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The differential cross section and polarization in p + p → d + π+
Volume 71 B, number 1
PHYSICS LETTERS
THE DIFFERENTIAL
7 November 1977
CROSS SECTION AND POLARIZATION
IN p + p ~ d + *r+
J.A. NISKANEN
Research Institute for TheoreticalPhysics, Universityof ttelsinki, Helsinki, PTnland Received 24 May 1977 Revised manuscript received 16 August 1977 The formalism originally developed for incorporating N&-configurations into the mechanism for the production of p-wave pions is now extended to s- and d-wave pions. The differential cross section and the polarization asymmetry are calculated for laboratory energies ranging from threshold to 750 MeV.
From several theoretical and phenomenological considerations it is well known that the reaction p + p ~ d + n+ and its inverse are dominated in the energy range 4 0 0 - 7 5 0 MeV by pion p-wave rescattering via the A(1236)-resonance. On the other hand, at threshold s-wave rescattering gives the dominant contribution. Recently two different approaches [1, 2] have been given to incorporate explicit NA-components into the nucleonic configurations. Both o f these calculations concentrate on the total absorption cross section, although the first one also gives an estimate o f the p-wave absorption differential cross section. Furthermore in the literature calculations of the differential cross section and the polarization asymmetry are scarce. In this work we shall use the formalism of ref. [ 1 ] to calculate also these quantities for which experimental data are continuously increasing. In ref. [1] we introduced a coupled channels formalism to incorporate the A-resonance in the reaction amplitude of the p-wave production in p + p ~ d + n+. There the NA-configuration was generated by coupled differential equations containing a tensor-like coupling interaction V2. The potential V2 includes also p-meson exchange and is given by [1,7]
V2=~I~TI.~
2 S12 V(t~r)-
-
V(Ar) \ J I
lah
\P! p A2 _02
+S1..2{VoO.lr)._(A)3Vo(Ar)+2(fo)2p[Vo(Or),_f; _(_~_)53 V0(Ar)]
A z .. 3,u z
J
(1)
tT) 5A _-p' \pl
X (1 - e ' X r ) 2, where p' = [O2 + p(A - M ) I i/2 ~ 1.170 and S[ 1 = 3S 1 .i112.i - S 1"112, Vo(xf) = exp(-x)/x and V(x) = (1 + 3Ix + 3/x2)Vo(x ). Here S and T a r e transition operators which transform a spin-~ object into a spin--} object. The extra short range cut-off parameter A is taken to be 7.6 fm -1 . The pion coupling constants are f2147r = 0.081 and f*2/4Tr = 0.35. The pNA-coupling constants are given [1] by the quark or strong coupling models using for the pNN-coupling constants the values [3] g2/4rr = 0.55, K = 6.6, where)' o = (mo/2M)g o (1 + K ) . To get a correct resonance form for the cross section it is necessary to make the mass of the A complex by adding its width - i P/2. This has the effect of attenuating the contribution from the NA-configurations at and above the resonance, which would otherwise be enormous. The greater success o f the perturbation approach [2] compared with the model 40
Volume 71B, number 1
PHYSICS LETTERS
7 November 1977
m~ sr
d' al,s
r
mb 15 40
"s'°
i
I
tt
0.5
1.0
1.5
I
2.0 q- o~.u
Fig. I. The p-wave total absorption cross section. Solid curve the non-Galilean invariant operator, dashed curve the Galilean invariant operator, and dotted curve "half-Galilean" operator is used. Also shown is the result when the contribution from s- and d-wave pions is added to the non-Galilean cross ruction. The experimental data are the same as in ref. [ 1].
Fig. 2. The differential cross section parameters 7o and 3'2. The notation is the same as in fig. 1. The experimental points are a collection from ref. [6] and the refs. given therein.
given in ref. [ 1 ] is largely due to our using an inappropriate form o f the width, which was too large. In this work we shall use the form given e.g. by Sugawara and von Hippel [4] r - f*2
24n
CA + M)2 _ A2
q3 /.t2
w(q)/2(M+
where q l "~ q(1 ~ ( q ) ) ) , q = r/g is the momentum o f the pion in the final state CM system, A , M and /a the masses o f the A, the nucleon and the pion respectively, a n d f * 2 / 4 n = 0.35 the coupling constant from the A decay vertex. The doubling o f the system o f equations in the complex case sets strong computational limits to the possible number o f channels we are, at present, able to accommodate. In fact in these calculations we have used only two dominant NA-components with L ~< 2. The effect of other channels is found to be rather small. The Hamiltonian operator for the pion production vertex is the already classic Galilean invariant form
H=f--i"{V""~(x)+2~[P't'It(x)+'c'n(x)P]lla where f2/4n = 0.08 I, p is the m o m e n t u m o f the nucleon, and t~ and n the pion conjugate isovector fields. Of
(3)
course, in the case of ANn-vertex f, a and x must be replaced by f * and by the operators S and T introduced by Sugawara and von Hippel [4]. Without taking a stand in the argument about the possible ambiguity o f the last term o f eq. (3), we shall also repeat the calculations with the non-Galilean operator, where the last term is simply omitted. This form has been used in ref. [2]. The s-wave rescattering o f the pion is described by the phenomenological Hamiltonian H s = 4n h l ~'11)+ 4rrh_2 , . ~ X n // /a2
(4) 1
where we take h 1 = - { (,u/q)(51 + 253) and X2 - - g
(la/~(q))~/q)(51 -
63) as given by Goplen [5], approxi41
Volume 71B, number 1
PHYSICS LETTERS
7 November 1977
Table 1 The polarization asymmetry parameters for a series of energies. The experimental points are a collection from the references given in ref. [6]. .
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Experiment Elab (MeV)
"0
425 480 540 600 660 720
1.00 1.22 1.43 1.63 1.81 1.99 .
.
.
.
.
Non-Galilean results ho hi h2 (mb/sr)
Galilean results ho hi
h2
Elab (MeV)
-1.37 1.68 -1.86 -1.54 -0.96 -0.64
0.86 0.80 -0.42 0.12 0.20 0.16
1.78 3.54 6.70 8.80 5.93 3.14
425 462 532 591 654 723
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0.15 -0.02 -0.52 -1.22 -1.26 -0.94 .
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2.34 4.67 8.72 11.34 7.79 4.37 .
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0.00 -0.16 -0.72 1.37 -1.16 -0.60 .
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ho
hi (mb/sr)
-1.43 ± 0.13 -0.79 ± 0.31 0.69 -2-_0.36 3.38±0.43 4.5 ± 0.8 2.6 -+0.8 .
.
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1.3 ± 0.7 2.1 ± 0.6 2.8± 0.7 4.9± 1.5 1.5 -+ 0.9 .
h;z
4.22 6.5 17.1 22.3 26.0 12.4
-* 0.55 ± 2.7 ± 3.0 ±2.3 ± 2.5 -+ 3.3
.
mated by ;k1 = 0.0054 and ~'2 = 0.0445 la/w(q). The propagator is taken to be that of Goplen et al. [5] in the two nucleon case. In the case when the rescattering pion originates from a decaying A, we suppose it to have already its final energy. The conventional way to parametrize the differential cross section and the polarization asymmetry is [6] do I polarized = 321r 1 {(3'0 + 72 c°s20 +"/4 c°s40) +PB "ti sin 0[• 0 + X2 cos20 ('A1 + ~'3 c°s20)] }"
(5)
Here PB is the beam polarization and ti the unit vector in the direction kp X q with kp and q being the momenta o f the incident proton and of the final state pion. Evaluating these parameters requires not only the amplitudes for p-wave production from 1 SO and 1 D2 pp-states but also at least the s- and d-wave production amplitudes from 3P 1 and 3P 2 states. These are the states we consider here. To this order 74 and ~'3 remain zero. A more detailed account o f the formalism together with complete formulae will be presented in a forthcoming paper [8]. Considering the numerical results, we see from figs. 1 and 2 that the non-Galilean operator gives an adequate description o f the cross section, whereas the Galilean invariant one is somewhat too low. The magnitude o f the cross section can be easily adjusted somewhat by altering the coupling potential, by changing the NAp-coupling within its experimental limits or by making the NA-interaction less repulsive than given in the model of ref. [ 1 ]. The values o f the polarization parameter ~'0 in table 1, although not impressively good, seem to favour the Galilean invariant operator. The correct qualitative behaviour arises from the s-wave production amplitude getting its main contribution near threshold from rescattering. Then it decreases and changes its sign for higher energies. It seems clear that the use of some form factors in the interaction H s would be beneficial here. In this way the reaction could give information about these s-wave scattering form factors. The forward-backward asymmetry parameter ~1 is not even qualitatively correct. Perhaps here higher partial waves are needed [8]. Also the troublesome spin-spin part in V2 arises here in the 3P2-state with more weight than in the other partial waves. In conclusion we might argue that the coupled channels formalism seems to give rather a reasonable description o f theprocess p + p ~ d + n ÷, even better than that found in our earlier work of ref. [ 1 ]. I would like to thank Doc. A.M. Green for a number o f discussions and for helpful comments on this work.
[ 1 ] A.M. Green and J.A. Niskanen, Nucl. Phys. A271 (1976) 503. [2] D.O. Riska, M. Brack and W. Weise, Phys. Lett. 61B (1976) 41 [3] G. Hbhler and E. Pietarinen, Nucl. Phys. B95 (1975) 210. [4} H. Sugawara and F. yon Hippel, Phys. Rev. 172 (1968) 1764.
42
[5] B. Goplen, W. Gibbs and E. Lomon, Phys. Rev. Lett. 32 (1974) 1012; B. Goplen, Ph.D. thesis, Los Alamos Scientific Laboratory report LA-5854-T, 1975. [6] D. Aebischer et al., Nucl. Phys. BI08 (1976) 214. [7] J.W. Durso, M. Saarela, G.E. Brown and A.D. Jackson, Nucl. Phys. A278 (1977) 445. [8] J.A. Niskanen, to be published.
PHYSICS LETTERS
THE DIFFERENTIAL
7 November 1977
CROSS SECTION AND POLARIZATION
IN p + p ~ d + *r+
J.A. NISKANEN
Research Institute for TheoreticalPhysics, Universityof ttelsinki, Helsinki, PTnland Received 24 May 1977 Revised manuscript received 16 August 1977 The formalism originally developed for incorporating N&-configurations into the mechanism for the production of p-wave pions is now extended to s- and d-wave pions. The differential cross section and the polarization asymmetry are calculated for laboratory energies ranging from threshold to 750 MeV.
From several theoretical and phenomenological considerations it is well known that the reaction p + p ~ d + n+ and its inverse are dominated in the energy range 4 0 0 - 7 5 0 MeV by pion p-wave rescattering via the A(1236)-resonance. On the other hand, at threshold s-wave rescattering gives the dominant contribution. Recently two different approaches [1, 2] have been given to incorporate explicit NA-components into the nucleonic configurations. Both o f these calculations concentrate on the total absorption cross section, although the first one also gives an estimate o f the p-wave absorption differential cross section. Furthermore in the literature calculations of the differential cross section and the polarization asymmetry are scarce. In this work we shall use the formalism of ref. [ 1 ] to calculate also these quantities for which experimental data are continuously increasing. In ref. [1] we introduced a coupled channels formalism to incorporate the A-resonance in the reaction amplitude of the p-wave production in p + p ~ d + n+. There the NA-configuration was generated by coupled differential equations containing a tensor-like coupling interaction V2. The potential V2 includes also p-meson exchange and is given by [1,7]
V2=~I~TI.~
2 S12 V(t~r)-
-
V(Ar) \ J I
lah
\P! p A2 _02
+S1..2{VoO.lr)._(A)3Vo(Ar)+2(fo)2p[Vo(Or),_f; _(_~_)53 V0(Ar)]
A z .. 3,u z
J
(1)
tT) 5A _-p' \pl
X (1 - e ' X r ) 2, where p' = [O2 + p(A - M ) I i/2 ~ 1.170 and S[ 1 = 3S 1 .i112.i - S 1"112, Vo(xf) = exp(-x)/x and V(x) = (1 + 3Ix + 3/x2)Vo(x ). Here S and T a r e transition operators which transform a spin-~ object into a spin--} object. The extra short range cut-off parameter A is taken to be 7.6 fm -1 . The pion coupling constants are f2147r = 0.081 and f*2/4Tr = 0.35. The pNA-coupling constants are given [1] by the quark or strong coupling models using for the pNN-coupling constants the values [3] g2/4rr = 0.55, K = 6.6, where)' o = (mo/2M)g o (1 + K ) . To get a correct resonance form for the cross section it is necessary to make the mass of the A complex by adding its width - i P/2. This has the effect of attenuating the contribution from the NA-configurations at and above the resonance, which would otherwise be enormous. The greater success o f the perturbation approach [2] compared with the model 40
Volume 71B, number 1
PHYSICS LETTERS
7 November 1977
m~ sr
d' al,s
r
mb 15 40
"s'°
i
I
tt
0.5
1.0
1.5
I
2.0 q- o~.u
Fig. I. The p-wave total absorption cross section. Solid curve the non-Galilean invariant operator, dashed curve the Galilean invariant operator, and dotted curve "half-Galilean" operator is used. Also shown is the result when the contribution from s- and d-wave pions is added to the non-Galilean cross ruction. The experimental data are the same as in ref. [ 1].
Fig. 2. The differential cross section parameters 7o and 3'2. The notation is the same as in fig. 1. The experimental points are a collection from ref. [6] and the refs. given therein.
given in ref. [ 1 ] is largely due to our using an inappropriate form o f the width, which was too large. In this work we shall use the form given e.g. by Sugawara and von Hippel [4] r - f*2
24n
CA + M)2 _ A2
q3 /.t2
w(q)/2(M+
where q l "~ q(1 ~ ( q ) ) ) , q = r/g is the momentum o f the pion in the final state CM system, A , M and /a the masses o f the A, the nucleon and the pion respectively, a n d f * 2 / 4 n = 0.35 the coupling constant from the A decay vertex. The doubling o f the system o f equations in the complex case sets strong computational limits to the possible number o f channels we are, at present, able to accommodate. In fact in these calculations we have used only two dominant NA-components with L ~< 2. The effect of other channels is found to be rather small. The Hamiltonian operator for the pion production vertex is the already classic Galilean invariant form
H=f--i"{V""~(x)+2~[P't'It(x)+'c'n(x)P]lla where f2/4n = 0.08 I, p is the m o m e n t u m o f the nucleon, and t~ and n the pion conjugate isovector fields. Of
(3)
course, in the case of ANn-vertex f, a and x must be replaced by f * and by the operators S and T introduced by Sugawara and von Hippel [4]. Without taking a stand in the argument about the possible ambiguity o f the last term o f eq. (3), we shall also repeat the calculations with the non-Galilean operator, where the last term is simply omitted. This form has been used in ref. [2]. The s-wave rescattering o f the pion is described by the phenomenological Hamiltonian H s = 4n h l ~'11)+ 4rrh_2 , . ~ X n // /a2
(4) 1
where we take h 1 = - { (,u/q)(51 + 253) and X2 - - g
(la/~(q))~/q)(51 -
63) as given by Goplen [5], approxi41
Volume 71B, number 1
PHYSICS LETTERS
7 November 1977
Table 1 The polarization asymmetry parameters for a series of energies. The experimental points are a collection from the references given in ref. [6]. .
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Experiment Elab (MeV)
"0
425 480 540 600 660 720
1.00 1.22 1.43 1.63 1.81 1.99 .
.
.
.
.
Non-Galilean results ho hi h2 (mb/sr)
Galilean results ho hi
h2
Elab (MeV)
-1.37 1.68 -1.86 -1.54 -0.96 -0.64
0.86 0.80 -0.42 0.12 0.20 0.16
1.78 3.54 6.70 8.80 5.93 3.14
425 462 532 591 654 723
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0.15 -0.02 -0.52 -1.22 -1.26 -0.94 .
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2.34 4.67 8.72 11.34 7.79 4.37 .
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0.00 -0.16 -0.72 1.37 -1.16 -0.60 .
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ho
hi (mb/sr)
-1.43 ± 0.13 -0.79 ± 0.31 0.69 -2-_0.36 3.38±0.43 4.5 ± 0.8 2.6 -+0.8 .
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.
1.3 ± 0.7 2.1 ± 0.6 2.8± 0.7 4.9± 1.5 1.5 -+ 0.9 .
h;z
4.22 6.5 17.1 22.3 26.0 12.4
-* 0.55 ± 2.7 ± 3.0 ±2.3 ± 2.5 -+ 3.3
.
mated by ;k1 = 0.0054 and ~'2 = 0.0445 la/w(q). The propagator is taken to be that of Goplen et al. [5] in the two nucleon case. In the case when the rescattering pion originates from a decaying A, we suppose it to have already its final energy. The conventional way to parametrize the differential cross section and the polarization asymmetry is [6] do I polarized = 321r 1 {(3'0 + 72 c°s20 +"/4 c°s40) +PB "ti sin 0[• 0 + X2 cos20 ('A1 + ~'3 c°s20)] }"
(5)
Here PB is the beam polarization and ti the unit vector in the direction kp X q with kp and q being the momenta o f the incident proton and of the final state pion. Evaluating these parameters requires not only the amplitudes for p-wave production from 1 SO and 1 D2 pp-states but also at least the s- and d-wave production amplitudes from 3P 1 and 3P 2 states. These are the states we consider here. To this order 74 and ~'3 remain zero. A more detailed account o f the formalism together with complete formulae will be presented in a forthcoming paper [8]. Considering the numerical results, we see from figs. 1 and 2 that the non-Galilean operator gives an adequate description o f the cross section, whereas the Galilean invariant one is somewhat too low. The magnitude o f the cross section can be easily adjusted somewhat by altering the coupling potential, by changing the NAp-coupling within its experimental limits or by making the NA-interaction less repulsive than given in the model of ref. [ 1 ]. The values o f the polarization parameter ~'0 in table 1, although not impressively good, seem to favour the Galilean invariant operator. The correct qualitative behaviour arises from the s-wave production amplitude getting its main contribution near threshold from rescattering. Then it decreases and changes its sign for higher energies. It seems clear that the use of some form factors in the interaction H s would be beneficial here. In this way the reaction could give information about these s-wave scattering form factors. The forward-backward asymmetry parameter ~1 is not even qualitatively correct. Perhaps here higher partial waves are needed [8]. Also the troublesome spin-spin part in V2 arises here in the 3P2-state with more weight than in the other partial waves. In conclusion we might argue that the coupled channels formalism seems to give rather a reasonable description o f theprocess p + p ~ d + n ÷, even better than that found in our earlier work of ref. [ 1 ]. I would like to thank Doc. A.M. Green for a number o f discussions and for helpful comments on this work.
[ 1 ] A.M. Green and J.A. Niskanen, Nucl. Phys. A271 (1976) 503. [2] D.O. Riska, M. Brack and W. Weise, Phys. Lett. 61B (1976) 41 [3] G. Hbhler and E. Pietarinen, Nucl. Phys. B95 (1975) 210. [4} H. Sugawara and F. yon Hippel, Phys. Rev. 172 (1968) 1764.
42
[5] B. Goplen, W. Gibbs and E. Lomon, Phys. Rev. Lett. 32 (1974) 1012; B. Goplen, Ph.D. thesis, Los Alamos Scientific Laboratory report LA-5854-T, 1975. [6] D. Aebischer et al., Nucl. Phys. BI08 (1976) 214. [7] J.W. Durso, M. Saarela, G.E. Brown and A.D. Jackson, Nucl. Phys. A278 (1977) 445. [8] J.A. Niskanen, to be published.