Small-χ structure functions and heavy flavour production

Nuclear Physics B (Proc. Suppl.) 18C (1990) 220-225 North-Holland

220

SMALL-x STRUCTURE FUNCTIONS AND HEAVY FLAVOUR PRODUCTION* S. CATANI+ Cavendish Laboratory, University of Cambridge, Cambridge CB3 OHE, U.K . M. CIAFALONI and F. HAUTMANN Dipartimento di Fisica, University di Firenze and Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, I-50125 Firenze, Italy The use of the kl -factorization theorem to compute high energy cross sections in resummed perturbation theory is reviewed. The results for heavy flavour photoproduction at small x are discussed in detail . We show that the resummation of the leading terms in the perturbative series gives large correction factors with respect to lowest order calculations . *Talk given by S. Catani. +On leave of absence from Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Italy. 1. INTRODUCTION In the present TeV energy range, massive particles are copiously produced in photon, lepton and hadron initiated processes . In this paper we are interested in studying cross sections for producing very massive states (M » A, A being the QCD scale) at high energy (s » M') . For instance one can consider heavy flavour photoproduction or hadroproduction, as well as Higgs bosons and W and Z particles produced in hadron collisions . In any case, we are mainly interested in massive states coupled to gluons, since these latter give the leading contribution at high energy. According to the QCD factorization theorem [1], the parton cross section C(p, M 2 /Q 2 ) for these processes is given in terms of gluon structure functions F(x, M2 /Q2) and a hard cross section contribution 4M2 Is. The gluon structure func0*hard(p)i where p tion is computed in perturbation theory by introducing a factorization scale Q2 (such as «,(Q2) < 1), acting as a lower cut-off on the transverse momentum ql. of the emitted partons . By definition it resums all the logarithmic contributions (a,(M 2) In M2/Q2)n due to

initial state collinear radiation . The hard cross section factor Qhard has no collinear singularities and is usually computed in fixed order perturbation theory. The coefficients of the perturbative expansions for the structure function and the hard cross section depend on the Bjorken variable x and the p-parameter respectively. In the kinematical regime s » 112 > A2 these coefficients are proportional to In x, In p and are therefore large [2] . It follows that these logarithmic contributions must be resummed in perturbation theory to all order in the QCD coupling constant a, in order to get reliable theoretical predictions . We stress that the resummation has to be performed for both the structure functions and the hard cross section factor. Whilst a renormalization group approach can be used for summing structure function contributions, new techniques are needed to handle the same problem in the hard cross section factor. In this paper we reviewed the method proposed in Ref. [3] . We shall show that the leading In p terms can be resummed by using a new form of the factorization theorem, which takes into account the transverse degrees of freedom . Then we shall discuss the resummed result

0920-aß32/91/$ .3.50 © lasevier Science fuhlisliers B.V. (North-liollawl)

221

S. Catani et al./Small-x structure functions for the heavy flavour photoproduction cross section. A different approach to the same problem has been discussed by J.C . Collins at this meeting [4].

(Fig. 2) and will generate, at higher orders, powers of Inp. All leading In p contributions can be explicitly evai"iated and resummed [3].

2. THE k1 -FACTORIZATION THEOREM Let us consider heavy flavour photoproduction. The parton cross section o,( °) in the Born approximation is obtained by computing the diagrams in Fig. 1. In the small-p limit the result is 4M 2Q( ° ) (P, M2) ^__ 27r a eQ aa p In 1 , (1) P where a is the fine structure constant and eQ is the heavy flavour charge . We note that Q(o) vanishes in the p --+ 0 limit. The factor p in Eq. (1) is due to the fermion exchange diagrams involved in the Born cross section, whilst the factor In 1/p is associated to the t-channel collinear singularity for s » M2 --+ 0.

Factorized structure of the heavy flavour photoproduction cross section in the high energy limit.

up,

P2

FIGURE 1 Diagrams for the photoproduction cross section in the Born approximation . The first order QCD corrections to the Born cross section is given by [5, 6] 0'ili(PI M2)

=

M2 aâ

2

ci(P) + Zl(W)log Q2 0

where the functions ci(p) and cl(p) are respectively the hard cross section contribution and the 1-loop gluon structurek function coefficient . For large energy or small p, one has the constant limits cl -> 27

CAeQ

FIGURE 2

,

cl -> 9 CAeQ

.

We see that the 1-loop cross section (2) dominates over the Born one at high energy. The constant behaviour in (3) is due to gluon exchange in the t-channel

The basic observation is that in leading logazithmic approximation, the photoproduction cross section satisfies a ki-dependent factorization formula [7], as follows (Fig. 2) 4MZQ(P,

Q2 )

o

= fd2k

fl dz

&('P,

4kM2 )

(z, k; Qô) (4)

Here F(x, k) is the unintegrated gluon structure function (defined below) and Q is the o$shell -r+g(k) -, Q+Q Born cross section, defined by coupling the external gluons to high energy partons with eikonal vertices . More precisely, if A"(pl ; k) is the -yg absorptive part at lowest order, we have 4M2 2 _ _ 4M2 PzP`~ s ' 4M 2 ) s k2 ``" ' s=2Pi»P2 1 Pi=Pi=O

'

(5)

while F(x, k) is the transverse momentum structure function, defined as the probability to have a gluon with an energy fraction x in association with an emitted total transverse momentum k. The gluon structure

S. Catani et al. /Small-x structure functions

222

function xF(x, M2 /Q02) = G(x, M2 /Q2) is obtained by integrating F(x, k) over the transverse momentum G(x,

M2/Q2)

introducing the N-moments 1 /Q2 ) _ f dP PN-1 U(P, M2 /Q 2 ) 1 dx xN-1 jr(x, k ; Qô) YN(k ; Qô) _ J0

o,N(M2

xF(x, M 2 /Q 2 ) 0

d2k

Y(x, k ;

The factorization structure in Eq. (4) is k l -dependent, thus generalizing the usual renormalization group statement . Its basis is in the high-energy, or Reggepole, factorization for exchanged gluons . In fact, if both sIM 2 and M 2 are large, the single gluon exchange diagram of Fig. 2 is dominant, in a physical gauge, for any fixed k 2 ^_- -k 2 . High energy factorization then implies that the gluon structure function factors out as function of k, and that the Born factor & is defined as a physical cross section for the high-energy process. Therefore Eq. (4) holds, and & is gauge invariant and calculable (at leading s level) despite its off-shell k1 dependence. All features of the large In p perturbative terms follow from Eq. (4). Note first that since & itself is small for p -+ 0 (& ti p log 1/p), the constant or logarithmic terms mentioned before are generated by the singuk lar z integration over the gluon structure function .F' . Furthermore, the small k1 behaviour of the structure function (T ti (k`) -1 for A « k1 « M) is responsible for the anomalous dimension terms, while large k 1 's are cut-off by & at k 1 - M and contribute to the hard cross section factor. Finally, since the small z behaviour of F is given by the Lipatov equation [8] .T(x,k) = 6(x - 1)5(2)(k) C 2~ dz [( x + ~ 7r Z,

I

Qô)

and studying the N -> 0 limit of UN(M2 /Q2) . Since Eq. (4) is a convolution in p-space with k_i-factorization, its N-moments can be written as (a, = CAa,/7r) 4M2 eN(02 ) = d2k Fiv(k ; Q0) J &N( 4M2 0 î+c , dY () M2 1 a f(~) -1 1 2^ra -,ri Q20 N '

1 FN(k; Q0 _ ; k2

The detailed analysis of the high energy limit (p -4 0) of the factorization formula (4) can be performed by

2 +_~

- 'OC 2i ( Q20

)-f

l1

N

9)

f(~)

f(7) = 23b(1) - 0(-y) - 0(1 - -y) ,

(10)

being the well-known [8] Lipatov's anomalous dimension function given in terms of Euler Vy-functions. The hard cross section & enters Eq. (9) through the weighted k l -average h N , defined by f(-y)

1 - hiv( 17

°° dk 2 _ k2 k2 (M2)7

Y) _ T

2

&N (4M2)

(11)

By explicitly computing &(p,k 2 /4M 2 ) in Eq. (5) for heavy flavour photoproduction, we find [3] that it is vanishingly small for p -> 0 and any k1 (small or large, i.e. ki ti M2/P) &(p, k 2 /4M 2 )

3. THE CROSS SECTION FOR HEAVY FLAVOUR PHOTOPRODUCTION

)

where we have expressed the solution .FN(k) of Eq. (7) in the form

2

( 7) and hence it is known to all orders, Eq. (4) can be used to sum all leading logs in the heavy flavour cross section .

(8)

^_" 27r

a eQ a, p In 1 P

1 _

Pk2

4M2 )

2

+

( pk2 )21

(12)

4M2

It follows that &N(k 2 /4M2 ) and hN(7) are both weakly N-dependent, i.e. hN = h(,y)[1+O(N)], and we can set N = 0 in Eq. (11) . The complete calculation of h(^y) gives [3] h( -y )

= 3 a a a eQ B(1- -y ,1- ,y ) 3 2~

where ß is Eulcr's beta function .

B(1 +-y, 1 --y) ,

(13)

S. Catani et al./Small-x structure functions The function h(,y) is analytic in the physical range 0 < y < 1 (Fig. 3). Note the finite value of h(0), corresponding to a simple pole in Eq. (11). It is due to the small k l region and given by the N = 0 moment of the Born cross section &(0) (p). Note also that h(-y) quickly increases as ,y -+ 1 and has a double pole singularity for -y = 1. This behaviour is due to the region of large k1 » M where, from Eq. (12), we have &N(k2/4M2)

13

a e4 a, k2 In M2 .

-y(a,/N) yN

We obtain

h(y)

h i- loop (7)

h(0)

GN(M2

/Qô)

= 10 =

dPk FN(k; Qô)

(M2)'Ym ~ N71 J Q2

f'(~yN)E]-i

(18)

28 =9 7raa.eQ ,

19

+ 21-Y) , (20) d according to 1-loop improoved renormalization group calculations [5, 6] .

FIGURE 3 Resummed (h), 1-loop (hi-loop) and Born (hBorn) hard cross section functions . 4. THE HIGH ENERGY BEHAVIOUR Let us now discuss the result of Eq. (9) in various regimes, on the basis of the explicit form (13) of h(-y) . The perturbati e regime is defined by N « 1 in moment space or by â, log .! « log in energy space. In Qô this region, the -y-integral can be performed by going to the pole 1 (15) f('Y) , which implicitly defines the anomalous dimension

(17)

Eq. (17) is the reummed result for the Ie g (aa/N)" contributions to the heavy flavour photoproduction cross section. In standard perturbative QCD calculations, the Reummed expression h(-y) for the hard cross section factor would be replaced in Eq. (17) by a a, fixed-order contribution . For instance, in the Born approximation one would have a constant factor

whilst

hBom ('Y)

=N

(16)

where GN are the gluon structure function N-moments

hBo~('Y) = h(O)

h(2)

.

4M2 QN(M2 /Qô) = GN(M 2 /QÔ) h(-yN)

(14)

The factor M2 /k 2 in Eq. (14) is due to the fermion exchange diagrams involved in the hard cross section (5) and the logarithmic enhancement Ink 2/M2 is related to the t-channel collinear singularity for M2 -., 0.

= N + 2((3)( N )4 + - - -

YN :--

hi-loop(-y) = h(0) (1

If &,/N is too large (Nlâ, < f(-ymi,,, = 1/2) = 4 In2) or p is too small (a. In 1/p > InM2 /Qû) the perturbative anomalous dimension (16) is no longer meaningful, the solutions of Eq. (15) being complex. In this case, it is more convenient to discuss directly the energy dependence of the cross section, which is obtained from Eq. (9) by inverse Mellin transform. By performing the N-integration first at the pole in Eq. (15), we obtain, for s » 4M2, 4M2 0,(P,

+i" d'y â,f(7) M2 ti h(-y) 2 ) fi -=o0 21ri Q0 7 (M2)'y (( .9 & .f(7)

Q2

(21)

4M2

The -y -integral is then evaluated by a saddle point method. The stationary phase condition is +I=0,

V=log

2

Qâ

, L=log

), (22)

S. Catani et al./Small-x structure functions

224 and yields

4M2 a(p, M 2 1Q0) = G(p, M2 1Q0) h(j) ,

(23)

where h(y) is the hard cross section factor and the gluon structure function is G(p, M2/Qô)

ti

Y

2

~aft7~

Q2 (4M2 0 6-f((r )

(24)

y 27ra,L f"(1) At intermediate energies, such that â,L « 1, the stationary point y ^_- (a,L/0112 , and we obtain 2 G(p , M /Qô) ~

~(âe

v

41 h(y) = 1 + 21

1 1/4 L_°/4exp(2 d,L1) ) , a, f

(a)

~, e

+O

(25) The gluon structure function (25a) has the exponential behaviour [9] obtained by solving the ÀltarelliParisi equation, or the Lipatov equation in the one-loop regime (f(-y) 1 /-y). h(y) takes a perturbative form so that the resummed result (25b) gives small corrections with respect to calculations at fixed order in a, . ^_r

In the opposite regime of extreme energies (ix,L » ß), the saddle point drifts towards the minimum of f(-y) at y = 1/2, so that the gluon structure function has a power dependence on s as given by the perturbative QCD pomeron [8, 9]

r

M2 86. In 2 s l 4&, ln 2 VQ2 [567rC(3)ix,L] 1 / 2 \4M2 (26) and the total cross section is G(p, M a

l Q2)

_

4M2 0. (p, M2 /Q2) = G(p, M2 /Q2) h(1 /2),

(27)

where, by Eq. (13), h(2)

3

= 2 aa,e2

.

(28)

5. DISCUSSION AND CONCLUSIONS We are now in a position to discuss the size of the large (a, In p)n corrections resurnmed in the asymptotic

formulas (26) and (27). The value h(1/2) on the righthand side of Eq. (27) is the result of the resummation for the hard cross section contributions . In lowest order perturbative calculations it is replaced by hBom(1/2) or hi _ loop (1/2) with the Born and 1-loop hard cross section function respectively given in Eqs . (19) and (20). We can see that the resummed value (28) is approximativly a factor of five larger than the Born cross section factor h(1/2) _ 27 2 (29) hBpm(1/2) 56 and still a factor of 2.5 larger than the corresponding 1-loop value h1_ioop(1/2)

81 2 7r 332 '

(30)

leading to a corresponding increase in the photoproduction total cross section (27) at very high energies . Obviously, the result (27) is valid only in the asympotic regime. Nevertheless a value of the resummed cross section larger than that predicted by lowest order calculations is expected even for lower energies . This trend can be argued by comparing the resummed, 1loop and Born cross section functions h for any values of y (Fig. 3). As discussed before, the steep behaviour of h(y) in (13) is due to the physical shape (14) of &(k2 /4M2 ), which does not provide a sharp cut-off for large k1 values. Therefore features similar to those illustrated so far are expected for any production cross section involving one initial state gluon [7]. One more point we want to emphasize is that the asymptotic value in Eq. (28) is just a finite constant, showing no trace of the leading powers of Inp it originated from. This is due to the structure of the perturbative series for large s. As we can see from the leading logarithmic result in Eq. (17), one can have at most one power of Inp for every power of a,. This result follows from the smoothness of h(y) for 0 < y < 1, which in turn derives from the fact that Q(p,k2/4M2 ) in Eq. (12) is uniformly small, of order p, even close to the phase space boundary of large kl = O(Vs-) . In contrast a point-like probe, as given by the current

S. Catani et al . /Small-x structure functions J = Fa,F"" a, would yield

REFERENCES

1 &poine(P)=PS 47N,paiac

=

1 +

225

1 P-1 2

)

k1

-Q2

-N

N =°1 ,

Qa

(31)

thus causing an additional singularity of Eq. (11) at ,y = N, and eventually double - log terms (r., as /N 2 ) in the coefficient function . The above remarks illustrate the fact that the QQ system is a non-local probe of the gluon k1 -dependence,

which cuts off the kl -integration around k1 ^-- M « f. For a local probe large values of k1 = O(Vs-)

are allowed, yielding a much stronger singularity of the coefficient of type a,/16x2 (as for time-like jet evolution [21) . It is rather peculiar that no such local probe seems

available in nature for gluons . Indeed, even produced jets, corresponding to emitted gluons, are screened by

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3. S. Catani, M. Ciafaloni and F. Hautmann., Phys . Lett. 242B (1990) 97. 4. J.C . Collins, and R.K . Ellis, these pro: 5. P. Nason, S. Dawson and R.K . Ellis, Nuci. Phys . B303 (1988) 607; R.K . Elks and P. N , Nucl. Phys . B312 (1989) 551 .

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preprint

strong virtual corrections [10], which do not allow val-

7. S. Catani, M. Ciafaloni and F. Haut

The results reviewd in this paper have being extended [7] to heavy flavour hadroproduction. Here the k1-factorization due to Regge behaviour involves the

8. L.N . Lipatov, Sov. J. Nucl. Phys . 2.3 (1976)

ues of k1 larger than the jet mass .

hard subprocess g(k1)+g(k2) --* Q+Q, with one gluon per incoming hadron . The form of the heavy flavour cross section is

4M2,,,('0' M2) =

J

d2kid2k2

i dzi

fo

zi

&( z Pz , ki, k2 ; M2 ) F(zi, ki) .F(z2, k2) i 2

(32)

where the off-shell Born cross-section & is again defined

by coupling the incoming gluons to external partons with eikonal vertices . The computation of &(P/zi -2,ki, k2 ; M2 ) is in progress .

ACKNOWLEDGEMENTS This research was supported in part by the U.K . Sci-

ence and Engineering Research Council and the Italian Ministero della Pubblica Istruzione.

in

; E.A . Kuraev, L.N. Lipatov and V.S . Fadin, Sov. Phys . JETP 45 (1977) 199; Ya. Balitskii and L.N . Lipatov, Sov. J. Nucl. Phys . 28 (1978) 6.

9. E.M . Levin and M.G . Ryskin, these proceedings . 10 . M. Ciafaloni, Nucl .

dz 2 z2

preparation .

Phys .

B2

(1987) 249 ; S. Catani, F. Fiorani and G. Marchesini, Nucl . Phys . B336 (1990) 18.