Forward scattering amplitudes in semi-hard QCD

Volume 213, number 2

PHYSICS LETTERS B

20 October 1988

FORWARD SCATTERING A M P L I T U D E S IN S E M I - H A R D QCD B. M A R G O L I S , P. V A L I N Department of Physics, McGill University, Montreal, Canada H3A 2T8

M.M. B L O C K Department of Physics, Northwestern University, Evanston, IL 60201, USA

F. H A L Z E N a n d R.S. F L E T C H E R Physics Department; University of Wisconsin, Madison, W153706, USA

Received 13 June 1988

We argue that the observation by UA4 of a high energy p value (real-to-imaginary ratio of the forward pO amplitude ) in excess of what had been predicted is accommodated by QCD. At x/s---0.1-1 TeV, typical hadron collisions become semi-hard and therefore calculable in perturbative QCD. An "effective" threshold develops in this energy region as a result of the "exploding" gluon content of the proton. We associate the increase with energy of the total cross section with this threshold and show that it will also result in a rapid increase ofp over and above what is expected from models which do not take into account this qualitative change in the structure of hadrons. The first signature of this "threshold" has arguably been seen by UA1 through the discovery of the minijet phenomena. We predict substantially higher cross section and p values at the Tevatron collider and the SSC.

The observation o f an unexpectedly large value o f p [ 1 ], the ratio o f the real-to-imaginary part o f the forward p~) elastic scattering amplitude, at x / ~ = 546 G e V has led to the speculation o f the presence o f a threshold [2 ]. The p u r p o s e o f this letter is to p o i n t out that such a threshold does not require new physics, i.e., the i n t r o d u c t i o n o f new particles, but is in fact a natural feature o f Q C D in this energy range. The r a p i d increase with energy o f the n u m b e r o f soft gluons inside h a d r o n s exhibits all the characteristics o f a " t h r e s h o l d " in the x f ~ = 0 . 1 - 1 TeV energy region, where the gluonic (rather than q u a r k ) structure o f high energy h a d r o n s fully develops. This is a laboratory for studying the transition region between perturbative and non-perturbative QCD, presenting new challenges to both theory and experiment. We m o l d this idea into a Q C D - i n s p i r e d p h e n o m e n o l o g y [3 ] which naturally allows for the large p value m e a s u r e d by UA4. The new " t h r e s h o l d " requires qualitatively different high energy forward a m p l i t u d e s from those o b t a i n e d previously using s m o o t h extrapolations o f lower energy d a t a [4] on atot, p, and B, the forward

slope o f the differential elastic scattering cross section. Before launching into detailed calculations, we m a k e the following observations: ( i ) M i n i m u m - b i a s physics at xf~> 0.1 TeV is d o m i n a t e d by gluon interactions. Indeed, a perturbative calculation shows that the inclusive cross section ( n ) a for g l u o n - g l u o n interactions in pO collisions at collider energies is 80 m b for Pv> 1 GeV. (ii) The sudden onset o f the d o m i n a n c e o f gluons in the interaction o f high energy h a d r o n s has all the features o f an energy threshold. This is the result o f the r a p i d growth o f the n u m b e r o f soft gluons inside the hadron. (iii) Gluons thus naturally can allow for a rising total cross section. The threshold effect can obviously account for large p values in the energy region where a~olvaries rapidly [ 2 ]. The above implies that the p value (as well as the rising cross section) at high energies is d o m i n a t e d by f l a v o r - i n d e p e n d e n t interactions and therefore relates features o f np, K p and pp scattering in the even amplitude. The data i n d e e d show that the p value o f the even a m p l i t u d e o f b o t h np and K p vanishes at the same value Elab ~ 85 GeV. This is particularly signif221

Volume 213, number 2

PHYSICS LETTERS B

icant since the forward amplitudes for n+p, n - p , K+p and K - p show strikingly different behavior [ 5 ]. It is well known [ 6 ] that the perturbative calculation of the cross section for producing jets with transverse momentum Pv is not only valid in the hard scattering regime with XT= 2Pv/x~SS~--1, but is in fact reliable for PT<< x/s provided the AocD
f

f

de

dp 2 dxldx2g(xl)g(x2)dp2,

i GeV

(1) with d6 9zra~ dp2 ( g g + g g ) = 2p4 •

(2)

For a toy gluon structure function g(x) = 3 ( 1-x)5/ x and as-~0.2, we calculate [7,8] that for x/~---630 GeV ( n g ) a,o~(pO) -~80 m b .

(3)

This jet cross section is of order of the total cross section. The fact that a 1 GeV gluon jet cannot be resolved by experiment is irrelevant. The situation is similar to that in e +e- interactions. The perturbarive result that R=3Zqeq2, obtained from the diagram e + e - ~ q q , is already valid for , ~ = 2 GeV, although the jet structure of the events does not become apparent until x / s > 7 GeV. The "jetty" behavior of minimum-bias interactions is clearly revealed by the mini-jet data of the UA1 experiment [9] [which are quantitatively consistent with eqs. ( 1 ) and (2) ] and becomes a feature of typical cosmic ray interactions above 100 TeV. Cosmic ray events in this energy range exhibit "multiple-core" (read "jet") structures with p v > 1 GeV [10,9]. It has also been argued that the onset of the rapidly evolving gluon structure of hadrons is resonsible for the violation of Feynman and KNO scaling as well as a plethora of other phenomena in the x ~ = 0.1-1 TeV energy range

[8]. The "threshold behavior" is due to the rapid rise with energy of the number of relatively soft gluons, which dictate the behavior of high energy cross sec222

20 October 1988

tions. In the previous calculation, typical fractional gluon momenta a r e x = 1 G e V / x / s = 10 -2-10-3. The gluon structure function increases faster than any power of x, with increasing energy (therefore reduced x-values). Solution of the Altarelli-Parisi equations, neglecting quarks, predicts

g(x)~ lexp[2x/3ln(ln O2/A2"~ 1] kin QZ/AZ] In

,

(4)

with b = ( 3 3 - 2 n r ) / 1 2 = (nf is the number of quark flavors), and QZ~p2. Q2 is the scale of Q2. The threshold behavior is further enhanced by the rapid evolution of the structure function with Q2 in the small-x regime. This explosive small-x behavior is a feature of explicit structure functions such as EHLQ [ 11 ] and Duke-Owens [ 12 ]. Screening of the large number of soft partons stacked inside a high energy hadron will modify the result of eq. ( 1 ) when x or PT at some fixed energy become too small [ 13 ]. This can be naturally taken care of by incorporating eq. ( 1 ) into an eikonal formalism [3 ], just the way Glauber theory is applied to the screening of nucleons in a nucleus. In such a picture the large inclusive cross sections will result in an increase of a, ot and eventually transform the proton into a black disc. We will work in the two-dimensional impact parameter space b, and introduce even and odd amplit u d e s f + a n d f -, such that fpp = f + - f -

and fp, = f + + f - .

(5)

We write

f+ (s, t ) = -k f dZbexp(iq.b)a+ (b,s) 7g

,

(6)

where the impact parameter amplitudes a + (b, s) are given by a + (b, s) = {exp [2iz -+(b, s)] - 1}/2i

(7)

in terms of the even and odd eikonals Z +. We divide the even eikonal into two pieces, one due to a valence-valence interaction, which accounts for both the constant (with s) portion of the cross section and the Regge-descending portion, and a rising portion, due to semi-hard gluon interactions. In the simplest model, which we will adopt and refer to as a quasifactorizable model, each eikonal factors into a piece

Volume 213, number 2

PHYSICS LETTERS B

dependent on b only and a piece dependent on s. We will assume that the b dependence is that due to a dipole form factor, i.e., K3(ltb)(flb) 3, ( K 3 is the modified Bessel function), where we allow for different /z values for different portions of the eikonal. Thus, for the even eikonal, we take z + ( b , s)

+ =Zvv(b, s) +z~(b, s),

(8)

where the valence portion of the even eikonal is given by

Zv+v(b, s ) = i [ C exp ( ¼ i n ) / ~ f s + avv] ×{3[(lz+)2/32n](It+b)3K3(lz+b)} ,

(9)

where the term with real, positive coefficient C accounts for the Regge term and the term with real, positive coefficient avv accounts for the constant cross section. The real, positive parameter/t + sets the scale of the variation in b. The semi-hard component of the even eikonal is given by 1

20 October 1988

The term 1/x ~+~ simulates the effect o f scaling violations "~ in the small x behavior of the gluon structure function, and is responsible for the "threshold" behavior. The structure function g ( x ) is normalized so that the gluons carry half of the proton's momentum. The odd eikonal, with its correct analyticity property, is given by

z - ( b , s ) = O [ e x P ( ¼izc)/s 1/2 ] ×{3[(lz-)2/32zc](It-b)3K3(lz-b)},

where the real coefficient D is the strength o f the odd Regge term, a n d / z - , a real positive coefficient, sets the scale for the b variation. This term is responsible for the difference between the lOp and the pp systems, and vanishes at high energy. The following quantities can now be calculated for both the pp and the lOp systems (since t h e f ' s are appropriately complex): the elastic scattering distribution da

z ~ ( b , s ) = i Wgg(b) f Fgg(rla~(g)dz, ag

r

(101

To

(17)

(18)

dt - k 2 If(s, t)12 ,

the total cross section

with (19)

Wgg(b) = ( l.tggb)3K3( i.tggb) /8 ,

( 11 )

atot = ~ I m f ( s , t = 0 ) ,

9~a~ ~ 2 agg(g) = ~ O(s-mo) ,

(12)

the ratio of the real-to-imaginary portion of the forward scattering amplitude

and, for our quasi-factorizable model [ see eq. (10) ],

g=rs.

(13)

The complex factors exp(~iz0 in eq. (9) and exp( - ½izr) in eq. (12) insure the correct analyticity properties for an even amplitude. The correct analyticity properties in the gluon term Xgg are ensured by substituting ro = m~/s exp( - ½it0

(14)

after analytically integrating over r. The convoluted structure function for gluon scattering is given by

Z

1

I

0

0

--

g(xl)g(x2)6(XiX2--~r) dxl d x 2 ' (15)

where

g(x)=Ngx-'-'(1-x)

s.

(16)

K

Re f(s, t = 0 ) P= Imf(s, t=0) '

(20)

and the nuclear slope parameter B=

Re f db ba(b, s) f db b3a*(b, s) 2[f dbba(b,s)[2

(21)

~ In the kinematic region of interest, a value of e ~ 0.1-0.15 effectively reproduces the x behavior of the structure functions [11,12] for Q2~ 10 (GeV). Note the most earlier work [3] took ¢=0. Table 1 Parameter values. l/ag=0.45 mb ~, cr~ = 5.0 mb, C= 2.8 GeV- t. D = - 4 . 5 GeV ~, Ng=2.27,

flgg= 0.77 GeV, p+ = 0.89 GeV, /2- =0.58 GeV, mo=0.49 GeV, e=0.11

223

Volume 213, number 2

PHYSICS LETTERS B x

220

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20 October 1988

p '

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~/~

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Fig. 1. Got versus x/s ~ r pp (solid) and ~p (dash) calculated using the model and parameters given in the text. Re~rences ~ r the data points shown in the figures can be ~ u n d in re~. [ 1,4,5 ].

x - ~ o - p

35

. . . . . . . .

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I/,/"/:''~''~-"

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~

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Fig, 2. Ratio of the real to imaginary parts of the forward scattering amplitude at t = 0 for pp (solid) and 10p (dash) versus x/s,

224

Volume 213, number 2

PHYSICS LETTERS B × -p

0

-

20 October 1988

p

25

t

:

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24 PROPOSED

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22 2~. ~4 I U

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16 t5

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10

1000

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Fig. 3. Forward elastic slope Bed0) for pp (solid) and ~p (dash) versus x/s.

D e p e n d i n g on the experiment, we use as d a t a input to o u r Z 2 fitting routine the quantities B, atot, a n d p, for both lop a n d pp, as well as A a = a ( l o p ) - a ( p p ) , A p = p ( l o p ) - p ( p p ) , and A B = B ( ~ p ) - B ( p p ) . We fit p a r a m e t e r s 1/ag, avv, C, D, /zg~,/x +, / t - , a n d rno for 15 ~
shown in figs. 1, 2, and 3, which are plots o f x/~ versus the fitted ato,, P, and B, respectively, for both 10p and pp. There is a direct correlation between mo [i.e., the value o f the integral in eq. ( 1 0 ) ] and the normalization 1/ag; mo is an effective glueball mass which parametrizes the b r e a k d o w n o f p e r t u r b a t i v e Q C D . The role o f m o is m a d e clear in the limit o f vanishing even Regge c o n t r i b u t i o n ( C = 0). N o w mo is uniquely det e r m i n e d by the energy value at which p . . . . vanishes

Table 2 Predictions for forward parameters. . ~ (TeV)

a,o, (mb)

p

B (GeV/c) -z

log2(S/So) model a~

0.54 1.8 40

63.1 + 0.72 80.8+ 1 . 3 4 138.2+3.50

0.147 + 0.004 0.148_+0.003 0.130+0.002

14.2 _+0.7 15.3_+ 1.0 19.8-+2.6

QCD-inspired model

0.54 1.8 40

69.5 -+0.9 92.6_+ 1 . 5 4 167.7_+3.02

0.190 -+0.004 0.178_+0.004 0.123_+0.002

15.4 _+0.2 17.3_+0.2 23.9_+0.4

a~ Ref. [4]. 225

Volume 213, number 2

PHYSICS LETTERS B

( ~ o o -~ 18 G e V ) . T h e zero o f Rezgg d e t e r m i n e s mo/ x//So" We find for E= 0.11, that mo = 4 GeV. T h e precise value is h o w e v e r sensitive to C:~ 0. T h e r e f o r e our fit is not i n c o n s i s t e n t with such a large value, especially as mo a n d 1/ag are strongly correlated. We n o t e that a s y m p t o t i c a l l y o u r m o d e l is a black disk w i t h

a,o, ~-2nb2,ax ~- 2n( e/ /t) 2 In 2 (S/So) , w h e r e So is a scale factor. T h e c o e f f i c i e n t o f the In 2 t e r m is 0.05 m b in c o n t r a s t to the 60 m b v a l u e w h i c h saturates the F r o i s s a r t b o u n d . O u r m o d e l allows us to calculate da/dt. W e o b t a i n a satisfactory d e s c r i p t i o n o f the d a t a at ~ s = 53 GeV. F o r x / ~ = 546 G e V the c a l c u l a t i o n exceeds the d a t a for Itl v a l u e s b e y o n d the b r e a k at t -~ - 0 . 8 G e V 2. It is k n o w n that this can be r e m e d i e d by r e a d j u s t m e n t o f the i m p a c t p a r a m e t e r profiles. T a b l e 2 c o n t a i n s p r e d i c t i o n s for h i g h e r energies which are c o n t r a s t e d w i t h the earlier fits o f Block a n d C a h n [4] w h i c h were not c o n s t r a i n e d by the Q C D f r a m e w o r k in this paper. T h e m o s t o u t s t a n d i n g feature o f table 2 is the m u c h h i g h e r v a l u e o f atot at the T e v a t r o n collider. In o u r m o d e l , this directly reflects the large p v a l u e o f U A 4 . T h i s r e s e a r c h was s u p p o r t e d in part by the U n i v e r sity o f W i s c o n s i n R e s e a r c h C o m m i t t e e w i t h funds g r a n t e d by the W i s c o n s i n A l u m n i R e s e a r c h F o u n d a tion, a n d in part by the U S D e p a r t m e n t o f Energy u n d e r contracts D E - A C 0 2 - 7 6 E R 0 0 8 8 1 a n d D E A C 0 2 - 7 6 E R 0 2 2 8 9 . A d d i t i o n a l s u p p o r t was r e c e i v e d f r o m the N a t i o n a l Science a n d E n g i n e e r i n g R e s e a r c h C o u n c i l o f C a n a d a and the D e p a r t m e n t o f E d u c a t i o n of Quebec.

226

20 October 1988

References [1] UA4 Collab., D. Bernard et al., Phys. Lett. B 198 (1987) 583. [2] E. Leader, Phys. Rev. Lett. 59 (1987) 1525; A.D. Martin, CERN Report CERN/TH-4852; P.H. Kluit and R.J. Timmermans, NIKHEF Report NIKHEF H/87-22. [3] P. L'Heureux, B. Margolis and P. Valin, Phys. Rev. D 32 (1985) 1681; B. Margolis, P. Valin and M.M. Block, in: High energy scenarios from constituent scattering, Proc. lind Intern. Conf. on Elastic and diffractive scattering (Rockefeller University, New York, October 1987), eds. R.L. Cool, K. Goulianos and N.N. Khuri; L. Durand and H. Pi, Phys. Rev. Lett. 58 (1987) 303. [4] M.M. Block and R.N. Cahn, Rev. Mod. Phys. 57 (1985) 563. [ 5 ] L.A. Fajardo et al., Phys. Rev. D 24 ( 1981 ) 46. [6] L.V. Gribov et al., Phys. Rep. 100 (1983 ) l. [7] F. Halzen and F. Herzog, Phys. Rev. D 30 (1984) 2326. [8] T.K. Gaisser and F. Halzen, Phys. Rev. Lett. 54 (1985) 1754; in: First Aspen Winter Physics Conf. ed. M.M. Block (New York Academy of Sciences, New York, 1986); T.K. Gaisser et al., Phys. Lett. B 166 (1986) 219; G. Pancheri and Y. Srivastava, Phys. Lett. B 159 (1985) 679; S. Rudaz and P. Valin, Phys. Rev. D 34 (1986) 2025. [9] UA1 Collab., CERN EP 88-29 (1988). [ 10] See e.g.G. Panchmi and Y.N. Srivastava, in: Physics simulations at high energy, eds. V. Barger et al. (World Scientific, Singapore, 1986 ); C.S. Kim et al., in: Physics simulations at high energy, eds. V. Barger et al. (World Scientific, Singapore, 1986); See also e.g.N. Yamdagni, in: Multiparticle dynamics 1984, eds. G. Gustafson and C. Peterson (World Scientific, Singapore, 1984). [ 11 ] E.J. Eichten, 1. Hinchliffe, K. Lane and C. Quigg, Rev. Mod. Phys. 56 (1984) 579. [12] D.W. Duke and J.F. Owens, Phys. Rev. D 30 (1984) 49. [ 13 ] See e.g.A. Mueller, in: Proc. Oregon DPF Meeting, ed. R.C. Hwa (World Scientific, Singapore, 1985 ).