научная статья по теме ON RESCATTERING EFFECTS IN THE REACTION π -D → π -D Физика

Текст научной статьи на тему «ON RESCATTERING EFFECTS IN THE REACTION π -D → π -D»

ЯДЕРНАЯ ФИЗИКА, 2004, том 67, № 4, с. 764-768

= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ ON RESCATTERING EFFECTS IN THE REACTION n-d — n-d

© 2004 V. V. Baru1),2), A. E. Kudryavtsev2), V. E. Tarasov2)*

Received January 21, 2003; in final form, April 20,2003

Rescattering corrections to the impulse approximation for the processes Yd — n0d and n-d — n-d are discussed. It is shown that the rescattering effects give nonnegligible contribution to the real part of these amplitudes. At the same time, the contributions from the imaginary parts of impulse and rescattering corrections drastically cancel each other. This cancellation means that the processes n-d — n0nn and Yd — nn/n-pp, when the nucleon pair is in the spin triplet state, are strongly suppressed near threshold as required by the Pauli principle.

1. INTRODUCTION

The study of the reactions Yd — n0d and n-d — — n-d near threshold has attracted continuous attention in the past few decades. Moreover, the new experimental data appeared due to recent success of the accelerator technologies stimulate increasing theoretical interest in this field. In this paper we would like to concentrate on the rescattering effects (RE) and their role for these reactions. Indeed, these effects are found to be important in many of theoretical investigations of the reaction Yd — n0d (see, e.g., [1—4]). However, recently in [5] the discussion about the role of these effects was renewed. In particular, it was emphasized in [5] that the contribution from the two-step process Yd — n-pp — n°d (see Fig. 1a) is totally compensated by the loop corrections to the impulse approximation (LCIA) (see Fig. 1b) according to the Pauli principle for the intermediate two-nucleon states. Thus, the rescattering effects in [5] do not contribute to the process of coherent n0 photoproduction on deuteron near threshold. Obviously, this conclusion of [5] disagrees with the results of other calculations performed, e.g., in [1—4]. Let us discuss the arguments of [5] in more detail:

(i) The final n°d state has quantum numbers JP = = l- at low energies, where pion is in the S wave with respect to the deuteron. However, the only possible state of the system ppn- with li = l2 = 0 is 0- (here, li is the orbital angular momentum of the pp system and l2 is the orbital momentum of the pion relative to the pp system). Therefore the S-wave intermediate

''Institut für Kernphysik, Forschungszentrum Jülich, GmbH, Germany.

2)Institute of Theoretical and Experimental Physics, Moscow, Russia.

E-mail:tarasov@heron.itep.ru

state ppn does not contribute to the process Yd —

— n0d.

(ii) In other words, the contribution of the diagram in Fig. 1a has to be compensated by the loop corrections to the impulse approximation (Fig. 1b) because of antisymmetry of the wave function for the pair of the intermediate nucleons.

Note, that the process Yd — n0wp — n°d is allowed by quantum numbers. However, the amplitude Yn — n°n which contributes to this reaction is ^20 times smaller than the corresponding amplitude for the charged pion production.

In this paper we are going to discuss the role of rescattering effects for the process of pion—deuteron elastic scattering at low energies. The diagrams corresponding to RE and LCIA for the nd scattering are very similar to the ones for the reaction Yd —

— nd (see Fig. 1 and Figs. 2b and 2c). Therefore we will investigate the relevance of RE and the problem of the cancellation of RE and LCIA performing the calculation of the nd-scattering amplitude.

The nd-scattering length was measured with a high accuracy [6, 7] and its value coincides with the theoretical predictions (see, e.g., [8—11]).

Fig. 1. Diagrams with intermediate negative pion rescattering contributing to the process Yd ^ n0d.

a

ON RESCATTERING EFFECTS

765

In all these theoretical calculations rescattering effects (including the two-step charge-exchange process n-p ^ n0n ^ n-p) give significant contribution to the value of the pion—deuteron scattering length.

In what follows we will directly demonstrate that the real part of the rescattering diagram (see Fig. 2c) gives nonnegligible contribution to the pion— deuteron scattering length. It is not compensated by the real part of LCIA (see Fig. 2b). However, the imaginary parts of RE and LCIA cancel each other. This cancellation means that there is no contribution to observables from the nNN states forbidden by the Pauli principle.

2. CALCULATION OF THE nd-SCATTERING AMPLITUDE

Below we use a simple potential approach for the calculation of the nN-scattering amplitude. This approach was already applied to the problem of the determination of the nN-scattering length in [9]. The model utilizes a pion—nucleon potential VnN(p, q), which is required for solving the Lippmann—Schwinger equation

T = V + VGT. (1)

The S-wave nN lengths b0 and b1 are related to the scattering length anN by the equation

anN = bo + bil • t, (2)

where I and t are isospin operators for pion and nucleon, bo and bi are isoscalar and isovector scattering lengths. The analyses [9, 10] of the experimental data [6, 7] show that the absolute values of b0 and bi are small compared to the typical scale of the problem (where ¡ is the pion mass). Note also that b0 < b1. Thus, the amplitude T in Eq. (1) may be perturbatively expanded in terms of the potential VnN (p, q).

Following [9] we choose VnN in the S wave in the separable form:

VMK q) = -(X°tXlI'T)9(k)g(Q), (3)

2mWN

where g(k) = (c2 + k2)-1, mnN = mj/(m + j), k and q are the 3-momenta of the pion, and m is the nucleon mass. The cut-off parameter c characterizes the range of the nN forces, and usually it is varied in the range 2.5j < c < 5j [9, 10]. The parameters Ao and \1 are chosen in such a way as to reproduce the scattering lengths b0 and b1. In what follows we will calculate the pion—deuteron scattering amplitude up to the second order in terms of the potential VnN. With this accuracy A0 and A1 are equal to

Ao = ^ + (4)

= ^ (l-^o-M).

Corrections to these expressions are of the order of ~O(b3, bf), what is negligible.

Let us calculate the pion—deuteron scattering length using the potential VnN (see Eq. (3)).

2.1. Single-Scattering Amplitude in the Born

Approximation The diagram corresponding to this amplitude is shown in Fig. 2a. The expression for the nd amplitude

f(1V corresponding to the sum of two diagrams with the scattering of pion on proton and neutron has the form:

(5)

f w _ _

Jnd —

(2n)(l + v/md)

J dpvd(p)K-p + Vn-n ].

Here, pd(p) is the deuteron wave function in the momentum space with the normalization condition / dp^d(p) = (2n)3. Neglecting the small corrections of the order of one may take out the potential

V in Eq. (5) of the integral and then get:

f i(v) _ 2

Jnd _ 2

b0--(b20 + 2bl)

(6)

This contribution is real as it should be in the Born approximation. Note also that the value fl^ depends on the value of the parameter c.

MEPHA^ OH3HKA TOM 67 № 4 2004

x

766

BARU et al.

2.2. Single Scattering in the One-Loop Approximation

The diagram for the one-loop correction to the Born approximation is shown in Fig. 2b. We have to calculate the sum of two diagrams with the scattering

of pion on proton and neutron, taking into account the sum over all intermediate states. The expression

for the amplitude f(1dVGV corresponding to this sum has the form:

f (1)VGV f nd

I (Am) =

dp

1 + ß/md

2 I mk — ßp\ 2

m + ß

^d(p)

2п

(2mnN )2

[(Ag + л2 )I (Am = 0) + a2i (Am)]

ds g2

mk — ßp m + ß

+ s

(k + s)2 p2 (p — s)2 л k2

--'— + — + —-'— + ed - Am---¿0

2ß 2m 2m 2ß

(7)

Here, k is the 3-momentum of the initial and final pion, ed is the deutron binding energy, Am = mn- + + mp — mno — mn = 3.3 MeV is the excess energy for the charge-exchange process n—p ^ n°n in the intermediate state. For the case of the elastic rescattering Am = 0.

The integral in Eq. (7) is calculated numerically for some values of the cut-off parameter c. In the limit of large c, i.e., when c » ß, and for ß/m < 1 the integral can be calculated analytically:

fïJVGV = Ф2 + 2b\) +

+ 2i [k0 (Am = 0)(b2 + b2x

+ ko(Am)b2]

Refd = f1V ' + Refü,VGV =2bo.

This is a naive but expected result for the real part of the amplitude corresponding to the impulse

approximation. The values of Re/^ycy for the charge-exchange process n-d

n°nn

П

(8)

where we introduced the notation k2 = k2 + 2/xAm — — 2^ed. Note that k < c near the threshold.

Thus, in the limit of large c the resulting contribution from the impulse approximation (see Figs. 2a and 2b) to the real part of the nd-scattering amplitude is

(9)

d are

presented in the table for different values of parameter c. Contrary to the real part of the loop amplitude,

the imaginary part of f(1JVGV (see Eq. (8)) does not depend on c as required by the unitarity.

Now let us discuss the contribution to the pion—deuteron scattering length from the double-scattering process.

2.3. Double-Scattering Contribution

Double-scattering diagram is shown in Fig. 2c. Performing the calculation, we have the following

integral for the double-scattering amplitude f^J (see [9] for details):

f (2) = f nd

4c4

(2n)5

[(bo — bj)J (Am = 0) — b2 J (Am)]

(10)

J (Am) =

dqi dq2^d(qi )^d(q2)g2 (k + qi — q2)

(k + qi — q2)2 + (ß/m)(qi + q%) + 2ß(ed — Am) — k2 — i0 '

In the limit of large c and for /i/m l this integral is reduced to the following expression:

ä = 2(bg — h2) x

g-ik-r+iko(Am=0)r -:-dT -

— 2b2

^d(r)

g—ik-r+iko(Am)r

dr,

(11)

where ^d(r) is the deuteron wave function in the coordinate space.

In the limit of small k and k0, i.e., near the threshold, for the real part of f^J we get

Re/$ = 2(62 - 262) (I

(12)

x

r

d

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This expression is well known as a static limit for the double-scattering amplitude, see, e.g., [12] and references therein.

The imaginary part of the amplitude f^ (11) in the same limit is

Imfd = 2ko(Am = 0)(b0 - b?) - 2ko(Am)bj.

(13)

Note that this contribution i

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