On the validity of the scaling law for partial-wave amplitudes

Nuclear Physics B94 (1975) 3 6 5 - 3 7 3 © North-Holland Publishing Company

ON THE VALIDITY OF THE SCALING LAW FOR PARTIAL-WAVE AMPLITUDES O. HAAN and K.H, MLITTER* lnstitut fiir Theoretisehe Physik, Universitiit Heidelberg Received 10 February 1975 (Revised 21 April 1975)

It is shown that the scaling law for partial-wave amplitudes Im ./4(s) holds for I > (x/)-/41s) (c~(4~z2) - 1) log s, if the leading Regge singularity c~(t) in the t-channel is bounded by

Re ~(t) <<.ed4u 2) + cd4u2) - 1 (x/'/- - 2~) 2u

and is smooth near t = 4u 2 in the sense of eq. (4).

In a previous paper [1] we have pointed out that t-channel unitarity at threshold t = 4/22 strongly suggests the validity of a scaling law for s-channel partial-wave amplitudes Im fl(S) = X / ~

e -4sx/~2'/26 t3(x) ,

(1)

s 3 x/Xfor s -+ oo, x =/(/2/x/s)3 = fixed. The structure function/3(x)** can be expressed in terms of the absorptive part Im F(s, t = 4/2 2) at the t-channel threshold [2]: e~

F

4 , / " ds 1 d s 2 e - 2xsas2/u 4 13(x) - (27r3x)~/24 4u j Im F(Sl, 4/22) I m F(s2, 4/22) 2 42

m

(2) In the papers of ref. [1] we have shown that the scaling law (1) with the structure function (2) is compatible with s-channel unitarity fl(s) ~< 1 if *Address after April 1, 1975: Gesamthochschule Wuppertal, Fachbereich 6. **As in paper 1 we only consider crossing symmetric scattering amplitudes for pseudoscalar particles with mass ~. Unfortunately, we have extended the summation in the partial wave expansion of eq. (14) ref. [ 1 ] over all integer/-values, instead of summing only the even ones. Therefore, the expression for the structure function 13(x) gets an additional factor 2.

O. Haan, K.H. Mi~tter / Scaling law

366

~s(~ ( 4 U2) l>~--~(

1)logs,

(3)

where c~(4tt2) is the position of the leading Regge singularity at the t-channel threshold. c~(4tt2) is assumed to be bigger than 1. It is the purpose of this paper to clarify under which conditions the scaling law really holds in this maximal domain. It turns out that essentially two conditions are sufficient: The absorptive part A(s, t) of the scattering amplitude must be smooth in the sense

A(s, t)~A(s, 4112)sa'(4u2)(t

4u2),

(4)

for

s ~ oo,

to(s) <~t <<.tl(s ) ,

to(S ) = 402 + 64/1~4

tl(S ) = 4U 2 + r(log s) e - 1

(5)

S - 4# 2 ' and all the Regge singularities an(t ) in the t-channel must be bounded by Re an(t ) <~c~(4#2) + ~(4U2) 2tt - !(x/~ - 2tz) '

t ~> 4tt 2 "

(6)

We are not able to derive our assumptions ( 4 ) - ( 6 ) from first principles. Nevertheless, they are fulfilled in a wide class of reasonable models. E.g. the first condition is satisfied in any Regge model

A(s, t) = ~i(t)sai(t)(log s) 8i(t)

to(S ) <~ t <. tl(S )

l

if the residues Hi(t) and the types 6i(t ) of the Regge singularities are continuous, the trajectories are differentiable near t = 4/.t2:

oti(t ) = o~i(4tt2) + o~i(4kt2)(t - 4U2) + O ((t

4U2) 1 + e).

In particular, a pomeron trajectory, with a correct t-channel unitarity cut for t 1 >~ 4/a 2 Im c~(t) ~ (t - 4/a2) (c~(4u2) + D/(~(4u2) + 1)

for t -+ 4/12

fulfills this conditions if c~(4~t2) > ½ + 6(4/~2). For the case 6(4tt 2) =- ~(4tt 2) - ½, which was discussed in ref. [1], our present proof of the scaling law does not hold. The second condition (6) is satisfied for example within the Gribov-reggeon calculus [3], if the input pomeron pole trajectory is bounded by (6) for t > 0 (cf. eqs. (28)-(32)). Obviously, linearly rising Regge trajectories are excluded by the in equality (6). Now let us turn to the proof of the scaling law under the conditions mentioned above. We start with the Froissart-Gribov representation for the absorptive part of

o. Haan, K.H. Mfitter / Sealing law

367

the s-channel partial-wave amplitudes oo

f

4

Imf/(s)-

rr (s ---4/32) to(S)

dtQlO+ 2t ~ p ( s , t ) , s - 4/32!

(7) '

where

to(S ) = 4/32 + 64/3_____L4 s - 4/32 is given by the boundary of the Mandelstam spectral functionp(s, t). In the following, we will analyze the contributions to the Froissart-Gribov integrals arising from four intervals: I:

to(S)<~t
lIl"

16/32 --.
II:

t l ( S ) < ~ t < 16/32 ,

IV:

t>~s.

(8)

In the first and second interval the Froissart-Gribov integrals can be evaluated by means of the elastic t-channel unitarity equation for the Mandelstam spectral function:

p(s, t)=~(t4 _ 4U2)) -'} f d s l d S z K - } A ( s

1, t - iO) A (s 2, t + i0) ,

-(9)

D

where

A(s, t)= ~i(F(s 1 + iO, t) - F(s - iO, t)) is the absorptive part of the s-channel amplitude.

SSlS 2

K =s 2 + s 2 + s 2 - 2 ( s s l + s s 2+sls2)

1

a t - /32

is the Kibble function. The domain D of integration is given by: D:

Sl,

s2 ~>4/32,

K~>O.

Theorem 1. If the asymptotic be]~aviour (s -+ oo) of the absorptive part A(s, t) near t = 4/32 is smooth in the sense of eq. (4), then the first Froissart-Gribov integral (cf. eqs. (7) and (8)) tl(s) 4 f dtp(s,t)Ql(l II - lr( s --4/32) to(S)

+

2t ) s - 4/32

(10)

368

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

yields in the limit s -+o° ,

l>2#a'(4uZ)x/~logs

(11)

the scaling law (1) with a slight modification of the structure function ~3(x) (cf. eq. (2)): /3(x) ~ fl (x - 2/~4a'(4/~2) l°-~--s) .

(12)

Comment to theorem 1. The modified structure function (12) reduces to the old one, if I > 0 (x/s log s). Furthermore, if l = 0(x/rs-log s), i.e., x - 2~u4a'(4/~ 2) logs s _ x/s log l s _ 2~ua'(4~2)/~3 logs s . 0 the asymptotic behaviour o f l I (cf. eqs. (10), (1) and (2)) in the limit s ~ oo is not changed: 1I ~ O(s~(4u2) - 1 - 4ul/x/~log s

(13)

Thus, the modification only affects the proportionality constant in the asymptotic formula (13). Proof o f theorem 1. In the kinematical domain under consideration s ~ oo, st,s2 ~> 4 / 1 2

to(S ) ~< t < tl(s ) ,

i.e. t ~ 4/~2 ,

K~S--{s(t-4112)-4SlS2}>O, t 4/~2

(14)

we are only interested in the leading behaviour of all quantities: A(Sl, t + iO)A (s2, t - i0) = I m F(Sl, 4/~2) Im F(s24ta2)(SlS2) (t - 4ta2)a'(4~2) ,

Ql(COS a) = ~/~_~ r(l +

1)

e ~(~+~) (1 + O(1/od)).

(15)

Combining eqs. (7), (9), (14) and (15) we find the following expression for the low-t part (I 0) of the Froissart-Gribov integral: ii _

8 s-~l ~ e_4U//x/s- f dsldS 2 Im F(Sl, 4/~2) . (2nU)~ (log s)~ S1 ' $2>/4U 2 Qa0 a

X Im F(s2, 4//2) e -4h(sl os=/s)logs f ~ o ' 0

t

,

e-aa

(16)

O. Haan, K.H. Mfttter / Scaling law

369

where l log sis 2 - s X - 2/i~/s log s - c((4/a2) - log '

(17)

o = r (log s) e - 4sls2 log s.

(18)

S

Indeed equality (16) yields the scaling law (1) with the modified structure function /3(x - 2t14c((4/12) log s/s) (cf. eqs. (2) and (12)) if we substitute in eq. (16): l X ~ X0 - 2/iX/s- log s

~x,(4/12) '

(19)

o ~ ~.

(20)

The second substitution (20) is justified since the leading behaviour in the limit s ~ oo comes from contributions of the integrand in the domain: s 1 - s2 - -

log s < G((log

s ) e) .

S

The first substitution (19) has no effect if l > O ( v ' s log s ) . If l/X/~ log s is finite, the leading c o n t r i b u t i o n to the integral (16) arises from s l, s 2 values with Sl "S21og s = c o n s t , S

where the substitution (19) is again correct. This completes the p r o o f of theorem 1. We are left with the problem of showing that the low t part I I of the FroissartGribov formula - with the a s y m p t o t i c behaviour (13) - is indeed the leading contribution in the limit (11) under consideration. Theorem 2. We assume that the Mandelstam spectral function is b o u n d e d by Ip(s, t)l<~O(e atc(t))

fort~

1,

(21)

S

where cosh c~t = 1 +

2s t - 4/12

and C(t) is given near t = 4/12 b y the position of the leading Regge singularity C(4/l 2) = ~x(4/.t2).

(22)

Then the following b o u n d s for the Froissart-Gribov integrals III, 1III, I I v in the t-in-

370

O. Haan, K.H. MMter / Scaling law

tervals 1I, Ill, IV are valid: Case A. If C(s) > O(x/~) ,

l/> O(log s C(s)) ,

IIii<~O(s4ul/x/-s-l°g s exp ( - ( l o g s) e

2~t V'~ log s

(23)

'

iiiiii ~< O(s-8lal/x/~log s) , Ilrwl ~ O(e -la3) ,

(24) (25)

cosh a 3 = a 3 .

Case B. If C(t) <~ a(4/~ 2) + 2 (X/~- _ 2/Q ,

t ~> 4/a 2 ,

and

v~

l>-~-

logs,

11ii I ~< O~-4vl/x/slog s+ot(4U2)- I exp

L-

r(x/ logl s

- ( l o g s) e - ~

---(23a)

Iili11 <~ 0 s - 8ul/xRlog s+a(4tz2) - I +4u/M

(24a)

IllvI ~ O(e -l~a) .

(25a)

The proof of the bounds (23), (23a), (24) and (24a) follows from a direct calculation of the Froissart-Gribov integrals with the assumption (21). Due to crossing symmetry there is a corresponding bound to (21) for t/s >1 1. This yields the inequalities (25) and (25a) . Thus we can conclude from theorems 1 and 2: Conclusion. The scaling law (1) with the structure function (2) holds under the conditions (4) and (21) for s ~ oo and A B

l >~ O(C(s) log s)

if

c(s) > o(x/7),

l > M~s log s ,

if

C(t) ~< ot(4~2 ) + 2 (x/t" - 2//),

t >1 4/.t2 ,

1 ~> 2~ta,(4/a2) "

(26)

There is a second lower bound on l/M, which comes from s-channel unitarity. As was mentioned already above and in ref. [1 ] the asymptotic behaviour of the scaling

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

371

law (cf. eq. [13] ) is compatible with s-channel unitarity only if the inequality (3) is satisfied that means, 1

a(4u 2) - 1

(27)

In other words, the scaling law (1) holds in the maximum domain (7) allowed by schannel unitarity, i.e. for s ~ o, l >~

(a(4/12) - 1) log s ,

if the absorptive part A(s, t) for s ~ oo and t = 4/12 is smooth in the sense of eq. (4) and all the Regge singularities are bounded by Re a(t) ~< C(t),

t >~4ta 2 ,

(28)

C(t) = a(4p2) + a(4//22) - 1 (x/~ - 2/.0 .

(29)

How realistic is this assumption? Let us consider for example the J-plane singularities within the Gribov reggeon calculus [3] where the pomeron pole J = a(t) is always accompanied by Regge cuts

J= an(t ) =ln(a(4t/n2) - 1) + 1 ,

n = 4, 6, ...

(30)

They arise from n-particle exchange contributions in the t-channel. One readily realizes, that all the Regge cut trajectories an(t ) (cf. eq. (30)) are bounded by Re an(t ) ~ C(t),

t> 0 ,

n = 4, 6, 8 . . . . .

(31)

if the pomeron pole trajectory a(t) is bounded in the same way. Re a(t) ~< C(t)

for t > 0 .

(32)

Obviously, this is related to the well-known fact that trajectories of the square root type near t ~ 0 yield self consistent solutions of the absorbed multiperipheral bootstrap which - by construction - fulfills s-channel unitarity [4]. Finally, let us discuss the next-order correction to the scaling laws in the limit a(4kt 2) - 1 s--> ~ ,

l>

4/1

x/slog s.

For this purpose, we compare the asymptotic behaviour of the scaling law (cf. eq. (13)) and the different parts of the Froissart-Gribov integral (cf. eqs. (7), (I 3), (23a), (24a) and (25a). (a) Further contributions from elastic t-channel unitarity (4/2 + r (log s) e - 1 < < t < 16/a2) decrease stronger than any power in log s (cf. eq. (23a))

O. Haan, K.H. Mi~tter / Scaling law

372

(log s) e ~

[11~-~ < 0 exp

V/}-log s

4p

(33)

"

(b) Above the n-particle threshold in the t-channel, the Froissart-Gribov integral

-

I~?

4

s

--

£,

dtp(s,t)Q

7r(s-4/12),..~. )2

l

(

1 +

2,) s

4p 2 .

n=4,6,

.

.

.

decreases like a power in s (relative to the scaling law)

I/l'?l

i/l__T ~< O(s-2(n-2)u[(l/vSlogs)-(~(4u2)-

1)/4uJ) ,

n = 4, 6 . . . . .

(34)

It is very interesting to see that all the n-particle exchange contributions in the tchannel become important at the same/-value, namely for l -

-

a(4/.t 2) - 1

-

-9

X/~ log s

4p

This situation has some similarity with the behaviour of a thermodynamical system at the critical point. The scaled angular momentum l/v~ log s corresponds to the temperature T, the value l

_-a(4/'t 2)- 1

V~ log s

4/.t

to the critical temperature T c .

For /

~> ee(4~t2) -- 1

v~-log s

4/1

i.e., away from the critical point where short range correlations are relevant - the partial wave amplitudes in the s-channel are determined by the nearest t-channel singularity at t = 4/~2. According to the scaling taw (1), they decrease like a power of s lm fl(s) -~ O(s-4U(/]v/7 log s - (c~(4U2) - 1)/4t*)) .

(35)

For l/v~ log s -+ (a(4/12) - 1)/40, i.e., at the critical point where long range correlations are dominant, all the n-particle exchanges in the t-channel (t > (n/~)2, n = 2, 4, become relevant. Therefore, we expect that the asymptotic behaviour of the partial wave amplitudes for 1 v~log s

~< o¢(4/22) - 1 4/.t

373

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

will drastically change. O f course it must be different from (35), due to s-channel unitary. The most appealing possibility w o u l d be that geometrical scaling [5] sets in*: Im ft(s) = g

> O.

That means the partial-wave amplitudes b e c o m e constant at high energies s ~ oo. In this case, the distribution o f the partial-wave amplitude p l o t t e d against the " t e m p e r a t u r e " T = / / x / s - l o g s develops a non-analytic behaviour at the "critical p o i n t " T = T c = (a(4/l 2) - 1)/4/1 in the limit s ~ oo (cf. fig. 1).

[m ~g) 2

I/

k

Fig. 1. Possible "phase diagram" for s-channel partial-wave amplitudes. Dashed line: s finite, solid line: s ~ ~. In domain II the scaling law (1) is valid, in domain I geometrical scaling is assumed to be valid. We w o u l d like to thank P. Grassberger ( C E R N ) w h o p o i n t e d out to us the scaling law for l > 0(s) can be derived from the Mandelstam representation.

References [1] O. Haan and K.H. Mfitter, Phys. Letters 52B (1974) 472; 53B (1974) 73. [2] A.K. Common, Phys. Letters 55B (1975) 318. [3] V.N. Gribov, I.Ya. Pomeranchuk and K.A. Ter-Martirosyan, Yad. Fiz. (USSR) 2 (1965) 361; Sov. J. Nucl. Phys. 2 (1966) 258. [4] D. Horn and F. Zachariasen, Hadron physics at very high energies (Benjamin, 1973) ch 15; J. Ball and F. Zachariasen, Phys. Letters 40B (1972) 41 i; J. Finkelstein and F. Zachariasen, Phys. Letters 34B (1971) 631. [5] J. Dias de Deus, Nucl. Phys. B59 (1973) 231. [6] G. Auberson, T. Kinoshita and A. Martin, Phys. Rev. D3 (1971) 3185.

*It has been shown in ref. [6] that geometrical scaling follows in the framework of axiomatic field theory if the Froissart bound is saturated.