Relativistic quantum mechanics, production reactions, and unitarized multiparticle amplitudes

293c

Nuclear Physics A508 (1990) 293~298~ North-Holland

RELATIVISTIC UNITARIZED

QUANTUM MECHANICS, PRODUCTION MULTIPARTICLE AMPLITUDES

REACTIONS,

AND

W. H. KLINK Department

of Physics and Astronomy,

The University

of Iowa, Iowa City, Iowa

52242*

A relativistic quantum mechanics for production processes is formulated and exactly solvable models arising from separable potentials are used to generate multiparticle scattering amplitudes. These multip~ticle amplitudes contain undetermined multip~ticle functions which can be used for fitting multiparticle data. A brief discussion of multiparticle unitarity is also given.

There particle

is a well-known

procedure

for converting

cross section data obtained

reactions

to phase shifts and ineIasticity

cedure for multiparticle

reactions

is lacking.

describing

reactions

require many more variables than the corresponding

multiparticle

particle amplitudes. of multiparticle

However,

reactions,

parameters;

from two

to two-particle

a similar pro-

This is mainly due to the fact that amplitudes

experimentalists

cannot hope to carry out complete

as this would require complicated

correlation

two-

analyses

experiments

over

large regions of phase space. Nevertheless, for the xN particle

there is considerable

and h’N

systems,

distribution

particular,

data.’

the kinematic

data available on multiparticle

generally

reactions, for example

in the form of total cross section data or single-

But there seem to be no models

region in which up to about

able to fit this data.

10 mesons are produced

In

is a region

where QCD models do not seem to work very well. One of the reasons it has been difficult to use model amplitudes that the amplitudes

are not unitary.

Yet a particularly

is the way in which channels open, contribute out as the beam energy is increased. model amplitudes It is always

data is

striking feature of multiparticle

significantly

data

to the cross section, and then die

In dealing with multip~ticle

data it is important

that

be unitary. possible

possible to compute

to take a model

the associated

amplitude

partial-wave

amplitude

of the n-particle

system, j its angular momentum,

internal configurations

associated

of the n-particle

and make it unitary,

amplitude.

the partial-wave

*Supported

to fit production

with the model amplitude.

system.

in part by DOE.

0375-9474 / 90 / $3.50 0 Elsevier Science PublishersB.V. (North-Holland)

provided

To see this let &:&,(sjy,) fi

it is be

is the invariant mass

and yn a set of variables describing

Consider the strongly interacting

the

scattering

W.H. Klink I Relativistic quantum mechanics

2942

operator

S, acting

on an appropriate

StS = SSt = I. For two-particle

Fock space, which satisfies

initial states this gives (2’IStSIP)

open channels are inserted as intermediate

Here 11d”‘“ll(sj)

the unitary

is the “length”

states, the unitarity

of a partial-wave

amplitude,

conditions

= (2’12), and when the

matrix

element becomes

defined by

ll~*-“11*(~~) = ~~~(~.)ld”“(~~,,)12; the measure dp(y,)

is given in the appendix

of I.2 I

(2)

is the inelasticity

parameter

= ml + . . . + m, is the sum of the rest masses of the particles in the n-particle

m(,) n,,,(s)

is the maximum

The unitarity wave

equation

amplitudes

a::&,

then &y&

with an appropriate

equation

a shortcoming

shift for the two-particle

(1).

reaction

are treated

with all the amplitudes

vector; by multiplying

to define a partial-wave

is that the inelasticity on a different whereby

automatically

The purpose of this paper is to show how to construct in the amplitudes

The basic idea is to obtain scattering

footing

satisfying

the unitarity

such amplitudes,

from exactly

it is instructive

functions

nonrelativistic &(P;,

. . . ,&)

spinless system,

rotate the entire n-particle P, the momentum

C = H - P2/2m(,), generators

condition.

in such a way that

described

data.

To construct a relativistic

by n-particle

to first consider an

momentum

space wave

E LZ(R3n). The natural invariance group for such a system is the

Galilei group, consisting

and H, the kinetic

reactions,

on

solvable models of a rela-

reactions.

n-particle

are treated

so as to be able to fit experimental

amplitudes

production

and phase

than the multiparticle

all the amplitudes

quantum

for particle

amplitude

parameter

tivistic quantum mechanics which describes production mechanics

an

Details are given in I.*

to have a procedure

there is still sufficient freedom

by imagining

represents a new channel. If the length

is a unit d~~&e,llld~~&,ll

of this procedure

It is desirable

the same footing,

=

can be satisfied

factor it is always possible

which satisfies the unitarity

amplitudes.

Such a condition

sphere in which each dimension

can be computed,

However,

allowed for a given beam energy.

(1) states that the squared lengths of the open channel partial-

must sum to one.

infinite-dimensional of A::&,

number of particles

while

channel.

of transformations system.

operator,

2,

The infinitesimal

the position

energy operator. (m(,)

which translate

in space and time, boost,

and

generators for such transformations

are

operator,

J’the

One of the Casimir

angular momentum

operators

is the total mass of the system),

of the Galilei

which commutes

operator, group is

with all the

of the Galilei group.

Interactions

come from a potential

there would be no scattering).

V which commutes with @, d, J’, but not H (otherwise

The simplest way to ensure such properties

of V is to make

W.H. Klink f Relativisticquantummechanics

a change of variables to Jacobi variables g’;, i = 2,. Then it is easy to show that space translations in the momentum Casimir

operator

space wave function

. . , n, along with p’, the total momentum.

and Galilei boosts act only on the @variable

. . . ,c&). Further, in these variables the

(P”(g), &,

has the form

$=c$.,

C=H-

n

(3) I

which is the reduced kinetic energy (pi are reduced masses). with space translations

The condition

and Galilei boosts, but not with time translations

to the kernel of V not depending In this group theoretical can be written

29%

that V commute is thus equivalent

on ji.

language

the Schrijdinger

and Lippm~n-S~w~ger

equations

as (C+V)pC,=EW (4)

Pb=Cb+GoVA where the free Green’s function To construct Poincare

a relativistic

is Go(z)

= (z - C)-‘.

quantum

mechanics

group; and to include particle production

is replaced by an appropriate uncharged

spinless particles

Pock space is I;“(R3),

the Galilei

group

the nonrelativistic

is replaced

Bilbert

Fock space. In this paper only a simplified of mass m will be considered,

with the action of the Poincare

by the

space L2(R3*)

world consisting of

so the one-particle

sector of the

group given by

(Ka,hcb)(P) = eip.a (5(A-$1, 9 E L2(R3); here Q is a four translation inner product

given by p

and A a Lorentz

tr~sfo~ation.

p,.

(5)

a is the Lorentz invariant

+a = Eao - p’v a’. The Fock space, S(L2(R3)),

S(L2(R3))=

is defined by

2 @[L”(R3)

LB.. . I$3LyR3)]y,

n=O

where

[ J”,y” means the n-fold symmetrized

subspace of S, a nonnormalizabie

tensor product

of L2(R3).

basis is given by n creation operators

On an n-particle at(p)

acting on the

vacuum state: IPlY...

, Pnf

= a+(Pl). -. Q’(Pn)lO),

where IO) is the vacuum state corresponding l%om the action infinitesimal operator

to the n = 0 subspace of S.

of a and A on n-particle

generators

of the Poincare

(7)

group.

subspaces of S, it is possible One of the Casimir

operators

to obtain

the

is the mass

defined by

M2 = PpPp,

PC the four momentum

operator.

(8)

296c

W.H. Klink I Relativistic quantum mechanics

As in the nonrelativistic to ensure relativistic transformations

case interactions

invariance,

will be governed

should commute

with

A, but not with the mass operator

constructed

by introducing

be a boost,

a Lorentz

M*.

Such potentials

the analogue of Jacobi variables for relativistic

transformation,

whose inverse carries the n-particle

vectors pi to the center of mass frame, where the momenta The action of the Poincark

group on wave functions

V, which,

by a potential

space translations

a’ and Lorentz are most easily

systems.

Let B(p)

four-momentum

are p r, satisfying

cr=‘=, p’r = 6.

over these new variables is given by (9)

where R, is a Wigner properties

defined by R,

rotation

of & under a Lorentz transformation,

mass momenta p’f are rotated angular momentum

remaining

An

of the n-particle

wave function

transformations

The

and describe

Details of these transformations

is now written

subspace that does not depend and Lorentz

transformations.

in which elements

Hilbert

axis in the center of mass frame.

under Lorentz

system.

of the n-particle

the

are given

3.

n-particle

translations

along a space-fixed

yn are then invariant

configurations

the n-particle

‘HP’

equation (9), i t is clear that all the center of

variables, consisting of j, the total angular momentum

variables

in reference

From the transformation

as a rigid body. This suggests making a change of variables to

system, and u, the spin projection

internal

= B-l(p)AB(h-‘p).

as

&($sj,

on p’or

y,).

u, it will be invariant

This suggests defining new “partial

u, have the norm given in equation

spaces generates

If u,, is an element

(2).

of

under space wave” spaces

The direct sum of these

a reduced Fock space in which the variables $7 are eliminated;

an

element u E &educed will then have norm

(10) In this reduced

Fock space V

will automatically

commute

Lorentz transformations,

but not with the mass operator

equation

as

can be written

CM2

and the corresponding

+

v>+

=

Lippmamr-Schwinger

s$,

where the free Green’s function

,7o(t)

=

r”’

space translations

1c, E &educed

equation

1c,= 4 + GoV$,

with

A4 *. Then a relativistic

and

Schrodinger

(11)

is

G,‘4

= 0

(12)

is now given by

G3(2)

=)

,

G&)

= (z-M*)-‘.

(13)

297~

W.H. K&k I Relativistic quantum mechanics

As in the nonrelativistic case exact solutions to equations such BS (12) can be obtained if V is chosen to be a separable potential. Consider, for example,

where u, E 7ic. The general solution of equation (12) IS worked out in II.4 The scattering amplitudes have the form

A2-2 =

(&

t'@) (15)

where #i and &? specify the initial and final states, respectively. D(sj) arising from the inversion of (I-

is a determin~t

GO%‘) and is of the form

k=3

The gk are matrix elements of free particle Green’s functions, namely j;;“,

~~~~~~k~~‘(~k~)l(~

f

kiZ -

gkfsj)

=

(Et,

Gkak)

=

Sk)-‘.

A striking feature of the amplitudes, equation (15), is their similarity with the amplitudes appearing in the unitarity equations, equation (1). If the d{ are chosen its delta functions in partial-wave variables, then the partial-wave amplitudes are d2+” = ~,(sjy,)LD-‘(sj)u;(sj)

where the exact unitarity of all the amplitudes is guaranteed by the terms involving the Green’s functions. Models can now be constructed corresponding to available data. Consider, for example, production reactions in which the angle and energy of one particIe (called c) is detected whiIe all the other outgoing particles are not. Such a reaction is written as a + fi -+ c + X, where X denotes the undetected particles. Then u, in equation (17) need only depend on s,

298c

W.H. Klink I Relativistic quantum mechanics

j, and s,, the invariant mass of the undetected cluster. A simple choice for u, which gives the correct d2+n

threshold properties is

E,* is the center of mass energy of particle c while R(s,) the undetected cluster. computed.

is the phase space volume of

The form of u, is chosen so that its length, equation (2) is easily

By putting in additional polynomial or exponential dependence, it should be

possible to fit existing distribution and total cross section data involving multiparticle reactions. More generally the vn variables are chosen relative to the type of multiparticle data being analyzed.

REFERENCES 1)

For older data see, for example, M. Perl, High Energy Hadron Physics (John Wiley, New York, 1974) especially Chaps. 7 and 8 and references cited therein.

2)

W. H. Klink, Analyzing Multiparticle Reactions: tudes, preprint, University of Iowa, January 1989.

3)

W. H. Klink, Phys. Rev. D4 (1971) 2260.

4)

W. H. Klink, Analyzing Multiparticle Reactions: els, preprint, University of Iowa, January 1989.

I. Unitarizing

Perturbative

Ampli-

II. Exactly Solvable Production Mod-