54 IZVESTIYA VUZ. FIZIKA

UPPER LIMIT OF THE ELASTIC SCATTERING CROSS SECTION AND THE RE- LATIONSHIP BETWEEN THE TOTAL CROSS SECTIONS AND THE S-AMPLITUDE

G. G. Arushanov

Izvest iya VUZ. Fizika, Vol. 10, No. 12, pp. 86-89, 1967

UDC 539.171.016

Proceeding from general principles, an upper limit is obtained for the differential elastic scattering cross section of zero-spin particles having small momentum transfer and arbitrary energies, in terms of the experimentally observable total cross sections and of the ratio of the real to the imaginary parts of the forward-scattering amplitude. An important relationship is also obtained between the total cross sections and the partial s-amplitude.

The asymptotic behavior of the in teract ion cross sect ions of hadrons (strongly in terac t ing par t ic les ) at high energies has recent ly a t t rac ted cons iderable a t - tention. Results are obtained by making use not only of such general pr inc ip les as analyt ic i ty and un i t a r i - ness , but also to models , e .g . , the hypothesis of Regge poles. F u r t h e r m o r e , the der ivat ion of these r e - sui ts entai ls a number of mathemat ica l difficulties, and the resu l t s themselves contain, as a ru le , inde- t e rmina te constants (e. g., F r o i s s a r t ' s l imi t for the total in te rac t ion cross sect ion and different ial c ross section). Great in te res t at taches, therefore , to r e - lat ionships following di rec t ly from genera l pr inc ip les of the theory, e .g . , un i t a r iness , and containing only observable quanti t ies .

Using only the un l ta r iness condition, a lower l imi t was obtained in [1] for the derivat ive of the imaginary par t of the t r a n s f e r r e d - m o m e n t u m sca t te r ing ampl i - tude in the case of forward sca t te r ing at high energ ies , this lower l imi t being expressed in t e rms of the ex- pe r imenta l ly observable total c ross sect ions. This resu l t was extended in [2] to der ivat ives of a r b i t r a r y order.

The presen t paper obtains an upper l imi t for the differential sca t te r ing c ross sect ion with smal l mo- mentum t r ans f e r s for a r b i t r a r y energies [Eq. (24')], the resu l t being expressed in t e rms of the total c ross sect ions and of the ra t io of the rea l to the imag ina ry par t of the forward sca t te r ing ampli tude. An i m p o r - tant re la t ionship [Eq. (21)] is also obtained between total c ross sect ions and the rea l and imaginary parts of the s -ampl i tude .

To obtain our resu l t s , in place of the usual expan- sion of the amplitude in a se r i e s of Legendre poly- nomials

c o

F(t)=iJ-----T~,(2l + l)az(K)PL + 2~

where ~ is the wave number of the coll iding par t ic les in their center-of-mass system, t = -2Kz(I - cos | is the square of the transferred 4-momentum, and | is the scattering angle, it will be convenient to con- sider the following integral representation:

co

F (t) = ipr~- 5 b (p) Jo (ltl '~ P) P dp, (2) 0

c o

Z o (p) = ( . p ) - ' (2l + 1) ada+,(2,~p), l=O

(3)

valid for a r b i t r a r y physical energies and sca t te r ing angles, cor responding to the regions u _ 0, -4K a _< <__ t _< 0. In t e r ms of the spect ra l function b(p), the different ial sca t te r ing c ross sect ion is equal to

c o

I! 12 dt==.~(t)=lF(t)l~=~ b(P)do(ltl~2p)pdp . (4)

Using Cauchy's inequality, f rom (4) we obtain an upper l imi t for the differential c ross sect ion

oo

(0 < g 7 [F, (0) j- bR (p) Jo ~ ([tl%) pdp-- 0

co

- - Pe (0)j'- bz (p) Jo ~ (l/imp) pdp "]. (5) 0

Here and below, the subscr ip t s R and I denote the rea l and imag ina ry par ts . Note that both t e r m s in (5) are posit ive.

we now solve the following i soper ime t r i c problem. Determine the ex t remum of the r ight side of (5), given the boundary condition b(0) = a 0 (a t r iv i a l consequence of (3)) and the i sope r ime t r i c conditions

oo

Fz(O)=]/'-~.f bR(P)pdp= : t o 4 1 / 7 ' (6)

c o

FR (o) = - ] / ~ j - b~ (p) p alp, (7) 0

co

~e, = 2,: 1 (b~(p) + b~ (p)) p dp. (8) 0

We shal l a ssume the quanti t ies a0, F(0), a t, and eel to be specified. Solving the problem formulated in the usual fashion, we obtain that the ex t remum is imp le - mented by the functions

be(p) = - (2x~)-' (h + & (0) J~ qtl'~p)), (9)

b, (p) = - - (2k~) - t (k 2 -- F~(O)Y~ (It[,12 p)), (1 O)

where kl, X2, and k 3 are Lagrangian multipliers de- pending on g and t. If we require that the physical

SOVIET PHYSICS JOURNAL 55

quanti t ies (cross sect ions and ampli tudes) be finite, it can be seen f rom (9) and (10) that a cutoff must be introduced in them by wri t ing, in place of (9) and (10),

bR(p) = {$%(P) 9 ~ R p : > R ' (9')

where 5 is the ra t io of the rea l to the imaginary part of the fo rward - sca t t e r ing amplitude.

By different iat ing Eqs. (13) and (14) with respect to t at the point t = 0, and using (19), we obtain

4~ aar~R (0) R' (0)Xa (0) = F~ (0), (22)

p > Rx ' (1o,)

where c#R(p) and ~oI(p) denote the r ight sides of (9) and (10), and R and R~ are functions of ~ and t which r ema in to be determined.

By the un i ta r iness condition, the Lagrangian m u l - t ip l ie rs must sa t is fy the inequali t ies

4~ a~oz R~ (0) R~' (0) h (0) = - - F~(0), (23)

where a p r ime denotes different iat ion with respec t to t. We requ i re the las t two formulas for obtaining an upper l imi t for the differential cross section. We shal l consider the case of smal l t r a n s f e r r e d momenta and shal l confine ourse lves to l inear t e rms with respect to t. In this approximation we have

O < -- (9<J -~ (X~ J- F~ (0)) ~< 2, (11)

-- 1 ~< -- (2k~)-' (k~ -- Fe (0)) < I. (12)

~ ~ (o) "-" aon-- & (0) Xg' (0) (~/2)"t, (9")

+,~ (p) "-" ao~ + Fn (0) X~ ~' (0) (p/2)-"t. (10")

The functions R, R~, and X a must satisfy the following three equations :

4~-I"~ F~(0) X3 = R ~" (2kaao~ + Ft (0) ~(R)),

4.~- 112FR(0) }'a = - - R~ (2<~ao,-- Fn (0) a (R0) ,

(13)

(14)

4~-1%t X] = R ~ (4ao~ X~ - - P} (0)) +

+ 2R~Fz (0) ~ (R) X (2x~ a0R + Fz(0)) + R

+ 2&~ t' Yg (Itl'@) p die + R~ (4a81 },] - - F~ (0)) - -

- 2 R; FR (0) ~ (R,) (2h no, - - FR (0)) + R~

+ 2Fk (0) S Y~ (Itf2p) p dp, (15) 0

where

( r~) = 1 - - Jg ( I t J , / 2R) - - . q (Itl,/~R), (10)

and X; and h z are expressed in t e rms of h a by the f o r m - ulas

)'1 = - - 2),a aoR - - F / ( 0 ) , (17)

~'2 = --2Xaaoz + FR (0). (18)

A t t = 0 Eqs. (13) and (14) give

R 2 (0) = 2F,, (0) R~(0) = 2F R (0) (19) V gao~ ' V'ga~t "

It follows from the second formula (19) that the s ign of the rea l par t of the forward sca t te r ing amplitude is always the same as the sign of the rea l par t of the sca t te r ing ampli tude in the s - s ta te . Substi tuting Eq. (1) in (19), we obtain an inequali ty for alI:

r

1 Z ( 2 / + l) atz > 1. (20) aoz l=l-

F r o m (15) at t = 0, and taking (19) into account, we obtain an important equation

2 % t % - x = a ~ - - ?~ act, (21)

Substi tut ing this in (5) and using (22) and (23), we ob- tain in the l inear approximat ion the following upper l imi t for the different ial sca t te r ing cross sect ion with smal l t r a n s f e r r e d momenta :

t. (24) (0) aoi

Since 5 and a0i have opposite s igns, the neglect of the t e r m with 6 in the las t brackets can only re inforce the inequality. F u r t h e r m o r e , if we also consider the in - equali ty

2%t~t-'>aoR, (21')

which follows from (21), and subst i tute it in (24), we finally obtain the upper l imi t

(J-) ~ 1 + ~t2 t. (24') o(0) 16~%~(1 + ~)

Equation (24') gives an upper l imi t for the differential c ross sect ion for smal l momentum t r a ns f e r s in t e rms of the values , observable by exper iment , of the total c ross sect ions a t and Get and of the rat io 5 of the rea l to the imaginary par t of the forward sca t t e r ing a m p l i - tude.

The results obtained are quite general and are

valid for arbitrary energies. (Throughout this paper we have not considered the ordinary and isotopic spins of the colliding particles, and therefore the results are useful in the analysis of collisions of high- energy particles when the spin and isotopic-spin de-

pendence of cross sections can be neglected [3]). Note that the inequalities (21), (24), and (24') reduce to

identities for ~r = 0. Let us consider the case of high energies, and let us assume that 5 vanishes as K ~ co (This assumption does not contradict experiment and follows from the dispersion formulas [4]). Then, at high energies, inequality (21) must be close to an

equation. Substituting ael/a t ~- 0.2 for the total-cross- section ratio [3], we obtain the estimate a0R -~ 0.4. In the case of substantial accelerating energies, the real

56 IZVESTIYA VUZ. FIZIKA

part of the fo rward - sca t t e r ing ampli tude has a nega- t ive sign [4]. Therefore , on the s t rength of (19), we can a s s e r t that at such energies a0i > 0. As regards (24'), we shal l note that at large energies the inequal - ity is close to an equation while, for very smal l values of 5 (5 ~ -0 .2 [4]) it differs l i t t le from the lower l imi t obtained in [5].

REFERENCES

i. S. W. MeDowell and A. Martin, Phys. Rev., 135, B966, 1964.

2. V. S. Popov and V. D. Mur , YaF, 3, 561, 1966.

3. V. S. Barashenkov, In teract ion Cross Sections of E lemen ta ry Par t i c les [in Russ ianl , Nauka, Moscow, 1966.

4. S. Lindenbaum, P rep r in t E-1802, OIYaI, 1964. 5. G. G. Arushanov, ZhETF, 51, 1402, 1966.

11 Apri l 1967 Tashkent State Univers i ty