- Email: [email protected]
Proton-proton and pion-proton inelastic collisions and the proton and pion form factors
Nuclear Physics B32 (1971) 205-213.
North-Holland Publishing Company
PROTON-PROTON AND PION-PROTON INELASTIC COLLISIONS AND THE PROTON AND PION FORM FACTORS R.SOSNOWSKI and W.W(!)JCIK Institute of Nuclear Research, Warsaw and Institute of Experimental Physics, University of Warsaw, Warsaw Received 2 November 1970 (Revised manuscript received 9 February 1971) Abstract: A model of proton-proton inelastic collisions at high energies is proposed. It assumes that the amplitude for a collision in which n secondary particles are produced is a product of n form factors identical with the charge form factor of the proton. The predictions of the model are compared with the experimental data. The main characteristics of the experimental distributions of secondary particles are well reproduced by the model. The experimental data on high-energy pion-proton collisions are also compared with the prediction of the model.
1. INTRODUCTION The fact that the differential cross section for the proton-proton elastic scattering is roughly proportional to the fourth power of the proton charge form factor [ 1,2] suggests that the space distribution of the “hadronic matter” of a proton is proportional to its charge density distribution. The proton-proton elastic scattering at high energies is usually interpreted as a diffraction of the incoming particles which are absorbed in an interaction volume. The shape of this volume is determined by the distribution of the hadronic matter in a single proton [2]. The absorption of colliding protons is due to inelastic processes in which new secondary particles are produced. In the present paper we propose a model in which secondary particles are produced coherently in the interaction volume identical with that which leads to the diffractive elastic scattering. In the model it is assumed that at each point of the interaction volume a spherically symmetric wave of each of secondary particles is created. Its amplitude is proportional to the intensity of the incoming wave and to the density of the hadronic matter at this point. As it follows from the Huyghens principle [3] the amplitude of this process is a Fourier transform, F(q2), of the interaction volume in which a new particle is created. The quantity q2 denotes the momentum transfer from the incoming to the created particle. In the case when n secondary particles are assumed to be produced independent-
RSosnowski and W.Wbjcik, p-p and n-p inelastic collisions
206
ly in the same interaction Fourier transforms
volume the amplitude
n
for this process is a product of the
n
&3 = lGA’(q)
A@,,P2, ... P,> = fi
’
(1)
where pl, p2, . . . pn are the four-momenta of the produced particles and qi is threemomentum transfer from the incident to ith secondary particle. Therefore the proposed model has not a multiperipheral character but belongs to the class of uncorrelated jet models [4] . We assume that in the collision of two protons, A and B, a secondary particle is produced by the proton A in the entire volume of the proton B and vice-versa. Therefore the amplitude for the production of a secondary particle is assumed to be
where qiA and qB are the momentum transfers to the produced particle from the primary protons, A and B respectively. The Fourier transform of the distribution of the hadronic matter for a single proton, F(q2), is taken equal to the proton charge form factor :
lq2
F(q2) =
2’
(, 1
(3)
l+o;71
For the reaction with n secondary particles in the final state A+B+1+2+...+n, the amplitude
(4)
will be:
where indices 1 and 2 correspond to the two secondary nucleons and the indices 3 9 --., n correspond to the produced pions, For the elastic scattering of protons (n = 2) the above amplitude is reduced to [flq2)12_ In the present paper the experimental data on p-p inelastic reactions are compared with the prediction following from the model based on the formula (5). The momentum transfers q were taken to be three-momentum transfers in the c.m. system of two colliding protons.
RSosnowski
and W. Wbjcik, p-p and n-p inelastic collisions
207
The proposed model relates the proton charge form factor to distributions of secondary particles produced in inelastic proton-proton collisions. It does not introduce any free parameter except for a normalization constant.
2. PROTON-PROTON
INELASTIC COLLISIONS
The amplitude given by the formula (5) was used to calculate the various distributions for proton-proton inelastic reactions at different energies and with different multiplicity of the produced pions. The calculations were done by the Monte-Carlo method using the programme FOWL,[S].In order to increase the efficiency of these calculations events were generated using the method proposed by Van Hove [6] . This method gives mainly the events with smaIl transverse momenta and avoides generating useless events with large transverse momenta which in practice do not enter into the calculated distributions as they have nearly vanishing amplitude. In figs. 1 and 2 some c.m.s. experimental distributions of particles produced in the p-p reactions at 10 GeV/c (ref. [7] ) are compared with the predictions of the model. It can be seen that the model describes well the main features of the experimental distributions. In particular it predicts correctly the difference between the c.m.s. angular distribution of secondary nucleons and secondary pions and the change in the angular distribution with the pion multiplicity. The reaction pp + A++(1236) prr- has been studied at 6.6 GeV/c (ref. [S] ), 16 GeV/c (ref. [9] ) and 28.5 GeV/c (ref. [ 121). It has been shown that this reaction can be well described assuming the multi-Regge-pole exchange mechanism. In particular it predicts correctly the distribution of the Treiman-Yang angle which is defined as @A ‘OS’=
x P2) -(PI
@Axp211plxhIf
’
%)
(6)
where PA, &, p1 and & are three-momentum vectors of the two primary protons, A+‘( 1236) and a secondary proton respectively (in the rest system of a secondary proton and r-)_ In fig. 3 the experimental distributions of the angle # are compared with the curves calculated from the model. The accuracy of the curves obtained by the MonteCarlo method is better than 10%. It can be seen that the obtained curves are in a good agreement with the experimental data although no special exchange mechanism has been assumed. To understand this we have to notice that the discussed model introduces the same cut of transverse momenta of secondary protons and pions. In the n-p rest system the angular distribution of n- cannot be isotropic if transverse momenta of pions and protons are on average nearly equal. This follows from simple kinematic consideration because the masses of proton and pion are different.
R.Sosnowski and W. Wbjcik, p-p and n-p inelastic collisions
208
-REACTION:
PP-PPTT+~IT~
60
30
40
20
20
10
TRANSVERSEMOMENTUM
AT
10 GeV/c
c.m. LONGITUDINAL
MOMENTUM, G&c
Fig. 1. Distribution of the c.m.s. longitudinal and transverse momenta for secondary particles from the reaction pp + pprr+n-no at 10 GeV/c. In the present figure and in all remaining figures the smooth curves represent the predictions of the model discussed in the text.
3. PION-PROTON
INELASTIC COLLISIONS
The discussed model can be extended to the collisions of other pairs of hadrons. Here we discuss n’p inelastic reactions. In this case our amplitude will be of the form n ACPl,P2,
.-.Pn> x Fp(4:*)FJQ2n) 2
i=3 l-I Fp(&)
+ &J&J1
9
Fp(q2) and F,(q2) are the proton and pion form factors respectively. The index i stands here for one of the secondary pions and the index 2 for the secondary
(7)
R.Sosnowski
and W. Wbjcik, p-p and n-p inelastic collisions
p-p INTERACTIONS pnn’
Fig. 2. C.m.s. angular distribution
AT
209
lOGeV/c
pplr*ll-
of secondary particles in inelastic p-p collisions at 10 GeV/c.
proton. The above formula takes into account the experimental fact that the primary pion and proton seem not to lose their identity in the collision and show the tendency to continue their movement after the collision (leading particles). In the calculations the pion form factor F,(q2) was taken to be identical with the proton form factor Fp(q2), which is suggested by the direct measurement of F,(q2) (ref. [lo] ). The predictions of the model are compared with the experimental data in figs. 4 and 5. It can be seen that the main features of the c.m.s. longitudinal momentum distributions of secondary pions and protons are well reproduced [ 1 l] .
RSosnowski
210
REACTION:
and W. Wbjcik, p-p and r-p inelastic collisions
p+p-A,?p+n26.5
16 GeV/c
6.6 GeV/c Mpfl>l.34GeV
Mpn- > 2GeN
-ta < 0.6 (
2
GeV/c
-ta < 0.5 GeV’
60
r
m Iz %
100
w
0’1
90”
1 160’
1 0”
90’
1 lso’
Q 1
1
90.
160” TREIMAN-YANG
ANGLE.
degrees
300
0.5 -t @++I
Fig. 3. Distribution
@9
0
0
-t
of the Treiman-Yang angle and the four-momentum pp -f ~++(1236)pn-.
0.5 -tdn”]
I GeV’
transfer in the reaction
4. CONCLUSIONS The discussed model reproduces the main trends in the distribution of secondary particles in both proton-proton ar.Zpion-proton inelastic collisions. We should not expect, howevrr, that the agreement between the predictions of the model and the experimental data will be very good. It does not take into account the effect of absorption of the incoming waves in the hadronic matter. This was found to be important for p-p elastic scattering [ 10, 131 and should influence also the inelastic reactions. For some inelastic channels discussed in this paper an abundant production of resonances has been observed experimerrtally. The calculated curves shown in figs. 1 and 2 and in figs. 4 and 5 were obtained neglecting this effect. Even with these simplifying assumptions the agreement between the calculated curves and the ex-
211
R.Sosnowski and W. Wbjcik, p-p and n-p inelastic collisions
rr*p INTERACTIONS
AT 8GeV/6
20-
- lo-
01 - IO-
-
o-
- lo-
,Kr
ccm. LONGITUDINAL
Fig. 4. Distribution
MOMENTUM
of the c.m.s. longitudinal momentum at 8 GeV/c.
-1 OF
0
PROTON.
of protons
1
GeV/c
for n+p inelastic reactions
perimental data is satisfactory. It should be stressed that the discussed model does not introduce any free parameters except one normalization constant for each reaction.
212
R.Sosnowski and W. Wbjcik, p-p and n-p inelastic collisions
r-r-p
1NTERACTlONS
AT
8 GeV/c
pTt+ll+li
p li+l?+liiliH-
c.m. LONGITUDINAL Fig. 5. Distributions
of the c.m.s. longitudinal
MOMENTUM
momentum 8 GeV/c.
GeVk
of pions for r+p inelastic
REFERENCES [l] [2] (31 [4] [S) [6]
OF PIONS.
T.T.Wu and C.N.Yang, Phys. Rev. 137 (1965) B708. T.T.Chou and C.N.Yang, Phys. Rev. 170 (1968) 1591. Chr.Huyghens, Trait6 de la lumiere, Leyden, 1690. L.Van Hove, Nuovo Cimento 28 (1963) 798. F.James, CERN computer program library w. 505. L.Van Hove, Nucl. Phys. B9 (1968) 331.
reactions
at
R.Sosnowski and W.Wbjcik, p-p and r-p inelastic collisions
213
[7] S.P.Almeides, J.G.Ruahbrooks, J.H.Scharenguivel, M.Behrens, V.Blobel, I.Borecka, H.C. Dehve, J.Diaz, J.Knies, ASchmitt, K.StrGmer and W.P.Swanson,‘Phys. Rev. 174 (1968) 1638. [8] E.Gellert, G.A.Smith, E.Colton and P.Schlein, Phys. Rev. Letters 20 (1968) 964. [9] J.G.Rushbrooke and J.R.Williams, Phys. Rev. Letters 22 (1969) 248. [lo] W.K.H.Panofsky, Proc. of the 14th Int. Conf. on high-energy physics, Vienna, 1968, p. 23. [ 111 Aachen-Berlin-CERN-Krakow-Warsaw Collaboration, paper submitted to the Heidelberg Int. Conf. on elementary particles, 1967. [12] E.L.Berger, Phys. Rev. 179 (1969) 1567. [ 131 W.Czyz, Lectures delivered at the Inaugural Conf. of the European Physical Society, Florence, 1969.
North-Holland Publishing Company
PROTON-PROTON AND PION-PROTON INELASTIC COLLISIONS AND THE PROTON AND PION FORM FACTORS R.SOSNOWSKI and W.W(!)JCIK Institute of Nuclear Research, Warsaw and Institute of Experimental Physics, University of Warsaw, Warsaw Received 2 November 1970 (Revised manuscript received 9 February 1971) Abstract: A model of proton-proton inelastic collisions at high energies is proposed. It assumes that the amplitude for a collision in which n secondary particles are produced is a product of n form factors identical with the charge form factor of the proton. The predictions of the model are compared with the experimental data. The main characteristics of the experimental distributions of secondary particles are well reproduced by the model. The experimental data on high-energy pion-proton collisions are also compared with the prediction of the model.
1. INTRODUCTION The fact that the differential cross section for the proton-proton elastic scattering is roughly proportional to the fourth power of the proton charge form factor [ 1,2] suggests that the space distribution of the “hadronic matter” of a proton is proportional to its charge density distribution. The proton-proton elastic scattering at high energies is usually interpreted as a diffraction of the incoming particles which are absorbed in an interaction volume. The shape of this volume is determined by the distribution of the hadronic matter in a single proton [2]. The absorption of colliding protons is due to inelastic processes in which new secondary particles are produced. In the present paper we propose a model in which secondary particles are produced coherently in the interaction volume identical with that which leads to the diffractive elastic scattering. In the model it is assumed that at each point of the interaction volume a spherically symmetric wave of each of secondary particles is created. Its amplitude is proportional to the intensity of the incoming wave and to the density of the hadronic matter at this point. As it follows from the Huyghens principle [3] the amplitude of this process is a Fourier transform, F(q2), of the interaction volume in which a new particle is created. The quantity q2 denotes the momentum transfer from the incoming to the created particle. In the case when n secondary particles are assumed to be produced independent-
RSosnowski and W.Wbjcik, p-p and n-p inelastic collisions
206
ly in the same interaction Fourier transforms
volume the amplitude
n
for this process is a product of the
n
&3 = lGA’(q)
A@,,P2, ... P,> = fi
’
(1)
where pl, p2, . . . pn are the four-momenta of the produced particles and qi is threemomentum transfer from the incident to ith secondary particle. Therefore the proposed model has not a multiperipheral character but belongs to the class of uncorrelated jet models [4] . We assume that in the collision of two protons, A and B, a secondary particle is produced by the proton A in the entire volume of the proton B and vice-versa. Therefore the amplitude for the production of a secondary particle is assumed to be
where qiA and qB are the momentum transfers to the produced particle from the primary protons, A and B respectively. The Fourier transform of the distribution of the hadronic matter for a single proton, F(q2), is taken equal to the proton charge form factor :
lq2
F(q2) =
2’
(, 1
(3)
l+o;71
For the reaction with n secondary particles in the final state A+B+1+2+...+n, the amplitude
(4)
will be:
where indices 1 and 2 correspond to the two secondary nucleons and the indices 3 9 --., n correspond to the produced pions, For the elastic scattering of protons (n = 2) the above amplitude is reduced to [flq2)12_ In the present paper the experimental data on p-p inelastic reactions are compared with the prediction following from the model based on the formula (5). The momentum transfers q were taken to be three-momentum transfers in the c.m. system of two colliding protons.
RSosnowski
and W. Wbjcik, p-p and n-p inelastic collisions
207
The proposed model relates the proton charge form factor to distributions of secondary particles produced in inelastic proton-proton collisions. It does not introduce any free parameter except for a normalization constant.
2. PROTON-PROTON
INELASTIC COLLISIONS
The amplitude given by the formula (5) was used to calculate the various distributions for proton-proton inelastic reactions at different energies and with different multiplicity of the produced pions. The calculations were done by the Monte-Carlo method using the programme FOWL,[S].In order to increase the efficiency of these calculations events were generated using the method proposed by Van Hove [6] . This method gives mainly the events with smaIl transverse momenta and avoides generating useless events with large transverse momenta which in practice do not enter into the calculated distributions as they have nearly vanishing amplitude. In figs. 1 and 2 some c.m.s. experimental distributions of particles produced in the p-p reactions at 10 GeV/c (ref. [7] ) are compared with the predictions of the model. It can be seen that the model describes well the main features of the experimental distributions. In particular it predicts correctly the difference between the c.m.s. angular distribution of secondary nucleons and secondary pions and the change in the angular distribution with the pion multiplicity. The reaction pp + A++(1236) prr- has been studied at 6.6 GeV/c (ref. [S] ), 16 GeV/c (ref. [9] ) and 28.5 GeV/c (ref. [ 121). It has been shown that this reaction can be well described assuming the multi-Regge-pole exchange mechanism. In particular it predicts correctly the distribution of the Treiman-Yang angle which is defined as @A ‘OS’=
x P2) -(PI
@Axp211plxhIf
’
%)
(6)
where PA, &, p1 and & are three-momentum vectors of the two primary protons, A+‘( 1236) and a secondary proton respectively (in the rest system of a secondary proton and r-)_ In fig. 3 the experimental distributions of the angle # are compared with the curves calculated from the model. The accuracy of the curves obtained by the MonteCarlo method is better than 10%. It can be seen that the obtained curves are in a good agreement with the experimental data although no special exchange mechanism has been assumed. To understand this we have to notice that the discussed model introduces the same cut of transverse momenta of secondary protons and pions. In the n-p rest system the angular distribution of n- cannot be isotropic if transverse momenta of pions and protons are on average nearly equal. This follows from simple kinematic consideration because the masses of proton and pion are different.
R.Sosnowski and W. Wbjcik, p-p and n-p inelastic collisions
208
-REACTION:
PP-PPTT+~IT~
60
30
40
20
20
10
TRANSVERSEMOMENTUM
AT
10 GeV/c
c.m. LONGITUDINAL
MOMENTUM, G&c
Fig. 1. Distribution of the c.m.s. longitudinal and transverse momenta for secondary particles from the reaction pp + pprr+n-no at 10 GeV/c. In the present figure and in all remaining figures the smooth curves represent the predictions of the model discussed in the text.
3. PION-PROTON
INELASTIC COLLISIONS
The discussed model can be extended to the collisions of other pairs of hadrons. Here we discuss n’p inelastic reactions. In this case our amplitude will be of the form n ACPl,P2,
.-.Pn> x Fp(4:*)FJQ2n) 2
i=3 l-I Fp(&)
+ &J&J1
9
Fp(q2) and F,(q2) are the proton and pion form factors respectively. The index i stands here for one of the secondary pions and the index 2 for the secondary
(7)
R.Sosnowski
and W. Wbjcik, p-p and n-p inelastic collisions
p-p INTERACTIONS pnn’
Fig. 2. C.m.s. angular distribution
AT
209
lOGeV/c
pplr*ll-
of secondary particles in inelastic p-p collisions at 10 GeV/c.
proton. The above formula takes into account the experimental fact that the primary pion and proton seem not to lose their identity in the collision and show the tendency to continue their movement after the collision (leading particles). In the calculations the pion form factor F,(q2) was taken to be identical with the proton form factor Fp(q2), which is suggested by the direct measurement of F,(q2) (ref. [lo] ). The predictions of the model are compared with the experimental data in figs. 4 and 5. It can be seen that the main features of the c.m.s. longitudinal momentum distributions of secondary pions and protons are well reproduced [ 1 l] .
RSosnowski
210
REACTION:
and W. Wbjcik, p-p and r-p inelastic collisions
p+p-A,?p+n26.5
16 GeV/c
6.6 GeV/c Mpfl>l.34GeV
Mpn- > 2GeN
-ta < 0.6 (
2
GeV/c
-ta < 0.5 GeV’
60
r
m Iz %
100
w
0’1
90”
1 160’
1 0”
90’
1 lso’
Q 1
1
90.
160” TREIMAN-YANG
ANGLE.
degrees
300
0.5 -t @++I
Fig. 3. Distribution
@9
0
0
-t
of the Treiman-Yang angle and the four-momentum pp -f ~++(1236)pn-.
0.5 -tdn”]
I GeV’
transfer in the reaction
4. CONCLUSIONS The discussed model reproduces the main trends in the distribution of secondary particles in both proton-proton ar.Zpion-proton inelastic collisions. We should not expect, howevrr, that the agreement between the predictions of the model and the experimental data will be very good. It does not take into account the effect of absorption of the incoming waves in the hadronic matter. This was found to be important for p-p elastic scattering [ 10, 131 and should influence also the inelastic reactions. For some inelastic channels discussed in this paper an abundant production of resonances has been observed experimerrtally. The calculated curves shown in figs. 1 and 2 and in figs. 4 and 5 were obtained neglecting this effect. Even with these simplifying assumptions the agreement between the calculated curves and the ex-
211
R.Sosnowski and W. Wbjcik, p-p and n-p inelastic collisions
rr*p INTERACTIONS
AT 8GeV/6
20-
- lo-
01 - IO-
-
o-
- lo-
,Kr
ccm. LONGITUDINAL
Fig. 4. Distribution
MOMENTUM
of the c.m.s. longitudinal momentum at 8 GeV/c.
-1 OF
0
PROTON.
of protons
1
GeV/c
for n+p inelastic reactions
perimental data is satisfactory. It should be stressed that the discussed model does not introduce any free parameters except one normalization constant for each reaction.
212
R.Sosnowski and W. Wbjcik, p-p and n-p inelastic collisions
r-r-p
1NTERACTlONS
AT
8 GeV/c
pTt+ll+li
p li+l?+liiliH-
c.m. LONGITUDINAL Fig. 5. Distributions
of the c.m.s. longitudinal
MOMENTUM
momentum 8 GeV/c.
GeVk
of pions for r+p inelastic
REFERENCES [l] [2] (31 [4] [S) [6]
OF PIONS.
T.T.Wu and C.N.Yang, Phys. Rev. 137 (1965) B708. T.T.Chou and C.N.Yang, Phys. Rev. 170 (1968) 1591. Chr.Huyghens, Trait6 de la lumiere, Leyden, 1690. L.Van Hove, Nuovo Cimento 28 (1963) 798. F.James, CERN computer program library w. 505. L.Van Hove, Nucl. Phys. B9 (1968) 331.
reactions
at
R.Sosnowski and W.Wbjcik, p-p and r-p inelastic collisions
213
[7] S.P.Almeides, J.G.Ruahbrooks, J.H.Scharenguivel, M.Behrens, V.Blobel, I.Borecka, H.C. Dehve, J.Diaz, J.Knies, ASchmitt, K.StrGmer and W.P.Swanson,‘Phys. Rev. 174 (1968) 1638. [8] E.Gellert, G.A.Smith, E.Colton and P.Schlein, Phys. Rev. Letters 20 (1968) 964. [9] J.G.Rushbrooke and J.R.Williams, Phys. Rev. Letters 22 (1969) 248. [lo] W.K.H.Panofsky, Proc. of the 14th Int. Conf. on high-energy physics, Vienna, 1968, p. 23. [ 111 Aachen-Berlin-CERN-Krakow-Warsaw Collaboration, paper submitted to the Heidelberg Int. Conf. on elementary particles, 1967. [12] E.L.Berger, Phys. Rev. 179 (1969) 1567. [ 131 W.Czyz, Lectures delivered at the Inaugural Conf. of the European Physical Society, Florence, 1969.