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Tests of vector-meson dominance relations in the electric Born model for pion photoproduction
PHYSICS
Volume 30B, number 2
IN
THE
TESTS OF ELECTRIC
VECTOR-MESON BORN MODEL
LETTERS
15 September 1969
DOMINANCE RELATIONS FOR PION PHOTOPRODUCTION
*
C. F. CHO and J. J. SAKURAI The Enrico Fermi Institute The University of Chicago,
Received
and the Department Chicago, Illinois,
of Physics, 60637 lJti
7 August 1969
Within the framework of the electric Born model and its hadronic analog we examine ious vector-meson dominance relations for single-pion photoproduction.
According to vector-meson amplitudes for
dominance
(1)
(where yv stands for the isovector r- + P -t &ansv.+
photon) and (2)
n
are related by a numerical factor obtainable from other experiments. In the past several years a number of relations have been proposed to test vector-meson dominance along this line. Because the p meson has a finite mass, it is desirable to examine the effect of m,, f 0 on these relations in some well-defined, calculable model. In this note we test the validity of the various vector-meson dominance relations using a simple model based on one-pion exchange supplemented by s- and u-channel nucleon exchange. Our model is based on the invariant matrix element
+ (Y.k)(Y.E) E*p;+ (Y’E)(Y.k) 2(s-m2)
u-m
2(u-m2)
1
u(P), (3)
which corresponds to one-pion exchange plus one-nucleon exchange with the yv m and pm vertices taken to be of the pure ycc type (‘the electric Born model’). The momenta k, q, fi and p’ are defined by k = photon momentum in (l), p momentum in (2) q = pion momentum * Supported in part by the US Atomic sion.
of var-
p
the
YV + P +rr++n
the validity
Energy Commis-
= initial nucleon momentum in (1)) final nucleon momentum in (2) p’ = final nucleon momentum in (l), initial nucleon momentum in 2). For large s and small 1t 1 the photoproduction cross section can be computed to be
This expression exhibits a sharp forward peak for 1t 1 <, ~2 in agreement with experiment [ 11. (Furthermore, even the magnitude of the observed cross section at 1t 1 = 0 agrees with (4) for a wide range of s; however, the model considerably overestimates the cross section for large ) t I.) The cross section for the p production process (2) with the final p spin-states summed over is given by
The first (second) term in the bracket is due to the production of transversely (longitudinally) polarized p in the helicity frame, and unless 1t 1 is close to zero, the second term is more important than the first term. When we project the transverse state, the rnz dependent term drops out, and we recover the vector-meson dominance relation [2]
(HI ‘d
Pll 0Zp
= (f,/e)2(lky12/1412)c
. m . ($7
YV
(6)
VU.is the 11 element of the p meson denwhere pll sity matrix element in the helicity frame. This means, among other things, that we expect a 119
PHYSICS
Volume 30B, number 2
forward peak in P($(do/dt)P even though the full cross section (5) has a forward dip *. In experimental study of the pion-pion interaction in the p meson region one often assumes the validity of the OPE peripheral formula not only at the pion pole but also in the physical region of p production. In the narrow width approximation this procedure corresponds to squaring just the first term of (3), which leads
LETTERS
15 September 1969
=
(minPl_l/Pll)
to
71
da
(dtp> =
&(9 2
4n
x ‘(
4a
slqlg.m.’
77
+I;
X
.m .
mp+/d2-q[(nzp-PP-t] 2m;
(4 (t - P2)2 (,)
This expression, unlike (5), vanishes at t = 0. Clearly it is unjustified to use the usual peripheral formula in a theory in which the p is coupled to a conserved source **. In practice, however, the difference is large only in the extreme forward direction 1t / <, 2 p2/mg. Recently there has been a great deal of discussion in the literature concerning the frame dependence of vector-meson dominance predictions. In particular vector-meson dominance has been tested in the Donohue-Hogaasen frame [6] in which Re(plO) vanishes [‘I]. In our model we may calculate pll(du/dt)p in various frames to see which frame is to be preferred. We obtain P
* While this work was in progress we became aware of the papers of Harari et al. [3] who argue that a forward peak is not expected in ~11 (do/dOp. According to our model this assertion is correct in the Gottfried-Jackson frame but not in the helicity frame. See eq. (8). This rather surprising result comes about because ~$7) receives a large contri-
(GJ) when we rotate from the Gottbution from p10
fried-Jackson frame into the helicity frame. ** The importance of supplementing the OPE diagram by the nucleon Born term in a theory with a conserved p source was previously emphasized by Achasov et al. [4] and by Horn and Jacob [5]. t The Donohue-Hogaasen frame is obtained from the helicity frame via rotation by angle q about the production normal using [6] tan 2 q = 2&Re
(1k, I2/I
A(yV) =
(d&)/dt),,
- (d&)/dt)
(d&;dt),v
+ (d&‘)/dt&
yv _ 2P2(-k)
(9)
t2 + /L’
It is amusing that this simple model gives A&V) = 1 at t = -p2, in good agreement with experiment [9] $; however, the model considerably underestimates the observed value for A at lar er values of It I . Note also that our is independent of t, which is not (d u?%‘dt),v surprising because a photon polarized in the direction normal to the production plane cannot excite the one-pion exchange term in (3). Coming back to vector-meson dominance, we can verify that the relations [lo] Ab,d
= (P~;/P~))~
(11)
(pCH))/(pCH) - pCH) - pCH) l-1 10 11 00
which has two solutions (depending on whether pl_l/pll is maximized or minimized).
120
in the Donohue-Hogaasen and in the GottfriedJackson frame. These expressions are very dif ferent from the photoproduction cross section (4) multiplied by (fp/c)2 q 12)c.m. . We thus see that no good argument can be advanced for testing vector-meson dominance in frames other than the helicity frame, in agreement with most authors [ 1,8]. Turning now to photoproduction by polarized y rays, we obtain
0 We are indebted to Professor J. D. Jackson teresting communication on this point.
for an in-
Volume
PHYSICS
30B, number 2
LETTERS
15 September
I 1.0_ E, = 4GeV
r E,
E,=
I
1969
I
16GeV
= 16GeV
-
PHOTO
------
HELICITY
. . . . . . . . . . -.. ,,H
----HELICITY . . . . . . . . . . . ,,H 0
(GeV/c)
G
exactly satisfied in our model for large s and small 1t 1 . To summarize, as s -t m with / t I small, the effect of mp # 0 completely disappears in all the vector-meson dominance relations provided we work in the helicity frame. Since experiments are performed at finite energies, it is worth studying the effect of mp f 0 on the cross section and the density matrix without making the approximation s - m. To this end we have attempted to test numerically the vector-meson dominance relations (6), (11) and (12) at various energies. Some typical results are shown in figs. 1 and 2 where the quantities B(e2, s) and BCfp2, s) are defined by are
(13)
2
fP 9 --____ 4a4n
,$,
.
.
1 s
.2
fi
Fig. 1. Test of eq. (6). B(e2,s) and BCfi,s) are defined by eq. (13). DH stands for the Donohue-HBgaasen frame in which p1_1/p11 is maximized (rather than minimized); with this choice the rotation angle @ at 1t lmin is 0 and &?‘rat 4 GeV and 16 GeV respectively.
B($,s).
.I
.
As expected at finite energies the vector-meson dominance relations are no longer satisfied exactly. (The relevant formulas will be given elsewhere.) Notice in particular that the curve for
.3
(GeV/c)
Fig. 2. Test of eq. (11).
p~~)1/p$~) at I&lab)
-_ 4 GeV lies considerably below the curvz for the photoproduction asymmetry parameter A. Analogous calculations are also done in the Donahue-Hogaasen frame; we see once again that there is not much point in testing vector-meson dominance in frames other than the helicity frame. We now briefly comment on the experimental situation. It has been reported by several authors that, although (6) is in good agreement with experiment [ll] with the standard value fE/Jn= 2, both (11) and (12) are badly vio-
lated [9,12] at _S~lab) = 4 GeV. As is clear from fig. 2, our model can accommodate a 40% discrepancy in the asymmetry relation (11) at E(lab) N 4 GeV; however, we find that the relatign (12) for (da(l)/dt) must be satisfied within a few % at the same e%%gy. We therefore feel that within the framework of the model it is difficult to understand the large discrepancy reported in ref. 12. Fortunately a good argument exists for believing that the ~1-1 used in ref. 12 is unreliable [ 131. If our model has something to do with reality, several conclusions can be drawn. (i) Vector-meson dominance must be tested in the helicity frame. 121
Volume 30B. number 2
PHYSICS
(ii) The effect of mp # 0 is important for the production of longitudinally polarized p mesons but disappears completely for transversally polarized p’s as s + 00with 1t / small. (iii) We expect a forward peak in $)(do/dt)o analogous to the forward peak observed in photoproduction. (iv) The usual peripheral formula 7r-p + pan is incomplete. (v) At low energies @tab) N 4 GeV) the asymmetry relation (11) can be violated by as much as 40%. However, the reported failure of vector-meson dominance is more likely due to the poorly determined pr\.
LETTERS
4. 5. 6. 7. 8.
9.
10. 11.
References A. M. Bovarski et al.. Phvs. Rev. Letters 20 (1968) 300; 21 (1968)‘1767; G. Buschhorn et al., Phys. Rev. Letters 1’7 (1966) 1027; 18 (1967) 571. D. S. Beder, Phys. Rev. 149 (1966) 1203. H. Harari and B. Horovitz, Phys. Letters 29B (1969) 314;
12. 13.
*****
122
15 September 1969
Y. Avni and H. Harari, Weizmann Institute preprint. N. N. Achasov, V. I. Belinicher and L. M. Samkov, JETP Letters 6 (1967) 103. D. Horn and M. Jacob, Nuovo Cimento 56A (1968) 83. J. T. Donohue and H. HGgaasen, Phys. Letters 25B (1967) 554. A. Biafas and K. Zalewski, Phys. Letters 28B (1969) 436. C. Iso and H. Yoshii, Ann. Phys. 47 (1968) 424; H. Fraas and D, Schildknecht, Nucl. Phys. B6 (1968) 395; M. LeBellac and G. Plaut, Nice preprint. Chr. Geweniger et al., Phys. Letters 28B (1968) 155; 29B (1969) 41. M. Krammer and D. Schildknecht, Nucl. Phys. B7 (1968) 583. A. Dar et al., Phys. Rev. Letters 20 (1968) 1261; M.Krammer, Phys. Letters 26B (1968) 633; R. Diebold and J. A. Poirier, Phys. Rev. Letters 20 (1968) 1532; I. Derado and Z. G. T. Guiragossian, Phys. Rev. Letters 21 (1968) 1556. R. Diebold and J. A. Poirier, Phys. Rev. Letters 22 (1969) 255 and 906; L. J. Gutay et al., Phys. Rev. Letters 22 (1969) 22. N. N. Biswas et al., Notre Dame preprint.
Volume 30B, number 2
IN
THE
TESTS OF ELECTRIC
VECTOR-MESON BORN MODEL
LETTERS
15 September 1969
DOMINANCE RELATIONS FOR PION PHOTOPRODUCTION
*
C. F. CHO and J. J. SAKURAI The Enrico Fermi Institute The University of Chicago,
Received
and the Department Chicago, Illinois,
of Physics, 60637 lJti
7 August 1969
Within the framework of the electric Born model and its hadronic analog we examine ious vector-meson dominance relations for single-pion photoproduction.
According to vector-meson amplitudes for
dominance
(1)
(where yv stands for the isovector r- + P -t &ansv.+
photon) and (2)
n
are related by a numerical factor obtainable from other experiments. In the past several years a number of relations have been proposed to test vector-meson dominance along this line. Because the p meson has a finite mass, it is desirable to examine the effect of m,, f 0 on these relations in some well-defined, calculable model. In this note we test the validity of the various vector-meson dominance relations using a simple model based on one-pion exchange supplemented by s- and u-channel nucleon exchange. Our model is based on the invariant matrix element
+ (Y.k)(Y.E) E*p;+ (Y’E)(Y.k) 2(s-m2)
u-m
2(u-m2)
1
u(P), (3)
which corresponds to one-pion exchange plus one-nucleon exchange with the yv m and pm vertices taken to be of the pure ycc type (‘the electric Born model’). The momenta k, q, fi and p’ are defined by k = photon momentum in (l), p momentum in (2) q = pion momentum * Supported in part by the US Atomic sion.
of var-
p
the
YV + P +rr++n
the validity
Energy Commis-
= initial nucleon momentum in (1)) final nucleon momentum in (2) p’ = final nucleon momentum in (l), initial nucleon momentum in 2). For large s and small 1t 1 the photoproduction cross section can be computed to be
This expression exhibits a sharp forward peak for 1t 1 <, ~2 in agreement with experiment [ 11. (Furthermore, even the magnitude of the observed cross section at 1t 1 = 0 agrees with (4) for a wide range of s; however, the model considerably overestimates the cross section for large ) t I.) The cross section for the p production process (2) with the final p spin-states summed over is given by
The first (second) term in the bracket is due to the production of transversely (longitudinally) polarized p in the helicity frame, and unless 1t 1 is close to zero, the second term is more important than the first term. When we project the transverse state, the rnz dependent term drops out, and we recover the vector-meson dominance relation [2]
(HI ‘d
Pll 0Zp
= (f,/e)2(lky12/1412)c
. m . ($7
YV
(6)
VU.is the 11 element of the p meson denwhere pll sity matrix element in the helicity frame. This means, among other things, that we expect a 119
PHYSICS
Volume 30B, number 2
forward peak in P($(do/dt)P even though the full cross section (5) has a forward dip *. In experimental study of the pion-pion interaction in the p meson region one often assumes the validity of the OPE peripheral formula not only at the pion pole but also in the physical region of p production. In the narrow width approximation this procedure corresponds to squaring just the first term of (3), which leads
LETTERS
15 September 1969
=
(minPl_l/Pll)
to
71
da
(dtp> =
&(9 2
4n
x ‘(
4a
slqlg.m.’
77
+I;
X
.m .
mp+/d2-q[(nzp-PP-t] 2m;
(4 (t - P2)2 (,)
This expression, unlike (5), vanishes at t = 0. Clearly it is unjustified to use the usual peripheral formula in a theory in which the p is coupled to a conserved source **. In practice, however, the difference is large only in the extreme forward direction 1t / <, 2 p2/mg. Recently there has been a great deal of discussion in the literature concerning the frame dependence of vector-meson dominance predictions. In particular vector-meson dominance has been tested in the Donohue-Hogaasen frame [6] in which Re(plO) vanishes [‘I]. In our model we may calculate pll(du/dt)p in various frames to see which frame is to be preferred. We obtain P
* While this work was in progress we became aware of the papers of Harari et al. [3] who argue that a forward peak is not expected in ~11 (do/dOp. According to our model this assertion is correct in the Gottfried-Jackson frame but not in the helicity frame. See eq. (8). This rather surprising result comes about because ~$7) receives a large contri-
(GJ) when we rotate from the Gottbution from p10
fried-Jackson frame into the helicity frame. ** The importance of supplementing the OPE diagram by the nucleon Born term in a theory with a conserved p source was previously emphasized by Achasov et al. [4] and by Horn and Jacob [5]. t The Donohue-Hogaasen frame is obtained from the helicity frame via rotation by angle q about the production normal using [6] tan 2 q = 2&Re
(1k, I2/I
A(yV) =
(d&)/dt),,
- (d&)/dt)
(d&;dt),v
+ (d&‘)/dt&
yv _ 2P2(-k)
(9)
t2 + /L’
It is amusing that this simple model gives A&V) = 1 at t = -p2, in good agreement with experiment [9] $; however, the model considerably underestimates the observed value for A at lar er values of It I . Note also that our is independent of t, which is not (d u?%‘dt),v surprising because a photon polarized in the direction normal to the production plane cannot excite the one-pion exchange term in (3). Coming back to vector-meson dominance, we can verify that the relations [lo] Ab,d
= (P~;/P~))~
(11)
(pCH))/(pCH) - pCH) - pCH) l-1 10 11 00
which has two solutions (depending on whether pl_l/pll is maximized or minimized).
120
in the Donohue-Hogaasen and in the GottfriedJackson frame. These expressions are very dif ferent from the photoproduction cross section (4) multiplied by (fp/c)2 q 12)c.m. . We thus see that no good argument can be advanced for testing vector-meson dominance in frames other than the helicity frame, in agreement with most authors [ 1,8]. Turning now to photoproduction by polarized y rays, we obtain
0 We are indebted to Professor J. D. Jackson teresting communication on this point.
for an in-
Volume
PHYSICS
30B, number 2
LETTERS
15 September
I 1.0_ E, = 4GeV
r E,
E,=
I
1969
I
16GeV
= 16GeV
-
PHOTO
------
HELICITY
. . . . . . . . . . -.. ,,H
----HELICITY . . . . . . . . . . . ,,H 0
(GeV/c)
G
exactly satisfied in our model for large s and small 1t 1 . To summarize, as s -t m with / t I small, the effect of mp # 0 completely disappears in all the vector-meson dominance relations provided we work in the helicity frame. Since experiments are performed at finite energies, it is worth studying the effect of mp f 0 on the cross section and the density matrix without making the approximation s - m. To this end we have attempted to test numerically the vector-meson dominance relations (6), (11) and (12) at various energies. Some typical results are shown in figs. 1 and 2 where the quantities B(e2, s) and BCfp2, s) are defined by are
(13)
2
fP 9 --____ 4a4n
,$,
.
.
1 s
.2
fi
Fig. 1. Test of eq. (6). B(e2,s) and BCfi,s) are defined by eq. (13). DH stands for the Donohue-HBgaasen frame in which p1_1/p11 is maximized (rather than minimized); with this choice the rotation angle @ at 1t lmin is 0 and &?‘rat 4 GeV and 16 GeV respectively.
B($,s).
.I
.
As expected at finite energies the vector-meson dominance relations are no longer satisfied exactly. (The relevant formulas will be given elsewhere.) Notice in particular that the curve for
.3
(GeV/c)
Fig. 2. Test of eq. (11).
p~~)1/p$~) at I&lab)
-_ 4 GeV lies considerably below the curvz for the photoproduction asymmetry parameter A. Analogous calculations are also done in the Donahue-Hogaasen frame; we see once again that there is not much point in testing vector-meson dominance in frames other than the helicity frame. We now briefly comment on the experimental situation. It has been reported by several authors that, although (6) is in good agreement with experiment [ll] with the standard value fE/Jn= 2, both (11) and (12) are badly vio-
lated [9,12] at _S~lab) = 4 GeV. As is clear from fig. 2, our model can accommodate a 40% discrepancy in the asymmetry relation (11) at E(lab) N 4 GeV; however, we find that the relatign (12) for (da(l)/dt) must be satisfied within a few % at the same e%%gy. We therefore feel that within the framework of the model it is difficult to understand the large discrepancy reported in ref. 12. Fortunately a good argument exists for believing that the ~1-1 used in ref. 12 is unreliable [ 131. If our model has something to do with reality, several conclusions can be drawn. (i) Vector-meson dominance must be tested in the helicity frame. 121
Volume 30B. number 2
PHYSICS
(ii) The effect of mp # 0 is important for the production of longitudinally polarized p mesons but disappears completely for transversally polarized p’s as s + 00with 1t / small. (iii) We expect a forward peak in $)(do/dt)o analogous to the forward peak observed in photoproduction. (iv) The usual peripheral formula 7r-p + pan is incomplete. (v) At low energies @tab) N 4 GeV) the asymmetry relation (11) can be violated by as much as 40%. However, the reported failure of vector-meson dominance is more likely due to the poorly determined pr\.
LETTERS
4. 5. 6. 7. 8.
9.
10. 11.
References A. M. Bovarski et al.. Phvs. Rev. Letters 20 (1968) 300; 21 (1968)‘1767; G. Buschhorn et al., Phys. Rev. Letters 1’7 (1966) 1027; 18 (1967) 571. D. S. Beder, Phys. Rev. 149 (1966) 1203. H. Harari and B. Horovitz, Phys. Letters 29B (1969) 314;
12. 13.
*****
122
15 September 1969
Y. Avni and H. Harari, Weizmann Institute preprint. N. N. Achasov, V. I. Belinicher and L. M. Samkov, JETP Letters 6 (1967) 103. D. Horn and M. Jacob, Nuovo Cimento 56A (1968) 83. J. T. Donohue and H. HGgaasen, Phys. Letters 25B (1967) 554. A. Biafas and K. Zalewski, Phys. Letters 28B (1969) 436. C. Iso and H. Yoshii, Ann. Phys. 47 (1968) 424; H. Fraas and D, Schildknecht, Nucl. Phys. B6 (1968) 395; M. LeBellac and G. Plaut, Nice preprint. Chr. Geweniger et al., Phys. Letters 28B (1968) 155; 29B (1969) 41. M. Krammer and D. Schildknecht, Nucl. Phys. B7 (1968) 583. A. Dar et al., Phys. Rev. Letters 20 (1968) 1261; M.Krammer, Phys. Letters 26B (1968) 633; R. Diebold and J. A. Poirier, Phys. Rev. Letters 20 (1968) 1532; I. Derado and Z. G. T. Guiragossian, Phys. Rev. Letters 21 (1968) 1556. R. Diebold and J. A. Poirier, Phys. Rev. Letters 22 (1969) 255 and 906; L. J. Gutay et al., Phys. Rev. Letters 22 (1969) 22. N. N. Biswas et al., Notre Dame preprint.