Threshold Resummed and Approximate NNLO results for Pair Production at the LHC
The next-to-leading order (NLO) QCD radiative corrections to production at hadron colliders are well understood. We combine NLO perturbative QCD calculations with soft-gluon resummation of threshold logarithms to find a next-to-next-to leading logarithmic (NNLL) prediction for the total cross section and the invariant mass distribution at the LHC. We also obtain approximate next-to-next-to-leading order (NNLO) results for the total cross section at the LHC which includes all contributions from the scale dependent leading singular terms. Our result for the approximate NNLO total cross section is the most precise theoretical prediction available. Uncertainties due to scale variation are shown to be small when the threshold logarithms are included. NNLL threshold resummation increases the invariant mass distribution by in the peak region for both and TeV. The NNLL threshold resummed and approximate NNLO cross sections increase the NLO cross section by for and TeV.
Exploring the Higgs and electroweak sector of the Standard Model is one of the primary goals of the LHC. The pair production of gauge bosons is important both as a test of the gauge structure and as a background for Higgs boson searches. Precise predictions for both total and differential cross sections are needed in order to understand the shape of the background to the Higgs signal and to search for anomalous three gauge boson couplings. Understanding the theoretical prediction for is particularly important for the measurement of the Higgs decay channel, , where there is no resonant structure. The background is estimated by a sideband analysis, where the cross section is normalized via a control region with a minimum dilepton invariant mass. Using Monte Carlo, the line shapes of the distributions are then extrapolated into the Higgs signal region CMS Collaboration (2013); ATLAS Collaboration (2013). A change in the invariant mass distribution will alter the dilepton invariant mass distribution, and consequently change the extrapolation of the background estimates in the Higgs signal region.
The production of pairs with a subsequent leptonic decay has been studied at the Tevatron Aaltonen et al. (2010); Abazov et al. (2009), while both ATLAS Aad et al. (2013) and CMS Chatrchyan et al. (2013a, b) have reported results at the LHC. The LHC results for the total cross section,
are slightly higher than the Standard Model predictions at next-to-leading order (NLO) in QCD Campbell et al. (2011),111The theoretical predictions have been evaluated at NLO using MCFM with MSTW2008nlo PDFs and a central scale choice of . The uncertainties shown in Eq. (2) result from varying the scale up and down by a factor of . The predictions of Eq. (2) include the next-to-next-to-leading order (NNLO) contribution from the initial state Campbell et al. (2011).
The slight differences between the measured LHC values and the MCFM NLO predictions have led to speculation that the measured cross section is a subtle sign of new physics Feigl et al. (2012); Curtin et al. (2013a, b).
The NLO QCD corrections to were computed in Refs. Frixione (1993); Ohnemus (1991), and then extended to include leptonic decays in Ref. Dixon et al. (1998). The NLO predictions for the total cross section have a uncertainty at the LHC due to the choice of parton distribution functions (PDFs) and renormalization/factorization scale variation Campbell et al. (2011). The contribution from is formally NNLO, but numerically contributes at TeV and at TeV Dicus et al. (1987); Glover and van der Bij (1989); Binoth et al. (2006, 2005). The NLO results have been interfaced with a shower Monte Carlo using the formalism of the POWHEG box Melia et al. (2011); Hamilton (2011); Hoche et al. (2011). The electroweak corrections and the contribution from the initial state are also known and contribute less than to the total cross section at the LHC Bierweiler et al. (2012); Baglio et al. (2013). These corrections are enhanced at large values of the invariant mass, but have opposite signs and largely cancel.
In this paper, we extend these results by including a resummation of threshold logarithms in the prediction of production. Previously, the resummation of large logarithms associated with gluon emission at low transverse momentum, , in production was considered Grazzini (2006). Unlike resummation which is normalized to the NLO cross section, the emission of soft gluons near threshold can potentially enhance the rate. We consider the next-to-leading logarithmic (NLL) and next-to-next-to-leading logarithmic (NNLL) resummation of threshold corrections. To accomplish this, we utilize the formalism of soft collinear effective theory (SCET) Bauer et al. (2000, 2001, 2002); Beneke et al. (2002) which allows the resummation to be performed directly in momentum space Becher et al. (2007); Becher and Neubert (2006). This formalism has been used for processes with colorless final states such as Drell-Yan Becher et al. (2008), Higgs production Ahrens et al. (2009a, b, 2011a), associated W/Z plus Higgs production Dawson et al. (2012), direct photon production Becher and Schwartz (2010) and SUSY slepton pair production Broggio et al. (2012).
The SCET formalism has also been applied to top quark pair production to resum the threshold corrections to the invariant mass distribution and to the total cross section Ahrens et al. (2010, 2011b, 2011c). The total cross section for top quark pair production using threshold resummation has been obtained using two different sets of threshold limits: one starting from the invariant mass distribution of the top quark pair (pair invariant mass kinematics) and the other beginning from the transverse momentum or rapidity distribution of the top quarkAhrens et al. (2010, 2011b); Kidonakis (2010). The total cross section is then obtained by integrating over the resummed distributions. Within the theoretical uncertainties, the total cross sections for top quark pair production obtained with the different starting points are in reasonable agreementAhrens et al. (2011b, c). The total cross section can also be obtained in the threshold limit, , where is the top quark velocity, and the terms of resummedBeneke et al. (2010). At the LHC and the Tevatron, however, the largest contributions to the total cross section for top quark pair production are not from the region. In this work, we will use pair invariant mass kinematics for the final state to obtain our resummed results.
The calculation of differential cross sections involves several scales. We consider the threshold limit which dynamically becomes important due to the fast decline of the parton luminosity function as increases Becher et al. (2008), with the invariant mass of the pair and the partonic center of mass energy. Near the partonic threshold, up to subleading powers of , the cross section factorizes into a soft function which describes the soft gluon emissions and a hard function which includes the virtual corrections to the cross section. We can combine the NLO soft and hard functions with their renormalization group (RG) evolution equations to give NNLL resummed results which resum large logarithms of the form with . Alternatively, the RG evolution of the hard function, known to NNLO, can be matched with exact NLO results for the hard function to obtain the approximate NNLO hard function which includes the leading scale dependent contributions. Combined with the known NNLO soft function Becher et al. (2008) for color singlet production, we are able to obtain the approximate NNLO result as an alternative to the NLO+NNLL resummed result. The advantage of the NLO+NNLL resummed results is that they contain powers of to all orders, while the advantage of the approximate NNLO results is that we used the soft function to one order higher (NNLO) and the results do not contain higher orders of which are sometimes not desired. In any case, the two results are extremely close to each other, and we recommend our approximate NNLO result as the most precise theoretical results available to be compared with future experiments, because it turns out to have smaller scale variations.
In Section II, we review the formalism and SCET resummation in the threshold region. This follows closely the approach of Ref. Ahrens et al. (2010). Section III contains results for the NNLL differential and total cross sections, along with approximate results for the NNLO cross section for . Brief conclusions are presented in Section IV.
In this section, we review the fixed order results for (II.1 and II.2), the SCET formalism used to derive the RG improved NNLL results for the differential and total cross sections, including the matching to the fixed order NLO result (II.3), and the derivation of an approximate NNLO result(II.4).
ii.1 Born Level Result
The Born level process arises through the annihilation process
This annihilation proceeds via -channel exchange and a -channel contribution, as shown in Figs. 1(a) and (b), respectively. The partonic cross section is
where the partonic level invariants are
with and . At the Born level we have . The two body phase space is
Finally, the color-averaged and spin-summed and averaged matrix element squared is
where is the -channel contribution, is the -channel contribution, and is from - and -channel interference. The results have been found in Ref. Frixione (1993) and are given in Appendix A for convenience.
Due to the collinear factorization Collins et al. (1988a); Bodwin (1985); Collins et al. (1985); Collins et al. (1988b); Aybat and Sterman (2009), the hadronic level cross section is obtained by convolving the partonic level cross section with PDFs. In general, the hadronic cross section can be written as
where the sum runs over all possible initial state partons, is the factorization scale,
The long-distance collinear physics is described by the parton luminosity
where is the PDF of a parton with flavor . The coefficient functions describe the hard partonic process and can be written as a power series in ,
We have chosen the normalization of Eq. (8) such that the leading order coefficient is
and the Born level cross section is recovered.
ii.2 NLO result
At NLO, the scattering coefficients of Eq. (8) receive corrections from virtual loops and real gluon emission in the channel, along with tree level contributions from . In dimensional regularization with , the one-loop virtual diagrams contribute
We note that since QCD does not renormalize electroweak couplings, all the UV divergences cancel leaving only IR divergences in . The one-loop corrections to the -channel exchange are given by , to the -channel exchange by , and the interference between the - and -channels by . Expressions for these terms can be found in Appendix A. As will be discussed in the next section, the real hard gluon emission contribution is not relevant for the resummation of the threshold logarithms and we therefore do not give it here, although it can be found in Ref. Frixione (1993).
ii.3 Threshold Resummation and Matching
We now discuss the resummation of the large logarithms in the partonic threshold limit, . In this limit, since there is no phase space available for hard gluon emission, the total phase space is well described by the Born process and Eq. (8) can be used. In addition to the collinear factorization, in the threshold limit the coefficient functions can be further factorized into hard, , and soft, , functions:222Since we are interested in a color singlet final state, the soft function has no dependence.
The soft function is given by the vacuum expectation values of soft Wilson loops Becher et al. (2007, 2008) and the hard function is calculated by matching the full QCD result onto the relevant SCET operator. It is this matching that integrates out the hard QCD modes and leaves the soft and collinear modes that comprise SCET.
where is the Wilson coefficient of the relevant SCET operator . The Wilson coefficient is calculated by matching the renormalized QCD and SCET amplitudes:
where is the renormalized QCD amplitude and is the SCET renormalization constant. In dimensional regularization, SCET loops are scaleless and vanish. This implies that the UV and IR singularities of SCET coincide and cancel. Since SCET and QCD describe the same low-scale physics and have the same IR pole structure, can be determined by the behavior of IR singularities in QCD Giele and Glover (1992); Kunszt et al. (1994); Catani and Seymour (1996, 1997); Catani (1998); Becher and Neubert (2009a, b). In the scheme, we have
The poles in and the NLO QCD squared amplitudes in Eq. (15) cancel. Hence, the one-loop contribution to the hard function is just the finite terms of Eq. (15) Manohar (2003); Becher and Neubert (2006).
Since the hard function is calculated in the perturbative region of QCD it can be expanded in powers of :
where , termed the hard scale, is the scale at which QCD and SCET are matched. The normalization of the hard function is such that
Now we have all the pieces to resum the threshold logarithms. As mentioned before, the hard function is calculated at the matching scale . Since the soft function describes the soft physics, it is evaluated at a soft scale, , associated with the scale of soft gluon emission. By using the RG equations (RGEs), the hard and soft functions can be evolved to the factorization scale. The RG evolution of the soft function resums logs of the form . By choosing the soft scale , the RGE running resums the large threshold logarithms. In Table 1 we list the accuracy of the resummation at a given order. The resulting coefficient function is Becher et al. (2008)
Finally, is can be expressed as a power expansion in logarithms,
We will present results both at NLL and NNLL. The corresponding order of the needed functions is given in Table 1. Explicit expressions for the functions and , and the anomalous dimensions , , and can be found in the appendices of Ref. Becher et al. (2008). The choice of the soft scale, ,is discussed in Section III.2.
The resummed results are only valid in the region . To extend these results to all , the resummed cross section needs to be matched with the full fixed order cross section. This allows the inclusion of the non-singular terms in which are present in the fixed order result but not the resummed result. For NNLL resummation, this means matching with the NLO cross section:
where is the threshold resummed cross section and contains only the NLO terms which are singular as ,
Subtracting prevents double counting of terms common to the resummed and fixed order results. Also, in the limit , the matched cross section corresponds to the resummed results, as desired.
ii.4 Approximate NNLO Results
The full NNLO cross section can only be determined from a complete calculation. However, the scale dependent terms that are singular as can be determined to NNLO accuracy via the known hard and soft functions and their respective RGEs. As will be shown in the next section, most of the NLO correction comes from the leading singular piece. Hence, we expect that including the scale dependent, leading singular pieces of the NNLO cross section is a good estimate of the full NNLO result. The inclusion of these pieces is known as approximate NNLO.
The coefficient function in Eq. (16) can be expanded in a power series,
The leading order, , and NLO, , contributions are fully known analytically. The NNLO contribution, , can only be approximately determined from the hard and soft functions. The soft function is known fully to NNLO Becher et al. (2008); hence the only approximation comes from the unknown scale-independent NNLO piece of the hard function. The approximate NNLO cross section is found by calculating the scale-dependent, leading singular pieces of and adding this contribution to the full NLO result.
We expand the hard function as a power series in :
where the full NLO hard function, , is given in Eq. (20). The NNLO hard piece, , contains only the scale dependent pieces at NNLO. We further expand in a power series of logs:
where . The coefficients can be found in Appendix B.
The approximate NNLO hard function in Eq. (29) is independent of scale to order . Hence, scale variation only contains the uncertainties, not taking into account the unknown NNLO scale independent and nonsingular in pieces at . Variation of the factorization scale may therefore underestimate the total uncertainty in the approximate NNLO result. However, this uncertainty can be further estimated by noting that there is an ambiguity in the logs used to expand . For example, introduce a scale . Then can be expanded in logs of the form instead of . The difference between these two schemes will be order one contributions to the scale independent piece. Hence, in addition to the variation of the factorization scale, the uncertainty associated with the unknown NNLO scale independent piece is estimated by making the replacement in Eq. (30) and varying the new scale around the central value , which is the natural choice from the RGEs.
We note that when including the scale independent pieces, the full NNLO hard function is independent of the scale . Similarly, since the soft function is known fully to NNLO there is no ambiguity in the choice of scales used in the log expansion.
iii.1 Soft Scale Choice
In the process of performing resummation in the SCET formalism, two additional scales are introduced: the hard scale , where the hard function is evaluated; and the soft scale, , where the soft function is evaluated. Since the hard function is calculated from matching QCD onto SCET at the scale of the hard scattering process, the central value of is naturally chosen to be the scale of the hard scattering process, .
The soft scale is chosen to be associated with the scale of the soft gluon emissions, such that the RG evolution resums large logs associated with soft gluon emission at threshold. Following Refs. Becher and Neubert (2006); Becher et al. (2007, 2008), we choose the soft scale to be related to the hadronic energy scale, avoiding the Landau poles that plague the traditional perturbative QCD resummation Catani et al. (1996); Becher and Neubert (2006). Hence, in the hadronic threshold limit we want . However, at hadron colliders most of the cross section is accumulated far from and the choice of soft scale away from this limit is less clear.
Another constraint on is that the soft function should be a well-behaved perturbative series. Hence, away from the threshold region, we choose such that the piece of the soft function is minimized relative to the LO piece. Figure 2 shows the ratio of the NNLL-resummed invariant mass distribution evaluated with only the piece of the soft function, , to the NNLL-resummed distribution evaluated with only the LO piece of the soft function, . In both distributions, all other pieces of the resummed cross sections (aside from the soft function) are evaluated to full NNLL order. This ratio is shown for TeV at various values of with and MSTW2008nnlo PDFs. The soft scale is chosen to correspond to the minimum of these ratios, which is found to be well described by the parameterization
For TeV, it is found that , , and . Performing a similar fit at TeV it is found that , , and . With this parameterization has the correct dependence on in the threshold region.
Figure 2 shows the minimum value of the ratio at TeV (solid) and TeV (dashed) as a function of . As can be clearly seen, the hadronic energy makes little difference to the choice of the soft scale. For simplicity, independent of the hadronic energy scale, all results presented here use the central value of the soft scale corresponding to the 8 TeV solution:
iii.2 Differential Cross Section
We begin by considering the validity of the matching of the NNLL results to the fixed order NLO results. In order for the matching of Eq. (26) to be valid, the sub-leading terms in must be small. In Fig. 3, we show the contribution of the leading singularity to the fixed order NLO differential cross section for TeV and TeV. We fix the central scale to be . From this figure, we see that the leading singularity captures of the NLO fixed order cross section. Hence, the threshold singularities contribute most of the NLO cross section and we may expect that by resumming the higher order logarithms we capture most of the higher order cross section.
In Fig. 4, we show versus for TeV, with MSTW2008 PDFs. The curves are LO, NLO, NLL (matched), and NNLL (matched), with , and varied up and down by a factor of from the central value of . It is apparent that the NNLL resummation slightly increases the rate at the peak.
A change in the invariant mass distribution may be consequential to the analysis of the decay channel. In the zero jet bin, the major background is the SM (non-Higgs) production of CMS Collaboration (2013); ATLAS Collaboration (2013). To estimate this background a sideband analysis is performed. A control region is defined with a minimum dilepton invariant mass, where the background strongly dominates the Higgs signal. The control region is used to normalize the cross section and then Monte Carlo is used to extrapolate the line shapes into the signal region. If higher order corrections alter the distribution, the dilepton invariant mass distribution will be changed and the extrapolation to the signal region will need to take this into account.
In Fig. 4 we explore the effect of higher order corrections on the invariant mass distribution by plotting the ratio of the NLO+NNLL matched and NLO distributions. The scales are set to be , , and . For the NLO cross section evaluated with NLO PDFs (solid), the resummation increases the invariant mass distribution by in the peak region for both and TeV and decreases it by and in the high mass region for and TeV, respectively. However, most of this change in the distribution is from the different PDFs used for the resummed and NLO results, as can be seen when the NLO cross section is evaluated with NNLO PDFs (dashed). In this case, the resummation only alters the invariant mass distribution by for a wider range . This indicates that the calculation of the cross section is firmly under theoretical control.
The factorization scale dependence of the invariant mass distributions is shown in Fig. 5 for (a) the NNLL resummed and NLO leading, and (b) the NLO and NNLL matched results. Here we present the factorization scale dependence as a percent difference from the central value. Using the definition of matching in Eq. (26), Fig. 5 indicates that there is a cancellation between the dependencies the NNLL resummed and NLO leading results for GeV. Comparing Figs. 5 (a) and (b), it is apparent that for GeV the NLO dependence also cancels against the NNLL resummed dependence. Hence, although the cancellation is not as efficient at lower , as the invariant mass increases the dependence of the NNLL matched result is less than that of the NLO result. This can be seen in Fig. 5, where we see that that for GeV the dependence of the NNLL matched result is lower than the NLO result. Hence, although the scale dependence of the NNLL matched result is larger than that of the NLO result at the peak of the invariant mass distribution, one can show that the resummation and matching procedure decreases the factorization scale dependence of the total cross section relative to the NLO result.
In Fig. 6, we show the deviation from the central scales for the NNLL resummed differential cross section, , versus for TeV, with MSTW2008nnlo PDFs. Again, we present the scale dependence as a percent difference from the central value. The central scales are , and and are separately varied up and down by a factor of . The hard and soft scale variations are of the order of and are relatively independent of . The factorization scale dependence near the peak is and is always greater than the hard and soft scale dependencies for the invariant mass range presented.
Finally, in Fig. 7 we show the variation of the NNLL matched cross section with the central scale choices , and at TeV. The cross section varies by less than as the scales are varied from the central values. Again, we see that there is a large cancellation of the factorization scale when computing the matched cross section. In contrast to the NNLL results in Fig. 6, the dependence of the matched result is less than (similar to) the hard and soft scale dependencies for (. Also, note that unlike the soft and factorization scales, the hard scale dependence, with a minimum near the central value, actually never decreases but always increases the total cross section as it is varied from the central value. This explains why in Fig. 6 the effect of the hard scale variation was always to increase the differential cross section value above the central value.
iii.3 Total Cross Section
In this section, we compile our final results for the total cross sections at the LHC. In Tables 2 and 3, we show successively improved results for the total cross sections using MSTW2008 PDFs. Both tables fix the central value of and . Table 2 takes the central factorization scale to be fixed at , while Table 3 uses a dynamical central scale, . The top line is the NLO result obtained from MCFM Campbell et al. (2011) (which includes the initial state) and is calculated using NLO PDFs. The second line of the tables is the contribution, , calculated using MCFM, but with NNLO PDFs (as is appropriate for combining with the NNLL and approximate NNLO results). The third and fourth rows contain the NNLL matched and approximate NNLO cross sections evaluated with NNLO PDFs but without the contribution, and , respectively. The fifth and sixth rows are the same as the third and fourth, but with the contribution now included. The uncertainties in the matched cross section correspond to taking the central values of the hard, soft, and factorization scales and varying each separately up and down by a factor of . The uncertainties in the approximate NNLO cross section correspond to varying the factorization and scales by a factor of two around their central values. The resulting uncertainties are added in quadrature.
As noted previously in the discussion of the previous subsection, the factorization scale dependence of the matched cross section is less than that of the NLO cross section. This cancellation is more extreme at and TeV. Hence, even with hard and soft scale variation taken into account the scale dependence of is less than that of . At and TeV, once the uncertainties associated with hard and soft scale variation are taken into account, the scale uncertainty of is similar to or greater than that of .
The scale dependence of the approximate NNLO cross sections, , at and TeV is reduced by at least a factor of three relative to the NLO cross section, while at and TeV the uncertainties of the NLO and cross sections are more similar. This is due to a cancellation of the factorization scale dependence of and at and TeV that is not present at and TeV. This also explains why the scale variation of approximate NNLO cross section without contribution is similar to (less than) that with the contribution at and TeV ( and TeV.) Finally, with at TeV, the zero in the scale uncertainty of indicates that the factor of of both the factorization and scales increase the cross section.
Although there is no significant difference between the cross section prediction of and , we consider to be our best prediction for the LHC production cross sections, since for the most part has less scale variation than . By comparing the NLO and approximate NNLO, , cross sections, it is apparent that the effect of the higher order corrections is to increase the cross section less than pb at TeV and less than pb at TeV, while reducing the theoretical uncertainty from scale variations. The matched NNLL cross section increases the NLO cross section by similar amounts, with a slightly increased scale uncertainty. There is very little difference between using a fixed factorization scale and a dynamic factorization scale. It appears that the prediction for the cross section is under good theoretical control.
Now that the Higgs boson is discovered, a full exploration of the electroweak sector has begun. An important signal for this exploration is the pair production of gauge bosons, in particular production. This signal is a major background to and is sensitive to the electroweak gauge boson triple coupling, which directly probes the mechanism of electroweak symmetry breaking and the gauge structure, respectively. In order to be sensitive to new physics in the signal and measure the decay channel well, it is important to have accurate and precise theoretical predictions for production cross section and differential distributions.
In this paper we resummed large logarithms associated with soft gluon emission at partonic threshold, , at NNLL order for pair production. This resummation was performed using the formalism of SCET Bauer et al. (2000, 2001, 2002); Beneke et al. (2002) which allows for the resummation directly in momentum space Becher et al. (2007); Becher and Neubert (2006). The NNLL resummed results were then matched onto the known NLO results Frixione (1993); Ohnemus (1991). We also calculated the approximate NNLO cross section. We thus obtain the most accurate cross sections and invariant mass distributions for production that have been calculated to date.
We found that the effect of the threshold resummation on the invariant mass distribution was to increase the differential cross section in the peak region for both and TeV. The matched NLO+NNLL and approximate NNLO cross section both increased the NLO cross section by for a factorization scale central value and for a central scale of , within the theoretical uncertainties. The theoretical uncertainties of the approximate NNLO cross section were generally decreased relative to those of the NLO cross section. These results indicate that the sideband analysis used for background estimation to signal CMS Collaboration (2013); ATLAS Collaboration (2013) is not significantly altered by higher order corrections. Also, the strong coupling constant perturbative expansion of the production cross section is firmly under theoretical control.
S.D. and I.L. are supported by the U.S. Department of Energy under grant No. DE-AC02-98CH10886. M.Z. is supported by the National Science Foundation, grant PHY-0969739. We thank George Sterman and Andrea Ferroglia for helpful conversations.