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New shadowing corrections to photon-deuteron total cross sections
Volume 61B, number 3
PHYSICS LETTERS
29 March 1976
NEW SHADOWING CORRECTIONS TO PHOTON-DEUTERON
TOTAL CROSS SECTIONS *
C.A. DOMINGUEZ Departamento de F(sica, Centro de Investigaeidn y de Estudios Avanzados del I.P.N., Apartado Postal 14-740, Mdxieo 14, D.F., Mexico
Received 30 April 1975 The Glauber correction to photon-deuteron total cross sections is reexamined in the light of the Extended Vector Dominance Model. If the photon couples to an infinite number of p-like vector mesons one obtains an extra shadowing correction of about 3 ub at high energy. Some of the consequences of this new contribution are briefly discussed.
It is well known that our present knowledge of photon-neutron scattering comes from measurements on deuterium targets. For this reason many efforts have been devoted in the past [1,2] to study photondeuteron interactions at high energy and the associated problem of the extraction of neutron cross sections from deuterium data. The first approximation to this problem consists in assuming that at high energies the deuteron cross section is given by the sum of the free nucleon cross sections, i.e., OT('Yd) = OT(Tp) + OT(Tn).
(1)
The justification of this assumption lies in the fact that the binding energy of the deuteron (~2.2 MeV) is very small compared to the energy scale in which one is interested (order of GeV). However, one knows that there are corrections to this approximation and that even though they are small they might be of crucial importance in some cases. Consider for example the calculation of the J - 0 neutron fixed pole which is already a very small quantity [3]. Concerning these corrections, the Glauber or shadowing correction [1 ] is perhaps the best known example. It arises from the shadowing between nucleons due to the propagation of a vector meson inside the deuteron. Numerically it turns out to be of the order of 4 - 5 % at high energy. Another correction, calculated by West some time Work supported in part by Consejo Nacional de Ciencia y Tecnologla (Mexico) under Contract # 540.
ago [2], due to the Fermi motion of the nucleons amounts to an extra 4 - 5 % contribution. One can convince oneself of the importance of ihe previous corrections from the following example. A fit to the "world-data" on OT(Tp) and Oy(Tn ) (Glauber correction performed) gives [3], in the asymptotic limit, OT(Tp) ~ (102.6 + 2.0) ~b and OT(Tn ) ~ (95.7 + 3.7)/lb. However, the inclusion of the West correction has the effect of increasing OT(Tn) to OT(Tn) ~ (98.6 + 3.7)/ab and this narrows the difference between OT(Tp) and OT(Tn) (which one would expect to be zero). In terms of statistical significance it has been pointed out [4] that the hypothesis eT(3,p ) = Oy(3'n) has a X2 probability of 0.3% if only the Glauber correction is performed, while it goes up to 5% if the West correction is included. In the present note we shall reexamine the Glauber correction in the light of the Extended Vector Dominance Model (EVDM) [5]. In other words we plan to find how does the infinite set of p-like vector mesons modify the results of the single-rho shadowing correction. This infinite number of heavy vector mesons was first predicted in the context of the Veneziano model and incorporated afterwards into the naive vector (rho) Dominance Model. The successful applications of EVDM have been quite many [5] and moreover there is now experimental confirmation of the existence of at least two of the o-daughters [6]. It is important to mention that the effect of these heavy vector mesons in the Glauber correction has an analogy in hadron-deuteron scattering. In fact, it has been pointed out by Gribov [7] and others [ 8 - 1 1 ] 297
Volume 61B, number 3
PHYSICS LETTERS
29 March 1976
that at high energy there should be an inelastic shadowing effect arising from the propagation of excited states of the projectile. This mechanism is capable of accounting for the breakdown of the Glauber picture at high energy [12]. The standard Glauber correction to eq. (1), obtained from the diagrams of fig. 1, can be written as
[11 OT(Td ) = OT(Tp ) + OT(Tn ) + OG,
(2)
where
°G : _ 4 [ 1
Fig. 1. Schematic diagrams relevant to the shadowing correction. The solid line represents the deuteron.
-- r/2~ do pN) !~1 + ,12 ] ~-~-(TN ~ t=0
v+x/~ ---~0 Xf
(3) dp p S(p) e x p ( - b p 2 ) .
2 - m2 p
In eq. (3), • = Re f, rN_.,oN/Imf, rN~oN, v is the photon energy, e x p ( - b p 2 ) is the m o m e n t u m transfer dependence of the differential cross section, and S(p) is the non-relativistic deuteron form factor, i.e.,
S(p) =f dr [if(r)] 2 exp(ip-r),
(4)
where if(r) is the deuteron wave function in coordinate space. If one allows now the photon to couple to an infinite number of p-like vector mesons, one obtains after a straightforward summation that eq. (3) becomes OG : _ 4 ( 1 -- r/2] do P°N) t=0 \ l +r/2/d-~ ('),N ~
(5) X ~
da(3,N ~ PnN)/dtlt=o
n=O da(TN ~ PoN)/dtlt=O where
~÷~/;,-2-m~, In =f
dp p S(p) exp(-bnp2)
(6)
-m2n and ~7has been assumed to be independent of n. An attractive feature of eq. (3) is that one can calculate o G in a more or less model independent way since the differential cross section is directly measurable (nevertheless, one has to keep in mind that 298
Vector Dominance has been assumed from the very beginning). A-priori this situation does not seem to be true anymore for eqs. (5) and (6) where a knowledge of the Pn photoproduction cross sections is required. However, it has been recently shown [13] within the most general version of EVDM (in which one takes into account diagonal as well as non-diagonal contributions) that the available photoproduction data force these cross sections to have very definite expressions. According to ref. [13] one has that do(TN -~ PnN)/dtl t =0 do(TN~PoN)/dtlt=0
-
1 (l+2n) l+2a'
(7)
where a is a small number (c~ "" 0.1) and the mass spectrum is given by m 2 = m2(1 + 2n), rn o being the mass of the 00(770) vector meson. We wish to emphasize the following two points: (a) it is the photoproduction amplitude for real On" mesons,f(TN -~ PnN), which appears in the shadowing correction, and (b) off-diagonal contributions to this amplitude, arising from the photon leg, have already been taken care of in deriving eq. (7). With regard to the form of the exponent b n we have calculated o G using the following two extreme choices: (a) b n = b o = constant as determined from pphotoproduction data, and (b) the ansatz suggested by Cocho et al. [14]
bn =
A 1 + 2mZ/p ''it-
(8)
Volume 61B, number 3
r
i
PHYSICS LETTERS
I
i
F
i
i
8
3 6 ::k
I
~4
Io,
n e u t r o n fixed pole. One w o u l d e x p e c t a change in the available result [3] since this q u a n t i t y is rather sensitive to small corrections to OT(3'n ). Therefore a recalculation o f this residue w o u l d be o f the o u t m o s t importance due to the theoretical implications that such a fixed pole has. The author wishes to thank Drs. M. Alexanian and G. Cocho for helpful discussions
2
0
29 March 1976
2
I 4
I 6
I 8
I I0
[ tZ
] t4
] t6
t8
References
v(GeV)
Fig. 2. Shadowing corrections versus incident photon energy. Curve (a) is the standard (single-rho) Glauber correction and curve (b) is the result of eqs. (5) and (6). A fit to the Gartenhaus wave function has been used in both cases.
Choosing e x p o n e n t i a l and Gartenhaus + wave functions for the d e u t e r o n the difference b e t w e e n the two choices o f b n were b e l o w 10% indicating that o G is rather insensitive to the particular parametrization o f b n . In any case we shall assign as usual [16] a 20% error to a G which includes theoretical uncertainties (e.g., neglect o f isoscalar c o n t r i b u t i o n s ) as well as experimental errors in the p - p h o t o p r o d u c t i o n cross section. The result o f the present calculation is shown in fig. 2 where we have drawn for comparison the standard Glauber correction. The effect o f the heavy vector mesons in the shadowing correction is thus quite sizeable, e.g., at t~ = 20 G e V t h e y add an extra 60% contribution. As to the consequences o f this n e w correction it is evident that the a s y m p t o t i c difference b e t w e e n OT(3,p) and OT(Tn ) will be n o w reduced due to the increase o f OT(3,n ). In o t h e r w o r d s the statistical significance o f the h y p o t h e s i s OT(?p ) = OT(?n ) (in the a s y m p t o t i c region) is b o u n d to increase. H o w e v e r , a question o f far more interest is h o w will the corrections f o u n d here for OT(Tn ) affect the value o f the residue o f the J = 0 We have used the following fit to the Gartenhaus wave function ~k(r) = const (e - a r - e - c r ) (1 - e-Cr)/r, where a = 0.232 fm -] and c = 1.59 fm -] [15].
[1] R.J. Glauber, Phys. Rev. 100 (1955) 242; V. Franco and R.J. Glauber, Phys. Rev. 142 (1966) 1195; S.J. Brodsky and J. Pumplin, Phys. Rev. 182 (1969) 1794. [2] G.B. West, Phys. Lett. 37B (1971) 509; Ann. Phys. (N.Y.) 74 (1972) 464; W.B. Atwood and G.B. West, Phys. Rev. D7 (1973) 773. [3] C.A. Dominguez, J.F. Gunion and R. Suaya, Phys. Rev. D6 (1972) 1404. [4] D.O. CaldweU et al., Phys. Rev. D7 (1973) 1362. [5] A. Bram6n, E. Etim,and M. Greco, Phys. Lett. 41B (1972) 607; M. Greco, Nucl. Phys. 63B (1973) 398; For a recent review of the subject see: M. Greco, lectures delivered at the International School of Subnuclear Physics, Erice, Sicily (1974). [6] For a review of the present experimental situation see the Rapporteur talks of Ph. Salin and K. Moffeit in: Proc. Sixth Intern. Symp. on Electron and photon interaction at high energies, eds. H. Rollnik and W. Pfeil (North Holland, Amsterdam, 1974). [7] V.N. Gribov, Zh. Eksp. Teor. Fiz. 56 (1969) 892. [8] J. Pumplin and M. Ross, Phys. Rev. Lett. 21 (1968) 1778. [9] G. Alberi and L. Bertocchi, Nuovo Cimento 61A (1969) 201. [10] D.R. Harrington, Phys. Rev. D1 (1970) 260. [11 ] S.A. Gurvits and M.S. Marinov, Phys. Lett. 32B (1970) 55. [12] D.P. Sidhu and C. Quigg, Phys. Rev. D7 (1973) 755. [13] C.A. Dominguez, Off-diagonal extended vector dominance at q2 = 0, Mexico preprint (1975) and Lett Nuovo Cimento (to be published). [14] G. Cocho, M. Gregorio, J. Le6n and P. Rotelli, Nucl. Phys. B78 (1974) 269. [15] M.J. Moravcsik, Nucl. Phys. 7 (1958) 113. [16] W.P. Hesse, Ph.D. thesis, University of California at Santa B~rbara (1971), (unpublished). See also ref. [4].
299
PHYSICS LETTERS
29 March 1976
NEW SHADOWING CORRECTIONS TO PHOTON-DEUTERON
TOTAL CROSS SECTIONS *
C.A. DOMINGUEZ Departamento de F(sica, Centro de Investigaeidn y de Estudios Avanzados del I.P.N., Apartado Postal 14-740, Mdxieo 14, D.F., Mexico
Received 30 April 1975 The Glauber correction to photon-deuteron total cross sections is reexamined in the light of the Extended Vector Dominance Model. If the photon couples to an infinite number of p-like vector mesons one obtains an extra shadowing correction of about 3 ub at high energy. Some of the consequences of this new contribution are briefly discussed.
It is well known that our present knowledge of photon-neutron scattering comes from measurements on deuterium targets. For this reason many efforts have been devoted in the past [1,2] to study photondeuteron interactions at high energy and the associated problem of the extraction of neutron cross sections from deuterium data. The first approximation to this problem consists in assuming that at high energies the deuteron cross section is given by the sum of the free nucleon cross sections, i.e., OT('Yd) = OT(Tp) + OT(Tn).
(1)
The justification of this assumption lies in the fact that the binding energy of the deuteron (~2.2 MeV) is very small compared to the energy scale in which one is interested (order of GeV). However, one knows that there are corrections to this approximation and that even though they are small they might be of crucial importance in some cases. Consider for example the calculation of the J - 0 neutron fixed pole which is already a very small quantity [3]. Concerning these corrections, the Glauber or shadowing correction [1 ] is perhaps the best known example. It arises from the shadowing between nucleons due to the propagation of a vector meson inside the deuteron. Numerically it turns out to be of the order of 4 - 5 % at high energy. Another correction, calculated by West some time Work supported in part by Consejo Nacional de Ciencia y Tecnologla (Mexico) under Contract # 540.
ago [2], due to the Fermi motion of the nucleons amounts to an extra 4 - 5 % contribution. One can convince oneself of the importance of ihe previous corrections from the following example. A fit to the "world-data" on OT(Tp) and Oy(Tn ) (Glauber correction performed) gives [3], in the asymptotic limit, OT(Tp) ~ (102.6 + 2.0) ~b and OT(Tn ) ~ (95.7 + 3.7)/lb. However, the inclusion of the West correction has the effect of increasing OT(Tn) to OT(Tn) ~ (98.6 + 3.7)/ab and this narrows the difference between OT(Tp) and OT(Tn) (which one would expect to be zero). In terms of statistical significance it has been pointed out [4] that the hypothesis eT(3,p ) = Oy(3'n) has a X2 probability of 0.3% if only the Glauber correction is performed, while it goes up to 5% if the West correction is included. In the present note we shall reexamine the Glauber correction in the light of the Extended Vector Dominance Model (EVDM) [5]. In other words we plan to find how does the infinite set of p-like vector mesons modify the results of the single-rho shadowing correction. This infinite number of heavy vector mesons was first predicted in the context of the Veneziano model and incorporated afterwards into the naive vector (rho) Dominance Model. The successful applications of EVDM have been quite many [5] and moreover there is now experimental confirmation of the existence of at least two of the o-daughters [6]. It is important to mention that the effect of these heavy vector mesons in the Glauber correction has an analogy in hadron-deuteron scattering. In fact, it has been pointed out by Gribov [7] and others [ 8 - 1 1 ] 297
Volume 61B, number 3
PHYSICS LETTERS
29 March 1976
that at high energy there should be an inelastic shadowing effect arising from the propagation of excited states of the projectile. This mechanism is capable of accounting for the breakdown of the Glauber picture at high energy [12]. The standard Glauber correction to eq. (1), obtained from the diagrams of fig. 1, can be written as
[11 OT(Td ) = OT(Tp ) + OT(Tn ) + OG,
(2)
where
°G : _ 4 [ 1
Fig. 1. Schematic diagrams relevant to the shadowing correction. The solid line represents the deuteron.
-- r/2~ do pN) !~1 + ,12 ] ~-~-(TN ~ t=0
v+x/~ ---~0 Xf
(3) dp p S(p) e x p ( - b p 2 ) .
2 - m2 p
In eq. (3), • = Re f, rN_.,oN/Imf, rN~oN, v is the photon energy, e x p ( - b p 2 ) is the m o m e n t u m transfer dependence of the differential cross section, and S(p) is the non-relativistic deuteron form factor, i.e.,
S(p) =f dr [if(r)] 2 exp(ip-r),
(4)
where if(r) is the deuteron wave function in coordinate space. If one allows now the photon to couple to an infinite number of p-like vector mesons, one obtains after a straightforward summation that eq. (3) becomes OG : _ 4 ( 1 -- r/2] do P°N) t=0 \ l +r/2/d-~ ('),N ~
(5) X ~
da(3,N ~ PnN)/dtlt=o
n=O da(TN ~ PoN)/dtlt=O where
~÷~/;,-2-m~, In =f
dp p S(p) exp(-bnp2)
(6)
-m2n and ~7has been assumed to be independent of n. An attractive feature of eq. (3) is that one can calculate o G in a more or less model independent way since the differential cross section is directly measurable (nevertheless, one has to keep in mind that 298
Vector Dominance has been assumed from the very beginning). A-priori this situation does not seem to be true anymore for eqs. (5) and (6) where a knowledge of the Pn photoproduction cross sections is required. However, it has been recently shown [13] within the most general version of EVDM (in which one takes into account diagonal as well as non-diagonal contributions) that the available photoproduction data force these cross sections to have very definite expressions. According to ref. [13] one has that do(TN -~ PnN)/dtl t =0 do(TN~PoN)/dtlt=0
-
1 (l+2n) l+2a'
(7)
where a is a small number (c~ "" 0.1) and the mass spectrum is given by m 2 = m2(1 + 2n), rn o being the mass of the 00(770) vector meson. We wish to emphasize the following two points: (a) it is the photoproduction amplitude for real On" mesons,f(TN -~ PnN), which appears in the shadowing correction, and (b) off-diagonal contributions to this amplitude, arising from the photon leg, have already been taken care of in deriving eq. (7). With regard to the form of the exponent b n we have calculated o G using the following two extreme choices: (a) b n = b o = constant as determined from pphotoproduction data, and (b) the ansatz suggested by Cocho et al. [14]
bn =
A 1 + 2mZ/p ''it-
(8)
Volume 61B, number 3
r
i
PHYSICS LETTERS
I
i
F
i
i
8
3 6 ::k
I
~4
Io,
n e u t r o n fixed pole. One w o u l d e x p e c t a change in the available result [3] since this q u a n t i t y is rather sensitive to small corrections to OT(3'n ). Therefore a recalculation o f this residue w o u l d be o f the o u t m o s t importance due to the theoretical implications that such a fixed pole has. The author wishes to thank Drs. M. Alexanian and G. Cocho for helpful discussions
2
0
29 March 1976
2
I 4
I 6
I 8
I I0
[ tZ
] t4
] t6
t8
References
v(GeV)
Fig. 2. Shadowing corrections versus incident photon energy. Curve (a) is the standard (single-rho) Glauber correction and curve (b) is the result of eqs. (5) and (6). A fit to the Gartenhaus wave function has been used in both cases.
Choosing e x p o n e n t i a l and Gartenhaus + wave functions for the d e u t e r o n the difference b e t w e e n the two choices o f b n were b e l o w 10% indicating that o G is rather insensitive to the particular parametrization o f b n . In any case we shall assign as usual [16] a 20% error to a G which includes theoretical uncertainties (e.g., neglect o f isoscalar c o n t r i b u t i o n s ) as well as experimental errors in the p - p h o t o p r o d u c t i o n cross section. The result o f the present calculation is shown in fig. 2 where we have drawn for comparison the standard Glauber correction. The effect o f the heavy vector mesons in the shadowing correction is thus quite sizeable, e.g., at t~ = 20 G e V t h e y add an extra 60% contribution. As to the consequences o f this n e w correction it is evident that the a s y m p t o t i c difference b e t w e e n OT(3,p) and OT(Tn ) will be n o w reduced due to the increase o f OT(3,n ). In o t h e r w o r d s the statistical significance o f the h y p o t h e s i s OT(?p ) = OT(?n ) (in the a s y m p t o t i c region) is b o u n d to increase. H o w e v e r , a question o f far more interest is h o w will the corrections f o u n d here for OT(Tn ) affect the value o f the residue o f the J = 0 We have used the following fit to the Gartenhaus wave function ~k(r) = const (e - a r - e - c r ) (1 - e-Cr)/r, where a = 0.232 fm -] and c = 1.59 fm -] [15].
[1] R.J. Glauber, Phys. Rev. 100 (1955) 242; V. Franco and R.J. Glauber, Phys. Rev. 142 (1966) 1195; S.J. Brodsky and J. Pumplin, Phys. Rev. 182 (1969) 1794. [2] G.B. West, Phys. Lett. 37B (1971) 509; Ann. Phys. (N.Y.) 74 (1972) 464; W.B. Atwood and G.B. West, Phys. Rev. D7 (1973) 773. [3] C.A. Dominguez, J.F. Gunion and R. Suaya, Phys. Rev. D6 (1972) 1404. [4] D.O. CaldweU et al., Phys. Rev. D7 (1973) 1362. [5] A. Bram6n, E. Etim,and M. Greco, Phys. Lett. 41B (1972) 607; M. Greco, Nucl. Phys. 63B (1973) 398; For a recent review of the subject see: M. Greco, lectures delivered at the International School of Subnuclear Physics, Erice, Sicily (1974). [6] For a review of the present experimental situation see the Rapporteur talks of Ph. Salin and K. Moffeit in: Proc. Sixth Intern. Symp. on Electron and photon interaction at high energies, eds. H. Rollnik and W. Pfeil (North Holland, Amsterdam, 1974). [7] V.N. Gribov, Zh. Eksp. Teor. Fiz. 56 (1969) 892. [8] J. Pumplin and M. Ross, Phys. Rev. Lett. 21 (1968) 1778. [9] G. Alberi and L. Bertocchi, Nuovo Cimento 61A (1969) 201. [10] D.R. Harrington, Phys. Rev. D1 (1970) 260. [11 ] S.A. Gurvits and M.S. Marinov, Phys. Lett. 32B (1970) 55. [12] D.P. Sidhu and C. Quigg, Phys. Rev. D7 (1973) 755. [13] C.A. Dominguez, Off-diagonal extended vector dominance at q2 = 0, Mexico preprint (1975) and Lett Nuovo Cimento (to be published). [14] G. Cocho, M. Gregorio, J. Le6n and P. Rotelli, Nucl. Phys. B78 (1974) 269. [15] M.J. Moravcsik, Nucl. Phys. 7 (1958) 113. [16] W.P. Hesse, Ph.D. thesis, University of California at Santa B~rbara (1971), (unpublished). See also ref. [4].
299