- Email: [email protected]
ω and ρ photoproduction with an effective quark model Lagrangian
17 September 1998
Physics Letters B 436 Ž1998. 42–48
v and r photoproduction with an effective quark model Lagrangian Qiang Zhao a , Zhenping Li a , C. Bennhold
b
a
b
Department of Physics, Peking UniÕersity, Beijing 100871, PR China Department of Physics, Center for Nuclear Studies, The George Washington UniÕersity, Washington D.C., 20052, USA Received 27 May 1998; revised 8 July 1998 Editor: W. Haxton
Abstract A unified approach to vector meson photoproductions is presented in the constituent quark model. The s- and u-channel resonance contributions are generated using an effective quark vector-meson Lagrangian. Taking into account p 0 and s t-channel exchanges for diffractive production in v and r 0 productions respectively, the available total and differential cross section data for v , r 0 , rq, and ry photoproduction can be well described with the same set of parameters in the quark model. Our results clearly indicate that polarization observables are essential to identify so-called ‘‘missing’’ resonances. q 1998 Elsevier Science B.V. All rights reserved. PACS: 24.85.q p; 12.39.-x; 13.60.Le
One of the main goals of the vector meson photoproduction experiments at TJNAF is to search for so-called ‘‘missing’’ resonances, which have been predicted by theory but have not been established experimentally w1,2x. One possible explanation for this long-standing puzzle is that these states couple weakly to the p N channel, which has provided most information for N ) states so far, but decay strongly into channels like r N and v N. Encouraged by recent successful descriptions of pseudoscalar meson photoproduction w3x we propose the parallel approach to vector meson photoproduction starting with the effective Lagrangian Leff s ycgm p mc q cgm e q A mc
ž
q c agm q
ibsmn q n 2 mq
/
fmmc q . . .
Ž 1.
where e q Ž m q . denote the quark charge Žmass., A m the photon field, and where the quark field c couples directly to the vector meson field 1
fm s
'2
r0q
1
'2
ry K )y
rq
v 1 y
'2
r0q K )0
K )q 1
'2
v
K )0
f
0
Ž 2.
with momentum q n . The coupling constants a and b in Eq. Ž1. allow for the two possible couplings of the quarks to the vector mesons; they are free parameters to be determined by the data. Unlike the large mass difference between the p and h in the pseudoscalar case, the v and r states have nearly equal masses, thus isospin violations for the v and r are relatively small. This encourages us to pursue an unified de-
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 8 9 4 - 6
Q. Zhao et al.r Physics Letters B 436 (1998) 42–48
scription of both v and r photoproduction with a single set of parameters, where the vector mesons couple directly to the quarks inside the baryon. We briefly outline our quark model approach to vector meson photoproduction below; a detailed derivation of the formalism is given in Ref. w4x. Based on the effective Lagrangian in Eq. Ž1., at tree-level there are s-, u- and t- channel contributions, thus the matrix element for the meson photoproductions can be written as Mfi s Mt q Ms q Mu .
Ž 3.
The derivation of the s- and u- channel contributions uses methods similar to previous calculations of pseudoscalar meson photoproduction w3x. We separate the s-channel contributions Ms in Eq. Ž3. into two parts; the s-channel resonances below 2 GeV and those above 2 GeV that are treated as continuum contributions. The electromagnetic transition amplitudes of s-channel baryon resonances and their mesonic decays have been investigated extensively in the quark model w2,5–7x in terms of helicity and the meson decay amplitudes. These transition amplitudes for s-channel resonances below 2 GeV have been translated into the standard helicity amplitudes w8x in Ref. w4x in the harmonic oscillator basis. The framework of vector meson photoproduction in terms of the helicity amplitudes has been thoroughly investigated w8x, and the various observables can be easily evaluated in terms of these amplitudes. The resonances above 2 GeV are treated as degenerate, since little experimental information is available on those states. Qualitatively, these resonances have larger total widths as their masses increases so that they could be effectively treated as background contributions except the high partial wave resonances. Thus, we write the total contribution from all states belonging to the same harmonic oscillator shell in a compact form, using the mass and total width of the high spin states, such as G 17 Ž2190. for the n s 3 harmonic oscillator shell. Such an approach have been proven to be very successful in investigating the structure of baryon resonances below 2 GeV in the pseudoscalar meson photoproductions. The u-channel contributions Mu in Eq. Ž1. include the nucleon, the D resonance for r production, whose transition amplitudes are treated separately, and all other excited states. The excited states are
43
treated as degenerate in this framework, allowing their total contribution to be written in compact form. This is a good approximation since contributions from u-channels resonances are not sensitive to their precise mass positions. The t-channel exchange, Mt , is proportional to the charge of the outgoing mesons and is needed to ensure the gauge invariance of the total transition amplitude. Therefore, from the effective Lagrangian in Eq. Ž1., t-channel vector meson exchange term will be derived in the charged vector meson photoproductions and only the s- and u-channel transitions will contribute to the amplitudes of the neutral vector meson photoproductions. However, in the neutral productions, the s- and u-channel amplitudes generated from the effective Lagrangian in Eq. Ž1. are not sufficient to describe the strong diffractive behavior. Such a phenomenon can be explained as contributions from the large background integral in the Regge trajectory expansion. As discussed in Ref. w9x and also in Ref. w10x and Ref. w11x about the diffraction duality, the non-resonant imaginary background amplitude in the neutral vector meson, such as v and r 0 photoproductions, have resulted in the large difference of the cross sections between the neutral and charged meson photoproductions, and it should be from the large contribution of the Pomeron singularity in the neutral photoproduction from high energies down to the threshold. While the quark model framework gives a very good description of the s- and uchannel resonance contributions, the non-resonance background amplitude from the Pomeron exchange is beyond the present quark model approach. Therefore, a t-channel p 0 exchange for the amplitude for v photoproduction and a s exchange term for the amplitude for r 0 photoproduction are introduced following the suggestions by Friman and Soyeur w12x who showed these two terms play dominant roles in v and r 0 productions respectively, over other meson exchange processes near the threshold in the Vector Meson Dominance ŽVMD. Model. The introduction of the p 0 and s exchanges for v and r 0 production, respectively, leads to two additional parameters, ap 0 and as , associated with the harmonic oscillator strength for the p 0 and s contributions at the corresponding vertices, which are determined by both corresponding meson and baryon wavefunctions. As the structure of pions and
44
Q. Zhao et al.r Physics Letters B 436 (1998) 42–48
s s is very different from those of the vector mesons, one should not expect the resulting parameters ap 0 and as have the same values as those in the vector meson coupling cases. A detailed derivation of the p 0 exchange is given in Ref. w13x. It is worth to note that some experimental observations w14–16x have shown that resonance contributions should have played important roles in the large t region in the differential cross sections, and the large t behavior can not be explained by Friman and Soyeur’s model. However, as will be shown in the following, using the effective Lagrangian proposed in Eq. Ž1., the sand u-channel resonance contributions are generated and the dominant role played by the resonances in the large t region is highlighted. Encouragingly, we have obtained a good description of the available
data with the same set of parameters for the four isospin channels. We assume that the relative strengths and phases of each s-, u- and t-channel term are determined by the quark model wavefunction in the exact SUŽ6. m O Ž3. limit. The masses and decay widths of the s-channel baryon resonances are obtained from the recent particle data group w17x. The quark masses m q and the parameter a for the harmonic oscillator wavefunctions in the quark model are well determined in the quark model, they are m u s m d s 0.33
GeV ,
a s 410 MeV.
Ž 4.
The coupling constants for the p 0 and s exchanges are taken from Ref. w12x. This leaves only the cou-
Fig. 1. The total cross section for Ža.: g p ™ v p, Žb.: g p ™ r 0 p, Žc.: g n ™ ry p, and Žd.: g p ™ rq n. The dotted line in Ža. and Žb. corresponds to the contributions from the transition matrix elements generated from the effective Lagrangian in Eq. Ž1., while the dashed line in Žc. represents to cross section for t F 1.1 GeV 2 . The data in Ža. and Žb. come from Ref. w14xŽtriangle. and Ref. w15xŽsquare.. The data in Žc. were taken with the restriction t F 1.1 GeV 2 given by Ref. w16x, and the data in Žd. come from Ref. w19x.
Q. Zhao et al.r Physics Letters B 436 (1998) 42–48
pling constants a and b, and the parameters ap and as to be determined by the data. Qualitatively, we would expect that ap Ž as . to be smaller than the parameter a s 410 MeV, since it represents the combined form factors for both p NN and vpg Ž s NN and rsg . vertices while the parameter a only corresponds to the form factor for the p NN or v NN Ž r NN . vertex alone. In Fig. 1, we compare total cross section data for g p ™ v p and the three channels in g N ™ r N with our calculations. We did not perform a systematic
45
fitting procedure due to the poor quality of the data. Our study suggests that a s y1.7 , bX s b y a s 2.5 , ap s 300 MeV ,
as s 250 MeV
Ž 5.
leads to good overall agreement with the available data. Our results for the s- and u- channel contributions alone are also shown for these reactions. In general, the contributions from the s- and u-channel
Fig. 2. The differential cross section for Ža.: g p ™ v p at Eg s 1.675 GeV, Žb.: g p ™ r 0 p at Eg s 1.730 GeV, Žc.: g n ™ ry p at Eg s 1.850 GeV, and Žd.: g p ™ rq n at Eg s 1.850. The dotted line in Ža. and Žb. denotes the contributions from the terms generated from the effective Lagrangian in Eq. Ž1.. while the dashed line denotes the contributions from the t-channel exchanges. The experimental data in Ža. and Žb. come from Ref. w14x, and in Žc. come from Ref. w16x.
46
Q. Zhao et al.r Physics Letters B 436 (1998) 42–48
resonances in v and r 0 photoproduction account for only 20 to 40 percent of the total cross section, demonstrating the dominance of diffractive scattering in these processes. Nevertheless, in the case of v photoproduction the quark model result exhibits some resonance structure around 1.7 GeV photon lab energy which comes from the F15 Ž2000. state. A similar structure also appears in r 0 photoproduction, and additional contributions from the F37 Ž1950., F35 Ž1905., P33 Ž1920. and P31Ž1910. resonances leads to a broader structure. Clearly, the presence of diffractive scattering complicates the extraction of the nucleon resonance contributions from the t-channel terms in the case of neutral vector meson photoproduction. Here, the photoproduction of charged vector mesons, ry and rq, presented in Fig. 1-c and 1-d, become very important. In these cases, the diffractive contributions are absent, and therefore, resonance contributions dominate the cross sections.
Therefore, the results for the charged meson production would be more important test to the model, as every term in the charged meson productions are generated by the effective Lagrangian in Eq. Ž1.. Our numerical results for charged r production are in good agreement with the few available data, even though the poor quality of the data has limited more accurate investigations. Note that the cross section for charged r production is smaller by about a factor of three compared to r 0 production. It shows that there really exists a large difference between the cross sections of the charged and neutral r meson photoproductions as analyzed in Ref. w10x and Ref. w11x. Once the t-channel terms are added as described above we also obtain a good description of the more numerous v and r 0 production data. The results for the differential cross sections for v and r production are shown in Fig. 2. We find that the overall agreement with the available data for the
Fig. 3. The target polarization for Ža.: g p ™ v p, Žb.: g p ™ r 0 p, Žc.: g n ™ ry p, and Žd.: g p ™ rq n at Eg s 1.7GeV. The dotted lines show the result without the contribution from the F15 Ž2000..
Q. Zhao et al.r Physics Letters B 436 (1998) 42–48
differential cross sections is quite good as well. As expected, the p 0 and s exchanges are responsible for the small-angle diffractive behavior, while the sand u-channel resonances dominate the large momentum transfer region. We point out that ry and rq production also shows some the diffractive behavior, although the size of the effect is smaller compared to v and r 0 production. This behavior can be explained by t-channel ry or rq exchanges, which are naturally generated by the effective Lagrangian in Eq. Ž1.. The data in the reaction g n ™ ryp are in very good agreement with the quark model predictions, indicating that the quark model wave functions appear to provide the correct relative strengths and phases among the terms in the s-, uand t-channels. While the shapes and magnitudes of the differential cross sections are well reproduced within our approach we find little sensitivity to individual resonances. For example, in the energy region of Eg ; 1.7 GeV, removing the F15 Ž2000. state – one of the ‘‘missing’’ candidates – changes the cross section very little, indicating the differential cross section may not be the ideal experimental observable to study the structure of the baryon resonances. In contrast to the cross sections, the polarization observables show a more dramatic dependence on the presence of the s-channel resonances. As shown in Ref. w18x, these polarization observables are equivalent to the usually used density matrix elements. To illustrate their effects we show, as an example, the target polarization for the four channels in v and r production with and without the contribution from the F15 Ž2000. resonance. We do not expect the quark model in the SUŽ6. m O Ž3. limit to provide a good description of these observables. However, it demonstrates the sensitivity of these observables to the presence of s-channel resonances. Fig. 3 shows that the F15 Ž2000. resonance has the most dramatic impact on the v channel while the effects on the r channels are smaller due to the contributions from the isospin 3r2 resonances, F37 Ž1950., F35 Ž1905., P33 Ž1920. and P31Ž1910., which reduce the significance of the F15 Ž2000. state. This shows that polarization observables are essential in analyzing the role of s-channel resonances. In summary, this investigation presents the first attempt to describe v and r meson photoproduction
47
in a quark model plus diffractive scattering framework. With p 0 and s exchange taken into account, the sizable contribution of the Pomeron singularity in the neutral vector meson photoproductions from high energies down to the threshold has been phenomenologically included, and this suggests that the duality hypothesis constrains the vector meson photoproductions as well. In this framework, the connection between the reaction mechanism and the underlying quark structure of the baryons resonances has been established. With the same set of parameters, we have obtained an overall description of the v , r 0 , rq and ry photoproduction. It shows that the intermediate resonance contributions have played important roles in the v and r meson photoproductions especially in the large t regions. The crucial role played by the polarization observables in determining the s-channel resonance properties is demonstrated. Data on these observables, expected from TJNAF in the near future, should therefore provide new insights into the structure of the resonance F15 Ž2000. as well as other ‘‘missing’’ resonances. Acknowledgements One author ŽZ. Li. acknowledges the hospitality of the Center for Nuclear Studies at The George Washington University. Discussions with F.J. Klein regarding the data are also acknowledged. This work was supported in part by the Chinese Education Commission and the US-DOE grant DE-FG02-95ER40907. References w1x N. Isgur, G. Karl, Phys. Letts. B 72 Ž1977. 109; Phys. Rev. D 23 Ž1981. 817. w2x R. Koniuk, N. Isgur, Phys. Rev. D 21 Ž1980. 1868. w3x Zhenping Li, Ye Hongxing, and Lu Minghui, Phys. Rev. C 56 Ž1997. 1099. w4x Q. Zhao, Z.P. Li, C. Bennhold, nucl-thr9711061, submitted to Phys. Rev. C. w5x L.A. Copley, G. Karl, E. Obryk, Nucl. Phys. B 13 Ž1969. 303; R.P. Feynman, M. Kislinger, F. Ravndal, Phys. Rev. D 3 Ž1971. 2706. w6x Le Yaouanc et al., Hadron Transitions in the Quark Model, ŽGordon and Breach, New York, 1988.; Phys. Rev. D 8 Ž1973. 2223; D 9 Ž1974. 1415. w7x F.E. Close and Zhenping Li, Phys. Rev. D 42 Ž1990. 2194; Zhenping Li, F.E. Close, ibid., Ž1990. 2207.
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Q. Zhao et al.r Physics Letters B 436 (1998) 42–48
w8x M. Pichowsky, C¸ . S¸ avklı, F. Tabakin, Phys. Rev. C 53 Ž1996. 593. w9x P.D.B. Collins, An Introduction to Regge Theory and Highenergy Physics, ŽCambridge University Press, 1977.. w10x P.G.O. Freund, Phys. Rev. Lett. 20 Ž1968. 235. w11x H. Harari, Phys. Rev. Lett. 20 Ž1968. 1395. w12x B. Friman, M. Soyeur, Nucl. Phys. A 100 Ž1996. 477. w13x Q. Zhao, Z.P. Li, C. Bennhold, nucl-thr9803021, submitted to Phys. Rev. C. w14x F.J. Klein, to appear on the Proceedings of the GWrTJNAF Workshop on N ) Physics, ŽWashington, D.C., 1997., F.J. Klein, Ph.D. thesis, University of Bonn Ž1996..
w15x H.R. Crouch et al., Phys. Rev. 155 Ž1967. 1468; Y. Eisenberg et al., Phys. Rev. D 5 Ž1972. 15; Y. Eisenberg et al., Phys. Rev. Lett. 22 Ž1969. 669; D.P. Barber et al., Z. Phys. C 26 Ž1984. 343; J. Ballam et al., Phys. Rev. D 7 Ž1973. 3150; W. Struczinski et al., Nucl. Phys. B 108 Ž1976. 45. w16x P. Benz et al., Nucl. Phys. B 79 Ž1974. 10. w17x E.J. Weinberg et al., Particle Data Group, Phys. Rev. D 54 Ž1996. 1. w18x W.M. Kloet, Wen-Tai Chiang and Frank Tabakin, nuclthr9803042. w19x D.P. Barber et al., Z. Phys. C 2 Ž1979. 1.
Physics Letters B 436 Ž1998. 42–48
v and r photoproduction with an effective quark model Lagrangian Qiang Zhao a , Zhenping Li a , C. Bennhold
b
a
b
Department of Physics, Peking UniÕersity, Beijing 100871, PR China Department of Physics, Center for Nuclear Studies, The George Washington UniÕersity, Washington D.C., 20052, USA Received 27 May 1998; revised 8 July 1998 Editor: W. Haxton
Abstract A unified approach to vector meson photoproductions is presented in the constituent quark model. The s- and u-channel resonance contributions are generated using an effective quark vector-meson Lagrangian. Taking into account p 0 and s t-channel exchanges for diffractive production in v and r 0 productions respectively, the available total and differential cross section data for v , r 0 , rq, and ry photoproduction can be well described with the same set of parameters in the quark model. Our results clearly indicate that polarization observables are essential to identify so-called ‘‘missing’’ resonances. q 1998 Elsevier Science B.V. All rights reserved. PACS: 24.85.q p; 12.39.-x; 13.60.Le
One of the main goals of the vector meson photoproduction experiments at TJNAF is to search for so-called ‘‘missing’’ resonances, which have been predicted by theory but have not been established experimentally w1,2x. One possible explanation for this long-standing puzzle is that these states couple weakly to the p N channel, which has provided most information for N ) states so far, but decay strongly into channels like r N and v N. Encouraged by recent successful descriptions of pseudoscalar meson photoproduction w3x we propose the parallel approach to vector meson photoproduction starting with the effective Lagrangian Leff s ycgm p mc q cgm e q A mc
ž
q c agm q
ibsmn q n 2 mq
/
fmmc q . . .
Ž 1.
where e q Ž m q . denote the quark charge Žmass., A m the photon field, and where the quark field c couples directly to the vector meson field 1
fm s
'2
r0q
1
'2
ry K )y
rq
v 1 y
'2
r0q K )0
K )q 1
'2
v
K )0
f
0
Ž 2.
with momentum q n . The coupling constants a and b in Eq. Ž1. allow for the two possible couplings of the quarks to the vector mesons; they are free parameters to be determined by the data. Unlike the large mass difference between the p and h in the pseudoscalar case, the v and r states have nearly equal masses, thus isospin violations for the v and r are relatively small. This encourages us to pursue an unified de-
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 8 9 4 - 6
Q. Zhao et al.r Physics Letters B 436 (1998) 42–48
scription of both v and r photoproduction with a single set of parameters, where the vector mesons couple directly to the quarks inside the baryon. We briefly outline our quark model approach to vector meson photoproduction below; a detailed derivation of the formalism is given in Ref. w4x. Based on the effective Lagrangian in Eq. Ž1., at tree-level there are s-, u- and t- channel contributions, thus the matrix element for the meson photoproductions can be written as Mfi s Mt q Ms q Mu .
Ž 3.
The derivation of the s- and u- channel contributions uses methods similar to previous calculations of pseudoscalar meson photoproduction w3x. We separate the s-channel contributions Ms in Eq. Ž3. into two parts; the s-channel resonances below 2 GeV and those above 2 GeV that are treated as continuum contributions. The electromagnetic transition amplitudes of s-channel baryon resonances and their mesonic decays have been investigated extensively in the quark model w2,5–7x in terms of helicity and the meson decay amplitudes. These transition amplitudes for s-channel resonances below 2 GeV have been translated into the standard helicity amplitudes w8x in Ref. w4x in the harmonic oscillator basis. The framework of vector meson photoproduction in terms of the helicity amplitudes has been thoroughly investigated w8x, and the various observables can be easily evaluated in terms of these amplitudes. The resonances above 2 GeV are treated as degenerate, since little experimental information is available on those states. Qualitatively, these resonances have larger total widths as their masses increases so that they could be effectively treated as background contributions except the high partial wave resonances. Thus, we write the total contribution from all states belonging to the same harmonic oscillator shell in a compact form, using the mass and total width of the high spin states, such as G 17 Ž2190. for the n s 3 harmonic oscillator shell. Such an approach have been proven to be very successful in investigating the structure of baryon resonances below 2 GeV in the pseudoscalar meson photoproductions. The u-channel contributions Mu in Eq. Ž1. include the nucleon, the D resonance for r production, whose transition amplitudes are treated separately, and all other excited states. The excited states are
43
treated as degenerate in this framework, allowing their total contribution to be written in compact form. This is a good approximation since contributions from u-channels resonances are not sensitive to their precise mass positions. The t-channel exchange, Mt , is proportional to the charge of the outgoing mesons and is needed to ensure the gauge invariance of the total transition amplitude. Therefore, from the effective Lagrangian in Eq. Ž1., t-channel vector meson exchange term will be derived in the charged vector meson photoproductions and only the s- and u-channel transitions will contribute to the amplitudes of the neutral vector meson photoproductions. However, in the neutral productions, the s- and u-channel amplitudes generated from the effective Lagrangian in Eq. Ž1. are not sufficient to describe the strong diffractive behavior. Such a phenomenon can be explained as contributions from the large background integral in the Regge trajectory expansion. As discussed in Ref. w9x and also in Ref. w10x and Ref. w11x about the diffraction duality, the non-resonant imaginary background amplitude in the neutral vector meson, such as v and r 0 photoproductions, have resulted in the large difference of the cross sections between the neutral and charged meson photoproductions, and it should be from the large contribution of the Pomeron singularity in the neutral photoproduction from high energies down to the threshold. While the quark model framework gives a very good description of the s- and uchannel resonance contributions, the non-resonance background amplitude from the Pomeron exchange is beyond the present quark model approach. Therefore, a t-channel p 0 exchange for the amplitude for v photoproduction and a s exchange term for the amplitude for r 0 photoproduction are introduced following the suggestions by Friman and Soyeur w12x who showed these two terms play dominant roles in v and r 0 productions respectively, over other meson exchange processes near the threshold in the Vector Meson Dominance ŽVMD. Model. The introduction of the p 0 and s exchanges for v and r 0 production, respectively, leads to two additional parameters, ap 0 and as , associated with the harmonic oscillator strength for the p 0 and s contributions at the corresponding vertices, which are determined by both corresponding meson and baryon wavefunctions. As the structure of pions and
44
Q. Zhao et al.r Physics Letters B 436 (1998) 42–48
s s is very different from those of the vector mesons, one should not expect the resulting parameters ap 0 and as have the same values as those in the vector meson coupling cases. A detailed derivation of the p 0 exchange is given in Ref. w13x. It is worth to note that some experimental observations w14–16x have shown that resonance contributions should have played important roles in the large t region in the differential cross sections, and the large t behavior can not be explained by Friman and Soyeur’s model. However, as will be shown in the following, using the effective Lagrangian proposed in Eq. Ž1., the sand u-channel resonance contributions are generated and the dominant role played by the resonances in the large t region is highlighted. Encouragingly, we have obtained a good description of the available
data with the same set of parameters for the four isospin channels. We assume that the relative strengths and phases of each s-, u- and t-channel term are determined by the quark model wavefunction in the exact SUŽ6. m O Ž3. limit. The masses and decay widths of the s-channel baryon resonances are obtained from the recent particle data group w17x. The quark masses m q and the parameter a for the harmonic oscillator wavefunctions in the quark model are well determined in the quark model, they are m u s m d s 0.33
GeV ,
a s 410 MeV.
Ž 4.
The coupling constants for the p 0 and s exchanges are taken from Ref. w12x. This leaves only the cou-
Fig. 1. The total cross section for Ža.: g p ™ v p, Žb.: g p ™ r 0 p, Žc.: g n ™ ry p, and Žd.: g p ™ rq n. The dotted line in Ža. and Žb. corresponds to the contributions from the transition matrix elements generated from the effective Lagrangian in Eq. Ž1., while the dashed line in Žc. represents to cross section for t F 1.1 GeV 2 . The data in Ža. and Žb. come from Ref. w14xŽtriangle. and Ref. w15xŽsquare.. The data in Žc. were taken with the restriction t F 1.1 GeV 2 given by Ref. w16x, and the data in Žd. come from Ref. w19x.
Q. Zhao et al.r Physics Letters B 436 (1998) 42–48
pling constants a and b, and the parameters ap and as to be determined by the data. Qualitatively, we would expect that ap Ž as . to be smaller than the parameter a s 410 MeV, since it represents the combined form factors for both p NN and vpg Ž s NN and rsg . vertices while the parameter a only corresponds to the form factor for the p NN or v NN Ž r NN . vertex alone. In Fig. 1, we compare total cross section data for g p ™ v p and the three channels in g N ™ r N with our calculations. We did not perform a systematic
45
fitting procedure due to the poor quality of the data. Our study suggests that a s y1.7 , bX s b y a s 2.5 , ap s 300 MeV ,
as s 250 MeV
Ž 5.
leads to good overall agreement with the available data. Our results for the s- and u- channel contributions alone are also shown for these reactions. In general, the contributions from the s- and u-channel
Fig. 2. The differential cross section for Ža.: g p ™ v p at Eg s 1.675 GeV, Žb.: g p ™ r 0 p at Eg s 1.730 GeV, Žc.: g n ™ ry p at Eg s 1.850 GeV, and Žd.: g p ™ rq n at Eg s 1.850. The dotted line in Ža. and Žb. denotes the contributions from the terms generated from the effective Lagrangian in Eq. Ž1.. while the dashed line denotes the contributions from the t-channel exchanges. The experimental data in Ža. and Žb. come from Ref. w14x, and in Žc. come from Ref. w16x.
46
Q. Zhao et al.r Physics Letters B 436 (1998) 42–48
resonances in v and r 0 photoproduction account for only 20 to 40 percent of the total cross section, demonstrating the dominance of diffractive scattering in these processes. Nevertheless, in the case of v photoproduction the quark model result exhibits some resonance structure around 1.7 GeV photon lab energy which comes from the F15 Ž2000. state. A similar structure also appears in r 0 photoproduction, and additional contributions from the F37 Ž1950., F35 Ž1905., P33 Ž1920. and P31Ž1910. resonances leads to a broader structure. Clearly, the presence of diffractive scattering complicates the extraction of the nucleon resonance contributions from the t-channel terms in the case of neutral vector meson photoproduction. Here, the photoproduction of charged vector mesons, ry and rq, presented in Fig. 1-c and 1-d, become very important. In these cases, the diffractive contributions are absent, and therefore, resonance contributions dominate the cross sections.
Therefore, the results for the charged meson production would be more important test to the model, as every term in the charged meson productions are generated by the effective Lagrangian in Eq. Ž1.. Our numerical results for charged r production are in good agreement with the few available data, even though the poor quality of the data has limited more accurate investigations. Note that the cross section for charged r production is smaller by about a factor of three compared to r 0 production. It shows that there really exists a large difference between the cross sections of the charged and neutral r meson photoproductions as analyzed in Ref. w10x and Ref. w11x. Once the t-channel terms are added as described above we also obtain a good description of the more numerous v and r 0 production data. The results for the differential cross sections for v and r production are shown in Fig. 2. We find that the overall agreement with the available data for the
Fig. 3. The target polarization for Ža.: g p ™ v p, Žb.: g p ™ r 0 p, Žc.: g n ™ ry p, and Žd.: g p ™ rq n at Eg s 1.7GeV. The dotted lines show the result without the contribution from the F15 Ž2000..
Q. Zhao et al.r Physics Letters B 436 (1998) 42–48
differential cross sections is quite good as well. As expected, the p 0 and s exchanges are responsible for the small-angle diffractive behavior, while the sand u-channel resonances dominate the large momentum transfer region. We point out that ry and rq production also shows some the diffractive behavior, although the size of the effect is smaller compared to v and r 0 production. This behavior can be explained by t-channel ry or rq exchanges, which are naturally generated by the effective Lagrangian in Eq. Ž1.. The data in the reaction g n ™ ryp are in very good agreement with the quark model predictions, indicating that the quark model wave functions appear to provide the correct relative strengths and phases among the terms in the s-, uand t-channels. While the shapes and magnitudes of the differential cross sections are well reproduced within our approach we find little sensitivity to individual resonances. For example, in the energy region of Eg ; 1.7 GeV, removing the F15 Ž2000. state – one of the ‘‘missing’’ candidates – changes the cross section very little, indicating the differential cross section may not be the ideal experimental observable to study the structure of the baryon resonances. In contrast to the cross sections, the polarization observables show a more dramatic dependence on the presence of the s-channel resonances. As shown in Ref. w18x, these polarization observables are equivalent to the usually used density matrix elements. To illustrate their effects we show, as an example, the target polarization for the four channels in v and r production with and without the contribution from the F15 Ž2000. resonance. We do not expect the quark model in the SUŽ6. m O Ž3. limit to provide a good description of these observables. However, it demonstrates the sensitivity of these observables to the presence of s-channel resonances. Fig. 3 shows that the F15 Ž2000. resonance has the most dramatic impact on the v channel while the effects on the r channels are smaller due to the contributions from the isospin 3r2 resonances, F37 Ž1950., F35 Ž1905., P33 Ž1920. and P31Ž1910., which reduce the significance of the F15 Ž2000. state. This shows that polarization observables are essential in analyzing the role of s-channel resonances. In summary, this investigation presents the first attempt to describe v and r meson photoproduction
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in a quark model plus diffractive scattering framework. With p 0 and s exchange taken into account, the sizable contribution of the Pomeron singularity in the neutral vector meson photoproductions from high energies down to the threshold has been phenomenologically included, and this suggests that the duality hypothesis constrains the vector meson photoproductions as well. In this framework, the connection between the reaction mechanism and the underlying quark structure of the baryons resonances has been established. With the same set of parameters, we have obtained an overall description of the v , r 0 , rq and ry photoproduction. It shows that the intermediate resonance contributions have played important roles in the v and r meson photoproductions especially in the large t regions. The crucial role played by the polarization observables in determining the s-channel resonance properties is demonstrated. Data on these observables, expected from TJNAF in the near future, should therefore provide new insights into the structure of the resonance F15 Ž2000. as well as other ‘‘missing’’ resonances. Acknowledgements One author ŽZ. Li. acknowledges the hospitality of the Center for Nuclear Studies at The George Washington University. Discussions with F.J. Klein regarding the data are also acknowledged. This work was supported in part by the Chinese Education Commission and the US-DOE grant DE-FG02-95ER40907. References w1x N. Isgur, G. Karl, Phys. Letts. B 72 Ž1977. 109; Phys. Rev. D 23 Ž1981. 817. w2x R. Koniuk, N. Isgur, Phys. Rev. D 21 Ž1980. 1868. w3x Zhenping Li, Ye Hongxing, and Lu Minghui, Phys. Rev. C 56 Ž1997. 1099. w4x Q. Zhao, Z.P. Li, C. Bennhold, nucl-thr9711061, submitted to Phys. Rev. C. w5x L.A. Copley, G. Karl, E. Obryk, Nucl. Phys. B 13 Ž1969. 303; R.P. Feynman, M. Kislinger, F. Ravndal, Phys. Rev. D 3 Ž1971. 2706. w6x Le Yaouanc et al., Hadron Transitions in the Quark Model, ŽGordon and Breach, New York, 1988.; Phys. Rev. D 8 Ž1973. 2223; D 9 Ž1974. 1415. w7x F.E. Close and Zhenping Li, Phys. Rev. D 42 Ž1990. 2194; Zhenping Li, F.E. Close, ibid., Ž1990. 2207.
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