γ∗γ∗ total cross section in the dipole picture of BFKL dynamics

ELSEVIER

Nuclear Physics B 555 (1999) 540-564 www.elsevier~/locate/npe

7* 7* total cross section in the dipole picture of BFKL dynamics Maarten Boonekamp ", Albert De Roeck b'c, Christophe Royon d, Samuel Wallon ~,f a Service de Physique des Particules, DAPNIA, CEA-Saclay, 91191 Cn'fsur Yvette Cede.v, France b Deutsehes Elektronen-Synchrotro~ DES]; Notkestrasse 85, D-22603 Hamburg, Germany c CERIV,, CH-1211 Geneva 23, Switzerland d Service de Physique des Particules, DAPNIA, CEA-Saclay, 91191 C-ifsur Yvette Cedex, France e Division de Physique :lhdorique, Institut de Physique Nucl~aire g'Orsay, 91506 Orsay, France 1 I Laboraloire de Physique 2hdorique des Particules Eldrnentaires, Uuiversitd P. ~ M. Curie, $ Place Jussieu, 75252 Paris Cedex 05, France

Received 4 January 1999; revised 23 April 1999; accepted4 June 1999

Al~act The total 7*7* cross section is derived in the leading order QCD dipole picture of BFKL dynamics, and compared with the one from two-gluon exchange. 'I]le double leading logarithm approximation of the DGLAP cross section is found to be small in the phase space studied. Cross sections are calculated for realistic data samples at the e+e - collider LEP and a future high energy linear collider. Next to leading order corrections to the BFKL evolution have been determined phenomenologically, and axe found to give very large corrections to the BFKL cross section, leading to a reduced sensitivity for observing BFKL effects. (~) 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 13.60.Hb; 13.65.+i; 13.85.Qk; 13.85.Ni

I. I n t r o d u c t i o n In this paper we study the possibility to investigate QCD pomeron effects in the high energy limit in virtual photon-photon scattering both at LEP and a future Linear Collider ( L C ) . In the past years, the BFKL pomeron [1] has been studied intensively in the 1Unit~ de P,echemhe des UniversitgsParis 11 et Paxls 6 Associde au CN1RS. 0550-3213/99/$ - see fron~mtter (~ 1999 Publishedby l~sevier Science B.V. All fights reserve& P]:I S0550-3213(99) 00345-4

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

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small-x regime at HERA both in the context of proton diffractive and fully inclusive structure functions [2,3] and of final state particle flow or forward jet production [4]. The coupling to the proton induces a non-perturbative scale in the structure function studies and the studies of final states suffer from non-perturbative hadronisation effects. The use of a purely perturbatively calculable process is much more favourable to establish effects of BFKL dynamics. The cross section of collisions of two objects with small transverse size is an ideal process where the BFKL approximation is expected to be most reliable. High energy virtual y ' y * interactions at e+e - colliders is such a process, and has been proposed in [5-7] as a laboratory to study BFKL. In this paper inclusive virtual photon scattering in e+e - collisions at LEP and a future LC is studied. In particular the LC, with an anticipated luminosity three orders of magnitude larger than the one presently at LEE and its larger centre of mass (CMS) energy of up to 1 TeV, offers an excellent opportunity to test BFKL dynamics. First we obtain the leading order (LO) BFKL cross section using the QCD dipole picture of BFKL dynamics. The double leading logarithm approximation of the DGLAP cross section [ 8] is compared with the BFKL one. The two-gluon approximation, where only two gluons are exchanged in the y ' y * interactions will turn out to be the dominant 'background' in the region of phase space studied in this paper. We will then consider phenomenologically the effects of higher order corrections to the BFKL cross section, and show that the cross sections for the two-gluon and BFKL-HO evolutions both at the LC and LEP colliders, are different by a factor of two to four.

2. ~/*?/* total cross section in the dipole picture of BFKL dynamics 2.1. B F K L cross section

We analyse the y ' y * subprocess in the framework of the colour dipole model [9-12]. As usual, in analogy with deep inelastic scattering (DIS) kinematics, we define the scaling variables which will describe the total cross section (see Ref. [5], and the scheme in Fig. 1 for the definitions of the variables) qlk2 Yt - k l k 2 '

(2.1)

qzkl Y2 = klk2

and xl -

Q~ 2 q l k2 '

x2 -

Qz2 2q2kl

,

(2.2)

the photons having virtualities Q~ = _q2 and Q22 = -q22. The total energy available in the s channel of the e+e - system is s = (kl + k2) 2. The energy available for the subprocess y ' y * is g = (ql + q2) 2 -----sylY2. We consider the domain of large Q2, Q~ (in order to be in the perturbative regime) and large ~ with the constraint Ql2, Q2 << ~,

(2.3)

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X

e+(k2) ~±(~'~) Fig. 1. y'y* subprocess in e+e - ---*e+e - + X collisions. in order to exhibit the energy dependence of the BFKL pomeron, which will be described in terms of a dipole cascade. Defining _

Xl-

_

Q2 2qlq2

, . ~

_

_

Q~

X2-

g '

Q2 Q2 2 ~ 2qlq2 ~ '

(2.4)

the total rapidity of the y ' y * subprocess is then given by l Y= I n - -

s ~ I n - V ~1~2

syl Y2

- In - -

(2.5)

V ~,~1~,~2

Let us first compute the y ' y * total cross section in the two-gluon exchange approximation. Note that the QED contribution which corresponds to a quark box coupled to four external virtual photons, is sub-leading in the high-energy limit, since t-channel exchange of two particles of spin J contributes like s ~2J-~), and is thus dominated by gluon contributions. In the dipole approach, one can view the two virtual photons as two dipoles, which can scatter through the exchange of a pair of soft gluons. This requires the knowledge of the photon wave function. The photon wave function squared ~r,L (x_, z ; Q 2) gives the probability distribution of finding a dipole configuration of transverse size x, z ( ( 1 - z ) ) being the light-cone momentum fraction carried by the quark (antiquark). The subscript T (L) corresponds to a transversally (longitudinally) polarized photon. Note that we neglect here the effect of quark masses. Heavy quark production will be considered in a future paper. Explicit expressions for these probability distributions can be found in [ 13,12]. 2 We will not use them here. The cross section is then obtained by convoluting the two probability distributions with the elementary dipole-dipole cross section. The latter corresponds to the scattering 2Note that our definition of the wave function is such that q~/;.L = (2v'C2/Nc)~,, z where q ~ z corresponding wave functions squared in Ref. [ 12].

are the

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of two colour neutral objects in the eikonal approximation [10,16]. It reads, if one averages over the angle between the two dipoles, 3

O'DD(~I X__2):oI: f The

9'*7* cross

d2~

(2-eik-'X-l -e-ik'x') (2-eik-'x2-e-ik-'x-2)

(2.6)

section in the two-gluon exchange approximation then reads

o_,.z,.(Ql2, Q2" 2, y ) =

f d2&dz, f d2x2dz2 xq~(Xl, zl; Q2) (P(X2, Z2; Q22) o'DO(Xl, X 2) -

Let us consider now the elementary Born cross section

d(x) g(k) --* d(x)

(2.7)

a-rd/k2 of the process (2.8)

for a dipole of transverse size x and a soft gluon of virtuality _k2, in light-cone gauge. In the high energy approximation, we only need its expression close to the physical pole k2 = 0. Thus its computation can be performed using the equivalent photon (in this case gluon) approximation of Weizs~icker and Williams [ 14], setting k 2 = 0 in the expression of the classical current of a dipole [2,15,16]. This gives, summing over colour and polarization of the emitted gluon and averaging over the colour of the dipole,

b-gd 2 asUc~ ( 2 - eik-'x--- e-ik'x)

~2"1

(2.9)

This squared quantity is related to the elementary dipole-dipole cross section since it can also be computed using the same equivalent gluon technique. Indeed, the elementary dipole-dipole cross section 0"Do reads (277")2 O'DD(X1,X__2) -- 4 N 2 __

- k2

(2.10)

k2 '

the factor (2zr)2/4N~ arising from the normalisation of the k integration and from the definition of &ud/_k2 as an averaged colour quantity. Thus, the T'y* total cross section in the two-gluon exchange approximation now reads

7")2/ °'~,*z,*(Q2,Q2;Y) = ( 3n-~c ×

-

d

d2x-, dz' - k

_k2 "

f

d2x_2dz2qS(X_l,Zl;Q2)~(x2,z2;Q2) (2.11)

This will give us later the hint to exhibit the relation with the calculation based on Feynman diagrams technique. 1,,2-NZ whereO'DD -NZ is the correspondingcross 3Note that this cross section is normalized so that O'DD = ~lVcO'DD section in Ref. 112].

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M. Boonekampet al./NuclearPhysicsB 555 (1999)540-564

Following Ref. [ 17], we define the Mellin-transform of the photon wave function as ~(T)

= J[ ~d2x__,d Z CrpT,L(X, z)(x2O2) 1-r ,

(2.12)

or equivalently

j" dz qO(x, z) = Jf ~d~-T 2~(T)At,B. X (x--2Q2)-I+Y'

(2.13)

Eq. (2.11 ) then reads o.r,r. (Q~, Q2; y) =

( ~c "a"2 ) 2 7 d2Xl dzl S d2x2dz2 ~(x I , El; Q~)~P(X_.2, Z2;Q2) -

=

(q'/"

k2 k 2

)2iSii

~2c2

X

d2x-I

di_x.2

d'y,

~

d')/22 ~)( ,)/1) 2 ~ (,)/2)

2-~ x_l

x_2

i d2k&gd-7(t-1x2/"12~-~1"~1-l+y, (x2/--12)2~2-l+ra Ogdk2

(2.14)

From Eq. (2.9) we have

¢r i ~-d2x^ = ~s f d2x-" (2 - eik'x- - e-iL'x) -~c X20"gd J X--.2

= 4¢rOesJ dx(1 x

Jo(kx))

(2.15)

The Mellin transform of this quantity then reads

~ f ---~&'g,.I(x d2x 2Q2 )-1+7 = 4~OesV(1-y)

( k 2 ) 1-7

(2.16)

_

where (see Refs. [2,15,16]) 2 -27-1 F(1 -- T)

v(~,) = - 7

(2.17)

v(1 +~)

Thus, the y ' y * total cross section now reads

2irr

×

=4"rr

~ J ~

4rrcrSv(1-yl)4rr°esv(1-y2)

to2) 4-'n-OZs@(y)zvrr v(1 - y)4-'rro~.@(l - y ) f ff.-~-Y

v(y) \ ~ j

O~, (2.18)

545

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

where in the last step the integration with respect to k has been performed, leading to 1

-

')/2 =

'Yl.

Let us compare this calculation, based on the dipole approach (light-cone quantization), with the calculation based on Feynman diagram calculation (covariant quantization). In the second approach one has to convolute the two off-shell Born cross sections &Tg/Q 2 and O'Tg/Q 2 of the processes Y(ql) g(k) --+ q gl and T(q2) g(k) ---+q gl. Here the gluon is off-shell, quasi transverse, with a virtuality k2 _ _k2. The 3'*7* total cross section in the two-gluon exchange approximation then reads in this scheme

1 O21O2°'r*r* (Ol2' 02; Y) = ~

1

d2k2J --Zl .s ~2

d2-kl

0

rg

zl ' 02

0

(--Xi,k~ xo'yg \ g Q ~ ) 82(ki - k 2 ) ,

(2.19)

where zl and z2 are the light-cone momentum fractions of the exchanged gluon respectively measured with respect to q2 and q+ (compare with DIS where xBj is the light-cone fraction with respect to the proton momentum). In the light-cone frame, we get

q¢=

(+

q ,-O /ZqL

,

=

(+ .) -Q~/2q+,q;, 0

(2.20)

with q+ >> q l and q~- >> q+. This kr factorization [ 18-20], allowed by the high energy regime Y >> 1 of the process, relies once more on the equivalent gluon approximation, which allowed us to compute the scattering in the dipole approach. Let us now define the double Mellin transformation in both longitudinal and transverse spaces, in order to deconvolute the integrations in Eq. (2.19). We set

8%

( -k2~ )

=

f

1

( k~22)

d x x °~& x,

,

(2.21)

0

or equivalently (2.22) for the longitudinal momenta, and

477"20~em(f~e~) hoj(9/)

(2.23) 0

or equivalently

O"w ^

=47"r2O~em

e} i dT (/2"~-7 2,--/7

(2.24)

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546

Fig. 2. Equivalence, at the order of two gluons exchange, between covariant quantization and light-cone quantization. for the transverse degrees of freedom. The index f runs over the light quark flavours, u, d, s. The flavour c will be considered in an incoming paper. Thus, we take ~-~f e} = 2/3 for the numerical studies in this paper. Eq. (2.19) now reads

Q1Q20-y.~,.(Q2,Q2;Y)=--~ d2kl / d2__k2(4~'2aemZ

e

f

x

x =~

f d3/! ~

/

ho(3/,) 3/1

\Q2f

d3/2 {_k~ "~-~'2 h0(3/2) ~2(__k1 _ k 2 )

~t~ \ Q 2 )

3/2

4"rrZ%m

e

Q~

f

x

d7 ho(y) h 0 ( 1 - y) ( Q 2 ~ ' 2i~" y -1---7 \-~92,] "

(2.25)

hr, L ~ h(o~=0)r,L were computed in Ref. [18] and are given by (hr)

hL

= as ( F ( 1 - y ) F ( I + T ) ) 3

(1( l + y ) ( 1 - 1 y ) ) __2 1 - 5Y 3/(1 3/)

3cry T - ( 2 - - 2 y - ~ 2 ~ i

.

(2.26)

Now one can check that both formalisms give the same result: in Ref. [17] it was checked that 47rOts q~(3/)u( 1 - 3/)= 2V/2teemh(3/) Z

l

e},

(2.27)

f

taking into account the difference of normalisation (see the footnotes at the beginning of this part). This is exactly what we need, since Eqs. (2.18) and (2.25) are identical when taking into account Eq. (2.27). The result (2.27) only means that extracting a soft gluon from the virtual photon can be equivalently computed by convoluting the distribution of dipoles in the photon with the elementary gluon dipole cross section or by calculating the Feynman diagram describing the off-shell 7(ql) g(k) ~ q gl born cross section. This result is summarized by Fig. 2. Let us now consider the computation of the 3/*7* total cross section when resumming the (Ceslogs)" ( n ) 0) contributions. In the dipole approach these terms appear when the relative rapidity of the two photons is large enough so that both photons can be

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

547

considered to be made of dipoles. At leading order, in the centre of mass frame, the two excited dipoles extracted from the two photons scatter through the exchange of a pair of soft gluons. But due to the frame invariance of the process, which is closely related to the conformal invariance of the dipole cascade [ 16], the process can also be viewed differently. Consider the frame where the right moving photon 1 has almost all the available rapidity while the left-moving photon has only enough rapidity to make it move relativistically (but not enough to add gluons to its wavefunction) [21,22,16]. In this frame, the photon 2, which makes the original dipole 2, scatters an excited dipole extracted from the fast right moving photon 1. Due to conformal invariance, the process can also be viewed as the scattering of the initial dipole 2 off the initial dipole 1, the only effect of the LO approximation being in multiplying the result by a matrix which is diagonal when using a unitary representation of the principal series of SL(2, C) (this is natural because of global conformal invariance, since Moebius transformation can be mapped in SL(2, C ) ) . Let us see this explicitly. Following Ref. [9,10], we define n ( x , x ' , ~') such that 1

U(x_', Y) : I d2x--i dzl ~/'(x__,Zl ) n(x, x__',~') d

(2.28)

, /

0

is the density of dipoles of transverse size x ~, where the momentum fraction of the softest of the two gluons (or quark or antiquark) which compose the dipole is larger or equal to e -Y. ~" is the relative rapidity with respect to the heavy quark given by = Y + In Zl. The leading order y'y* total cross section then reads

× n(_xI , x_'~,~ ) n(x__2, x_L, ?2) o'oo (x_~,x_2),

(2.29)

where O'oD(X__l, Xe) is the unaveraged (with respect to orientation) elementary dipoledipole cross section (see Eq. (A.33) of Ref. [16]), which has a non-trivial angular dependence to be taken into account if one is interested in azimutal distributions. The rapidities ~ and ~'2 are such that ~" = ~'l + Y2. In order to get the expression for n(x__,x__',~'), one relies on the global conformal invariance of the dipole emission kernel, related to the absence of scale. This distribution can be expanded on the basis of conformally invariant three points holomorphic and antiholomorphic correlation functions [23,24]. Introducing complex coordinates in the two-dimensional transverse space

P = (Px, Py), P=px+ipy and P * = p x - i p y ,

(2.30) (2.31)

the complete set of eigenfunctions E n'~' of the dipole emission kernel is ' ~Plo, P2o) = ( - - 1

-

-

\PioP2o./'

\PToP~oJ '

(2.32)

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

548

with the conformal weights

h-

1 -n

2 + iv,

h-

l+n

2 + iv,

(2.33)

where n is integer and v real. This set constitutes a unitary irreducible representation of SL(2,C) [25]. The Mellin transform of n(x_,_x', f') with respect to f" is defined by

n(x,x'_ , Y) = f 2i----~,O?n,o(x_,x_').

(2.34)

In this double Mellin space (one for longitudinal degrees of freedom, one for transverse), the dipole distribution is diagonal. One gets (see Eq. (2.65) in Ref. [16])

n(x, x', ~') =

+co +?dr Z

ix I {x*x')"12 x'-2iv (2;Ncx(n,v)~,) \xx'*J x exp

2~ Ix'l

11:--0000

(2.35) where

x(n'v)=+(1)--½gt(In]-~l + i v ) - l + (

Inl+12

iv)

)

+ iv . The

3/*7* total

(2.36)

cross section then reads

o-z,.~,.(Q1,Q2;Y)=

d2x__ldzl d2x2dz2~(X__l,Zl;Q~)~(x_.2, z2;Q2) 2 X

+oo

+oo

7 r2a s ~ o o L d v ~(

(n-l] 1+iil)n

_

,,_-

\xlx2*)

+

Ixi

(n+i] ) +

"

(2.37)

The elementary dipole-dipole cross section o'Do can be expressed as (see t e e [16] ) O-DD(XI,X2)

+oo+T =4as21x'llx2l ~ J d~ 16

"=-°°-oo

(XlX~

n/2 Xl --2iv

× t,x;x2)

x7

1 ÷ (-1)"

(v2_i_(E.~)2) (v2_i_(.q_~) 2) (2.38)

Defining -.-oo

+oo

n(xl, Xz) =

~ n{,,#} (xz,x2) exp t/m--OOlOo

(2.39)

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549

and

+oo +oo O'DD(XI'X--2)---- Z ~O-DD{n'v} (XI'x2)' nm--(X~ C¢3

(2.40)

Eq. (2.37) can be written as

O'T*,y*(Q;,Q2;Y)----/

d2Xl

dz,/d2x__2dz2(Z)(X_l,Zl;Q21) (i~(_x.2,z2;Q 2)

+oo +oo X S

-2-~°-nn{na'}(x--l'x-2) n{n,~} ( x l ' x - l )

: i .2._,.z, q-oo -too

Y~ f ~---~P~°nD{n,~}(x"x2)exp(2°~;NCx(n'~')f') " (2.41) In the case where angular-averaged cross sections are considered, one can make n = 0 in the previous expression. Setting y = ½ + iu in order to write down expressions in term of the anomalous dimension, one gets for the averaged elementary cross section

-boo 0-DD (Xl , X_.2)---- ~-~O'DD{0,v}(Xl, X2) --oo

'/½+i~o

O[s 2

dy

1 1 - 3,) 2,

(xl)r(x22)l-r

(2.42)

--ioo

and thus

OrT*y'(Q~,Q2;[) = f d2x_.,dz, f d2x__2dz2q~(x__l,Zl;Q;)rP(x2,z2; Q2) x ~(x2)~'(x2 ;) l-~ y: (i--_-y) 2 exp 4q'r3OLs 2

l fQ~) ~'

x e x p ( 2 ~ N c x ( y ) f ") with

X(7) = x(O, i( 1 - 7)).

t~2Q2°-

k¢l 2 r*~,*

(2.43)

Using Eqs. (2.27) and (2.17) one immediately obtains

(Q~, Q22; yl, y2) = 32ote2m~"

e

dy ho~p(y)

2i~

Y

ho~.( 1 - 7 ) 1- Y

550

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

exp 2cesNcx(y)Y'Tr

\~22 j

(2.44)

We have made the approximation Y = Y', neglecting the rapidity taken by the quarks (see Ref. [26] for an interesting discussion of this effect). Note that the integrand is symmetrical with respect to 3 / ~ 1 - y. Moreover one can easily check that h ( y ) / y = h( 1 - Y ) / ( 1 - 3') for each of the two polarizations. Defining the flux factors tl -

l+(l-yl) 2

2 ,

1, = 1 - Yl ,

(2.45)

t2 =

1 + (1 - y2) 2 2 ,

12 = 1 - Y2,

(2.46)

the contribution of this y'y* subprocess to the e+e - total cross section is

do-~, e- (Q~, Q2; yl, y2) = (ere m ]2 (~--.~, e~'] 2 dQ 2 dQ2 dyl dy2

\-~-J

\'-7

/

--~r.~.(Q~,Q~;Y) 02 a2 yl y2

4 32Ce4m dQ~ dQ 2 dy, dy2

9

~

Q2 Q~ yl y2

x [/2-~_

+t2

dy ~

~_

ll

hL(y) + t, T

e-~ \Q22J

exp

3/ "77" X(Y)Y'

(2.47)

where Y, yl, and Y2 are related by formula (2.5). This result agrees with that of Ref. [5,6]. In the kinematical domain where Ql and Q2 are of the same order, the y integration can be performed by a saddle point approximation, the saddle being located very close to 1/2 (on the left if Qx > Q2, on the right if Q1 < Q2). Finally we get d o ' ~ e - (Q~, Q22; yl, y2)

= _4 {ff2m'~ 2 2 w2x/~ dQ 2 dQ 2 dyl dy2 9 \ 16 J x exp

c~s

a~

Q~ yl

1

exp[ (4cesNc/~)Yln 2]

y2 01 02

ln2(Q21/Q2 ) [2h + 9 h ] ( 56a~Nc/7"r) Y( ( 3 )

[2/2+9t2] ,

(2.48)

where we have neglected the dependence of hr, L (ys)/Ys with respect to Q l/Q2, setting 3' = 1/2. This formula agrees with the one calculated in [5] where the original BFKL technique (covariant quantization) was used. This equivalence between the original BFKL approach and the dipole approach for 3'* -3'* scattering is summarized by Fig. 3. Formula (2.48) will be used in the following to obtain the BFKL cross sections after integration over the kinematical variables.

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

551

Fig. 3. Equivalence, at leading order, between covariant quantization and light-cone quantization.

2.2. Double leading log and two-gluon cross sections Let us compare this cross section with the cross section obtained in the double leading log approximation of the DGLAP cross section, valid for Q1/Q2 far from 1. It corresponds to replacing X(Y) and hr.L by their dominant singularity at y = 0, corresponding to the collinear singularity respectively of the BFKL kernel and of impact factors, the last one reducing then to the usual coefficient functions of the operator product expansion. The dominant singularities of hr,L when y --~ 0 are given by

l)

In the double logarithmic approximation, one sums up terms of the type

Z(ozslnQ~/Q~Y)P, p)O

neglecting terms with higher powers in as of the type o~p in(P-,) QI/Q2 2 2 Yn p>/n>~O

which would correspond to next to leading order in Q2. Thus, we only keep here the contribution corresponding to the exchange of two transversally polarized photons, since the longitudinal contribution (as well as the constant term in the expansion of hT (see Eq. (2.58)) is less singular in y space, which leads to a decrease of the power in In Q12/Q~ (by 1 for one longitudinal photon, by 2 for two longitudinal photons). Taking into account these terms could be done consistently when including NLQ 2 (if one includes one longitudinal photon) and NNLQ 2 (if one includes two longitudinal photons). This will be discussed in a forthcoming paper. Thus, this yields d

4 /aern'~ 2 dQ~ dQ 2 dy, dy2 (QZ, QZ;y) t ~ , , Q~; yl, y2) =-~ I "rr .' Q2 Q2 yl y20".y.~,.

DGLAP-DLL./~,2

°'e,e

'28 m d012 --

97r

dy, dy2 ;

012 Q22 yl y2

J

ZtTT

t 29~2"} --s '4

552

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

1 (Q12~" × Q---~ \~22 j

c~,Ncy, exp Try

(2.50)

A saddle point approximation gives for the y integration d DGLAP-DLLasymp/ t32 Q2; yl, y2) tJel ek~51 ~

-

(

80QmOls "~ 9~2 j

dQ 2 dQ 2 dy, dy2 1 ~

Q~ Q2 y, y2 Q~

xexp [2~/~s-~NCyln(Q21/Q2)] (in(Q~/Q~))5/4 × tlt2

(2.51)

(_~y)7/4

the saddle-point being located at Ys = V/OrsNc/,n-Y In ( Q12/ Q22). This asymptotic formula requires Ys to be very small. In fact this region is far from being reached experimentally, and one faces the same problem as in DIS (see Refs. [ 15,27] for discussion of the DIS case). In the experimental regime which can be reached by LEP and LC, the correct way is to write down an expansion of the exponential part of formula (2.50) in terms of Bessel functions, exp yln

+ 2/

~'V

Y = Jo(z) + ~ k=l

+

Jk(z),

(2.52)

where

. [ c=t,t.

asNcY

,7XY772 ' V 7r ln(Q1/Q2)

(2.53)

i

This allows us to separate the integral in y from the integrals over the kinematical variables. Formula (2.50) then reads 2 128CeemCrs dQ~ dQ~ dyl dy2 f d.Ytlt2 1 1 42 dcre, ~- (QZ, Q2;y,,y2)_ 81~3 QI2 Q~ yl y2 z:t,n- y4Q12

x Jo(z) +~--1

+

Jk(Z) •

(2.55)

Closing the contour of the 9' integration to the left and neglecting higher twist contributions arising from the remaining integration from Yo - icx~ to Y0 + ioo (with Y0 < 0), and using the Cauchy theorem, one gets 2 128Crem°ls dQ12dQ2dyl dY2tlt2__j3(z)(_c)-3 do-e, e- (Q 2,Q2;yl,Y2)81q.r3 a2 O2 yl y2 12

_(16ae2mOe ' 2 1 \ j 2-Q12

aQ aQ aylay2tlt2 I3 y, y2

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

553

× (2,/asNcYlnQ-~2) {ln(Q2/Q22)'~ 3/2 \ V rr Q2,] k ~¢~ Y /

(2.56)

We will use this expression in the following to evaluate the DGLAP double leading log cross section. It is also useful to compare these results with the cross section corresponding to the exchange of one pair of gluons (see Eq. (2.25)) 2 2. d°-e+e- (QI, Q2, yl, Y2) = ( aem ] 2 dQ21dQ~ ,.9 dyl °'7",* ( Q~, Q2; y) \ "rr / ~T ~2 Yl Y2 128O~e4mdQ~ dQ 2 dyl dy2 d3/ I hL(3/) 9~ Ql2 Q2 y, y 2 . , ~ L(1--Yi) 3/

/ x [(l--y2)hL(l--3/) l+(l--y2)2hT(l--3/)] +

1 - 3/

2

fCy-

lq-(l~yl)2hT~)] 1 (Q~)r (2.57) Q-~ \ ~ J "

Using the expansion of hr and hL around 3/= 0, 9'

-3~-

y-~+6--yy+

~

hL(3/) a s [ 1 1 (34 3/ - 3¢r - 3 + -

3 6 J y+0(3/2)

+\27 62 )

3/+ 0(3/2)

'

]

(2.58)

(2.59)

and the crossing symmetry

hzL(1 -- T) 1-Y

h~L(y) Y

(2.60)

together with the Cauchy theorem, one gets

dQ 2 dQ 2 dyl dy2 64(a2mO~s)2 1 do_ere_(Q~,QZ;yl,y2) : Q~ Q2 y, y2 243¢r3 Q~ × t't21n3 Q--~2+ (7t,t2 + 3t112 + 3tfl,) In2 Q---~ -- 277-2 tlt2 + 5(t112 + till) + 61112 In Q-~

+

+

9

)

1

7r2 tlt2+(46-27"r2)(G12+t211) -41112 . (2.61)

Note that the previous formula was already obtained in Ref. [6] in the transverse case. The two-gluon cross section is an exact calculation in the high energy approximation and contains terms up to the NNNLO. The leading order part of the two-gluon cross 2 2 section consists in taking only the In3 Q1/Q2 t e r m into account.

554

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

3. Numerical results in the leading log approximation In this section, results based on the calculations developed above will be given for LEP (190 GeV centre-of-mass energy) and a future linear collider (500-1000 GeV centre-of-mass energy), y'y'interactions are selected at e+e - colliders by detecting the scattered electrons, which leave the beampipe, in forward calorimeters. Presently at LEP these detectors can measure electrons with an angle 0tag down to approximately 30 mrad. For the LC it has been argued [5] that angles as low as 20 mrad should be reached. Presently [29] angles down to 40 mrad are foreseen to be instrumented for a generic detector at the LC. Apart from the angle the minimum energy Etag for a detectable (tagged) electron is important, which is generally dictated by the background conditions at the experiment. The pair production background at the LC will make it difficult to measure single electrons with an energy below 50 GeV. At LEP electrons down to about half of the beam energy can be measured. The energy of the photons Er determine the hadronic energy of the collision W~, = 4E~lE~,2, which should be as large as possible for the test of BFKL dynamics. In particular the energy dependence of the cross section is of interest. The virtuality Q2 of the photon is related to the energy and angle of the scattered electron as Q2 = 4EbEtag sin2(0tag/2), with Eb the beam energy. After having specified a region of validity for our calculations, we will give the accessible integrated cross section as a function of the detector acceptance, in terms of the energy and angular range of the tagged leptons. As a starting point we will assume detection down to 30 GeV and 33 mrad at LEP, and 50 GeV and 40 mrad at the LC.

3.1. Kinematical constraints Let us first specify the region of validity for the parameters controlling the basic assumptions made in the previous section. The main constraints are required by the validity of the perturbative calculations. The "perturbative" constraints are imposed by considering only photon virtualities Q~, Q22 high enough so that the scale /1,2 in a s is greater than 3 GeV 2. /z 2 is defined using the Brodsky-Lepage-Mackenzie (BLM) 2 2 scheme [ 2 8 ] , / z 2 = e x p ( - ~ ) ~ [6]. In this case O~s remains always small enough such that the perturbative calculation is valid. In order for gluon contributions to dominate the QED one, Y (see Eq. (2.5)) is required to stay larger than ln(K) with K = 100. (see Ref. [6] for discussion). Furthermore, in order to suppress DGLAP evolution, 2 2 while maintaining BFKL evolution we will constrain 0.5 < Q1/Q2 < 2 for all nominal calculations.

3.2. BFKL and DGLAP differential cross sections In this section, we will consider the DGLAP and BFKL differential cross sections in Yl, y2, Q~, and Q~. It is often assumed [5] that the Born cross section (the exchange of one pair of gluons) is comparable in magnitude to the DGLAP prediction, since we

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564 g lO-e _ _ BFKL i n t e g . . . . . . . . . . :u ..... 10-9 . . . . . BFKL HO

555

-11 10 DGLAP i n t e 9 . 2 gluons integ.

io-'2

o10 Io

10~u

s~ ,''-

/~//;is

......................................................

10-12

,,g;

..............v~8.5

• 1o 8

!

.....

2 g l u a n s LO

......... DGLAP integ. __

DGLAP saddle

Y=85

f

,,.-......... _ _ BFKL i n t e g . . . . . . . . . . DGIJ~P i n t e g .

10-9 . . . . . BFKL HO

.....

2 gluons integ

,o I 10

lO ~o

13

lO~11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I°'2

.....

2 gluar's

.....

2 g l u a r l s LO

........

DGLAP integ.

__

DGLAP saddle

............................. Y = 6 .

nleg.

Y-&

10-" --BFKL ~O-9

;nteg ..........

. . . . . BFKL HO

.....

DGLAP i n t e g . 2 gluons integ.

10 -13 lo

lO-I0

16151

lo le

o

........................ v ~ : ~

2

....

3 Loq ( o , ' / o ~ ~)

1

2

3

n

log ( 0 , ~ / 0 , ')

Fig. 4. Differential cross sections vs. In Q~/Q2, for different values of Y. Exact values are shown as well as saddle-point approximations. The dashed vertical line on the left-hand side is the value Q21/Q~ = 2. (On the left: full line: BFKL integral calculation, dashed line: BFKL-HO, dotted line: DGLAP-DLL integral calculation, dashed-dotted line: two-gluon exact cross section. On the right: full line: DGLAP-DLL saddle point approximation, dotted line: DGLAP-DLL integral calculation, dashed line: LO two-gluon cross section, dashed-dotted line: two-gluon exact cross section). generally select regions where Q2/QZ (_9(1) in order to observe a large BFKL over l/ 2 = D G L A P cross-section ratio. In this domain the DGLAP prediction is expected to be low, as the kr ordering required by the DGLAP evolution equation will force the DGLAP cross section to vanish if Q1 ~ Q2. Fig. 4 shows the differential cross sections in the BFKL, DGLAP double leading logarithm ( D L L ) and two-gluon approximation, as a function of In QZ/Q2 and for three values of Y. The cross sections on the left-hand side are calculated using the unintegrated exact formulae ( 2 . 4 7 ) , ( 2 . 5 6 ) , and (2.61) for respectively the BFKL, DGLAP (in the double leading log approximation) and two-gluon exchange cross sections. Also the phenomenological HO-BFKL cross sections, as detailed in Section 4, are given. We note that the two-gluon cross section is almost always dominating the D G L A P one in the double leading log approximation. The saddle point approximation turns out to be a very good approximation for the BFKL cross section and is not displayed in the figure (saddle-point results are close to the exact calculation up to 5% in the high Y region, and up to 10% at lower Y. A similar conclusion was reached in [5] ). We note that the difference between the BFKL and two-gluon cross sections increase with Y. On the right side of Fig. 4, curves for the exact LO and saddle-point (Eq. ( 2 . 5 1 ) ) DGLAP calculations are shown, as well as the full N N N L O (Eq. ( 2 . 6 1 ) ) result and the

556

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

LO (Eq. (2.61), In3 Qi/Q2 2 2 term only) result for the two-gluon cross section. Unlike for the BFKL calculation, for the DGLAP case the saddle-point approximation appears to be in worse agreement with the exact calculation, and overestimates the cross section by one order of magnitude, which is due to the fact that we are far away from the asymptotic regime. The comparison between the DGLAP-DLL and the two-gluon cross section in the LO approximation shows that both cross sections are similar when Q1 and Q2 are not too different (the dashed line describes the value QI/Q2 2 2 = 2), so precisely in the kinematical domain where the BFKL cross section is expected to dominate. However, when Q2/Q2 1 / 2 is further away from one, the LO two-gluon cross section is lower than the DGLAP one, especially at large Y. This suggests that the two-gluon cross section could be a good approximation of the DGLAP one if both are calculated at NNNLO and restricted to the region where Q2/Q2 1 / 2 is close to one. In this paper we will use the exact NNNLO two-gluon cross section in the following to evaluate the effect of the non-BFKL background, since the two-gluon term appears to constitute the dominant 2 2 part of the DGLAP cross section in the region 0.5 < Q1/Q2 < 2.

3.3. Integrated cross sections In this section, we will study the integrated cross section over the four kinematical variables Yl, Y2, Q2, and Q22, for the exact two-gluon calculation and the saddle point approximation for the BFKL one. First we study the effect of the choice of parameters to define the perturbative region for our calculations: Table 1 shows the effect of varying the cut on #2. At the LC, no effect is seen: scattering the incoming leptons above 40 mrad requires high photon virtualities so that the selected region is always in the perturbative domain. Table 2 contains the BFKL and two-gluon cross sections for different values of K. The BFKL to two-gluon ratio is enhanced at high K. Table 3 contains the BFKL and two-gluon cross sections for different values of the range in ratio QZ/Q2 1 / 2" The BFKL to two-gluon ratio is rather insensitive to this restriction. The cut on 1/2 ~< Q2/Q2 1/ 2 ~ 2 also guarantees that the DGLAP contribution can be well approximated by the two gluon contribution. We note that for the parameter choice in this paper the maximum ratio between the two-gluon and BFKL cross sections is about 20 for the nominal energy and angle cuts at the LC and 40 at LEE Next we study the effect of the tagged electron energy and angle. Fig. 5 shows the importance of tagging electrons down to low energies (see also Table 4). Reaching 10% of the beam energy or less allows to enhance the counting rates significantly; the difference between the BFKL and two-gluon predictions also increases, improving the detectability of BFKL dynamics. The effect of increasing the LC detector acceptance for electrons scattered at small angles is illustrated in Fig. 6 and in Table 5. The plateau seen at low angles results from the kinematical constraints (see Section 3.1). The results found for the BFKL and two-gluon cross sections are in good agreement with Refs. [5,6] when the same kinematical cuts are made. The two-gluon cross section is smaller than the [6] results and we checked explicitly that this is due to different

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

557

Table 1 Variation of the cut o n / . , 2 ( G e V 2 ) , for LEP and the LC. The detector acceptance is taken into account ]./2

o.LEP BFKL

o-LEP 2g

Ratio

LC O'BFKL

o'L~C

Ratio

2 3 4

2.89 0.57 0.18

3.78x 10 - 2 1.35 × 10 - 2 6.14x 10 - 3

76.5 42.2 29.3

6.2x 10 - 2 6.2× 10 - 2 6.2x10 - 2

2.64x 10 - 3 2.64x 10 - 3 2.64x 10 - 3

23.5 23.5 23.5

Table 2 Variation of the cut on x, for LEP and the LC. The detector acceptance is taken into account K

_LEP OBFKL

o-LEP 2g

Ratio

o-LC BFKL

o-LC 2g

Ratio

10 50 I00

1.23 0.81 0.57

9.02X 10 - 2 2.81×I0 -2 1.35X 10 - 2

13.6 28.8 42.2

8.8X 10 - 2 7.2×10 - 2 6.2X 10 - 2

9.63×10 - 3 4.17× 10 - 3 2.64X 10 - 3

9.1 17.3 23.5

Table 3 Integrated cross sections (pb) for different ranges of Q~/Q2, at LEP and LC energies. Electrons are detected between 30 and 95 GeV, down to 33 mrad at LEE and between 50 and 250 GeV, down to 40 mrad at the LC

Q~/Q2

trLEP BFKL

_LEP ~'2g

ratio

LC °'BFKL

LC Or2g

Ratio

0.5-2 0.1-10 0.Ol-100

0.57 1.71 2.00

1.35X l0 - 2 3.94X l0 - 2 4.59× 10 - 2

42.2 43.4 43.6

6.2× 10 - 2 0.123 0.128

2.64× l0 - 3 5.65X l0 - 3 6.03X l0 - 3

23.5 21.8 21.2

Table 4 Integrated BFKL and two-gluon cross sections at the LC for different lower cuts on the electron tagging angle 8ta~, for the kinematic range defined in the text (tagging energy between 50 and 250 GeV)

kinematical

0tag

BFKL

Two-gluon

Ratio

10 15 20 25 30 35 40

6.7 6.6 3.3 1.1 0.37 0.14 6.18x 10 - 2

8.1 x 7.9x 4.0x 1.7x 8.4x 4.5× 2.6×

82.7 83.5 82.5 64.7 44.0 31.1 23.8

c u t s in p a r t i c u l a r o n / z 2 ( s e e S e c t i o n 3 . 1 ) .

F i n a l l y , in T a b l e 6 w e g i v e t h e m e a s u r a b l e beam

10 - 2 10 - 2 10 - 2 10 - 2 10 - 3 10 - 3 10 - 3

energy. Although

cross sections for different values of the

the total cross sections

i n c r e a s e w i t h Ebeam, t h e o p p o s i t e

is

o b s e r v e d a f t e r t a k i n g i n t o a c c o u n t t h e d e t e c t o r a c c e p t a n c e a n d t h e f a c t t h a t at c o n s t a n t photon virtuality, the scattered electron aligns more and more with the beam direction w h e n Ebeam i n c r e a s e s . Assuming

an integrated luminosity of 200 pb -t

per experiment

at L E P w i t h c e n t r e -

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

558

- -

BFKL

........

2

¢ .t

gluon

- -

B~KL

.......

2

glJOn

,~ 1 0 '

~o"

LC

LEP I0'

...........................................................

-*4 5 ~ =

~@

....

i ....

i

,,,i

,

"~ 40

35

~s,;,

,~,

:'o ....

~'o ' ' '.~;'

~', ....

E~,

~o

I~

'~', ' '3'0' ";~"

,I,,, 40

.... 45

(GeV)

,I,,,,r,,, 50

55

60

Etat (GeV)

Fig. 5. Integrated BFKL and two-gluon cross sections, at LEP and LC as a function of the tagging energy• Leptons are tagged from Etag up to the beam energy. We take /:)tag > 33 mrad at LEP, 0tag 3> 40 mrad at the LC. At the bottom of the figure the ratio between the BFKL and two-gluon cross sections is also shown.

¢

~ 10 -I

- -

BFKL

........

BFKL HO

'"'"

2-gluon

ri::::::::::::::::::::::::::::::::::::::::: ....................

10 -2

LC

10 -3

_~o

~oo

....

I ....

........................................ ::.:::::::: I , i l h l l l l l l l l l , l l l l • l

' ' ' ' l ' ' '

F

60

~3o 20 10

15

20

25

30

35"

40

01.,(m~d)

Fig. 6. Integrated BFKL-LO, BFKL-HO and two-gluon cross sections at the LC, as a function of the tagging angle• Leptons are tagged between 50 and 250 GeV. At the bottom of the figure is plotted the ratio between the BFKL-LO and the two-gluon cross sections.

of-mass energies around 190 GeV, BFKL predicts roughly 200 events provided forward leptons are tagged down to 10 GeV, the two-gluon prediction being 42 times lower. At the LC, with 50 fb -1 at v/s = 500 GeV, Etag > 20 GeV and 0tag > 40 mrad, we can expect 7500 BFKL events, compared to 24 times less for the two-gluon contribution. The final results for the cross sections are also given in Table 9.

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

559

Table 5 Integrated BFKL and two-gluon cross sections at the LC for different lower cuts on Etag for the kinematic range defined in the text

Etag

BFKL

Two-gluon

Ratio

60-250 50-250 45-250 40-250 35-250 30-250 25-250 20-250

5.1 x 10-2 6.2x 10-2 6.9x 10-2 7.7x 10-2 8.8x 10 -2 0.10 0.12 0.14

2.4x 10-3 2.6× 10-3 2.7x 10-3 2.8x 10-3 2.9x 10-3 3.1 x 10-3 3.2× 10-3 3.3 x 10-3

21.2 23.8 25.6 27.5 30.3 32.3 37.5 42.4

Table 6 Integrated cross sections (pb) for different values of Ebeam (GeV), after imposing 0tag > 40 mrad and Etag > 50 GeV Ebeam

O'BFKL

O'two_gluon

Ratio

250 500 1000

6.18X 10 -2 7.00X 10 -3 8.77X 10 -4

2.64X 10-3 5.21X 10-4 9.92X 10 -5

23.4 13.4 8.8

4. Phenomenological approach of HO effects in BFKL equation In this section w e adopt a p h e n o m e n o l o g i c a l approach to estimate the effects o f higher orders. We will g e n e r i c a l l y label these has " H O - B F K L " calculations,

4.1. Variation o f the scale f o r rapidity A t leading order, the rapidity Y is not uniquely defined. In the f o r m u l a ( 2 . 5 ) , it is possible to add a multiplicative constant ( in front o f ~. Only a N L O calculation can fix this constant. Taking the f o l l o w i n g definition o f the rapidity: Y=ln

(----------~s

(4.1)

we can study the variation o f the B F K L and D G L A P cross sections for different values o f (. T h e p a r a m e t e r ~: [26]

sets the time scale for the formation o f the interacting

dipoles. It defines the effective total rapidity interval which is l n ( 1 / v ~ X 2 ) + ln(, b e i n g not predictable (but o f order o n e ) at the leading log approximation. T h e results

are given in Table 7. We note a large d e p e n d e n c e o f the cross sections on this parameter, and also o f the ratio b e t w e e n the B F K L and t w o - g l u o n predictions which vary b e t w e e n 24 and 2.3! A p h e n o m e n o l o g i c a l way to d e t e r m i n e this factor s~ was used in Ref. [ 3 ] , w h e r e a f o u r - p a r a m e t e r fit o f the proton F2 structure function m e a s u r e d by the H1 Collab-

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

560

oration [31] was performed using the QCD dipole picture of BFKL dynamics. The parameter ( was found to be 1/3. For this particular value, we note that the BFKL to two-gluon ratio prediction is reduced to a value of 12. This result is in fairly good agreement with what has been found in Ref. [6] where the authors vary the ( parameter by a factor 4.

4.2. Effective value for as It has recently been demonstrated that the NLO corrections to the BFKL equation are large [32]. The main effect is a reduced value of the so-called Lipatov exponent in formula (2.48) [30]. A phenomenological way to approach this is to introduce an effective value of the coupling constant which allows to reduce the value of the Lipatov exponent. In the same four-parameter fit described above, used to fit inclusive and diffractive data at HERA, as described in [3,2], the value of the Lipatov exponent ap

asNc

ap = 4 In 2 - -

77"

(4.2)

was fitted and found to be 0.282. In this fit, as was kept constant. This low value of a; leads to a low value of as close to 0.11. This low value can be explained phenomenologically by the decrease of the Lipatov exponent due to large NLO corrections. The same idea can be applied phenomenologically for the y'y* cross section. We first studied the variation of the BFKL cross section by setting the scale /.~2 i n as in the exponential of formula (2.48) to a constant number and consequently taking as fixed. The values of as and of the BFKL cross sections are given in Table 8. The cross section is calculated for/z 2 = 10, 100, 1000, and 10000 GeV 2 (note that for this study 2 2 is suppressed in the expression of/z2). The decrease of the BFKL the term v/QIQ2 cross section is quite significant. The last effect studied was to use a varying as, and at the same time taking into account the HO effects described above. For this purpose, we modify the scale in as so that the effective value of as for Q~ = Q~ = 25 GeV 2 is about as(Mz). The scale /x2 for as in the exponential is then expressed as follows: /,2 = ( ~ .

(4.3)

The variation of the BFKL cross section as a function of sr are given in Table 8 for the LC. Finally, the results of the BFKL and two-gluon cross sections are given in Table 7 if we assume both r = 1000, and ( = 1/3 (see Section 4.1). The sr value corresponds to a value of as = 0.11 for /1, 2 = 10 GeV 2. The ratio BFKL to two-gluon cross sections is reduced to 2.3 if both effects are taken into account together. 4 4 After having submitted our paper, we noticed that Ref. [33] gives an evaluation of the BFKL pomeron intercept at NLO using the BLM scheme. This scheme reduces considerably the Q2 dependence of the

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

561

In Table 9, we also give these effects for LEP with the nominal selection and at the LC with a detector with increased angular acceptance. The ratio given is the comparison of the H O - B F K L and two-gluon cross section. In both cases the sensitivity to BFKL effects is increased. The effect on the cross section from the angular cut for the LC is shown in Fig. 6. The column labelled 'LEP*' gives the results for the kinematic cuts used by the L3 Collaboration who have recently presented preliminary results [ 35 ]. The cuts are Etag = 30 GeV and 0tag > 30 mrad a n d / z z > 2 GeV 2. For this selected region the difference between HO-BFKL and two-gluon cross section is only a factor of 2.4. A cut on Q2/Q2, l / 2 as done for the other calculations in this paper, would help to allow a more precise determination of the two-gluon 'background'. In Fig. 4 the differential cross section calculated with the BFKL-HO parameters sr = 10000, and ( = 1/3 is shown. In Fig. 6, the result of the HO calculation as a function of 0tag is also displayed for the same values of ( and (. An important observation is that the difference between the HO calculation and the LO BFKL calculation in Fig. 4 increases significantly with increasing Y. Hence to establish the BFKL effects in data, a study of the energy or Y dependance, rather than the comparison with total cross sections itself, will be crucial. To illustrate this point, we calculated the BFKL-HO and the two-gluon cross sections, as well as their ratio, for given cuts on rapidity Y (see Table 10). We note that we can reach up to a factor 5 of difference ( Y ) 8.5) keeping a cross section measurable at LC. The cut Y ~> 9. would give a cross section hardly measurable at LC, even with the high luminosity possible at this collider. Cuts on Y will be hardly feasible at LEP because of the low cross sections obtained already without any cuts on Y. Note that an additional uncertainty in the cross-section calculations is the number of active flavours considered. For this paper, three quark flavours were considered both for BFKL and two-gluon calculations. Including charm only as an additional flavour, without taking into account any mass effect, would increase both cross sections by a factor 2.56. Including charm can be justified for the LC, but some more sophisticated approach could be necessary for LEP energies.

5. Conclusion

In this paper, we have studied the differences between the Two-gluon and BFKL and D G L A P 3"*3"*cross sections both at LEP and LC. Both the longitudinal and transverse two-gluon cross sections have been calculated, and the transverse part is found to be identical as in [6]. It turns out that the double leading logarithmic approximation of the D G L A P cross section is much lower than the two-gluon one, calculated to N N N L O (see intercept, in accordance with a recent result obtained in axiomatic quantum field theory [34]. The obtained intercept is of the order of 0.2. In our phenomenological approach where the ~r value has been extracted from the proton structure F2 measured at HERA (see above), we neglected the effect of the scale at the proton coupling. Taking for instance an effective scale to be about 1 GeV 2 for this coupling, we get sr N 105. Using this value for y ' y * total cross section then gives a similar intercept of the order of 0.2-0.25.

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

562

Table 7 Variation of the scale for rapidity (see text). The last number (referred by 1/3") takes also HO effects on as in the BFKL equation (

BFKL

Two-gluon

Ratio

I 0.1 0.01 1/3 1/3"

6.2× 1.6× 6.2x 3.1 x 6.2x

2.64× 2.64x 2.64x 2.64x 2.64x

23.5 6.1 2.3 11.7 2.3

10 -2 10 . 2 10 . 3 10 - 2 10-3

10 -3 10 -3 10 . 3 10 -3 10-3

Table 8 Variation of the scale for as (the change is made in the exponential only). In the left table are given the results for a fixed a s (the scale /~2 is given) and in the second table, a s is running with different values of the parameter ( (see text). For comparison, the two-gluon cross section is 2.64 l0 -3 pb /~2

OlS

BFKL

(

BFKL

10 100 1000 10000

0.28 0.20 0.15 0.12

8.0X 2.4× 1.3x 9.4x

e -5/3 10 100 1000

6.2x 1.3× 9.4x 6.2x

10 -2 10 . 2 10. 2 10 -3

10 -2 l0 -2 10 . 3 10 . 3

Table 9 Final cross sections (pb), for selections described in the text

LEP LEP* LC 40 mrad LC 20 mrad

BFKLLo

BFKL,q o

Two-gluon

Ratio

0.57 3.9 6.2x 10 -2 3.3

3.1 x l0 -2 0.18 6.2x 10 -3 0.11

1.35x 10 -2 6.8x 10 - 2 2.64x 10 -3 3.97x 10 - 2

2.3 2.6 2.3 2.8

Table 10 Final cross sections (pb), for selections described in the text, after different cuts on Y Y cut

BFKLHo

Two-gluon

Ratio

No cut Y /> 6. Y ~> 7. Y /> 8. Y ~ 8.5 Y/> 9.

1.1 × 10 -2 5.34x 10- 2 2.54x 10 - 2 6.65x 10 -3 1.67x10 -3 5.36x 10 -5

3.97x 10 -2 1.63x10 -2 6.58x 10-3 1.43x10 -3 3.25x10 -4 9.25x 10 -6

2.8 3.3 3.9 4.7 5.1 5.8

formula ( 2 . 6 1 ) ) . The LO BFKL cross section is much larger than the two-gluon cross section. Unfortunately, the higher order corrections of the BFKL equation (which we estimated phenomenologically, assuming results from F2 and F~9 fits made at HERA can be transported to LC) are large, and the two-gluon and BFKL-HO cross-section ratios are reduced to two to four. The Y dependence of the cross section remains a powerful tool to increase this ratio and is more sensitive to BFKL effects, even in the presence

M. Boonekamp et al./Nuclear Physics B 555 (1999) 540-564

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o f large h i g h e r o r d e r c o r r e c t i o n s . H o w e v e r , the h i g h e r o r d e r c o r r e c t i o n s to the B F K L e q u a t i o n s w e r e t r e a t e d h e r e o n l y p h e n o m e n o l o g i c a l l y , a n d w e n o t i c e d that e v e n a s m a l l c h a n g e o n the B F K L p o m e r o n i n t e r c e p t i m p l i e s big c h a n g e s o n the c r o s s sections. T h e u n c e r t a i n t y o n t h e B F K L c r o s s s e c t i o n after h i g h e r o r d e r c o r r e c t i o n s is thus q u i t e big. H e n c e t h e m e a s u r e m e n t p e r f o r m e d at L E P or at L C s h o u l d b e c o m p a r e d to the p r e c i s e c a l c u l a t i o n o f the t w o - g l u o n c r o s s section after the k i n e m a t i c a l cuts d e s c r i b e d in this paper, a n d the d i f f e r e n c e c o u l d s u g g e s t B F K L effects. A fit o f t h e s e cross s e c t i o n s will b e a w a y to d e t e r m i n e the B F K L p o m e r o n i n t e r c e p t i n c l u d i n g h i g h e r o r d e r effects.

Acknowledgements W e like to t h a n k R. P e s c h a n s k i a n d J. B a r t e l s a n d C. E w e r z for m a n y u s e f u l disc u s s i o n s . S.W. t h a n k s the A l e x a n d e r v o n H u m b o l d t F o u n d a t i o n a n d the II. I n s t i t u t f~ir T h e o r e t i s c h e P h y s i k , w h e r e the first stage o f this w o r k h a s b e e n d o n e , for support.

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