PHYSICAL REVIEW D VOLUME 46, NUMBER 8 15 OCTOBER 1992

Calculation of scattering with the light-cone two-body equation in 43 theories

Chueng-Ryong Ji Department of Physics, North Carolina State University, Raleigh, North Carolina 27695

Yohanes Surya Department of Physics, The College of William and Mary, Williamsburg, Virginia 23185

(Received 6 April 1992)

The analysis of the light-cone two-body bound-state equation is extended to the scattering problem. The rotational invariance is violated in the light-cone quantization method when the Fock space is trun- cated for practical calculations. Using a simple scalar field model, we investigate the explicit rotation dependence of the two-body scattering phase shifts in the light-cone quantization approach. We find the regions of coupling constant and c.m. momentum where the rotation dependence in the phase shift is negligible. We also make a connection of our analysis with the light-cone scattering formalism recently presented by Fuda.

PACS number(s): 11.80.Et, 1 l.lO.St, 11.20.Dj, 11.80.-m

I. INTRODUCTION

The accurate calculation of scattering amplitudes and bound-state properties of systems of strongly interacting particles requires the inclusion of the fundamental rela- tivistic effects such as the correct relativistic energy- momentum relation, retardation effects, particle and an- tiparticle pair production. A conventional tool for deal- ing with the relativistic two-body problem in quantum field theory is the Bethe-Salpeter formalism [I] utilizing the Green's functions of covariant perturbation theory. Both scattering [2] and bound-state [3] problems have been analyzed for the system of two particles interacted by a third scalar particle in the ladder approximation of the Bethe-Salpeter formalism. However, this formalism has difficulties with the relative time dependence and in systematically including higher-order irreducible kernels such as crossed diagrams and vacuum fluctuations [4].

An alternative approach that can remove these difficulties and restore a systematic perturbative calcula- tion for obtaining higher accuracy is the reformulation of the covariant Bethe-Salpeter equation in the light-cone frame [ 5 ] . The light-cone quantization method provides a Hamiltonian formalism and a Fock-state representation at equal light-cone time ~ = t +z/c, which retains all of the simplicity and utility of the Schrijdinger nonrelativis- tic many-body theory [6]. This method not only suppresses the vacuum fluctuations but also systematical- ly includes cross diagrams when higher Fock-state contri- butions are taken into account.

The first step in solving the full set of coupled Fock- state equations on the light-cone is to find a simple, analytically tractable equation for the lowest-particle- number sector, and to develop a systematic perturbation theory for obtaining higher particle number states and greater accuracy. These requirements are satisfied by the simplest approximation corresponding to the lowest- order irreducible kernel without including self-energy corrections [7], i.e., the light-cone ladder approximation

(LCLA). This approximation also incorporates the im- portant relativistic effects we wish to maintain.

Some time ago, we analyzed the bound-state problem in LCLA [8]. The purpose of the present paper is to ex- tend the previous analysis of the bound-state problem to the scattering problem [9]. For simplicity, we consider a scalar field model (Wick-Cutkosky model) [3] which de- scribes the interaction of two scalar particles 4,$ with equal mass m exchanging another scalar particle x of mass h. Then the interaction Lagrangian is given by

In LCLA, the included Fock states are only two-body and three-body states and all higher Fock states are trun- cated [lo]. Such truncation of Fock space causes the problem of violating the rotational invariance [ l 11 be- cause the transverse components of the angular momen- tum (J, and J,,) in the light-cone Poincare algebra [12] contain interactions changing particle numbers in equal t +z/c. This is analogous to the problem in the cannoni- cal equal-time approach where the truncation of equal- time Fock space causes violation of the boost invariance because the boost operators contain interactions chang- ing particle numbers in equal t . From our point of view, the two-body bound-state wave function in ground state (1s) suggested by Karmanov [13] may be regarded as an example of the violation of the rotational invariance be- cause it depends not only on the relativistic relative momentum q but also on a certain unit vector 6;

where K = t m a [the dimensionless coupling constant a=g2/(16.rrm2)] and e ( q ) = d r n 2 + q 2 . In order to inter- pret Eq. (1.2) as a rotationally invariant wave function, one needs to introduce a new angular momentum around

46 3565 - @ 1992 The American Physical Society

3566 CHUENG-RYONG JI AND YOHANES SURYA 46

the unit vector fi which is not physically measurable in addition to the physical angular momentum associated with the relative momentum q. These variables q and fi are connected with the usual light-cone variables k, and x (transverse projection and the portion of the momentum of one of the particles) as follows:

In terms of k, and x , Eq. (1.2) is given by

As shown in the above example, the light-cone bound- state wave function depends on the choice of the axis 6 and loses the characteristic of rotational invariance of the wave function. This seems an unavoidable problem in the light-cone quantization method with the Fock-space truncation. On the other hand, the physical quantities calculated from the light-cone bound-state wave function such as the form-factor and bound-state spectrum are ex- pected to be rotationally invariant [6]. For example, the form factor in the lowest Fock-state contribution can be given by [14]

as a convolution of two wave functions before and after the scattering by a virtual photon with momentum q ~ = ( q +=0,q - =q:,q,). Also, one can get a relation [8] between coupling constant g and the bound-state spec- trum Musing a variational principle:

where the kernel K (x,k,;y,l,) is given in the LCLA [lo] [See Sec. I1 for the explicit expression of K (x,k,;y,l,).] Since the physically measurable quantities are obtained by integrating x and k, as shown in these examples [Eqs. (1.8) and (1.911, the results for physical quantities are ro- tationally invariant, more generally, Poincare invariant.

Similarly, in the scattering problem, a physical quanti- ty such as the phase shift is expected to be independent from the quantization axis 6. As we will see in the next section, the light-cone two-body scattering amplitude T(q,ql , f i ) possesses the unit vector fi as well as the c.m. momenta, q and q', before and after the scattering, re- spectively. This is not difficult to expect since we already discussed the 6 dependence in the bound state wave func- tion [see Eq. (1.211. Thus, if one defines the partial wave scattering amplitude (T,) just like one does in the conven- tional scattering formalism, then T, has the 6 depen- dence:

where the physical two-body scattering constrains / ql = / q'l and ^q.^ql= cose. Subsequently, the calculated phase shift from T,(q,qr,6) defined in Eq. (1.10) would have the 6 dependence, contrary to the above expectation that the physical phase shift should not depend on 8. Re- cently, the light-cone scattering formalism which avoids this problem has been presented by Fuda [15]. He used the bipolar harmonics in the partial wave expansion and removed the 6-dependence in the physical partial wave

I scattering amplitudes by summing all possible partial waves around the axis fi. In the next section, we show that the physical partial wave scattering amplitude T,( /ql ) independent from 6 in the bipolar harmonics for- malism is given by a rotational average of TJ(q,ql,fi) given by Eq. (1.10) over the fi direction:

where I q / = Iq'l. Thus, in principle, one can use the con- ventional method of phase-shift analysis developed for the nonrelativistic Schrodinger equation in order to find the &independent physical phase shift, even though it would require the calculation of T,(q,q',fi) in all fi direc- tions.

In this paper, we consider the three particular fi- directions in order to present the explicit &dependence

h

of T,(q,q1,6), i.e., n=?, 9, or 9, when the scattering plane is perpendicular to ^x direction and the initial c.m. momentum direction is in 9 direction. Two of these three directions give the extrema (maximum and minimum) of T,(q,q1,6). By the explicit calculation of the phase shifts for all three directions in various coupling constants and kinematic regions, we find the regions of the coupling constant and the c.m. momentum where the 6 depen- dence is negligible. In these regions, the results from the bipolar harmonics method should agree with ours. Out- side these regions, our results provide the boundaries (maximum and minimum) of phase shifts and the physi-

46 CALCULATION OF SCATTERING WITH THE LIGHT-CONE . . . 3567

cal phase shifts from the bipolar harmonics method should lie within these boundaries. Thus, our work indi- cates the regions where the number of coupled equations in the bipolar harmonics formalism is not needed to be large to obtain converged numerical results. We ana- lyzed the S- and P-wave phase shifts as a function of the coupling constant and the c.m. momentum. We com- pared our results with those obtained by Schwarz and Zemach [2]. They used the covariant Bethe-Salpeter equation rather than the light-cone formalism. We also compared our results with the nonrelativistic results ob- tained by Schwarz [16] and found that the relativistic effects included in our analysis give the repulsive force between two scalar particles. The same effect was found in our previous analysis of the bound-state problem [8].

In the next section, we present the details of our light- cone formulation including the connection with the bipo- lar harmonics formalism. In Sec. 111, the numerical re- sults are presented including the comparison with other relativistic phase shifts obtained by Schwarz and Zemach [2] as well as the nonrelativistic results given by Schwarz [16]. The summary and conclusions follow in Sec. IV. In

the appendix, we present the calculations of the S- and P-wave kernels for 6=%, 9, and 2.

11. FORMALISM

The derivation of the light-cone two-body equation has been presented in the Appendix of the paper by Brodsky, Ji, and Sawicki [lo]. Using the usual light-cone variables, the light-cone two-body equation in the LCLA is given by

where P P is the total four-momentum of the two-body system, P*=P0fP3 (P,=O frame is chosen), x and kl are the light-cone momentum fraction, and the transverse momentum of one of the two particles and the kernel K (x, kl;y,Il) is given by

In the bound-state problem, P'P- is given by M~ where M is the mass of the bound state ( M < 2m). However, in the scattering problem, P'P- is given by s =PPPp where s 1 (2m12. Now, let us consider the c.m, scattering system where the initial and final momenta of the first (second) particle are k( - k ) and I ( -I), respectively. In this frame, the light-cone variables are given by [see Eqs. (1.5) and (1.611

Using Eq. (2.3), one can rewrite Eq. (2.1) in terms of the kinetic variables in the c.m. system:

i s -

where the Jacobi factor appears as 1 /€(I ) and the kernel K (k , 1 , s ) is given by

Here, 6 is the direction of the spatial part chosen in the its perpendicular direction k X 1 is equivalent to the effect definition of the light-cone time T = t +r.%/c; i.e., if 6 = B , of rotating k X I in a given direction of the light-cone time then ~ = t + z / c . Because choosing different directions evolution, e.g., r=t + z / c . However, the point is that the for 6 corresponds to choosing different dynamics of the dynamics changes if the relative angle between 3 and system, the scattering kernel K(k , l ,B) in Eq. (2.5) is k X I changes. Because we are interested in the depen- changed by rotating the direction 6. The effect of rotat- dence of the phase shift on the direction 6, we fix the ing the direction 6 in a given scattering plane defined by scattering plane as the plane made by ^y and 2 and the

3568 CHUENG-RYONG JI AND YOHANES SURYA

direction of initial momentum k as 2 and then vary the direction 3.

As discussed in Sec. I, the light-cone kernel given by Eq. (2.5) is not rotationally invariant because of the 6 dependence. Thus, the light-cone wave function also has the 6 dependence. Nevertheless, the light-cone two-body equation in terms of the kinetic variables in the c.m. sys- tem is the Schrodinger-type equation in which the relativ- istic effects that we wish to maintain are contained in the kernel [Eq. (2 .5)] , Jacobi factor, and the Green's function. This allows us to attempt to apply the conventional tech- nology developed in the phase-shift analysis of the Schrodinger formulation to the present light-cone formu- lation. In the limit m >> k2, 1 the above equations [Eqs. (2.4) and (2.5)] are reduced to the Schrodinger equation with the Yukawa potential:

where the reduced mass D = m /2 , E = v'; - 2m > 0 for

and

The partial wave expansion of the scattering amplitude T ( k , 1 ) is given by

where k . l = k / 111 cose. Nowby defining V j ( / k / , / l / ) as

one can obtain the following equation from Eq. (2.7):

T , ( / k I , 111 ) = V , ( / k l , l l l )

the scattering problem. From the optical theorem [18] , one can relate the phase The conventional method to solve Eq. (2.6) in the shift 6 , ( k ) as

scattering problem is to set up an equivalent Lippman- I & ] .

Schwinger equation [17]: gr2 e sin6, T,( lk1,lk )=-- 3

(2.13)

T ( L , I I = v ( k , i i - J % V ( L , ~ ) G ( ~ ) T ( ~ , ~ ) , (2.7) p Iki

( 2 ~ ) The nonrelativistic scattering formalism given by Eqs.

where (2.6)-(2.13) is now turned to the light-cone scattering for- malism. In the light-cone formulation, the Lippmann-

- 4n-a V ( k , l ) = (2 ,8 ) Schwinger equation corresponding to Eq. (2.7) is given

( k - 1 ) 2 + h 2 by

where

and

1 G ( q ) =

q 2 - ( s / 4 - m 2 ) + i e '

Using the partial wave expansion given by

~ , ( k , i , i i ) = J ~ R T ( ~ , I , ^ ~ ) P , ( C O S ~ )

and similarly defining v,( k , l , 6 ) as

v , ( k , i , i i ) = J ~ R v ( k , i , i i ) ~ , ( c o s e ) ,

we obtain

Once the partial scattering amplitude is obtained by solving Eq. (2.19), then again using the optical theorem, the phase shift is obtained by the equation

46 CALCULATION OF SCATTERING WITH THE LIGHT-CONE . . . -

where we define T,( lkl,^n)=T,(k,l,^n) when Ikl=111. Recently, Fuda [15] introduced the partial wave scattering amplitude Tjh,lgh, as

using the bipolar harmonics yzj(^k,^n 1,

which are angular momentum eigenstates of S* and S3 with eigenvalues j ( j + 1 ) and m, respectively, where

A similar equation is valid with the T ' s replaced by the V's in Eq. (2.21). Then, the partial wave integral equations be- come

However, the expansion using the bipolar harmonics con- tains many spurious amplitudes and the only physical partial wave amplitudes are given by

~ , ( l k l ) = ~ j b , , ~ ( I k l , l k l ) . (2.25)

Using Eqs. (2.17), (2.21), and (2.22), we find that the phys- ical amplitude is nothing but the rotational average of TI ( 1 kl ,̂ n ) [Eq. (11-20)] over the 6 direction:

~ , ( I k l ) = ~ i j ~ , ~ ~ ( l k l , I k l )

= J d k d d ~ Y~~*(^L,B)T(L,I,^.)Y~~(Z,^~)

Equation (2.26) provides the connection between our analysis and the bipolar harmonics formalism presented by Fuda [15].

In this paper, we solve Eq. (2.19) and obtain the phase shift using Eq. (2.20) for three particular directions of 3, i.e., 6=?, 9, or 2. Two of our chosen directions give the extrema (maximum and minimum) of Ti( lkl ,$) and, therefore, our results provide the boundaries of the phase shifts so that the physical phase shifts from the bipolar harmonics method are located within these boundaries. We can find numerically the regions of coupling constant and c.m. momentum that give negligible 3-dependence. In these regions, our results should agree with the results from the bipolar harmonics method.

and 1 = 11 I (sin@+ cos& ), respectively. We consider the three particular cases: $=%, 9 , and 4. In order to solve Eq. (2.19), one needs to find Vl(k,l,^n) for these three cases. For $=%, the potential given by Eq. (2.15) be- comes Eq. (2.8) and the relativistic effects are coming only from the Jacobi factor and the Green's function. Thus, this case involves the least relativistic effect and the calculation of V, is the simplest of the three cases. For 6=9, the terms proportional to 3 .k vanish in Eq. (2.15) and the calculations are simpler than those in the case h n=$ where one has to deal with the comparison between two momenta Ik/ and 11 I inside the absolute magnitude in Eq. (2.15). Details of these calculations are presented in the Appendix.

We calculated both S-wave (j =0) and P-wave (j = 1) phase shifts for various coupling constants (P=a/a) and c.m. momenta. To compare our results with the relativis- tic results obtained by Schwarz and Zemach [2] using the covariant Bethe-Salpeter formalism, we fix the exchanged particle mass h = m . In Fig. 1, the S-wave results for P=0.1, 0.32, and 0.7 are shown. The dimensionless c.m. momentum sauare k2 /m2 varies from 0 to 1.2. As one can see in Fig. 1, the light-cone results for 6=?, 9 , and 2 are almost same if p L 0.3 even though the results certain- ly deviate from the nonrelativistic results as k 2 / m 2 gets larger. We also find that the relativistic result obtained by Schwarz and Zemach [2] using the covariant Bethe- Salpeter formalism is the same with the light-cone results in this region. However, as the coupling constant gets larger, one finds a significant difference in the three light- cone results for $=$, 9, and 3. For example, at P=O.7, the result for 6=2 indicates a formation of the bound state because the phase shift jumps by 7 and the curve passes 6 = a / 2 but the results for ii =^y and 2 do not ex- hibit such behaviors. As discussed above, the a=? case has the least relativistic effect and the potential is more attractive than the cases of $=^y and 2 . In fact, the non-

111. NUMERICAL RESULTS relativistic result had the bound state for this coupling as shown in Fig. 1. The results by Schwarz and Zemach [2]

AS indicated in Secs. I and 11, the c.m. momenta before in Fig. 1 are also comparable with the light-cone results and after the scattering of a particle are given by k = lkl2 of 6=9 and 2. As gets even larger, the nonrelativistic

3570 CHUENG-RYONG JI AND YOHANES SURYA 46 -

I " : " ' I " ' ' '

- - - - LC X-DIR - A - - -- - - - - LC Y-DIR LC 2 -DIR

- - - - - BETHE-SALPETER I SCHRODINGER -

FIG. 1. S-wave phase shifts with the coupling constants a

P=-=0.1, 0.32, and 0.7 and the exchanged particle mass 5'

A=m. The dot-dashed, dot-double-dashed, and solid curves are the light-cone scattering results with ^n=R 9, and 4, respective- ly. These results are compared with the covariant Bethe- Salpeter result [2] and the nonrelativistic results [16] are given by dashed and dotted curves, respectively.

results are so much different from our light-cone results and the covariant Bethe-Salpeter results. Therefore, we do not include the nonrelativistic results in Fig. 2 which shows the results for p= 1, 5, and 10. For each coupling in Fig. 2, we find that the results for ^n=? and ^y give the upper and lower bounds, respectively, in the phase shift. The result for 6 ="iies between two boundary curves and is similar to the covariant Bethe-Salpeter results.

In Fig. 3, we also present the P-wave phase shift results for p= 1 and 3. Similar observation discussed for S-wave phase shift can be made in Fig. 3. For the P wave, even at p= 1, one cannot find a noticeable 6 dependence. Ex- cept the nonrelativistic result, all the light-cone results of A n=%, 9, and 2 and the covariant Bethe-Salpeter result

LC X-DIR LC Y-DIR 1 LC Z-DIR BETHE-SALPETER - - - - -

1 0 -

FIG. 2. S-wave phase shifts with the coupling constants a D=-= 1, 5, and 10. 'T

t

0 8 - LC X-DIR - - - . - - . - - . - - LC Y-DIR 5 i LC Z-DIR t B E T H E - S I U P E T E R

0 4 C . . . . . . . . SCHRODINGER C

FIG. 3. P-wave phase shifts with the coupling constants

0 = ~ = 1 and3. 'T

agree with each other if P < 1 As the coupling gets larger than p= 1, one begins to see the 6 dependence; the cases ^n=% and ^n=? yield the maximum and the minimum of the phase shifts, respectively, and the result from $ 2 2 and the covariant Bethe-Salpeter results are inside these two boundaries.

In Fig. 4, the P-wave phase-shift results are given for even larger coupling constants P=5, 10, and 20. In this figure, we again drop the curve for the nonrelativistic re- sults because of the large difference from the relativistic curves and compare only the light-cone results and the covariant Bethe-Salpeter result. One sees the P-wave res- onance phenomena due to the centrifugal barrier in the potential as the coupling constant reaches p=5, because

7T the curves for ^n=? and ^n=?? pass 6= - but do not show

2 the jump by 6= 7~ at k2/m 2=0. As shown in this figure, at p=20 and k2/m 2 1, we find a dramatic difference in the light-cone prediction especially for the ^n=P case be-

2 5 I " / " " I I '

- - - - t - - - - - - - - LC X-DIR LC Y-DIR LC Z-DIR - - - - - BETHE-SALPETER -

4

FIG. 4. P-wave phase shifts with the coupling constants a 0= - = 5, 10, and 20. 'T

46 CALCULATION OF SCATTERING WITH THE LIGHT-CONE . . . 3571

cause the result for 6=& drops significantly in this region and is even much lower than the results from 6=9. Thus, we see the significant 6 dependence as the coupling constant and the c.m. momentum become significantly large (01 20, k 2 / m 2 1 1). This seems to indicate that the number of coupled equations in the bipolar harmonics formalism would be larger to obtain converged numerical results in these regions.

IV. SUMMARY AND CONCLUSION

In this paper, we investigate the scattering problem in the light-cone formalism using a simple scalar field model presented by Wick and Cutkosky [3]. Practical computa- tions using the light-cone quantization method require, in general, the truncation of the higher Fock states. How- ever, the calculated scattering amplitude in the truncated Fock space is not rotationally invariant because the transverse angular momentum operator whose direction is perpendicular to the direction of the quantization axis in the light-cone quantization method involves the in- teraction that changes the particle number. Thus, one needs to develop the light-cone scattering formalism which yields the rotationally invariant physical quantities because the physical quantity should be completely Poin- care invariant.

Recently, Fuda [15] presented such formalism using the bipolar harmonics expansion that generates the rota- tionally invariant partial-wave scattering amplitude, T,( 1 kl ). As shown in Eq. (2.261, T,( k ) is given by a ro- tational average of the 6-dependent partial-wave scatter- ing amplitude T.(lkl ,$) over the 6 direction. The calcu- lation of T,( lkl,6) can be given by the conventional partial-wave expansion method analogous to the nonrela- tivistic scattering formalism. In Sec. 11, the light-cone two-body scattering equation is transformed to the Schrodinger-type equation and the calculation of Ti( / kl , 6 ) becomes straightforward.

In the present paper, we chose three particular direc- tions of 3 , two of which are supposed to give the extrema of the partial-wave scattering amplitudes. In Sec. 111, we presented the numerical results of the phase shifts for the considered scalar field model. For the small coupling constant (p 5 0.3 for the s wave and p i 1 for the p wave), the results do not show any observable 6 dependence and our results completely agree with the covariant Bethe- Salpeter results obtained by Schwarz and Zemach [2] while thev show a noticeable deviation from the nonrela- tivistic results. However, as the coupling constant gets larger (P > 0.3 for s wave and 8 > 1 for p wave), one be- gins to see a larger deviation for different3 directions.

As presented in Sec. 111, in general, the results with h n = 2 and 6= j i give the maximum and the minimum of the light-cone phase shifts, respectively, and the result with 6=2 and the covariant result are inside these two

boundaries. On the other hand, if the coupling is very large (for example P=20 in the p wave), the phase shift with 6=2 shows a dramatic falloff in the large c.m. momentum region (k2/m > 1) and gives the minimum of phase shifts in the light-cone scattering calculation. This indicates the large 6 dependence in the large coupling and large c.m. momenta. Thus, we expect that the num- ber of coupled equations in the bipolar harmonics formal- ism would be larger to obtain converged numerical re- sults as the coupling constant and c.m, momentum are larger.

More detailed calculation using the bipolar harmonics method to confirm such expectation is under investiga- tion.

ACKNOWLEDGMENTS

This work was supported by the Department of Energy under Contract Nos. DE-FG05-90ER40589 and SURA- 90-A90 12SD. The North Carolina Supercomputing Center is also acknowledged for the grant of Cray Y-MP time. One of us (C.R.J.) thanks M. Fuda for the useful discussion at Adelaide.

APPENDIX

To solve Eq. (2.19), one need to find Vj(k, l , 6 ) first. In this appendix, we present the calculations of V,( k , l , 6 ) for 6 = 2 , 9 and 2. The initial and final momenta of k and 1 are chosen as k = I kl2 and 1 = 11 I (sin@+cos&). In the following, we do not put -4n-a in our potential for sim- plicity.

(1) 6=% case: Since the potential in this case is given by

one can easily find Vj(k,l,%):

where the second kind of Legendre polynomial is given by Q, which has a numerical advantage in handling the singularity of V,.

(2) 6 =9 case: In this case, V( k, 1 ,9) is given by

Thus, we have

3572 CHUENG-RYONG JI AND YOHANES SURYA

where

(3) ̂ n=2 case: Since the potential in this case is given by

one needs to split the angle integration into two regions depending on whether cos0 is smaller or greater than y r ( k i d 1 )/I1 le( k ) , However, the splitting of the angle integration is required only when y < 1. Note that y L 1 if I k l L I l I a n d y = l i f m = O .

When y L 1, the potential is simplified by

where

Thus, one can easily find V,( k, 1,2) if y 1 1:

On the other hand, if y < 1, then one needs to split cos8 integration into two regions and the potential in each region is given by

~ - l ( k , 1 , 2 ) = ~ - B c o s 0 , if - 1 5 c o s 0 5 y ,

v - ' ( ~ , ~ , ~ ) = A ' - B ' c o s ~ , if y L c o s 0 5 1 , (A 10)

where

Thus, if y < 1, then we have

where t = A / B and t' = A ' / B 1 . If B = O or B1=O, then one should use the first equality in Eq. (Al2). Depending on the signs o f t and t', one can further simplify the integration in Eq. (A12). The first integration can be

simplified for j = O and 1 as follows. For j =0,

CALCULATION OF SCATTERING WITH THE LIGHT-CONE . . .

Likewise, the second integration is given as follows. For j =0,

For j =1,

However, the signs of t and t' depend on the kinematic variables s , Ikl, and 11 1. The calculations to find the domain of the kinematic variables in which the signs o f t and t ' are determined are straightforward. Here, we present the result of Vj( k, 1 , 2 ) for each domain.

Using the variables

the results of Vo(k,1,2) and V1 (k , l,P) are given by as follows.

(i) 0 < k2 5 k::

(a) p2<pCD,

3574 CHUENG-RYONG JI A N D YOHANES SURYA

(fl pZ > p:,; same as Eq. (A20). (A22)

(ii) k: < k2 5 12:

(a) pZ < p:,, same as Eq. (A20); (b) p2 = p?+,, same as Eq. (A2 1); (c) P:D < pZ 5 p?+,,,, same as Eq. (A22); (d) p2 < p:,, same as Eq. (A20).

[I] E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232 (1951); for review up to 1969, see N. Nakanishi, Prog. Theor. Phys. Suppl. 43, 1 (1969).

[2] C. Schwarz and C. Zemach, Phys. Rev. 141, 1454 (1966). [3] G. C. Wick, Phys. Rev. 96, 1124 (1954); R. E. Cutkosky,

ibid. 96, 1135 (1954). [4] An approximate summation of the crossed-ladder dia-

grams has been considered by E. Brezin, C. Itzykson, and J. Zinn-Justin, Phys. Rev. D 1, 2349 (1970); for a form of the eikonal approximation to include the effect of crossed graphs, see S. J. Wallace, in Nuclear and Particle Physics on the Light Cone, Proceedings of the Workshop, Los Alamos, New Mexico, 1988, edited by M. B. Johnson and L. S. Kisslinger (World Scientific, Singapore, 1988), pp. 477-488; V. B. Mandelzweig and S. J . Wallace, Phys. Lett. B 191, 469 (1984); S. J . Wallace and J. A. McNeil, Phys. Rev. D 16, 3565 (1977).

[5] P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949); S. Wein- berg, Phys. Rev. 150, 1313 (1966); L. Susskind, ibid. 165, 1535 (1968); K. Bardakci and M. B. Halpern, ibid. 176, 1686 (1968); S. D. Drell, D. Levy, and T. M. Yan, ibid. 187, 2159 (1969); J. B. Kogut and D. E. Soper, Phys. Rev.

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[6] For a recent review of the light-cone quantization method, see S. J . Brodsky and H. C. Pauli, Report No. SLAC- PUB-5558 (1991) (unpublished); C.-R. Ji, in Nuclear and Particle Physics on the Light Cone [4], pp. 477-488 (1988).

[7] Renormalization procedure in the light-cone quantization method has been discussed by R. J. Perry, A. Harin- dranath, and K. G. Wilson, Phys. Rev. Lett. 65, 2959 (1990); R. J. Perry and A. Harindranath, Ohio State Uni- versity report, 1991 (unpublished).

[8] C.-R. Ji, Phys. Lett. 167B, 16 (1986); C.-R. Ji and R. J. Furnstahl, ibid. 167B, 11 (1986).

[9] The light-cone scattering analysis in the same scalar field model without addressing the rotation problem discussed in the present paper can be seen in P. Danielewicz and J . M. Namyslowski, Phys. Lett. SIB, 110 (1979); J . M. Namyslowski and P. Danielewicz, Acta Phys. Pol. B12, 95 (1981).

46 CALCULATION OF SCATTERING WITH THE LIGHT-CONE . . .

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1111 A violation of rotational invariance in the perturbative one-loop and two-loop calculations using the Yukawa model based on the light-cone quantization was discussed by M. Burkardt and A. Langnau, Phys. Rev. D 44, 3857 (1991). However, what we discuss in the present paper is different from theirs because we are not involved with the renormalization procedure but with the truncation of the Fock space at the tree level.

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