Solvable one-dimensional models of particle production

Nuclear Physics B81 (1974) 301-316 North-Holland Pubhshmg Company

SOLVABLE ONE-DIMENSIONAL MODELS OF PARTICLE PRODUCTION R.D. PECCEI * Rutherford Laboratory, Chtlton, DMcot, Berkshtre Recewed 3 April 1974 Abstract We discuss a variety of simple one-dimensional models for particle production For each of these models we derive a closed form for the generating functional which allows one to calculate the totahty of mcluswe, semi-inclusiveand exclusive distributions The models &scussed include an n-trajectory multlpenpheral model and a general cluster model. In this latter model we derive a parameter independent relation between semi-reclusiveand inclusive cross sections

1. Introduction Although partacle production processes are of quate a complex nature they show a remarkable number of simple features. Among these we may mention approxamately constant cross sections and scalmg propertaes of reclusive dastributions Reahstlc models for particle production, if they aam to describe things an detail, are quite unwieldy and generally cannot provide one wath analytical answers. Thus some of the sample global features of production processes emerge only after quite lengthy calculations In this paper we would hke to take a somewhat different approach. We would hke to consider a variety of models, whach whale stall aimang to describe at least the gross features of particle production processes, are solvable by analytical means. In most cases, except for the last example, we shall restrict ourselves to models m only one damenslon. That as, we shall neglect all transverse variables an the (fond q) hope that, because the transverse m o m e n t u m is bruited, ttus is not a bad approxlmataon. The models we shall consider are exclusive models. Hence, we shall make a varaety of assumptions for what is the matrix element for the productaon o f n particles Our mare object will be to derive expressions for the lnclusave dlstrabutions, which follow for each of our excluswe models. The techmque we shall use to obtain these mclusave (and sema-inclusave) distributions as that of the generating functaonal. Specafically we shall compute closed form expressions for the generating functional for * On leave of absence from Stanford Unwerslty, Stanford, Cahfornla, USA

R.D Peccet,One-dtmenstonalmodels

302

multiparticle processes for' (1) the Chew-Plgnotti model; (ii) A general multlperlpheral model, (lii) A general cluster model, (iv) A coherent state (three-dimensional) model. In each of these cases, we shall discuss, when appropriate the form of the inclusive and semi-inclusive distributions. We should remark that a number of the results we shall derive are known in the literature. However, we shall obtain then here using a common formalism for all these models, the one of generating functionals, to which we now turn.

2. Generating functional for multiparticle processes A very compact and elegant way of calculating the totality of inclusive and exclusive distributions is provided by the generating functional for multiparticle processes. This functional was Introduced a few years ago in particle physics by Lowell Brown [1 ]. It IS a generalization of the usual formula for the total cross section *

o= s o

r rl

d3p/7¢~4(p_~pl)lL(p;pl...pn),2 n

] ~d3pl '

.

_

,

=

(1)

One extends the above formula by multiplying the matrix element squared ITn 12 by the product of arbitrary functions 4~(P~) for each of the momenta, thereby obtaining the exclusive generating funcnonal 1

d3pl

Clearly by setting ~b(p) = 1 one recovers the total cross section formula

(3)

E l l ] =o.

Exclusive, or inclusive, distributions are obtained by taking functional derlv~atlves orE[q)] and setting qS(p) = 0, or q~(p) = 1, respectively. That is /7

El"'End°n =84(p_ t~lpt)lTn(p, pl...pn)12 d3pl ...

d3pn

=

- El" "En83E[cp]

¢=0'

(4)

while for the Inclusive case one has

El""EndEn- E ~ J1 d3pl ... d3pn m=n

-d3pn+l

d3pm Em

* We consider for simplicity the case of particles of only one type. Extension to more reahstlc cases can be easily incorporated.

R D. Peccel, One-dunenstonal models

303

m

X ~ ( P - ,=1 ~ P ' ) ' T m ( P ' Pl '" 'Pn'Pn+l . . . . Pm )]2

E 1 ... E n 6 3E [~] -

(5)

83~6o0...~3¢(&) ,=1

To calculate E [q~] one has to specify ITn [2 Hence the use of tins excluswe funcnonal lies in the possibxhty of calculating mcluswe distribution via eq (5). One can alternanvely consider an mcluswe functional I [~b] = E [q~ + 1 ],

(6)

winch can be calculated by specifying only the incluave cross sections i[~b]= n~

1 ~d3pl~ dapn(Et'"EndEn~En ~.. / ~.~J ... \d3E: d--3p~._ q~(pl)... ~(pn ).

(7)

g/

By functionally dlfferentiahty the above and setting q~(p) - - 1 one can calculate exclusive cross secnon from a knowledge of the lncluswe cross sections. One can use these generating functlonals to calculate integrated, or semi-integrated, quantities. For example if we set m eq. (2) the function ¢(p) equal to a variable, z, we obtain the generating funcnon for partial cross sections

%z n.

E[~] = ~

(8)

/,/

KnowingE[z] one can calculate on via

°n = ~ 1r - - d n E Izl z=0'

(9)

dz n

winch Is the (integrated) analogue ot eq. (4). Alternatwely, one can obtain the mcluslve moments, (n(n-1) ... ( n - k + l ) ) = l ~ n ( n - 1 ) . . . o

(n-k+l) a

n

(10)

as

d- kk E [z] z=l (n(n-1) ... (n-k+l)) = o1 -dz

(11)

m direct analogy to eq. (5). Semi-reclusive distributions, that is inclusive distributions of fixed multiphclty events, can also be calculated readily. For example, E1 "'" Ek d(k)°n 1 ~d3pk+l d3pn n

X ITn(P;pl ..... Pk, Pk+l' ...,p,)[2

(12)

R D. Peccet, One-dtmenstonal models

304 is given by

El... E n d(k)on d3pl

. d3pk

(n-k)!

{

dn k [ El'''Ek-63E[O]

ka3o(pl)., 63qS(pk)

0=zJ}z=0

(13)

If one considers an arbitrary exclusive, or lncluswe, model one cannot in general succeed in calculating the generating functional. There exist, however, a small class of one-dimensional models for which it is in fact possible to obtain a closed form expression for E[qS]. In this paper we would like to discuss a variety of these models. We shall begin by considering the simplest of these models which is the multiperlpheral model in the Chew-Plgnotti approximation [2, 3].

3. Chew-Pignotti MPM We ignore here transverse momentum and assume that the matrix element squared takes the strong ordering, factorized form [2,3]

ITn(Y,Y 1. .Yn)l 2 = e-V ~

perm

(Ar[e2~(r-yn)o(Y-yn)]g[e2~[Yn-Yn-1

l

O(Yn-y n _ 1)] g... [e 2c~Iy2-y1) O(Y2-y])]g[e 2ayI O(y 1)] A}

(14)

Here we have fixed the two end particles at the imtlal raplditles 0 and Y ~nd shall henceforth ignore them for simplicity. There is no problem in incorporating them in the model If one wishes to do so. A and g are, respectwely, external and internal couphng constants m the model; c~ is the intercept of the input Regge trajectory, and e -g is a flux factor. We have written eq. (14) in a cumbersome fashion since we want later on to generalize our results readily to a many-trajectory model. In fact we may rewrite eq. (14) simply as

Tn ,2c p -- e -Y e 2aYe--,2 j~ gn ~[] O(}_Yn)OfYn_Yn_l)...O(Y2_Yl)O(Yl)" perm

(15)

The phase space factor appropriate to the model is just [2, 3] de = e - r dy I ... dy n,

(16)

Hence we can write the generating functional for the Chew-Plgnottl model (CP model) as E CP [qS] = ~ 1 , [ d y .... d y . gnlAI2 e (2a-2)Y ~ O(y-yn)O(yn-Yn_l) n rt: ,¢ • ,, penn "'" 0(Yl) ~b@n) ~(Yn-1 ) "'" q~(Yl)

(17)

R.D. Peccel, One-dtmenslonalmodels

305

Since the ~perm cancels the 1In! one is left with, simply,

Y

Yn Y2 ECP [(b]: ~ g n l A I 2 e(2C~-2)Y f dYnC)(Yn)f dYn-t¢a(Yn-l)"" f dYl~(Yl) (18) n 0 0 0 The above expression can be formally summed if one introduces a rapidity ordering symbol T(q$(yl)~b(Y2)) =0(y 1 -y2)4b(yl)~b(y2) + 00e2-Yl)q$(y2)4b(yl).

(19)

Then the quantity m braces in eq. (18) becomes

y

f

Yn dYn~b(Yn)

0

f %

0

Y2

1)--f

1 Y dYl (~@1)=-nT dYl"'dYn T(dP(Yt)"'Cb~n))"

f

0

0

(20)

Using the above identity eq. (18) becomes ~

ECP [~1 = ~ n

IAI2 e(2~-2)Y

V

T(fddy

O(y)g) n n!

'

(21)

which ymlds Y ECP[~b] =,A, 2 e (2~-2)Y T ( e x P [ o f dy g4b(y)]).

(22)

The total cross section for the model is just o CP = IAI 2 e (2~-2+g)Y

(23)

If one reqmres that it be a constant, o, this fixes 2 ( l - a ) =g,

(24)

IAI2 = e.

(25)

With this boundary condition the generating functional for the Chew-Plgnotti model becomes

ECP[o]=oT(expI?dygO(y)-gY]).

(26)

0 Setting q$(y) = z yMds the generating function for the model E CP [z] = o

egg(z-1) ,

(27)

which one recognizes as the generating function of a Poasson distribution. Hence, as is well known [2,3], oCP _ o(g Y) n

n

n!

e-gY"

(28)

R D. Peccet, One-dimenstonalmodels

306

The inclusive distributions can now be readily calculated using eqs. (5) and (26) d~ CP

_

hyx.., dy,,

6 E CP [0]

¢,=1

5q~(Vl)... ~,~0',,) Y

= o T(g n expf : g~(v) dy - gY ]) ~=1'

(29)

- o g n"

(30)

which gives dE CP

dy ... d)' n

We remark that thxs 1s preosely the answer one would get from a Mueller-Regge model [4] which included only pomerons. This is a simple example of the correspondence between multipenpheral exclusive models and Mueller-Regge lncluswe models *

4. n-trajectory MPM Our work of sect. 3 can be straightforwardly generahzed to a multlperlpheral model with n-trajectories. In this case instead of eq. (14) one has

ITn(Y;y 1 ...yn)J 2 = e -r ~_i {A r [R(Y-Yn)O(Y-Yn)lg[R(Yn-Yn_ffOO'n-Yn_l)] perm

× g ... g[R(Yl)O(Yl) ] A},

(31)

where now A=

A2 [m 1)

(32)

n

is a vector of external couphng constants and g l l " " "gln g=

(33)

L1 £ .

is a matrix of internal coupling constant, t h e input Regge propagators are now characterlzed by a diagonal matrix

* This correspondence has been noted by a number of people. For the earhest references see ref. [5]. For an exhausting dascusslon see [13].

R D. Peccet,One-dtmenstonalmodels

307

e2Cq(Yt-Yt-O e 2c~2(Yt-Yl-1) (34)

R('yt-Yt-1) = I • " " e2~n (Yt-Yt- 1) ' We note that

R0,, -y,_ ~) -- ROy,)R(-&_ 1) = R0',) R-~0~,_ 1)"

(35)

Using this fact we can write the generating functional for this n-trajectory MPM as

y

EMPM(q~)= n~

Yn A T S(Y) f dYnS(-yn)gS(yn)¢Oen) f dYn_lS(-Yn_l)gS(_Yn_l)~@n_l) 0 0

... f dy I S(-y~)gSO'I)00'~) A, where

(36)

0 e(2~1-2)y.

S(y) =

[ ".

.

(37)

e(2~n - 2)y This equation can be again formally summed yielding Y EMPM(¢) = A r S ( Y ) T ( e x p f dy S(-y)gS(y)¢(y))A. (38) 0 which, of course, reduces to eq. (22) in the case of one trajectory. Eq. (38) is only a formal answer and it is not terribly useful because of the rapidIty ordering instruction T, which now is important since we are dealing with matrices. A more useful version of eq. (38) can be obtained by considering instead of EMPM(¢) the matrix/~[¢] defined by EMPM(~b) = A T if'[C] A.

(39)

We note that the functional Y

I(f; Y)= T(exp ffO')dY)

(40)

0 obeys the integral equation Y

I(fl; Y) = 1 + f dyf(y)I(f;y).

(41)

0 Hence if7 [¢]

=S(y) I(S-lg SO; IT)

obeys the integral equation Y if[C; Y] =SCv) + f S(Y) S-I(y)gS(y) 0

(42)

q~)S(-y)EI¢,y] dy,

(43)

308

R.D Peccet, One-dimenstonal models

or Y

"ff[49,Y] =S(Y) + f S(Y-y) g49(2)ff [49;y] dy.

(44) 0 The above integral equation Is a functional analogue of the multiperipheral integral equation obtained some time ago by Chew et al. [6]. Along with eq. (5), it allows us to calculate all the inclusive distributions m the model. For example, setting 49 = 1 in eq. (44) gives the following integral equation for the total cross section in the model (actually o = ATe'A) Y ~(Y) =S(Y) + f S(Y-y)g "~(y) dy. (45) 0 The solution to eq. (45) lS readily obtained by Laplace transforms. Defining

"o(0) = f C oY "6(Y)dY,

(46)

0

S(O) = f e-°Y S(Y) dY, 0

(47)

one has

"5(o) -- s(o) + s(o) g'6(o),

(48)

~(0) = [s-l(0) - g ] - l .

(49)

or

Hence, finally, C+to~

o M P M ( Y ) = ~1/

f

dO

C_too

eOY AT - - 1 A. S-l(O)-g

(50)

Eq. (50) reforms us of the well known result that

n oMPM(Y) : l_~l72 eCa,- 1 ) r ,

(51)

where a t - 1 ls the location of the ith zero in the O-plane o f S - ! - g and 3,2 its appropriate residue. We should note that in general ~i is distinct from %; that Is, the output poles in the multlperipheral model are different from the input poles. The generating function for partial cross section E [z ] can be obtained in precisely the same manner as the total cross section. One only needs to set 49 -- z instead of 49 -- 1 in eq. (44). Hence one has

H E MPM [z] = G %2(Z) e [al(z) 11 Y, /=1 "

(52)

tLD Peccet,One-dtmenstonalmodels

309

where az [ z ] - l , and 72(z) are the location of the zeros and the residue of S-l(O)-gz. In terms of this notanon a z [1 ] = ~t, 7~ [1 ] = 7~. To calculate the one particle inclusive distribution in the model let us take the functional derivative of eq. (44) with respect to ~b(ya) Y 8ff[~b; Y] _S(Y_yt)g~[~,yl] + f dy S(Y-y)gg)(y) 8~(yl) ~ ~b(Y1) 0 Setting ~b= 1 yields the following integral equanon for the one particle inclusive distribution (again really d22/dy 1 = AT(d'~/dyl) A) ~

r

d2

dX (r, yl) =S(Y_yl)g~(yl) +f dy S(Y-y)g~f-~lO,;Yl). dYl

0

(53)

This equation can also be readily solved by taking Laplace transforms. Define P(Ox, 02) =

fd(Y-y) e-oI(Y-y)

0 Then eq. (53) becomes simply

f dy e -Ozv ~d -21 r , y ] . 0

~(01, 02) = S(01) go(02) + S(01) g~(O1, 02) ,

(54)

(55)

which, on using eq. (49), gives ~'(01, 02) = ~(01) g~(O2)"

(56)

Upon inverting this equation one obtains for the inclusive distribution a not unexpected [5] Mueller-Regge answer dY--ldZ(y, Yl)

= ~ ~¢1e(~t-I)(Y-Yl) Tq e(~l-1)Yl 71-

(57)

t]

It is clear that if one wants Information for higher inclusive distributions one can obtain it by using a procedure similar to the one just exposed.

5. Cluster models

As a third class of solvable models we would like to consider here cluster models. These models are again of a multIperlpheral variety, but what is assumed to be produced multaperlpherally is not a single particle but a cluster of particles. In general one assumes that the clusters themselves are produced independently (CP producnon mechanism) and that the probability of a cluster decaying into a fixed number of particles is both energy independent and contains no correlations among the produced particles. The matrix element squared for a cluster model IS then given by (neglecting transverse variables)

310

R.D Peccet,One-dtmenstonalmodels n

1

[Tn(y;y 1 ...yn)[2 = 1 ~O ]zT. fdYlC '

.dyC[Tl(Y;yC.."

yC)12p

X XLn [yC... yCl ,yl... Ynl " Here X,, ( y f , .. yC;y,... Yn) IS the

(58)

probability that 1 clusters of rapidity y r . . . yC decay into n particles o f rapidity Y 1-'-Yn, whale ITn(Y, yC.. yC)12 P is the matrix element squared for producing these clusters according to the Chew-Plgnottl mechanism [eq. (14)]. It ~s useful to introduce a generating functional for cluster decay j,..

--

•

,

e[q~;yc] : k~

1

fdYl...dy k Wk(Yc,Yl...yk)dp(yl)..c~(yk) '

(59)

where Wk(Yc; y 1"" Yk) represents the probablhty that a cluster of rapidity Yc decays into k particles of rapidity y 1 "" Yk. Note that by definmon e [1 ;Yc] = 1.

(60)

Consider now the product of the generating functlonals for decay of clusters of rapidity yC... yC

e[cp;yC]...e[(3;YC]={~kl +fdYll...dYkIWkl (YC;YII'Ykl)qb(Yll)"'(9(Yk I) ... ~ ~_~.f dYll. dYkjWkl(yC,ylf.. ykl) (y(yll) . ~(y kl) } . 1,1 j.

(61)

It is easy to convince oneself that, if there are no final state interactions, one can rewrite the right-hand side of eq. (61) in terms of the probabilities forj clusters to de. . . . Al, n k.V e..cl . . . .r.,c.,, v ) Namely, cay into n particles UI, ] ,.r I ""-"n " e [gb,yC] ... e [qi, yC] = n~ l~T.fdYl.-.dYn

)~,,n(yC...yC;yl'"Yn)¢(Yl)"'(P(Yn)" (62)

Bearing this in mind, it as now trivial to deduce the generating functional for a cluster model

E[4,] = ~ l f d y rt

I ... dy. e-YIr.(r;yl ...Yn)[ 2 q~CVl) .-. ~b(.Vn)

/'/"

l fdyC1...dyC . . . n

.

n n'

=

e

_y1 _.y C ~TII} ( ;Yl

X Xl,n(y C...yC;y I ""Yn) ~(Yl)"'" 4~(Yn)'

2 2 iC )ICP (63)

R D Peceet, One-dtmenslonal models

311

which on using eq. (62) yields E[q~] =

~-~lfdyC...dyCe-rlT/(Y; ylC ""YiC'I2 ) CP e [q~'YT] "'" e [¢; YC]' s I. s

(64)

Eq. (64) is precisely of the form of the generating function for a CP multlpenpheral model with the replacement q~(y) ~ e [4, v ] . Hence, using eq. (26), we can write immediately * Y /;,cluster = o T (e xp [ ! d y g e ( ~ ; y ) - g Y ] ) . --N,I

(65)

(66)

Note that if the clusters are forced to have only one particle then e(¢;y) = ¢(v) and eq. (66) reduces back to the CP expression, eq. (26). If in eq. (66) we set ~ = z we obtain the generating function for multiplicity distributions for the cluster model Ecluster[z] = o d Ye(z)- 1),

(67)

where, of course,

e[z] =~k [ -~!fdyl'''dyk Wk@c;Yl"'Yk)]zk=-~#kPkzk"

(68)

The form o f the generating function shown in eq. (67) is that of a compound Polsson distribution [7, 9]. We remark that this is also the form of the multaperipheral generating function EMPM(z), of eq. (52), If we retain the contribution of only the leading output trajectory. This correspondence had been remarked upon earlier [9]. Although eq. (66) is the generating functional for an arbitrary cluster model, in what follows we shall speclahze to a model in which the cluster decay is uncorrelated and the probability of a cluster decaying into a given number of particles is energy independent **. Thus we shall assume that

Wk@; y 1 ... yk ) = k! Pk P(Y-Yl) P(.v-Y 2) "'" P(Y-Yk ),

(69)

where the dlstrlbunons p(y-yt) are arbitrary but normalized to one

f dy t p(.v-y,) = l.

(70)

Also, o f course,

Pk = 1.

(71)

* An equation similar to this has been written down by Caneschl [8]. ** These kind of models have been of interest lately. See for instance ref [ 10] for a particularly simple example of such a model.

R D Peecet,One-dimensionalmodels

312

It is interesting to compute the inclusive distributions in this model. Using eq. (5) we have dy 1

_

8q~(Vl)

= = og T

(/

8e[Gy]

dy 8~b(yl)

expl

fdyge[Gy]-gY

0

IOl

(72) Since we are now dealing with functions, not matrices, the T instruction is unnecessary Using eq. (60) and 8e [qS;y] = ~[~ 1 ~b(Y1) qS=l k=l ~ d d Y 2 d" 'Y" k f

Wk [F;Y l ... Yk ]

= ~ kP k p(y-yl) = (k) P(Y-Yl)' k

(73)

one obtams [11] - E cluster Y 9 dy I (Y;yl) =g o(k) f dy p(V-yl). 0 In a similar fashion one can show that m this model the correlation function C2(VlY2)=l

dY, dYl dY2

( 1 dN~{1 d 2 ; )

(74)

(75)

dYl]\° G

is given by [11] Y ~21uster(ylY2) =g(k(k- 1)) f dy p0~-yl) O(.v-Y2). (76) 0 The semi-incluswe distrlbutxons in this model are also obtained qmte straightforwardly. One needs first to know the functional derivative of E[q~] evaluated at q~= z. One finds for example, that in the model, Y 6E[~I (77) 8~(.v1) ~=z 0 Y 82E [4)1 ~ k(k-1) ek zk-2 E [z] / P(Y-Yl) P(Y-Y2) dY2 8~b(Yl)SC~(Y'2) 4=z =g 0 Y Y + ( g ~ k P k z k - 1 ) 2 E [ z ] ;O(y-Yl)dYZPO~-Y2)dY. (78)

=g~ kP~zk-lE[zlf oO,-Yl)dY,

0

0

Hence, using eq. (13), one now can evaluate the one and two-particle semi-inclusive distributions. After a httle calculation one obtains * * Caneschl [8] obtains similar equations in a model where PlY-Y1) = 8(Y-Y1).

R.D. Peccet,One-dtmenstonalmodels d(l)on

313

Y

dye--i-=g ~ kPk °n-k(Y) f dy P(Y-Yl )' 0 d(2)On = dYldY 2. g ~ k ( k

(79)

Y

1)PkOn_k(Y) f dyp(y-yl)P(v-Y2) 0 Y Y +g2~klk2PkiekEon_ki_k2(Y) f dyo(v-y 1) f dyp(y-y2). 0 0

(80)

Recalhng eqs. (74) and (76) one can rewme the above in terms of the inclusive distributions: dy l

(k)

d(2)on dy 1dy 2

dy 1 ]

~ k(k-1)P k an_k (k(k- 1)) C2CVl' Y2)

klk2 klk2PkIPkEOn-kI-k:~ (1 dE ' ~ ( l d E "~ 4 (k)(k) \o dYl]\O dY2]"

(82)

Actually the above expressions can be further simplified by using the sum rules d (1) on fdy 1 dy I -/7 on, (83)

fdYl

d(2)On dY2 dy 1dY2

-

n(n-1)o n.

(84)

Using this information one can rewrite eq. (81) as d(1)On nOn 1 dE dy 1 (n) o dy1 "

(85)

This equation relates, in a parameter-independent manner, semi-Inclusive one particle distributions to inclusive distributions in cluster models. &milady one has that d(2)°n _n(n-l)°n (1 dE ~{1 dE ) + C , ( y l Y 2 ) [ k~ k(k-1)P k On_k ] dYl dY2 (n)2 (k(k- 1)) where C*(ylY2) is a correlation function whose integral is ldentmally zero: C,@,y2)=C2CvlY2)

f2 ( 1 d E ~(1 d E )

- (/7)-----2

dy 1 , ] \ o

~Y2

'

(86)

(87)

where, of course,

f2 = fdYl dY2 C2(YlY2) = (n(/7--1)) -- (n) 2 .

(88)

R.D.Peccet,One.dtmensionalmodels

314

From these formulas one can compute other quantities of interest, if desired, like

6. Coherent state models As a final example of a model that can lead to a closed form for the generating functional, even though it is not one-dimensional, we would hke to consider the coherent state models of the Santa Barbara School [12]. In these models one assumes that one can replace the density operator P =~(P-

~Pt)TLP)(PIT+'

(89)

by a diagonal outer product of coherent states 19 -~ffTr[Tr) (Trl e -F[~r] .

(90)

Here e - F M is a weighting function which contains the physics and 17r) is a coherent state:

d3p/Elrr(p)l 2

17r)=exp-~-

exp

~--

which IS an eigenstate of the annihilation operator a(p) a(P)l 7r) = 7r(p)l 7r)

(92)

The functional integral in eq. (90) IS over the space of functions 7r(p). With this approximation one can write the exclusive cross sections ?/

fl""fn d°n =5(e- t~lpt)lZn(e;Pl...pn)12 d3pl

...d3pn

=

n

= 6(P - I~=lPt)(PI ...PnITIP) (PIT+IPl ...P n) =(Pl...Pn[OlPl...Pn )

(93)

as

El... E n do n d3pl...d3pn But

y67r (Pl "" Pn 17r)(Trip1 "'" Pn ) e-FITrl "

(94)

R D. Peccet,One-dtmenslonalmodels

315

Hence the exclusive cross sections in these models are given by

d3pl ..- d3pn ) (96) The generating functional E[q~] can be easily calculated using eq. (96). One has

EI ] =2

'

r d3,' d%( E1 ..E

~"J ~1 "'" En

\d3pl... •

d3p. ,I ~(Pl) "" ~(P.)

= ~n f 67re-F[M exp[- f dgp/Eln(p),2]l (fd3p/Elrr(p),20(.p)) n,

(97)

which gives, finally

E~ls)= f6n e-Fl'd expf -d3p - ~ I~r(p)l 2 (~b(p)- 1)

(98)

Using eq. (98) one can calculate readily the inclusive distributions m the model. One finds

o =f6rr

e -F[M ,

E n dX n d3pl ... d3pn

E1

(99)

- f 6 r r e -FIn] ]fr(p 1 )12 ... )~.(pn)12.

(100)

These equations suggest a natural analogy with statistical mechanism with the corres. pondence a*--* Z (partition function),

El... E n d~ n *-* P (Pl "'"Pn) d3pl ... d3pn

(probability densmes).

(101)

I would like to thank R.G. Roberts and D.P. Roy for some very useful and encouraging conversations. I would also like to thank Roger Phillips for his hospitality at Rutherford Laboratory.

References [1] [2] [3] [4] [5]

L.S. Brown, Phys Rev. D5 (1972) 748. G.F Chew and A. Plgnottl, Phys. Rev. 176 (1968) 2112. C. de Tar, Phys. Rev. D3 (1971) 128. A.H. Mueller, Phys. Rev D2 (1970) 2963 W.A. Bardeen and R.D Peccel, Phys. Letters 45B (1973)'353; S.Y Mak and C.-I Tan, Phys. Rev D8 (1973)4061,

316

[6] [7] [8] [9] [10] [11] [12] [13]

R.D. Peecel, One-dtmemsonal models F DUlmlO and G Marchesml, Nucl Phys B59 (1973) 222; M.B. Green and B.R. Webber, Nucl Phys B61 (1973) 370. G.F. Chew, M.L. Goldberger and F.E. Low, Phys. Rev. Letters 22 (1969) 208. W. Feller, Introduction to probab'hty theory and Its apphcatlons (Wiley, New York, 1960) vol. '1. L. Caneschl, CERN preprmt Th 1826. W.R. Frazer, R.D. Peccel, S.S. Pmsky and C I. Tan, Phys. Rev. D7 (1973) 2647. E Berger, CERN preprmt Th. 1816. M. Le Bellac, H.I Mmttmen and R.G. Roberts, Phys. Letters B48 (1974) 115 D.J. Scalapmo and R L. Sugar, Phys. Rev. D8 (1973) 2284, J.C. Botke, D.J Scalapmo and R L. Sugar, Phys. Rev. D9 (1973) 813. S.S. Pmsky and G.H. Thomas, Phys. Rev. D9 (1974) 1350