Inclusive processes and shadow effects in deuteron

Nuclear Physics B56 (1973) 90-108. North-Holland Publishing Company

INCLUSIVE SHADOW

PROCESSES

EFFECTS

AND

IN DEUTERON

A.B. K A I D A L O V and L.A. K O N D R A T Y U K

Institute for Theoretieal and Experimental Physics, Moscow, USSR Received 13 July 1972 (Revised 22 January 1973) Abstract: Using the data on inclusive spectra we analyze the contribution of inelastic intermediate states to the shadow correction A for high-energy (E > 10 GeV) hadron-deuteron scattering. We note that the intermediate states with large masses M 2 ~ ER-1 may contribute appreciably to the shadow correction, but the estimate of this contribution in the framework of the Regge-pole model for inclusive processes shows that it is small. The main contribution to the A is due to the small masses M < 2 GeV, which are produced by diffraction dissociation. We calculate the value Ainel for pd, rrd and Kd scattering and estimate the energy dependence of A. The results are in agreement with the existing experimental data.

1. I n t r o d u c t i o n The aim o f this paper is to estimate c o n t r i b u t i o n s o f different inelastic mechanisms to the shadow correction A = Op + o n -- o d for high-energy (Ela b > 10 GeV) hadrond e u t e r o n scattering. The shadow correction A m a y be represented as a sum o f two terms A = Ael + Ainel '

(1.1)

where A e l - OpOn 4n R 2

(1.2)

is the usual Glauber correction [1] which is described by the diagram a in fig. 1, and ZXinel is the c o n t r i b u t i o n o f the inelastic processes (see diagram b in fig. 1) w h i c h was first discussed in paper [2]. With the help o f the optical m o d e l [3] or o f the graph technique [4] the following expression for Ainel m a y be derived (see also the appendix)

ai~el = - I m f dt ds' F(t) [fpn(Sl, s', t) + f n p (Sl, s', /)],

(1.3)

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

ct

91

b

Fig. 1. The diagrams for the (a) elastic and (b) inelastic shadow corrections. where s 1 = (Pa + Pb) 2 ; Pa is the momentum of the incident particle, Pb is the momen turn of the nucleon in deuteron, x/~ is the mass of the hadron system in the intermediate state, F(t) is the deuteron form-factor [5]

F( t ) = eat,

a = ~1R2d ~ 40 (aeV/c) -2

lmfpn (Sl, S ' , t ) = ~ l l m ~ dFvf;aV)(sl +ie, s' +ie, kl ..... k ) 32rr2 s 2 v

(1.4)

(1.5)

X f(nav) (sl+ie, s'-ie, k 1 ..... kv), d3kl dFv

(2zr)3 2co 1

d3kv

(27r) 4 6 ( k l + - . ' + k v - P v ) ,

(27z)3 2 ~ v

p 2 = s' = M 2. Let us note that the amplitudes f(paV) and f(nav) in the right-hand part of the expres, sion (1.5) are on different sides of the cut in s'. This fact can easily be checked from the derivation of formula (1.3) in the framework of the graph method [4] or optical potential model (see the appendix)*. In the following we shall neglect the spins. The cross section of inclusive reaction a + N -> X + N may be expressed through the amplitudes fN(a v) d2°N= 1 ~dpvlf(NaV)(Sl, S,,k 1..... ku)]2" dtds' 327r2s 2 v

(1.6 t

In the calculations of Ainel it is usually assumed that the amplitude

f(NaV)(sl, s', kl, ..., kv) is purely imaginary for large s 1 [ 3 - 5 , 7]. In this case -Ira [fpn(Sl'S"t)+fnp(SrS"t)]=2

d2Crp d2on - =2--, dt ds' dtds'

(1.7)

and the correction Ainel is positive. But this assumption is justified only for the diffraction-dissociation part of inelastic processes which are generated by the exchange of the vacuum singularity in the t-channel. This diffractive part of Ainel we denote by Ap. It can be shown (see paper [8]) that the diffraction-dissociation cross section * Generally speaking, the expression (1.5) may describe the integrand fpn (sb s', t) in (1.3) only if in the integration over s' in (1.3) the finite region in s' is essential when s l~ ~. If the region s' ~ s I is also important special consideration is needed [6]. We shall discuss this case in sect. 6

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

92

Op = f Op (Sl, M 2) d_M2 is a small part of the total inelastic cross section Oinel : Op/Oinel ~ 10%. The main part of Oinel is connected with processes of the other type which we call the inelastic background O"B = Oinel -- O'p

I,

where

o B = f o B ( S l , M 2) d m 2

is the total cross section o f the inelastic background, I is the interference term. The inelastic background may be produced by the non-vacuum reggeon exchanges or b y the multiperipheral mechanism [9]. For these processes the phase of the amplitude, the magnitude and sign o f the background contribution to the shadow correction Ainel are different from the diffractive part and should be considered separately. For example, if the background in the inclusive reaction a + N ~ X + N is described by the contribution o f non-interfering channels with amplitudes

f~ (s 1, s', t) = fin (Sl , s', t)= f i (sl, s', t), the corresponding inelastic correction may be written in the form (assuming the main contribution to be given by the finite region in s' (see also sect. 6)):

~n =2 f dt ds' F(t) ~ i i

d2d

dtds' '

(1.8)

where 1 - a2 ~i = - '

1 + a2

~i =

Re fi(sl, s', t) , "

Imfi(sl , s, t)

We see from (1.8) that the contribution of a channel with a purely real amplitude has the negative sign opposed to the sign of the diffractive-dissociation contribution. The interesting qualitative analysis o f different inelastic contributions to the shadow correction for Nd scattering was made in the framework of the Regge-pole model and duality conjecture in paper [10]. In the present paper, using existing experimental information on inelastic spectra in inclusive reactions, we give the quantitative estimate of Ap and A B. We calculate Ap and the upper limit on A B using only some general properties of inclusive spectra, deduced from experiment. It is important in the following that the cross section OB(Sl, s') is distributed smoothly over all allowed mass interval [11, 12], up to the values o f the squared mass s' ~ s 1 in contrast to the diffraction-dissociation cross section ap(S') which is concentrated near threshold [8]. This leads to the additional small factor 1/rn,v/a in the correction A B in comparison with the corrections AXp and Ael. The small factor 1 / m v ~ a p p e a r s due to the cut-off on the longitudinal component of m o m e n t u m transfer P

2"

q2 ~ S o ~ - ~ a - i s1 which is equivalent to the cut-off on the mass squared s ' < ~ 1/mn/~ s I But the single factor 1/mn/~ is insufficient for neglecting the correction A B with respect to the cor-

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

93

rection died because the ratio AB/Ael is proportional to 1

°inel

mx/~ Oel and is not small due to the large value of Oinel/Oel. Such supression of A B would be more appreciable for heavy nuclei. As the upper limit on A B is high enough it is very important to know the phase o f the background amplitude. In the framework of Regge-pole phenomenology the inelastic background in reaction a + N --* X + N in the region s'/s 1 "~ 1 is produced mainly through the exchange of P', co, O, A 2 and n poles. The, contributions of these poles do not interfere with each other and are small. The contributions o f co, P', p and A 2 poles are found to be proportional to a(0) - 1 and are small due to the closeness of their intercepts to 1. The n-contribution has the additional small factor 1/mx/d. The final result is IAB/Ael] ~ 1/mx/m. This means that the main contribution to the inelastic screening effect is due to diffraction dissociation. This contribution should be determined by extrapolation of the experimental spectrum d2o/dtds ' for each fixed value o f s ' = M 2 to large values o f s 1 [8]. The calculations o f refs. [3, 7], where the integration was carried out over all the mass spectrum, gave an overestimated value of Ainel.

2. Kinematical r e l a t i o n s Let us write down the necessary kinematical relations. The invariant cross sections o f the reaction a+b-+X+c,

(2.1)

where X is an arbitrary hadronic state, is d2 °(Sl , Pc)

f(sl, Pc) = 2 E c

(2.2)

d3pc

In general, the cross section f(s 1, Pc) depends on three variables Sl = (Pa + Pb )2'

t = (Pa - Pc )2'

M2 = (Pa + Pb - Pc )2"

(2.3)

In our case the target nucleon is denoted before and after collision by b and c, m b = m e = rn, a is the incident hadron n +, K, K, p o r ~ . The region of the variables considered is s 1 ~ m 2,

M2

~

1

s 1,

t ~
(2.4)

where a is defined b y formula (1.4). Other variables used for the description of the inclusive reaction (2.1) are

p?,

x= - , Pb

P± _ - p ±c,

(2.5)

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

94

where Pb and Pc are the momenta of particles b and c in the rest system of a. The relations between the variables (2.3) and (2.5) have the form p2 t=

-

m2(1 _ x) 2

x

x

M 2 = Sl(1 - x) + m 2 + t,

(2.6)

where in the right-hand parts the terms of the order of m2/s as compared with 1 are neglected. In the region (2.4) the invariant phase volume for particle c is d3pc_ 1 2E c 2x dx d2p± = ~ l d M 2 dr.

(2.7)

Instead of the invariant cross section f(s r Pc) we shall use the invariant function p(s r t, M2):

p(s 1 , t, M 2) -

f(sl, Pc) Oinel

,

(2.8)

where Oinel is the total inelastic cross section. The function p(s 1, t, M 2) may be expressed through the differential cross section as follows:

p(s 1, t, M 2) -

st

d2a

7rOinel d t m d M

-

2x

d2o

rrOineldxdp2 "

3. Diffraction dissociation and the inelastic background in the region

(2.9)

s'/s I <~ 1/mx/a

The differential cross section o f the inclusive reaction a + N ~ X + N may be written as d 20(a)

d 2oP(a) - - -

dt ds'

dt ds'

d 2oB(a) +-

+I,

(3.1)

dt ds'

where I is the interference term. As it was mentioned in the introduction the diffrac tion-dissociation contribution may be extracted from the experimental data by the extrapolation of d2o(a)/dt ds' at each given value s' = M 2 to the large values o f s 1 . Such extrapolation of missing-mass data for pp collisions [ 14] was done in paper [81. In calculation o f d 2 a ~ ) / d t dM for t = 0 the following parametrization o f the differential cross section was used d 2 ° - A ( M ) e b(M)t, dtdM

A ( M ) = Ap(M) + B(M) . Sff (M)

(3.2)

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

dt~

t6eV 7

95

• - 9.9 C,-~V ;c _ ~tO &e¢

( - ~ - o . o ~ G~rs)

o

4/.,*5-

-

~¢"

1'6

~.~

,.~

1.6

~o

~~

~~

3.~

I . ) c.~v)

Fig. 2. The mass spectra in the reaction p+p --, p+X. The calculated value of d2op(P)/dt dM for t = - 0.04 (GeV/c) -2 is given in fig. 2, where the whole spectrum d2o(P)/dt dM is also given. The dotted curves in fig. 2 are obtained by the smooth extrapolation of experimental data on the mass region M ~> 2.2 GeV. A smooth dependence of d2o(P)/dt dM on M in the region M ~> 2.2 GeV is confirmed by the data on reactions pp -~ pX for 19 and 24 GeV [1 i], 30 GeV [14], and 7r-p -~ r r - X for 8 and 16 GeV [15]. Then the cross sections given in fig. 2 were extrapolated to t = 0 in correspondence with the expression (3.2) and table 1 for the slope parameter b(M) which was taken from paper [14]. The average value ofT(M) over the mass interval 1 . 5 - 2 GeV was found to be ~ = 1.2 • 0.2. The form d2op/dt dM indicates that the dominant mass region for diffraction dissociation is 1 . 2 - 1 . 8 GeV [8] and the contribution of masses M ~> 2 GeV may be neglected. In the language of Regge phenomenology it means that three-pomeron vertex r/ppp is equal to zero or is very small*. The corresponding invariant function pp(s 1, x, P l ) is mainly concentrated in the region 1 - x <~mg/sl , m 0 ~ 2 GeV, where it has 6-functional bumps describing the contributions of the N(1400) and N(1688) resonances. Unlike diffraction dissociation the mass spectrum for the inelastic background is distributed almost uniformly over the whole mass interval Mmi n ~
A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

96

Table 1 M(GeV)

1.2

1.3

1.4

1.5

1.6

1.7

b(M)(GeV -2)

22

25

21

11

8

6

• •

x

1.8-2.0

5.7

2.0-2.8

5.5

9.9 a-eY

t5.1 &eV - 20 Gels

-

n -.~q C-elf

tl

i

O

h

Fig. 3. The invariant function OB(Sl, M, t) at p± = 0 for the reaction p+p ~ p+X. For additional illustration of this statement we give in fig. 3 the value p(P)(sx, t, M) - p(pP)(Sl, t, M) = p(P)(Sl, t, M) + Ap (p)

(3.3)

for t = tmin where AO is the interference term. Neglecting the interference term Ap we may write down the inelastic shadow correction in the form Ainel = Ap + AB.

(3.4)

The energy dependence o f the cross section d2oB/dt dM for fixed value of M may be described by the contributions o f non-vacuum reggeons. The application of the Regge-pole model in the region s'/s I ~ 1 seems to be quite reasonable [17] (see also review [18] on inclusive reactions dominated by the non-vacuum reggeon exchanges). The contributions o f P', p, co and A 2 poles in the absence o f interference with the P pole give 3' = 1. The calculated value ~ > 1 may be explained by the influence of the n pole. The analysis of the inelastic background contribution in the framework of the Regge-pole model is given in sect. 6.

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

97

4. Contribution of diffraction dissociation The contribution of diffraction dissociation may be represented in the form A(pa) = 2 f

dt dM F(t) dd;~df4 .

(4.1)

Using the following parametrization of the cross section: d2 °(a) b (a)( ~ -A(pa)(M) e P ,M)t, and integrating over t we obtain A(pa) =

2faMexp {[a + b(a)(M)] tmi n } a + b(pa)(M) ,

(4.2)

where m2(M 2 _ m2) 2 tmin

=-

(sl-m2a-m2)(Sl

" m2-M2)

(4.3)

In order to calculate the correction A~P) for pd scattering we use the function Ap(P)(M) given in fig. 2 and the data on bp(P)(M) from paper [14] (we assume that for M ~< 2 GeV, the parameter bp(M) is equal to the slope parameter of all the spectrum, b(M) given in table 1). We obtain A(pP) = 0.40 -+ 0.04 rob, 0.48 -+ 0.05 mb and 0.50 -+ 0.06 mb for E = 10 GeV, 15 GeV and 20 GeV, respectively. The magnitude OfAel = Op On/87r(a + bel ) for o p = o n = 39 mb, bel = 10 GeV -2 is 3 •1 mb . () () The errors in A~P are connected mainly with bad knowledge o f A ~ p in the mass region M ~ 2 GeV. The function contains extrapolation errors besides experimental ones [8]. The former are unavailable in absence of experimental information in the energy region E ~ 102 GeV. In order to obtain a possible variation of Ap with a further increase of energy we assume that the function Ap(M) decreases for M/> 2 GeV as follows: 1 Ap(M) = const - - . M in 2 M

(4.4)

(In accordance with results o f papers [19], due to the vanishing of three-reggeon r/ppp and four-reggeon r/pppp vertices for small momenta, the function Ap(M) for large values of M satisfies the inequality Ap(M) ~< const/M In 2 M 2. If 7?ppp ~ t for t -+ 0 then there is a non-vanishing contribution to Ap as s 1 ~ o o from triple-pomeron exchange [10]. We shall neglect it in the following because it is very small.) Using for A ~ ) ( M ) at M ~< 2 GeV the values of fig. 2 and at M > 2 GeV the expression (4.4) we get, for E / > 20 GeV,

A(pp) = 6(0P) + 6~p),

(4.5)

98

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

where o; p) = 0.47 -+ 0.04 mb

(4.6)

is the contribution of masses M ~< 2 GeV, and [ In 4 o~p) = (0.15 + 0.15)~[1 - I n S l ~ - x / a ]

(4.7)

is the contribution of masses M > 2 GeV. It follows from the expressions ( 4 . 5 ) - ( 4 . 7 ) that the correction A(pP) for E ~> 20 GeV increases slowly from 0.50 -+ 0.06 mb f o r E = 20 GeV to 0.62 -+ 0.15 mb for E = 200 GeV. The difference of Ap(P) f o r E = 200 GeV and for E ~ 10 3 - 10 4 GeV is not greater than 10%1 The growth o f £Xp is connected with the decrease of longitudinal momentum transfer, i.e. with decreasing of tmi n when s 1 increases and M is fixed. Remembering that only finite masses contribute to Ap we may replace the exponent exp {[a + bp(M)] tmin} in the integral (4.2) by 1 for s 1 >> m3x/a. Then we have the expression for Ap bp Ap = 2Op - a+bp

,

(4.8)

where b-p is the mean value of bp(M) and Op is the cross section of diffraction dissociation. It is interesting to note that the usual Glauber correction may be represented also in the form analogous to (4.8)* bel Ael = 2Oel a~+ be 1 ,

(4.9)

where bel is the diffraction peak width for elastic scattering, d % l / d t = A e belt. Using expressions ( 4 . 8 ) - ( 4 . 9 ) we may write down the ratio Ap/Ael in the form Ap

opbp -

Ael

(4.10)

Crelbel •

As the main features of spectra d 2 o / d t dM are very similar for the reactions p + p ~ p + X, 7r + p ~ X + p and K + p ~ X + p the expressions (4.8) and (4.10) can be used also for 7rd and Kd scattering. In accordance with paper [8] bp = bel, O(pp) = 1 . 7 +- 0.4 mb** for reaction p + p ~ p + X, O(p7r'K) = 1.5 -+ 0.5 mb for reactions 7r + p ~ X + p and K + p ~ X + p. Taking into account these values o f bp and Op, as well as a possible energy dependence of Ap, we represent the ratio Ap/Ael for E ' > 10 GeV in the form * The representation of the shadow correction in the form analogous to (4.8) or (4.9) was used earlier in paper [ 7 ] . . ** The cross section o1~7 ) given in paper [8] should be increased by 20%.

A.B. KaMalov, L.A. Kondratyuk, Inclusive processes Ap (1 Ae1 - C 1

In

C2 (Sl% 3N/~)),

99

(4.11)

where C] p) = 0.20 + 0.04, Ct ~r) = 0.38 -+ 0.13, C] K) = 0.50 -+ 0.17 and an expected value o f C 2 is C 2 < 1. For pd scattering C(2p) = 0.35 -+ 0.35. As follows from expressions (4.4) - (4.7) the energy dependence of Ap/Ael is determined by the mass dependence of Ap(M) = d2°p

t=0

at large values of M. This dependence is not known now and may be found when the data on inclusive spectra in the energy region E > 30 GeV will appear. The dependence 1 const/ln (sl/m3x/a) given by formula (4.4) is the weakest dependence possible from a theoretical point of view and is determined by the suggestion (4.4). For a power law Ap(M) ~ I/M l+'r, 7 > 0 the energy dependence of Ap/Ael would be of the power form 1 - const/sl'r. Let us mention that the energy-dependent part of Ap is not large. For example, in pd scattering its relative contribution to Ap is not greater than 20% for E = 20 GeV and 10% for 200 GeV.

5. Limit on A B from the form of inelastic spectrum The upper limit on A B is given by the expression (d2aB d2 oBch'ex"1 AB <~2 f d t d M F ( t ) k d t d M + dtdM l '

(5.1)

or in variables p± and x:

x [PB(s1'x'P±)+PB

tSl,X,p±)] exp

x(p± + m2(1 - x ) 2

(5.2) Here o B and PB describe the reaction a + p ~ X + p, and 0~3h'ex" and O~h'ex" describe the charge-exchange inclusive reaction a + p ~ X + n. The charge-exchange reaction is absent for the case of diffractive dissociation which is generated by the exchange of vacuum quantum numbers in the t-channel. The main contribution to the integral (5.2) is given by the x-region 1 - x ~ 1/mn/~ or the mass-squared region, M 2 ~ 1/mv;d s 1. Taking into account that the value of 1 - x is small and the functions PB and O~h.ex. change weakly when the variables Pi and x change in the intervals Ap± ~ 1/x/a and Ax ~ l/rex~a, we obtain after the integration over p± and x in (5.2) AB

~--- 7rN/~-

--~ ma3/~--~ ~ Oina,

(5.3)

A.B. Kaidalov,L.A. Kondratyuk,Inclusiveprocesses

100

where = I - ch.ex.-, e , 2(PB t P B )lpi=O;x=l mx/a

e ~ 1.

(5.4)

The important feature of the expression (5.3) is the inverse proPortionality o f A B to the volume of the nucleus A B ~ 1/ma3/2 [13]. The parameter a - ~ is sufficient to neglect the large mass contribution to shadow corrections for heavy nuclei. In the framework of the Regge-pole model [16] or of the limiting-fragmentation model [20] the function p at large s 1 depends on only two variables x and p± and does not depend on s 1 . The upper limit on A B does not depend on energy in that case. For proton spectra the s 1 -dependence of p was analyzed in paper [ 11 ] for the 1 2 - 2 4 GeV energy interval and was found to be very weak. The value of p ( p l = 0, x = 0.9) given in this paper is equal to 2 GeV -2. Choosing p ch.ex- ~ ~p [21] and using the expression (5.2) we obtain A B < 1.2 rob.

(5.5) 1

The constant h-~ is proportional to the range o f inelastic interactions and describes the p± dependence of d 2 o/d2p±dx. For example, if we parametrize the cross section o f the inclusive reaction a + p ~ X + p in the form [12] dZ°B e -xf~p ± - D Oinel dp 2 dx

(5.6)

and take into account that o B ~ Oinel and mean multiplicity for proton and neutron is equal to ½ we obtain D = 51 B C = ¼C,

(5.7)

~-

~(X)lp±= 0 =~

cx

c ~-4~x.

(5.8)

Thus in model (5.6) we have

AB Ae 1 ~<

8~

C°inel belt/el

(5.9)

For pp interaction at the energy of ~ 10 GeV, c ~ 2bp [12] and AB < 1 °inel Ae~ ~ 2mx/a Oel

(5.10)

The limit on the ratio A~P)/A~) obtained from (5.10) agrees with the expression (5.5). For a purely-real or purely imaginary background amplitude the value o f A B would be the largest and would be given by the right part of (5.3). In these cases the contribution of the inealstic background to the shadow correction A would be appreciable. But as we shall show in the following section on the basis o f the Regge-pole

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

101

model, A B is significantly less than the upper limit (5.3) and the contribution of A B to the shadow correction may be neglected.

6. Limit on A B from the Regge-pole model

In the framework of the Regge-pole model the amplitude of the transition

Pa + Pb ~ Pc + kl + "'" + kv (see (1.5)) may be written in the form . . .[st ~ i ( t ) f(aV)(s 1, s', t, k 1...... kv) = ~ gi(t)rl(~i)t~o] 7}aV)(s', t, k 1..... kv) i

(6.1)

where r/(°ti) -

1 +- e -inai(t) - sin 7roti(t)

is the signature factor, gi(t) is the reggeon coupling constant for the vertex

Pb -+ Pc + ai and 7!aV)(Sl , s', t, k I ..... kv) is the amplitude of the transition o~. + Pa ~ kl + "'" + kv" The function fNl N2 (Sl, St, t) which is the integrand in formula (1.3) may be represented as

1 fN1N2 (Sl, s,, t) = ~167rZs

(Sl)ai(t)+aj(t) ~ij ~(~i)~(~j)gN'(t)g~i2(t) ~0

(6.2)

1

x Dj(s" t), where fi] (s', t) is the crossing-symmetric amplitude of the transition oti(t ) + Pa -~ 0~](t) + Pa (see fig. 4). The discontinuity of this amplitude on the right-hand cut in s' has the form 1

Im fij (s', t) = ~ ~

dFvT~ a~) (s', t, k 1 ..... k~)7} av)* (s', t, k t ..... k~).

(6.3)

v

The amplitude fii(s', t) has the same amalytic properties as the usual two-particle amplitude [19]. Let us represent it in the form*

fij (S:') = -- 77p.(t)g~(O) L\SS'oir(~c~p(O,-c~i(t,-aj(t)+ °i°J \-( ~]s'lap(O'-ai( t, + Ris (s; t) + ....

:;'"l (6.4)

where the first term describes the pomeron contribution, Rij describes the resonance contributions and 77P is the real three-reggeon vertex for ap(O)~i(t ) ~j(t) coupling, o i is the signature. * The expression for fij(s', t) which we use includes only dynamical singularities and is different from the representations used in papers [10, 16] which do not possess the necessary symmetry under the exchange s' --. - s'.

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

102 o.

a

R~j'

_

. . . . .

Fig. 4 . T h e i n e l a s t i c s h a d o w c o r r e c t i o n in t h e R e g g e - p o l e

model.

The real analytic properties of the amplitude ft'/"(s', t) and its crossing-symmetric form determine the phase of the pomeron contribution in (6.4). The t-dependence of the residue function ~P.(t) in general is not known. As the discontinuity of the amphtude fij(s, t) m s for t =1 is positive, the residue rlii(t) should be proportional to [sin ½n(~Xv(0) - 2ai(t))] n where n is equal to +1 or 1. I f n = 1 the amplitude would have the ghosts at ap(0) - 2ai(t) = 2k where k is an integer. Therefore, we choose n = +1. Then •

r

-

r

.

.

.

.

.

.

V

r/P.(t) ~ sin ½7r (ap(0) - 2ai(t)),

(6.5)

and Im fii(s', t) goes to zero when av(0) - 2ai(t ) = 2k, in particular for the reggeons with ai(O ) = 1 the discontinuity offii(s', t) in s' is equal to zero for t = 0. This means that the cross section of the inclusive reactions a + N -~ X + N, d2o/dt dM 2 cannot be constant as a function o f M 2 when t = 0 a n d M 2 is large. For fixed value o f M 2 this cross section should decrease faster than 1/s 1 with the increasing of s 1 . In principle, one may construct the amplitude fi/"(s', t) with nonzero discontinuity for ap(0) - 2ai(t ) = 0 and without ghosts in t. For example, the following expression for fij(s', t):

fij(S', t) = C(t) [(S')aP-ai(t)-a/. (t) + (--S')aP ai(t) aj(t) _ 2] sin rr (ap-- ai( t ) -- a/.( t ) )

(6.6)

satisfies all these conditions. But the amplitude (6.6) has non-Regge asymptotic behaviour at large s' which is due to non-analytic (or fixed pole) terms in j-plane. This situation takes place for example, in the dual models [22]. The conclusions reached in sect. 3 are equivalent to the statement that in calculating the diffractive-dissociation contribution Ap only the resonance term in (6.4) should be taken into account. On the contrary, in the calculation of the inelastic background A B only the first term in (6.4) should be considered because the resonance term Ri/, decreases rapidly with increasing energy. Therefore we have f~.B)(s', t) = r/P.(t) g~ap(0) {-- 1 -- exp [--i#(otp(0) -- oti(t ) -- a/"(t))] ) ×

~p(0)-~i(t) c~/"(t).

(6.7)

To estimate £./.(B) we take into account the P' and co poles which have large coupling constants with nucleons and the ~ pole which may contribute to the charge-exchange reaction. All these poles do not interfere with each other. The interference o f p ' with co and rr is forbidden by G-parity conservation. The contributions of co

103

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

NN and n do not interfere due to different spin structure of the vertices gNN and gTr and due to isospin conservation at the three-reggeon vertex. Therefore, formula (6.2) for the contribution of the inelastic background takes the form

IS 1 \ 2 a i ( t ) ~ . ,

, 1 ~] JN1N2 ~(B) (Sl, S, t ) - 16n2s 2 i

~2(o~/)gNil(,)gNi 2(t) t~0 )

(6.8)

Jii(s, t),

1 (sS~)2ai(t) 2 lmfii(s,,t)=d2°(i) 16zr2 s2 g2 (t) l~?(ai(t))[ ds' d~

(6.9)

where o (i) is the contribution of i-reggeon exchange to the cross section of the reaction a + N ~ X + N. Note that in the model considered the invariant function J(Sl, s', t) depends only on two variables, s/s 1 and t. Actually, from the expression (2.9) and (6.7) and due to the proportionality Im fii ~ s', we have p(0(Sl, S', t) = sl

|--~-j/Sl~2ai(t)-i= ¢(t) (1 ~

d2°(0

x ) l ~ 2 ~ i ~t )

nOinel dtds' -~O(t)\ s'!

where the function ~0(t) depends on t only. This property o f p (t) was first pointed out in paper [17]. Using the expressions (1.3), (6.8) and (6.9) we may represent the contribution of the inelastic background to the shadow correction in the form i d2o (i) a B = + - - - -ch.ex.~ (6.10) i~sp \ d t d s dt ds' I '

2fdtds'F(t) 2 ~i(d2°¢)

where

1 ~i =

sin ½n(Otp(0) - 2oti(t)) '

(6.1 1)

and the sum includes all Regge pole terms i 4= P. Let us note that formula (6.10) completely coincides with formula (1.8) for two cases which seem to be the most important from the practical point of view: when c~i(0) = 1 or 0, i.e. when the production amplitude is purely imaginary or purely real. We shall consider the contributions of P', ~ and 7r poles to the shadow correction. (i) Poles P' and co. The contributions of these poles are negligible because d 2 o(i)/dtds ' ~ sin 2 ½n (~p - 2ai(t)) and ai(O ) are nearly equal to ½, ai(0 ) = ~-(1 - e) where e ~ 0.1 [23]. So the additional small factor for the P' and o~ contributions is e or u'(O)/a ~ 1/m2a. (ii) n pole. For the n pole, e i = - 1 and its contribution has the opposite sign with respect to the diffractive-dissociation contribution. The n contribution has, however, the small factor arising from the proportionality of d2o/dtds ' to s'/s 1 for small s'/s 1. As the integration over s Is hmlted from the above by s --~ 1/rnx/a s 1 the addmonal small factor is about I/mx/a.Using the experimental estimate o c~.ex. ~ ~1 °inel [21] and assuming that the n contribution to this reaction is of the form •

,

t

-

.

-

t<~

.

(~"

<

.

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

104 d2 °(c~!ex.

s'

dp 2 ds-- = ~ °inel s21 b~r e-blrP~ '

(6.12)

we obtain the following estimate

A~ ~) Ael

-

be °inel

1

(6.13)

bel Oel 2m2a '

where we also took into account the isospin factor O(c~!ex.= 2o ('). For b~/bel <~ 1 we have I A(B~r)/AelI ~ 0.1. This estimate of the contribution is crude enough. The data on the mass spectrum in the charge-exchange reaction a + p ~ X + n for small s'/s <~ 1/mx/a are needed for more accurate calculation. 7, Conclusion Let us state the main conclusions which follow from our consideration. (i) The main contribution to the inelastic shadow correction on deuteron is due to diffraction dissociation. At the energy region E >2>m2x/a the ratio of elastic to inelastic screening is practically energy independent and is equal to Ap/Ael = 0.20 + 0.04 for pd scattering, 0.38 + 0.13 for 7rd scattering and 0.5 + 0.2 for Kd scattering. (ii) The contribution o f the inelastic background A B in the region E >> m2x/~ is about - 0 . 1 Ael or less. The numerical estimate for pd scattering gives A ~ ) ~< --0.3 mb at the energy E ~ 1 0 - 1 0 2 GeV. In order to make the value o f A B more definite the experimental data on charge exchange reaction a + p ~ X + n are needed. (iii) In the processes o f particle scattering on heavier nuclei the contribution o f the inelastic background is less important even than on deuteron because its contribution is inversely proportional to the volume of a nucleus. The relative contribution of the diffraction dissociation at the energies E >>m2R is the same as for deuteron. In order to check the theoretical predictions on the inelastic shadow correction one should have the experimental measurements of the total correction A with an accuracy higher than 0.5 mb. Such measurements at the energy E > 10 GeV exist only for 7rd scattering [24]. According to the results of this work the shadow correction AOr) = °POn 4¢r in the energy region 1 5 - 6 0 GeV can be described by the formula = r~ 2 (1 + bp) where r~ 2 = (2.8 + 0.3) X 10 -2 mb -1, b = (6 + 2) X 10 -3 GeV -1. As it was pointed out in sect. 5 in the case when PB(S1, X, Pi) for the reaction 7r + N -~ X + N does not depend on s 1 the correction A B does not depend on energy. Because Ael does not increase with the energy the experimental growth of the shadow correction may be only due to the growth of Ap. The energy dependence

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

(R-2) = C 1

105

C2 ln(s 1/m3,v/a)

which follows fron (4.1 1) is consistent with the experiment when C 2 ~ ~ Cj. If we neglect the contribution of the inelastic background, then at 2mE = s 1 ~ mJv~ A~)eor : A(]r) + A~)7r) : (1.38 + 0.13) × 1.2 m b : 1.66 + 0.16 mb, which is in a good agreement with the experiment. For example, at E = 55 GeV, A(ex) = (1.68 + 0.12) mb. So, the experimental data on A (~) confirm the conclusior) that the contribution of intermediate states with high masses to the shadow correction is small and its absolute value is of the order of 10 -1 mb. It should be pointed out that the contributions of the same order of magnitude may be connected with radiation correction [25] and relativistic effects in deuteron structure (for example, the mixture of nuclear resonance in the wave function of the deuteron [26] and so on). It is a pleasure for us to thank V.V. Anisovich, M.S. Marinov, K.A. Ter-Martirosyan, I.S. Shapiro and P.E. Volkovitski for useful discussions. Appendix Let us consider the derivation of formulae (1.3), (1.5) in the framework of the potential model using the eikonal approximation. In order to discuss the processes of fragmentation we consider hadrons as composite systems with wave functions ~n(~l ..... ~N) which are dependent on the coordinates of the partons ~i" The amplitude of the elastic scattering which is a consequence of the double rescattering can be written in the form 42)(A)=_

~l ~

d3q v (o~) (A f (2")3

q)

2

1

v(vO)(q),

(A.1)

Pv -- (Pa + q)2 + ie

where

v(nm)(r) =f~0n*(~l ..... ~N ) ~ V/(r - ~ i ) i X tPrn (~1 ..... ~N ) d3~l "'" d3~N; Vi(r - ~ i ) is the potential of interaction of the i-parton and target nucleon, p2 = p2 + m 2 _ M 2, M is the mass of the intermediate state. In the eikonal approach we have f(02)(a) =

X 6 (qz

iPa ff~ 1 f d2q V(0V)(A - q) v(vO)(q) "~ v,M 167r2pvp a

(A.2)

M2 - m2 2Pa )"

In general, the potential V (°v) is complex. Besides the phase, which is connected with

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

106

the complex nature of the potential Vi(r - {i), it contains the phases which are connected with the rescattering of hadrons inside a produced system u. These rescattering effects are described by the wave function ¢v({1, "-', {N)- According to these remarks we represent the potential V (Ou) in the form

v(Ov)(q) = (1 + iT?v) u(Ov)(q) eiSv,

(A.3)

where U(°V)(q) is a real function, ir/~ is the imaginary part connected with the complexity of the potential Vi(r - { i ) and 6 v is the phase describing the contribution of rescattering effects in a produced system. As

v(vO)(q) = (1 + i~v) u(Ov)(q) e -i6v,

(A.4)

the expression (A.2) for A = 0 may be transformed to the form

iPa ~ f~02 )(0) : - ~ u,M

J/~d2q± d2ov(q±, M) I + ir~v 1 --i77v " d2q±

(A.5)

This expression is in agreement with the Pumplin-Ross results [3]. Thus, calculating the contribution of the inelastic channel u in amplitude (A.2) we must take into account the complexity connected with the rescattering effects in a produced system in the same manner as while calculating the differential cross section for the channel v. It means that the amplit, udes fp andfn in the right-hand part or (1.5) are on different sides of the cut in s. But, as was mentioned in the introduction, all these considerations may be applied if, in the sum over M, the finite region is only essential when s 1 ~ ~. If the region M 2 ~ S is also essential the situation is different. For the latter case the expression (A.2) is not, in geneTal, applicable as we cannot neglect the principal value of the integral over qz and, therefore, cannot transform (A. 1) to (A.2). If the potential is described by the sum of the reggeon contributions

v(Ov)(q) = ~ V (Or) (q, M2), i ai we may express formula (A. 1) for 42)(0) through the amplitude of reggeon-particle scattering f/j(M 2, q2):

v(O v)(_ q)

1 P b2' - - (Pa + q)2 +

/)

~

V (vO)(q)

(A.6)

ie

ni(e~i)B/(a/) (M 2 )eq(q2)+e~/(q 2) fi/.(M 2, q2 ).

// Now, in general, the expression f o r 4 2 ) ( 0 ) is different from (A.5) and depends on the properties off//. This case is discussed in detail in sect. 6. Now let us consider the scattering of the particle a on the nucleus. For the contribution of the double-scattering term in the amplitude we have in eikonal approximation

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes A

iPa v,M ~A) X/~}k (A

107

q ~ k ) =1

q,q)8

z

167r2pvpa

- ma 2p• - ,

(A.7)

describes the interaction with the k-nucleon in the nuwhere the potential v(Ov)(q) k cleus: b~A)(A - q, q) = f eiqR/+ i(A - q)R k i 0( R 1..... R A)I 2

X d 3 R l ... d3RA 8

( R1 + + R A ) A

"

For the d e u t e r o n F ( d ) ( q q) = F ~ d ) ( q q) = F ( q 2 ) . Using the variables t = np ' p ' q2 _ q2, s' = M 2 = 2 Paqz - m2a and taking into account that

(AL8) q2 =

A = --__4TO lm fd~2)(0 )

Pa we see that formula (A.7) can really be transformed to the expressions (1.3) and (1.5) if we change the transition potentials V (°") for the transition amplitudes f(av) with the corresponding n o r m a l i z a t i o n factors. References [11 [21 [31 [4] [5] [6] [71 [8] [9] [10] [11] [12] [13] 14] 15] 16] 17] 18] 19] 20] 21] 22] 23]

R. Glauber, Phys. Rev. 100 (1955) 242. E.A. Abers, tt. Burkhardt, V.L. Teplitz and C. Wilkin, Nuovo Cimento 42 (1966) 365. J. Pumplin and M. Ross, Phys. Rev. Letters 21 (1968) 1778. V.N. Gribov, JETP (Sov. Phys.) 56 (1969) 892. G. Fiildt, Nucl. Phys. B29 (1971) 16. V.V. Anisovich, L.G. Dakhno and P.E. Volkovitski, Yad. Fiz. 15 (1972) 168. S.A. Gurvits and M.S. Marinov, Phys. Letters 32B (1970) 55. A.B. Kaidalov, Yad. Fiz. 13 (1971)40l. K.A. Ter-Martirosyan, Proc. of the 6th Winter Physical School (LPTI), Leningrad, 1971, p. 334. O.V. Kancheli and S.G. Matinyan, Yad. Fiz. 13 (1971) 143. J. Allaby et al., CERN preprint (May, 1971). G. Cocconi, Nucl. Phys. B28 (1971) 341. A.B. Kaidalov and L.A. Kondratyuk, JETP (Soy. Phys.) Letters 15 (1972) 170. E.W. Anderson et al., Phys. Rev. Letters 16 (1967) 855. E.W. Anderson et al., Phys. Rev. Letters 25 (1970)699. J.M. Wang and L.L. Wang, Phys. Rev. Letters 26 (1971) 1287. R.P. Feynman, Phys. Rev. Letters 23 (1969) 1415. C. Quigg, Stony Brook preprint (1971). V.N. Gribov and A.A. Migdal, Yad. Fiz. 8 (1968) 1002. J. Benecke, T.T. Chou, C.N. Yang and E.Yen, Phys. Rev. 188 (1968) 2153. H. Bogglid et al., Nucl. Phys. B27, (1971) 285. G. Veneziano, MIT preprint (1971). K.G. Boreskov, A.M. Lapidus, S.T. Suldaorukov and K.A. Ter-Martirosyan, ITEP preprint N865 (1971).

108

A.B. Kaidalov, L.A. Kondratyuk, Inclusive processes

[24] Yu.P. Gorin et al., IPHE preprint, SEP 7 1 - 4 9 (1971). [25] L.A. Kondratyuk and V.B. Kopeliovich, Yad. Fiz. 13 (1971) 609. [26] A.K. Kerman and L.S. Kisslinger, Phys. Rev. 180 (1969) 1483.