Threshold resummation for Drell-Yan production: theory and phenomenology
TThreshold resummation for Drell-Yan production:theory and phenomenology
Marco Bonvini ∗ Dipartimento di Fisica, Università di Genova and INFN, Sezione di Genova, ItalyE-mail: [email protected]
We present a phenomenological study of Drell-Yan pair production at hadron colliders basedon the NNLO fixed order calculation and on NNLL resummation of threshold logarithms. Wegive an argument to prove that resummation effects are relevant also for values of x = M / s farfrom threshold. We compare different prescriptions for the calculation of resummed quantities,emphasizing the differences coming from subleading terms, which are important when x is small.We present phenomenological predictions for Drell-Yan rapidity distributions at the LHC, westudy the ambiguity related to the resummation prescription, and we compare it to that comingfrom scale variation. XVIII International Workshop on Deep-Inelastic Scattering and Related SubjectsApril 19 -23, 2010Convitto della Calza, Firenze, Italy ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - ph ] O c t hreshold resummation for Drell-Yan production: theory and phenomenology Marco Bonvini
1. Introduction: threshold resummation in the Drell-Yan process
A generic (differential) parton model cross section σ at hadron colliders can be written as aconvolution σ ( x ) = (cid:90) x dzz L (cid:18) xz (cid:19) ˆ σ ( z ) (1.1)where x = M / s ( M is the Drell-Yan pair invariant mass, √ s the hadronic cms energy), L ( z ) is aparton luminosity and ˆ σ ( z ) is the parton-level cross-section. In ˆ σ ( z ) logarithms of 1 − z appear atall orders in α s α n s (cid:20) log l ( − z ) − z (cid:21) + , l = n − , . . . , , (1.2)and in the partonic threshold limit z →
2. Relevance of resummation of log ( − z ) at small x We see from eq. (1.1) that the partonic threshold region z → z → N -space, where N is the Mellin-conjugate variable to x . Eq. (1.1) canbe rewritten as a Mellin inversion integral σ ( x ) = π i (cid:90) c + i ∞ c − i ∞ dN x − N L ( N ) ˆ σ ( N ) = π i (cid:90) c + i ∞ c − i ∞ dN e N log x + log L ( N )+ log ˆ σ ( N ) (2.1)where L ( N ) , ˆ σ ( N ) are the Mellin transforms of L ( z ) , ˆ σ ( z ) . This inversion integral is dominatedby the region where the exponent has a minimum in the positive real axis (saddle point N ). Bya general argument, one can show [3] that a saddle point always exists. We show in Fig. 1 theposition of the saddle point N as a function of x , where ˆ σ ( N ) is the DY q ¯ q contribution at NLOand the parton luminosity is computed using CTEQ6.6 in pp collisions. The saddle point N ismonotonically increasing with x , meaning that at large x the contribution to the cross section mainlycomes from the large N region, which corresponds to the partonic threshold region. For smaller N ( s add l e po i n t ) xNLO in pp collisions Figure 1:
Position of the saddle point N as a function of x for the order- α s Drell-Yan q ¯ q cross section. hreshold resummation for Drell-Yan production: theory and phenomenology Marco Bonvini x , the saddle point N decreases; to evaluate when the large logarithms still give the dominantcontribution, we use a fixed order computation to compare the logarithmic terms with the fullresult. The DY q ¯ q contribution at the NLO is given by (up to an electroweak normalization factor)ˆ σ ( z ) = δ ( − z ) + α s π C ( z ) + O ( α ) (2.2) C ( z ) = C F (cid:26)(cid:20) log ( − z ) − z (cid:21) + − log √ z − z − ( + z ) − z √ z + (cid:18) π − (cid:19) δ ( − z ) (cid:27) . (2.3)In Fig. 2 we show the Mellin transform of C ( z ) (black solid curve) and its logarithmic part, thefirst term in eq. (2.3) (blue dotted curve). The red dashed curve corresponds to the first two termsin eq. (2.3): indeed, the term log √ z − z has the same dynamical origin of the logarithm inside the plusdistribution, and may be included as well in resummation (see Sect. 3). Inspection of Fig. 2 leads to Figure 2:
NLO Drell-Yan coefficient function as a function of N , and its logarithmic approximations. the conclusion that, down to values of N around 2, the logarithmic part dominates the partonic NLOcross section. Hence, if the saddle point N (cid:38)
2, the integrand defining the hadronic cross section σ ( x ) in eq. (2.1) is dominated by the logarithmic part of the partonic cross section. Comparingwith Fig. 1 we can conclude that for values of x (cid:38) .
003 resummation of the logarithms is relevant,in the sense that it includes to all orders the dominant terms of the perturbative coefficients. Notethat this value is rather small, and usually considered too small for threshold resummation to havea sizable effect.
3. Comparison between the Borel prescription and the minimal prescription
Threshold resummation is performed in N space [1], and the generic resummed quantity canbe written as a function of log N ( h k are coefficients which depend on α s )ˆ σ res ( N ) ≡ Σ (cid:18) ¯ α log 1 N (cid:19) = ∞ ∑ k = h k ¯ α k log k N , ¯ α = β α s . (3.1)This series has a radius of convergence 1 in ¯ α log N , since the sum has a branch cut in e α ≤ N ≤ + ∞ due to the Landau pole of α s . For this reason, the Mellin inversion integral (2.1) is not defined,because the parameter c should be to the right of all the singularities of the integrand, but the cutmakes this condition impossible to be satisfied. Otherwise stated, the series obtained performing3 hreshold resummation for Drell-Yan production: theory and phenomenology Marco Bonvini the Mellin inversion of (3.1) term by term is divergent [4, 3]. Hence a prescription is needed toobtain a resummed cross section in the physical x space.In [5] the minimal prescription has been proposed: the idea is simply to choose the integralpath in (2.1) to the left of the cut. This choice has some good properties, one of them being that theresult is asymptotic to the divergent series. However, it also has some undesired features: first it isdefined only at the hadron level, and the integral gets a contribution from an unphysical region ofthe parton densities [5]; second, there is no way to control subleading terms. To clarify this secondstatement, we apply the minimal prescription to the k -th term of the series (3.1), i.e. log k N . In thiscase the integral is an exact Mellin inversion, and the result is a sum of terms of the form (cid:34) log j log z log z (cid:35) + , j = k − , . . . , , (3.2)which are an approximation of the physical logarithms (1.2) in the limit z →
1. There is no way toforce the minimal prescription to reproduce the physical logarithms.In [4] a different prescription, based on the Borel summation of the divergent series (3.1), hasbeen proposed. We present here a simpler proof of the Borel prescription formula, following [3].First, we rewrite the Mellin inversion of log k N as12 π i (cid:90) c + i ∞ c − i ∞ dN z − N log k N = k !2 π i (cid:73) d ξξ k + Γ ( ξ ) (cid:20) log ξ − z (cid:21) + (3.3)where the integral path in the r.h.s. is any closed contour which contains ξ =
0. Next, we use theidentity k ! = (cid:90) + ∞ dw e − w w k (3.4)to eliminate the k ! in (3.3). The Borel method consists in exchanging the w integral with the seriesin (3.1), obtainingˆ σ res ( z ) = π i (cid:73) d ξξ Γ ( ξ ) (cid:90) + ∞ dw ¯ α e − w / ¯ α Σ (cid:18) w ξ (cid:19) (cid:20) log ξ − z (cid:21) + . (3.5)However, this expression is still divergent. The Borel prescription is obtained putting an uppercutoff C to the w integral; this choice amounts to the inclusion of an higher twist term. It is shownin [4] that this result is asymptotic to the divergent series. Most importantly, the z dependence isexplicit, and can be modified in order to better reproduce the physical logarithms. Since in theprevious section we have seen that the inclusion of the log √ z term can help to better approximatethe shape of the full fixed order, we include also these logs and obtain our final expression:ˆ σ resBP ( z ) = π i (cid:73) d ξξ Γ ( ξ ) (cid:90) C dw ¯ α e − w / ¯ α Σ (cid:18) w ξ (cid:19) (cid:104) ( − z ) ξ − (cid:105) + z − ξ / . (3.6)The main features of the Borel prescription are that it is a parton level expression and that we cancontrol the z dependence. Furthermore, the parameter C can be used to estimate the ambiguity ofthe prescription. 4 hreshold resummation for Drell-Yan production: theory and phenomenology Marco Bonvini d s / d M / d Y [ pb / G e V ] Y (cid:214) s = 7.00 TeVM = M Z m R /M < 20.5 < m F /M < 2x = 0.00017 LONLONNLOBorel LL+LOBorel NLL+NLOBorel NNLL+NNLO 0 5 10 15 20 25 30 35 40 45 -4 -3 -2 -1 0 1 2 3 4 d s / d M / d Y [ pb / G e V ] Y (cid:214) s = 7.00 TeVM = M Z m R /M < 20.5 < m F /M < 2x = 0.00017 LONLONNLOMinimal LL+LOMinimal NLL+NLOMinimal NNLL+NNLO 0 5e-07 1e-06 1.5e-06 2e-06 2.5e-06 3e-06 3.5e-06 -1.5 -1 -0.5 0 0.5 1 1.5 d s / d M / d Y [ pb / G e V ] Y (cid:214) s = 7.00 TeVM = 1000 GeV0.5 < m R /M < 20.5 < m F /M < 2x = 0.02041 LONLONNLOBorel LL+LOBorel NLL+NLOBorel NNLL+NNLO 0 5e-07 1e-06 1.5e-06 2e-06 2.5e-06 3e-06 3.5e-06 -1.5 -1 -0.5 0 0.5 1 1.5 d s / d M / d Y [ pb / G e V ] Y (cid:214) s = 7.00 TeVM = 1000 GeV0.5 < m R /M < 20.5 < m F /M < 2x = 0.02041 LONLONNLOMinimal LL+LOMinimal NLL+NLOMinimal NNLL+NNLO Figure 3:
Rapidity distribution of neutral Drell-Yan pairs of invariant mass M = m Z (upper plots) and M = pp collisions at √ s = α s ( m Z ) = .
4. Results: Drell-Yan rapidity distributions at hadron colliders
In Fig. 3 we show the theoretical predictions for the production of a neutral DY pair in pp collisions at √ s = Y < Y > C = M . The pdf uncertainty is not included.The upper plots refer to the production of a DY pair of invariant mass M = m Z . In this case x ∼ − is rather small (with respect to the value 0 .
003 found in Sect. 2), meaning that the large z region of the partonic cross section does not give the dominant contribution to the cross section. In-deed, there is a quite large difference between the Borel prescription and the minimal prescriptionresults, coming from the different way of including subleading terms. We believe that this dis-crepancy can be used in order to better estimate the ambiguity due to the unknown higher orders.Moreover, the resummed contribution does not reduce the scale dependence, since resummationonly affects the q ¯ q channel, and the reduction of scale dependence is a combined effect of the q ¯ q , qg and gg contributions.The lower plots refer to the production of a DY pair of invariant mass M = x ∼ .
02 is in the region for which resummation is relevant. At the NNLO+NNLL level ofaccuracy, the inclusion of the resummed term gives a small improvement: the scale uncertainty ismildly reduced.
5. Conclusions
In conclusion, we have shown by a quantitative argument that inclusion of threshold logarithmsat all orders does improve the accuracy of QCD perturbative predictions even when x = M / s is not5 hreshold resummation for Drell-Yan production: theory and phenomenology Marco Bonvini close to 1 (down to x ∼ − − − ). Furthermore, we have presented a realistic phenomenologicalapplication of the Borel prescription for the computation of resummed cross sections. References [1] S. Catani and L. Trentadue, Nucl. Phys. B (1989) 323; G. Sterman, Nucl. Phys. B (1987)310; S. Forte and G. Ridolfi, Nucl. Phys. B (2003) 229 [arXiv:hep-ph/0209154].[2] S. Catani, D. de Florian and M. Grazzini, JHEP (2002) 015 [arXiv:hep-ph/0111164].[3] M. Bonvini, S. Forte and G. Ridolfi, arXiv:1009.5691 [hep-ph].[4] S. Forte, G. Ridolfi, J. Rojo and M. Ubiali, Phys. Lett. B (2006) 313 [arXiv:hep-ph/0601048];R. Abbate, S. Forte and G. Ridolfi, Phys. Lett. B (2007) 55 [arXiv:0707.2452 [hep-ph]].[5] S. Catani, M. L. Mangano, P. Nason and L. Trentadue, Nucl. Phys. B (1996) 273[arXiv:hep-ph/9604351].(1996) 273[arXiv:hep-ph/9604351].