- Email: [email protected]
The Q2-dependence of γ*N → Δ transition form factors
_A
~
m ~
Nuclear Physics A663&664 (2000) 405c-408c
ELSEVIER
www.elsevier.nl/locate/npe
The Q2-dependence of "y*N ++ ~ transition form factors S. S. Kamalov
a*
and Shin Nan Yang a
aDepartment of Physics, National Taiwan University, Taipei 10617, Taiwan Recent experimentsll] indicate that the behavior exhibited by the ratios E1+/M 1+ and Sl+/M1+ of the ,*N +-+ Ll transition remain small and negative for Q2 :::; 4.0 GeV 2. It implies that the perturbative QCD (pQCD) is still not applicable at these momentum transfers. In this Talk, we want to show that this data can be understood from the dominance of the pion cloud contribution at low Q2, in both EW 2) and Si~2), as predicted by a dynamical model [2,3] for electromagnetic production of pion, together with a simple scaling assumption for the bare " N Ll form factors. The main feature of the dynamical approach to the pion photo- and electro-production [2,3] is that the unitarity is built in by explicitly including the final state 1r N interaction in the theory, namely, t-matrix is expressed as (1) where v-y" is the transition potential operator for ,*N --t 1rN and, t"N and 90 denote the 1rN t-matrix and free propagator, respectively, with E == W the total energy in the eM frame. In the (3,3) channel the transition potential v-y" consists of two terms
(2) where v~" is the background transition potential which includes Born terms and vector mesons exchange contributions, as described in Ref. [4]. The second term of Eq. (2) corresponds to the contribution of bare Ll. We decompose Eq. (1) in the following way,
(3) where
(4) The advantage of such a decomposition is that all the processes which start with the electromagnetic excitation of the bare .6. are summed up in t~". We evaluate t~" with t"N matrix elements obtained in a meson-exchange model [5]. Note that to make principal value integration associated with v~" convergent, we introduce an off-shell dipole form factor with cut-off parameter A=440 MeV. The gauge invariance, 'Permanent address: Laboratory of Theoretical Physics, JINR Dubna,141980 Moscow region, Russia 0375-9474/00/$ see front matter ~ 2000 Elsevier Science B.Y. All rights reserved. PII 80375-9474(99)00628-4
s.s. Kama/au, S.N
406c
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Yang /Nuclear Physics A663&664 (2000) 405c-408c
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violated due to the off-shell rescattering effects, is restored by the substitution -+ J: -klJk·JB /k 2 , where is the electromagnetic current corresponding to the background contribution v~". and k the photon momentum. For the the .0. resonance contribution t~.., the corresponding multipole amplitudes A~". is evaluated using a Breit-Wigner form, as was done in the isobar model of Ref. [4]'
J:
AI.\(W, Q2) = AI.\(Q2)
{"II.\ r 1.\ M~ !".I.\
MI.\ - W 2 - lMl.\r 1.\
ei
(5)
where ! ".I.\(W) is the usual Breit-Wigner factor describing the decay of the .0. resonance with total width r I.\(W) and physical mass MI.\=I232 MeV. The expressions for !"II.\ , !".I.\ and r 1.\ are taken from Ref. [41. The phase if>(W, Q2) in Eq. (5) is to adjust the phase of A 1.\ to be equal to the corresponding pion-nucleon scattering phase 0(33 ). At the resonance energy, if>(MI.\, Q2) = 0 and it does not affect the Q2 dependence of the electromagnetic vertex. The main parameters in the bare 'Y'N.0. vertex are the _,P's in Eq. (5). For t he magnetic dipole MI.\ and elect ric quadrupole tl.\ transit ions they are related to the convent ional electromagnetic helicity amp litudes Ab2 and A~2 by -1.\(
£
Q2)
=
1 1.\) 2"I( -A 1.\ l / 2 + J3A 3/ 2 .
(6)
At the photon point Q2 = 0, the bare amplitudes MI.\(O) and tl.\(O) of Eq. (5) were extracted from the best fit to the results of the recent analyses of Mainz [6J and VPI group [7J as shown in Fig . 1 by solid curves. The dashed curves denote the contribution from t~". only. The dotted curves represented the K-matrix approximation to t~"., namely, without the principal value integral term included.
S.S. Kamalou S.N Yang/Nuclear Physics A663&664 (2000) 405c-408c
407c
The numerical values for M~ and t~ and the helicity amplitudes, at Q2 = 0, are given in Table. 1. Here we also give" dressed" values obtained using K-matrix approximation for t~1r' One notices that the determined above bare values for the helicity amplitudes, which amount to only about 60% of the corresponding dressed values, are close to the predictions of the constituent quark model (CQM), as was pointed out by Sato and Lee [9]. The large reduction of the helicity amplitudes from the dressed to the bares ones results from the fact that the principal value integral part of t~1r' which represents the effects of the off-shell pion rescattering, contributes approximately for half of the Ml+ as indicated by the dashed curves in Fig. 1. Table 1 Comparison of the "bare" and "dressed" values for the amplitudes A~, At2 and At2 (in 10- 3 GeV- 1/ 2 ) . Amplitudes
"bare"
"dressed"
PDG
M~
158± 2 0.4 ± 0.3 -80±2 -136 ± 3
289± 2 -7±0.4 -134± 2 -256 ± 2
293±8 -4.5 ± 4.2 -140 ± 5 -258 ± 6
t~
At2 At2
We now turn to the Q2 evolution of the multipoles in the (3,3) channel. In the present work, we parametrize the Q2 dependence of the dominant M~ amplitude by
(7) where G D is the nucleon dipole form factor. For the small t~ and S~ amplitudes, following Refs. [4,8], we assume that they have the same Q2 dependence as M~. This is motivated by the scaling law which has been observed for the nucleon form factors. Using the fJ and 1 in Eq. (7) as free parameters, we fit the recent experimental data [1] as well as old one quoted in Ref. [4] on the Q2 dependence of the Mi~2) multi pole or equivalently, the G'M form factor defined by Eq. (24) from Ref. [4]. (so called" Ash" definition [10]). Our result is shown in Fig. 2a. The obtained values for the (3 and 1 parameters are: (3 = 0.44 GeV- 2 and 'Y = 0.38 GeV- 2 . Here the dashed curve corresponds to contribution from the bare Ll, i.e., t~1r of Eq. (4). The results for the ratios REM = Ei~2) 1Mi~2) and R SM = Si~2) 1Mi~2) are shown in Fig. 2b. It is seen that they are in good agreement with the results of the model independent analysis of Ref. [1] up to Q2 as high as 4.0 GeV 2 . Note that since the bare values for the electric and Coulomb excitations are small, the absolute values and shape of these ratios are determined, to a large extent, b~ the pion rescattering contribution. The bare Ll excitation contributes mostly to the Ml~2) multi pole. In summary, we calculate the Q2 dependence of the ratios El+IMl+ and 8 1+1Ml+ in the 1* N ---+ Ll transition, with the use of a dynamical model for electromagnetic production of pions and a simple scaling assumption for the bare 1* N ---+ Ll transition
408c
SS Kamaloo, SN. Yang/Nuclear Physics A663&664 (2000) 405c-408c
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0.6
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-5 r--.J.-ri-'''-:;----_ -10
3.6 4.2
1.8 2.4
3.0
3.6 4.2
Q2 (GeV/c)2
Figure 2. The Q2 dependence of (a) ImMJ~2) with corresponding G Mform factor normalized on the nucleon dipole form factor GD, and (b) ratios REM = El+/Ml+ and RSM = Sl+/Ml+ in the isospin 3/2 channelat W=1232 MeV. The full and dashed curvesare the results obtained with the "dressed" and "bare" IN.6. vertexes, respectively. Experimental data as quoted in Ref. [4]. The new data at Q2=2.8 and 4.0 (GeV/c)2 are from Ref. [1].
form factors. We find that both ratios El+/Ml+ and Sl+/Ml+ remain small and negative for Q2 :::; 4.0 GeV 2 . Our results agree well with the recent measurement of Frolov et a!' [1], but deviate strongly from the predictions of pQCD. Our results indicate that the bare Ll is almost spherical and hence very difficult to be directly excited via electric E2 and Coulomb C2 quardrupole excitations. The experimentally observed E~~2) and S~~2) multipoles are, to a very large extent, saturated by the contribution from pion cloud, i.e., pion rescattering effects. This work is supported in part by the NSC/ROC under the grant no. NSC 88-2112M002-015. REFERENCES 1. V. V. Frolov et at., Phys. Rev. Lett. 82 (1999) 45.
2. 3. 4. 5. 6. 7. 8. 9. 10.
S. N. Yang, J. Phys. G 11 (1985) L205. H. Tanabe and K. Ohta, Phys. Rev. C 31 (1985) 1876. D. Drechsel, O. Hanstein, S. S. Kamalov and L. Tiator, Nucl. Phys. A 645 (1999) 145. C. T. Hung, S. N. Yang and T.-S. H. Lee, J. Phys. G 20 (1994) 1531; C. Lee, S. N. Yang and T.-S. H. Lee, ibid. G 17 (1991) L131. O. Hanstein, D. Drechsel, and 1. Tiator, Nuc!' Phys. A 632 (1998) 561 R. A. Arndt, 1. 1. Strakovsky and R. 1. Workman, Phys. Rev. C 53 (1996) 430 (SP97 solution of the VPI analysis). J. M. Laget, Nucl. Phys. A 488 (1988) 765. T. Sato and T.-S. H. Lee, Phys. Rev. C 54 (1996) 2660; T.-S. H. Lee, in N* Physics, eds. T.-S.H. Lee and W. Roberts, p. 19, (World Scientific, Singapore 1997). W. W. Ash, Phys.Lett. 24B (1967) 165.
~
m ~
Nuclear Physics A663&664 (2000) 405c-408c
ELSEVIER
www.elsevier.nl/locate/npe
The Q2-dependence of "y*N ++ ~ transition form factors S. S. Kamalov
a*
and Shin Nan Yang a
aDepartment of Physics, National Taiwan University, Taipei 10617, Taiwan Recent experimentsll] indicate that the behavior exhibited by the ratios E1+/M 1+ and Sl+/M1+ of the ,*N +-+ Ll transition remain small and negative for Q2 :::; 4.0 GeV 2. It implies that the perturbative QCD (pQCD) is still not applicable at these momentum transfers. In this Talk, we want to show that this data can be understood from the dominance of the pion cloud contribution at low Q2, in both EW 2) and Si~2), as predicted by a dynamical model [2,3] for electromagnetic production of pion, together with a simple scaling assumption for the bare " N Ll form factors. The main feature of the dynamical approach to the pion photo- and electro-production [2,3] is that the unitarity is built in by explicitly including the final state 1r N interaction in the theory, namely, t-matrix is expressed as (1) where v-y" is the transition potential operator for ,*N --t 1rN and, t"N and 90 denote the 1rN t-matrix and free propagator, respectively, with E == W the total energy in the eM frame. In the (3,3) channel the transition potential v-y" consists of two terms
(2) where v~" is the background transition potential which includes Born terms and vector mesons exchange contributions, as described in Ref. [4]. The second term of Eq. (2) corresponds to the contribution of bare Ll. We decompose Eq. (1) in the following way,
(3) where
(4) The advantage of such a decomposition is that all the processes which start with the electromagnetic excitation of the bare .6. are summed up in t~". We evaluate t~" with t"N matrix elements obtained in a meson-exchange model [5]. Note that to make principal value integration associated with v~" convergent, we introduce an off-shell dipole form factor with cut-off parameter A=440 MeV. The gauge invariance, 'Permanent address: Laboratory of Theoretical Physics, JINR Dubna,141980 Moscow region, Russia 0375-9474/00/$ see front matter ~ 2000 Elsevier Science B.Y. All rights reserved. PII 80375-9474(99)00628-4
s.s. Kama/au, S.N
406c
'1
!~
30 20
Yang /Nuclear Physics A663&664 (2000) 405c-408c
...
r-:-~--r-"-""""-""""''---~''''''''''-'''' Re l.l,.''1')
b)
E
7-
0
0
10
H
~ - 1
a I---~-T-'-'---::...-""------~ S -t-2 -;t-10
~
w
~
-20 ........~---I-,ol<:"'--+--t--~+-----+-~+-i . ,
. 30
.
% 20 -t ~
~
0 1--- --:'--:1J--''------'--
-
-
-
• I
---i
S -t -1 w
10
~
0~':"----':""-----':~,-----_"":"::::~=-9
-2
- - - - -:-=-
J:
violated due to the off-shell rescattering effects, is restored by the substitution -+ J: -klJk·JB /k 2 , where is the electromagnetic current corresponding to the background contribution v~". and k the photon momentum. For the the .0. resonance contribution t~.., the corresponding multipole amplitudes A~". is evaluated using a Breit-Wigner form, as was done in the isobar model of Ref. [4]'
J:
AI.\(W, Q2) = AI.\(Q2)
{"II.\ r 1.\ M~ !".I.\
MI.\ - W 2 - lMl.\r 1.\
ei
(5)
where ! ".I.\(W) is the usual Breit-Wigner factor describing the decay of the .0. resonance with total width r I.\(W) and physical mass MI.\=I232 MeV. The expressions for !"II.\ , !".I.\ and r 1.\ are taken from Ref. [41. The phase if>(W, Q2) in Eq. (5) is to adjust the phase of A 1.\ to be equal to the corresponding pion-nucleon scattering phase 0(33 ). At the resonance energy, if>(MI.\, Q2) = 0 and it does not affect the Q2 dependence of the electromagnetic vertex. The main parameters in the bare 'Y'N.0. vertex are the _,P's in Eq. (5). For t he magnetic dipole MI.\ and elect ric quadrupole tl.\ transit ions they are related to the convent ional electromagnetic helicity amp litudes Ab2 and A~2 by -1.\(
£
Q2)
=
1 1.\) 2"I( -A 1.\ l / 2 + J3A 3/ 2 .
(6)
At the photon point Q2 = 0, the bare amplitudes MI.\(O) and tl.\(O) of Eq. (5) were extracted from the best fit to the results of the recent analyses of Mainz [6J and VPI group [7J as shown in Fig . 1 by solid curves. The dashed curves denote the contribution from t~". only. The dotted curves represented the K-matrix approximation to t~"., namely, without the principal value integral term included.
S.S. Kamalou S.N Yang/Nuclear Physics A663&664 (2000) 405c-408c
407c
The numerical values for M~ and t~ and the helicity amplitudes, at Q2 = 0, are given in Table. 1. Here we also give" dressed" values obtained using K-matrix approximation for t~1r' One notices that the determined above bare values for the helicity amplitudes, which amount to only about 60% of the corresponding dressed values, are close to the predictions of the constituent quark model (CQM), as was pointed out by Sato and Lee [9]. The large reduction of the helicity amplitudes from the dressed to the bares ones results from the fact that the principal value integral part of t~1r' which represents the effects of the off-shell pion rescattering, contributes approximately for half of the Ml+ as indicated by the dashed curves in Fig. 1. Table 1 Comparison of the "bare" and "dressed" values for the amplitudes A~, At2 and At2 (in 10- 3 GeV- 1/ 2 ) . Amplitudes
"bare"
"dressed"
PDG
M~
158± 2 0.4 ± 0.3 -80±2 -136 ± 3
289± 2 -7±0.4 -134± 2 -256 ± 2
293±8 -4.5 ± 4.2 -140 ± 5 -258 ± 6
t~
At2 At2
We now turn to the Q2 evolution of the multipoles in the (3,3) channel. In the present work, we parametrize the Q2 dependence of the dominant M~ amplitude by
(7) where G D is the nucleon dipole form factor. For the small t~ and S~ amplitudes, following Refs. [4,8], we assume that they have the same Q2 dependence as M~. This is motivated by the scaling law which has been observed for the nucleon form factors. Using the fJ and 1 in Eq. (7) as free parameters, we fit the recent experimental data [1] as well as old one quoted in Ref. [4] on the Q2 dependence of the Mi~2) multi pole or equivalently, the G'M form factor defined by Eq. (24) from Ref. [4]. (so called" Ash" definition [10]). Our result is shown in Fig. 2a. The obtained values for the (3 and 1 parameters are: (3 = 0.44 GeV- 2 and 'Y = 0.38 GeV- 2 . Here the dashed curve corresponds to contribution from the bare Ll, i.e., t~1r of Eq. (4). The results for the ratios REM = Ei~2) 1Mi~2) and R SM = Si~2) 1Mi~2) are shown in Fig. 2b. It is seen that they are in good agreement with the results of the model independent analysis of Ref. [1] up to Q2 as high as 4.0 GeV 2 . Note that since the bare values for the electric and Coulomb excitations are small, the absolute values and shape of these ratios are determined, to a large extent, b~ the pion rescattering contribution. The bare Ll excitation contributes mostly to the Ml~2) multi pole. In summary, we calculate the Q2 dependence of the ratios El+IMl+ and 8 1+1Ml+ in the 1* N ---+ Ll transition, with the use of a dynamical model for electromagnetic production of pions and a simple scaling assumption for the bare 1* N ---+ Ll transition
408c
SS Kamaloo, SN. Yang/Nuclear Physics A663&664 (2000) 405c-408c
'+ 200
r--....-r-~~--,-~~---.-,~~---,
E" 160
7-
a)
5
~ 120 c:;O
.::-~
::l
~
J
80 40 / 0 I-+-+-t-~-t-+---+-+-+--I-->-+--jf-+-O-+--+--+-j
b)
2 ~
~
-0
-5 -10
t
t::~====:r==~ I-+-+-t---t-+---+-+-+--I-->-+--j~o-+-----l
5 c:;O
O.8
~
I')
>;. c:;
~
---
0.4
O·~.O
0.6
1.2
1.8 2.4 3.0 Q2 (GeV/cf
~Vl
0
1-----:~-----------1
-5 r--.J.-ri-'''-:;----_ -10
3.6 4.2
1.8 2.4
3.0
3.6 4.2
Q2 (GeV/c)2
Figure 2. The Q2 dependence of (a) ImMJ~2) with corresponding G Mform factor normalized on the nucleon dipole form factor GD, and (b) ratios REM = El+/Ml+ and RSM = Sl+/Ml+ in the isospin 3/2 channelat W=1232 MeV. The full and dashed curvesare the results obtained with the "dressed" and "bare" IN.6. vertexes, respectively. Experimental data as quoted in Ref. [4]. The new data at Q2=2.8 and 4.0 (GeV/c)2 are from Ref. [1].
form factors. We find that both ratios El+/Ml+ and Sl+/Ml+ remain small and negative for Q2 :::; 4.0 GeV 2 . Our results agree well with the recent measurement of Frolov et a!' [1], but deviate strongly from the predictions of pQCD. Our results indicate that the bare Ll is almost spherical and hence very difficult to be directly excited via electric E2 and Coulomb C2 quardrupole excitations. The experimentally observed E~~2) and S~~2) multipoles are, to a very large extent, saturated by the contribution from pion cloud, i.e., pion rescattering effects. This work is supported in part by the NSC/ROC under the grant no. NSC 88-2112M002-015. REFERENCES 1. V. V. Frolov et at., Phys. Rev. Lett. 82 (1999) 45.
2. 3. 4. 5. 6. 7. 8. 9. 10.
S. N. Yang, J. Phys. G 11 (1985) L205. H. Tanabe and K. Ohta, Phys. Rev. C 31 (1985) 1876. D. Drechsel, O. Hanstein, S. S. Kamalov and L. Tiator, Nucl. Phys. A 645 (1999) 145. C. T. Hung, S. N. Yang and T.-S. H. Lee, J. Phys. G 20 (1994) 1531; C. Lee, S. N. Yang and T.-S. H. Lee, ibid. G 17 (1991) L131. O. Hanstein, D. Drechsel, and 1. Tiator, Nuc!' Phys. A 632 (1998) 561 R. A. Arndt, 1. 1. Strakovsky and R. 1. Workman, Phys. Rev. C 53 (1996) 430 (SP97 solution of the VPI analysis). J. M. Laget, Nucl. Phys. A 488 (1988) 765. T. Sato and T.-S. H. Lee, Phys. Rev. C 54 (1996) 2660; T.-S. H. Lee, in N* Physics, eds. T.-S.H. Lee and W. Roberts, p. 19, (World Scientific, Singapore 1997). W. W. Ash, Phys.Lett. 24B (1967) 165.