Eikonal Approximation

897 Progress of Theoretical Physics, Vol. 56, No. 3, September 1976

Eikonal Approximation --Backward and Transverse Effects-Takayuki MATSUKI

Department of Physics, Tokyo Institute of Technology 1\deguro-ku, Tokyo 152 (Received February 9, 1976)

§ I.

Introduction

Several years ago, Glauber 1l presented useful high-energy approximation-the eikonal approximation. Subsequently, Saxon and Schiff 2l and Wu 3l derived correction terms to Glauber's formula. Their purpose was to make Glauber's theory applicable to large momentum transfers. Considerable efforts in this direction have been devoted by many authors!} The most familiar formalism proposed until now is the one by Sugar and Blankenbecler (S-B). 5l They expanded a momentum operator p about a constant vector k and wrote the T-matrix as a perturbation series in H= (p-k)"/2m. The eikonal approximation is obtained in zeroth order in H. The correction terms derived by Saxon and Schiff and Wu are given by the first order terms in H. The characteristic of the S-B formalism is that one can at least formally calculate correction terms in each order of approximation. In fact, W allace 6l calculated corrections up to third order along almost the same line of approach as Sugar and Blankenbecler. Although the S-B formalism has the above merits, it contains approximate Green's functions which propagate only in the forward direction. Consequently, the perturbation H contains the effects of propagation in both the backward and transverse directions since the exact Green's function contains all the effects. It is possible to construct a formalism in which one separately takes into account these effects (forward, backward and transverse directions). The merits of separation of the above three effects are that we can study high-energy problems in detail, e.g., the Fresnel approximation, crossing relation, etc.

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We propose a high-energy approximation method in nonrelativistic and relativistic quantum mechanics. In our formalism, we take into account the effects of propagation in the forward, backward and transverse directions. With the backward and the transverse effect neglected, the present method gives eikonal approximation, while the Fresnel approximation is obtained from our formalism with only the backward effect neglected. In quantum field theory, as an application of our formalism, we investigate the s-u crossing problem which is closely related to backward scattering.

T. lvlatsulci

898

of the Dirac particles.

§ 2.

Nonrelativistic potential scattering

\Ve make the explicit separation of the forward, backward and transverse effects. We first subtract the effects of propagation in the transverse direction from the exact Green's function by replacing a momentum operator p by (p · nj). Here nj is usually a dimensionless, unit vector. The trajectory**) on which p is projected is characterized by nj. After such replacement, the forward and backward effects are separated according to Andrianov and Ishihara. 8),g) We construct the formalism following the above prescriptions: The T-matrix JS

g1ven by

(2·1)

T=V+ VgV,

vcvhere V(q) 1s a potential and the total Green's function g 1s written as***J

g= [k 2 /2m-p 2 /2m- V(q)

+ier

To separate the transverse effects from g, -vve project p on =

g = ~ g j (Njg j)"

(2· 2)

1 •

ni

and expand g as

(2· 3)

n=O

with

(2 4) °

*J It must be noticed that a "backward" direction here is meant in space for the nonrelativis· tic case, while in time for the relativistic case, **J Here we consider a linear classical trajectory. ***l \Ve follow the notation of Ref. 13).

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The first attempt to develop such a formalism 7l was done in one dimension by Andrianov 8J and Ishihara9J (A-I). They showed that the T-matrix in one dimension can be described in terms of forward and backward propagators. Ishihara also applied his formalism to nonrelativistic potential scattering. 10J However, his application is not so general that it is not easy how to develop the A-I formalism for a relativistic case. In this paper, we propose a general method. By this method, we can separate the Green's function into three parts representing the effects of propagation in the fonvard, backward and transverse directions. *J A combination of forward and backward effects yields the A-I formalism for both nonrelativistic and relativistic cases. It contains Ishihara's approximation10l as a special case. On the other hand, we have the Fresnel approximation by a combination of forward and transverse effects. In §§2 and 3, we construct the formalism for the nonrelativistic and relativistic cases, respectively. In § 4, we consider the s-u crossing problem for eikonal amplitudes using the results in § 3. Section 5 is devoted to discussion of the results. In § 5, we also mention many applications of our formalism, especially to scattering

899

Eikonal Approximation 1~ =

(2· 5)

p'/2m- (p · nJ '/2m .

(A)

A-I formalism

Replacing g in the T-matrix by gh Eq. (2 · 4), we obtain the approximate T-matrix which is decomposed due to Andrianov and Ishihara,sl. 9l

Tj=V+ VgjV =

(1 + Vg/- 1 ) V!lij(1 +g/+ 1 V)

(2· 6)

=

(1 + Vg/+ 1 )@j V(1 +g/- 1 V),

(2 ° 7)

where

g/j_ 1 =[vi~+ v(p ·nj)- V(q) +ie] - 1 , v=kjm,

(2·8)

!lij= [1-g/+ Vg/-)VJ1

mj= [1- Vg/- Vg/+ 1

1 ,

(2·9)

1] - 1 •

(2·10)

As expressed in Eq. (2 · 8), g/~ 1 are the total Green's functions if one takes into account only the forward or backward scattering effects. The operators l].ii and represent the effects of to-and-fro motions in a potential Let us calculate the amplitude (k1 lTilki) which includes backward scattering effects as vvell as a forward one. We use Eq. (2 · 6) for the case ni = kd k, and Eq. (2 · 7) for the case n 1 = k1/ k. Here we give the amplitude for the case nm = (ki+kf)/2k cos'(8/2).

mj

v.

I

ve must notice that s-u crossing symmetry is satisfied only superficially by the former, but actually it is satisfied by the latter. This point is discussed in detail in Appendix B.

§ 5.

Discussion

In this paper, we have proposed a general method by which the Green's function can be separated into three parts representing the effects of propagation in the forward, backward and transverse directions. The main basic idea of our method is the projection of a momentum operator on some straight line trajectory.

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Fig. 2. s-u crossed diagram in fourth order. The symbols G;l denote the propagating directions in time in each channel.

Fig. 3. By s-u crossing, GC+J goes to GH. By eikonal approximation (E.A.), GC+J goes to G; + G/->. If there is a vector field AP (x), we must also replace AP (x) by A (x) · u1 .

Acknowledgements The author would like to thank Dr. M. Yamazaki for valuable discussions of the results in § 4. He also wishes to appreciate helpful discussions with his colleagues.

Appendix A In this Appendix, we give the explicit expressions for nh Nj and G/±> for the modified Abarbanel-ltzykson (AI) and Wallace (W) approximations, and also for our approximation in § 3 to make a comparison easy. We have for our approximation,

-;k

nm=k

e

cos 2 2'

Nm=p 2 /2m- (p·nm) 2 /2m,

Gmc±J= [ (k/ m

cos 2 ~) (k=fp)

+ ier1,

(A·1)

*> Andrianov•> and Ishihara•> derived independently Eq. (2·6) in one dimension. Andrianov decomposed, however, the T-matrix further and expressed the scattering amplitude in terms of a phase function. Recently this extension in the nonrelativistic case was done by Andrianov him· self. See Ref. 20).

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G=- [P·r-m+ie]- 1

T. lvfatsuki

906

where k=(ki+k1)/2, for \V approximation,

I

nw = k k cos 312

~

,

Nw= p 2/2m- (p ·nw)'/2m -!?}(1- cos

~)/2m, (A·2)

and for AI approximation, =k-; !?cos

28 ,

(A·3) vV e can consider the relation among these approximations as follows: The velocities along h"t +k1 are v/cos (8/2), v and vcos (8/2) for our, Wand AI approximations, respectively. The approximation (A· 3) was obtained by Ishihara. 10 l

Appendix B In this Appendix, we g1ve the explicit expresswns for uj, ]\Ti and G/=l that make the eikonal amplitude in § 4 crossing symmetric. In order to do so, it is necessary to take j = j' =AI or m. vVe ha,·e for AI(Abarbanel-Itzykson) approximation,*) UA]

= Pk! )2J5k 2 '

1VAI=P 2 -

(P·uAIY-m'+ Pk 2,

. ]~1 c+) G AI pk ' + ZS pk · P - 2= - [ ± 2where !? = 1, 2 and

pk =

,

(B·1)

(pk +p/) /2, and for our approximation,

N"' =P'- (P · um)',

Gmu = - [ ± 2m(um · P) -2m' +is]~~.

(B·2)

References 1) 2) *l

R. ]. Glauber, Lectures in Theoretical Physics (lnterscence Publishers, Inc., New York, 1958), p. 315. D. S. Saxon and L. I. Schiff, Nuovo Cim. 6 (1957), 614. This approximation is modified so as to contain a backward propagator.

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nA 1

Eikonal Approximation 3) 4)

5) 6) 7)

11) 12) 13) 14) 15) 16)

17) 18) 19)

20)

T. T. Wu, Phys. Rev. 108 (1957), 466. Many references are found in: C. J. Joachain and C. Quigg, Rev. Mod. Phys. 46 (1974), 279. However, the following interesting paper is dismissed in this paper: M. Obu, Prog. Theor. Phys. 50 (1973), 147. R. L. Sugar and R. Blankenbecler, Phys. Rev. 183 (1969), 1387. S. J. Wallace, Phys. Rev. Letters 27 (1971), 622; Ann. of Phys. 78 (1973), 190. W. Tobocman and M. Pauli, Phys. Rev. D5 (1972), 2088. T. W. Chen and D. W. Hoock, Phys. Rev. D12 (1975), 1765. In the former paper, backscattering was already introduced. In the latter paper, they proposed a formalism in which they took into account only one effect of backward scattering. Their equation (15) corresponds to (1 + VG,H) V(1 +g;'+> V) in our formalism. Here G, denotes a free backward propagator. Compare this expression with our equation (2·6). A. A. Andrianov, Yad. Fiz. 20 (1974), 589 [Soviet J. Nucl. Phys. 20 (1975), 316]. T. Ishihara, Prog. Theor. Phys. 54 (1975), 1106. T. Ishihara, TIT preprint TIT/HEP-18 (1975) [to be published in Prog. Theor. Phys. 56 No. 5 (1976)]. H. D. I. Abarbanel and C. Itzykson, Phys. Rev. Letters 23 (1969), 53. E. Kujawski, Phys. Rev. D4 (1971), 2573; Ann. of Phys. 74 (1972), 567. Y. Tikochinsky, Phys. Letters 29B (1969), 270. T. Matsuki, Prog. Theor. Phys. 55 (1976), 751. K. Gottfied, Ann. of Phys. 66 (1971), 868. W. E. Frahn and B. Schiirmann, ibid. 84 (1974), 147. M. Levy and J. Sucher, Phys. Rev. 186 (1969), 1656. Many references about the eikonal approximation are found in Sov. Phys.-Theor. Math. Phys.; B. M. Barbashov and V. N. Pervushin, Soviet Phys.-Theor. Math. Phys. 3 (1970), 537. B. M. Barbashov, S. P. Kuleshov, V. A. Matveev and A. N. Sisakyan, Soviet Phys.-Theor. Math. Phys. 3 (1970), 555. B. M. Barbashov and V. V. Nesterenko, Soviet Phys.-Theor. Math. Phys. 4 (1970), 841, etc. A. A. Andrianov, Teor. Mat. Fiz. 17 (1973), 407 [Soviet Phys.-Theor. Math. Phys. 17 (1973)' 1234]. J. P. Hamad, Ann. of Phys. 91 (1975), 413. V. N. Pervushin, Teor. Mat. Fiz. 9 (1971), 264 [Soviet Phys.-Theor. Math. Phys. 9 (1971), 1127]. Q. Bui-Duy, Phys. Rev. Dll (1975), 1635. These authors derived the high-energy approximation for a Dirac particle. A. A. Andrianov, Yad. Fig. 22 (1975), 385 [Soviet J. Nucl. Phys. 22 (1976), 198].

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8) 9) 10)

907