I present results for two-loop soft anomalous dimensions for heavy quark
production which control soft-gluon resummation at
next-to-next-to-leading-logarithm (NNLL) accuracy. I derive an explicit
expression for the exact result and study it numerically for top quark
production via e+ e- -> t tbar, and I construct a surprisingly simple but very
accurate approximation. I show that the two-loop soft anomalous dimensions with
massive quarks display a simple proportionality relation to the one-loop result
only in the limit of vanishing quark mass. I also discuss the extension of the
calculation to single top and top pair production in hadron colliders.Comment: 10 pages, 6 figures; improved form of the analytical result; equation
  adde

Nikolaos Kidonakis

S. V. Ivanov

English

arXiv

I present results for two-loop soft anomalous dimensions for heavy quark production which control soft-gluon resummation at next-to-next-to-leading-logarithm accuracy. I derive an explicit expression for the exact result and study it numerically for top quark production via e(+) e(-) -\u3e t (t) over bar t, and I construct a surprisingly simple but very accurate approximation. I show that the two-loop soft anomalous dimensions with massive quarks display a simple proportionality relation to the one-loop result only in the limit of vanishing quark mass. I also discuss the extension of the calculation to single top and top pair production in hadron colliders

Kidonakis, Nikolaos

DigitalCommons@Kennesaw State University

Kennesaw State UniversityDigitalCommons@Kennesaw State UniversityFaculty Publications6-12-2009Two-Loop Soft Anomalous Dimensions and Next-to-Next-to-Leading-Logarithm Resummation forHeavy Quark ProductionNikolaos KidonakisKennesaw State University, nkidonak@kennesaw.eduFollow this and additional works at: https://digitalcommons.kennesaw.edu/facpubsPart of the Elementary Particles and Fields and String Theory Commons, and the NuclearCommonsThis Article is brought to you for free and open access by DigitalCommons@Kennesaw State University. It has been accepted for inclusion in FacultyPublications by an authorized administrator of DigitalCommons@Kennesaw State University. For more information, please contactdigitalcommons@kennesaw.edu.Recommended CitationKidonakis N. 2009. Two-loop soft anomalous dimensions and next-to-next-to-leading-logarithm resummation for heavy quarkproduction. Phys Rev Lett 102(23):232003.arXiv:0903.2561v3  [hep-ph]  18 May 2009Two-loop soft anomalous dimensions and NNLLresummation for heavy quark productionNikolaos KidonakisKennesaw State University, Kennesaw, GA 30144, USAAbstractI present results for two-loop soft anomalous dimensions for heavy quark productionwhich control soft-gluon resummation at next-to-next-to-leading-logarithm (NNLL) accu-racy. I derive an explicit expression for the exact result and study it numerically for topquark production via e+e− → tt̄, and I construct a surprisingly simple but very accurateapproximation. I show that the two-loop soft anomalous dimensions with massive quarksdisplay a simple proportionality relation to the one-loop result only in the limit of vanish-ing quark mass. I also discuss the extension of the calculation to single top and top pairproduction in hadron colliders.The calculation of higher-order corrections to heavy quark cross sections and differentialdistributions is needed for increasing the accuracy of the theoretical predictions. Given theimportance of the top quark [1] to electroweak and Higgs physics, the top quark cross sectionat the Tevatron and the LHC, both in top pair [2] and single-top [3] production modes, as wellas at a future e+e− collider needs to be calculated with the highest possible precision. Bottomand charm quark cross sections are also important in understanding QCD and calculatingbackgrounds to new physics.Soft-gluon emission is an important contributor to higher-order corrections, particularlynear partonic threshold. The soft-gluon corrections can be formally resummed to all orders inperturbation theory. The resummation arises from the factorization of the cross section into ahard-scattering function H and a soft function S that describes noncollinear soft-gluon emissionin the process [4]. The evolution of the soft function is controlled by a process-dependentsoft anomalous dimension ΓS. The resummed cross section can be written in moment spaceas σres(N) = exp[Ei(N, µ)]H(µ)S(N) exp[∫m/Nm (dµ/µ) 2 ΓS(µ)], with µ the factorization scale,m the heavy quark mass, and N the moment variable which is conjugate to a kinematicalvariable measuring distance from partonic threshold. The exponents Ei resum collinear andsoft radiation from any massless partons in the scattering. The logarithms of N exponentiate.At leading-logarithm accuracy we determine the coefficients of the highest power of ln N at eachorder in the strong coupling αs, at next-to-leading logarithm (NLL) accuracy we also determinethe second highest power of ln N , etc.For heavy quark pair, such as tt̄, hadroproduction the soft anomalous dimensions are ma-trices in color space due to the complicated color structure of the scattering processes, andwere calculated at one loop for both qq̄ → tt̄ and gg → tt̄ channels in Ref. [4], thus allowingresummation at next-to-leading logarithm. From the calculation in [4] one can also derive thecorresponding result for heavy quark production at e+e− machines, a simple function relatedto the cusp anomalous dimension of Ref. [5]. For single top production at hadron colliders theone-loop soft anomalous dimensions were presented for the t, s, and tW channels in Ref. [6].To increase the accuracy of the resummation to next-to-next-to-leading logarithms (NNLL) forall these processes one needs to derive the soft anomalous dimensions at two loops.The calculations of soft anomalous dimensions involve diagrams with eikonal (Wilson) linesrepresenting the heavy quarks. The eikonal approximation is valid for describing the emissionof soft gluons from particles in the hard scattering and leads to a modified form of the Feynmanrules and thus of diagram calculations (see e.g. [4-9]). When the gluon momentum goes to zero,the quark-gluon vertex reduces to gsTcF vµ/v · k, with gs the strong coupling, v a dimensionlessvelocity vector, k the gluon momentum, and T cF the generators of SU(3) in the fundamentalrepresentation. Below we calculate explicitly the soft anomalous dimension through two loopsfor the process e+e− → tt̄, study its properties in detail and then discuss the extension toprocesses with more complicated color structure, i.e. single top and top pair hadroproduction.The eikonal diagrams are calculated in dimensional regularization with n = 4 − ǫ and inFeynman gauge in momentum space. The one-loop soft anomalous dimension, Γ(1)S , can beread off the coefficient of the ultraviolet (UV) pole of the one-loop diagrams in Fig. 1. Writing2(a) (b)Figure 1: One-loop diagrams with heavy-quark eikonal lines.ΓS = (αs/π)Γ(1)S + (αs/π)2Γ(2)S + · · ·, we find the one-loop expressionΓ(1)S = CF[−(1 + β2)2βln(1 − β1 + β)− 1](1)with CF = 4/3, β =√1 − 4m2/s, and s = (pe+ + pe−)2.We now continue with the two-loop diagrams. A few results for some of these diagramshave appeared in [8, 9, 10]. In Fig. 2 we show graphs with vertex corrections and in Fig. 3graphs with heavy-quark self-energy corrections. There are three more graphs, not shown, withvanishing contributions: two graphs with gluon loops involving a four-gluon vertex, and onegraph involving a three-gluon vertex with all three gluons attaching to a single eikonal line.The calculations are challenging due to the heavy quark mass and involve multiple complicatedintegrals and delicate separations of infrared and ultraviolet poles (which by construction ofthe soft function are opposites of each other [4]). The analytical structure involves logarithmsand polylogarithms. It is understood in the results below that for each diagram we includethe appropriate one-loop counterterms for the divergent subdiagrams (but note that graph2(b) does not have a divergent subdiagram). Typically there are large cancellations of termsbetween the individual diagrams and their respective counterterms.We now present the UV poles of the kinematic terms for the two-loop diagrams, which laterwill have to be combined with color and symmetry factors, and an overall factor α2s/π2. Thesum of the UV poles of diagrams 2(a) and 2(b) is minus one-half the square of the UV pole ofthe one-loop diagram of Fig. 1(a): I2a + I2b = [(1 + β2)2/(8β2)](−1/ǫ2) ln2[(1 − β)/(1 + β)].The UV poles of the crossed diagram, Fig. 2(b), areI2b =(1 + β2)28β21ǫ{− ln(1 − β1 + β)[Li2((1 − β)2(1 + β)2)+ ζ2]− 13ln3(1 − β1 + β)+ Li3((1 − β)2(1 + β)2)− ζ3}.(2)The diagram in Fig. 2(c) represents corrections from quark, gluon, and ghost loops. Thequark-loop contribution is I2cq = [nf (1 + β2)/(6β)][1/ǫ2 − 5/(6ǫ)] ln[(1 − β)/(1 + β)] with nfthe number of light quark flavors, while the gluon plus ghost contribution is I2cg = (5/24)[(1 +3(a) (b) (c)(d) (e) (f)Figure 2: Two-loop vertex diagrams with heavy-quark eikonal lines.(a) (b) (c)Figure 3: Two-loop heavy-quark self-energy diagrams with eikonal lines.4β2)/β][1/ǫ2 − 31/(30ǫ)] ln[(1 − β)/(1 + β)]. Diagram 2(d) is given byI2d =(1 + β2)4β{− 1ǫ2ln(1 − β1 + β)+1ǫ[12ln2(1 − β1 + β)+ ln(1 − β1 + β)+ ln(1 − β1 + β)ln((1 + β)24β)− 12Li2((1 − β)2(1 + β)2)+ζ22]}(3)and the result for diagram 2(e) is its negative: I2e = −I2d. The diagram of Fig. 2(f) givesI2f =1ǫ{−14[2ζ2 + ln2(1 − β1 + β)] [(1 + β2)2βln(1 − β1 + β)+ 1]+(1 + β2)12βln3(1 − β1 + β)}. (4)Diagram 3(a) represents two graphs with overall C2F color factor, I3a1 = −3/(2ǫ2) + 1/(2ǫ),and another graph I3a2 = 1/ǫ2 − 1/(2ǫ). The quark-loop contribution to graph 3(b) is I3bq =(nf/3)[1/ǫ2 − 5/(6ǫ)] while the gluon plus ghost loop contribution is I3bg = (5/12)[1/ǫ2 −31/(30ǫ)]. Note the relation d/dβ[2βI2c/(1 + β2)] = [−2/(1 − β2)]I3b. Finally diagram 3(c)gives I3c = [(1 + β2)/(2β)](−1/ǫ2) ln[(1 − β)/(1 + β)].Combining the above kinematic results with color and symmetry factors, the contributionof the diagrams in Figs. (2) and (3) to the two-loop soft anomalous dimension isC2F [I2a + I2b + 2 I2d + 2 I2e + I3a1 + I3a2 + I3c]+ CF CA[−12I2b + I2f − I2cg − I2e − I3bg −12I3a2]+12CF [I2cq + I3bq]= − 12ǫ2(Γ(1)S)2+β04ǫ2Γ(1)S −12ǫΓ(2)S (5)where we have cancelled out factors of α2s/π2 on both sides of the equation. On the right-handside of Eq. (5) in addition to Γ(2)S , which appears in the coefficient of the 1/ǫ pole, there alsoappear terms from the exponentiation of the one-loop result and the running of the coupling,with β0 = (11/3)CA − 2nf/3, CA = 3, which account for all the double poles of the graphs.From Eq. (5) we solve for the two-loop soft anomalous dimension:Γ(2)S ={K2+CA2[−13ln2(1 − β1 + β)+ ln(1 − β1 + β)− ζ2]+(1 + β2)4βCA[Li2((1 − β)2(1 + β)2)+13ln2(1 − β1 + β)+ ζ2]}Γ(1)S+ CF CA{12+12ln(1 − β1 + β)+13ln2(1 − β1 + β)− (1 + β2)28β2[−Li3((1 − β)2(1 + β)2)+ ζ3]− (1 + β2)2β[ln(1 − β1 + β)ln((1 + β)24β)− 16ln2(1 − β1 + β)− Li2((1 − β)2(1 + β)2)]}. (6)We have written the two-loop result Γ(2)S in Eq. (6) in the form of a term which is a multipleof the one-loop soft anomalous dimension Γ(1)S plus additional terms. The well-known two-loopconstant K [11] is given by K = CA(67/18− ζ2)− 5nf/9. Note that the color structure of Γ(2)Sinvolves only the factors CFCA and CF nf ; all C2F terms cancel out.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1β012345Γ S (2)0.94 0.96 0.98 101020304050Figure 4: Two-loop soft anomalous dimension Γ(2)S for top quark production via e+e− → tt̄.The two-loop soft anomalous dimension for e+e− → tt̄ is plotted as a function of β withnf = 5 in Fig. 4. Note that Γ(2)S vanishes at β = 0, the threshold limit, and diverges at β = 1,the massless limit. The inset in Fig. 4 shows more clearly the sharp increase of Γ(2)S as βapproaches 1. Figure 5 also displays Γ(2)S but in a logarithmic plot which makes the small βregion more clear.It is instructive to study the small and large β behavior of Γ(2)S in more detail. We firstexpand around β = 0 and findΓ(2)S exp = −227β2 [CFCA(18ζ2 − 47) + 5CFnf ] + O(β4) . (7)Note that Γ(2)S is an even function of β and hence only even powers of β appear in the expansion.In Fig. 6 we see that just keeping the first term in the expansion as shown in Eq. (7) providesa good approximation to the complete result for small β. The ratio of the expansion to thecomplete Γ(2)S remains within a few percent of unity for values of β up to around 0.3. Keepingmore powers of β in the expansion results of course in a better approximation. Keeping powersthrough β12 results in an excellent approximation to Γ(2)S over a wide range of β, starting todiffer by a few percent above β = 0.8 as seen from the corresponding curve in Fig. 6.As observed in Ref. [12], the two-loop soft anomalous dimension for processes with mass-less quarks is proportional to the one-loop result of Ref. [13] by the simple factor K/2, i.e.Γ(2)S massless = (K/2) Γ(1)S massless. This property was further explored and generalized in Refs.[14, 15, 16]. We see this K/2 factor for heavy quarks as the first term in Eq. (6). It is clearthat for the massive case the situation is rather more complicated than for massless processes:although all the CFnf terms of Γ(2)S are included in (K/2) Γ(1)S , there are many additional CF CA60.001 0.01 0.1 1β10-610-510-410-310-210-1100101Γ S (2)Figure 5: Logarithmic plot of Γ(2)S for top quark production via e+e− → tt̄.terms. It is interesting to check however that in the massless limit of our Γ(2)S this relation issatisfied. In Fig. 6 we also plot the ratio of (K/2) Γ(1)S to Γ(2)S . We see that indeed the ratio is 1at the limit β = 1, i.e. the massless limit, so the massless proportionality relation is recovered.However, for any other value of β the ratio is not one; in fact it is 1.144 near β = 0, thethreshold limit, gradually decreases with increasing β, and falls steeply to 1 in the region veryclose to β = 1.Given that the expansion around β, Eq. (7), gives very good approximations to Γ(2)S atsmaller β while the expression (K/2) Γ(1)S is a better approximation at large β, we can derivean approximation to Γ(2)S for all β values by starting with the β expansion of Γ(2)S , and thenadding (K/2) Γ(1)S and subtracting from it its β expansion:Γ(2)S approx = Γ(2)S exp +K2Γ(1)S −K2Γ(1)S exp=K2Γ(1)S + CFCA(1 − 23ζ2)β2 + O(β4). (8)As seen from Fig. 6, adding just the β2 terms to (K/2) Γ(1)S as in Eq. (8) provides an excellentapproximation to the exact result for Γ(2)S for all β, never more than half a percent away fromthe exact value. It is quite remarkable that such a simple expression provides this good anapproximation to a rather long and complicated expression for Γ(2)S . If we keep additionalterms through order β12 then the approximation is truly impressive, not differing by more thanone per mille anywhere in the β range, and in fact not differing by more than a few parts permillion for most β. At a linear collider with√s = 500 GeV, β = 0.726 for a top mass of 172GeV. For this value of β the expansion through order β12 of Eq. (7) as well as the approximate70 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1β0.951.001.051.101.15(K / 2) ΓS (1) / ΓS (2)           ΓS approx (2)      (Ο(β2)) / ΓS (2) ΓS approx(2) (Ο(β12)) / ΓS (2) ΓS exp (2)  (O(β12)) / ΓS (2)ΓS exp (2)  (O(β2)) / ΓS (2)Figure 6: Expansions and approximations to Γ(2)S for e+e− → tt̄.expression at order β2 or higher of Eq. (8) give excellent approximations. However, (K/2)Γ(1)Sby itself is 11% larger than Γ(2)S .The result in Eq. (6) can be written in terms of the cusp angle γ = ln[(1 + β)/(1 − β)] asΓ(2)S =K2Γ(1)S + CF CA{12+ζ22+γ22− 12coth2 γ[ζ3 − ζ2γ −γ33− γ Li2(e−2γ)− Li3(e−2γ)]− 12coth γ[ζ2 + ζ2γ + γ2 +γ33+ 2 γ ln(1 − e−2γ)− Li2(e−2γ)]}, (9)where Γ(1)S = CF (γ coth γ−1), and is consistent with the cusp anomalous dimension of Ref. [5].Finally we consider the two-loop soft anomalous dimensions for single top and for top pair(or bottom or charm pair) production at hadron colliders. Since the two-loop soft anomalousdimensions for massless quarks obey the simple proportionality relation to one loop, and thediagrams for single top and top pair processes involve eikonal graphs with both massless andmassive eikonal lines, it is clear that for β near 1 the relation Γ(2)S = (K/2) Γ(1)S will hold forboth single top and tt̄ production. For smaller β, however, this relation is no longer valid.For reference, at the Tevatron and the LHC the tt̄ cross section receives most contributions inthe region around 0.3 < β < 0.8 which peak roughly around β ∼ 0.6. However, since evena 15% discrepancy between the exact Γ(2)S and (K/2) Γ(1)S may result in less than 1% effectin the total cross section it may turn out that using the massless proportionality relation isadequate for many purposes (see also the related discussion in [17]). Nevertheless, a rigoroustreatment requires that we include the extra terms for the massive case. More details will begiven elsewhere.8This work was supported by the National Science Foundation under Grant No. PHY0555372.References[1] W. Wagner, Rept. Prog. Phys. 68, 2409 (2005) [hep-ph/0507207];A. Quadt, Eur. Phys. J. C 48, 835 (2006);R. Kehoe, M. Narain, and A. Kumar, Int. J. Mod. Phys. A 23, 353 (2008), arXiv:0712.2733[hep-ex];T. Han, Int. J. Mod. Phys. A 23, 4107 (2008), arXiv:0804.3178 [hep-ph];W. Bernreuther, J. Phys. G. 35, 083001 (2008), arXiv:0805.1333 [hep-ph];D. Wackeroth, in ICHEP 2008, arXiv:0810.4176 [hep-ph].[2] CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 74, 2626 (1995) [hep-ex/9503002];D0 Collaboration, S. Abachi et al., Phys. Rev. Lett. 74, 2632 (1995) [hep-ex/9503003].[3] D0 Collaboration, V.M. Abazov et al., arXiv:0903.0850 [hep-ex];CDF Collaboration, T. Aaltonen et al., arXiv:0903.0885 [hep-ex].[4] N. Kidonakis and G. Sterman, Phys. Lett. B 387, 867 (1996);Nucl. Phys. B505, 321 (1997) [hep-ph/9705234].[5] S.V. Ivanov, G.P. Korchemsky, and A.V. Radyushkin, Yad. Fiz. 44, 230 (1986) [Sov. J.Nucl. Phys. 44, 145 (1986)];G.P. Korchemsky and A.V. Radyushkin, Phys. Lett. B 171, 459 (1986);Nucl. Phys. B283, 342 (1987); Phys. Lett. B 279, 359 (1992) [hep-ph/9203222].[6] N. Kidonakis, Phys. Rev. D 74, 114012 (2006) [hep-ph/0609287].[7] E. Eichten and R. Jackiw, Phys. Rev. D 4, 439 (1971);A. Sen, Phys. Rev. D 24, 3281 (1981);S. Aoyama, Nucl. Phys. B194, 513 (1982);J.G.M. Gatheral, Phys. Lett. 133B, 90 (1983);J. Frenkel and J.C. Taylor, Nucl. Phys. B246, 231 (1984);D. Knauss and K. Scharnhorst, Annalen Phys. 41, 331 (1984);A. Bassetto, I.A. Korchemskaya, G.P. Korchemsky, and G. Nardelli, Nucl. Phys. B408,62 (1993) [hep-ph/9303314];I.A. Korchemskaya and G.P. Korchemsky, Nucl. Phys. B437, 127 (1995) [hep-ph/9409446];C.F. Berger, Phys. Rev. D 66, 116002 (2002) [hep-ph/0209107].[8] N. Kidonakis, hep-ph/0208056; in DIS 2003 (PNPI, Gatchina, 2003), p. 541[hep-ph/0307145].[9] N. Kidonakis, A. Sabio Vera, and P, Stephens, arXiv:0802.4240 [hep-ph];N. Kidonakis and P. Stephens, in DIS 2008, arXiv:0805.1193 [hep-ph].[10] N. Kidonakis, in PANIC 2008, arXiv:0901.2155 [hep-ph].9[11] J. Kodaira and L. Trentadue, Phys. Lett. 112B, 66 (1982).[12] S.M. Aybat, L.J. Dixon, and G. Sterman, Phys. Rev. Lett. 97, 072001 (2006)[hep-ph/0606254]; Phys. Rev. D 74, 074004 (2006) [hep-ph/0607309].[13] N. Kidonakis, G. Oderda, and G. Sterman, Nucl. Phys. B531, 365 (1998)[hep-ph/9803241].[14] T. Becher and M. Neubert, arXiv:0901.0722 [hep-ph]; arXiv:0903.1126 [hep-ph].[15] E. Gardi and L. Magnea, JHEP 03, 079 (2009), arXiv:0901.1091 [hep-ph].[16] L.J. Dixon, arXiv:0901.3414 [hep-ph].[17] N. Kidonakis and R. Vogt, Phys. Rev. D 68, 114014 (2003) [hep-ph/0308222];Phys. Rev. D 78, 074005 (2008), arXiv:0805.3844 [hep-ph].10

Two-loop soft anomalous dimensions and NNLL resummation for heavy quark production

Abstract

Similar works

Full text

Available Versions

DigitalCommons@Kennesaw State University

Crossref