High energy pp data and the reggeon theory of diffraction scattering

Nuclear Physics B70 (1974) 229-236. North-Holland Publishing Company

HIGH ENERGY pp DATA AND THE REGGEON THEORY OF DIFFRACTION SCATTERING* U.P. SUKHATME and J.N. NG Department of Physics, University of Washington, Seattle, Washington 98195, U.S.A. Received 29 October 1973 (Revised 19 November 1973)

Abstract: Recent proton-proton scattering experiments from the CERN ISR show several striking features. There is strong evidence that the total cross section is rising. The elastic differential cross section shows a break in the forward slope, as well as a pronounced dip-bump structure, with the dip located at t = -1.3 (GeV/c)2 . We demonstrate that all these features and their energy dependence can be simultaneously and quantitatively very well described within the framework of Gribov reggeon calculus.

Recent experiments performed at the CERN intersecting storage rings facility have yielded several very important high energy scattering results. There are strong indications that the proton-proton total cross section has a significant rise of approximately 4 mb when s (the square of the centre of mass energy) increases from 548 (GeV/c) 2 to 2776 (GeV/c) 2 [ 1-3]. The predominant behaviour of the differential cross section is to fall approximately exponentially through many decades [4]. There is a noticeable change in slope (break) at t ~- - 0 . 1 5 (GeV/c) 2 [1, 5]. For [tl ~< 0.15 (GeV/c) 2 , the average slope varies from 11.8 (GeV/c) - 2 at s = 548 (GeV/c) - 2 to 13.1 (GeV/c) - 2 at s = 2776 (GeV/c) 2, whereas the average slope for 0.15 (GeV/c) 2 ~< It I ~< 0.3 (GeV/c) 2 is approximately constant at the value 10.8 ( G e V / c ) - 2 . At large It I, the differential cross section shows a dip-bump structure, with the dip location ( t ~- - 1 . 3 (GeV/c) 2) remaining approximately independent of energy [4]. However, the bump height decreases with increase in energy. In this paper, we demonstrate that the rising total cross section, the dip-bump structure in the differential cross section, as well as the break in slope at Itl~- 0.15 (GeV/c) 2 can all be simultaneously very well described within the framework of a reggeon theory of diffraction scattering (Gribov reggeon calculus)** [6,7]. * Work supported in part by the US Atomic Energy Commission. ** In ref. [6], we have shown that the structure of da/dt could be reasonably ~,xplainedby Gribov theory. However, the parameters of ref. [6] were such as to be consistent with an approximately constant total cross section as given by the earlier ISR measurements quoted in ref. [8]. The present article is an update of ref. [6] and includes the rising total cross section and the correct normalizations for da/dt at t = 0.

230

U.P. Sukhatme, J.N. Ng, High energy pp data

(a)

(b)

(c)

(0)

(e)

(f)

Fig. 1. Reggeon diagrams giving dominant contributions to the elastic scattering amplitude. In Gribov theory, the elastic scattering amplitude for large s and fixed t is expressed as a power series in ~ - 1 , where ~ = in (S/So). The contributions to the amplitude up to O(~ - 1 ) come from the six reggeon diagrams shown in fig. 1 [6,7]. In this paper we discuss scattering at ISR and higher energies, and hence only pomeron exchanges are important [6]. (Contributions from other lower-lying trajectories are 1 typically less than the single pomeron exchange term by a factor of (sis O)-~ .) We will parameterize the pomeron couplings (see fig. 1.) g, N, P and C by

g( t) = g(O) e R2t , N ( t ) = N ( 0 ) e ¢R2t ,

(1) (/3~> 0 ) ,

P(t, t l , t2) = a [ a ( t l ) + a ( t 2 ) -

2] - b a ' t ,

C(t) = c , where we have assumed a linear pomeron trajectory a ( t ) = 1 + a't, R 2 corresponds to the Regge eadius, and a, b, c and/3 are constants. Note that the triple-pomeron coupling I" vanishes for t = t 1 = t 2 = 0, as is required for internal consistency in Gribov theory [7]. The imaginary part of the elastic amplitude, P, at high energy is given to 0 ( ~ - 1 ) by [6,7] t : t The 1 --* 3 pomeron contribution ImP3 does not appear in eq. (2) since it is of order ~-2. This is discussed in ref. [7].

U.P. Sukhatme, ZN. Ng, High energy pp data

231

ImP(s, t) = I m P 1 + I m P 2 + IMP4, ImP 1

= e-2Zg2(t),

ImP 2

=e-2Zg2(t)

=

b't b'+ a + b ' z +

(

-

ImP4

(2)

iny~+fdue_~1

11 °

- 4zb'2g2(t) + N(t) +ag(t) 2

_e-~Zg2(t)c2 4a'~

[i

+

~{(

,

3

e-~ z - I)

- e-~Z(2 - {z)(ln3,~ + where z - - Sa ~t, b = b - a, and tn7 = Euler's constant = 0.5772. The real part of the amplitude is given by*

ReP =-~

(3)

I m P ( s , t) .

Eqs. (2) and (3) are strictly valid for large s and small fixed t. We will use them to describe the pp scattering data at ISR energies for Itl~< 2.3 (GeV/c) 2. The differential an total cross sections are related to the amplitude P by the following equations:

~=

4n

((Imp) 2 +(ReP)2),

(4)

Otot = 8 lr ImP(s, t = O) = 8n

2(0) - ~

((N(0) + ag(0)) 2 + c2g2(0))+ O(~ - 2 )

.

(5)

The forward slope is given by

_dF

do']

b(s, t = O) = ~ Lln d-tJt=O

(6)

= 4R 2 + 2a'~ - 2 ] ' / ( 1 + f ) , where * For small t and large s, eq. (3) readily follows from a series expansion of the even signature factor for the pomeron in the Sommerfeld-Watson representation for the amplitude. For more details, see Gribov and Migdal (ref. [7]).

232

U.P. Sukhatme, J.N. Ng, High energy pp data

f = _ (602 + C2)/4C~,~ , ft

=

1

6o2

t

- ~b ~ l n ? ~

-

--

8

(1 -- 3)R2coN(0) -

2 o,'~ g(0)

3c2 (21n'),~- 1), +

oo = a + N(O)/g(O).

In our fits, we choose the pomeron trajectory slope a ' = 0.37, as given in ref. [5], and a conventional scale parameter s O = 1 (GeV/c) 2. F o r simplicity, we also choose c = 0. The parameter 3 is timed to be ~, 1 in accordance with the single pole approximation for N ( t ) discussed by Ter-Martirosyan [9]. It should be noted that there is no compelling reason for making the simple choice of parameters given above, and we could have left them free in obtaining fits to the data. However, we found that such extra freedom was not necessary, and the accuracy o f currently available data does not justify the use o f an excessive number o f free parameters. We found a very good fit to the ISR pp data at the available energies (s = 5 4 8 , 9 3 2 , 2005, 2776 (GeV/c) 2) for Itl ~ 2.3 (GeV/c) 2, with the following values o f parameters: g2(0)=

3.75

(nab)½ (GeV/c) - 1 1

N(0) = b'

4.72

,

R2 =

,

a

1.54(GeV/c) - 2

,

3

(mb)~ ( G e V / c ) - ~

=-4.2

(GeV/c) - 1

,

= - 0.213 (GeV/c) - 1

A sample o f our results is shown in figs. 2-6. Let us now discuss how Gribov's theory gives rise to the various features of the data.

60

i

'

I

. . . .

I

I

I

I

I

I I I [

I

'

O'TO T

i

1

i

vl

(CO)

55

50 E

40

35 10 2

i

iO 3

I

i 04

s (GeVlc) t

Fig. 2. Proton-proton total cross section as a function ors. (e) indicates data taken from ref. [1 ], (x) from ref. (2), ( ) from ref. [ 10], (t0 from ref. [11], and (D) from ref. [12]. The solid line corresponds to Gribov theory with the parameterization given in the text.

U.P. Sukhatme, J.N. Ng, High energy pp data

14

I

I

233

I c)z

b~

III

i

Io o

)

0.1

I

0.2 - t (GeV/c) z

0,3

0.4

Fig. 3. The slope parameter b is plotted as a function -t for energy s = 2776 (GeV/c)2 . A rising total cross section reaching an asymptotically constant value is a characteristic feature of Gribov theory (see eq. (5)). Our fit gives Otot(S ~ ~ ) = 58.8 mb. The complete s-dependence o f oto t and a comparison with experimental data [ 1, 2, 1 0 - 1 2 ] is shown in fig. 2. The existence o f the dip is essentially caused by a zero in ImP. This is a consequence o f the characteristic negative sign of the cut in Gribov theory. In our fit, the zero in I m P is predominantly due to the cancellation o f the positive contribution o f figs. 1 (d) and (e) with the negative contribution o f fig. 1 (b). The location o f this zero is almost energy-independent. Consequently, from eq. (3), R e P also has a zero near the same location. Thus, there is a pronounced dip which remains almost fixed at t ~- - 1 . 3 (GeV/c) 2 over the entire ISR energy range. (At s = 105 (GeV/c) 2, the dip is at t ~- - 1 . 2 (GeV/c)2). The height of the bump at t ~- - 2 (GeV/c) 2 decreases with increase in s, more or less in agreement with the data. In the near forward region, our fit has a continuously decreasing slope b(s, t) as Itl increases (see fig. 3), thus simulating the break in the experimental data. The

16,

'

'

'

'

'''I

'

'

'

'

'

'''

I

I

I

I

I

14' -

.

>

I0

0.15
I

I

I

I

I

I

?0 ~

I I I

10 3 s

(GeVlc)

I

_

I I

10 4

z

Fig. 4. The slope parameter b is plotted as a function of s. (e) denotes data taken from ref. [5 ], and (x) is taken from ref. [ 1 ].

234

U.P. Sukhatme, £N. Ng, High energy pp data

I0 z

I I 2 s= 2776 (GeV/c)

I0

L

i0-1

10-4

LO-a I I t 2 - t (GeV/c) ~

: 0

Fig. 5. C o m p a r i s o n o f G r i b o v f o r m u l a w i t h p p e l a s t i c I S R d a t a a t s = 2 7 7 6 ( G e V / c ) 2 . T h e d a t a is t a k e n f r o m refs. [ 4 , 5 ] .

8

t

I

-~r

et

"_o

o

0.5 -t,

1.0 (GeV/cl t

1.5

F i g . 6. A p l o t o f t h e real p a r t o f t h e a m p l i t u d e as a f u n c t i o n o f -t a t s = 2 7 7 6 ( G e V / c ) 2 . T h e value of ReP(t = 0) = 0.197 and ImP(t = 0) = 2.75.

U.P. Sukhatme, J.N. Ng, High energy pp data

235

break is produced by the presence of diagrams 1 (b) to 1 (e), which modify the simple exponential behaviour of the pole term (fig. 1 (a))*. The break is not a characteristic feature of Gribov reggeon calculus. The magnitude of the break, and in fact even its presence, is dependent on the choice of parameter values. Note that the break is produced by O(~- 1) corrections to the pole term. Therefore, the reggeon calculus predicts a vanishing break for very large ~. The s-dependence of the slope b(s,t) is shown in fig. 4. Although we have included ReP in our formula, its effects are relatively small compared to ImP. It is easy to see from eq. (3) that ReP(s, t = 0) is positive for large s, and tends to zero like ~- 2. Away from the forward direction, ReP(s, t) has the t-dependence shown in fig. 6 [13]. As emntioned earlier, the parameterizations used in this article are not unique, but are the simplest possible choice **. Nevertheless, it is important to note that intrinsic characteristics of Gribov theory, like Otot rising to an asymptotically value, a shrinking forward diffraction peak, a dip-bump structure etc., will always be present independent of the parameterization used. Note that for ~ -> o% our model has Otot -~ constant from below and Oel/Otot ~ 0. This is in sharp contrast to models having an asymptotically growing total cross section and non-zero limits for the ratio Oel/Otot. (In particular, the model of Cheng, 1 Walker, and Wu [14] has Otot ~ ~2 and Oel/Otot -~ ~). In order to emphasize the important features of the theory at high energies, we have restricted ourselves to exchange of pomerons only. It is well known that the same reggeon theory framework can account for pre-ISR pp scattering data at lower energies [ 15]. This requires the inclusion of lower-lying Regge trajectories (p, f, co, A2) , as is obvious from the fact that Otot (pp) is decreasing for s ~ 300 (GeV/c) 2 . Gribov reggeon calculus can also be applied to other reactions besides pp scattering. Note that many parameters (a, b, c, a', So) in the theory are reaction-independent, and they have already been timed by the pp data. The currently available data on reactions other than pp is at low energies, where lower-lying Regge trajectories are important, and must be taken into account [ 15]. Measurements on other reactions at the highest NAL energies will provide important checks of the general features of Gribov theory. This work would not have been possible without the constant encouragement, stimulating discussions, and helpful suggestions provided by Professor Marshall Baker,

* Note that it is impossible to produce a break using only pole and pure cut terms (figs. 1 (a) and 1 (b)) with exponential form factors, since an explicit computation shows that db (s, t)/dl t l is always positive. ** We have very recently received a preprint from Pajares and Schiff describing a fit to the ISR data, which corresponds to using a small scale parameter s o ~- 0.03 (GeV/e) 2 , and c = fl = 0. We wish to thank Dr. Pajares and Dr. Schiff for communicating their results to us.

236

U.P. Sukhatme, J.N. Ng, High energy pp data

We also w i s h t o t h a n k all o u r o t h e r colleagues in t h e t h e o r y g r o u p at t h e U n i v e r s i t y o f W a s h i n t o n for n u m e r o u s useful discussions.

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