- Email: [email protected]
On the treatment of NN interaction effects in meson production in NN collisions
20 May 1999
Physics Letters B 454 Ž1999. 176–180
On the treatment of NN interaction effects in meson production in NN collisions C. Hanhart
a,b,c,1
, K. Nakayama
b,d,2
a
Institut fur ¨ Theoretische Kernphysik, UniÕersitat ¨ Bonn, D-53115 Bonn, Germany Institut fur GmbH, D-52425 Julich, Germany ¨ Kernphysik, Forschungszentrum Julich ¨ ¨ c Department of Physics and INT, UniÕersity of Washington, Seattle, WA 98195, USA 3 d Department of Physics and Astronomy, UniÕersity of Georgia, Athens, GA 30602, USA b
Received 24 September 1998; received in revised form 2 March 1999 Editor: J.-P. Blaizot
Abstract We clarify under what circumstances the nucleon–nucleon final state interaction fixes the energy dependence of the total cross-section for the reaction NN ™ NNx close to production threshold, where x can be any meson whose interaction with the nucleon is not too strong. It is shown that the results obtained from the procedure used recently by several authors to include the final state interaction in the reactions under discussion should be interpreted with caution. In addition, we give a formula that allows one to estimate the effect of the initial state interaction for the production of heavy mesons. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 25.40.y h; 13.75.Cs Keywords: Final state interaction; Initial state interaction; Meson production
As early as 1952 K. Watson pointed out under what circumstances one expects the final state interaction to strongly modify the energy dependence of the total production cross-section NN ™ NNx w1x given by
sN N ™ N N x Ž h . A
1
m xh
H0
d r Ž qX . < A Ž E, pX . < 2 .
E-mail: [email protected] E-mail: [email protected] 3 Present address. 2
In the above equation qX is the momentum of the outgoing meson and AŽ E, pX . is the NN ™ NNx transition amplitude, which, for future convenience, is expressed as a function of the total energy E and the relative momentum of the two nucleons in the final state pX Žnote, that pX and qX are related to each other via energy conservation.. The phase space is denoted by d r Ž qX . and h denotes the maximum momentum of the emitted meson in units of its mass. Watson w1x argues that if there is a strong and attractive force between two of the outgoing particles, as is the case for the reactions under consideration, the energy dependence of the total cross-section is determined
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 3 7 7 - 9
C. Hanhart, K. Nakayamar Physics Letters B 454 (1999) 176–180
177
Fig. 1. Decomposition of the production amplitude in the final state NN interaction and the production part. Here C denotes the nucleon–nucleon wave function and T stands for the NN T-matrix.
by the phase space and the energy dependence of the relevant attractive interaction, i.e.,
sN N ™ N N x Ž h . A
A
m xh
H0
m xh
H0
d r Ž qX . < T Ž pX , pX . < 2
d r Ž qX .
ž
sin d Ž pX . pX
2
/
,
Ž 1.
In the above equation T Ž pX , pX . is the on-shell NN T-matrix, d Ž pX . denotes the NN phase shifts at the energy, 2 EŽ pX ., of the final NN subsystem Žhere restricted to s-waves., where EŽ pX . ' pX 2r2 m, with m denoting the nucleon mass. When data for the reaction pp ™ ppp 0 close to threshold became available w2x Eq. Ž1. indeed turned out to give the correct energy dependence of the total cross-section w3x. Several authors w4–9x concluded from this observation that it is appropriate to calculate the transition NN ™ NNx to lowest order in perturbation theory and just include the final state interaction ŽFSI. by using a formula of the type in Eq. Ž1.; they implement the FSI by use of just the on-shell NN T-matrix, not only to get the right energy dependence of the cross-section, but also to get the strength of the matrix elements. In this letter we criticize this procedure. We shall demonstrate that the observation that the energy dependence of the cross-section is given by the on-shell FSI does not necessarily imply that the strength of the matrix elements is also determined by the on-shell NN interaction. We also show that Watson’s requirement that the FSI be attractive in order to obtain the energy dependence of the cross-section given by Eq. Ž1. is unnecessary. Finally we give an expression that allows one to estimate the
effect of the initial state interaction ŽISI. on the reaction NN ™ NNx, with x any meson heavier than the pion, in terms of the Žon-shell. NN scattering phase shifts and inelasticities. The starting point of the present investigation is the decomposition of the total transition amplitude into a production amplitude, hereafter called M, and the NN FSI Žsee also Fig. 1.. The decomposition is to be done in such a way that all the NN interactions taking place after the meson is produced are regarded as part of the FSI Ža more formal definition of the FSI can be given based on the last cut lemma w10x.. As M is not specified, no approximation is involved in this decomposition. Schematically we can write A s M q TGM . An integration over the intermediate momenta is needed to evaluate the second term on the right hand side. This is actually the term where the off-shell information of both the NN T-matrix and M enters, as will become clear below. To be concrete, we use non-relativistic kinematics for simplicity. The generalization to a fully relativistic treatment is straightforward and does not provide any new insights. In addition, since we only want to investigate effects of the FSI on the energy dependence of the total crosssection, overall constant factors are dropped. Using w11x
G Ž E,k . s P
1 E y 2 EŽ k .
y i pd Ž E y 2 E Ž k . . ,
C. Hanhart, K. Nakayamar Physics Letters B 454 (1999) 176–180
178
where P denotes the principal value, we write the total transition amplitude A in the form
½
X
i
P Ž E, pX .
apX
5
.
Ž 2.
P Ž E, p . s
apX
`
X
kŽ p . 2a
P
H0
`
dk
H0
s
p
dk
,
X
i =
X
X
Ž h Ž pX . e i d Ž p . y eyi d Ž p . .
1 apX
P Ž E, pX . .
Ž 6.
For energies near the production threshold energy one has h Ž pX . s 1, so that, X
A Ž E, pX . s M Ž E, pX . e i d Ž p .
y
sin Ž d Ž pX . . apX
P Ž E, pX . .
Ž 7.
Using the effective range expansion X
X
p cot Ž d Ž p . . s y
T Ž p ,k . M Ž E,k . X
1 y
Ž 3.
X
X
X
= Ž h Ž pX . e i d Ž p . q eyi d Ž p . .
= cos Ž d Ž pX . .
with the function f defined as f Ž E,k . s
A Ž E, pX . s 12 M Ž E, pX . e i d Ž p .
E y 2 EŽ k .
pX 2 y k 2
Ž 5.
where h Ž pX . denotes the inelasticity. Substituting Eq. Ž5. into Eq. Ž2., we get for the transition amplitude
k 2 f Ž E,k .
k 2 f Ž E,k . y pX 2
X
Ž h Ž pX . e 2 i d Ž p . y 1 . , 2
X
X
where k Ž p . s p p mr2 is the phase space density; the factor of 1rapX , with a denoting the low-energy NN scattering length, has been introduced for further convenience. Also, for convenience, we display only those arguments of M that are relevant for the present discussion, that is the total energy E and the relative momentum pX of the two nucleons in the final state. As pointed out in Ref. w1x, M depends weakly on E if the production mechanism is short ranged. In the above equation, all the off-shell effects are contained in the function P Ž E, pX ., whose explicit form is X
i
k Ž pX . T Ž pX , pX . s
A Ž E, pX . s M Ž E, pX . 1 y i k Ž pX . T Ž pX , pX .
= 1q
latter formula use has been made of the fact that the on-shell T-matrix and the phase shifts are related by
X
1
1 q
a
2
`
L
2
Ý rn ns0
T Ž p , p . M Ž E, p .
pX 2
nq 1
ž /
Ž 8.
L2
Eq. Ž7. can be further reduced to X
s
K Ž p ,k . M Ž E,k . X
X
X
K Ž p , p . M Ž E, p .
.
Ž 4.
The last equality in the above equation follows from the half-off-shell unitarity relation of the NN T-matrix, namely X
T Ž pX ,k . s 12 Ž h Ž pX . e 2 i d Ž p . q 1 . K Ž pX ,k . with the K-matrix real by definition. Therefore, all of the imaginary part of f – and therefore of P – is introduced by the production amplitude M. In the
X
A Ž E, pX . s yM Ž E, pX . e i d Ž p .
ž
sin Ž d Ž pX . . apX
/
= P Ž E, pX . q 1 y 12 ar o pX 2 y . . . .
Ž 9. This is the central formula of the present discussion. It reveals a number of important features. First of all, it shows that the energy dependence of the total cross-section is, indeed, given by Eq. Ž1. as has been shown by Watson, provided the production amplitude M Ž E, pX . and the function P Ž E, pX . have
C. Hanhart, K. Nakayamar Physics Letters B 454 (1999) 176–180
a weak energy dependence compared to that due to the FSI. Secondly, it is not necessary that the FSI be attractive in order for the total cross section to have the energy dependence given by Eq. Ž1.: as long as M Ž E, pX . and P Ž E, pX . have a weak energy dependence, the energy dependence of the total cross-section will be given by the on-shell FSI times phase space for pX 2 < Ž ar 0 .y1 . Thirdly, and most relevant to the present discussion, the above formula also shows that the strength of the amplitude AŽ E, pX . depends on the function P Ž E, pX .. As has been mentioned before, the function P Ž E, pX . summarizes all the off-shell effects of the FSI and production amplitude. As such, it is an unmeasurable and model-dependent quantity. In particular, it depends on the particular regularization scheme used. For example, in conventional calculations based on meson-exchange models, where the regularization is done by introducing form factors, the function P Ž E, pX . is very large and cannot be neglected 4 . Other regularization schemes, however, may yield a vanishing function P Ž E, pX .. Since the total amplitude AŽ E, pX . should not depend on the particular regularization scheme, the production amplitude M Ž E, pX . in Eq. Ž9. must depend on the regularization scheme in such a way to compensate for the regularization dependence of P Ž E, pX .. The above consideration shows that results from calculations aimed at quantitatiÕe predictions, such as those using the procedure of evaluating M in the on-shell tree level approximation and multiplying it with the on-shell NN T-matrix w4–9x, without consistency between the NN scattering and production amplitudes should be interpreted cautiously. All the above considerations are not restricted to the NN final states; whenever there is a strong two-particle correlation in the final state, the energy dependence of a total production cross-section is given by the on-shell phase shifts of two of the outgoing particles. This condition is for example also met in the reaction pp ™ pK L, as demonstrated in Ref. w12x. The situation is very different for the effect of the NN interaction, responsible for the initial state distor-
tions. Since the kinetic energy of the initial state has to be large enough to produce a meson, the NN ISI is evaluated at large energies. Therefore, in this regime we expect the variation with energy of the ISI to be small. 5 At least in the case of meson-exchange models this implies a flat off-shell behavior of the NN T-matrix at a given energy, in which case the principal value integral is expected to be small, as can be seen from Eq. Ž3.. It is this observation that allows us to use Eq. Ž6. to estimate the effect of the ISI on the total production cross-section for the production of heavier mesons. The ISI therefore leads to a reduction of the total cross-section of the order of
l s 12 e i d LŽ p. Ž hLŽ p . e i d LŽ p . q eyi d LŽ p . .
We checked this numerically.
2
s hLŽ p . cos 2 Ž d LŽ p . . q 14 1 y hLŽ p . F 14 1 q hLŽ p .
2
,
2
Ž 10 .
where p denotes the relative momentum of the two nucleons in the initial state with the total energy E. The index L indicates the quantum numbers of the corresponding initial state. Note that, for production reactions close to threshold, selection rules strongly restrict the number of allowed initial states. In the literature there is one example that quantifies the effect of the ISI for meson production reactions, namely Ref. w13x, where the reaction pp ™ pph is studied. The inclusion of the ISI in this work leads to a reduction of the total cross-section by roughly a factor of 0.3. At threshold only the L s 3 P0 state contributes to the ISI. The phase shifts and inelasticities given by the model used in w13x for the ISI are d LŽ p . s y60.78 and hLŽ p . s 0.57 w14x at TLab s 1250 MeV. These values agree with the phase shift analysis given by the SAID program w15x. Both phase shifts and the inelasticity vary by 10% only over an energy range of 500 MeV w15x. Using the above mentioned values for d LŽ p . and hLŽ p . we get
5
4
179
In case of pion production the phase shifts of the 3 P0 partial wave, which is the initial state for the s-wave p 0 production, still vary reasonably rapidly with energy. Therefore we do not expect the principal value integral to be small.
180
C. Hanhart, K. Nakayamar Physics Letters B 454 (1999) 176–180
for the reduction factor l, defined in Eq. Ž10., a value of 0.2. Therefore, in the case of the kinematics of the ISI for the h production, the principal value integral – within the meson-exchange model used – indeed turns out to be a correction of the order of 20% compared to the leading on-shell contribution. In summary, our primary point has been to demonstrate that, for the purpose of achieving quantitative predictions, FSI must be treated cautiously and in a way which is consistent with the corresponding production amplitude. Our criticism is not to the result of Ref. w1x. In fact, our function P Ž E, pX . appearing in Eq. Ž2. is related to the factor Ž f Ž r ., R . in Eq. Ž32. of Ref. w1x, where f Ž r . accounts for the short-range behavior of the strongly interacting particles in the final state. Note that Watson w1x does not give a prescription how to calculate the overlap integral Ž f Ž r ., R ., which would be required to fix the overall normalization. We emphasize that we do not claim that off-shell effects are measurable w16x. The result of this paper is the demonstration of the necessity to properly account for loop effects of the FSI in situations where the latter strongly influences the energy dependence of the total cross-section as in meson production in NN collisions. In addition, we have given a compact formula that allows one to estimate the effect of the ISI in terms of the phase shifts and inelasticities of NN scattering. This formula should prove to be useful for theoretical investigations of the production of heavy mesons close to their production threshold.
Acknowledgements We thank J. Durso, J. Haidenbauer, Th. Hemmert and N. Kaiser for useful discussions and W. Melnitchouk for careful reading of the manuscript. One of the authors ŽC.H.. is grateful for the financial support by COSY FFE–Project Nr. 41324880.
References w1x K. Watson, Phys. Rev. 88 Ž1952. 1163. w2x H.O. Meyer et al., Phys. Rev. Lett. 65 Ž1990. 2846; Nucl. Phys. A 539 Ž1992. 633. w3x G. Miller, P. Sauer, Phys. Rev. C 44 Ž1991. 1725. w4x A. Moalem, Nucl. Phys. A 589 Ž1995. 649. w5x R. Shyam, U. Mosel, Phys. Lett. B 426 Ž1998. 1. w6x A. Sibirtsev, W. Cassing, nucl-thr9802025. w7x A. Sibirtsev, W. Cassing, Eur. Phys. J. A 2 Ž1998. 333. w8x E. Gedalin et al., nucl-thr9803028. w9x V. Bernard, N. Kaiser, U.-G. Meißner, Eur. Phys. J. A, accepted for publication, and nucl-thr9806013. w10x J.G. Taylor, Phys. Rev. 150 Ž1966. 1321. w11x M. Goldberger, K. Watson, Collision Theory, Wiley, New York, 1964. w12x J.T. Balewski, Eur. Phys. J. A 2 Ž1998. 99. w13x M. Batinic, A. Svarc, T.-S.H. Lee, Phys. Scripta 56 Ž1997. 321. w14x T.-S.H. Lee, private communication. w15x Extracted from the VIRGINIA TECH PARTIAL-WAVE ANALYSES ON-LINE Ž http:rrclsaid.phys.vt.edur CAPSrsaid_branch.html. w16x For a recent discussion, see H.W. Fearing, Phys. Rev. Lett. 81 Ž1998. 758.
Physics Letters B 454 Ž1999. 176–180
On the treatment of NN interaction effects in meson production in NN collisions C. Hanhart
a,b,c,1
, K. Nakayama
b,d,2
a
Institut fur ¨ Theoretische Kernphysik, UniÕersitat ¨ Bonn, D-53115 Bonn, Germany Institut fur GmbH, D-52425 Julich, Germany ¨ Kernphysik, Forschungszentrum Julich ¨ ¨ c Department of Physics and INT, UniÕersity of Washington, Seattle, WA 98195, USA 3 d Department of Physics and Astronomy, UniÕersity of Georgia, Athens, GA 30602, USA b
Received 24 September 1998; received in revised form 2 March 1999 Editor: J.-P. Blaizot
Abstract We clarify under what circumstances the nucleon–nucleon final state interaction fixes the energy dependence of the total cross-section for the reaction NN ™ NNx close to production threshold, where x can be any meson whose interaction with the nucleon is not too strong. It is shown that the results obtained from the procedure used recently by several authors to include the final state interaction in the reactions under discussion should be interpreted with caution. In addition, we give a formula that allows one to estimate the effect of the initial state interaction for the production of heavy mesons. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 25.40.y h; 13.75.Cs Keywords: Final state interaction; Initial state interaction; Meson production
As early as 1952 K. Watson pointed out under what circumstances one expects the final state interaction to strongly modify the energy dependence of the total production cross-section NN ™ NNx w1x given by
sN N ™ N N x Ž h . A
1
m xh
H0
d r Ž qX . < A Ž E, pX . < 2 .
E-mail: [email protected] E-mail: [email protected] 3 Present address. 2
In the above equation qX is the momentum of the outgoing meson and AŽ E, pX . is the NN ™ NNx transition amplitude, which, for future convenience, is expressed as a function of the total energy E and the relative momentum of the two nucleons in the final state pX Žnote, that pX and qX are related to each other via energy conservation.. The phase space is denoted by d r Ž qX . and h denotes the maximum momentum of the emitted meson in units of its mass. Watson w1x argues that if there is a strong and attractive force between two of the outgoing particles, as is the case for the reactions under consideration, the energy dependence of the total cross-section is determined
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 3 7 7 - 9
C. Hanhart, K. Nakayamar Physics Letters B 454 (1999) 176–180
177
Fig. 1. Decomposition of the production amplitude in the final state NN interaction and the production part. Here C denotes the nucleon–nucleon wave function and T stands for the NN T-matrix.
by the phase space and the energy dependence of the relevant attractive interaction, i.e.,
sN N ™ N N x Ž h . A
A
m xh
H0
m xh
H0
d r Ž qX . < T Ž pX , pX . < 2
d r Ž qX .
ž
sin d Ž pX . pX
2
/
,
Ž 1.
In the above equation T Ž pX , pX . is the on-shell NN T-matrix, d Ž pX . denotes the NN phase shifts at the energy, 2 EŽ pX ., of the final NN subsystem Žhere restricted to s-waves., where EŽ pX . ' pX 2r2 m, with m denoting the nucleon mass. When data for the reaction pp ™ ppp 0 close to threshold became available w2x Eq. Ž1. indeed turned out to give the correct energy dependence of the total cross-section w3x. Several authors w4–9x concluded from this observation that it is appropriate to calculate the transition NN ™ NNx to lowest order in perturbation theory and just include the final state interaction ŽFSI. by using a formula of the type in Eq. Ž1.; they implement the FSI by use of just the on-shell NN T-matrix, not only to get the right energy dependence of the cross-section, but also to get the strength of the matrix elements. In this letter we criticize this procedure. We shall demonstrate that the observation that the energy dependence of the cross-section is given by the on-shell FSI does not necessarily imply that the strength of the matrix elements is also determined by the on-shell NN interaction. We also show that Watson’s requirement that the FSI be attractive in order to obtain the energy dependence of the cross-section given by Eq. Ž1. is unnecessary. Finally we give an expression that allows one to estimate the
effect of the initial state interaction ŽISI. on the reaction NN ™ NNx, with x any meson heavier than the pion, in terms of the Žon-shell. NN scattering phase shifts and inelasticities. The starting point of the present investigation is the decomposition of the total transition amplitude into a production amplitude, hereafter called M, and the NN FSI Žsee also Fig. 1.. The decomposition is to be done in such a way that all the NN interactions taking place after the meson is produced are regarded as part of the FSI Ža more formal definition of the FSI can be given based on the last cut lemma w10x.. As M is not specified, no approximation is involved in this decomposition. Schematically we can write A s M q TGM . An integration over the intermediate momenta is needed to evaluate the second term on the right hand side. This is actually the term where the off-shell information of both the NN T-matrix and M enters, as will become clear below. To be concrete, we use non-relativistic kinematics for simplicity. The generalization to a fully relativistic treatment is straightforward and does not provide any new insights. In addition, since we only want to investigate effects of the FSI on the energy dependence of the total crosssection, overall constant factors are dropped. Using w11x
G Ž E,k . s P
1 E y 2 EŽ k .
y i pd Ž E y 2 E Ž k . . ,
C. Hanhart, K. Nakayamar Physics Letters B 454 (1999) 176–180
178
where P denotes the principal value, we write the total transition amplitude A in the form
½
X
i
P Ž E, pX .
apX
5
.
Ž 2.
P Ž E, p . s
apX
`
X
kŽ p . 2a
P
H0
`
dk
H0
s
p
dk
,
X
i =
X
X
Ž h Ž pX . e i d Ž p . y eyi d Ž p . .
1 apX
P Ž E, pX . .
Ž 6.
For energies near the production threshold energy one has h Ž pX . s 1, so that, X
A Ž E, pX . s M Ž E, pX . e i d Ž p .
y
sin Ž d Ž pX . . apX
P Ž E, pX . .
Ž 7.
Using the effective range expansion X
X
p cot Ž d Ž p . . s y
T Ž p ,k . M Ž E,k . X
1 y
Ž 3.
X
X
X
= Ž h Ž pX . e i d Ž p . q eyi d Ž p . .
= cos Ž d Ž pX . .
with the function f defined as f Ž E,k . s
A Ž E, pX . s 12 M Ž E, pX . e i d Ž p .
E y 2 EŽ k .
pX 2 y k 2
Ž 5.
where h Ž pX . denotes the inelasticity. Substituting Eq. Ž5. into Eq. Ž2., we get for the transition amplitude
k 2 f Ž E,k .
k 2 f Ž E,k . y pX 2
X
Ž h Ž pX . e 2 i d Ž p . y 1 . , 2
X
X
where k Ž p . s p p mr2 is the phase space density; the factor of 1rapX , with a denoting the low-energy NN scattering length, has been introduced for further convenience. Also, for convenience, we display only those arguments of M that are relevant for the present discussion, that is the total energy E and the relative momentum pX of the two nucleons in the final state. As pointed out in Ref. w1x, M depends weakly on E if the production mechanism is short ranged. In the above equation, all the off-shell effects are contained in the function P Ž E, pX ., whose explicit form is X
i
k Ž pX . T Ž pX , pX . s
A Ž E, pX . s M Ž E, pX . 1 y i k Ž pX . T Ž pX , pX .
= 1q
latter formula use has been made of the fact that the on-shell T-matrix and the phase shifts are related by
X
1
1 q
a
2
`
L
2
Ý rn ns0
T Ž p , p . M Ž E, p .
pX 2
nq 1
ž /
Ž 8.
L2
Eq. Ž7. can be further reduced to X
s
K Ž p ,k . M Ž E,k . X
X
X
K Ž p , p . M Ž E, p .
.
Ž 4.
The last equality in the above equation follows from the half-off-shell unitarity relation of the NN T-matrix, namely X
T Ž pX ,k . s 12 Ž h Ž pX . e 2 i d Ž p . q 1 . K Ž pX ,k . with the K-matrix real by definition. Therefore, all of the imaginary part of f – and therefore of P – is introduced by the production amplitude M. In the
X
A Ž E, pX . s yM Ž E, pX . e i d Ž p .
ž
sin Ž d Ž pX . . apX
/
= P Ž E, pX . q 1 y 12 ar o pX 2 y . . . .
Ž 9. This is the central formula of the present discussion. It reveals a number of important features. First of all, it shows that the energy dependence of the total cross-section is, indeed, given by Eq. Ž1. as has been shown by Watson, provided the production amplitude M Ž E, pX . and the function P Ž E, pX . have
C. Hanhart, K. Nakayamar Physics Letters B 454 (1999) 176–180
a weak energy dependence compared to that due to the FSI. Secondly, it is not necessary that the FSI be attractive in order for the total cross section to have the energy dependence given by Eq. Ž1.: as long as M Ž E, pX . and P Ž E, pX . have a weak energy dependence, the energy dependence of the total cross-section will be given by the on-shell FSI times phase space for pX 2 < Ž ar 0 .y1 . Thirdly, and most relevant to the present discussion, the above formula also shows that the strength of the amplitude AŽ E, pX . depends on the function P Ž E, pX .. As has been mentioned before, the function P Ž E, pX . summarizes all the off-shell effects of the FSI and production amplitude. As such, it is an unmeasurable and model-dependent quantity. In particular, it depends on the particular regularization scheme used. For example, in conventional calculations based on meson-exchange models, where the regularization is done by introducing form factors, the function P Ž E, pX . is very large and cannot be neglected 4 . Other regularization schemes, however, may yield a vanishing function P Ž E, pX .. Since the total amplitude AŽ E, pX . should not depend on the particular regularization scheme, the production amplitude M Ž E, pX . in Eq. Ž9. must depend on the regularization scheme in such a way to compensate for the regularization dependence of P Ž E, pX .. The above consideration shows that results from calculations aimed at quantitatiÕe predictions, such as those using the procedure of evaluating M in the on-shell tree level approximation and multiplying it with the on-shell NN T-matrix w4–9x, without consistency between the NN scattering and production amplitudes should be interpreted cautiously. All the above considerations are not restricted to the NN final states; whenever there is a strong two-particle correlation in the final state, the energy dependence of a total production cross-section is given by the on-shell phase shifts of two of the outgoing particles. This condition is for example also met in the reaction pp ™ pK L, as demonstrated in Ref. w12x. The situation is very different for the effect of the NN interaction, responsible for the initial state distor-
tions. Since the kinetic energy of the initial state has to be large enough to produce a meson, the NN ISI is evaluated at large energies. Therefore, in this regime we expect the variation with energy of the ISI to be small. 5 At least in the case of meson-exchange models this implies a flat off-shell behavior of the NN T-matrix at a given energy, in which case the principal value integral is expected to be small, as can be seen from Eq. Ž3.. It is this observation that allows us to use Eq. Ž6. to estimate the effect of the ISI on the total production cross-section for the production of heavier mesons. The ISI therefore leads to a reduction of the total cross-section of the order of
l s 12 e i d LŽ p. Ž hLŽ p . e i d LŽ p . q eyi d LŽ p . .
We checked this numerically.
2
s hLŽ p . cos 2 Ž d LŽ p . . q 14 1 y hLŽ p . F 14 1 q hLŽ p .
2
,
2
Ž 10 .
where p denotes the relative momentum of the two nucleons in the initial state with the total energy E. The index L indicates the quantum numbers of the corresponding initial state. Note that, for production reactions close to threshold, selection rules strongly restrict the number of allowed initial states. In the literature there is one example that quantifies the effect of the ISI for meson production reactions, namely Ref. w13x, where the reaction pp ™ pph is studied. The inclusion of the ISI in this work leads to a reduction of the total cross-section by roughly a factor of 0.3. At threshold only the L s 3 P0 state contributes to the ISI. The phase shifts and inelasticities given by the model used in w13x for the ISI are d LŽ p . s y60.78 and hLŽ p . s 0.57 w14x at TLab s 1250 MeV. These values agree with the phase shift analysis given by the SAID program w15x. Both phase shifts and the inelasticity vary by 10% only over an energy range of 500 MeV w15x. Using the above mentioned values for d LŽ p . and hLŽ p . we get
5
4
179
In case of pion production the phase shifts of the 3 P0 partial wave, which is the initial state for the s-wave p 0 production, still vary reasonably rapidly with energy. Therefore we do not expect the principal value integral to be small.
180
C. Hanhart, K. Nakayamar Physics Letters B 454 (1999) 176–180
for the reduction factor l, defined in Eq. Ž10., a value of 0.2. Therefore, in the case of the kinematics of the ISI for the h production, the principal value integral – within the meson-exchange model used – indeed turns out to be a correction of the order of 20% compared to the leading on-shell contribution. In summary, our primary point has been to demonstrate that, for the purpose of achieving quantitative predictions, FSI must be treated cautiously and in a way which is consistent with the corresponding production amplitude. Our criticism is not to the result of Ref. w1x. In fact, our function P Ž E, pX . appearing in Eq. Ž2. is related to the factor Ž f Ž r ., R . in Eq. Ž32. of Ref. w1x, where f Ž r . accounts for the short-range behavior of the strongly interacting particles in the final state. Note that Watson w1x does not give a prescription how to calculate the overlap integral Ž f Ž r ., R ., which would be required to fix the overall normalization. We emphasize that we do not claim that off-shell effects are measurable w16x. The result of this paper is the demonstration of the necessity to properly account for loop effects of the FSI in situations where the latter strongly influences the energy dependence of the total cross-section as in meson production in NN collisions. In addition, we have given a compact formula that allows one to estimate the effect of the ISI in terms of the phase shifts and inelasticities of NN scattering. This formula should prove to be useful for theoretical investigations of the production of heavy mesons close to their production threshold.
Acknowledgements We thank J. Durso, J. Haidenbauer, Th. Hemmert and N. Kaiser for useful discussions and W. Melnitchouk for careful reading of the manuscript. One of the authors ŽC.H.. is grateful for the financial support by COSY FFE–Project Nr. 41324880.
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