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Resummation of next-to-leading logarithms in top quark production
ELSEVIER
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 64 (1998) 402-405
Resummation of next-to-leading logarithms in top quark production Nikolaos Kidonakis ~ * ~Department of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, Scotland We discuss the resummation of next-to-leading logarithms (NLL) in heavy quark production near threshold. Results are presented for top quark production at the Fermilab Tevatron.
1. I n t r o d u c t i o n There has been a lot of interest recently in the resummation of soft gluon radiation near threshold for heavy quark production in hadronic collisions [1-4]. This resummation is a direct consequence of the factorization of short- from longdistance effects in QCD scattering [5]. In the perturbative expansion for the heavy quark cross section one finds logarithmic terms that can be resummed to all orders of perturbation theory. These logarithms come from distributions that are singular for z = 1, where z = Q2/s, with Q2 and s the invariant mass squared of the produced heavy quark pair and the partons in the incoming hadrons, respectively. One of the interesting features of singular distributions in QCD-induced cross sections is their sensitive dependence on the color exchange in the hard scattering. This feature as well as the presence of final state interactions are the main complications relative to the Drell-Yan process [6]. We will show that these effects contribute only at NLL. As we discuss below, resummation is conveniently carried out in terms of moments, and logarithmic dependence on the moment variable exponentiates. The moments have to be inverted to derive the physical cross section. Previous formalisms [1-4] resum only leading logs (and some NLL) and there has been a debate about which subleading logs one should keep. In our analysis (in moment space) we have for the first time resummed all the NLL so we have effectively pushed the debate in [1-3] to the next-to-next-to-leading logarithmic level. In *This work was supported by the PPARC under grunt GR/K54601.
0920-5632/98/$19.00© 1998 Elsevier Science B.V. PII S0920-5632(97)01097-9
All rights reserved.
Section 4 we invert moments using the method in Ref. 1. 2. R e s u m m a t i o n f o r Q C D h a r d s c a t t e r i n g For the hadronic process hl(Pl ) d-h2(p2) --4 Q(~ + X the cross section ~, can be written as a convolution of parton distribution functions ¢, with the partonic hard scattering ~, =
f=q,q,g
This convolution becomes a product in terms of moments, =
) f=q,#,,g
(2.2) where
=
fl
=
f~ dzzN-l&(z), and ¢(N) = f : d x x N - l ¢ ( x ) . Here ~- = Q2/S where S is the invariant mass squared of the incoming hadrons. Singular distributions in z give rise under moments to logarithmic N-dependence, as in
fOOI dz z N-1 [ l n m - ( l ~ z ) ] o ~ l n m + l N . L
l-z
(2.3)
+
Thus, the large-N behavior is a diagnostic for singular distributions in 1 - z. We can derive the resummation of soft gluon contributions by reexpressing moments of the cross section (with hi = f, h2 = f) in a form in which collinear gluons are factorized into alternate parton distributions ¢, and soft gluons into a function S ji,
cr]f_~QO~ = ¢1/] ® h*jhi ® SjI ® e f / f .
(2.4)
N. Kidonakis/Nuclear Physics B (Proc. Suppl.) 64 (1998) 402-405 This factorization takes into account the color exchange at the hard scatterings, hi and h~ in the amplitude and its complex conjugate, respectively, where I and J are color tensor indices. From Eqs. (2.1) and (2.4) we find ~(N) = [~b(N)/¢(N)]'~h*j SjI(N) hi. The ratio of the universal ¢ and ¢ has been analyzed in Drell-Yan production [6]. The soft. matrix Sjs(Q/(Np)) requires tenorrealization: S(f; = (Z?s)jBSBAZs,A,, where Zsj1 is a matrix of renormalization constants [7]. Hence SjI satisfies the renormalization group equation
dSs
Sa(rS)A .
P dp -
(2.5)
In a minimal subtraction scheme, and with ¢ = 4 - n, the anomalous dimension matrix Fs is Vs(g)-
~ 0 Res~-+0Zs(g,,). 2 0g
(2.6)
At the level of leading logarithms of N in S jr, and therefore at NLL of N in the cross section as a whole, we choose a color basis in which the anomalous dimension matrix is diagonal, with eigenvalues A1 for each basis color tensor labelled by I. Then, the solution to Eq. (2.5) is
X exp
--
[)H(Ols(~))
+ ~)(O~s(/12))]
/N
(d),~ ,,, ] +g3 [a,,(t~-z)2@),0] , (2.9) where EDY is the Drell-Yan exponent, and
o] :
]
.
(2.7) Thus, we find for the partonic cross section [8,9]
&.ff_~qo(N) = A' eE-''(N'°'Q') h*j (1, a.,(Q 2)
o] + - CF, A.
(2.10)
3. T h e Fs m a t r i c e s for qq --~ QQ a n d gg -+ OQ We now give explicit results for the anomalous dimension matrices Fs. We begin with quarkantiquark annihilation, q(p~) + q(Pb) ~ Q(pl) + Q(p2). The explicit calculation is given in [9]. Here we give the results for the anomalous dimension matrix in a color tensor basis consisting of singlet and octet, exchange in the s channel, 1
=
1.
+
(3.1)
The anomalous dimension matrix (shifted by CFa,/Tr as in Eq. (2.10)) in this color basis and in an axial gauge A ° = 0 is [9] Fs,11
-
a" CF(L9 + 1 + ~ri), 71"
Fs,21 Fs,22 --
'~.lI(1, o/s(Q2))=,~.lI(l, O's([Q]2))
403
-'~r In
, F~12=2-~A., s,21,
a~{CF[41n(~)Tr -L~-l-~ri]
where s = (p~ -I-pb) 2, tl = (p~ --pl) ~ -- m 2, ua = (Pb - - P l ) 2 -- rn2, and rn is the heavy quark mass. Also LO- 1-2m2/s /3
( lnl-/3
+
)
,
(3.3)
where A' is an overall constant and 0 is the centerof-mass scattering angle. The exponent, is
where /3 = V / 1 - 4m2/s. Fs is diagonalized in this singlet-octet basis for arbitrary ~ when the scattering angle is 0 = 900 (where Ul = tl). It is also diagonalized at./~ = 0. Next we discuss gluon fusion, g(Pa) + g(Pb) ~)(Pl) + Q(p2). We choose the color basis
E jr(N, 0, Q~) = EDy(N, Q2)
cl = 6 ~b321, c 2 = d ~cT$l, c3=i.f a~cT$l.
1
zN-1
(3.4)
Then in the A ° = 0 gauge the anomalous dimension matrix (shifted by CAa,/Tr as in Eq. (2.10))
N. Kidonakis /Nuclear Physics B (Proc. Suppl.) 64 (1998) 402~105
404
is [9]
Is[
Fs,11
:
FN,21
=
<'[-CF(Lis +
7F 0, Ps, l~ : 0,
Ps,:u Cs,22
1) -
i ....
i ....
J ....
~. _
c~, { - C F ( L z
+ 1)
7r
\rn2s) + Le
--
--7ri]}
N u -4 1F -~/~ FS,31, Fs,13 = ~ S,31, i
,
,5o
Fs,,~a
i ....
CATFi],
-f-~ [ln¢/lUl'~ ['S,32
....
-
CA 4 Fs,31, rs,aa = rs,~2.
4. Results for top quark p r o d u c t i o n at the Fermilab Tevatron We are now interested in the m a g n i t u d e of the NLL terms, particularly the 93 contribution, relative to the LL results of [1]. Therefore we use a similar cutoff scheme, modifying the definitions of the exponent in Eq. (2.9) to work directly in mom e n t u m space. This m e t h o d uses the correspondence between l o g a r i t h m s of the m o m e n t variable N and logarithmic t e r m s in the m o m e n t u m space variable 1 - z. We work at 0 = 900 to avoid the diagonalization of the a n o m a l o u s dimension m a trices. T h e r e s u m m e d partonic cross section is [11]
s--2msl/2 ITab(8 , rrl 2)
:
ds4 feb
-d Sou t
x dg~°b)(s's4'm2) (4.1) ds4 where ab = qg or gg and s4 = 2m 2 (1 - z) at threshold, d-~!°b)/ds4is the differential of the Born cross section [1]. After replacing z N-1 _ 1 in Eq. (2.9) by - 1 and introducing w = 1 - z, the exponential function for the q~ channel is [11] S4
fqCt ( ~ - - ~ m 2 ) :
exp[ED~
+
Eqq(,~octet)]
,
(4.2)
,
i
.
.
.
.
i
1 so
.
.
.
.
~ 70
Top quark
(3.5)
We note t h a t the m a t r i x is diagonalized at 0 = 900 and also a t / 7 = 0. We have checked t h a t the one-loop expansion of our results for both channels are consistent with [10].
,
i 1 ao
mail
.
.
(GiVI~ t)
.
.
i
.
.
.
.
190
200
Fig. 1. T h e r e s u m m e d top quark p r o d u c t i o n cross section at 0 = 90 o in the MS scheme. We show the r e s u m m e d results w i t h o u t the g3 t e r m s (lower solid line) and the r e s u m m e d results with all the NLL t e r m s (Q = m, upper solid line; Q = 2m, lower dashed line; Q = m/2, upper dashed line).
where EqDY is the Drell-Yan part. T h e colordependent ga contribution in Eq. (2.9) leads to Eqq(Ai)=-
dw o -D--
hi
a<,
,0=900 t
where i denotes singlet or octet. Since our calculation is not done in m o m e n t space, the w integral is cut off at w0 = s4/2m 2. Because the running coupling constant diverges when w2Q2/A2 ~ 1, the m i n i m u m cutoff in eq. (4.3) is scut = S4,min "~ 2m2A/Q, where A is the Q C D scale p a r a m e t e r . In general we choose a larger value for the cutoff consistent with the s u m of the first few t e r m s in the p e r t u r b a t i v e expansion; here we cut off the z-integration at z = 1 - 10A/Q. T h e t r e a t m e n t of the gluon-gluon channel in the MS scheme is very similar but now we have three distinct color structures. However, only two of t h e m are independent. We define fgg for each eigenvalue so t h a t fgg,i = e x p [ E DY -t- Egg(A,)],
N. Kidonakis/Nuclear Physics B (Proc. SuppL) 64 (1998) 402~05
in complete analogy with the relations for the qq channel. For details see Ref. 11. In Fig. 1 we show results for the resummed top quark cross section at the Fermilab Tevatron with x/~ : 1.8 TeV (with and without the NLL 93 terms) versus the top quark mass. We used the MRSD-' parton densities [12]. We also show the variation of the cross section with the factorization scale. Then for m = 175 GeV/e 2 and Q : m the total t{ resummed NLL MS cross section at 0 = 900 is 5.3 pb. Without the NLL g3 contribution the cross section is 3.7 pb. Thus the NLL g3 terms enhance the total MS cross section considerably. 5. Conclusions We have given explicit results for the resummation of next-to-leading logarithms in top quark production. We have shown that the NLL terms are numerically significant. Our methods have been applied to b-quark production at HERA-B, with similar conclusions [11], and they can also be applied to jet production [13]. We have recently completed the evaluation of the anomalous dimension matrices for q(t --~ 9g, qg -+ q9, and the more complicated case of g9 -+ g9, all relevant in jet production [13].
5. 6.
7. 8.
9.
10. 11. 12. 13.
587; hep-ph/9506253, 1995; J. Smith and R. Vogt, Z. Phys. C75 (1997) 271. H. Contopanagos, E. Laenen, and G. Sterman, Nucl. Phys. B484 (1997) 303. G. Sterman, Nucl. Phys. B281 (1987) 310; S. Catani and L. Trentadue, Nucl. Phys. B327 (1989) 323; B353 (1991) 183. J. Botts and G. Sterman, Nuc1. Phys. B325 (1989) 62. N. Kidonakis and G. Sterman, Phys. Lett. B387 (1996) 867; in Les Rencontres de Physique de la Vall~e d'Aoste, La Thuile, March 4-9, 1996, ed. M. Greco, p. 333, hepph/9607222; in DIS 97, Chicago, IL, April 14-18, 1997, hep-ph/9708353. N. Kidonakis, SUNY at Stony Brook Ph.D. Thesis (1996), hep-ph/9606474, UMI-9636628-mc (microfiche); N. Kidonakis and G. Sterman, hep-ph/9705234, 1997, to appear in Nucl. Phys. B. R. Meng, G. A. Schuler, J. Smith and W. L. van Neerven, Nucl. Phys. B339 (1990) 325. N. Kidonakis, J. Smith, and R. Vogt, Phys. Rev. D56 (1997) 1553. A. D. Martin, R. G. Roberts, and W. J. Stirling, Phys. Lett. B306 (1993) 145. N. Kidonakis, G. Oderda, and G. Sterman, in preparation.
Acknowledgements
Questions
The work in Sections 2 and 3 was done in collaboration with George Sterman. The work in Section 4 was done in collaboration with Jack Smith and Ramona Vogt. I would like to thank them for their help and useful conversations.
L. Trentadue, Parma:
REFERENCES 1. E. Laenen, J. Smith, and W. L. van Neerven, Nucl. Phys. B369 (1992) 543; Phys. Lett. B321 (1994) 254. 2. E. L. Berger and H. Contopanagos, Phys. Rev. D54 (1996) 3085; hep-ph/9706206, 1997. 3. S. Catani, M. L. Mangano, P. Nason, and L. Trentadue, Nucl. Phys. B478 (1996) 273; Phys. Lett. B378 (1996) 329. 4. N. Kidonakis and J. Smith, Phys. Rev. D51 (1995) 6092; Mod. Phys. Lett. A l l (1996)
405
You have shown some numerical outputs of your calculation. You have used the method of Laenen, Smith, and van Neerven to cut-off the z --+ 1 limit. Could not. the results be affected by this by hand limiting of the final phase space? N. l(idonakis: We were interested in the relative size of the NLL. Of course the result depends somewhat on the procedure used and the value of the cutoff. However, we calculated the cross section with different cutoffs and found that the NLL contribution is always significant, and that is our main conclusion.
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 64 (1998) 402-405
Resummation of next-to-leading logarithms in top quark production Nikolaos Kidonakis ~ * ~Department of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, Scotland We discuss the resummation of next-to-leading logarithms (NLL) in heavy quark production near threshold. Results are presented for top quark production at the Fermilab Tevatron.
1. I n t r o d u c t i o n There has been a lot of interest recently in the resummation of soft gluon radiation near threshold for heavy quark production in hadronic collisions [1-4]. This resummation is a direct consequence of the factorization of short- from longdistance effects in QCD scattering [5]. In the perturbative expansion for the heavy quark cross section one finds logarithmic terms that can be resummed to all orders of perturbation theory. These logarithms come from distributions that are singular for z = 1, where z = Q2/s, with Q2 and s the invariant mass squared of the produced heavy quark pair and the partons in the incoming hadrons, respectively. One of the interesting features of singular distributions in QCD-induced cross sections is their sensitive dependence on the color exchange in the hard scattering. This feature as well as the presence of final state interactions are the main complications relative to the Drell-Yan process [6]. We will show that these effects contribute only at NLL. As we discuss below, resummation is conveniently carried out in terms of moments, and logarithmic dependence on the moment variable exponentiates. The moments have to be inverted to derive the physical cross section. Previous formalisms [1-4] resum only leading logs (and some NLL) and there has been a debate about which subleading logs one should keep. In our analysis (in moment space) we have for the first time resummed all the NLL so we have effectively pushed the debate in [1-3] to the next-to-next-to-leading logarithmic level. In *This work was supported by the PPARC under grunt GR/K54601.
0920-5632/98/$19.00© 1998 Elsevier Science B.V. PII S0920-5632(97)01097-9
All rights reserved.
Section 4 we invert moments using the method in Ref. 1. 2. R e s u m m a t i o n f o r Q C D h a r d s c a t t e r i n g For the hadronic process hl(Pl ) d-h2(p2) --4 Q(~ + X the cross section ~, can be written as a convolution of parton distribution functions ¢, with the partonic hard scattering ~, =
f=q,q,g
This convolution becomes a product in terms of moments, =
) f=q,#,,g
(2.2) where
=
fl
=
f~ dzzN-l&(z), and ¢(N) = f : d x x N - l ¢ ( x ) . Here ~- = Q2/S where S is the invariant mass squared of the incoming hadrons. Singular distributions in z give rise under moments to logarithmic N-dependence, as in
fOOI dz z N-1 [ l n m - ( l ~ z ) ] o ~ l n m + l N . L
l-z
(2.3)
+
Thus, the large-N behavior is a diagnostic for singular distributions in 1 - z. We can derive the resummation of soft gluon contributions by reexpressing moments of the cross section (with hi = f, h2 = f) in a form in which collinear gluons are factorized into alternate parton distributions ¢, and soft gluons into a function S ji,
cr]f_~QO~ = ¢1/] ® h*jhi ® SjI ® e f / f .
(2.4)
N. Kidonakis/Nuclear Physics B (Proc. Suppl.) 64 (1998) 402-405 This factorization takes into account the color exchange at the hard scatterings, hi and h~ in the amplitude and its complex conjugate, respectively, where I and J are color tensor indices. From Eqs. (2.1) and (2.4) we find ~(N) = [~b(N)/¢(N)]'~h*j SjI(N) hi. The ratio of the universal ¢ and ¢ has been analyzed in Drell-Yan production [6]. The soft. matrix Sjs(Q/(Np)) requires tenorrealization: S(f; = (Z?s)jBSBAZs,A,, where Zsj1 is a matrix of renormalization constants [7]. Hence SjI satisfies the renormalization group equation
dSs
Sa(rS)A .
P dp -
(2.5)
In a minimal subtraction scheme, and with ¢ = 4 - n, the anomalous dimension matrix Fs is Vs(g)-
~ 0 Res~-+0Zs(g,,). 2 0g
(2.6)
At the level of leading logarithms of N in S jr, and therefore at NLL of N in the cross section as a whole, we choose a color basis in which the anomalous dimension matrix is diagonal, with eigenvalues A1 for each basis color tensor labelled by I. Then, the solution to Eq. (2.5) is
X exp
--
[)H(Ols(~))
+ ~)(O~s(/12))]
/N
(d),~ ,,, ] +g3 [a,,(t~-z)2@),0] , (2.9) where EDY is the Drell-Yan exponent, and
o] :
]
.
(2.7) Thus, we find for the partonic cross section [8,9]
&.ff_~qo(N) = A' eE-''(N'°'Q') h*j (1, a.,(Q 2)
o] + - CF, A.
(2.10)
3. T h e Fs m a t r i c e s for qq --~ QQ a n d gg -+ OQ We now give explicit results for the anomalous dimension matrices Fs. We begin with quarkantiquark annihilation, q(p~) + q(Pb) ~ Q(pl) + Q(p2). The explicit calculation is given in [9]. Here we give the results for the anomalous dimension matrix in a color tensor basis consisting of singlet and octet, exchange in the s channel, 1
=
1.
+
(3.1)
The anomalous dimension matrix (shifted by CFa,/Tr as in Eq. (2.10)) in this color basis and in an axial gauge A ° = 0 is [9] Fs,11
-
a" CF(L9 + 1 + ~ri), 71"
Fs,21 Fs,22 --
'~.lI(1, o/s(Q2))=,~.lI(l, O's([Q]2))
403
-'~r In
, F~12=2-~A., s,21,
a~{CF[41n(~)Tr -L~-l-~ri]
where s = (p~ -I-pb) 2, tl = (p~ --pl) ~ -- m 2, ua = (Pb - - P l ) 2 -- rn2, and rn is the heavy quark mass. Also LO- 1-2m2/s /3
( lnl-/3
+
)
,
(3.3)
where A' is an overall constant and 0 is the centerof-mass scattering angle. The exponent, is
where /3 = V / 1 - 4m2/s. Fs is diagonalized in this singlet-octet basis for arbitrary ~ when the scattering angle is 0 = 900 (where Ul = tl). It is also diagonalized at./~ = 0. Next we discuss gluon fusion, g(Pa) + g(Pb) ~)(Pl) + Q(p2). We choose the color basis
E jr(N, 0, Q~) = EDy(N, Q2)
cl = 6 ~b321, c 2 = d ~cT$l, c3=i.f a~cT$l.
1
zN-1
(3.4)
Then in the A ° = 0 gauge the anomalous dimension matrix (shifted by CAa,/Tr as in Eq. (2.10))
N. Kidonakis /Nuclear Physics B (Proc. Suppl.) 64 (1998) 402~105
404
is [9]
Is[
Fs,11
:
FN,21
=
<'[-CF(Lis +
7F 0, Ps, l~ : 0,
Ps,:u Cs,22
1) -
i ....
i ....
J ....
~. _
c~, { - C F ( L z
+ 1)
7r
\rn2s) + Le
--
--7ri]}
N u -4 1F -~/~ FS,31, Fs,13 = ~ S,31, i
,
,5o
Fs,,~a
i ....
CATFi],
-f-~ [ln¢/lUl'~ ['S,32
....
-
CA 4 Fs,31, rs,aa = rs,~2.
4. Results for top quark p r o d u c t i o n at the Fermilab Tevatron We are now interested in the m a g n i t u d e of the NLL terms, particularly the 93 contribution, relative to the LL results of [1]. Therefore we use a similar cutoff scheme, modifying the definitions of the exponent in Eq. (2.9) to work directly in mom e n t u m space. This m e t h o d uses the correspondence between l o g a r i t h m s of the m o m e n t variable N and logarithmic t e r m s in the m o m e n t u m space variable 1 - z. We work at 0 = 900 to avoid the diagonalization of the a n o m a l o u s dimension m a trices. T h e r e s u m m e d partonic cross section is [11]
s--2msl/2 ITab(8 , rrl 2)
:
ds4 feb
-d Sou t
x dg~°b)(s's4'm2) (4.1) ds4 where ab = qg or gg and s4 = 2m 2 (1 - z) at threshold, d-~!°b)/ds4is the differential of the Born cross section [1]. After replacing z N-1 _ 1 in Eq. (2.9) by - 1 and introducing w = 1 - z, the exponential function for the q~ channel is [11] S4
fqCt ( ~ - - ~ m 2 ) :
exp[ED~
+
Eqq(,~octet)]
,
(4.2)
,
i
.
.
.
.
i
1 so
.
.
.
.
~ 70
Top quark
(3.5)
We note t h a t the m a t r i x is diagonalized at 0 = 900 and also a t / 7 = 0. We have checked t h a t the one-loop expansion of our results for both channels are consistent with [10].
,
i 1 ao
.
.
(GiVI~ t)
.
.
i
.
.
.
.
190
200
Fig. 1. T h e r e s u m m e d top quark p r o d u c t i o n cross section at 0 = 90 o in the MS scheme. We show the r e s u m m e d results w i t h o u t the g3 t e r m s (lower solid line) and the r e s u m m e d results with all the NLL t e r m s (Q = m, upper solid line; Q = 2m, lower dashed line; Q = m/2, upper dashed line).
where EqDY is the Drell-Yan part. T h e colordependent ga contribution in Eq. (2.9) leads to Eqq(Ai)=-
dw o -D--
hi
a<,
,0=900 t
where i denotes singlet or octet. Since our calculation is not done in m o m e n t space, the w integral is cut off at w0 = s4/2m 2. Because the running coupling constant diverges when w2Q2/A2 ~ 1, the m i n i m u m cutoff in eq. (4.3) is scut = S4,min "~ 2m2A/Q, where A is the Q C D scale p a r a m e t e r . In general we choose a larger value for the cutoff consistent with the s u m of the first few t e r m s in the p e r t u r b a t i v e expansion; here we cut off the z-integration at z = 1 - 10A/Q. T h e t r e a t m e n t of the gluon-gluon channel in the MS scheme is very similar but now we have three distinct color structures. However, only two of t h e m are independent. We define fgg for each eigenvalue so t h a t fgg,i = e x p [ E DY -t- Egg(A,)],
N. Kidonakis/Nuclear Physics B (Proc. SuppL) 64 (1998) 402~05
in complete analogy with the relations for the qq channel. For details see Ref. 11. In Fig. 1 we show results for the resummed top quark cross section at the Fermilab Tevatron with x/~ : 1.8 TeV (with and without the NLL 93 terms) versus the top quark mass. We used the MRSD-' parton densities [12]. We also show the variation of the cross section with the factorization scale. Then for m = 175 GeV/e 2 and Q : m the total t{ resummed NLL MS cross section at 0 = 900 is 5.3 pb. Without the NLL g3 contribution the cross section is 3.7 pb. Thus the NLL g3 terms enhance the total MS cross section considerably. 5. Conclusions We have given explicit results for the resummation of next-to-leading logarithms in top quark production. We have shown that the NLL terms are numerically significant. Our methods have been applied to b-quark production at HERA-B, with similar conclusions [11], and they can also be applied to jet production [13]. We have recently completed the evaluation of the anomalous dimension matrices for q(t --~ 9g, qg -+ q9, and the more complicated case of g9 -+ g9, all relevant in jet production [13].
5. 6.
7. 8.
9.
10. 11. 12. 13.
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Acknowledgements
Questions
The work in Sections 2 and 3 was done in collaboration with George Sterman. The work in Section 4 was done in collaboration with Jack Smith and Ramona Vogt. I would like to thank them for their help and useful conversations.
L. Trentadue, Parma:
REFERENCES 1. E. Laenen, J. Smith, and W. L. van Neerven, Nucl. Phys. B369 (1992) 543; Phys. Lett. B321 (1994) 254. 2. E. L. Berger and H. Contopanagos, Phys. Rev. D54 (1996) 3085; hep-ph/9706206, 1997. 3. S. Catani, M. L. Mangano, P. Nason, and L. Trentadue, Nucl. Phys. B478 (1996) 273; Phys. Lett. B378 (1996) 329. 4. N. Kidonakis and J. Smith, Phys. Rev. D51 (1995) 6092; Mod. Phys. Lett. A l l (1996)
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You have shown some numerical outputs of your calculation. You have used the method of Laenen, Smith, and van Neerven to cut-off the z --+ 1 limit. Could not. the results be affected by this by hand limiting of the final phase space? N. l(idonakis: We were interested in the relative size of the NLL. Of course the result depends somewhat on the procedure used and the value of the cutoff. However, we calculated the cross section with different cutoffs and found that the NLL contribution is always significant, and that is our main conclusion.