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Pionization effects in nucleon-nucleus scattering
Volume 51B, number 3
PHYSICS LETTERS
5 August 1974
PIONIZATION EFFECTS IN NUCLEON-NUCLEUS SCATTERING D.R. HARRINGTON*
Institut flir TheoretischeKernphysik, UniversitiitKarlsruhe, Germany Received 4 January 1974 Revised manuscript received 18 April 1974 A simple unitary model for meson production in nucleon-nucleon scattering is extended to nucleon-nucleus scattering. In the high energy limit the produced mesons remain coherent with the projectile over the entire depth of the nucleus and tend to increase the total cross sections. A crude estimate of the size of the effect gives an increase of the total cross section for proton-deuteron scattering of 13 mb at NAL energies.
In this note we wish to show how a straightforward extension of a simple unitary model [1,2] for pionization in nucleon-nucleon scattering ~ can be used to study the effects of pionization in nucleon-nucleus scattering. In this model the mesons are produced independently for fixed impact parameter b and nuclear coordinates {r,}: The amplitude for producing n mesons with momenta k 1 , k 2 .... , k n when the incident proton receives a transverse m o m e n t u m transfer A and the nucleus makes a transition from its initial state li) to a final state If) is Ffi(A ; k 1 , k 2, ...,
X
kn) = (27ri) -1
fd2b exp ( - i
a-b)
[i(2~o])-l/2s(~;b,{%})lexp[-~E(b,{r})]
li>.
(1)
Here
s(k; b, { r } ) = fd4x
exp (ik-x)
S(x; b, {r~}),
(2)
where
S(x; b, {ra})= ~,TL V { [7~(Z--#L 02
+ (r±-b)211/2} ~
V(Ir-ral),
(3)
with ko= t~(k) = 0a2+k2) I/2 and 7L = (1-/32) - I/2. The source of the mesons in thus the overlap of a Lorentz-contracted cloud, moving with the projectile speed/3 L, with the sum of the corresponding clouds for all the nucleons in the nucleus. (We work in the rest system of the nucleus and define the z-axis parallel to the incident momentum). The quantity E in (1) is essentially the probability that a single meson is emitted and then re-absorbed:
E(b,{Pa})=(21t)-3 fd3k(2w)-lls(k;b,{ra})12=½ fd4xd4x'S(x';b,{r
})A1(x'-x)S(x;b,{r
}).
(4)
This quantity plays a fundamental role in the model. From it we can calculate directly the quasi-elastic scattering amplitudes
* Permanent address: Department of Physics, Rutgers University, New Brunswick, N.J. 08903, USA. Many other related but more complicated models have been suggested. See, for example, the papers listed in ref. [3].
287
Volume 51B, number 3 Ffi (At = (2~ri)-1 f
PHYSICS LETTERS
d2b exp ( - i
A'b) (fr exp [-½E(b, (?"))] - 1 li),
5 August 1974
(5)
the total cross section ow = 2fd2b
(ill - e x p [ -
½E(b, { r
))] li),
(6)
and the average number of mesons produced
(n) =fd2b
(il E(b, (ra)) li)/o T.
(7)
As has been shown in ref. [1], E grows as In 7L in the high energy limit. In our notation
E(b, { r ) )
~ (4~) -1 ~2 In 7L f d2b ' h2(b'-b)
~, h(b'-b,)
h(b'-b),
(8)
where oo
h(b) = f dz V[(z2+b2)l/21.
(9)
--oo
In the high energy limit there is thus no dependence on the z-components of the nucleon coordinates. The explanation of this limiting behaviour is quite simple: When n mesons are emitted and m absorbed the longitudinal momentum transfer to the target in the high energy limit is n m
Qz=i~=lO3Ll~--~z)--i~=l(~Ll~i--kiz).
(10)
In our model the mesons are produced with a fixed distribution in transverse momentum but with a distribution in kz which expands with increasing energy. For most events each term in the sums in (10) become small and the coherence length Qz I will thus grow as the energy increases. The highest energy mesons prodveed have kz ~ 7L]a, where a is the range of V(r), so that if only these were involved the coherence length would increase as 7L. Considering the entire distribution, however, a substantial fraction of collisions are coherent over a length l only if In 7L >> 1 + In (l/a). (An analagous increase in the coherence length with increasing energy is found in bremsstrahlung and pair production by very high energy electrons passing through matter [4] ). The possibility of such an effect has apparently been ignored in several papers [5] which estimate the amount of meson production when very high energy protons are incident on nuclei: It is assumed that if the mesons are produced via an "incoherent" mechanism in nucleon-nucleon collisions then they cannot interact with each other in the nucleus and can therefore only cascade. Unfortunately our model does not provide a realistic description of cascade-like processes since in the mesons produced cannot act as secondary projectiles but can only interact in cooperation with the incident particle. The effect of pionization on proton-nucleus total cross sections can be studied by separatingE into its diagonal and non-diagonal terms:
E(b, (r )) ~ E(b, ( b ) )
= ~a
El(b-b) +,~<~E2(b-b, b-b,),
(11)
where E l ( b ) = (47r)-1 ~,2 In
VLfd2b ' h2(b ') h2(b'-b)
(12)
is the analogue of E for interactions on a single nucleon, while E2(bl, b2 ) = 2(4zr)-i ),2 In 288
"YLf
d2o'h2(b') h(b'-bl) h(b-b2)
(13)
Volume 51B, number 3
PHYSICS LETTERS
5 August 1974
is essentially the probability that a meson produced on a nucleon centered at b 1 relative to the projectile will be absorbed by another nucleon centered at b 2. In other words the El'S represent that part of the pionization effect included in the elastic nucleon-nucleon scattering amplitudes while the E2's represent the additional effects in nuclear targets due to mesons produced on one nucleon and absorbed on another. The function E 1 is of course positive, and E 2 will be also if we make the reasonable assumption that h(b) has the same sign for all b. It is then clear that the effect of pionization will be to increase the total cross sections beyond the predictions of Glauber theory using only the elastic nucleon-nucleon amplitudes. For the case of a deuteron target the pionization effects on the total cross section are easily isolated. If the proton and neutron are at -+rd[2 then
E(b, bd) = E 1( b - ~l b d) +El(b+½b d) +E2(b-½bd,b+½bd).
(14)
The deuteron's total cross section can thus be written tr(d) = 2t~(TN)-- 8Oel + 6Opion
(15)
where o (TN ) -- 2 J/~d2b ( l - e x p [ - ½ E l ( b ) ] )
(16)
is the nucleon-nucleon total cross section,
6Oel=2fd2b(dl{1-expt-½El(b-½bd)])(1-exp[-½El(b+~bd)])
Id)~ aT(N)2(dl(41rr2d)-i Id)
(17)
is the usual shadow scattering defect, and 5Opion = 2
fd2b
(dl e x p t -
½El(b-
½b d) -
~El(b+½bcl)]
{ 1- exp[-
½E2(b- ~b d, b+½bd)l )
Id)
(18)
is due to inelastic double scattering and is clearly positive i f E 2 is positive. The magnitude of ~Opion at high energy will depend strongly on the nature of the large-b tail of h(b). As 7L ~ o. there will be no contribution to 8oni,, from values o f b near +-b~/2because of the essentially complete absorption . regions . . will . increase . . with increasing energy. As a result, while O(d) (N) , there, and the areas of these absorbing T , oT and 50el will eventually all increase with the same leading dependence on 7L (e.g. 21rm -2 [In (In ~/L)] 2 for h(b) ~ exp (-rob)), 8Opion will receive contributions only from the limbs of the interaction regions and will increase less rapidly, or not at all, in this limit. We have made a crude estimate of 8Opion at NAL energies (TL = 500), taking Gaussian potentials V(r) = Vo exp (-r2/2a 2) and a Gaussian deuteron wave function and using the asymptotic from (13) for E 2 even though this may not be very accurate at these "low" energies. We have taken a = 3.64 (GeV/c)-1 to give (kip = 345 MeV in the inclusive meson cross section [6], and ~ln ('),L/2)(lr~ka3 V2) 2 = 1.55 which gives a protonproton total cross section of 42 mb. With these parameters the asymptotic inclusive distribution do/dy ly =0 = 9 mb is probably too small [7] by about a factor of 4, but this is not too surprising in view of the simplicity of the model. With these parameters we f'md 8Opion ~- 13 mb which can be compared to 8ael ~ 5 mb. We should emphasize that this value for 5apion is at best an order-of magnitude estimate since our model does not even fit proton-proton collisions particularly well. It seems likely, however, that the increase in proton-nudeus total cross sections due to pionization may be of considerable importance at NAL energies and above. This increase is consistent with general result emphasized recently by Blankenbeder [8] : non-diffractive channels tend to increase total cross section while diffractive channels tend to decrease them. Our estimate indicates that the two effects may be of comparable magnitude at NAL energies for proton-deuteron scatteringand it would be desirable to have more reliable models for both. It would be especially useful to have a unified model describing both diffraction dissociation and pionization in proton-proton scattering, perhaps along the lines suggested by Skard and Fulco [9]. 289
Volume 51B, number 3
PHYSICS LETTERS
5 August 1974
The author would like to thank Dr. H. H6gaasen and Prof. H. Pilkuhn and their colleagues at the Universities o f Oslo and Karlsruhe, respectively, for their hospitality. He would also like to thank Prof. Pilkuhn for pointing out the related phenomena discussed in ref. [4] and Dr. D. Julius for helpful discussions.
References [1] G. Calucci, R. Jengo and C. Rebbi, Nuovo Cimento 4A (1971) 330; 6A (1971) 601. [2] G.G. Zipfel, Jr., Phys. Rev. Lett. 24 (1970) 756. [3] R. Aviv, R.L. Sugar and R. Blankenbecler, Phys. Rev. D5 (1972) 352; S. Auerbach, R. Aviv, R.L. Sugar and R. Blankenbecler, Phys. Rev. D6 (1972) 2216; G. Calucci, Nucl. Phys. B44 (1972) 629; L.B. R6dei, Nucl. Phys. B60 (1973) 141; D.J. Scalapino and R.L. Sugar, Phys. Rev. D8 (1973) 2284. [4] M.L. Ter-Mikaelian, High-energy electromagnetic processes in condensed media (Wiley-Interscience, New York, 1972). [5] E. Lehmann and G. Winbow, Phys. Lett. 46B (1973) 372; P.M. Fishbane, J.L. Newmeyer and J.S. Treffl, Phys. Rev. D7 (1973) 3324; A. Dar and J. Vary, Phys. Rev. D6 (1972) 2412. [6] M. Banner et al., Phys. Lett. 41B (1972) 547. [7] S.R. Choudury, Phys. Lett. 48B (1974) 246. [8] R. Blankenbecler, Phys. Rev. Lett. 31 (1973) 964. [9] J.A. Skard and J.R. Fuleo, Phys. Rev. D8 (1973) 312.
290
PHYSICS LETTERS
5 August 1974
PIONIZATION EFFECTS IN NUCLEON-NUCLEUS SCATTERING D.R. HARRINGTON*
Institut flir TheoretischeKernphysik, UniversitiitKarlsruhe, Germany Received 4 January 1974 Revised manuscript received 18 April 1974 A simple unitary model for meson production in nucleon-nucleon scattering is extended to nucleon-nucleus scattering. In the high energy limit the produced mesons remain coherent with the projectile over the entire depth of the nucleus and tend to increase the total cross sections. A crude estimate of the size of the effect gives an increase of the total cross section for proton-deuteron scattering of 13 mb at NAL energies.
In this note we wish to show how a straightforward extension of a simple unitary model [1,2] for pionization in nucleon-nucleon scattering ~ can be used to study the effects of pionization in nucleon-nucleus scattering. In this model the mesons are produced independently for fixed impact parameter b and nuclear coordinates {r,}: The amplitude for producing n mesons with momenta k 1 , k 2 .... , k n when the incident proton receives a transverse m o m e n t u m transfer A and the nucleus makes a transition from its initial state li) to a final state If) is Ffi(A ; k 1 , k 2, ...,
X
kn) = (27ri) -1
fd2b exp ( - i
a-b)
[i(2~o])-l/2s(~;b,{%})lexp[-~E(b,{r})]
li>.
(1)
Here
s(k; b, { r } ) = fd4x
exp (ik-x)
S(x; b, {r~}),
(2)
where
S(x; b, {ra})= ~,TL V { [7~(Z--#L 02
+ (r±-b)211/2} ~
V(Ir-ral),
(3)
with ko= t~(k) = 0a2+k2) I/2 and 7L = (1-/32) - I/2. The source of the mesons in thus the overlap of a Lorentz-contracted cloud, moving with the projectile speed/3 L, with the sum of the corresponding clouds for all the nucleons in the nucleus. (We work in the rest system of the nucleus and define the z-axis parallel to the incident momentum). The quantity E in (1) is essentially the probability that a single meson is emitted and then re-absorbed:
E(b,{Pa})=(21t)-3 fd3k(2w)-lls(k;b,{ra})12=½ fd4xd4x'S(x';b,{r
})A1(x'-x)S(x;b,{r
}).
(4)
This quantity plays a fundamental role in the model. From it we can calculate directly the quasi-elastic scattering amplitudes
* Permanent address: Department of Physics, Rutgers University, New Brunswick, N.J. 08903, USA. Many other related but more complicated models have been suggested. See, for example, the papers listed in ref. [3].
287
Volume 51B, number 3 Ffi (At = (2~ri)-1 f
PHYSICS LETTERS
d2b exp ( - i
A'b) (fr exp [-½E(b, (?"))] - 1 li),
5 August 1974
(5)
the total cross section ow = 2fd2b
(ill - e x p [ -
½E(b, { r
))] li),
(6)
and the average number of mesons produced
(n) =fd2b
(il E(b, (ra)) li)/o T.
(7)
As has been shown in ref. [1], E grows as In 7L in the high energy limit. In our notation
E(b, { r ) )
~ (4~) -1 ~2 In 7L f d2b ' h2(b'-b)
~, h(b'-b,)
h(b'-b),
(8)
where oo
h(b) = f dz V[(z2+b2)l/21.
(9)
--oo
In the high energy limit there is thus no dependence on the z-components of the nucleon coordinates. The explanation of this limiting behaviour is quite simple: When n mesons are emitted and m absorbed the longitudinal momentum transfer to the target in the high energy limit is n m
Qz=i~=lO3Ll~--~z)--i~=l(~Ll~i--kiz).
(10)
In our model the mesons are produced with a fixed distribution in transverse momentum but with a distribution in kz which expands with increasing energy. For most events each term in the sums in (10) become small and the coherence length Qz I will thus grow as the energy increases. The highest energy mesons prodveed have kz ~ 7L]a, where a is the range of V(r), so that if only these were involved the coherence length would increase as 7L. Considering the entire distribution, however, a substantial fraction of collisions are coherent over a length l only if In 7L >> 1 + In (l/a). (An analagous increase in the coherence length with increasing energy is found in bremsstrahlung and pair production by very high energy electrons passing through matter [4] ). The possibility of such an effect has apparently been ignored in several papers [5] which estimate the amount of meson production when very high energy protons are incident on nuclei: It is assumed that if the mesons are produced via an "incoherent" mechanism in nucleon-nucleon collisions then they cannot interact with each other in the nucleus and can therefore only cascade. Unfortunately our model does not provide a realistic description of cascade-like processes since in the mesons produced cannot act as secondary projectiles but can only interact in cooperation with the incident particle. The effect of pionization on proton-nucleus total cross sections can be studied by separatingE into its diagonal and non-diagonal terms:
E(b, (r )) ~ E(b, ( b ) )
= ~a
El(b-b) +,~<~E2(b-b, b-b,),
(11)
where E l ( b ) = (47r)-1 ~,2 In
VLfd2b ' h2(b ') h2(b'-b)
(12)
is the analogue of E for interactions on a single nucleon, while E2(bl, b2 ) = 2(4zr)-i ),2 In 288
"YLf
d2o'h2(b') h(b'-bl) h(b-b2)
(13)
Volume 51B, number 3
PHYSICS LETTERS
5 August 1974
is essentially the probability that a meson produced on a nucleon centered at b 1 relative to the projectile will be absorbed by another nucleon centered at b 2. In other words the El'S represent that part of the pionization effect included in the elastic nucleon-nucleon scattering amplitudes while the E2's represent the additional effects in nuclear targets due to mesons produced on one nucleon and absorbed on another. The function E 1 is of course positive, and E 2 will be also if we make the reasonable assumption that h(b) has the same sign for all b. It is then clear that the effect of pionization will be to increase the total cross sections beyond the predictions of Glauber theory using only the elastic nucleon-nucleon amplitudes. For the case of a deuteron target the pionization effects on the total cross section are easily isolated. If the proton and neutron are at -+rd[2 then
E(b, bd) = E 1( b - ~l b d) +El(b+½b d) +E2(b-½bd,b+½bd).
(14)
The deuteron's total cross section can thus be written tr(d) = 2t~(TN)-- 8Oel + 6Opion
(15)
where o (TN ) -- 2 J/~d2b ( l - e x p [ - ½ E l ( b ) ] )
(16)
is the nucleon-nucleon total cross section,
6Oel=2fd2b(dl{1-expt-½El(b-½bd)])(1-exp[-½El(b+~bd)])
Id)~ aT(N)2(dl(41rr2d)-i Id)
(17)
is the usual shadow scattering defect, and 5Opion = 2
fd2b
(dl e x p t -
½El(b-
½b d) -
~El(b+½bcl)]
{ 1- exp[-
½E2(b- ~b d, b+½bd)l )
Id)
(18)
is due to inelastic double scattering and is clearly positive i f E 2 is positive. The magnitude of ~Opion at high energy will depend strongly on the nature of the large-b tail of h(b). As 7L ~ o. there will be no contribution to 8oni,, from values o f b near +-b~/2because of the essentially complete absorption . regions . . will . increase . . with increasing energy. As a result, while O(d) (N) , there, and the areas of these absorbing T , oT and 50el will eventually all increase with the same leading dependence on 7L (e.g. 21rm -2 [In (In ~/L)] 2 for h(b) ~ exp (-rob)), 8Opion will receive contributions only from the limbs of the interaction regions and will increase less rapidly, or not at all, in this limit. We have made a crude estimate of 8Opion at NAL energies (TL = 500), taking Gaussian potentials V(r) = Vo exp (-r2/2a 2) and a Gaussian deuteron wave function and using the asymptotic from (13) for E 2 even though this may not be very accurate at these "low" energies. We have taken a = 3.64 (GeV/c)-1 to give (kip = 345 MeV in the inclusive meson cross section [6], and ~ln ('),L/2)(lr~ka3 V2) 2 = 1.55 which gives a protonproton total cross section of 42 mb. With these parameters the asymptotic inclusive distribution do/dy ly =0 = 9 mb is probably too small [7] by about a factor of 4, but this is not too surprising in view of the simplicity of the model. With these parameters we f'md 8Opion ~- 13 mb which can be compared to 8ael ~ 5 mb. We should emphasize that this value for 5apion is at best an order-of magnitude estimate since our model does not even fit proton-proton collisions particularly well. It seems likely, however, that the increase in proton-nudeus total cross sections due to pionization may be of considerable importance at NAL energies and above. This increase is consistent with general result emphasized recently by Blankenbeder [8] : non-diffractive channels tend to increase total cross section while diffractive channels tend to decrease them. Our estimate indicates that the two effects may be of comparable magnitude at NAL energies for proton-deuteron scatteringand it would be desirable to have more reliable models for both. It would be especially useful to have a unified model describing both diffraction dissociation and pionization in proton-proton scattering, perhaps along the lines suggested by Skard and Fulco [9]. 289
Volume 51B, number 3
PHYSICS LETTERS
5 August 1974
The author would like to thank Dr. H. H6gaasen and Prof. H. Pilkuhn and their colleagues at the Universities o f Oslo and Karlsruhe, respectively, for their hospitality. He would also like to thank Prof. Pilkuhn for pointing out the related phenomena discussed in ref. [4] and Dr. D. Julius for helpful discussions.
References [1] G. Calucci, R. Jengo and C. Rebbi, Nuovo Cimento 4A (1971) 330; 6A (1971) 601. [2] G.G. Zipfel, Jr., Phys. Rev. Lett. 24 (1970) 756. [3] R. Aviv, R.L. Sugar and R. Blankenbecler, Phys. Rev. D5 (1972) 352; S. Auerbach, R. Aviv, R.L. Sugar and R. Blankenbecler, Phys. Rev. D6 (1972) 2216; G. Calucci, Nucl. Phys. B44 (1972) 629; L.B. R6dei, Nucl. Phys. B60 (1973) 141; D.J. Scalapino and R.L. Sugar, Phys. Rev. D8 (1973) 2284. [4] M.L. Ter-Mikaelian, High-energy electromagnetic processes in condensed media (Wiley-Interscience, New York, 1972). [5] E. Lehmann and G. Winbow, Phys. Lett. 46B (1973) 372; P.M. Fishbane, J.L. Newmeyer and J.S. Treffl, Phys. Rev. D7 (1973) 3324; A. Dar and J. Vary, Phys. Rev. D6 (1972) 2412. [6] M. Banner et al., Phys. Lett. 41B (1972) 547. [7] S.R. Choudury, Phys. Lett. 48B (1974) 246. [8] R. Blankenbecler, Phys. Rev. Lett. 31 (1973) 964. [9] J.A. Skard and J.R. Fuleo, Phys. Rev. D8 (1973) 312.
290