relativistic eikonal formula for production processes - progress of

Progress of Theoretical Physics, Vol. 51, No. 3, March 1974

Relativistic Eikonal Formula for Production Processes

Hide yuki KoYAMA

Department of Mathematical Sciences University of Osaka Prefecture, Sakai

(Received July 28, 1973)

865

Eikonal formulae for the production process NN-7NN+mc are studied on the basis of a relativistic field-theoretical model in which the potential between nucleons, pion emission and absorption are all induced by a scalar field A. The functional integral method is used to derive formulae. Within the " straight path" approximation for the initial particles, it is shown that an exact impact parameter representation for production amplitude can be. obtained, including all internal interactions, in particular, nucleon loops. The amplitude is properly represented with the aid of three variables; i.e., transverse vector and two longitudinal coordinates. The latter two correspond to two moving intermediate nucleons respectively. These variables compose a vector in a four-dimensional space called "eikonal frame", and a modified rapidity is introduced. Multiple scattering effects by the A field are investigated and a cluster expansion with respect to correlated A-lines is performed for eikonal phase of production amplitude.

§I. Introduction

In the present paper we study the high-energy production process NN~NN + mr on the basis of a relativistic field-theoretical model. This problem has been discussed by many authors using solvable and unitary models/l in which an unknown function called chain is properly introduced. The chain induces internal potential between two nucleons as well as productions and absorptions of pions. In these papers, however, criterion of introducing the unknown function is not very definite.

On the other hand, purely field-theoretical studies2l have been done, using functional derivative method. In these cases, the chain is of some field and so its functional form is uniquely given. These are generalizations of the relativistic impact parameter representation for elastic scattering8l to production processes. For production processes, however, the impact parameter has been introduced in an approximate sense. It has not been shown successfully that the impact parameter representation can be exactly constructed also for inelastic scattering within the framework of straightforward approximation for initial particles.

In this work w:e introduce a field A which behaves as chain. This is necessary in order to produce the large pionization effects as well as the large imaginary part of the eikonal phase for elastic scattering. Within the straightforward' approximation for initial particles, it will be shown that an exact impact parameter representation for production amplitude is constructed, including all

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866 H. Koyama

internal interactions, in particular, nucleon loops. It will also be shown that characteristic features of this approximation are properly represented by introducing three new variables, transverse vector and two longitudinal coordinates, the latter two corresponding to two straightforwardly-moving intermediate nucleons, respectively. In order to express these features, we propose a new four-dimensional space called "eikonal frame" and a modified rapidity.

Throughout this work, we use the functional integral method4l which will simplify calculations and give a perspective rather than the functional derivative method, although the two formulae can be transformed mutually in any step of calculations. For example, by mere transformations of c-number :fields which are functional integral variables, one can easily observe propagations of eikonal features of the initial particles to the internal lines and the produced particles.

In the next section we will construct the model and S operator. In § 3 the straightforward approximation for initial particles will be formulated. After introducing an "eikonal frame" in § 4, the impact parameter representation for production amplitude will be formulated in § 5. The modified rapidity will be introduced there. In § 6 multiple scattering effects will be discussed and eikonal phase for production process will be expanded in a series of clusters of correlated A chains. Section 7 will be devoted to discussion.

§ 2. Construction of S operator

In the present work we shall neglect spin and internal quantum numbers of nucleon (N) and pion (n) and introduce a scalar :field A which induces productions and absorptions of pions. It is assumed that pions do not couple. directly with nucleons.

The action functional W=fd4x.l'(x), .l'(x) being the Lagrangian density, is given by

W[N, A,¢] =iNKNN+tfNNA+ Wt[A, ¢],

Wt[A, ¢] =tAKAA+t¢Kq,¢+tgAA¢, (2·1)

where N, A, ¢ are the c-number :fields corresponding to N, A, n respectively, and KN = 0-M 2 , KA = 0- m 2 , Kq, = 0- ,ll2 are Klein-Gordon operators. In Eq. (2 ·1), x- integration is not written explicitly. Through this paper, fg is defined by f d 4x f (x) g (x), being the inner product in the functional space, and x stands for (x,u Xs, Xo).

By making use of the generating functional

r[v, n] =(exp(iW[N, A,¢] +iNv+i¢n))NAq,, (2·2) S operator IS defined by

S = exp {NHj__} exp {¢<-lj__} r[KNJi, Kq,n] exp {N<+ll__} exp {¢<+ll__} I , OJi on O)i on v=n=O

(2·3)

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Relativistic Eikonal Formula for Production Processes 867

where <F[N, A, ¢])NA.p represents the functional integral with respect to the cnumber :fields N, A, ¢, and is defined as4>

In Eq. (2 · 3), the negative frequency :field NC-l, for example, IS defined as

NH (x) = 1 J dap bt e-ip:c (2nY12 v2Ep p

(2·4)

with the commutation relation [bp, b~,] =Epo3(p-p'). In Eq. (2·2), an Aintegration, for example, means all the possible contractions with respect to the internal A-lines .at one time. One can transform the functional integral formula to the functional derivative formula by making use of the relation4>

at any stage of calculations. Notice that this formula also shows a method to calculate the functional integral.

§ 3. Eikonal approximation

In this section, carrying out the N-integration in Eq. (2 · 2), we apply Abarbanel and Itzykson's eiko~al formula. 3l Two initial nucleons will be treated as non-identical particles.

We write Eq. (2 · 2) in the form

r[v, n] =<exp(iWl[A, ¢] +i¢n)rN[v, A])A.p, (3·1)

where

Making use of the transformation (KN + f AY12N-4N and the formula4l o (Mq) =I exp Tr In MJ oq' M being any linear operator, we have

rN[v,A] =PN[A]exp[- ~ vKN-1v],

where KN=KN+fA and

PN[A] =Jexp[ -t Tr ln(1+fKN-1A)]J,

(3·2)

(3·3)

which represents the summation of contributions from all N-loop diagrams involving A-terminals.

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868 H. Koyama

Fig. 1. Production diagram representing internal straightforward nucleon lines.

Let us take expectation value of S operator between the initial and final two nucleon states. Their momenta are labelled as represented in Fig. 1. We write the expectation value in the form

So=<P/P{ISIP1P2)

=exp{,p<-ll_~<exp(iW1 [A, ¢] +i¢K¢n)r0 [A])A¢ exp{,p<+ll_}l , (3·4) On" ! On" n=oO

where

Taking into considerations only the connected diagrams, we get

ro[A] = <P/P/IP1P2) +i2 CP1'I (KNKN - 1KN) IP1) CP2'I (KNKN - 1KN) IP2)PN[A],

(3·5) where IP;x) =exp(ipx)/v2(2n-)812 and K.N- 1 = (KN+fA+ie)- 1• Use of the eikonal formula of Abarbanel and Itzykson3l gives

r [A] = <P 'P 'iP P) + l sd4x d4x ei<P,-P,'Jxa+i(p,-p,'Jx• o 1 2 1 2 4 (27r)6 a b

where

fa<P<(x;; y) = (f/2P;o)B(2P,oa;-xiO+Yo)02(x,j_ -y;j_)o(x;s-Ys-{3;(x;o-Y;o))

(3·7)

with Pa = (Pt + Pr') /2, h = CP2 + P2') /2, i3a = Pas/Pao and {3b = hs/ho· In Eq. (3 · 6), faP (x) A can also be written in an alternative form

Hereafter we assume i3a>O and {3b<O. Equation (3 · 4) is now written as

So= <P1'P2'IP1P2) +To,

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To= exp {¢<-)j__} sd4xad4xbei(P,-P,')Xa+i(p,-p,')Xb iJn 4 (2n)6

X _a_ --0-rr [7C; aa, ab] I exp {¢<+)1_} I ' 8aa 8ab a.~a.~o iJn ~=0

(3·8)

where

S 0 still works as S operator in pion subspace.

§ 4. Impact parameter representation for production amplitudes

Let us study the n pion production amplitude

(4·1)

where I 0) stands for the no-pion state, and T 0 is given by Eq. (3 · 8). It is convenient to in~roduce the variables (xa + xb) /2 =X and Xa- xb = B- 2Pa<J + 2pb(J', where B is taken so as to have two-dimensional components, i.e., B = (B, 0, 0) and B will stand for impact parameter.

Let us show that (J and <J' integrations in Eq. (3 · 8) can be carried out and impact parameter representation is constructed. To see this, we transform the argument of ¢<-l(x) as follows: x~x+X1 where Xl=X+Pa<J+Pb<J'. We also use the relation

IT iJ F[g(O)]I =IT iJ F[g(Xl)]l ' (4·2) J~liJn(x1 +Xl) ~=o J~ 1 iJn(x1) ~=o

where g(XI) =f f(x)n(x-XI)d4ll:, f(x) being arbitrary function. Applying Eq. ( 4 · 2) to Eq. (3 · 8), 7C (x) in r 1 beqomes 7C (x +XI). Then making the transformations ¢ (x +XI)~¢ (x) and A (x + X 1) ~A (x), which are the functional integral variables, we have the translated r1 written as

(4·3)

Notice that Eq. ( 4 · 3) depends on aa + (J and ab + (J' and is independent of X. So 82/oaaoab in Eq. (3 · 8) can be replaced by 82/o<Jo<J'. The result is

T = _J __ J d4Xd 2 Bd<J d<J' ei(p, +Po-p,'- p,') X-iJ(B-2Pa~+2P.~') 0 4 (2n)6 •

xexp[Jd4x¢H(x+Xl)-iJ-J_§__§_rr[7C;<J,<J']I , (4·4) iJn (x) o<J o<J' ~=o

where A= CP1'- P1) /2- CP2'- P2) /2 and J = 4PaoPbo (/3a- /3b). Making use of the relation

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870 H. Koyama

¢/-) (x + X1) = exp { (Pa(J + h(J') 0~} ¢<-l (x +X) exp {- (Pa(J + Pb(J') 0~} and of the fact that Pa CP1'- P1) = Pb CP2'- P2) = 0, one finally gets

T = J sd4Xei(P,+P•-P{-Pz')X 0 4 (2n)6

x sd2B exp( -i4j_B)exp[ sd4x¢<-l(x+X) 0 JI[n]l , on (x) ~=o where

I[n] = d(Jd(J'- -z"r[n; (J, (J']. Ss"' () () -oo O(J ()(J'

(4·5)

(4·6)

In the expectation value of Eq. ( 4 ·1), X integration is carried out and we have the energy momentum conserving a-function. Note that all the internal interactions including N-loops are contained in the functional I[n]. Thus, the impact parameter representation ( 4 · 5) has been obtained owing to only the straightforward approximation with respect to the initial nucleons.

Now Eq. (4·6) is written as

(4·7) where

(4·8)

(4·9) fb (x) = La2 (_!_B + x l_) iJ (xa- {3bxo)

2ho 2

from Eq. (3 · 7). The function fa represents the N-A vertex where the longitudinal coordinate moves with velocity f3a. This exhibits the characteristic feature of the eikonal approximation and the effects will be properly represented introducing an eikonal frame in the next section.

§ 5. Eikonal frame

To see the propagations of the eikonal effects to pion productions, let us introduce the new four-dimensional space called an eikonal frame, which is given through the transformation

x1_ =b, (5·1)

and we denote the four-vector in the eikonal frame as x= (b, zb, za). At high

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energy limit, f3a = - {3b = 1, Eq. (5 ·1) denotes 45° rotation in the x 0 - x 3 plane m

the Minkowsky space. The reverse relation is

x = (b f3azb- {3bza zb- Za ) ' ' (f3a- f3bY12 ' (f3a- f3bY12 •

(5·2)

Let us write Eq. ( 4 · 8) with the aid of the variables in the eikonal frame.

This can be done through the transformations A(x) -'>A(x) and ¢(x) -"¢(.X) in

Eq. ( 4 · 8) , which yields

Iab[n] =fab[n] =(exp[ ~ A(KA +9¢)A+i(]a + jb)A+ ~ ¢K~¢ +i¢K~n ]PN)A¢,

(5·3)

where

Jb(x) =La2 (b+l_ B)aczb), 2Pbo 2

(5·4)

n(x) =n(x),

KA:==D-m2 , K~=D-;i, (5·5)

D = (f3a- f3b>-![ ra -2 a~:2 + 2 (1- f3af3b) az~;Zb + rb -2 a~:2 J + 8~2 (5. 6)

with ra-2=:=1-f3a2 and rb-2=1-{3b2, PA being obtained by replacement of Din

PA by D. Thus Eq. (4·5) is written as

T =-J __ sd4Xei(Pl+Po-P1'-Po')X 0 4 (27!")6

x Jd2Bexp(-id_j_B)exp[Jd4x¢H(x+X). _ 0_ Jl[rr]l-, (5·7) , 01!" (x) n=O

where l[rr] =I[n] and

¢<-l(x+X) = 1 s ~~k at(k)e-ik(x+X). (2nY12 v 2wk

Writing kx=kx in Eq. (5·8), we have

k - (k k k ) - (k _ ko - f3 aka _ ko - {3 bks ) - _!_> s, 0 - _!_> (f3a- f3bY12 ' (f3a- f3bY12

and the pion mass shell relation becomes

ra-2ko2 -2(1-f3af3b)koks+rb-2ks2 + (f3a-f3b) (k_j_2 +;i) =0,

which is also obtained from the equation K~e-ilix = 0.

(5·8)

(5·9)

(5 ·10)

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Notice that the zeroth and the third components appear in symmetric feature in the eikonal frame. So let us introduce a new variable "fJ (modified rapidity) defined as

rj=_!_ln !o =_!_ln ko-f3bks, 2 ks 2 ko- f3aks

(5·11)

which is in agreement with rapidity at high-energy limit f3a = - {3b = 1. With "fj, ¢J<-> is redefined as follows:

(5·12)

where a is normalized as

(5·13) and

The Lorentz invariant phase space volume is given by d 2kj_dn/a. At asymptotic limit f3a =- {3b = 1, a is equal to 1. The n pion state is defined by

The production amplitude ( 4 ·1) is written as

J fl'(P/+P2'+'f:,k,-p1-P2) sd2Bexp(-iJj_B) 4(2n/ J=l

X _1 ii: s d'xi e-ikJZJ_(]_J[n]l , ../ n! J=l .J2 (2rrY12 (]Tt (x1) ;?"so

(5·15)

where k is given by Eq. (5·9) and l[Tt] =l[rr] can be computed from Eqs. (4·7), ( 4 · 8) and (5 · 3). The elastic amplitude is given by

':To= J 2 (]'(PI'+ Pz'-Pl-PE) sd2B exp ( -iJj_B) l[Tt=O] 4(2rr) ·

from Eqs. (4·1) and (5·7).

§ 6. Contributions of correlated A chains

In this section, let us investigate only contributions ·of A-lines which bridge the two nucleons and correlate each other through pion propagators, and all loop diagrams due to N and A will be neglected. In this case it will be shown that l[Tt] is in the familiar eikonal form l[Tt] = exp (ix) -1 and the phase X is expanded in a series of correlated clusters of the A-lines.

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We start from Eq. (5 · 3) being written as

Jab[i't] =\exp( ~ ¢Kq,¢ + i¢Kq,i't )-rA[¢]) ~, (6·1)

where

If PN=l, Eq. (6·2) can be integrated, so that

(6·3)

where

(6·4)

with CAi1 =if;(KA+9¢)-Ij1 and PA[if;]=iexp[-tTrln(l+gKA·-1¢)]1 stands for

contributions from the loop diagrams with respect to A. In Eq. (6 · 4) C A ab = C A ba

since (KA +9¢)-1 is the symmetric operator and ori.ly CAab gives the chain effects

in which we are interested.

Making use of Eq. (5 · 4), we have

cAab=_.L__ sd4xo2 ( b- _l B)o(za) Jab 2

(6·5)

where Jab =J =4PaoPbo (f3a- (3b) and d4x= d 2bdzadzb. Let us calculate the operator

.T=[i(KA+9¢(x)+ie)r] with the aid of the operator identity exp(A+B)

= T exp [n dteAt Be-At] exp A and the Fourier transform of ¢. We get the result

where

and (k*k)=kl_ *kl_ +ks*ka+k0*k0 with

k*= (r.j_, rb-2ka- (l-f3af3b)ko, ra-2ko- (l-f3af3b)ka). f3a- {3b f3a- {3b

Cn (k, t) represents the momentum correlations between pions connected with one

A-line. If T operates on the function exp [iql_ (b + B/2) + iq8zb] which is the

Fourier transform of (2nYo2 (b+B/2)o(zb) in Eq. (6·5), 8/ax becomes iq where

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87 4 H. Koyama

q = (qj_, qs, 0). The eigenvalue of T obtained in such a way is denoted by T(q), which can be expressed by a cluster series of pion momentum correlations in the following way. We introduce T defined by T=ln[T(q)exp{-i(KA +i.s)-r}] and expand Tin a power series of T(q)exp{-i(KA+i.s)-r}-1. Then making use of Eq. (6 · 6) and exchanging the order of the summations, we get the cluster senes T = :E:~o T n, where To involves no correlation and Tr is given by

where

¢ (k, t) = (ig/ (2n)4) rjJ (k) exp (ikx-it<k*k)- 2itk*q). It can be easily proved that if there is no correlation, that is Cn = 1 for any n, each T n = 0 except for T 0•

This formulation gives a way of taking account of the momentum correlations of one after the other. However we shall study the case with no correlation in the remaining part of the present paper, for the correlation effect is relatively small from experimental facts. 5l Contributions of To yield

where

with

Gkb =-9- (rdt sd4k (2n)4 Jo

xexp{-ikx+i(kszb+! kj_B) +itZ12 -2it(ks*qs+kj_qj_)}

Zi2 = -<k*k)

= - (/1a- /1b)-l [T a - 2ko2 - 2 (1- l1al1b) koks + Tb - 2k/] - k /,

(6·7)

(6·8)

(6·9) which should be equal to the pion mass square ;i if the mass relation (5 ·10) holds.

Here we turn to discussion of the straightforward approximation of Abarbanel and Itzykson.3l In their eikonal formulation, one treats the operator similar to T and so gets a calculated result similar to Eq. (6 · 6). If one neglects Zi2 as well as the momentum correlations in Eq. (6 · 6), one gets the straightforward type of formula. Conversely our formulation taken here will give a way improving the straightforward approximation. Now let us proceed to investigate correlations between A-lines. For this purpose, we write Eq. (6 · 7) in the simplified form

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(6 ·10)

where

and

E(~) = -P exp(iq_]_B+iqszb)exp[ -ir({J~~~b qs2+q_j_2+m2-ie) J. In the present case cAaa=C}b=O and PA=1, so that Eq. (6·3) becomes rA[¢] = exp (- C A ab). Since effectively I a= Ib = 10 = 1 in Eq. ( 4 · 7) from the pion mass relation, we obtain

(6 ·11)

where

(6 ·12)

Equation (6 ·11) can be integrated if one expands it in a power series of E. Following the procedure similar to the way to calculate the momentum correlations in Eq. (6 · 6), we get

where

lab [n] = exp [if; XN], N=l

. • N (-;- )M M 1 XN=z :E -- :E n -Rnj'

M=l M {nj} f=l nj!

(6 ·13)

(6·14)

:E{nJ} representing the summations with respect to n1, n2, • .. , nM in the restrictions that :E n1 =N and n1>1 for any j, and

where

F(~) = E(~) exp ( -in'Gab (~)) exp (-!Gab (~)Kq, -lGab (~)),

dinC~1,~2; ... ,~n) = IT exp[ -iG(~J)Kq,-1G(~1)]. J>l=l

(6·15)

(6 ·16)

(6 ·17)

XN represents the contributions from the connected N A-lines. For example, we have

ix1 = J d~F(~), iX2 =_!_ sd~1d~2FC~1)F(~2) [di2(~1, ~2) -1],

2!

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876 H. Koyama

Xa represents the contributions from the generalized pion ladder diagrams without any correlations between pion momenta and with ability emitting pions from two A-lines.

§ 7. Discussion

In the present paper we have studied the case where the field A as the bridge is scalar, so that each N-A vertex has asymptotic behavior of the form 1/Po"'-'1/../s. In the vector meson case, how_ever, the vertex behaves as constant at high energy, as is well known. Thus, in general we have no theoretical reason to choose only a few terms in the cluster expansion Eq. (6 ·14) and the total numerical analysis is almost impossible. Alternative formulation such as strong coupling approximations6> should be constructed in the framework of field t;heory.

If we put ad hoc a restriction that only one pion is connected with each A-line as in the solvable model/' the numerical result can be obtained7' and the Feynman scaling is easily seen in our eikonal frame. Moreover in this model we can show that it is forbidden to emit pion from an A-line related only with a single N-line (contribution from C A aa or C /b in Eq. (6 · 4)) because of the straightforward approximation with respect to N-line.

It would be expected that our modified rapidity r; given by Eq. (5 ·11) is a proper quantity to analyze high-energy experimental facts, in comparison with the usual rapidity. This is a remaining problem to be examined.

Acknowledgements

I would like to thank Professor T. Kotani and Mr. M. Doi for valuable suggestions and helpful discussions.

References

1) G. Calucci, R. Jengo and C. Rebbi, Nuovo Cim. 4A (1971), 330; 6A (1971), 601. R. Aviv, R. L. Sugar and R. Blankenbecler, Phys. Rev. D5 (1972), 3252. S. Auerbach, R. Aviv, R. Sugar and R. Blankenbecler, Phys. Rev. Letters 29 (1972), 522; Phys. Rev. D6 (1972), 2216.

2) H. Abarbanel, NAL-THY-77. H. M. Fried, Lectures presented at the Winter School for Theoretical Physics, Univ. of Wroclaw, 1967. S. Tanaka, Phys. Rev. D4 (1971), 2419.

3) H. Abarbanel and C. ltzykson, Phys. Rev. Letters 23 (1969), 53. H. M. Fried, preprint.

4) J. Rzewuski, Field Theor:y (Iliffe Books, LTD., London, 1967), vol. 2. 5) P. W. Ko, Phys. Rev. Letters 28 (1972), 935.

W. D. ·Shepherd et al., Phys. Rev. Letters 28 (1972), 703. 6) S. Hori, Nucl. Phys. 30 (1962), 644. 7) M. Doi and T. Kotani, preprint.

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relativistic eikonal formula for production processes - progress of

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