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1EDINBURGH 96/30ITP-SB-96-32LBNL-39025Resummed heavy quark production cross sections to next-to-leading logarithm N.Kidonakis 1Department of Physics and Astronomy University of Edinburgh Edinburgh EH93JZ,Scotland J.Smith 2Deutsches Electronen-Synchrotron,DESY Notkestrasse 85,D-22603,Hamburg,Germany and Institute for Theoretical Physics State University of New York at Stony Brook Stony Brook,NY 11794-3840R.Vogt 3Nuclear Science Division,Lawrence Berkeley National Laboratory,Berkeley,CA 94720
and
Physics Department,
University of California at Davis
Davis,CA 95616
January 1997
2
Abstract
We study how next-to-leading logarithms modify predictions from leading logarithmic soft-gluon resummation in the heavy quark pro-duction cross section near threshold.Numerical results are presented
for top quark production at the Fermilab Tevatron and bottom quark production at?xed-target energies.
3 1Introduction
The calculation of hadronic cross sections in the elastic limit(i.e.,near threshold)in perturbative QCD involves contributions from the emission of soft gluons.In n-th order QCD one encounters leading logarithmic(LL) contributions proportional to(?αn s/n!)[ln2n?1((1?z)?1)/(1?z)]+.There are also terms with next-to-leading logarithmic(NLL)contributions proportional to(?αn s/n!)[ln2n?2((1?z)?1)/(1?z)]+.These“plus”distributions are large near threshold,z=1,(the precise de?nition of z will be given in the next section)and resummation techniques were originally developed to sum them in Drell-Yan production[1].As the Drell-Yan process involves electroweak interactions it has a rather simple color structure.Gluons are only radiated from the incoming quark-antiquark pair in the hadronic collision process so the amplitude is a color singlet.
The?rst resummation of LL terms(as well as some NLL terms)in the heavy(top)quark production cross section was discussed[2,3]prior to the discovery of the top quark at the Fermilab Tevatron[4].The analysis was based on the fact that the LL terms are identical to those in the Drell-Yan process and,even though a cuto?was used to de?ne the resummed perturba-tion series,this paper pointed out the importance of incorporating resumma-tion e?ects in threshold production of heavy quarks in QCD.Recently other groups have applied more sophisticated LL resummation methods to the top quark production cross section,c.f.Refs.[5,6],which avoid the use of an explicit cuto?,and lead to slightly di?erent values for the top quark cross section.At present the experimental results cannot discriminate between them and there is an ongoing discussion as to which method is theoreti-cally superior.The quark-antiquark annihilation channel is the dominant partonic process for top quark production in p
S=1.8TeV.However the gluon-gluon channel is more important in bottom and charm production near threshold in?xed-target experiments with proton and pion beams.The LL resummation method of[2]has also been applied to b-quark production at HERA-B[7]and more generally for ?xed-target bottom and charm production by hadron beams[8].
The orderα2s corrections to the Drell-Yan process in[9]have allowed a check of the NLL terms in the resummation formulae,thus the theory is in excellent shape.However,this information cannot be used in heavy-quark resummation since some of the NLL terms are a consequence of the more
4 complicated color structure in heavy quark production.There is no exact calculation of the heavy-quark production cross section at orderα4s.Even though the NLL terms at orderα3s for heavy quark production near thresh-old were available in1990[10],at the time of the heavy quark resummation analysis[2]it was not clear how to resum them.Therefore all work has con-centrated on the LL terms.Now this situation has changed.A resummation formalism which correctly incorporates NLL resummation near threshold has recently been presented by Kidonakis and Sterman[11,12].The authors an-alytically compared the order-by-order expansion of their results with the known NLL terms[10]and found agreement at threshold in both the quark-antiquark channel and the gluon-gluon channel.
In this paper we will apply this new NLL resummation formalism to calculate the top quark production cross section at the Fermilab Tevatron and the bottom quark production cross section at?xed target energies.We are particularly interested in the size and therefore the phenomenological importance of the NLL terms with respect to the LL terms.To determine the relative importance of the NLL terms,we will use the cuto?method proposed originally in[2].There are several reasons for this,which we now discuss.First,as already stated,we are primarily interested in the corrections the NLL terms make to the LL terms.Next,by varying the cuto?the failure of the perturbative expansion and the onset of the nonperturbative region may be studied directly,bypassed in the newer methods.Finally,given the complexity of the formulae it is best to use the simplest approach to incorporate the NLL terms in a?rst test of their magnitude.An analysis of the NLL terms is very important since the present LL resummations are based either on neglecting them entirely[5]or retaining only some of them [6].These terms are not universal and could possibly be of importance in several areas of perturbative QCD besides the analysis of the heavy quark cross section,including the production of large transverse energy jets and the production of supersymmetric particles.The conclusions of our study should not be generalized to these other reactions since each process is di?erent and requires a separate study.
5 2General Formalism
Here we present some basic formulae which we need for our analysis,based on the theory developed by Kidonakis and Sterman[11,12].For heavy quark production to be kinematically allowed,the square of the partonic center of mass energy,s=x a x b S,must be larger than the threshold value of the invariant mass of the heavy quark-antiquark pair,Q2=4m2.When the heavy quarks are produced with zero velocity,the true threshold and the partonic threshold are equivalent.In either case,the plus distributions which must be resummed are functions of
Q2
z=
= ab IJ dx a x bφa/h1(x a,Q2)φb/h2(x b,Q2) dQ2d cosθ?dy
×δ y?1x b ?σ(IJ)ab Q2
Q pair,andθ?is the scattering angle in the pair center of mass system.Theφ’s are parton densities,evaluated at the mass fac-torization scale Q2,in the DIS or
6
s -channel singlet and octet.The functions h I and h ?J have no collinear or soft divergences at the partonic threshold since these terms have been fac-tored into the exponent E IJ in eq.(2.3).The exponential in (2.3),called S IJ in [11,12],satis?es a renormalization group equation with an anomalous dimension matrix ΓIJ .
The exponents of eq.(2.3)are given by
E (ab )
IJ (n,θ?,Q 2)=? 10dz z n
?1
?1
1?y g (ab )1[αs ((1?y )2?r (1?z )r Q 2)]+g (ab )
2[αs ((1?z )Q 2)]
+g (I )3[αs ((1?z )2Q 2),θ?]+g (J )?3[αs ((1?z )2Q 2),θ?] ,(2.4)
where g 1,g 2and g 3are functions of the running coupling constant αs with z -(and y )-dependent arguments.The parameter r ,1in the DIS scheme and 0in the MS scheme (c.f.[15]and our later discussion).As it stands,eq.(2.4)is ill-de?ned since it incorporates arbitrarily soft gluon radiation at the point where the QCD perturbation expansion diverges.There are di?erent opinions as to how to de?ne the exponent in a manner which clearly separates the perturbative and non-perturbative regions.We will return to this point shortly.To go to NLL in the exponents E IJ ,we need g 1up to and including O (α2s )and g 2and g (I )3to O (αs ).Refs.[5,6]do not include the NLL g 3term,treated numerically for the ?rst time here.
The functions g 1and g 2depend on the mass factorization scheme and the identity of the incoming partons but are essentially independent of the color structure.The function g 1is scheme dependent because of the r dependence of the y integration.It also depends on the identity of the incoming partons so that
g (ab )1[αs ]=(C a +C b ) αs
2K αs
q =C F and C g =C A .The constant K is (c.f.[16])
K =C A 676 ?5
7
in the DIS scheme [15],
g
(q 2
C F αs
q )
q (p b )→
πC F (L β+1+iπ),
Γ21=2αs
t 1 ,
Γ12=αs C A
ln
u
1
π C F 4ln
u 1
2 ?3ln u 1
t 1u 1 +L
β+iπ .
(2.9)where
L β=1?2m 2/s
1+β +iπ ,(2.10)
8
and β2=1?4m 2/s .The Mandelstam invariants for the reaction are s =(p a +p b )2,t =t 1+m 2=(p a ?p i )2,and u =u 1+m 2=(p b ?p i )2.ΓIJ is diagonalized in this basis when the parton-parton c.m.scattering angle θ?=90?(u 1=t 1=?s/2at threshold)with eigenvalues
λ1=λsinglet =?
αs π ?C F (L β+1+iπ)+C A
t 21
+iπ .
It is also diagonalized at partonic threshold s =4m 2for arbitrary θ?.In the partonic channel g (p a )+g (p b )→π[C F (L β+1)+C A iπ],Γ21=0,
Γ31=
2αs t 1 ,Γ12=0,
Γ22=αs 2
?ln m 2s
4N
Γ31,Γ13=14Γ31,Γ33=Γ22.(2.12)At θ?=90?the anomalous dimension matrix becomes diagonal with eigenvalues
λ1=?
αs π
?C F (L β+1)+C A t 21
?iπ ,λ3=λ2.(2.13)We note that ΓIJ is also diagonal at partonic threshold for arbitrary θ?
.
9 3Heavy-quark production cross sections
As we have already stated,we are primarily interested in the magnitude of the NLL terms,particularly the g3contribution,relative to the LL results of [2,3,7,8].Therefore we use a similar cuto?scheme,modifying the de?nitions of the exponents in eqs.(2.3)and(2.4)to work directly in momentum space and avoid working in Mellin space.This method,introduced in[2],exploits the correspondence between the moment variable n and logarithmic terms in the momentum space variable1?z.To directly compare with other recent LL resummation schemes in Refs.[5,6]would,in principle,require three separate studies since each group of authors de?nes eq.(2.4)di?erently. Therefore,in this paper we only consider the approach based on[2].
First,we identify
s4
1?z=
Q production.The invariants satisfy s+t1+u1=s4so that in the elastic limit s4→0or z→1.Note that eq.(3.1)di?ers from the identi?cation1?z=s4/m2in Refs.[2,5].The factor of two is necessary to compare the NLL terms[12] with the expressions previously given in[10].With the identi?cation in eq.
(3.1)all the NLL terms in the q
Q invariant mass and the Qg invariant mass are di?erent invariants).A similar observation holds for the gg channel.
Further we exploit the fact that a heavy quark-antiquark pair will be pref-erentially produced back-to-back in the parton-parton center-of-mass frame near threshold.We also choose the case ofθ?=90?to use the results of eqs.(2.11)and(2.13).This choice avoids both the diagonalization of the three dimensional matrix in the gluon-gluon channel and an extra numerical integration overθ?.
The resummation is done using eqs.(3.17)and(3.19)of[2]where the “plus”distributions are removed through integration by parts.Therefore
10 the resummed partonic cross section is de?ned as
σab(s,m2)=? s?2ms1/2
s cut
ds4f ab s4σ(0)ab(s,s4,m2)
q or gg and s cut will be de?ned below.The other inputs needed here are the function f ab,in either the DIS scheme or the
q channel.There is only one kinematic structure in this channel so that the singlet and octet eigenvalues in eq.(2.11)multiply the same function.At this order in perturbation theory however,only the octet component contributes,c.f.the discussion in[11].Therefore in the DIS scheme atθ?=90?,the exponential function is
f DIS
q2m2 =exp[E DIS q q(λ2)],(3.3) where
E DIS
q q (g1)+E DIS
q
ω′
ω′Q2/Λ2ω′2Q2/Λ2dξπ αs(ξ)+1
2
C F
Λ2 .(3.4)
Since our calculation is not done in moment space,theω′integral is cut o?atω0=s4/2m2.Because the running coupling constant diverges when ω′2Q2/Λ2～1,the minimum cuto?in eq.(3.2)is s cut=s4,min～2m2Λ/Q.If we choose the scale Q2=m2,s cut～2mΛ.In general we choose a larger value
to stay away from the point of divergence.Once we have chosen a reasonable
11 cuto?,consistent with the sum of the?rst few terms in the perturbative expansion,we can study the size of the NLL terms with respect to the LL terms(the order-by-order expansion does not require a cuto?).
We use the two-loop running coupling constant,
αs(ξ)=
1
a
ln(lnξ) 12π
,
b=?6
17C2A?(6C F+10C A)T f n f
MS scheme,
f q s4MS q q(λ2)],(3.6) where now
E q=E q
ω′ Q2/Λ2ω′2Q2/Λ2dξπ αs(ξ)+1
q
(λi)=? 1ω0dω′Λ2 ,θ?=90? +λ?i αs ω′2Q2
12 The di?erential function in eq.(3.2)follows from the kinematic behaviour of the Born cross section,represented by
F B q
s2+
2m2
F(0)q
2s2
F B q
F(0)q ds4=?
1
q s4σ(0)q ds4=πα2s K q F(0)q ds4
×exp[E q q(λ2)],(3.12) where K q
σ(0)q q/ds4in eq.(3.20) of Ref.[2]where the angleθ?was analytically integrated before di?erentiating with respect to s4.
The treatment of the gluon-gluon channel is very similar but now we have three distinct color structures.However,only two of them are independent. Therefore we de?ne f gg for each eigenvalue so that f gg,i=exp(E gg+E gg(λi)) with a one-to-one correspondence between E gg(λi)and the eigenvalues of eq.
(2.13)so that
E gg(λi)=? 1ω0dω′Λ2 ,θ?=90? +λ?i αs ω′2Q2
13
where i =1,2.In the
ω′ Q 2/Λ2ω′2Q 2/Λ2dξπ αs (ξ)+1
σ(0)gg /ds 4has two contri-butions:
F B gg,I (s,t 1,u 1)
=t 1t 1+4m 2s t 1u 1
F B gg,II (s,t 1,u 1)= 1?4t 1u 1F (0)gg,i =[(s ?s 4)2?4sm 2]1/2
F (0)gg,I (s,s 4,m 2)
2s 2
s ?s 4+
320m 4s 2(s ?s 4)5 (3.17)d
ds 4
=d ds 4?
1(s ?s 4)2?4sm 2 512m 6s s 2?64m 42m 2 d
ds 4=π
2m 2
(3.19)+C F (N 2?4)f gg,2 s 4F (0)gg,I
2m 2 d ds 4
,
14 where K gg=(N2?1)?2is a color average factor.One can numerically compare the coe?cients of the color terms in dσ(0)gg/ds4in eq.(3.21)of Ref.
[2]with the color terms near threshold in d
MS scheme whereΛn
f?μ0?μandμis the renormalization scale. It was assumed that the integral over the nonperturbative region below s0
was negligible compared to the integral over the perturbative region.The ?nal resummed cross section was obtained by comparison with the order-by-order approximate cross section up to O(α4s)as a function ofμ0.Since each term in the perturbative result is positive,μ0was chosen such that the resummed cross section was slightly larger than the sum of the approximate cross sections.It was found that forμ=m in the DIS schemeμ0≈(0.05?0.1)https://www.doczj.com/doc/044899630.html,rger values were needed in the
q channel,μ0≈(0.1?0.2)m,whileμ0≈0.35m in the gg channel.Theμ0needed in the gg channel was larger because of the color factor C A in E gg,see eq.(3.14).As before,we will use the cuto?method since increasing s cut away from s4,min reduces the scale and scheme dependence of the resummed perturbative cross section,as we discuss in detail in the next section and was also shown previously for the LL resummation[2,3].In the next section,we will calculate theμ0appropriate for our chosen s cut.
4Numerical results
We?rst present our results for top quark production.In?g.1(a)we plot the
exponents contributing to f DIS
q
s=351GeV.Since only the octet component contributes to g3here,only E q
15 the one-loop calculation of g1,E DIS,given in[2].The one-loop and two-loop results are quite similar until s4approaches s4,min.Asω′Q/Λ→1,E DIS
q
q
+E q
q (g2)|is shown).The power of1?z inαs
determines the slope of the exponents at small s4/2m2.The g2contribution, linear in1?z,is the?attest as s4/2m2→0.The g1component shows more growth,diverging from E DIS near s4,min,and the g3contribution,quadratic in1?z,grows fastest.The growth of the sum is intermediate to that of
E DIS
q q (λ2).
To better illustrate the importance of these contributions to the partonic cross section,we show the relative enhancements as a function of s4/2m2in ?g.1(b).The negative contribution and the slow growth of g2near s4,min are re?ected in the near threshold behavior of f q q(g1)where
e.g.f q
q (g1)).In the intermediate range of s4/2m2,E DIS is
larger than E DIS
q q (λ2)dominates f DIS
q
MS scheme,shown in?g.
2.The g2term vanishes in this scheme.The change in the upper limit of the ξintegral increases the phase space of E q.Both these changes enhance the
MS scheme.Note that the real parts of theλi’s are the same in eqs.(2.11)and (2.13)so that the strength of the g3contributions are the same in the q
q channel.The E gg(λ1)component is quite small and negative near threshold,making the sum E gg+E gg(λ1) indistinguishable from E gg.The E gg(λ2)contribution is the same as in the q
16 found also that,for the same value of s4/2m2,the exponents are larger for smaller values of Q2,as expected.
We have checked the energy dependence of the g3contributions.We?nd that E gg(λ1)is always negative,decreasing the NLL correction for all s.Its energy dependence is quite strong and thus above threshold it serves to damp the resummation.On the other hand,there is very little energy dependence of E IJ(λ2)close to threshold.As s grows large,the ln(m2s/t21)term in eqs.(2.11)and(2.13)dominates the energy dependence,causing E IJ(λ2)to change sign,eventually leading to a strong damping of the NLL component (and probably leading to a smooth energy variation of the resummed cross section above threshold).This logarithmic term also has the only explicit s′4 dependence of the g3contribution atθ?=90?apart from the argument of αs.Sinceλ1only depends on s′4throughαs,these contributions have similar behaviours in s.
In?g.4we plot the partonic t q and gg channels.Atη≈1the gg contribution is numerically smaller than the corresponding q
MS scheme and0.13m in the DIS scheme when Q2=m2,similar to the values used previously.Changing s cut changes the e?ectiveηrange.However,note that forη≥1,the q q result in the
q contributions with and without the NLL components are nearly the same but atη≤0.02the g3contribution makes the partonic cross section up to a factor of two larger. Both with and without the NLL terms,the scheme dependence increases as ηdecreases.
We note that for?xed angle,θ?=90?,the partonic cross section in the gg channel can become negative aroundη～1?2.Because d
F(0)gg,I/ds4,when it grows larger than d
17 the partonic gg cross section becomes negative.In our calculation of the hadronic cross section,we exclude the negative region if it occurs.For the particular values of s cut and Q2shown here,the partonic cross section is always positive.However,for other values of s cut and Q2the change in sign can occur,both with and without the g3contribution and thus is not just a feature of the NLL contribution at this angle.
The results in?g.4imply that the relative importance of the NLL con-tributions to the hadronic cross section depend on which regions inηare weighted most heavily by the convolution with the parton densities.With equivalent parton luminosities,ifη≈0.5,then we expect the gg contribution to be comparable to that of the q
q channel to dominate.Thus the parton luminosity in the x space probed by the hadronic cross section determines the ultimate importance of the NLL contribution.A comparison with previous results obtained by integrating over the angleθ?[2,3]is inconclusive since here we work at?xedθ?.Finally we note that previously[2,3,7,8]di?erent cuto?s were chosen in the q
x f h1i(x,μ2)f h2j(
τ
m2+s cut)2/S.We evaluate the parton densities atμ2=Q2.Since the parton densities are only available at?xed order,the application to a re-summed cross section introduces some uncertainty.We have used the MRS D?′DIS densities for the q MS densities [18,19]for both channels in this scheme4.Note thatΛ5=0.1559GeV for both sets.The result is not strongly dependent on the parton distributions in the x range probed at this energy,x～0.2at y=0for m=175GeV/c2, entering primarily through the value ofΛ5since di?erent sets of parton den-sities have a di?erentΛ5,changing s4,min as well as the value ofαs.Takingαs to only one loop,as in[2],increasesαs and thus the hadronic cross section.
18 For a more complete discussion of the parton density andαs dependence,see [8].
In?g.5(a)we plot the t
S=1.8TeV in the DIS and q channel and the
MS cross section atθ?=90?is5.3pb at m=175GeV/c2,similar to the angle-integrated results of Ref.
[5].Previous NLO results showed that for m=100GeV/c2,even though ηmax=80the total hadronic cross section was obtained already forη≈3 (c.f.?g.1in[3]).When m=175GeV/c2,ηmax=https://www.doczj.com/doc/044899630.html,paring?gs.4a and 5a,we can infer that in this case the importantηregion isη≈0.3.At this ηthe q
t cross section at?xed angle due to the NLL terms,in?g.5(b)we show our results with the same values of s cut and Q2but without the NLL contributions,i.e.no g3component.The NLL contribution increases with top mass at this energy.The DIS cross section is enhanced between25and30%as the top mass increases from140to200 GeV/c2.The enhancement is somewhat larger for the
q channel and47to55%in the gg channel.The change in the amount of enhancement is,in part,due to the energy dependence of the g3contribution.For our choice of s cut,the t
MS cross section atθ?=90?by43%at this mass and value of s cut in our calculation.The relatively large contribution from the NLL terms veri?es that the lowerηregion is most important.
We have further investigated the s cut and scale dependence of our cal-culation.In Tables1-3we give the numerical values of the hadronic top production cross sections at LL and NLL with s cut=10s4,min,5s4,min and 20s4,min respectively.For each value of s cut we show the results for Q2=m2, 4m2and m2/4.In the tables we have also indicated the equivalent value of the cuto?μ0de?ned in[2]in both the DIS and
q contributions.When s cut=10s4,min,the gg cross section changes41% between Q2=m2/4and4m2at m=175GeV/c2.In contrast,the q MS cross section changes47%and the DIS cross section by65%.At the same mass but with s cut=20s4,min,the gg cross section changes10%between the highest and lowest scales while the q MS cross section changes35%and the
19 DIS cross section52%.On the other hand,for the lowest s cut the gg cross section has the largest scale dependence,more than a factor of two change while the q channel.
Note that for a?xed value of s cut,the gg cross section is actually largest when Q2=4m2than at lower Q2.This result is counterintuitive since one generally expects that increasing the scale decreases the cross section.This is true in the q
F(0)gg,I/ds4and d
MS scheme when Q2=m2.At such low values ofμ0in the gg channel,the gg cross section can become large,see e.g.?gs.12-14in[2].Additionally,as s cut is reduced,all the contributions to the hadronic cross section increase because as s cut approaches s4,min the exponents grow more rapidly.The fastest growth of the exponents in this region occurs when Q2=m2/4.
A comparison of the NLL and LL results in the tables helps to clarify where the NLL enhancements are largest.As discussed above,the enhance-ment for a?xed s cut is largest for Q2=4m2since the larger scale increases the e?ective s4probed at larger Q2.The enhancement is also larger for lower s cut due to the running of the coupling constant.Finally we note that the NLL enhancement increases with mass since t
MS
q q ,particularly near s4,min,see?gs.1and2
and also e.g.?gs.12and13in Ref.[2].For s cut=10s4,min and Q2=m2, the
20 49-55%larger at LL.The scheme dependence is largest for small s cut and large Q2because smaller values of s cut are increasingly sensitive to the upper limit of theξintegral in E ab(g1),see the appendix.The scheme dependence increases slightly with mass at?xed energy.Thus close to threshold the scheme dependence is unavoidable.Even though the partonic cross section is only weakly scheme dependent forη>1,as seen in?g.4,some scheme dependence will remain until new sets of parton densities which have incor-porated resummation e?ects before being?tted to data are available.In the absence of such densities we favor the
q and gg channels can be treated consistently only in this scheme.
Sinceαs depends strongly on the heavy quark mass scale,we repeat our calculations for bottom quark production with m=4.75GeV/c2at √
b production.We note that the energy dependence of g3is stronger for the lighter quark mass.While E gg(λ1)is smaller near threshold,it increases faster with energy than at the top quark mass.In general,the bottom quark cross section is more sensitive to s cut and Q2than the much more massive top quarks,see also[7,8].
The partonic cross sections with s cut=1.4s4,min and Q2=m2are shown with and without the NLL g3terms in?g.9.This value of s cut corresponds toμ0≈0.37m and0.51m in the DIS and
c production,s4,min already corresponds to theμ0values use
d in[8],suggesting that a stabl
e cross section is even more di?cult to obtain for charm production.
In?g.10we show the b