector-meson dominance model ŽVDM. prediction w6x d s T0 dt
Ž W 2 ,Q 2 . g ) p™V p
m4V
s
ŽQ
2
ds 0
2 q m 2V
.
ŽW 2. .
dt
Ž 6.
g p™V p
For longitudinally polarized virtual photons, as a consequence of the coupling of the vector meson V to a conserved source as required by electromagnetic current conservation, the result w6x d s L0 dt
off-diagonal transitions of the diffraction-dissociation type was investigated in Ref. w16x. For definiteness, in Ref. w16x, the calculation of vector-meson production was based on a spectrum of an infinite series of vector-meson states coupled to the photon in a manner that assures duality 6 to quark–antiquark production in eqey annihilation. Under the fairly general assumption of a power law for the diffraction-dissociation amplitudes Žat zero t . in terms of the ratios of the masses of the diffractively produced vector states T w V p ™ VN p x s c 0 T w V p ™ V p x
s
ŽQ
2
2
2 q m 2V
.
j V2
Q ds m 2V d t
0
ŽW 2.
Ž 7.
g p™V p
was obtained. Both relations Ž6. and Ž7. contain SCHC. The parameter j V denotes the ratio of the imaginary forward scattering amplitudes for the scattering of longitudinally and transversely polarized vector mesons and may in principle depend on the vector meson V under consideration and on the energy W. The value of j V s 1 corresponds to the conjecture of helicity independence of vector-meson nucleon scattering in the high-energy limit. The predictions Ž6. and Ž7. for vector-meson production by virtual photons are based on the idealization that the propagation of the single vector meson V is responsible for the Q 2 dependence of the diffractive electroproduction of that vector meson V. This idealization is by no means true in nature. Time-like photons also couple to the continuum of hadronic states beyond r 0 , v , f , etc., resulting from eqey annihilation into quark–antiquark pairs, and vector-meson forward scattering need not necessarily be ‘diagonal’ in the masses of the ingoing and outgoing vector mesons. The process of diffraction dissociation, corresponding in the present context to ‘off-diagonal’ transitions such as r 0 p ™ r X 0 p etc., is in fact well known to exist in hadron–hadron interactions, as explicitly observed in proton–proton scattering w15x. The modification of the vector-meson electroproduction cross-section resulting from the inclusion of
2 pq1
ž / mN
Ž N s 1,2,3, . . . . ,
Ž W 2 ,Q 2 . g ) p™V p
m4V
m1
Ž 8.
an intuitively very simple and satisfactory result was obtained. The sum of the poles in the transverse amplitude for g ) p ™ V p was shown to sum up approximately to a single pole, the pole mass m V of the vector meson V being changed, however, to a value of m V ,T different from m V . Prediction Ž6., taking into account off-diagonal transitions as embodied in GVD, thus becomes w16x 7 d s T0 dt
Ž W 2 ,Q 2 . g ) p™V p
m 4V ,T
s
ŽQ
2
d s T0
2 q m 2V ,T
.
dt
ŽW 2. .
Ž 9.
g p™V p
For destructive interference among neighbouring vector-meson states, incorporated in Ref. w16x through an alternating-sign series of vector-meson states, one finds m V ,T - m V .
Ž 10 .
The precise value of m V ,T in Ž9. depends on the details of the strong amplitude, i.e. on the strength c 0 and the exponent p of the power-law ansatz Ž8. for Žspin-conserving. diffraction dissociation. 6
Compare w17x for a recent examination of the validity of duality to qq production in eq ey annihilation at energies below 3 GeV. 7 The simple result Ž9. is an approximation that coincides with the full GVD result at Q 2 s 0 and Q 2 ™`, but may vary by ;10% at intermediate Q 2 values.
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
331
The alternating-sign assumption leading to Ž10. was originally motivated w18x by GVD investigations on the total photo-absorption cross-section. For a recent analysis in this context, see e.g. w19x. Destructive interference among neighboring states, characteristic for alternating signs, was independently introduced in Ref. w20x, and more recently in a QCD-based analysis in Ref. w14x in order to explain the interference pattern observed in eqey annihilation into pqpy in the r X , r XX region between 1 GeV and 2 GeV. Diagonal vector-meson dominance would require the mass spectrum observed in diffractive pqpy photoproduction to resemble the one seen in eqey annihilation. Experimentally, there are significant differences, constructive versus destructive interference effects w20,14x. From the point of view of GVD, these differences qualitatively support the presence and importance of off-diagonal transitions: when passing from eqey annihilation to photoproduction, each specific vector-meson photon coupling is to be replaced by a sum of contributions due to the transition of the Žspacelike. photon to various vector-meson states subsequently scattered on the proton. This, in general, will lead to the observed differences in the mass spectra. The destructive interference pattern specifically observed in eqey™ pqpy together with the somewhat different pattern seen in g p ™ pqpyp , support our ansatz for vector-meson production with alternating signs and off-diagonal contributions. With Ž9., the asymptotic behaviour of the transverse forward-production cross-section in Žoff-diagonal. GVD becomes
terms with destructively interfering amplitudes imply multiplication of each s V p by a specific correction factor somewhat smaller than unity w16x. The result Ž9. Žor rather the underlying amplitude. with the constraint Ž10. in Ref. w16x was obtained by straightforward summation of an alternating series. In view of the ensuing extension to longitudinal photons, we note that the transverse amplitude may, to a good approximation, be represented by a sum of dipole terms 8 by combining neighbouring terms in the series. Switching to an equivalent continuum formulation, we obtain the following representation of the transverse amplitude as an integral over dipoles
d s T0
A L ,g ) p ™ V p Ž W 2 ,Q 2 ,t s 0 .
dt s
Ž W 2 ,Q 2 ™ ` .
A T ,g ) p ™ V p Ž W 2 ,Q 2 ,t s 0 . s m 2V ,T
d m2
Hm
2 V ,T
Ž Q 2 q m2 .
Q
4
dt
ŽW 2. .
Ž 11 .
Note that the modified pole mass m V ,T of the discrete formulation has turned into an effective threshold in Ž12.. Upon integration and squaring we immediately recover Ž9.. The impact of off-diagonal transitions on the result for longitudinally polarized virtual photons Ž7. was not explored in Ref. w16x. The representation Ž12. for the transverse production amplitude as a continuous sum over dipole contributions, abstracted from the assumed destructive interference between production amplitudes from neighbouring states, is well suited for a generalization to longitudinal photons. Taking into account the coupling of the photon to a conserved source as transmitted to the hadronic amplitude, we have
s j V m2V ,L
g p™V p
While the power of Q 2 in Ž9. and Ž11. remains unchanged with respect to Ž6., the normalization of the asymptotic cross-section relative to photoproduction is affected by the fourth power of m V ,T . Concerning sum rule Ž1.: it is unaffected by the introduction of off-diagonal terms, since the initial photon remains, when passing from the left-hand side to right-hand side of Ž1.. In relation Ž2., off-diagonal
Ag p ™ V p Ž W 2 ,t s 0 . .
Ž 12 .
g ) p™V p
m4V ,T d s 0
2
Hm
2 V ,L
(
Q2
d m2
m2
Ž Q 2 q m2 .
=Ag p ™ V p Ž W 2 ,t s 0 . .
2
Ž 13 .
In deriving Ž13., we have taken j V to be m-independent. We expect the threshold mass of the longi8
Although always possible, given the result Ž9. of the series, the dipole approximation of two neighbouring terms in the series is most natural for the choice ps 0 in Ž8., the value supported by diffraction-dissociation data w15x.
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
332
tudinal amplitude, m V ,L , to be larger than mT , i.e. m2V ,T - m2V ,L - m2V . As a consequence of the alternating signs, this is certainly true if the occurrence of an additional inverse mass, associated with the extra Q 2 factor in A L ,g ) p ™ V p , is the only difference between the m-dependence of A T,g ) p ™ V p and A L ,g ) p ™ V p . A priori, the transverse and longitudinal Žstrong-interaction. diffraction-dissociation amplitudes TT r L w V p ™ VN px may possess different m-dependences Ž p L / p T in Ž8.., thus affecting the ratio m2V ,Lrm2V ,T . Integration of Ž13. yields
(
A L ,g ) p ™ V p Ž W 2 ,Q 2 ,t s 0 .
p m2V ,L
sjV
y
2 Q m2V ,L 2 Q
™ 23 j V
(Q
2
jV
(Q
2
m V ,L
(Q
2
ŽQ
2
q m2V ,L
.
Q2
2
Ag p ™ V p Ž W ,t s 0 .
for Q 2 ™ 0
s T ,g ) p ™ V p Ž W 2 ,Q 2 .
ŽQ
2
2 q m 2V ,T
.
sg p ™ V p Ž W 2 . .
p m2V ,L 2
m 3V ,L
y
(Q
4
2
Q2 y
Ž Q 2 q m2V ,L .
Q2
™ 49 j V2
m2V ,L
Q2
arctan
m V ,L
(Q
2
2
for Q 2 ™ 0
m4V ,L
j V2
m 2V ,L
m 4V ,T
for Q 2 ™ ` .
Ž 16 .
sg ) p ™ V p Ž W 2 ,Q 2 . ' s T ,g ) p ™ V p q e s L ,g ) p ™ V p
for Q ™ ` .
The above predictions for transverse and longitudinal production amplitudes are valid for high-energy Ž x s Q 2rŽW 2 q Q 2 . < 1. forward Ž t , 0. production. It would be preferable to compare the predictions with forward-production data, thus eliminating the influence of a possible Q 2 dependence of the slope of the t-distribution. No reliable data for forward production have been extracted from the experiments so far. Accordingly, in order to be able to compare at all with data available at present, we ignore a possible Q 2 dependence of the t-distribution by putting bŽ0.rbŽ Q 2 . , 1, where b is the slope parameter in the t-distribution, expŽyb < t <.. From Ž9., the transverse production cross-section integrated over t then becomes
m 4V ,T
m 4V ,T
j V2
s s T ,g ) p ™ V p Ž 1 q e R V Ž W 2 ,Q 2 . .
2
Ž 14 .
s
2
The approach to the large-Q 2 limit is rather slow, but note the enhancement factor Ž m V ,Lrm V ,T . 4 in Ž16.. For completeness, we quote also the total virtual-photon cross-section and its asymptotic limit
2
m2V ,L
s
™
Ag p ™ V p Ž W 2 ,t s 0 .
Ag p ™ V p Ž W 2 ,t s 0 .
Ž Q 2 q m2V ,T .
p2
m 3V ,L
y
arctan
m V ,L
p ™
2
From Ž14. and Ž15. we obtain for the longitudinalto-transverse ratio R V s L ,g ) p ™ V p R V Ž W 2 ,Q 2 . s s T ,g ) p ™ V p
Ž 15 .
™
m4V ,T Q4
ž
p2 1qe
Ž Q 2 ™ `. .
4
j V2
m4V ,L m 4V ,T
/
sg p ™ V p Ž W 2 .
Ž 17 .
We note that our simple ansatz for diffraction dissociation does not lead to the change of the W dependence of r 0-meson production with increasing Q 2 for which there is some experimental indication w21,22x. Such an effect can be incorporated into GVD by modifying the W dependence of diffraction dissociation. In essence this amounts to replacing Ag p ™ V p ŽW 2 ,t s 0. on the right-hand-sides of Ž12. and Ž13. by an appropriate ŽRegge. ansatz in terms of W 2rm 2 . We have found that this modification may lead to the change in the W dependence with increasing Q 2 indicated by the data w21,22x on r 0 production. A detailed discussion is beyond the scope of the present note. A remark on SCHC is appropriate at this point. From photoproduction measurements at lower energies it is known w5x that SCHC is not strictly valid. It is violated Žat non-zero t . at the level of approximately 10%. In vector dominance this amount of
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
helicity-flip contributions is traced back to helicityflip contributions in diffractive hadron reactions which occur at approximately the same rate. Extrapolating the results from lower to the higher HERA energies quantitatively may seem somewhat problematic. If helicity-flip contributions are associated with non-pomeron exchange, one may expect such contributions to be diminished with increasing energy, at least as long as Q 2 is kept fixed. The situation may be different, if W and Q 2 are increased at the same rate, keeping x f Q 2rW 2 small and constant. In this case very massive vector states become important with increasing Q 2 and contributions different from helicity-conserving pomeron exchange may still be present in the limit of very high energies. If an average over a large range in Q 2 is performed, one may accordingly expect helicity flip contributions to be present at HERA energies at a rate similar to the one found at lower energies. In the comparison of our predictions with experiment, we proceed in two steps. In a first step we consider the experimental evidence for the validity of SCHC, before we turn to a comparison of Ž15. – Ž17. with HERA and Fermilab data. The validity of SCHC is not only of interest in itself, due to the presence of the longitudinal degree of freedom of the virtual photon in electroproduction, but is as well a prerequisite for the determination of R V , as long as data are lacking for a direct separation of s T,g ) p ™ V p and s L ,g ) p ™ V p . A recent measurement by the ZEUS collaboration w23x of the full set of 15 density matrix elements determining the vector-meson Ž r 0 and f . decay distribution w24x can be analyzed in terms of helicity-conserving and helicity-flip amplitudes. Using parity invariance as well as natural-parity exchange, the number of independent helicity amplitudes determining the density matrix elements can be reduced to ten. This number is reduced to five, if nucleon helicity-flip amplitudes are assumed to vanish. The normalized density matrix elements, accordingly, depend on four ratios of amplitudes, if one takes the amplitudes to be purely imaginary. In the notation M Ž lg , lr . for g ) p ™ r 0 p, these ratios of helicity amplitudes can be chosen as M Ž00.rM Žqq ., M Žq0.rM Žqq ., M Žqy .r M Žqq ., and M Ž0 q .rM Ž00.. The first one of the above ratios measures the relative magnitude of lon-
333
gitudinal and transverse helicity-conserving transitions. The other three ratios quantify helicity-nonconservation. The four ratios can be determined w25x from the measured r 0 density-matrix elements. According to w25x, violations of SCHC are small and of the order of magnitude measured in photoproduction at the lower energy of W s 9.4 GeV, where w5 x M Žq0.rM Žqq . s 0.14 " 0.02 and M Žqy .r M Žqq . s y0.05 " 0.02. As a consequence, the longitudinal-to-transverse production ratio
Rr s
1q
< M Ž 00 . < 2 < M Ž qq . < 2
1q
2 < M Ž 0 q. < 2 < M Ž 00 . < 2
< M Ž q0 . < 2 < M Ž qq . < 2
q
< M Ž qy . < 2 < M Ž qq . < 2
Ž 18 . is found to be well approximated by Rr ,
< M Ž 00 . < 2 < M Ž qq . < 2
1 ,
04 r 00
04 e 1 y r 00
,
Ž 19 .
04 i.e. by the SCHC expressions for Rr . Here r 00 0 denotes the r density matrix element solely determined by the dependence on the polar-angle of the r 0 ™ pq py decay distribution. The result w25x for the four ratios of the helicity amplitudes in Ž18., obtained from the nine measured density matrix elements, thus justifies the widely used procedure Ž19. of determining Rr from the r 0 decay distribution under the assumption of helicity conservation. The experimental results on Rr to be given below are based on this procedure. We now turn to the Q 2 dependence and compare predictions Ž15. – Ž17. with experimental data from HERA w22,26x at an average g ) p c.m. energy of W s 80 GeV Ž50 GeV. for f Ž r 0 . production 9. For a given vector meson V, our predictions depend on four parameters, the two effective vector-meson masses m V ,T and m V ,L , the ratio j V of the longitudinal-to-transverse strong-interaction amplitudes, and the photoproduction cross-section, i.e. Ž15. at Q 2 s 0. The solid lines in Figs. 1–3 show the result of a simultaneous four-parameter fit to the data for
9
At HERA energies, we may take the polarization parameter e s1.
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
334
and sg p ™ r p s 11.1 mb, sg p ™ f p s 1.2 mb. The statistical errors in the parameters are small compared with the estimated systematic ones. The quality of the fits strongly supports the underlying picture: the propagation of hadronic spin-1
Fig. 1. Data for sg ) p ™ f p Žin Ža.. and for sg ) p ™ r p Žin Žb.. from HERA compared with the GVD prediction Ž17.. Solid lines: Four-parameter fit with the values Ž20. of the fit parameters. Dashed line: Two-parameter fit with the values Ž21. of the fit parameters.
sg ) p ™ V p and R V , performed separately for the r 0 and the f meson. The data are well described by the fits, with the parameters jr s 1.06 ,
mr2 ,T s 0.68 mr2 ,
mr2 ,L s 0.71 mr2 ,
jf s 0.90 ,
mf2 ,T s 0.41 mf2 ,
mf2 ,L s 0.57 mf2 ,
Ž 20 .
Fig. 2. Ža. GVD prediction Ž16. for the longitudinal-to-transverse ratio Rf . Solid line: Four-parameter fit with the values Ž20. of the fit parameters. Dashed line: Two-parameter fit with the values Ž21. of the fit parameters. Žb. Data for f production by transversely polarized photons, s T,g ) p ™ f p , extracted from the measured values of sg ) p ™ f p by using the two-parameter Rf fit shown in Ža.. Dashed line: GVD prediction Ž15. with the twoparameter fit values Ž21.. Dotted line: VDM prediction, i.e. Ž15. with mf ,T ' mf .
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
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The values of R V obtained in the fit seem somewhat low with respect to the central values of the data at large Q 2 . This is of course merely a consequence of the fact that the large-Q 2 R V data hardly contribute to the overall x 2 , owing to their large errors. Varying the four fit-parameters within one standard deviation from their best-fit values, we find that a considerable spread in R V is allowed. In other words, with current data a precision determination of our parameters is not yet possible. In fact, a twoparameter fit results in a similar x 2 Ždashed lines in Figs. 1–3. as the four-parameter fit. In the twoparameter fit, obtained by fixing j V s 1 and m2V ,L s 1.5 m2V ,T Žcorresponding to an asymptotic value R V ™ 5.5., we find mr2 ,T s 0.62 mr2 ,
Fig. 3. As Fig. 2, but for r 0-meson production including HERA data for Rr extracted from the r-decay distribution using SCHC.
states and destructive interference govern the Q 2 dependence of exclusive electroproduction of vector mesons at small x and arbitrary Q 2 . Both the asymptotic 1rQ 4 behaviour of the cross section, see Ž17., and the flattening of R V , see Ž16., are clearly visible in the data. Moreover, the fitted values Ž20. are in accordance with theoretical expectation. The value of j V , 1, i.e. helicity independence of diffractive vector-meson scattering, is very gratifying indeed. The mass parameters, m V ,T and m V ,L , show the theoretically expected ordering m 2V ,T - m2V ,L - m 2V .
mf2 ,T s 0.40 mf2 ,
Ž 21 .
and sg p ™ r p s 11 mb, sg p ™ f p s 1.0 mb. With respect to the results of the fits given in Ž20. and Ž21. it may be worth quoting the estimate 0.41 m 2V Q m2V ,T Q 0.74 m2V from Ref. w16x, based on a reasonable choice of the diffraction-dissociation parameters in Ž8.. In Figs. 2b and 3b, we show the transverse crosssection, s T,g ) p ™ V p . The data in Figs. 2b and 3b were extracted from the data on sg ) p ™ V p in Fig. 1 with the help of our two-parameter fit 10 for R V . Figs. 2b and 3b demonstrate the dramatic difference at large Q 2 between the data and the GVD prediction Ž15. with m V ,T - m V on the one hand, and the VMD prediction Ž6., or rather Ž15. with m V ,T ' m V , on the other hand. Comparing the dotted VMD predictions in Figs. 2b and 3b for the transverse cross-section s T,g ) p ™ V p with the data for sg ) p ™ V p in Fig. 1a and 1b, one notices that the dotted curves would approximately describe the data for sg ) p ™ V p s s T ,g ) p ™ V p q sL,g ) p ™ V p . This, at first sight paradoxical, coincidence of fits of s T,g ) p ™ V p q s L ,g ) p ™ V p , entirely based on the transverse VMD formula, was in fact observed previously w21,27,28x in fits that vary the
10 No other procedure to extract s T,g ) p ™ V p suggests itself, as the number of data points for R V is very small, and the Q 2 values for sg ) p ™ V p and R V are not identical.
336
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
power of Ž Q 2 q m 2V . at fixed mass m V . Implicitly the fits obviously assume s L ,g ) p ™ V p s 0, and, disregarding the information from vector-meson decay indeed seem to confirm s L ,g ) p ™ V p s 0. This conclusion is inconsistent, however, with the results of the above analysis of the r 0 density matrix elements. This analysis establishes that longitudinal r 0 mesons are almost exclusively produced by longitudinal Žvirtual. photons. The mentioned approximate coincidence of fits based on the VMD formula for s T ,g ) p ™ V p with the data for s T ,g ) p ™ V p q s L ,g ) p ™ V p appears as a numerical accident. We turn to a comparison of our results with the r 0-production data from the Fermilab E665 Collaboration w29x taken at energies 10 - W - 24 GeV and at four-momentum transfers 0.15 - Q 2 - 10 GeV 2 . Fig. 4 shows the E665 data for Rr and for s T r taken from Tables 7 and 13 of Ref. w29x. The theoretical curves are based on the parameters from Ž21., i.e. jr s 1, mr2 ,L s 1.5 mr2 ,T , mr2 ,T s 0.62 mr2 and sg p ™ r p s 11 mb. Recent theoretical work on the electroproduction of vector mesons has been concentrated on attempts to deduce the cross-sections from perturbative w9–13x and non-perturbative w14x QCD. For the production cross-section by transversely polarized vector mesons, the pQCD calculations typically lead to a strong asymptotic decrease, as 1rQ 8 , modified sometimes by additional corrections to become 1rQ 7. It may be argued w11x that the region of Q 2 Q 30 GeV 2 explored at present, in which experiments find a fall-off rather like 1rQ 4 , is not sufficiently asymptotic for pQCD to yield reliable results. Further experiments at still larger values of Q 2 will clarify the issue. As for the longitudinal-to-transverse ratio, R V , pQCD calculations led to the same result of a linear rise in Q 2 as the simple VDM predictions, compare Ž6. and Ž7.. Such a linear rise is always obtained, if electromagnetic current conservation is the only source of the Q 2 dependence of R V . For large Q 2 , this linear rise is in conflict with experimental results. A behaviour of the cross-section for Q 2 4 mr2 , for both the production of longitudinally as well as transversely polarized r 0 mesons, somewhat closer to the experimental data, was obtained in Ref. w12x; the calculation was based on open qq production and parton-hadron duality. It is interesting to note
Fig. 4. As Fig. 3, but for E665 data.
that the resulting cross-sections have a VDM form 11 multiplied by correction factors depending on the scaling variable x. The asymptotic form for R V derived in Ref. w12x has recently been reproduced in a calculation based on r 0- meson wave-functions w13x. In Ref. w13x, also pQCD calculations of the helicity-flip amplitudes have been presented. While the magnitude of the helicity-flip amplitudes is ap-
11
Compare Ž37. and Ž38. in Ref. w12x.
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
proximately reproduced, a detailed comparison w25x shows that the x 2 of these predictions is not better than the x 2 for a representation of the density matrix elements under the assumption of SCHC. The coincidence of the relative magnitude of the helicity-flip amplitudes at large Q 2 with the helicity-flip amplitudes in photoproduction and diffractive hadron physics remains unexplained in the pQCD approach. In summary, we have investigated electroproduction of vector mesons in GVD. We have shown that destructive interference between neighbouring vector states naturally leads to the spectral representations ) Ž12. and Ž13. of the Žzero-t . amplitudes for g T,L qp ™ VT,L q p. Both predictions, the asymptotic 1rQ 2 fall-off of the transverse amplitude and the approach of R V towards a constant value, are in good agreement with the experimental data. The expected hierarchy, m2V ,T - m2V ,L - m2V , of the pole masses m V ,T , m V ,L and the helicity independence of the strong-interaction amplitudes Žreflected in j V , 1. strongly support the GVD picture: the propagation of hadronic vector states determines, for arbitrary Q 2 , the Q 2 dependence of vector-meson production by virtual photons in the diffraction region of x , Q 2rW 2 < 1. Moreover, SCHC is experimentally violated at the order of magnitude of 10%, a value also observed in diffractive hadron-hadron scattering and photoproduction at lower energies. Returning to our starting point, the photoproduction sum rules Ž1. and Ž2., the present analysis strengthens their dynamical content, which is to reduce photoproduction to vector-meson-induced reactions. More generally, in conjunction with the experimental observation of states with masses up to about 20 GeV w30x in diffractive production in DIS at small x and up to large Q 2 , the present investigation supports the point of view w31x that propagation and diffractive scattering of hadronic vector states is the basic dynamical mechanism in DIS at small values of the scaling variable.
Acknowledgements It is a pleasure to thank Sandy Donnachie, Teresa Monteiro and Gunter Wolf for useful discussions. ¨
337
References w1x L. Stodolsky, Phys. Rev. Lett. 18 Ž1967. 135. w2x H. Joos, Phys. Lett. B 24 Ž1967. 103; K. Kajantie, J.S. Trefil, Phys. Lett. B 24 Ž1967. 106; P.G.O. Freund, Nuovo Cimento A 44 Ž1966. 411. w3x A. Donnachie, P.V. Landshoff, Phys. Lett. B 296 Ž1992. 227; C. Caso et al., Particle Data Group, Review of Particle Properties, Eur. Phys. J. C 3 Ž1998. 1. w4x J.J. Sakurai, D. Schildknecht, Phys. Lett. B 40 Ž1972. 121; B. Gorczyca, D. Schildknecht, Phys. Lett. B 47 Ž1973. 71. w5x D. Leith, in: A. Donnachie, G. Shaw ŽEds.., Electromagnetic Interactions of Hadrons, vol. 1, Plenum Press, New York, 1978, p. 345. w6x H. Fraas, D. Schildknecht, Nucl. Phys. B 14 Ž1969. 543. w7x C.F. Cho, G.J. Gounaris, Phys. Rev. 186 Ž1969. 1619. w8x Aachen–Berlin–Bonn–Heidelberg–Munchen collaboration, ¨ Phys. Rev. 175 Ž1968. 1669; W.G. Jones et al., Phys. Rev. Lett. 21 Ž1968. 586. w9x M.G. Ryskin, Z. Phys. C 57 Ž1993. 89. w10x S.J. Brodsky, L. Frankfurt, J.F. Gunion, A.H. Mueller, M. Strikman, Phys. Rev. D 50 Ž1994. 3134. w11x J.R. Cudell, I. Royon, Phys. Lett. B 397 Ž1997. 317. w12x A.D. Martin, M.G. Ryskin, T. Teubner, Phys. Rev. D 55 Ž1997. 4329. w13x D.Yu. Ivanov, R. Kirschner, Phys. Rev. D 58 Ž1998. 114026. w14x H.G. Dosch, T. Gousset, G. Kulzinger, H.J. Pirner, Phys. Rev. D 55 Ž1997. 2602; G. Kulzinger, H.G. Dosch, H.J. Pirner, hep-phr9806352. w15x K. Goulianos, Phys. Rep. 101 Ž1983. 169. w16x H. Fraas, B.J. Read, D. Schildknecht, Nucl. Phys. B 88 Ž1975. 301. w17x A.B. Clegg, A. Donnachie, Z. Phys. C 62 Ž1994. 455. w18x H. Fraas, B.J. Read, D. Schildknecht, Nucl. Phys. B 86 Ž1975. 346. w19x L. Frankfurt, V. Guzey, M. Strikman, Phys. Rev. D 58 Ž1998. 094039. w20x A. Donnachie, H. Mirzaie, Z. Phys. C 33 Ž1987. 407. w21x T. Monteiro for the ZEUS Collaboration, in: J. Repond, D. Krakauer ŽEds.., Proc. 5th Int. Workshop on Deep Inelastic Scattering and QCD, Chicago, American Institute of Physics, New York, 1997, p. 626. w22x ZEUS Collaboration, J. Breitweg et al., DESY-98-107, 1998. w23x ZEUS Collaboration, papers submitted to XXIX Int. Conf. on High-Energy Physics, Abstracts 792, 793 and Addendum, Vancouver, 1998. w24x K. Schilling, G. Wolf, Nucl. Phys. B 61 Ž1973. 381. w25x D. Schildknecht, G.A. Schuler, B. Surrow, hep-phr9810370. w26x ZEUS Collaboration, M. Derrick et al., Phys. Lett. B 377 Ž1996. 259; B 380 Ž1996. 220; H1 Collaboration, C. Adloff et al., Z. Phys. C 75 Ž1997. 607; ZEUS Collaboration, J. Breitweg et al., Eur. Phys. J. C 2 Ž1998. 247. w27x F. Gade for the H1 Collaboration, in same Proc. as w21x, p. 631. w28x M. Erdmann, plenary talk at the XXIX Int. Conf. on High Energy Physics, Vancouver, 1998; hep-exr9812030.
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Physics Letters B 449 Ž1999. 328–338
Vector-meson electroproduction from generalized vector dominance Dieter Schildknecht a
a,b,1
, Gerhard A. Schuler
a,2
, Bernd Surrow
c
Theoretical Physics DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland Fakultat ¨ fur ¨ Physik, UniÕersitat ¨ Bielefeld, D-33501 Bielefeld, Germany Experimental Physics DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland
b c
Received 20 October 1998; revised 5 January 1999 Editor: R. Gatto
Abstract Including destructively interfering off-diagonal transitions of diffraction-dissociation type, we arrive at a formulation of GVD for exclusive vector-meson production in terms of a continuous spectral representation of dipole form. The transverse cross-section, s T,g ) p ™ V p , behaves asymptotically as 1rQ 4 , while R V ' s L ,g ) p ™ V prs T,g ) p ™ V p becomes asymptotically constant. Contributions violating s-channel helicity conservation stay at the level established in low-energy photoproduction and diffractive hadron–hadron interactions. The data on r 0-meson production from the Fermilab E665 Collaboration and on f- and r 0-meson production from HERA are found to be in agreement with these predictions. q 1999 Elsevier Science B.V. All rights reserved.
The key role played by the vector mesons in the dynamics of hadron photoproduction on nucleons, at energies sufficiently above the vector-meson production thresholds, became clear in the late sixties and early seventies. Indeed, the total photoproduction cross-section on protons, sg p ŽW 2 ., was found to be
related to forward vector-meson photoproduction, d s 0rdt < g p ™ V p ŽW 2 ., extrapolated to t s 0 w1x 3 ,
sg p Ž W 2 . s
Vs r 0 , v , f ,Jr c
=
1
Supported by the BMBF, Bonn, Germany, Contract 05 7BI92P and the EC-network contract CHRX-CT94-0579. 2 Heisenberg Fellow; supported in part by the EU Fourth Framework Programme ‘‘Training and Mobility of Researchers’’, Network ‘‘Quantum Chromodynamics and the Deep Structure of Elementary Particles’’, contract FMRX-CT98-0194 ŽDG 12MIHT..
'16p
Ý
ž
ds 0 dt
(
ap g V2 1r2
/
,
Ž 1.
and to the total cross-sections for the scattering of transversely polarized vector mesons on protons,
3
A precision evaluation of Ž1. requires a correction for the Žsmall. ratio of real to imaginary forward scattering amplitudes to be inserted in the right-hand side of Ž1..
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 0 5 2 - 0
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
s V p , obtained w2x by applying the additive quark model for hadron–hadron interactions ap
sg p Ž W 2 . s
Ý
g V2
0
Vs r , v , f ,Jr c
sV p Ž W 2 . .
Ž 2.
The factor aprg V2 in Ž1. and Ž2. denotes the strength of the coupling of the Žvirtual. photon to the vector meson V, as measured in eqey annihilation by the integral over the vector-meson peak:
ap g V2
1 s
4p 2a
Ý Hse
q y
e ™V™ F
Ž s. d s ,
Ž 3.
F
or by the partial width of the vector meson:
G V ™ eq eys
a 2 mV 12 Ž g V2r4p .
.
Ž 4.
The sum rules Ž1. and Ž2. are based on Ži. the direct couplings of the vector mesons to the photon and on Žii. subsequent strong-interaction diffractive scattering of the vector mesons on the proton. Relations Ž1. and Ž2. accordingly provide the theoretical basis for applying concepts of strong-interaction physics, such as Regge-pole phenomenology, to the interaction of the photon with nucleons. Compare w3x for a recent analysis of the experimental data for the total photoproduction cross-section in terms of Regge phenomenology. The sum rule Ž1. is an approximate one. The fractional contributions of the different vector mesons to the total cross-section, sg p , were found to be w4x 4 rr s 0.65 ,
rv s 0.08 ,
rf s 0.05 ,
Ž 5.
adding up to approximately 78% of the total crosssection. An additional contribution of rJr c , 1–2% has to be added for the Jrc vector meson. To saturate the sum rule Ž1., the contributions of the leading vector mesons have to be supplemented by more massive contributions also coupled to the pho-
4
Compare also the review w5x.
329
ton, as observed in eqey annihilation. From the point of view of generalized vector dominance ŽGVD. w4x, the sum rules Ž1. and Ž2. appear as an approximation that is reasonable for the Q 2 s 0 case of photoproduction, while breaking down with increasing space-like Q 2 , the role of r 0 , v and f being taken over by more massive states. Relations Ž1. and Ž2. implicitly contain the propagators of the different vector mesons. Being evaluated for real photons at Q 2 s 0, no explicit propagator factors appear in Ž1. and Ž2., and the photon vector-meson transition with subsequent vector-meson propagation is reduced to a multiplication of the various cross-sections by coupling constants characteristic of the vector-meson photon junctions. It was pointed out a long time ago w6x 5 that an experimental study of vector-meson electroproduction would provide an additional and particularly significant test of the underlying photoproduction dynamics. The presence of the vector-meson propagators in the respective production amplitudes for the various vector mesons would be explicitly tested in vectormeson electroproduction. In addition, vector-meson production by virtual photons, at values of Q 2 4 m2V , would allow to test the expected dominance of the production by longitudinal photons over the production by transverse ones. Moreover, the Žapproximate. hypothesis of helicity conservation with respect to the centre-of-mass frame of the reaction g ) p ™ V p, the hypothesis of ‘s-channel helicity conservation’ ŽSCHC., introduced in Ref. w6x by generalizing experimental results from photoproduction w8x to electroproduction, would become subject to experimental tests. More recently, it was conjectured w9–14x that vector-meson electroproduction at large values of Q 2 was calculable in perturbative QCD ŽpQCD. and would provide experimental tests of it. We will comment on the results from the pQCD approach below. Expressing the cross-section for forward Ž t , 0. production of vector mesons on nucleons by transversely polarized virtual photons in terms of the
5
See also Ref. w7x.
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
330
respective real-photon cross-sections, we have the vector-meson dominance model ŽVDM. prediction w6x d s T0 dt
Ž W 2 ,Q 2 . g ) p™V p
m4V
s
ŽQ
2
ds 0
2 q m 2V
.
ŽW 2. .
dt
Ž 6.
g p™V p
For longitudinally polarized virtual photons, as a consequence of the coupling of the vector meson V to a conserved source as required by electromagnetic current conservation, the result w6x d s L0 dt
off-diagonal transitions of the diffraction-dissociation type was investigated in Ref. w16x. For definiteness, in Ref. w16x, the calculation of vector-meson production was based on a spectrum of an infinite series of vector-meson states coupled to the photon in a manner that assures duality 6 to quark–antiquark production in eqey annihilation. Under the fairly general assumption of a power law for the diffraction-dissociation amplitudes Žat zero t . in terms of the ratios of the masses of the diffractively produced vector states T w V p ™ VN p x s c 0 T w V p ™ V p x
s
ŽQ
2
2
2 q m 2V
.
j V2
Q ds m 2V d t
0
ŽW 2.
Ž 7.
g p™V p
was obtained. Both relations Ž6. and Ž7. contain SCHC. The parameter j V denotes the ratio of the imaginary forward scattering amplitudes for the scattering of longitudinally and transversely polarized vector mesons and may in principle depend on the vector meson V under consideration and on the energy W. The value of j V s 1 corresponds to the conjecture of helicity independence of vector-meson nucleon scattering in the high-energy limit. The predictions Ž6. and Ž7. for vector-meson production by virtual photons are based on the idealization that the propagation of the single vector meson V is responsible for the Q 2 dependence of the diffractive electroproduction of that vector meson V. This idealization is by no means true in nature. Time-like photons also couple to the continuum of hadronic states beyond r 0 , v , f , etc., resulting from eqey annihilation into quark–antiquark pairs, and vector-meson forward scattering need not necessarily be ‘diagonal’ in the masses of the ingoing and outgoing vector mesons. The process of diffraction dissociation, corresponding in the present context to ‘off-diagonal’ transitions such as r 0 p ™ r X 0 p etc., is in fact well known to exist in hadron–hadron interactions, as explicitly observed in proton–proton scattering w15x. The modification of the vector-meson electroproduction cross-section resulting from the inclusion of
2 pq1
ž / mN
Ž N s 1,2,3, . . . . ,
Ž W 2 ,Q 2 . g ) p™V p
m4V
m1
Ž 8.
an intuitively very simple and satisfactory result was obtained. The sum of the poles in the transverse amplitude for g ) p ™ V p was shown to sum up approximately to a single pole, the pole mass m V of the vector meson V being changed, however, to a value of m V ,T different from m V . Prediction Ž6., taking into account off-diagonal transitions as embodied in GVD, thus becomes w16x 7 d s T0 dt
Ž W 2 ,Q 2 . g ) p™V p
m 4V ,T
s
ŽQ
2
d s T0
2 q m 2V ,T
.
dt
ŽW 2. .
Ž 9.
g p™V p
For destructive interference among neighbouring vector-meson states, incorporated in Ref. w16x through an alternating-sign series of vector-meson states, one finds m V ,T - m V .
Ž 10 .
The precise value of m V ,T in Ž9. depends on the details of the strong amplitude, i.e. on the strength c 0 and the exponent p of the power-law ansatz Ž8. for Žspin-conserving. diffraction dissociation. 6
Compare w17x for a recent examination of the validity of duality to qq production in eq ey annihilation at energies below 3 GeV. 7 The simple result Ž9. is an approximation that coincides with the full GVD result at Q 2 s 0 and Q 2 ™`, but may vary by ;10% at intermediate Q 2 values.
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
331
The alternating-sign assumption leading to Ž10. was originally motivated w18x by GVD investigations on the total photo-absorption cross-section. For a recent analysis in this context, see e.g. w19x. Destructive interference among neighboring states, characteristic for alternating signs, was independently introduced in Ref. w20x, and more recently in a QCD-based analysis in Ref. w14x in order to explain the interference pattern observed in eqey annihilation into pqpy in the r X , r XX region between 1 GeV and 2 GeV. Diagonal vector-meson dominance would require the mass spectrum observed in diffractive pqpy photoproduction to resemble the one seen in eqey annihilation. Experimentally, there are significant differences, constructive versus destructive interference effects w20,14x. From the point of view of GVD, these differences qualitatively support the presence and importance of off-diagonal transitions: when passing from eqey annihilation to photoproduction, each specific vector-meson photon coupling is to be replaced by a sum of contributions due to the transition of the Žspacelike. photon to various vector-meson states subsequently scattered on the proton. This, in general, will lead to the observed differences in the mass spectra. The destructive interference pattern specifically observed in eqey™ pqpy together with the somewhat different pattern seen in g p ™ pqpyp , support our ansatz for vector-meson production with alternating signs and off-diagonal contributions. With Ž9., the asymptotic behaviour of the transverse forward-production cross-section in Žoff-diagonal. GVD becomes
terms with destructively interfering amplitudes imply multiplication of each s V p by a specific correction factor somewhat smaller than unity w16x. The result Ž9. Žor rather the underlying amplitude. with the constraint Ž10. in Ref. w16x was obtained by straightforward summation of an alternating series. In view of the ensuing extension to longitudinal photons, we note that the transverse amplitude may, to a good approximation, be represented by a sum of dipole terms 8 by combining neighbouring terms in the series. Switching to an equivalent continuum formulation, we obtain the following representation of the transverse amplitude as an integral over dipoles
d s T0
A L ,g ) p ™ V p Ž W 2 ,Q 2 ,t s 0 .
dt s
Ž W 2 ,Q 2 ™ ` .
A T ,g ) p ™ V p Ž W 2 ,Q 2 ,t s 0 . s m 2V ,T
d m2
Hm
2 V ,T
Ž Q 2 q m2 .
Q
4
dt
ŽW 2. .
Ž 11 .
Note that the modified pole mass m V ,T of the discrete formulation has turned into an effective threshold in Ž12.. Upon integration and squaring we immediately recover Ž9.. The impact of off-diagonal transitions on the result for longitudinally polarized virtual photons Ž7. was not explored in Ref. w16x. The representation Ž12. for the transverse production amplitude as a continuous sum over dipole contributions, abstracted from the assumed destructive interference between production amplitudes from neighbouring states, is well suited for a generalization to longitudinal photons. Taking into account the coupling of the photon to a conserved source as transmitted to the hadronic amplitude, we have
s j V m2V ,L
g p™V p
While the power of Q 2 in Ž9. and Ž11. remains unchanged with respect to Ž6., the normalization of the asymptotic cross-section relative to photoproduction is affected by the fourth power of m V ,T . Concerning sum rule Ž1.: it is unaffected by the introduction of off-diagonal terms, since the initial photon remains, when passing from the left-hand side to right-hand side of Ž1.. In relation Ž2., off-diagonal
Ag p ™ V p Ž W 2 ,t s 0 . .
Ž 12 .
g ) p™V p
m4V ,T d s 0
2
Hm
2 V ,L
(
Q2
d m2
m2
Ž Q 2 q m2 .
=Ag p ™ V p Ž W 2 ,t s 0 . .
2
Ž 13 .
In deriving Ž13., we have taken j V to be m-independent. We expect the threshold mass of the longi8
Although always possible, given the result Ž9. of the series, the dipole approximation of two neighbouring terms in the series is most natural for the choice ps 0 in Ž8., the value supported by diffraction-dissociation data w15x.
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
332
tudinal amplitude, m V ,L , to be larger than mT , i.e. m2V ,T - m2V ,L - m2V . As a consequence of the alternating signs, this is certainly true if the occurrence of an additional inverse mass, associated with the extra Q 2 factor in A L ,g ) p ™ V p , is the only difference between the m-dependence of A T,g ) p ™ V p and A L ,g ) p ™ V p . A priori, the transverse and longitudinal Žstrong-interaction. diffraction-dissociation amplitudes TT r L w V p ™ VN px may possess different m-dependences Ž p L / p T in Ž8.., thus affecting the ratio m2V ,Lrm2V ,T . Integration of Ž13. yields
(
A L ,g ) p ™ V p Ž W 2 ,Q 2 ,t s 0 .
p m2V ,L
sjV
y
2 Q m2V ,L 2 Q
™ 23 j V
(Q
2
jV
(Q
2
m V ,L
(Q
2
ŽQ
2
q m2V ,L
.
Q2
2
Ag p ™ V p Ž W ,t s 0 .
for Q 2 ™ 0
s T ,g ) p ™ V p Ž W 2 ,Q 2 .
ŽQ
2
2 q m 2V ,T
.
sg p ™ V p Ž W 2 . .
p m2V ,L 2
m 3V ,L
y
(Q
4
2
Q2 y
Ž Q 2 q m2V ,L .
Q2
™ 49 j V2
m2V ,L
Q2
arctan
m V ,L
(Q
2
2
for Q 2 ™ 0
m4V ,L
j V2
m 2V ,L
m 4V ,T
for Q 2 ™ ` .
Ž 16 .
sg ) p ™ V p Ž W 2 ,Q 2 . ' s T ,g ) p ™ V p q e s L ,g ) p ™ V p
for Q ™ ` .
The above predictions for transverse and longitudinal production amplitudes are valid for high-energy Ž x s Q 2rŽW 2 q Q 2 . < 1. forward Ž t , 0. production. It would be preferable to compare the predictions with forward-production data, thus eliminating the influence of a possible Q 2 dependence of the slope of the t-distribution. No reliable data for forward production have been extracted from the experiments so far. Accordingly, in order to be able to compare at all with data available at present, we ignore a possible Q 2 dependence of the t-distribution by putting bŽ0.rbŽ Q 2 . , 1, where b is the slope parameter in the t-distribution, expŽyb < t <.. From Ž9., the transverse production cross-section integrated over t then becomes
m 4V ,T
m 4V ,T
j V2
s s T ,g ) p ™ V p Ž 1 q e R V Ž W 2 ,Q 2 . .
2
Ž 14 .
s
2
The approach to the large-Q 2 limit is rather slow, but note the enhancement factor Ž m V ,Lrm V ,T . 4 in Ž16.. For completeness, we quote also the total virtual-photon cross-section and its asymptotic limit
2
m2V ,L
s
™
Ag p ™ V p Ž W 2 ,t s 0 .
Ag p ™ V p Ž W 2 ,t s 0 .
Ž Q 2 q m2V ,T .
p2
m 3V ,L
y
arctan
m V ,L
p ™
2
From Ž14. and Ž15. we obtain for the longitudinalto-transverse ratio R V s L ,g ) p ™ V p R V Ž W 2 ,Q 2 . s s T ,g ) p ™ V p
Ž 15 .
™
m4V ,T Q4
ž
p2 1qe
Ž Q 2 ™ `. .
4
j V2
m4V ,L m 4V ,T
/
sg p ™ V p Ž W 2 .
Ž 17 .
We note that our simple ansatz for diffraction dissociation does not lead to the change of the W dependence of r 0-meson production with increasing Q 2 for which there is some experimental indication w21,22x. Such an effect can be incorporated into GVD by modifying the W dependence of diffraction dissociation. In essence this amounts to replacing Ag p ™ V p ŽW 2 ,t s 0. on the right-hand-sides of Ž12. and Ž13. by an appropriate ŽRegge. ansatz in terms of W 2rm 2 . We have found that this modification may lead to the change in the W dependence with increasing Q 2 indicated by the data w21,22x on r 0 production. A detailed discussion is beyond the scope of the present note. A remark on SCHC is appropriate at this point. From photoproduction measurements at lower energies it is known w5x that SCHC is not strictly valid. It is violated Žat non-zero t . at the level of approximately 10%. In vector dominance this amount of
D. Schildknecht et al.r Physics Letters B 449 (1999) 328–338
helicity-flip contributions is traced back to helicityflip contributions in diffractive hadron reactions which occur at approximately the same rate. Extrapolating the results from lower to the higher HERA energies quantitatively may seem somewhat problematic. If helicity-flip contributions are associated with non-pomeron exchange, one may expect such contributions to be diminished with increasing energy, at least as long as Q 2 is kept fixed. The situation may be different, if W and Q 2 are increased at the same rate, keeping x f Q 2rW 2 small and constant. In this case very massive vector states become important with increasing Q 2 and contributions different from helicity-conserving pomeron exchange may still be present in the limit of very high energies. If an average over a large range in Q 2 is performed, one may accordingly expect helicity flip contributions to be present at HERA energies at a rate similar to the one found at lower energies. In the comparison of our predictions with experiment, we proceed in two steps. In a first step we consider the experimental evidence for the validity of SCHC, before we turn to a comparison of Ž15. – Ž17. with HERA and Fermilab data. The validity of SCHC is not only of interest in itself, due to the presence of the longitudinal degree of freedom of the virtual photon in electroproduction, but is as well a prerequisite for the determination of R V , as long as data are lacking for a direct separation of s T,g ) p ™ V p and s L ,g ) p ™ V p . A recent measurement by the ZEUS collaboration w23x of the full set of 15 density matrix elements determining the vector-meson Ž r 0 and f . decay distribution w24x can be analyzed in terms of helicity-conserving and helicity-flip amplitudes. Using parity invariance as well as natural-parity exchange, the number of independent helicity amplitudes determining the density matrix elements can be reduced to ten. This number is reduced to five, if nucleon helicity-flip amplitudes are assumed to vanish. The normalized density matrix elements, accordingly, depend on four ratios of amplitudes, if one takes the amplitudes to be purely imaginary. In the notation M Ž lg , lr . for g ) p ™ r 0 p, these ratios of helicity amplitudes can be chosen as M Ž00.rM Žqq ., M Žq0.rM Žqq ., M Žqy .r M Žqq ., and M Ž0 q .rM Ž00.. The first one of the above ratios measures the relative magnitude of lon-
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gitudinal and transverse helicity-conserving transitions. The other three ratios quantify helicity-nonconservation. The four ratios can be determined w25x from the measured r 0 density-matrix elements. According to w25x, violations of SCHC are small and of the order of magnitude measured in photoproduction at the lower energy of W s 9.4 GeV, where w5 x M Žq0.rM Žqq . s 0.14 " 0.02 and M Žqy .r M Žqq . s y0.05 " 0.02. As a consequence, the longitudinal-to-transverse production ratio
Rr s
1q
< M Ž 00 . < 2 < M Ž qq . < 2
1q
2 < M Ž 0 q. < 2 < M Ž 00 . < 2
< M Ž q0 . < 2 < M Ž qq . < 2
q
< M Ž qy . < 2 < M Ž qq . < 2
Ž 18 . is found to be well approximated by Rr ,
< M Ž 00 . < 2 < M Ž qq . < 2
1 ,
04 r 00
04 e 1 y r 00
,
Ž 19 .
04 i.e. by the SCHC expressions for Rr . Here r 00 0 denotes the r density matrix element solely determined by the dependence on the polar-angle of the r 0 ™ pq py decay distribution. The result w25x for the four ratios of the helicity amplitudes in Ž18., obtained from the nine measured density matrix elements, thus justifies the widely used procedure Ž19. of determining Rr from the r 0 decay distribution under the assumption of helicity conservation. The experimental results on Rr to be given below are based on this procedure. We now turn to the Q 2 dependence and compare predictions Ž15. – Ž17. with experimental data from HERA w22,26x at an average g ) p c.m. energy of W s 80 GeV Ž50 GeV. for f Ž r 0 . production 9. For a given vector meson V, our predictions depend on four parameters, the two effective vector-meson masses m V ,T and m V ,L , the ratio j V of the longitudinal-to-transverse strong-interaction amplitudes, and the photoproduction cross-section, i.e. Ž15. at Q 2 s 0. The solid lines in Figs. 1–3 show the result of a simultaneous four-parameter fit to the data for
9
At HERA energies, we may take the polarization parameter e s1.
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and sg p ™ r p s 11.1 mb, sg p ™ f p s 1.2 mb. The statistical errors in the parameters are small compared with the estimated systematic ones. The quality of the fits strongly supports the underlying picture: the propagation of hadronic spin-1
Fig. 1. Data for sg ) p ™ f p Žin Ža.. and for sg ) p ™ r p Žin Žb.. from HERA compared with the GVD prediction Ž17.. Solid lines: Four-parameter fit with the values Ž20. of the fit parameters. Dashed line: Two-parameter fit with the values Ž21. of the fit parameters.
sg ) p ™ V p and R V , performed separately for the r 0 and the f meson. The data are well described by the fits, with the parameters jr s 1.06 ,
mr2 ,T s 0.68 mr2 ,
mr2 ,L s 0.71 mr2 ,
jf s 0.90 ,
mf2 ,T s 0.41 mf2 ,
mf2 ,L s 0.57 mf2 ,
Ž 20 .
Fig. 2. Ža. GVD prediction Ž16. for the longitudinal-to-transverse ratio Rf . Solid line: Four-parameter fit with the values Ž20. of the fit parameters. Dashed line: Two-parameter fit with the values Ž21. of the fit parameters. Žb. Data for f production by transversely polarized photons, s T,g ) p ™ f p , extracted from the measured values of sg ) p ™ f p by using the two-parameter Rf fit shown in Ža.. Dashed line: GVD prediction Ž15. with the twoparameter fit values Ž21.. Dotted line: VDM prediction, i.e. Ž15. with mf ,T ' mf .
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The values of R V obtained in the fit seem somewhat low with respect to the central values of the data at large Q 2 . This is of course merely a consequence of the fact that the large-Q 2 R V data hardly contribute to the overall x 2 , owing to their large errors. Varying the four fit-parameters within one standard deviation from their best-fit values, we find that a considerable spread in R V is allowed. In other words, with current data a precision determination of our parameters is not yet possible. In fact, a twoparameter fit results in a similar x 2 Ždashed lines in Figs. 1–3. as the four-parameter fit. In the twoparameter fit, obtained by fixing j V s 1 and m2V ,L s 1.5 m2V ,T Žcorresponding to an asymptotic value R V ™ 5.5., we find mr2 ,T s 0.62 mr2 ,
Fig. 3. As Fig. 2, but for r 0-meson production including HERA data for Rr extracted from the r-decay distribution using SCHC.
states and destructive interference govern the Q 2 dependence of exclusive electroproduction of vector mesons at small x and arbitrary Q 2 . Both the asymptotic 1rQ 4 behaviour of the cross section, see Ž17., and the flattening of R V , see Ž16., are clearly visible in the data. Moreover, the fitted values Ž20. are in accordance with theoretical expectation. The value of j V , 1, i.e. helicity independence of diffractive vector-meson scattering, is very gratifying indeed. The mass parameters, m V ,T and m V ,L , show the theoretically expected ordering m 2V ,T - m2V ,L - m 2V .
mf2 ,T s 0.40 mf2 ,
Ž 21 .
and sg p ™ r p s 11 mb, sg p ™ f p s 1.0 mb. With respect to the results of the fits given in Ž20. and Ž21. it may be worth quoting the estimate 0.41 m 2V Q m2V ,T Q 0.74 m2V from Ref. w16x, based on a reasonable choice of the diffraction-dissociation parameters in Ž8.. In Figs. 2b and 3b, we show the transverse crosssection, s T,g ) p ™ V p . The data in Figs. 2b and 3b were extracted from the data on sg ) p ™ V p in Fig. 1 with the help of our two-parameter fit 10 for R V . Figs. 2b and 3b demonstrate the dramatic difference at large Q 2 between the data and the GVD prediction Ž15. with m V ,T - m V on the one hand, and the VMD prediction Ž6., or rather Ž15. with m V ,T ' m V , on the other hand. Comparing the dotted VMD predictions in Figs. 2b and 3b for the transverse cross-section s T,g ) p ™ V p with the data for sg ) p ™ V p in Fig. 1a and 1b, one notices that the dotted curves would approximately describe the data for sg ) p ™ V p s s T ,g ) p ™ V p q sL,g ) p ™ V p . This, at first sight paradoxical, coincidence of fits of s T,g ) p ™ V p q s L ,g ) p ™ V p , entirely based on the transverse VMD formula, was in fact observed previously w21,27,28x in fits that vary the
10 No other procedure to extract s T,g ) p ™ V p suggests itself, as the number of data points for R V is very small, and the Q 2 values for sg ) p ™ V p and R V are not identical.
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power of Ž Q 2 q m 2V . at fixed mass m V . Implicitly the fits obviously assume s L ,g ) p ™ V p s 0, and, disregarding the information from vector-meson decay indeed seem to confirm s L ,g ) p ™ V p s 0. This conclusion is inconsistent, however, with the results of the above analysis of the r 0 density matrix elements. This analysis establishes that longitudinal r 0 mesons are almost exclusively produced by longitudinal Žvirtual. photons. The mentioned approximate coincidence of fits based on the VMD formula for s T ,g ) p ™ V p with the data for s T ,g ) p ™ V p q s L ,g ) p ™ V p appears as a numerical accident. We turn to a comparison of our results with the r 0-production data from the Fermilab E665 Collaboration w29x taken at energies 10 - W - 24 GeV and at four-momentum transfers 0.15 - Q 2 - 10 GeV 2 . Fig. 4 shows the E665 data for Rr and for s T r taken from Tables 7 and 13 of Ref. w29x. The theoretical curves are based on the parameters from Ž21., i.e. jr s 1, mr2 ,L s 1.5 mr2 ,T , mr2 ,T s 0.62 mr2 and sg p ™ r p s 11 mb. Recent theoretical work on the electroproduction of vector mesons has been concentrated on attempts to deduce the cross-sections from perturbative w9–13x and non-perturbative w14x QCD. For the production cross-section by transversely polarized vector mesons, the pQCD calculations typically lead to a strong asymptotic decrease, as 1rQ 8 , modified sometimes by additional corrections to become 1rQ 7. It may be argued w11x that the region of Q 2 Q 30 GeV 2 explored at present, in which experiments find a fall-off rather like 1rQ 4 , is not sufficiently asymptotic for pQCD to yield reliable results. Further experiments at still larger values of Q 2 will clarify the issue. As for the longitudinal-to-transverse ratio, R V , pQCD calculations led to the same result of a linear rise in Q 2 as the simple VDM predictions, compare Ž6. and Ž7.. Such a linear rise is always obtained, if electromagnetic current conservation is the only source of the Q 2 dependence of R V . For large Q 2 , this linear rise is in conflict with experimental results. A behaviour of the cross-section for Q 2 4 mr2 , for both the production of longitudinally as well as transversely polarized r 0 mesons, somewhat closer to the experimental data, was obtained in Ref. w12x; the calculation was based on open qq production and parton-hadron duality. It is interesting to note
Fig. 4. As Fig. 3, but for E665 data.
that the resulting cross-sections have a VDM form 11 multiplied by correction factors depending on the scaling variable x. The asymptotic form for R V derived in Ref. w12x has recently been reproduced in a calculation based on r 0- meson wave-functions w13x. In Ref. w13x, also pQCD calculations of the helicity-flip amplitudes have been presented. While the magnitude of the helicity-flip amplitudes is ap-
11
Compare Ž37. and Ž38. in Ref. w12x.
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proximately reproduced, a detailed comparison w25x shows that the x 2 of these predictions is not better than the x 2 for a representation of the density matrix elements under the assumption of SCHC. The coincidence of the relative magnitude of the helicity-flip amplitudes at large Q 2 with the helicity-flip amplitudes in photoproduction and diffractive hadron physics remains unexplained in the pQCD approach. In summary, we have investigated electroproduction of vector mesons in GVD. We have shown that destructive interference between neighbouring vector states naturally leads to the spectral representations ) Ž12. and Ž13. of the Žzero-t . amplitudes for g T,L qp ™ VT,L q p. Both predictions, the asymptotic 1rQ 2 fall-off of the transverse amplitude and the approach of R V towards a constant value, are in good agreement with the experimental data. The expected hierarchy, m2V ,T - m2V ,L - m2V , of the pole masses m V ,T , m V ,L and the helicity independence of the strong-interaction amplitudes Žreflected in j V , 1. strongly support the GVD picture: the propagation of hadronic vector states determines, for arbitrary Q 2 , the Q 2 dependence of vector-meson production by virtual photons in the diffraction region of x , Q 2rW 2 < 1. Moreover, SCHC is experimentally violated at the order of magnitude of 10%, a value also observed in diffractive hadron-hadron scattering and photoproduction at lower energies. Returning to our starting point, the photoproduction sum rules Ž1. and Ž2., the present analysis strengthens their dynamical content, which is to reduce photoproduction to vector-meson-induced reactions. More generally, in conjunction with the experimental observation of states with masses up to about 20 GeV w30x in diffractive production in DIS at small x and up to large Q 2 , the present investigation supports the point of view w31x that propagation and diffractive scattering of hadronic vector states is the basic dynamical mechanism in DIS at small values of the scaling variable.
Acknowledgements It is a pleasure to thank Sandy Donnachie, Teresa Monteiro and Gunter Wolf for useful discussions. ¨
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