A scaling property of shrinking diffraction peaks

Nuclear Physics B117 (1976) 322-332 © North-Holland Publishing Company

A SCALING PROPERTY OF SHRINKING DIFFRACTION PEAKS G. AUBERSON and S.M. ROY * CERN, Geneva

Received 16 August 1976

We prove that b ( s ) ~ - -dt -dl n l - ~/daA s,t)l \ - (dt

]' t=0

<~ - -1 IIn ( s fd~tA (s,O) )1 2 , s ~ 4to

where ( d , / d t ) A ( s , t) is the absorptive contribution to the elastic unpolarized differential cross section for particles with arbitrary spins at c.m. energy and momentum transfer x/s, x / - t , and t o is the right extremity of its Lehmann-Martin ellipse for s ---,~. If this inequality is saturated apart from a constant factor, then there must exist sequences of s n -~- ~ such that

I-do'A/ ,,,+~L dt \

r

\

[do A

-1

where f(r) is an entire function of order 7-

1. Introduction There is considerable interest in the possibility that the elastic diffraction peak possesses a scaling property at high energies [ 1,2]. The first rigorous results on this question were obtained by Auberson, Kinoshita and Martin [3] for a class of scattering amplitudes including those saturating the Froissart bound, and slightly generalized by Cornille and Simao [4] and Roy [5]. Substantially new results have been obtained recently by Cornille and Martin [6]. Here we obtain from unitarity and axiomatic analyticity properties an asymptotic upper bound on b ( s ) -d@-ln dOA~

(1)

-dTt~' t) I,=o '

* On leave from Tara Institute of Fundamental Research, Bombay, India. 322

G. Auberson, S.M. Roy /Shrinking diffraction peaks

323

where (do/dt)A(s, t) is the absorptive contribution to the elastic differential cross section for particles of arbitrary spin at c.m. energy and momentum transfer X/s and x / - t , respectively. We show that if this bound is saturated apart from a constant factor, (do/dt) A must have a non-trivial scaling property with a scaling variable r - -tb(s). In the case of dominantly absorptive amplitudes the present results (i) represent a substantial generalization of those of ref. [3], and (ii) imply a non-trivial scaling property of the differential cross section provided the diffraction peak width shrinks as 1/0n s) 2 for s -+ ~o. Such a shrinkage is compatible with, but not implied by, the present high-energy data [7].

2. Basic results Our starting point is the partial wave expansion, for arbitrary spins,

dt

(2)

~=0

which converges for physical s for t within the Lehmann-Martin ellipse [8] of rightextremity to(S), with to - lim to(S)

(3)

,

(e.g., to = 4rn~ for 7rTrand lrN scattering), k being the c.m. momentum. A fundamental consequence of unitarity proved only recently by Mahoux [9] generalizing an earlier result of Cornille and Martin [6] is that

ol(s) >1 0

for l = 0, 1,2 . . . . .

(4)

It is known in the spinless case that

2r<,

ge4~oe 1

<~b(s) s ~<~

1 E(o,tt)] 2(toZe) In

e > 0,

(s)

where the left-hand side is due to McDoweU and Martin [10] and the right-hand side due to Singh [1 1] is an improvement of previous results [12]. The left-hand side has been shown by Cornille and Martin [6] to be valid for arbitrary spins provided that 1. the factor 92-is replaced by g, they also show that ifsoZot/Oel -~ o% and if the lefthand side of (5) is saturated apart from a constant factor, (do/dt) a must have a "weak scaling" property. Here we generalize to arbitrary spins the right-hand side of the bound (5) and prove that a "strong scaling" property must hold if the resulting bound is saturated apart from a constant factor. Our main results are summarized

G, Auberson, S.M, Roy /Shrinking difJ~action peaks

324

by the following theorems, valid for elastic scattering of particles with arbitrary spin; e will denote a positive number which can be chosen arbitrarily small.

Theorem 1: Upper bound on (do/dt)A(s, t) for complex t For [tl < t o , do A

~< I o

d(lA

s--+~

- co(s) \ [ t o -- e '

(6)

-aT (s, o) where Io is tile modified Bessel function of order zero, and 2 doA

co(s) = ln [s /~7-(s, O)l .

(7)

Theorem 2: Upper bound on b(s) b(s) ~

[co(s)]2 ~-bmax(S) 4(to e)

s~

(8)

Theorem 3: Bound on curvature of diffraction peak d2,

j-doA~

.71

at 2 ,nL--~-ts, t)jl,=o ,~<

b(s)[cot(s)] 2 8(to - e) '

(9)

where ,

1- 2 / d / d o A

,:o I .

(10)

Remark. Theorems 1 to 3 are generalizations to arbitrary spin of Singh's results in the spinless case [11,13]. Note that co(s) ~ const. In s, for s -+ ~, because the lower bound of Jin, Martin and Cornille [ 14] which readily generalizes to arbitrary spins using the amplitudes of Mahoux and Martin [8], gives 2 Oto t ~> const d°'A(s, 0)~>-~-dt 16rr s ~

S-- 12

'

(11)

and hence * * Strictly speaking, the right-hand side of the inequality (11) has only been proved for some sequences of values of s -~ ~. The precise asymptotic behaviour of oJ (s) is not needed in the derivation of the theorems given here.

G. Auberson, S.M. Roy /Shrinking diffraction peaks

(2-e)lns

~<

w(s)

325

(12)

~< 1 4 l n s .

Theorem 4." Bounds on physical region cross sections

For - 4 k 2 ~< t ~< 0 we have do A, ./do A. t2b(s)[cOl(S)] 2 1 + tb(s) <~ -ffi-ts, t)/--~i-(s , O) s-~= <~ 1 + tb(s) 4 16(t o - e)

(13)

Remark. The left-hand side of this inequality is due to Cornille and Martin [6]; the right-hand side is presumably new. Theorem 5." Strong scaling theorem Let

_do A[

7" ) /do A.

f(s, ~)=-~- i,s, t :

~

b(s)/brrrax(S) >1 bo

4=0,

/T

(14)

cs, o).

If

(15) t

where bmax(S) is defined by eq. (8), then every sequence Sn -* oo must contain a subsequence s n ~ oo such that lim f ( s n,

r) =f(r),

(16)

where (i) the limit is uniform in any bounded set of the complex r plane, ( i i ) f ( r ) is an entire function of order half obeying f(O) = 1,f'(O) = - 1 and the representation

/x/be f(r) =

(17)

dp(X) Jo(XX/r) , k=O

where dp(X)is a positive measure obeying 2l~/b 0

f k=O

2/N/b 0

dp(X) = 1,

f

dp(X)X 2 = 4 .

(18)

k=O

Remarks

(i) In ref. [3], an analogous scaling property has been proved under a condition, which for purely absorptive amplitudes reads ato t > const(ln s) 2" then the McDowell-Martin bound implies that the condition (15) for validity of theorem 5 also

G. A uberson, S.M. Roy/Shrinking diffraction peaks

326

holds. On the other hand, the condition (15) places no restriction on the behaviour of Otot allowing ato t ~ s 7, (3' > 0), as well as Otot ~ (ln s) 2. Thus, for purely absorptive amplitudes, theorem 5 is of more general applicability. (ii) The scaling variable r = - t b ( s ) is not necessarily a constant multiple of t(da/dt)A(s, 0)/Oel because the condition (15) allows b(s)Oel/(da/dt)a(s, O)-+ o o for s -+ ,~. Correspondingly, the asymptotic behaviour of our scaling function can be quite different from that in refs. [3,6], as discussed later. (iii) As in ref. [3], uniqueness of the scaling function is not proved.

Theorem 6." Upper bound on do/dtA (s, t} at finite energies in terms Of Oel and da/dt A (S, O) For any physical s and for - 1 ~< cos 0 -= 1 + t/2k 2 <~ 1, L-l

dOA(s, t) ~ Oel [ ~ (2•+ 1)(1 +l(I+ 1)sin20) -l/a dt 4k 2 /=o + (2L + 1)eL (1 +L(L + 1)sin20) -1/4 ] ,

(19)

where the integer L and the fraction e L are given by L--1

dOA(s, 0) = Oel [ ~ (2•+ 1) + ( 2 L + 1)eLl , dt 4k 2 o Further, if s(da/dt)A(s, O)/oel

do a ,

do A t)/qi- (s, o) <.

0~
(20)

>~, then we have the asymptotic bound (1 + 4 r ' ) 3/4 - I

for r , > 0

(21)

r' fixed

where ,

do A .

r --- (-t) ~ -

ts,

3. P r o o f o f t h e o r e m s

0)/ool.

(22)

1 to 4

To prove theorem 1, we pose the problem of finding an upper bound on

(do/dt)A(s, t) for t within the Lehmann-Martin ellipse (in particular Itl < to - e), given (do/dt)A(s, 0), and the information that ot ~> 0 and d°A(s, to dt

e) < const s z s-+~ "

(23)

G. Auberson, S.M. Roy / Shrinkingdiffraction peaks

327

Using the facts that IPt(l +(t/2k2))l
~PL(s)(l+'t[.~d°A(s,O)+PL (1+ [t[~ d°A (l+t_O e ~ 2 ~ : ~ 7(s)+l ~7]-~7-(S, to-e)/PL(s)+, 2k 2 ], and, since

P,(z) - < :o((2l + 1)x/~

-

1))

f o r z ~ > l , l = 0 , 1 , 2 .... (ref. [5]p. 193), --~dcrA" Is, t) [

,<:o((2L(s)+l)

V~-)

d°A O)[l +o(1)], -~-(s,

V~-Co(s) L (s) -= 2 ~ '

(24)

which is equivalent to theorem 1. Theorems 2 and 3 follow exactly similarly, and we omit their proof. For theorem 4, we use the inequality, valid for -1 ~< cos 0 ~ 1, l= 0,1,2, ... , 1 +(cos 0 - 1) [Pt'(cos 0)] ~
1)e;(cos 0 = l) + l(cos 0 - 1)2p7(cos 0 = l)

(25)

(whose left-hand side is due to Singh [11] and right-hand side to Cornille [6]), and obtain, after inserting theorem 3, the desired result.

4. Proof of theorem 5

From theorem 1 and assumption (15), we see that for S large enough, (£(s, r) ls > S) is a family of analytic functions of r in the disc Irl < bo(ln S) 2, uniformly bounded in this disc by

[2| L/TrT't

If(s,r)l < Io,\ i/~o-o]. s>S

(26)

328

G. Auberson, S.M. Roy /Shrinking diffraction peaks t

We may thus repeat the arguments of ref. [3] to conclude that every sequence Sn -+~ must contain a subsequence Sn ~ oo such thatf(Sn, r) converges (uniformly in any bounded region of the r plane) to an entire function f ( r ) of order ~<½. We know from the uniformity of the convergence, and from theorem 4 that f ( 0 ) = 1, and 1 -r<.f(r)<.

1 -r+--

7-2

for r ~> 0 ,

4bo

(27)

and hence f ( r ) cannot be identically equal to one. Further, from analyticity inside a circle C of radius R, around r = 0, df(r)dr

_ df(sn, r)]drr=O =]1~

~ dr' f ( r ' ) - f ( S n ' r ' ) C rt2

_<1 --~j~ max IfCr') - f C s n , r')l T'E C

>0.

(28)

Sn'-+~

Hence f ' ( r = 0) = - 1 ; further, since IPl(cos 0)1 ~< 1 for - 1 ~< cos 0 ~< 1, f(s, r) <- 1 ,

f ( r ) <~ 1

for all r/> 0 .

(29)

If the order o f f ( z ) were less than half, the Phragm4n-Lindel6f theorem [ 15] and eq. (29) would imply that f ( r ) is bounded everywhere and hence a constant; this is not the case. Hence f ( r ) must be of order half. Integral representation. As in proof of theorem 1, we show easily that, uniformly for - T < ~ r < . O, fors -->°o, L (s)

f(s,r)=o(1)+

~ t:o

,

L (s)

(2l+,)al(S)P,(1 2kZ.£b(s)/II2(2l'+1)o,'(+). It'=o

(30)

with L (s) given by e q. (24); further for l ~
lx/~- 1

0 ~ 1o((21 + l)~/~(z - i ) > - Pl(z) < 1o((21+ 1) 2 1 - ~ - I ) ) - I o I l ~ z Z ~ l - - I \ z +x/z 2 - 1/

< [(+Z(s) + 1)+q~- t) r(+)~ 7+ ~ uniformly for - T ~

1~ , J I0((2~+(+)+ 1)v~+'-(z- 1)) = o(1),

r ~ 0. Hence we may approximate P/(z) by

l o ( ( 2 l + 1) ½ ~ - ( z - 1)) to obtain

G. A uberson, S.M. Roy /Shrinking diffraction peaks

f(s,

r) =

~

329

]x/bo

dUs(X)Io(XVC~) + o(1) ,

- T ~ < r ~< 0 ,

S -->- o o

,

(31)

h=O

1~(s)

d~s(X) _

~ (2I+I)o,(s)6(X

dk (2/'+

2l+1

)

(32)

1)of(s)

/'=0

2/x/bo Ilusll ~ f

dus(X) = 1 .

(33)

X=0

Consider a sequence si~ -+ '~ such that f(s',, r)~f(r). It is known " that for every sequence of positive measures d;ts;z(X ) of unit n o r m on )t = [0, 2/x/bo], there exists a subsequence Sn and a positive measure d/~(X) of unit norm such that, for every continuous function g(X), 2/x/b 0

2/~/b 0

lim

f

dl-tsn(X)g()k)= f

Sn~

0

0

d~(X)g(X).

(34)

Choosing g(X) = 1 0 ( X x / ~ ) , we have

2/x/bo f

f ( r ) = lira Sn~

j/x/bo dUsn(k)lo(XXf~)=

0

dU(X)Io(XX/~),

(35)

0

first for - T ~ < r ~< 0, and by analytic continuation, for all complex r. Finally, f ' ( 0 ) = - 1 yields eq. (18).

5. Proof of theorem 6

From or(s)

=~ f

~ d o A ,-

1

d(cos 0)Pt(cos v ) - - - ~ t s , t = - 2 k 2 ( 1 - cos 0 ) ) ,

(36)

-1

the positivity

of(do/dt)A(s, t), and OeAII

Oel

ot(s) <~Oo(S) . . . . ~< . . . . , 4k 2 4k 2

IPl(cos 0)[~< 1, we have (37)

* This follows directly from lhe compactness of the set of positive measures wilh norm -<<1 in the vague topok~gy. See e.g. ref. [16].

G. Auberson, S.M. Roy /Shrinking diffraction peaks

330

where 17A denotes the absorptive contribution to 17el. Further from ref. [17] IPl(cos 0)[ ~< [1 + l ( l + 1)sin20] -1/4 ,

- 1 ~< cos 0 < 1 ,

(38)

we have d°a (s, t) ~< ~ ( 2 / + 1) 171(s)[ 1 + l(l + 1) sin 20]- 1/4 dt t=o

(39)

We seek then an upper bound oll the right-hand side of this equation given (d17/dt)A(s, 0), and the constraints 0 ~< Ol(S) <~ Oej(4k2), and readily derive theorem 6.

6. Zeros and asymptotic behaviour

(i) Exactly as in ref. [3], we deduce that f ( r ) has infinitely many zeros in a small neighbourhood of the positive r axis (i.e., negative t axis). (ii) Unlike ref. [3], our assumption (15) allows the left-hand side of the equation A / d17A ~ 4k2b(s) b(s) 17el/~-[s, O) = / d'c f(s, 7")

(40)

o

to be unbounded for s -+ 0% and hence allows f ( r ) to be non-integrable in r = [0,oo]. This is most easily seen from the following example in the spinless case, with at(s) denoting partial waves of the absorptive part: al(s) = 1, l ~ [0, L1] and [L2, L3] al(s) = 0 otherwise. L1-X/~ 1~(1--

~c.2)

L3=-cV~lns,

'

L2-cx/~lns[1

647rc 4°b

_1)23

(Ins

'

c < 1/(2X/to).

b<2c 2,

(41)

Then, for s ~ 0% OAel(S) = O t o t ( S ) ~ O }

b(s) -+ b s) 2

,

(ha

.f(s,r)--+f(7")= s~

1-

b + b Jo 2c 2 2c 2

Thus f ( r ) can approach a constant for ~-~ oo.

b(s) °A

. - + -b

(In s Otot) 2

N/7"

---+ r "~

17'

1

~

.

(42)

G. Auberson,S.M.Roy /Shrinkingdiffractionpeaks

331

For comparison, note that in the (spinless) strong scaling case of ref. [3], f(r) is not only integrable on r E [0, °°] but obeys the local bound I f ( r ) l < c/x/r for r-+oo. In the weak scaling case of ref. [6], it was shown that

f

7" ~-

dr' f(r') < const,

d°A " - / " ~ { S , 0)/Oel .

(43)

0

From unitarity

f dr'f2(r ') < ?

o

dr'f(r')

< const,

(44)

o

and hence we have the Plancherel formula [3] f(r') = ~ ?

duh(u)Jo(xt~Tu),

duh2(u) = ? 0

o

d r ' f 2 ( r ') < c o n s t . (45)

0

Further, from theorem 6, we deduce that

f(r') < (1 + 4r') 3/4 3r'

1

,

r ' ~> 0 .

(46)

for r ' ~ oo

(47)

Here

f(r') < const(r') - j / 4

Thus tile Hankel transform representation of our scaling function f(r) and its asymptotic behaviour are quite different from the previously known cases of strong and weak scaling. We wish to thank A. Martin for useful discussions.

References

[1 ] J. Dias de Deus, Nucl. Phys. B59 (1973) 231; V. Singli and S.M, Roy, Phys. Rev. D1 (1970) 2638. [2J V. Barger, Proc. 17th Conf. on high-energy physics, London, 1974; V. Barger, J. Luthe and R.J.N. Phillips, Nucl. Phys. B88 (1975) 237; P.P. Divakaran and A.D. Gangal, TII"R preprint (June 1976). [31 G. Auberson, T. Kinoshita and A. Martin, Phys. Rev. D3 (1971) 3185. [4] 1t. Cornille, Nuow) Cimenlo Letters 4 (1970) 267; I1. Cornille and I".R.A. Simao, Nuow~ Cimento 5A (1971) 138. [5~ S.M. Roy, Phys. Reporls 5 (1972) no. 3.

332

G. Auberson, S.M. Roy /Shrinking diffraction peaks

[6] H. Cornille and A. Martin, CERN preprint TH. 2130 (1976); H. Cornille, Saclay preprint D. Ph. T. 76/14, Phys. Rev. D, to be published; H. Cornille and A. Martin, Saclay preprint D. Ph. T. 76/72. [7] G. Giacomelli, Phys. Reports 23 (1976) 123. [81 H. Lehmann, Nuovo Cimento 10 (1958) 579; A. Martin, Nuovo Cimento 42 (1966) 930; G. Mahoux and A. Martin, Phys. Rev. 174 (1968) 2140. [9] G. Mahoux, to be published. [10] S.W. McDowell and A. Martin, Phys. Rev. 135 (1964) B960. [1t] V. Singh, Phys. Rev. Letters 26 (1971) 530. [12] T. Kinoshita, Lectures at the Boulder Summer School, 1966; J.D. Bessis, Nuovo Cimento 45A (1966) 974. [13] V. Singh, Ann. of Phys. 92 (1975) 377. [14] Y.S. Jin and A. Martin, Phys. Rev. 135 (1964) B1369; H. Cornille, Nuovo Cimento 4A (1971) 549. [15] R.P. Boas, Entire functions (Academic Press, NY, 1954) p. 4. [16] M. Reed and B. Simon, Methods of modern mathematical physics. I: Functional analysis (Academic Press, NY, 1972) subsect. 4.5. [17J A. Martin, Phys. Rev. 129 (1963) 1432.