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Large-angle pD scattering, the form factor and the NN∗ content of the deuteron
16 February 1976
PHYSICS LETTERS
Volume 60B, number 5
LARGE-ANGLE
pD SCATTERING,
THE NN”
CONTENT
THE FORM
OF THE
FACTOR
AND
DEUTERON*
S.A . GURVITZ Weizmann Institute of Science, Rehovot, Israel
and S.A. RINAT Theoretical Division, Los Alamos Scientific Laboratory. University of California, Los Alamos, New Mexico 87544, USA and Weizmann Institute of Science, Rehovot, Israel Received 28 December 1975 Single scattering and n-exchange are shown to account for all large-q* pD data (kL >, 1.5 GeV/c) our to 9’ < 8.5 factor we predict the deuteron charge form factor which agrees with the recent SLAC data. The agreement invalidates previous estimates of the NN* content of the deuteron. (GeV/c)* . From the extracted body-form
Large-angle pD scattering is usually thought to proceed by an exchange mechanism rather than by multiple scattering which dominates small angle cross sections [l] . However, n-exchange alone falls short of the observed intensities [2]. It has therefore been conjectured that the exchange of excited I= l/2 baryons [2] (or an alternative mechanism [3]) governs pD scattering at large q2 which assumption leads to an estimate of the pN* content of the deuteron [2]. In the following we reopen the discussion of multiple scattering, based on an approach [4] which is free of typical eikonal approximations [S] . The derived amplitudes are expected to be correct out to the largest angles and this has been verified in a comparison of predictions with observed angular distributions of p and rr elastically scattered from D, 3He and 4He [6]. In view of its implications for the form factor and the NN” content of the deuteron, we focus in the following on large-angle pD scattering. We start with two salient remarks (full details can be found in refs. [4,6]). i) In Glauber theory one has for the single and double scattering amplitudes [5] (S is the deuteron body form factor, assumed here to be scalar)
* Work partially performed under the auspices of the U.S. Energy Research and Development Administration.
F~~(E,q2.)=~(~q2){Fpp(E.q2)+Fpn(E,q2)}, (la> F~~(E.q2)aFpp(E.)q2)Fpn(E,fq2). (lb) If pN amplitudes decrease exponentially as they do for small q2 the corresponding cross sections u(l), ~(~1 will also fall and ultimately a(l)@ uc2) [5]. Actually pN cross sections rise again for increasing q2 [7]. One thus has q2 regions where FpN(q2) in eq. (la) increases with q2 while FpN (q2/4) in (1 b) decreases. We thus expect and indeed found that in spite of a falling form factor S, uc2) decreases faster than u(l), and the latter may ultimately dominate [6]. ii) A non-eikonal calculation of multiple scattering amplitudes leads to expressions formally equal to their eikonal limits. For instance the single scattering amplitude F(l) can, like in (la), be shown to factorize into a form factor and a sum of, approximately onshell elementary amplitudes FpN [4]. The latter contain recoil effects which grow with q2 and which have been neglected in (la). As a result, the effective FpN to be used should be evaluated at E’ which differs from E in a q2-dependent manner. In fact, one finds tg941
We have verified that for the reason mentioned
in i), 405
Volume 60B, number 5
PHYSICS LETTERS
16 February 1976
I
kL= 1.7GeV/c
,o,
/
kL--2D3Ge~ ~kL=?_.26/6eV/ckL= 2.87 6eV/c
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8.5
q Z(GeV/c) 2
Fig. 1. Large q2 pD data [ 11 ] with predictions. Drawn and dashed lines have been calculated with respectively the HumberstonWal/ace [12] and an a l t e r n a t i v e form factor (fig. 2). Heavy bars give contributions due to n-exchange at OCM = 180 ° [9] (point indicated by arrow).
IFp(2D)I"¢ IFp(~[ for k L >~ 1.5 GeV/c and sufficiently large angles. We thus find for the cross section due to multiple scattering dOpD(E, q2) dq 2
do(pl)D(E,q2 ) _ _ dq 2
- [Ss2(¼q2) + S~(¼q2)]
(3) X IFpp(E'(q2),q 2) +Fpn(E'(q2),q2)12 . Its calculation requires knowledge of the effective pN amplitudes, which have to be extracted from free pN amplitudes for a range of energies E' > E and further standard scalar and quadrupole deuteron body form factors SS, SQ [1,3]. We stress that the calculation leading to the form factors in (3) employs Dpn vertex functions and not deuteron wave functions; these are only related in the non-relativistic limit [3]. To the cross section (3) we now add the n-exchange contribution as for instance calculated by Sharma, Bhasin and Mitra [9, 10]. This is a permitted procedure since F (1) and F exch differ by a phase ~½ rr. The small exchange cross sections for 0CM = 180 ° equal ~ 9 , 3 and 2/lb for k L = 1.7, 2 and 2.25 GeV/c respectively and fall rapidly with k L . 406
In fig. 1 we compare all available, large-q 2 pD data [ 11 ] with the prediction o (1) + ~xch. The drawn lines correspond to deuteron form factors (fig. 2) as calculated from the Humberston-Wallace wavefunctions [12]. Although totally wrong in magnitude the shape of the spectra is reproduced through the mechanism described in i). We then ask whether the data for all E can be made to fit by means of one choice for S 2 + S 2 *. The dashed form factor in fig. 2 produces the dashed cross section do(1)/dq 2, eq. (3), in fig. 1. It is obvious that when o exch (in fig. (1) shown as a heavy bar for 0CM = 180 °) is added to o (1) close agreement with the pD is obtained except for k L = 1.2 GeV/c. We found numerically that for large q2, 0(2) grows relative to o (1) for decreasing k t . In particular for k L = 1.2 GeV/c the poorly known input prevents a reliable calculation o f F (1), F (2) (IF21 ~ IF 1 I). The extracted deuteron body form factor is related to the corresponding charge form factor as measurable in eD scattering experiments. In the absence of * A similar attempt by Bertocchi and Capella [8] failed due to an i n c o r r e c t treatment of recoil [6].
Volume 60B, number 5 I
[
f
PHYSICS LETTERS I
l I I I I D body form foctor
16 February 1976 10O
,
-
"--
•
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-
~
)
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-r
r~
D Eleclrlcform foclor
~':m~.G~?ls~,@
i0 k IO'Z) ~ _
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162
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•
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164
N
16:
16'
I
i
2
I
2; I
I
i
4 6 qZ(GeV/e) a
I
I
I
exchange current's one has [e.g. 13]
d°eo(E, q2) [.do] Mott
(4)
X [A'(q 2) + B(q 2) {(2+q2/4M2) -2 + tan2(0/2)}], with
an2E,
.
(5)
A measurement of the e.m. deuteron form factors A', B and knowled~ge of the nucleon form factors G p'n thus yields S~+ S~ needed in eq. (3). Conversely, knowledge of the latter predicts A', eq. (5). In fig. 3 we show these predictions for A', using a fit for G p [14] and three different forms for G~/GPE [15] (7" Eq2/aM2)
o G~(q2)/GP(q2) = 1.9131" t 1.913~'(1+5 -67-)-1
2
3
4
q2 (GeV/c)2
8
Fig. 2. Squared D body formfactor calculated with HumberstonWallace wave functions ( - - - ) . Dashed line fits all pD data.
A'= (azo+
~ " - -
z
o,
\\\X
(x o) (X1), (X2)
(6)
After these predictions have been made and circulated in May '75, data have been published which extended the measurements of A' out to q2 = 4(GeV/c)2
Fig. 3. Squared electric D form factor predictions with legend as in fig. 2. X i corresponds to different choices for G~, eq. (6). Data are from ref. [161.
[16]. These data shown in fig. 3, agree with the conjectured behaviour o f A ' for a neutron-form factor lying between XO, X 1 , as seems indeed to be the most reasonable choice. Summarizing we find that large angle pD scattering for k t ~ 1.5 GeV/c appears to be accounted for by single p scattering and normal n-exchange provided the deuteron body form factor has a postulated behaviour. From it one predicts the electro-magnetic form factor out to - 8 . 5 (GeV/c) 2 which prediction essentially agrees with the new SLAC data extending to q2 ~ 4(GeV/c)2. The agreement a fortiori confirms the description of the elastic pD cross section and in particular necessitates a re-evalution of the NN* content of the deuteron wave function, in as much as derived from N* exchange in ND reactions [2]. We finally remark that the available range of energies and detection techniques strongly favour pD scattering over eD scattering as a source of information on form factors for large q2. It may well be that pD data will produce the first tests for the asymptotic behavior of the deuteron form factor. Beyond q2 ~ 2(GeV/c)2 the latter seems already to scale like q-10 as predicted [17].
407
Volume 60B, number 5
PHYSICS LETTERS
References [1] D.R. Harrington, Phys. Rev. Lett. 21 (1968) 1946; V. Franco and R.J. Glauber, Phys. Rev. Lett. 22 (1969) 370. [2] A.K. Kerman and L.S. Kisslinger, Phys. Rev. 180 (1969) 1483. [3] e.g.G. Barry, Ann. of Phys. (N.Y.) 73 (1972) 482. [4] S.A. Gurvitz, Y. Alexander and A.S. Rinat, Ann. of Phys. 94 (1975) to be published. [5] R.J. Glauber, Proc. of Rehovot Conf. on Elementary Particles and Nuclear Physics, NHPC (1968). [6] S.A. Gurvitz, Y. Alexander and A.S. Rinat, Phys. Letters 59B (1975) 22; Ann. of Phys., to be published. [7] NN and ND interactions, Particle Data Group; UCRL20000 NN. [8] See also K. Gabathuler and C. Wilkin, Nucl. Phys. B70 (1974) 215; L. Bertocchi and A. Capella, Nuovo Cim. 51A (1967) 369. [9] J.S. Sharma, V.S. Bhasin and A.N. Mitra, Nucl. Phys. B35 (1971) 466.
408
16 February 1976
[10] J.V. Noble and H.J. Weber, Phys. Lett. 50B (1974) 233. (The calculation reported there contains an error). [ 11 ] k L = 1.2 GeV/c; E.T. Boschitz et al., Phys. Rev. C6 (1972) 457; k L = 1.7 GeV/c; G.W. Bernett et al., Phys. Rev. Lett. 19 (1967) 387; k L = 2.03 GeV/c; E. Colemann et al., Phys. Rev. Lett. 16 (1966) 761; k L = 2.25; 2.87; 3.3 GeV/c; L. Dubal et al., Phys. Rev. D9 (1974) 597. [12] T.W. Humberston and J.B.G. Wallace, Nucl. Phys. A141 (1970) 362. [13] M. Gourdin, Physics Reports l l C (1974) 30. [14] F. IacheUo, A.D. Jackson and A. Lande, Phys. Lett. 43B (1973) 191. [15] S. Galster et al., Nucl. Phys. B32 (1971) 221. [16] R. Arnold et al., Phys. Rev. Letters 35 (1975) 776. [17] D. Amati et al., Phys. Letters 27B (1968) 38; S.J. Brodsky and G.R. Farrar, Phys. Rev. Lett. 31 (1973) 1153; V.A. Matveev, R.M. Marudyan and A.N. Tavkhelidze, Lett. al Nuovo Cim. 7 (1973) 419.
PHYSICS LETTERS
Volume 60B, number 5
LARGE-ANGLE
pD SCATTERING,
THE NN”
CONTENT
THE FORM
OF THE
FACTOR
AND
DEUTERON*
S.A . GURVITZ Weizmann Institute of Science, Rehovot, Israel
and S.A. RINAT Theoretical Division, Los Alamos Scientific Laboratory. University of California, Los Alamos, New Mexico 87544, USA and Weizmann Institute of Science, Rehovot, Israel Received 28 December 1975 Single scattering and n-exchange are shown to account for all large-q* pD data (kL >, 1.5 GeV/c) our to 9’ < 8.5 factor we predict the deuteron charge form factor which agrees with the recent SLAC data. The agreement invalidates previous estimates of the NN* content of the deuteron. (GeV/c)* . From the extracted body-form
Large-angle pD scattering is usually thought to proceed by an exchange mechanism rather than by multiple scattering which dominates small angle cross sections [l] . However, n-exchange alone falls short of the observed intensities [2]. It has therefore been conjectured that the exchange of excited I= l/2 baryons [2] (or an alternative mechanism [3]) governs pD scattering at large q2 which assumption leads to an estimate of the pN* content of the deuteron [2]. In the following we reopen the discussion of multiple scattering, based on an approach [4] which is free of typical eikonal approximations [S] . The derived amplitudes are expected to be correct out to the largest angles and this has been verified in a comparison of predictions with observed angular distributions of p and rr elastically scattered from D, 3He and 4He [6]. In view of its implications for the form factor and the NN” content of the deuteron, we focus in the following on large-angle pD scattering. We start with two salient remarks (full details can be found in refs. [4,6]). i) In Glauber theory one has for the single and double scattering amplitudes [5] (S is the deuteron body form factor, assumed here to be scalar)
* Work partially performed under the auspices of the U.S. Energy Research and Development Administration.
F~~(E,q2.)=~(~q2){Fpp(E.q2)+Fpn(E,q2)}, (la> F~~(E.q2)aFpp(E.)q2)Fpn(E,fq2). (lb) If pN amplitudes decrease exponentially as they do for small q2 the corresponding cross sections u(l), ~(~1 will also fall and ultimately a(l)@ uc2) [5]. Actually pN cross sections rise again for increasing q2 [7]. One thus has q2 regions where FpN(q2) in eq. (la) increases with q2 while FpN (q2/4) in (1 b) decreases. We thus expect and indeed found that in spite of a falling form factor S, uc2) decreases faster than u(l), and the latter may ultimately dominate [6]. ii) A non-eikonal calculation of multiple scattering amplitudes leads to expressions formally equal to their eikonal limits. For instance the single scattering amplitude F(l) can, like in (la), be shown to factorize into a form factor and a sum of, approximately onshell elementary amplitudes FpN [4]. The latter contain recoil effects which grow with q2 and which have been neglected in (la). As a result, the effective FpN to be used should be evaluated at E’ which differs from E in a q2-dependent manner. In fact, one finds tg941
We have verified that for the reason mentioned
in i), 405
Volume 60B, number 5
PHYSICS LETTERS
16 February 1976
I
kL= 1.7GeV/c
,o,
/
kL--2D3Ge~ ~kL=?_.26/6eV/ckL= 2.87 6eV/c
+o, ,S
.¢3
:k
." I0 , "
,
-I
,r
Ill I
I I0
I I :--
f/
fir ~16' I
/~,1 I
s ! )/
Ij /
,/
,,I ,ly
-
ill i t
1
I0 ~
-
::
~l/I
kL= 3...'36eV/¢
/
iI II
/
,
.
10
I?.4'
2
I
.75
I
I,/
325'
r /i
?,'
ill
/
I
35
*1
I~
4.g: ,~0
I
I
~ j
5.0
~i /
I~//
' lY 1
60
[
]
7.0 '
]
Z5
]
I
8.5
q Z(GeV/c) 2
Fig. 1. Large q2 pD data [ 11 ] with predictions. Drawn and dashed lines have been calculated with respectively the HumberstonWal/ace [12] and an a l t e r n a t i v e form factor (fig. 2). Heavy bars give contributions due to n-exchange at OCM = 180 ° [9] (point indicated by arrow).
IFp(2D)I"¢ IFp(~[ for k L >~ 1.5 GeV/c and sufficiently large angles. We thus find for the cross section due to multiple scattering dOpD(E, q2) dq 2
do(pl)D(E,q2 ) _ _ dq 2
- [Ss2(¼q2) + S~(¼q2)]
(3) X IFpp(E'(q2),q 2) +Fpn(E'(q2),q2)12 . Its calculation requires knowledge of the effective pN amplitudes, which have to be extracted from free pN amplitudes for a range of energies E' > E and further standard scalar and quadrupole deuteron body form factors SS, SQ [1,3]. We stress that the calculation leading to the form factors in (3) employs Dpn vertex functions and not deuteron wave functions; these are only related in the non-relativistic limit [3]. To the cross section (3) we now add the n-exchange contribution as for instance calculated by Sharma, Bhasin and Mitra [9, 10]. This is a permitted procedure since F (1) and F exch differ by a phase ~½ rr. The small exchange cross sections for 0CM = 180 ° equal ~ 9 , 3 and 2/lb for k L = 1.7, 2 and 2.25 GeV/c respectively and fall rapidly with k L . 406
In fig. 1 we compare all available, large-q 2 pD data [ 11 ] with the prediction o (1) + ~xch. The drawn lines correspond to deuteron form factors (fig. 2) as calculated from the Humberston-Wallace wavefunctions [12]. Although totally wrong in magnitude the shape of the spectra is reproduced through the mechanism described in i). We then ask whether the data for all E can be made to fit by means of one choice for S 2 + S 2 *. The dashed form factor in fig. 2 produces the dashed cross section do(1)/dq 2, eq. (3), in fig. 1. It is obvious that when o exch (in fig. (1) shown as a heavy bar for 0CM = 180 °) is added to o (1) close agreement with the pD is obtained except for k L = 1.2 GeV/c. We found numerically that for large q2, 0(2) grows relative to o (1) for decreasing k t . In particular for k L = 1.2 GeV/c the poorly known input prevents a reliable calculation o f F (1), F (2) (IF21 ~ IF 1 I). The extracted deuteron body form factor is related to the corresponding charge form factor as measurable in eD scattering experiments. In the absence of * A similar attempt by Bertocchi and Capella [8] failed due to an i n c o r r e c t treatment of recoil [6].
Volume 60B, number 5 I
[
f
PHYSICS LETTERS I
l I I I I D body form foctor
16 February 1976 10O
,
-
"--
•
-
-
~
)
[
-r
r~
D Eleclrlcform foclor
~':m~.G~?ls~,@
i0 k IO'Z) ~ _
\
162
¢k¢ O"
•
6
N\
'dSi
164
N
16:
16'
I
i
2
I
2; I
I
i
4 6 qZ(GeV/e) a
I
I
I
exchange current's one has [e.g. 13]
d°eo(E, q2) [.do] Mott
(4)
X [A'(q 2) + B(q 2) {(2+q2/4M2) -2 + tan2(0/2)}], with
an2E,
.
(5)
A measurement of the e.m. deuteron form factors A', B and knowled~ge of the nucleon form factors G p'n thus yields S~+ S~ needed in eq. (3). Conversely, knowledge of the latter predicts A', eq. (5). In fig. 3 we show these predictions for A', using a fit for G p [14] and three different forms for G~/GPE [15] (7" Eq2/aM2)
o G~(q2)/GP(q2) = 1.9131" t 1.913~'(1+5 -67-)-1
2
3
4
q2 (GeV/c)2
8
Fig. 2. Squared D body formfactor calculated with HumberstonWallace wave functions ( - - - ) . Dashed line fits all pD data.
A'= (azo+
~ " - -
z
o,
\\\X
(x o) (X1), (X2)
(6)
After these predictions have been made and circulated in May '75, data have been published which extended the measurements of A' out to q2 = 4(GeV/c)2
Fig. 3. Squared electric D form factor predictions with legend as in fig. 2. X i corresponds to different choices for G~, eq. (6). Data are from ref. [161.
[16]. These data shown in fig. 3, agree with the conjectured behaviour o f A ' for a neutron-form factor lying between XO, X 1 , as seems indeed to be the most reasonable choice. Summarizing we find that large angle pD scattering for k t ~ 1.5 GeV/c appears to be accounted for by single p scattering and normal n-exchange provided the deuteron body form factor has a postulated behaviour. From it one predicts the electro-magnetic form factor out to - 8 . 5 (GeV/c) 2 which prediction essentially agrees with the new SLAC data extending to q2 ~ 4(GeV/c)2. The agreement a fortiori confirms the description of the elastic pD cross section and in particular necessitates a re-evalution of the NN* content of the deuteron wave function, in as much as derived from N* exchange in ND reactions [2]. We finally remark that the available range of energies and detection techniques strongly favour pD scattering over eD scattering as a source of information on form factors for large q2. It may well be that pD data will produce the first tests for the asymptotic behavior of the deuteron form factor. Beyond q2 ~ 2(GeV/c)2 the latter seems already to scale like q-10 as predicted [17].
407
Volume 60B, number 5
PHYSICS LETTERS
References [1] D.R. Harrington, Phys. Rev. Lett. 21 (1968) 1946; V. Franco and R.J. Glauber, Phys. Rev. Lett. 22 (1969) 370. [2] A.K. Kerman and L.S. Kisslinger, Phys. Rev. 180 (1969) 1483. [3] e.g.G. Barry, Ann. of Phys. (N.Y.) 73 (1972) 482. [4] S.A. Gurvitz, Y. Alexander and A.S. Rinat, Ann. of Phys. 94 (1975) to be published. [5] R.J. Glauber, Proc. of Rehovot Conf. on Elementary Particles and Nuclear Physics, NHPC (1968). [6] S.A. Gurvitz, Y. Alexander and A.S. Rinat, Phys. Letters 59B (1975) 22; Ann. of Phys., to be published. [7] NN and ND interactions, Particle Data Group; UCRL20000 NN. [8] See also K. Gabathuler and C. Wilkin, Nucl. Phys. B70 (1974) 215; L. Bertocchi and A. Capella, Nuovo Cim. 51A (1967) 369. [9] J.S. Sharma, V.S. Bhasin and A.N. Mitra, Nucl. Phys. B35 (1971) 466.
408
16 February 1976
[10] J.V. Noble and H.J. Weber, Phys. Lett. 50B (1974) 233. (The calculation reported there contains an error). [ 11 ] k L = 1.2 GeV/c; E.T. Boschitz et al., Phys. Rev. C6 (1972) 457; k L = 1.7 GeV/c; G.W. Bernett et al., Phys. Rev. Lett. 19 (1967) 387; k L = 2.03 GeV/c; E. Colemann et al., Phys. Rev. Lett. 16 (1966) 761; k L = 2.25; 2.87; 3.3 GeV/c; L. Dubal et al., Phys. Rev. D9 (1974) 597. [12] T.W. Humberston and J.B.G. Wallace, Nucl. Phys. A141 (1970) 362. [13] M. Gourdin, Physics Reports l l C (1974) 30. [14] F. IacheUo, A.D. Jackson and A. Lande, Phys. Lett. 43B (1973) 191. [15] S. Galster et al., Nucl. Phys. B32 (1971) 221. [16] R. Arnold et al., Phys. Rev. Letters 35 (1975) 776. [17] D. Amati et al., Phys. Letters 27B (1968) 38; S.J. Brodsky and G.R. Farrar, Phys. Rev. Lett. 31 (1973) 1153; V.A. Matveev, R.M. Marudyan and A.N. Tavkhelidze, Lett. al Nuovo Cim. 7 (1973) 419.