fulltext

Probability description of final-state distribution

in independent fragmentation model (*)

G. J. WANG(1)(**) and Y. D. HE(2)(3)(***)

(1) Department of Physics, Wayne State University - Detriot, MI 48202, USA (2) Department of Physics and Space Science Laboratory, University of California

Berkeley, CA 94720, USA

(3) Nuclear Science Division and Institute for Nuclear and Particle Astrophysics

Lawrence Berkeley National Laboratory - Berkeley, CA 94720, USA

(ricevuto il 5 Maggio 1997; approvato il 30 Giugno 1997)

Summary. — We present a probability account of the density distribution for final states in a class of independent fragmentation models. The formalism leads to a simple but exact expression for the single-particle distribution of the final states. As an example, we discuss a particular model used in high-energy parton-parton and e1_e2 _{scatterings—the independent jet fragmentation model. The universality and} simplicity of the probability description allow us to apply our formalism to various physical processes of fragmentation nature.

PACS 13.87.Fh – Fragmentation into hadrons.

PACS 02.50 – Probability theory, stochastic processes, and statistics.

1. – Introduction

Fragmentation processes, cascading phenomena, and their corresponding inverses—connectivity and clustering problems—are of fundamental interest in many different fields. In physics, examples are the jet fragmentation process of quarks and the bremsstrahlung process of charged particles. As an example, the process of jet fragmentation plays an important role in high-energy e1_e2 _{scatterings and}

high-energy parton-parton scatterings in pp, p p, pA, and AA collisions. However, there is no first-principle theory that can be used to derive the final-state distribution for the jet fragmentation process. The reason is that the fragmentation of quarks into hadrons is a non-perturbative phenomenon and is not calculable within the framework of the strong interaction theory—QCD. Therefore, one has to develop phenomenological

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail: wanggjHrhic2.physics.wayne.edu

(***) E-mail: yudongHphysics.berkeley.edu

models to describe the density distribution for final states. The scaling behavior involved in the jet fragmentation is certainly a useful feature that simplifies the problem. In the last two decades, various models for jet fragmentation have been proposed [1].

Models studied so far can be classified into three categories: independent fragmentation, clustering and strings [1]. The independent fragmentation model was first proposed by Field and Feynman [2]. This simple model has served as building block in jet fragmentation study. Recently, much more sophisticated models such as the string model have been studied in the literature for e1_e2 _{scatterings [1]. Even}

though the independent fragmentation model is no longer widely used in the current version of Monte Carlo generators for jet fragmentation in e1_e2_{scatterings, it is still}

used for parton-parton scatterings in pp, pp, pA, and AA collisions. Furthermore, the independent jet fragmentation model has its own merits because of its simplicity and universality.

In this paper, we propose a probability description of the density distribution for final states in a class of independent fragmentation models. The focus of our study has been on the development of a simple but complete probability description for the independent fragmentation process. The approach we developed has a sound mathematical ground and is an extension of the early work by Field and Feynman. It should be pointed out that our method works not only for a particular model—jet fragmentation, but also for other physical processes with fragmentation nature such as bremsstrahlung.

2. – Independent fragmentation

A) Description of the model

The independent fragmentation model assumes that quark jets can be analyzed on the basis of a recursive principle as a way to parameterize jet fragmentation (1_{). The}

ansatz is based on the idea that a quark of flavor a separating from an antiquark and with momentum p0 in the beam direction creates a color field in which new

quark-antiquark pairs are produced. Then quark a combines with an antiquark b from a new pair b b to form a meson of flavor a b leaving the remaining b flavor quark to be combined with further antiquarks c. A hierarchy of mesons is thus formed of which a b is first in rank, b c is second in rank, c d is third in rank, etc. This chain fragmentation ansatz assumes that, if the meson in rank 1 carries away a fraction of momentum x1

from a quark jet of flavor a with momentum p0, the remaining cascade starts with a

quark of flavor b and momentum p14 p0( 1 2x1), and the remaining hadrons share the

remaining momentum in precisely the same way as the hadrons which come from a jet originated by a quark of flavor b with momentum p1. It is further assumed that all

distribution density functions (p.d.f.) at each rank, f (x), are identical. This scaling function f (x) is often called splitting function. At some stage the quark left has an

(1_{) Field commented on the model in ref. [3] that “A simple mathematical model due to Feynman} and myself (called the FF parameterization) can be used to parameterize the non-perturbative aspects of quark jets. It is not meant to be a theory. It is simply a parameterization that incorporates many of the expected features of fragmentation.”

energy so low that it cannot be expected to form a jet and the chain stops. The minimum fractional momentum that controls the end point of the chain, r, is determined by the mass of meson (mp) and the initial energy of the jet (in the center-mass-system, ks ) as

r Camp/ks with a A2. As a result, the parameter r determines the mean multiplicity of the particle production. Because the fragmentation process takes place independently at each rank and the fractional momentum distributions scale each other, the model is called independent jet fragmentation model. For simplicity, the evolution of quarks in the final states is ignored, so that no Q2_{-dependence needs to be}

implemented into the jet fragmentation function. For the same reason, the flavor distinction is also ignored in this model. It should be pointed out that flavors can be easily incorporated into the model based on their relative suppression factors in phase space due to different masses. In addition to meson production, baryon production can be implemented into the model in somewhat complicated ways [1].

The overall single-particle distribution, F(z), is defined as the p.d.f. of fractional momentum normalized to the initial momentum of the system. Field and Feynman [2] found that F(z) is related to the splitting function f (x) in each rank by the following integral equation: F(z) 4f(z)1



z 1 f ( 1 2t) F

g

z t

h

dt t ; (2.1)

here we use f ( 1 2t) instead of f(h) as used in their paper. As our convention in the following, we use F(z) not F(x) and f (x) not f (z) if they do not appear in the same formula. The problem of getting F(z) becomes to solve the integral equation (2.1). This has been done by taking a specially defined “Fourier-like” transformation by Field and Feynman [2].

A very important asymptotic behavior of F(z) is found to be

F(z) K R z , as z K0 , (2.2) with R 42 1

s

01f ( 1 2t) ln (t) dt . (2.3)

Consequently, F(z) cannot be normalized in the work of Field and Feynman. Moreover, no distinction was made in their work between two cases: one is the ideal case where the fragmentation keeps going forever, the other is the practical case where the fragmentation eventually stops at a certain stage. In our probability description below, we will see that for any practical fragmentation process the asymptotic behavior of F(z) as z K0 may be quite different from the above 1/z form and the normalization condition for F(z) can be satisfied by introducing the control parameter r mentioned above.

B) Probability description

We begin our probability description by assuming that f (x) is the scaling p.d.f. that controls the fractional momentum distribution at any rank. P(n) is the probability of an

event which has n particles in the final state (n 41, 2, 3, R) and ]xi( (i 4 1 , 2 , R , n) is a set of random variables obtained by sampling f (x) n times. Evidently, the fractional momenta normalized to the initial momentum are:

zi4 xi

»

j 41 i 21

( 1 2xj) (i 41, 2, R , n) , (2.4)

with x04 0 . Now we define new variables:

yi4 1 2

!

j 41

i

zj (i 40, 1, 2, R , n) . (2.5)

From eqs. (2.4) and (2.5), we have

xi4 1 2 yi y_{i 21} (i 41, 2, R , n) (2.6) and zi4 yi 212 yi (i 41, 2, R , n) . (2.7)

Since ]xi( (i 4 1 , 2 , R , n) are n independent random variables from f (x), the probability of getting this set of random variables ]xi( is

P(x1, x2, R , xn)

»

i 41 n dxi4

»

i 41 n f (xi) dxi. (2.8)

Using Jacobi determinant, the p.d.f. of ]yi( is

P(y1, y2, R , yn) 4

»

i 41 n f

g

_{1 2} yi y_{i 21}

h

dyi

N

i 41

»

n 21 yi. (2.9)

In the following, we use the abbreviations:

(1) P(y) represents P(y1, y2, R , yn), while dy stands for dy1dy2Rdyn; (2) (C) represents the constraint 0 EynE r E yn 21E R E y1E 1;

(3) (Ci) represents both the constraint (C) and constraint z Eyi 212 yiE z 1 dz (i 41, 2, R , n).

For an event with n particles, the fragmentation stops when condition (C) is met. The p.d.f. of a n-particle event is given by

P(n) f P(n , r) 4



R



(C)

P(y) dy .

(2.10)

In an event in which there are n particles produced in the final states, the conditional probability of the i-th particle with momentum in the interval (z , z 1dz) is

P(zNn; i) dz4



R



(Ci)

P(y) dyOP(n) .

(2.11)

Note that P(zNn; i) is normalized to 1 as can be seen from the definition of (Ci). For an event of n particles in the final state, the conditional probability of any one particle with

momentum in (z , z 1dz) is P(zNn) dz4

!

i 41 n P(zNn; i) dz . (2.12)

Since P(zNn; i) is normalized to 1, P(zNn) is normalized to n, the multiplicity of the event. Therefore, the p.d.f. of any particle with fractional momentum z in all events having n 41, 2, R is F(z) 4

!

n 41 Q P(n) P(zNn)

N

!

n 41 Q P(n) 4

!

n 41 Q

!

i 41 n P(n) P(zNn; i)

N

!

n 41 Q P(n) . (2.13)

It is easy to prove that P(n) defined in eq. (2.10) is normalized, i.e.

!

n 41 Q

P(n) 41 .

(2.14)

From eq. (2.11), we have

F(z) 4

!

n 41 Q

!

i 41 n



R



(Ci) P(y) dy f

!

n 41 Q

!

i 41 n Pni(z) , (2.15) where Pni(z) f P(zNn; i) P(n), i.e. Pni(z) f P(zNn; i) P(n, r) 4



R



(Ci) P(y) dy . (2.16)

Note that F(z) is not normalized to 1; the function normalized to 1 is F(z) Oan(r)b, where an(r)b is the average multiplicity

an(r)b 4



0 1 F(z) dz . (2.17) C) Expression for F(z)

It is clear from eq. (2.15) that F(z) contains a series of sums of Pni(z). Now we derive expressions for Pni(z) and hence for F(z) under different conditions.

1) n 4i41, i.e. there is only one particle in the final state. Since the constraint in the integral now becomes

(C1): . / ´ 0 Ey1E r , z G12y1G z 1 dz , (2.18) we obviously have P11(z) dz 4



(C1) f ( 1 2y1) dy14 . / ´ f (z) dz 0 (z D12r) , (z E12r) . (2.19)

2) n D1, i.e. there is more than one particle in the final state. When we consider the first one, we have

(C1): . / ´ 0 EynE r E yn 21E R E y1, z G12y1G z 1 dz (2.20) and Pn1(z) dz 4



R



(C1) P(y) dy (2.21) becomes (2.22) Pn1(z) dz4



12z2dz 12z f ( 12y1) dy1 y1



R



0 EynE r E R E y1

»

i42 n f

g

_{1 2} yi y_{i 21}

h

dyi

N

i 42

»

n 21 yi4 4



1 2z2dz 1 2z f ( 1 2y1) dy1 y1



R



0 Eyn8 E (rOy1) EREy28 E 1

»

i 42 n f

g

1 2 y 8i y_{i 21}

h

dy 8i

N

i 42

»

n 21 y 8i4 4



1 2z2dz 1 2z f ( 1 2y1) dy1P

g

n 21, r y1

h

, where P(n 21, rOy1) is defined in eq. (2.10). Consequently,

!

n 42 Q Pn1(z) dz 4



1 2z2dz 1 2z f ( 1 2y1) dy1

!

n 41 Q P

g

n , r y1

h

(2.23) becomes

!

n 42 Q Pn1(z) dz 4 . / ´ f (z) dz 0 (z E12r) , (z D12r) , (2.24) where

!

n 41 Q P

g

n , r y1

h

4 1 . (2.25)

From eqs. (2.19) and (2.24) we have

!

n 41 Q

Pn1(z) 4f(z) . (2.26)

This is a natural result since z of the first particle is sampled directly from f (z). 3) i 4nD1, i.e. the last particle. In this case, with the constraint

(Cn): . / ´ 0 EynE r E R E y1, z Eyn 212 ynE z 1 dz , (2.27)

we have

Pnn(z) dz 4



R



(Cn)

P(y) dy .

(2.28)

Suppose the range of variables y_n21 is within [a(z), b(z) ], then the constraint (Cn) gives

a(z) 4

.

/

´

r , z , z , b(z) 4

.

/

´

r 1z r 1z 1 (z Er), (r EzE12r), ( 1 2rEzE1) . (2.29)

Here we have assumed r E0.5. In the case of rF0.5, all the derivations are similar. Integrating eq. (2.28) over the possible range of ynfirst, we obtain

Pnn(z) dz 4



R



a(z) Eyn 21E b(z) yn 21E Ry1E 1 f

g

z y_{n 21}

h

dzi 41

»

n 21 f

g

_{1 2} yi y_{i 21}

h

dyi

N

i 41

»

n 21 yi. (2.30)

We define a new function

G(z , y) f

!

n 41 Q



R



y EynE R E y1E 1 f

g

z yn

h

»

i 41 n

k

f

g

_{1 2} yi y_{i 21}

h

dyi

l

N

i 41

»

n yi, (2.31)

where y04 1 is used. Note that G(z , y) 4 0 if y F 1. From eq. (2.30) we have

!

n 42 Q

Pnn(z) 4G

(

z , a(z)

)

2 G

(

z , b(z)

)

. (2.32)

4) n DiD1, i.e. inclusion of intermediate particles. In this case, we have

Pni(z) dz 4



R



(Ci)

P(y) dy ,

(2.33)

where (Ci) in eq. (2.33) is the originally defined constraint (Ci): . / ´ 0 EynE r E yn 21E R E y1E 1 , z Eyi 212 yiE z 1 dz . (2.34)

Similar to the derivation of eq. (2.32), we integrate eq. (2.33) over yifirst. Assuming the range of y_{i 21}to be [c(z), d(z) ], then from (Ci) we have

c(z) 4

.

/

´

r 1z, r 1z, A, d(z) 4

.

/

´

1 1 A (z Er), (r EzE12r), ( 1 2rEzE1), (2.35)

where A means it does not exist. Finally we have

!

i 42 Q

!

n 4i11 Q Pni(z) 4G

(

z , c(z)

)

2 G

(

z , d(z)

)

. (2.36)

Combining eqs. (2.19), (2.24), (2.28) and (2.36), the final expression for F(z) is (2.37) _{F(z) 4}

!

n 41 Q

!

i 41 n Pni(z) 4

!

i 41 Q

!

n 4i Q Pni(z) 4 4 f (z) 1 G

(

z , a(z)

)

2 G

(

z , b(z)

)

1 G

(

z , c(z)

)

2 G

(

z , d(z)

)

. In eq. (2.37), the first term comes from the contribution of the first particle, the second and the third terms come from the contribution of the last particle, and the last two terms come from the contribution of intermediate particles. It is seen from eqs. (2.29) and (2.35) that b(z) 4

.

/

´

r 1z, r 1z, 1 , c(z) 4

.

/

´

b(z) b(z) A (z Er), (r EzE12r), ( 1 2rEzE1). (2.38)

i.e. b(z) coincides with c(z) in the case z G12r and G

(

z , d(z)

)

4 0 since d(z) 4 1. In the case of z D12r, G

(

z , b(z)

)

4 0 since b(z) 4 1. Also in this case, c(z) and d(z) do not exist. We have 2 G

(

z , b(z)

)

1 G

(

z , c(z)

)

2 G

(

z , d(z)

)

4 0 , (2.39) and hence F(z) 4f(z)1G

(

z , a(z)

)

(2.40)

in all cases. Therefore we obtain

F(z) 4./ ´ f (z) 1G(z, r) f (z) 1G(z, z) (z Er), (z Dr). (2.41)

Clearly, a different behavior of F(z) at z 4r manifests. For zGr, F(z) is determined by

G(z , r); for z Fr, F(z) is determined by G(z, z) which is independent of r! In what

follows we shall see that the latter is just the F(z) in the work of Field and Feynman.

So far we have obtained a general expression for F(z) in eq. (2.41), where G(z , y) is determined by eq. (2.31). If G(z , y) can be directly calculated from eq. (2.31), then we can obtain the explicit expression for F(z).

D) Integral equation approach

Another way to find F(z) is to solve an integral equation. From eq. (2.31)

(2.42) _{G(z , y) 4}



y 1 f ( 1 2y1) f

g

z y1

h

dy1 y1 1 1



y 1 f ( 1 2y1) dy1 y1 n 42

!

Q



R



y EynE R E y1 f

g

z yn

h

»

i 42 n f

g

_{1 2} yi y_{i 21}

h

dyi

N

i 42

»

n yi,

where the sum on the right-hand side can be written as

!

n 41 Q



R



(yOy1) EznE R E z1E 1 f

g

z zny1

h

i 42

»

n f

g

1 2 zi z_{i 21}

h

dzi

N

i 42

»

n zi4 G

g

z y1 , y y1

h

. (2.43)

That is, G(z , y) satisfies the following integral equation:

G(z , y) 4



y 1 f ( 1 2y1)

k

f

g

z y1

h

1 G

g

z y1 , y y1

h

l

dy1 y1 . (2.44)

Substituting it into eq. (2.41), in the case of z Dr, we have (2.45) _{F(z) 4f(z)1}



z 1 f ( 1 2t)

k

f

g

z t

h

1 G

g

z t , z t

h

l

dt t 4 f (z) 1



z 1 f ( 1 2t) F

g

z t

h

dt t .

This is exactly the same as eq. (2.1) obtained by Field and Feynman in ref. [2].

Similarly, we can get the corresponding integral equation for an(r)b by referring to eq. (2.17): an(r)b 411



r 1 f ( 1 2t)

o

n

g

r t

h

p

dt . (2.46)

This equation can also be solved by taking the “Fourier-like” transformation.

In the limiting case where r K0 (or equivalently, ksKQ), the contribution of F(z) to an(r)b comes mostly from z Fr, and eq. (2.2) is valid. With the help of eq. (2.3), we can find an approximated expression for an(r)b:

an(r)b C ln r

s

01f ( 1 2t) ln (t) dt

, (2.47)

as r K0. A better approximation can be obtained by incorporating the fact that as

r K1, an(r)b K1. Therefore, we have

an(r)b C ln r

s

01f ( 1 2t) ln (t) dt

1 1 (2.48)

for any r. It should be mentioned that in the case that r is not very small, even though the average multiplicity from z Dr might be very different from eq. (2.47), eq. (2.48) may still be a very good approximation as can seen in the two examples below.

3. – Application: jet fragmentation

In the jet fragmentation model, as long as the scaling function f (x) is specified, the fragmentation function F(z) can be obtained using the above formalism. The scaling function is usually taken to be

f (x) 4 (m11)(12x)m

(m F0) . (3.1)

In this case, we have (3.2) _{G(x , y) 4}

!

n 41 Q



R



y EynE R E y1E 1 (m 11)n 11

g

_{1 2} x yn

h

m

»

i 41 n

g

yi y_{i 21}

h

m dyi

N

»

i 41 n yi4 4

!

n 41 Q



R



y EynE R E y1E 1 (m 11)n 11_(y n2 x)m

»

i 41 n dyi

N

»

i 41 n yi4 4

!

n 41 Q



y 1 (m 11)n 11 (yn2 x)m yn (2ln yn)n 21 (n 21)! dyn4



y 1 (t 2x)m tm 12 (m 11) 2 dt 4 4 (m 1 1 )

y

( 1 2x) m 11 x 2

g

1 2 x y

h

m 11 ₁ x

z

.

From eq. (2.41), we obtain the fragmentation function

F(z) 4 (m11)

y

( 1 2z) m z

z

.

/

´

1

{

1 2

y

r 2z r( 1 2z)

z

m

g

r 2z r

h

}

(z Dr), (z Er). (3.3)

When m 40, f(x) is a uniform distribution. The fragmentation function in this case is

F(z) 41Oz (with zDr).

If f (x) 42x, a monotonical increasing function in (0, 1), the expression for F(z) becomes F(z) 4

.

`

/

`

´

2 3

g

1 r2 1 2 r

h

z 2 3

g

1 z 1 2 z 2

h

(z Er) , (z Dr) . (3.4)

Apparently, in the above two cases, F(z) near zC0 either behaves like F(z)Cconst4 (m 11)2Or in the case of f (x) 4 (m 1 1 )( 1 2 x)mor F(z) Pz in the case of f(x) 42x, far from being F(z) P (1Oz). Hence, eq. (2.2) can only be satisfied in the limiting case of

z KrK0 or equivalently ks KQ.

If f (x) decreases monotonically in ( 0 , 1 ), e.g., f (x) Pe2x_{, it is difficult to get the}

analytic expression for F(z) directly from eq. (2.31) or eq. (2.44), but the numerical calculation is always feasible.

4. – Conclusion and discussions

The independent fragmentation model provides a useful explanation of many features of fragmentation processes. We worked out a particular example—jet independent fragmentation. Our probability description is valid for a number of physical processes of fragmentation nature such as bremsstrahlung because our approach is simple and universal. However, there are some difficulties with this model for jet fragmentation. The time ordering in which this has been described and in which

it is implemented in computer programs is misleadingly wrong. The first place the color field breaks, as seen in the laboratory, is at the low-energy end of the jet. The first step described above: the separation of the rank-1 meson containing the original quark actually happens last. This difficulty does not exist for bremsstrahlung process. It is noted that the independent fragmentation model is so simple that it can be used in many other physical processes of fragmentation in nature. In particular, as long as the scaling splitting function f (z) is assumed, we can readily derive the fragmentation function F(z).

Since eq. (2.1) can also be interpreted as an equation for f (x) in terms of F(x):

f (x) 4F(x)2



x 1 F

g

x t

h

f ( 1 2t) dt t , (4.1)

so the inverse problem, i.e. to solve for f (x) for a given F(x) can be proceed in a similar way of solving F(x) in terms of f (x). In practice, it is not straightforward to solve for

f (x) based on eq. (4.1). However, for a class of functional forms of F(x) and for a set of

realistic values of r (C1023_{) used in practice, it can be easily verified by Monte Carlo}

simulation that

f (x) CxF(x)

(4.2)

is fairly a good approximation. For some certain systems such as those described by eq. (3.1), formula (4.2) is exactly valid. In most cases, we can use it as a good approximation.

It is worth mentioning that the formalism we discussed above provides a powerful technique in realistic Monte Carlo sampling for a given fragmentation function F(z) [4]. This method assures the conservation of energy and momentum for the fragmenting system. For a large number of functional forms, we have verified eq. (4.2) by Monte Carlo simulation.

It can be shown that, if the scaling function f (x) takes the form of eq. (3.1) and the parameter r is specified, the multiplicity distribution follows:

P(n , r) 4rm 11 [2(m11) ln r]

n 21

(n 21)! .

(4.3)

This is a Poisson distribution with mean

an(r)b 42 (m11) ln r11 . (4.4)

When f (x) takes other forms, similar results will appear with an(r)b given by eq. (2.48). Equation (2.48) is exact in the case of f (x) 4 (m11)(12x)m, even though in this case the contribution from z Er region may not be small as rK0. In the case of

f (x) 42x, by using eq. (2.17), we can get

an(r)b 4

g

22 3ln r 11

h

2 2 9 ( 1 2r 3_{) ,} (4.5)

where the first expression within parentheses is the prediction from eq. (2.48). Therefore, eq. (2.48) is a very good approximation for the second example.

center-mass-system, we expect

anb Pln s , (4.6)

which has been experimentally established for e1_e2_{scattering. For jet fragmentation}

in e1_e2 _{scattering, the experimentally observed multiplicity distribution is not}

inconsistent with the Poisson distribution. However, for pp collisions the experimentally measured multiplicity distribution is broader than the Poisson distribution. For hadron-hadron collisions at a fixed energy, using different values of r for different multiplicity events, the experimentally determined distribution would be reproduced by a superposition of a series of P(n , r). In the geometrical model suggested by Chou and Yang [5], different multiplicities are believed to relate to different impact parameters [6]. Therefore, the distribution of r discussed in this paper is relevant to the distribution of impact parameters.

* * *

This work was supported in part by the Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. DOE under Contract No. DE-FG02-94ER40831 (GJW) and DE-AC03-76SF00098 (YDH).

R E F E R E N C E S

[1] See, e.g., BARLOWR., Rep. Prog. Phys., 56 (1993) 1067, and references therein. [2] FIELDR. D. and FEYNMANR. P., Nucl. Phys. B, 136 (1978) 1, and references therein. [3] FIELD R. D., Applications of Perturbative QCD (Addison-Wesley Publishing Co.) 1989,

p. 58.

[4] WANGG. J. et al., Chin. J. Comput. Phys., 5 (1988) 1. [5] CHOUT. T. and YANGC. N., Phys. Rev. D, 32 (1985) 1692.