Scaling laws for inelastic partial-wave amplitudes and the high-energy behaviour of multiparticle production

Nuclear Physics B88 (1975) 171-188 © North-Holland Publishing Company

SCALING LAWS F O R INELASTIC PARTIAL-WAVE AMPLITUDES AND THE HIGH-ENERGY BEHAVIOUR O F MULTIPARTICLE PRODUCTION O. HAAN and K.H. MUTTER Institut fiir Theoretische Physik, Universiti~t Heidelberg Received 2 October 1974 (Revised 19 November 1974)

t-channe! unitarity equations are derived for n-particle overlap functions. Together with s-channel unitarity they lead to scaling laws for the inelastic s-channel partialwave amplitudes fl(n)(s) in the limit s ~ 0%l ~ ~ x = l(iz/~/s) 3 = fixed. Assuming the validity of the scaiin~ law in the whole range, allowed by s-channel unitarity - i.e. for l > L (s) = (c~(4t~) - 1) (~/'s-/4~)log (S/Sl) we obtain constant production cross sections a(n)(s) at high energies s ~ ~ up to log s factors.

1. Introduction The most striking fact of high-energy collision data is the approximate constancy o f all the cross sections with vacuum quantum number exchange. It is well known that the description o f this simple behaviour within the framework o f the analytic S-matrix theory is indeed very complicated. The reason for this is the following: The familiar picture that nearby singularities in the t-channel, e.g. the elastic t-channel unitarity cut, govern the high-energy behaviour in the s-channel seems not t o be con, sistent with s-channel unitarity if the leading J plane singularity (pomeron) has intercept one [ 1 ]. This can be seen explicitly from the results o f a previous paper [3] where we have investigated the properties o f the s-channel partial wave amplitudes, which enforce the t-channel unitarity behaviour at threshold t = 4/~2 for all the tchannel partial waves J > a (4~ 2). a (4# 2) is the position o f the leading Regge singularity at the t-channel threshold. We have found that the absorptive parts o f the schannel partial-wave amplitudes have to obey a scaling law in the limit s ~ 0% l-~ ~° x = l(I.t/~/s)3 = fixed Ira f / ( s ) = V"f~-e -4sx/u2/'t6 /3(x) s3

~/x

The behaviour o f the structure function ~(x) for x ~ 0 is determined b y a(4/l 2) and 5 (41a2), the position and the type o f the leading pomeron singularity at t = 4#2:

172

O. Haan, K.H. Miitter/Scaling laws

[. 1 ~ 26(4#2) +1 /5 0 ¢) ~ X -c~(4~u2)-~ ~ l o g ~ ) The scaling law can be proven rigorously for l > O(s) as has been pointed out to us by Grassberger. If we extrapolate the scaling law to smaller values of l, we find that the s-channel unitarity bound Im j~(s) ~< 1 is satisfied for l t> L (s) = (a(4~u 2) -1)(x/~//a) log (s/s 1 ) but violated for l < L (s). Therefore, inelastic t-channel cuts have to contibute in order to save s-channel unitarity for the lower partial-waves l < L (s). On the other hand there is no problem with s-channel unitarity, if we assume that the high partial-waves l/> L (s) are governed by the elastic t-channel unitarity cut. In ref. [4] we found some interesting consequences, assuming the validity of the scaling law in the maximal range allowed by s-channel unitarity i.e. for l > L (s). The pomeron intercept ol(0) is always one irrespective of the position a(4/~ 2) and the type 5 (4/a 2) of the leading J-plane singularity at the t-channel threshold. Arguments for the stability of the pomeron intercept, have been given also by Veneziano, Finkelstein and Harari [2]. In this paper, we want to show, that scaling laws, also hold for the inelastic partial waves, i.e. for the partial waves ~21(s,z ) of the "grand canonical partition function" [5]

~2(s,t,z)= ~ znf(n)(s,t)= ~ ( 2 1 + n l=0

1)~2l(S,z)P l ( 1 +

2t ) s-4/a 2 '

where f(n)(s,t) is the n-particle overlap function. The reason for this is that ~2(s,t,z) obeys exactly the same t-channel unitarity equations (for 4/a 2 ~< t ~< 16/a 2) as the absorptive part Im F(s,t) = ~2(s,t, z = 1). The proof is given in sects. 2 and 3 and in the appendix A. The scaling laws are derived in sect. 4. Assuming again the validity of the scaling laws for 1/> L (s) i.e. in the maximal range allowed by s-channel unitarity, we find also a constant highenergy behaviour of the production cross section o(n)(s) (up to log s factors).

2. t-channel unitarity for overlap functions and for the grand canonical partition function Let us briefly review the analyticity properties of the n-particle overlap function * n f(n)(s,t)= ~zr f d k l . . . d k n 5 (4) ( ~ k i - P) X (kl . . . k n iT] plP2 ) (k 1 . . . k n ITI p'lp'2 ), *

(1)

• The n-particle states are normalized to one. The energy-momentum conservation 6-function is separated from the T-matrix elements.

O. Haan, K.H. Matter/Scaling laws

173

where s = P 2=(t71 + p 2 ) 2 = ( p l

t

t

+p'2 )2,

t = ( p 1 - P l )2,

dk dk = 2k ° •

O)f(n)(s,t) n = 4, 6 . . . . is analytic in the Lehmann-Martin ellipse in the cos 0 plane cos0=l+

2t t<~to(S ) , s_4/a2 '

s=fixed,

(2)

where to(S ) > 4/a 2 is the boundary of the Mandelstam spectral function. This follows immediately from the convergence of the partial-wave decomposition

f(n)(s,t) =

(1+ )

l=0

s_4/a 2

Notice that the partial waves ft(n)(s) are bounded

O<~ft(n)(s)<,Imft(s),

n=4,6

(4)

by the partial waves of the absorptive part Im F(s,t) of the four-point function. (ii) In a theory with pseudoscalar mesons, f(2)(s,t) is analytic in a bigger ellipse with a major axis 0 ~< t ~< tl (s), tl(S) > 16~u2. This follows from the fact that any Feynman graph with two particles in the intermediate state of the s-channel conrains at least four particles in the t-channel. In the following the properties of the n-particle overlap function f(n)(s,t) (cf. eq. (1)) on the elastic t-channel cut 4bt2 ~< t ~< 1@2 will play a very important role. The main goal of this section and of appendix A is to prove the following discontinuity formula:

f(n)(s,t + iO) --f~)(s,t-iO) = 2i disc j4n)(s,t) = i 8 ( t ( t - 4 p 2 ) ) -½ /I

t n

X ~ ] = 2,4 . . . .

f dSldS2K-½f(D(s,t+iO)f(n-D(s,t-iO), K(S,Sl,S2,t)>~0

(5)

where s > 4/a 2, 4# 2 ~< t <: 16/a 2 , n/> 4 and K is the well'-known Kibble function: SSlS 2 g(s, Sm,S2,

t) = s 2 +s~ + s~

- 2(SS1

+

SS2

+ SlS2)

i t _/12 .

(6)

Indeed, eq. (5) looks very similar to the corresponding formula which follows for the discontinuity of the absorptive part Im F(s,t) from elastic t-channel unitarity. This similarity becomes even more transparent if we consider the quantity

[2(s,t,z) =

~ n=2,4,

z n f(n)(s,t), . . .

(7)

O. Haan, K.H. Miitter /Scaling laws

174

which might be interpreted as a "grand canonical partition function" [5]. One readily realizes that the unitarity relation (5) for ~2 (s,t,z) is diagonal in z:

~2(s,t + iO, z) - ~2(s,t-iO, z) = i 8 (t(t - 4/a2)) --~ ~r

X f dSldS 2 K - ÷ ~(s 1, t + i0, z) g2(s2, t-iO, z). (8) K~>0 For z = 1, this equation reduces to the well-known equation [6] for the discontinuity of the absorptive part

Im F(s,t) = ~2(s,t,z = 1). The validity of eq. (8) is obvious in any model which obeys elastic t-channel unitarity - , e.g., the mulfiperipheral model [7] - where the n-particle overlap, function depends on a coupling constant X in such a way that

f(n)('h') = g(~k', ~,)n f(n)(~,). The function g(h', X) does not depend on n. In this case the grand canonical partition function (7) with z = g('h', k) can be interpreted as the absorptive part of an elastic scattering amplitude Im FCh'), which follows from the original one Im F('h) by changing the coupling constant ~, -+ X'. Then eq. (8) is nothing else but the tchannel unitarity equation for the new scattering amplitude F 0 , ' ). The model-independent proof of eq. (5) is rather long. Here we will treat only the simplest case n = 4. The general proof will be performed in appendix A by complete induction. Proof of the case n = 4: Going back to the definition (1), one readily realizes that the discontinuity on the cut 4/a 2 ~< t < 16/~2 can arise only from pole contributions in the T-matrix elements, e.g. (kl...

=

1

2!

k4[T[plp 2) Vc~zr ~ .

[ ( P l - K 2 )2

/a2]-I

X (klk2[TlPl, K 2 - p l ) (k3 k 4 [TIP2 , P l - K2>'

(9)

where K2 = k I + k2 . This means: disc~4)(s,t)=disc___~l 4! f d k l t t 16rr (2!)2

"'"

dk48(4 )

ki_p

(2!) 2 4!

x [(p~ - / q ) 2 _.21 -~ [(p, _/q)2 _ .21 -~ t

t

~

!

t

X (klk 2 [TlPlK2-Pl)(klk 2 [Tip 1 K 2 - P l > (k3k 4 ITIP2,Pl-K2~(k3 k 4 ITIp 2, Pl - K2>*.

(10)

O. Haan, K.H. M~tter/Scaling laws

175

Fig. 1. Contribution to the discontinuity of the four-particle overlap function (cf. eq. (10)). A graphic representation of this contribution is shown in fig. 1. The factor 4 !/2 !2 in eq. (10)results from the number of possibilities how to arrange the four particles 1,2, 3 , 4 into two clusters - each cluster containing two particles. The factor cancels exactly the normalization factor in eq. (9). The pole denominators in eq. (10) only depend on K 2 = k 1 + k 2. Therefore, it is useful to introduce the total cluster m o m e n t u m K 2 as a new variable. The integrals over the momenta of the particles within one cluster

~TrfdkI dk 2 6 (4)(k 1 + k 2 -

K2) ( k l k 2 ITlPl, K 2 - p2 ) (k 1k21T[P'l, K 2-p'l )*

= f(2)(K~, t),

~fdk3dk4

6(4)(k3 + k4 - P + K2) < k3k4[T[P2' Pl - K2) (k3k41TIP'2' P'I-K2)*

= f ( 2 ) ( ( p - K2)2, t) can be expressed in terms of the two-particle overlap function. Thus we find disc f(4)(S,t)=~ t

disc t

f d 4 K2

f(2)(K2, t) f(2)(( P-K2)2,t) ((P 1 - g 2 )2 _/d2 )((p '1- g 2 )2 _//2 )

This leads us to eq. (5) with n = 4, if we substitute 1

d4K 2 = ds 1 ds 2 ~ - ( ( s 1 + s2-s)2-4SlS2 )½ d cos 02 d~ 2, where

s 1= K2'2 s2 = (P_K2)2, K2 = ig21 (sin02 cos ¢ 2 , s i n 0 2 sinO2, c o s 0 2 ) and calculate the integrals over the angles 02, 4)2

di?c fd4K2 -

rr2

((Pl-K2)2-

~2)-1 ((P'I - K2 )2 - / a 2 ) - I • • "

-i f dSldS2K(S, Sl, S2, t)-~ ... (t(t - 4/a2) ~ K(S,Sl,S2, t)>0

(11)

O. Haan, K.H. Miitter /Scaling laws

176

3. t-channel unitarity for the partial waves of the grand canonical partition function The t-channel unitarity relation (8) can be diagonalized by introducing "t-channel partial wave amplitudes" oo

coj(t,z) = 4 ( t - 4 / / 2 ) -1

f

ds~2 (s,t,z)Q3

(1+

4u 2

2s ) t_4//2

,

coj(t + iO, z) - ¢oj(t - iO, z) = 2 0 ( 0 toj(t + iO, z) coj(t - iO, z).

(12)

(13)

If z = I, this is exactly the well-known elastic unitarity relation for the t-channel patial wave amplitudes of the four-point function. The equivalence of eqs. (8) and (13) is well established in one direction [6]. From the unitarity relation (13) and the Froissart-Gribov formula (12), one can derive the discontinuity formula (8). This has been shown explicitly only for the case z = 1. However, since z is a mere parameter, which does not appear explicitly in eqs. (8), (12) and (13), the statement is true for all z and J values, where the Frissart-Gribov formula (12) converges. Here we want to show that the equivalence of (8) and (13) also holds in the other direction. That means we will derive the validity of the unitarity relation (13) for the partial wave amplitudes from eq. (8) and the Froissart-Gribov formula (12). For the proof let us start from the right-hand side of eq. (13). Inserting the FroissartGribov formula (12) and the dispersion relation for the Legendre functions 1

Q'(x)= 1

f --1

X --X

~

ooj(t + i0, z) ooj(t - iO, z) = ~

, = 1,2,

" " "'

(t - 4//2) -2

00

×f

dSldS2 ~2(s 1, t+iO, z)~2(s 2, t-iO, z ) G j ( s 1, s 2, t),

(14)

4t~2 where

Gj(Sl, s2, t) =

i f 1 dXld, , 2 (xpj(x,x)Pj(x,2) 1 _ Xl ) (x 2 _ x2 )

(15)

-1

(15a)

K(s,s 1 ,s2,t)>O

~

2Si Xi-- 1 +t - 4//2 '

i = 1,2

and K = K(s, s 1 , s 2, t) is the Kibble function (6).

'

O. Haan, K.H. Miitter /Scaling laws

177

Eq. (15a) is derived in appendix B. Now eq. (13) follows immediately if we combine eqs. (14), (15a) and (8):

~oj(t + iO, z) ~oj(t - iO, z) Qj (I + 2s t 1 6 / > ds dSldS 2 t-/22] ~2 (Sl, t + i0, z) ~2(s2, t =~-~__~0 x / K ( t - 4/22) 2

2rr (t - 4/22) } _1

4u 2

i0, Z)

t --

4/22

1

2i p(t) (wj(t + iO, z) - coj(t - iO, z)).

4. The scaling law for the s-channel partial-wave amplitudes of the grand partition function Let us summarize the properties of the grand canonical partition function

~2(s,t,z) defined in eqs. (7) and (1): O) ~2(s,t,z) is analytic in the Lehmann-Martin ellipse, i.e. for t <~to(S ) where to(S ) ) 4/22 is the boundary of the Mandelstam spectral function. (ii) The t-channel partial-wave amplitudes coj(t,z) defined by the Froissart Gribov formula (12) obey elastic unitarity for 4/22 ~< t < 16/22. In particular near the threshold t = 4/22 , the effective range expansion holds:

7~(t, z, J) = ( 4u-~_ t ) J ~s(t, z) = eej(z) + a (1) (z) (t - 4122) + . . . 4

c~j(J+ 1) (z) (d + 1)!

( t - 4/22) J+l + . . .

(16)

aj(z), a(j1)(z) are the analogues to the scattering length and the effective range. (iii) The constraints of s-channel unitarity can be most easily seen in the s-channel partial-wave expansion

~2(s,t,z)= ~

l=0,2 ....

(2l+l)~21(s,Z)Pl(1 + 2t i, s_4/22 ]

(17)

O. Haan, K.H. Miitter/Scaling laws

178

~-J . . . . z n fl(n)(s), ~2l(S'Z) = n=2,4

(18)

Im ft(s) =

(19)

~

f/(n) (s), [f/(s)[2 =.f/(2) (s).

n = 2 , 4 ,

.

.

.

Thus the s-channel partial waves I2t(s,z ) are bounded by one, if Izl ~ 1 and we can conclude: The grand canonical partition function ~2(s,t,z) (for Izl ~< 1) has exactly the same s- and t-channel unitarity and analyticity properties as the absorptive part of the four-point function Im F(s,t) = ~2(s,t,z = 1). Due to this complete analogy, all the results which are obtained for the absorptive part lm F(s,t) from the analyticity and unitarity properties can be extended to the grand canonical partition function. In particular, in ref. [3] we have derived sufficient conditions for the s-channel Partial-wave amplitudes Im fl(s)which guarantee t-channel unitarity at threshold t = 4/a 2. The result - extended to the grand canonical partition function - can be summarized as follows: in order to ensure that, for all the t-channel partial-wave amplitudes ~ (t,z,ar), J > 1 + (4#/M),M > 0 (cf. eqs. (12) and (16)), the derivative of the order J + 1 oS+l .'Z,t- - - . . , 1 r ( a r + ~ ) OlJ(Z)2 w tr, z,a) -z-, ~tJ+l z# 4a~/rr (4/a2_t)~

+ const,

t ~ 4/a 2

(20)

has a square root singularity with a residue proportional to the scattering length squared ag(z) 2, it is sufficient to assume the following properties of the s-channel partial.wave amplitudes ~2l(S,Z) (cf. eq. (17)): (i) There is no constraint for the partial waves

~2l(S,z) for l < L ( s ) = - ~

logs~

•

0i) The square root singularity (20) is produced for all J values J > 1 + 41ffM, M > 0, if the partial waves ~2t(s,z), l >t L (s) obey the scaling law:

~21(s,z) = ~

e-4Sx/u2/a 6 /3(x,z) s 3 x/x

(21)

III

for s ~ 0% l ~ 0% x = l(ta/~/s) 3 = f'xxed (iii) The residue of the square root singularity in eq. (20) is proportional to the scattering length squared aj(z) 2 (cf. also eqs. (12) and (16)), if the moments of the structure function ~(x,z) are related to the moments of the grand canonical partition function:

fax o

x J+÷ # ( x , z ) -

P(J+I)

J

ds

~2(s, t

=

4/a 2, z)

2

.

(22)

O. Haan, K.H. MUtter/Scaling laws

179

5. The high-energy behaviour of production cross sections It has been shown in ref. [4] that the scaling law (21) - for z = 1 - explains in a model-independent way the experimentally observed constant behaviour of the total and elastic cross sections at high energies (s ~ oo). In this section we will see that the extended version (21) of the scaling law for the partial waves of the grand canonical partition function [2 (s, t = 4/~2, z) leads indeed to constant production cross sections

o(n)(s) "~ const

for

s~

up to log s factors. The proof goes as follows: First we expand the scaling law (21) and eq. (22) a t z = 0 (cf. eqs. (7) and (18)):

~ln)(s) = N / ~ e-4SX /~ 2 //6 ~(n)(x). s3

x/x

'

n=4,6

....

(23)

where s ~ % l ~ ~, x = lOa/~/s) 3 = fixed. oo

f dxx J+½ ~(n)(x)=-----12~2-2~jr3(½//4)JP(J+1) 0

x ~3 f ~ .t~)(s,t=4//2)f ds p+q=n 4 2

~

f(q)(s,t

= 4//2).

(24)

4 2 sJ

Notice that there is no scaling law for the partial waves f/(2) (s) of the two-particle overlap function f(2)(s,t) since the latter has no elastic t-channel unitarity cut at 4//2 ~< t < 16//2. There is however a lower bound: f/(2) = Iftl 2 > (Ira f/) 2 = 2n

e -8sx/u2

/112 ~2(x ) s6 x

(25)

in terms of the scaling law for the partial waves Im ]~(s) of the absorptive part. The inequality (25) provides us with a lower bound for the two-particle overlap function f(2)(s,t=4#2)~>

~ l= L (s)

x(s) where

(2l+l)(Imfl)2Pl(l+

x/x '

8//2 ) s - 4//2

(26)

O. Haan, K.H. Miitter/Scalinglaws

180

logs~"

/~3 x (s ) _- -Mi

(27)

1

The inequality (26) can be inserted into eq. (24) (for n = 4) to calculate a lower bound for/3(4)(x) - the structure function which appears in the scaling law (23) for the partial waves of the four-particle overlap function. In this way, we get a lower bound for (a) the four-particle production cross section

0(4)($)/> !6zr

~

S I=L(s)

(2l + 1)f/(4)(s)

oo

} f dx/3(4)(x)vtx-e-4~/~2

(28)

x(s) (b) the four-particle overlap function at threshold t = 4/a 2

f(4)(s,t = 4~ 2) 1>

I=L (s)

(2l+ l)fl(4)(S)Pl (l + 8112 ) s-4/12

oo

/>-~$ f

dx 13(4)(x).

(29)

x(s) The bounds (26) and (29) can be inserted again into eq. (24) (for n = 6) to calculate a lower bound for/3(6)(x) and so on. Finally, we end up with lower bounds on all the structure functions ~(n)(x) (cf. eq. (23)) in terms of the structure function /3(x), which appears in the scaling law for the absorptive parts Im 3~(s) of the s-channel partial wave amplitudes. As was pointed out in ref. [4], the behaviour of/3(x) at x ~ 0 is directly related to the high-energy behaviour of the absorptive part Im F(s,t = 4/a2). A Regge behaviour Im

F(s,t = 4/~2) ~ s'~(4"2)(1og s~) 6(4t~2)

(30)

induces - via eq. (22) for z = 1, j3(x, z = 1) =/3 (x) and ~ (s, t = 4# 2 , z = 1) = Im F(s,t = 4 # 2 ) - a singular behaviour oft3(x) at x ~ 0 ~(x) ~ x - 3-c~(4t~2) (log 1 ) 2~(4~2)+1

(31)

Now, the crucial point is that the Regge singularity J = a(4rr 2) also determines the lower bound l ~>L (s) = (V~/M) log (s/s 1) for the validity of the scaling law. In order to ensure t-channel unitarity at threshold t = 4/a 2 for all the partial waves with J > a (~/~2) > 1, it is sufficient to assume the validity of the scaling for

O. Haan, K.H. Miitter/Scaling laws I/>L(s)= (ot(4//2) - 1)4-~

log-~l ,

181

(32)

which fixes the lower bound (27) in the integral (26): /12 logs~ x(s) = (a(4//2) - 1)~-~-

(33)

With eqs. (3 I) and (33)we find for the high-energy behaviour of the two-particle overlap function: f(2)(s,t = 4//2)/> 0 {sC~(4#2)(log s ] 48(4~t2)-2~(4~*2)-'}]

t

t

Tll

(34)

I

via eq. (24) (for n = 4), this leads to a lower bound on the structure function ~(4)(x) near x -+ 0

Finally we end up with lower bounds for (a) the four-particle production cross section [cf. eq. (28)1

((logs 148(4/~2)-3~(4~2)-~

0(4)(8)/> 0 \ \

-~1!

)

,

(36)

(b) the four-particle overlap function [cf. eq. (29)]

f(4)(s,t= 4//2)/> O(s a(4u2) (log s ] 4~(4u2)-3a(4u2)-2 t ,,

(37)

Proceeding in this way, we find that the lower bounds (36) and (37) are also true for the n-particle production cross section u(n)(s) and overlap function f(n)(s,t = 4//2).

6. Concluding remarks We have seen that analyticity and unitarity properties in the s- and t-channel lead to important consequences for the multiparticle production: (i) The inelastic partial wavesj¢ n)(s) obey scaling laws [cf. eqs. (23) and (24]. (ii) Assuming the validity of the scaling laws in the maximal range l t> L (s) = (V's-/4//) (a(4//2) - 1) log (Sis 1 ) allowed by s- and t-channel unitarity, the highenergy behaviour of the overlap functionf(n)(s,t = 4//2) at the t- channel threshold t = 4//2 is determined by the porneronj4n)(s,t = 4//2) ~ sa(4u ) (log s terms).

182

o. Haan, K.H. Miitter/ Scaling laws

Furthermore the n-particle production cross sections o(n)(s) become constant (up to log s powers)at high energies, irrespective of the exact position c~(4p 2) > 1 of the pomeron singularity at the t-channel threshold. In other words, the scaling laws for partial-wave amplitudes which guarantee the correct s- and t-channel unitarity properties (for s ~ ¢¢ and t ~ 4/22) provide a natural explanation of the experimentally observed diffractive component in the multi-particle production. It is not so simple to get information on the multiperipheral component i.e., on the high multiplicity cross sections o(n)(s), where n increases with s. The reason for this can be seen most easily if one considers the grand canonical partition function ~2(s, t,z) [cf. eq. (7)]. High multiplicity events determine the behaviour of ~2(s, t,z) for large z values (Izl > 1). However, for Izl > 1 we lose our s-channel unitarity constraints. The partial waves [2l(S,Z ) [cf. eqs. (17)-(19)] are no longer bounded. The boundedness of the s-channel partial waves is crucial in order to decide which partial waves produce the t-channel cut, i.e. for which partial waves the scaling law (21) holds.

Appendix A. Proof of the t-channel unitatity equation (5) for the n-particle overlap function According to the definition (1) the discontinuity disc f(n)(s,t) = 1 odn)(s,t +iO)-f(n)(s,t-iO)), t

4# 2 ~ t < 16/a2

gets contributions only from the OPE graphs [cf. fig. (2)] of the T-matrix elements. The only relevant part of the n-particle phase space F n is the neighbourhood of the OPE poles ("pole-dominance regions"): A(/): I ( P l - g ( I ) ) 2 - p 2 ] < e, 1(/9'1 - g(/))2-/a21 < e,

(A,1)

where K(I) = ~ielki is the total momentum of the particles i E/, I is a cluster of l particles in the n-particle intermediate state.

I= (i 1 . . . i l } C {1 . . . . . n } .

(A.2)

Since there is no trilinear coupling in a theory with pseudoscalar mesons, the cluster I always contains an even number of particles (l = 2, 4 . . . . . n - 2). In order to separate the relevant pole dominance regions (A. 1) from the irrelevant remainder, we prove the following lemma. Lemma: There exists the following decomposition of the n-particle phase space integral: K f = r=~l (--1)r=l ~ f +f F l~rnl< . . . ~ m r ~ ; K r K n ]~1 A(/]) ]~1 C(A(/]))

(A.3)

O. Haan, K.H. Mt~tter/Scaling laws K,

K~-K,

183

p+p_K r

Fig. 2. Cluster decomposition of the n-particle production amplitude (cf. eq. (A.7)). for K = I , . . . , N c , and.N c is the total number of possibilities to arrange the n particles into two clusters: n-2

N= ~

l=2,4..,

n!

l!(n--l)!

(A.4)

Notice that there are n! [l!(n-l)!] - I possibilities to arrange n particles into two clusters with l and (n - l) particles, respectively. C(A(I)) is the complement of A(/) with respect to the whole n-particle phase space:

A (1) U C(A(I))

=

rn .

Lemma (A.3) will be proved by complete induction. It is correct for K = 2:

f= f rn

a(/1)

+f

=f

C(A(/1))

+f

/,(/1 )

Y=f +J - f r n

a(l 1)

a(l 2)

+f

a(/2)nC(a(ll ))

a ( l l ) n a ( l 2)

(A.5)

C(a(ll)nC(a(12))

+f

C(a(I1))nC(a(I2))

Now let us assume that (A.3) is correct for some K. The last integral on the righthand side of eq. (A.3) can be decomposed as follows:

f

=f (K

}

+f

K+I ) K C(A(Ir)) C(a(/r)) na(IK+ 1) n C(a(/r) r=l ',r=l / r=1 n

K = ~ (-1) r ..~< r=0 1~ ml< ... mr
f + f (s n 1A(Ims))nA(IK+l ) K+I = r =n l

(A.6)

c(acp

Insertion of (A.6) into (A.3) shows immediately the validity of the decomposition (A.3) for K + 1 which completes the proof of the lemma. Next we have to evaluate that part of the n-particle phase space integral (1) which covers the pole dominance regions (cf. the decomposition (A.3)). Within

O. Haan, K.H. Miitter/Scaling laws

184

the intersection N1=1 r . A (I.) 1 of the r-pole dominance regions A(//.), ] = 1, "" "' r, we can insert for the n-particle production amplitude (cf. fig. 2) the OPE approximation:

(k 1.. knlTIPlP:) = X

(÷

.ll!(12-11)!'"(n-lr)!

(K 1 IT~o1, K 1- P l ) ( K 2 - K 1 [TkoI - K 1 - p l ) • . . ((/91 - K 1)2 _ , 2 ) ((p 1 - K 2 ) 2 " 2 ) " "

((/91 -Kr )2 2

)

.

(h.7)

According to fig. 2 the outgoing particles 1 , . . . , n are grouped together in r + 1 clusters, with ll, 12-I 1. . . . n-I r particles in the first, second . . . . (r + l)th cluster, respectively. The cluster states (Kll , ( K 2 - K l l . . . . are characterized by the total cluster momenta. Of course, they also depend on the individual momenta of the particles in the cluster. The integrations with respect to these individual momenta can be performed, e.g. l1

t'*

~rJdkl . . . d k n 6(K 1 -i~=l ki) OgllTlPl, KI-Pl)(KI[TIP'I, K 1 - p'l )* (A.8) = f(ll)(K~, t). There are n! [ll ] (/2- ll)! • • • (n - lr)!] -1 combinations to arrange the n ougoing particles (cf. fig. 2) into r + 1 clusters, where the first, second . . . . and (r + 1) th cluster contain ll, l 2 - l l , . . , n - 1r particles, respectively. Having all this in mind, one finds for the discontinuity of the n-particle overlap function:

disc f(n)(s,t)=disc t t

~ r=l

(-1)r-114-\ rt 3

fd4Kl 2~l 1

... <1r A1

. . . fd4Kr

Ar

X Z(K1, t ) . . . Z(K r, t)fql)(K~, t)/12-11)((K2-K1 )2, t ) . . . X f(n-lr)((pl +P2 - Kr)2, t)}

o

(A.9)

Here we have introduced the abreviation for the pole denominators

Z(K, t) = ((Pl - K)2 - " 2 ) - 1 ( ( p ' 1 - K)2 - 2 ) - 1 .

(A.10)

Obviously, the discontinuity (A.9) only receives contributions from the 6-function in Z(K, t + i0) = Z2(K) + Zl(K), such that we can substitute in eq. (A.9)

O. Haan, K.H. Miitter/ Scaling laws z ( / ¢ 1, t + i 0 ) . . ,

z ( K r, t + i0) ~ z l ( / q ) . . ,

185

z~ (Kr),

Z(K1, t-iO)... Z(K r, t-iO) ~ ( - 1 ) r Z1 (K 1 ) . . . Z1 (Kr)"

(A. 11)

Now we are in the position to prove the discontinuity formula (5) by complete induction with respect to n. The validity for the case n = 4 has been shown already in sect. 3. Let us assume that eq. (5) is correct for n = 6 . . . . . N - 2 . We will show that under this assumption eq. (A.9) (for n = iV) is equivalent to eq. (5) (for n = N. The discontinuity of the first term with r = 1 can be written as N-2 2i disc fCW)(s,t)=~ 1~2 d4Kl Z 1 (K1) t 1= &l

f

X {f(ll)(K 2,_ t

+ iO)f(N-ll)((pl -K1 )2, t + i0) •

+f(ll)(K2, t- i 0 ) f (N-ll) ((P1-K1)2, t - i0)} +

... r=2

2i disc

f

8 ll~ 2=2 'Xl d4K1 f(ll)(K~ ' t + iO)f(N-12)((Pl-K1)2 , t-iO) f(N)(s,t) =~-~

N-2 X ZI(K1) + 4 ~ f d4gl 7r3 ll 2 /x1

Zl(gl)(f(ll)(g~, t +iO) 2idisc t

X f (N-/1) ((Pl - K1 )2, t) - 2i disc t +

f(ll)(K~, t)f~-ll)((pl - K 1)2, t-iO) }

. ..

(A.12)

r=2 The first term on the right-hand side of eq. (A.12) is exactly identical with the fight-hand side of eq. (5) for n = N. Thus we are left with the problem of showing that the remaining terms vanish. According to our assumption, the discontinuity formula (S) for disc F - / ) ( ( P l - K 1 ) 2 , t), disc f(/)(K 2, t), l = 2 . . . . . N - 2 t t can be inserted into eq. (A.12) and we find for the remaining contributions which have to be shown to vanish: 2

() = _

f

~

2~11<12~N-2 /~1

4

7

2

23

d4K1

f

d4K2ZI(K1)ZI(K2)'L +

A2 fd4K1

2<11<12~N-2 41

f d4X2Zm(k)Z%+

42

... r=2 -1

,=3

""

(A.13)

186

O. Haan, K.H. MUtter/Scaling laws

L =fll(K~, t + iO)fq2-11)((K2-K1)2 , t + iO)f~-12)((P-K2 )2, t- iO) -- f(ll)(K~, t + iO) f(12-ll)((K 2-K 1)2, t - iO) fOV-12)((P-K 2)2, R = 2i (disc t

t - i0),

f(ll)(K2, t) f(12-11)((K 2- K 1)2, t- iO) f(N-lz)((P-K 2)2, t-iO)

-- fqt) (K~, t + iO) disc fq2 -ll )((K2_K1 )2, t) f(N-12) ((p-K 2 )2, t-iO) t

+ fql)(K 2, t + iO)fq2-ll)((K2_K1

)2, t + i0) disc

t

f~-12)((P-K2)2, t)).

In order to derive the right-hand side o f e q . (A.13), one has to combine on the left-hand side the first two terms with the next term r = 2 in the sum ~ N - I which is completely given in eq. (A,9). Next, one has to apply the discontinuity formula (5) on the three discontinuities on the right-hand side of eq. (A.13): disc t

f(ll)(K~, t),

disc t

f(12-ll)((K 2-

K1)2, t), disc t

fOV-12)((P-K2)2, t).

The resulting contribution now contains four overlap functions and can be arranged together with the next term r = 3 in the sum Z~N3-1 . . . in such a way that each term contains the product of three overlap functions and one discontinuity of an overlap function. Performing this procedure ½N times, we are left with a decomposition where each term consists of a product of (½N - 1) overlap functions f(2) and one discontinuity ditsc f(2), which of course is zero. This completes our proof.

Appendix B. Proof of eq. (15a) We use the addition theorem for the Legendre polynomials

27r

Pj(x'I)Pj(x'2)=-2~Tr f 0

d(~Pj(X'lX'2+

lx/~-x'211~-~-x'2 2 cos~b),

(B.1)

and write the integral (15) in the following rotational invariant form:

Gj(S 1, S 2,

1

1

t) - 4 (2rt)2

fd~2 n 1 d~2n2

where

n i = (cos ~i sin 0 i, sin 0 i sin 0 i, cos Oi),

Pj(n 1 "n 2) (xl--n. n l ) ( X 2 - - n , rt2)'

(B.2)

O. Haan, K.H. Miitter/Scaling laws X

i=cosOi ' ~b=~bl - - $ 2 '

187

n=(0,0,1).

t

Substituting the rotated variable t

n 2 = R ( n l ) n 2,

R ( n l ) n l =n,

we obtain 1

1

Gj(S1, $2, t) - 4 (2rr)2

fd~2n~Pj(n.n,2) /

dg2nt (x I - n . n l ) ( x 2 - n ; • n l ) "

(B,3)

t

Going back to the variables x 1 = cos 01, x~ = cos 0 2, we can perform the integrations over q~l, ~2: 1

1

If Gj(SI, S 2, t) =-~

, f -1

dx I

,

1

Pj(x2)

t dx 2 Xl - X l! N/k(x2 ' Xl' x2t )

"

-1

(B,4)

)

where -

,

, --

2

,2

'2

' '

kl.x2, Xl, X 2 ) - x 2 + X 1 + X 2 -- 1 -- 2X2XlX 2. In eq. (B.4) we insert the discontinuity formula for the Legendre functions:

--= (Oj(x2-tO) - Qj(x 2 + iO)) = Pj(x2). 7rl The contour along this cut in the x~ plane can be deformed in such a way that we pick up the square root singularity k--'= :

%(SI, S 2, t ) = ~ f f /

Xl_X, 1

-

C(xl-ie)

C(x '1 +ie)

dx 2 N/k

,

(B,5)

where ' = x 2 x '1 + ( x ] - 1)~(x C(X1) ' ' 2 - 1)}. •

0o

t

t

¢

1

t

.

.

.

.

t

The function fc(x:, dx2 k(x2' Xl' x2) ~ aj(x2) is analytm m the complex x 1 plane with a cut frl~m - 1 to 1. Thus, using Cauchy's formula in eq. (B,5), we end up with eq. (15a). References

[1] S. Mandelstam, Nuovo Cimento 30 (1963) 1127. [2] G. Veneziano, Phys. Letters 43B (1973) 413; J. Finkelstein, Phys. Rev. D8 (1973) 4176); H. Harari, Phys. Letters B51 (1974) 479.

188 [3] [4] [5] [6] [7]

O. Haan, K.H. Matter/Scaling laws O. Haan and K.H. Miitter, Phys. Letters 52B (1974)472. O. Haan and K.H. MUtter, Phys. Letters 53B (1974) 73. A.H. MueUer, Phys. Rev. D4 (1971) 150. °~ S. Mandelstam, Phys. Rev. 112 (1958) 1344; 115 (1959) 1741, 1752. D. Amati, A. Stanghellini and S. Fubini, Nuovo Cimento 26 (1962) 896.