High precision predictions for exclusive production at the LHC
We present a resummation-improved prediction for jets at the Large Hadron Collider. We focus on highly-boosted final states in the presence of jet veto to suppress the background. In this case, conventional fixed-order calculations are plagued by the existence of large Sudakov logarithms for which lead to unreliable predictions as well as large theoretical uncertainties, and thus limit the accuracy when comparing experimental measurements to the Standard Model. In this work, we show that the resummation of Sudakov logarithms beyond the next-to-next-to-leading-log accuracy, combined with the next-to-next-to-leading order calculation, reduces the scale uncertainty and stabilizes the perturbative expansion in the region where the vector bosons carry large transverse momentum. Our result improves the precision with which Higgs properties can be determined from LHC measurements using boosted Higgs techniques.
With the confirmation of the existence of a Higgs particle by both the CMS and ATLAS collaborations AtlasHiggs (); CMSHiggs () at the Large Hadron Collider (LHC), one of the main objectives of the LHC is to test as many of its properties as possible. The experimental discovery channel is based on its bosonic decay only, and it becomes crucial to probe its couplings to fermions directly to verify whether they agree with Standard Model predictions. However, the standard Higgs production channel of gluon fusion suffers from large QCD backgrounds in its fermion decay modes. The so-called Higgsstrahlung process provides another important way of Higgs measurement at hadron colliders, where the Higgs is produced in association with either a or vector boson (we will refer to them generally as ). It was the main search channel for a light Higgs at the Tevatron. At the LHC, the large background from the semi-leptonic decay of the process obscures the Higgs signal and leads to a poor signal to background ratio. However, recent studies Butterworth:2008iy () have shown that with techniques such as jet substructure and large transverse momentum cuts on the Higgs and the vector boson, the Higgsstrahlung process offers a viable alternative to study the fermion decay channel Chatrchyan:2013zna (); Chatrchyan:2012ww (); TheATLAScollaboration:2013lia (); Aad:2012gxa () as well as possible invisible decays of the Higgs CMS:2013yda (); ATLAS:2013pma (). To further suppress the unwanted backgrounds, both CMS and ATLAS adopt a jet veto procedure allowing no extra jets with greater than .
Though effective in the experimental analysis, on the theory side, imposing a jet veto with much less than the center of mass energy at which the hard processes take place leads to large Sudakov logarithms that can destabilize the perturbative series in conventional perturbative QCD calculations. It has been shown that the impact of QCD corrections is sizable for events with and boosted at large in the presence of jet veto111The imposed jet veto makes the lower cut on the vector boson equivalent to that on the Higgs in the boosted region. in Ref. Ferrera:2011bk (), where the fully exclusive cross section for production up to next-to-next-to-leading order (NNLO) in QCD was studied. The jet veto reverses the sign of the higher-order QCD correction compared to the inclusive case, and reduces the total cross section. Recently there have been works on the transverse-momentum resummation of Dawson:2012gs () and on the jet veto resummation Shao:2013uba () for the Higgsstrahlung process. Neither study addressed the impact of the jet veto as a function of the transverse momentum of or , which is required to understand the QCD corrections in the boosted phase-space region.
In this manuscript, we present the fully exclusive cross section for production at NNLO in QCD matched with the resummation of jet veto logarithms at the partial next-to-next-to-leading-log prime () accuracy and study the relevant theoretical uncertainty. We follow the counting of logarithmic accuracy according to Ref. Stewart:2013faa (), and the subscript stands for “partial”, indicating that we have not included the non-logarithmically enhanced contributions at , the effect of which is found to be small. In the small-jet-radius limit, our calculation is expected to be comparable with the accuracy. Compared to Ref. Shao:2013uba (), we have achieved higher accuracy in both the resummation and the fixed order calculation. In addition, we follow the experimental analysis more closely by studying the transverse momentum dependence of the cross section and focusing on the impact of the resummation in the boosted region of and .
Early effort in resuming jet veto was first investigated in Ref. Berger:2010xi () using the global variable beam thrust. The resummation of jet veto logarithms to here relies on recent developments in the soft collinear effective theory (SCET) Bauer:2000ew (); Bauer:2000yr (); Bauer:2001ct (); Bauer:2001yt (); Bauer:2002nz (), which has been applied successfully to the production Becher:2012qa (); Becher:2013xia (); Stewart:2013faa (). Using the effective theory, the case was studied in Ref. Liu:2012sz () and Ref. Liu:2013hba () for and was extended to the challenging low jet region in Ref. Boughezal:2013oha (). A complete framework for the combination of resummed results for production processes in different exclusive jet bins was also presented in Ref. Boughezal:2013oha (). A systematic study on the jet veto clustering effects up to in the small-jet-radius limit was carried out in Ref. Alioli:2013hba ().
The NNLO QCD correction is obtained by modifying the numerical code FEWZ Gavin:2010az (); Gavin:2012sy (); Li:2012wna (), originally used to calculate the DY process. The heavy-quark induced process is not included. Higgs production in association with a vector boson is dominated by DY like processes up to next-to-leading order (NLO) in QCD. At NNLO, there are small contribution from the process where the Higgs is produced via heavy quark loop induced by gluon pairs. Its contribution is found to be around 1%(1%) for and 5%(9%) for inclusive cross section for an 8 TeV (14 TeV) LHC Brein:2012ne (). This is expected to be further suppressed in the region where the vector boson and Higgs carry large transverse momentum222for the process, this subleading process actually is enhanced below the top pair threshold, but here we will only focus on the resummation on its DY like contribution for the purpose of this manuscript.. Furthermore, the decay of the vector boson into lepton final states is available as a result of the nature of the original FEWZ. The modification is validated via a series of numerical checks against MCFM Campbell:2010ff () for differential observables at NLO, Sherpa Gleisberg:2003xi () for plus 1 or 2 jet(s) exclusive results, as well as VH@NNLO Brein:2012ne () for inclusive cross sections. In this study, we integrate over the lepton phase space inclusively for better numerical stability. We found that the scale uncertainty gets reduced and the convergence of the perturbative series is improved dramatically after resumming the jet veto logarithms. The results of our analysis should stay largely unaffected in the presence of standard experimental acceptance cuts due to the absence of new phase space singularities at the given perturbative order.
Our manuscript is organized as follows. In section II, we briefly review the theoretical set-up in the resummation of jet veto logarithms. In section III, we present the numerical consequence of the resummation for both and production at the LHC. We adopt cuts similar to the current experimental analyses and concentrate on the highly-boosted region. Finally, we conclude in Section IV. All necessary technical details are given in the Appendix.
Ii Review of the theoretical formalism
In this section, we briefly review the theoretical framework for understanding jet vetoed cross sections at hadron colliders. The formalism has been established in a series of works for both jet and exclusive jet based on either the QCD coherence argument Banfi:2012yh (); Banfi:2012jm () or the effective theory analysis Becher:2012qa (); Becher:2013xia (); Stewart:2013faa (); Liu:2012sz (); Liu:2013hba (); Boughezal:2013oha (). Our approach relies on the recently developed works in SCET Bauer:2000ew (); Bauer:2000yr (); Bauer:2001ct (); Bauer:2001yt (); Bauer:2002nz (). Here we will only highlight the formulae used in our calculation and we refer the readers to Refs. Becher:2012qa (); Becher:2013xia (); Stewart:2013faa (); Liu:2012sz (); Liu:2013hba () for a detailed discussion and derivation of the factorization theorem for jet vetoed cross sections and to Refs. Campbell:2013qaa (); Heinemeyer:2013tqa () for short reviews. We group all the necessary ingredients for resummation in the Appendix.
ii.1 factorization, resummation and matching
When is much less than the energy scale which characterizes the production process, the cross section can be approximated by a factorized piece with an additional term,
up to contributions suppressed by , where is the jet size parameter for the clustering algorithm. Here is the Born level partonic cross section initiating the relevant process. For instance for production the Born level cross section comes from the tree level process and . Here and are the invariant mass square and rapidity of the entire final state, respectively. The Bjorken scale parameter is given by
with denoting the machine center of mass energy. Here , and are the hard function, beam function for describing the collinear radiations in the forward region and soft function for the radiations with low energies. Their field theoretic definition can be found in Ref Becher:2012qa (). Their explicit form up to is given in the Appendix.
Due to the homogenous expansion in where the soft and the collinear modes share the same scaling in their virtualities, other than the normal divergence regulated by the renormalization scale in the dimensional regularization, there exists additional rapidity divergences which are regulated by a new fictitious scale in both the beam and soft functions Chiu:2011qc (); Chiu:2012ir (). The regularization scheme gives rise to a new renormalization group equation in SCET, which eventually allows us to sum up the full set of large logarithms of the form to all orders, by evolving each function in Eq. (II.1) from its natural scales to the common scales . The scales are estimated by demanding that in each function the perturbative series behaves properly. Therefore, we have
Here choosing the imaginary scale allows us to sum up a tower of large terms for time-like processes Ahrens:2008qu ().
Eq. (II.1) also contains a non-factorizable correction term,
which contributes at the level especially when . As argued in Refs Banfi:2012yh (); Banfi:2012jm (), the overall coefficient determines the complete logarithmic series coming from this term. Therefore, we follow the procedure in Ref. Stewart:2013faa () to include this piece in the resummed cross section at by multiplying it with the total evolution factor.
The SCET cross section Eq. (II.1) only gives the exact result when . When away from zero, the missing power-suppressed term may have a sizable effect on the cross section. In order to recover its contribution, we have to match the SCET calculation onto the fixed-order QCD cross section. Here we adopt the most straightforward matching scheme, in which we subtract the most singular terms in predicted by SCET up to two loops, from the NNLO QCD cross section, and replace that by the along with resummed one (see, section III). The final result for the resummation-improved cross section is
Here is the NNLO cross section obtained based on a modification of the FEWZ code, detailed in Appendix A. The SCET expanded cross section contains all the order singular contributions to , and denotes the cross section for with resummation. All the building blocks needed for and can be found in the Appendix.
ii.2 dependent contributions
For a complete -resummation, all the matrix elements should be computed up to (see for instance, Ref. Stewart:2013faa ()). Starting from the order, clustering effects will give rise to dependent contributions, which, for small jet radius, dominate over the other non- enhanced terms of the two loop matrix elements. The corrections at appearing in both the soft and beam functions comes from the clustering of two correlated radiations coming from a single collinear splitting of distance roughly into two different jets. They can therefore be extracted from the strongly ordered collinear behavior of QCD, given by
where, is the one loop soft or beam function with properly regularizing the divergence in the effective theory and is the one loop splitting function. The transverse momentum is with respect to the mother particle before splitting and is the phase space measure which accounts for the jet algorithm. We note that an additional symmetry factor should be included when evaluating or with the gluon or the quark further splitting, respectively, in the beam function.
Using the fact that in the small limit,
and writing out explicitly the measure , we get
Using the equation above, all the order terms, either in the anomalous dimensions or in the matrix elements, can be calculated in a straightforward way. For our purposes, we have the following independent terms in the beam function:
where is the splitting function which can be found in the appendix.
We note that other than the contributions, we also need the non- enhanced independent terms at for claiming accuracy. The analytic formula for the full two loop soft function including these contributions is known Stewart:2013faa (), and is listed in the appendix. For the beam function, those terms can be obtained easily by comparing the fixed two loop SCET expansion with the full QCD calculation. We found that for , these contributions are numerically negligible for both large and small , therefore in this work we do not include their contributions and denote our accuracy as .
Iii Numerical results
We start the section by demonstrating the validity of the SCET factorization theorem numerically. Later on, we will introduce our scale choices and our strategy for the theoretical uncertainty estimation.
In Fig. 1, we show the comparison of the cross section with a jet veto between the QCD prediction and the SCET calculation up to NNLO in . The results are given as a function of for both and production at the LHC with machine energy (LHC14). The jets are constructed using anti-k jet algorithm with . For small , the two calculations differ by a non-logarithmically enhanced constant piece (independent of ) at , in addition to power suppressed terms. The numerical results presented in Fig. 1 shows almost no differences between them for small . The differences are found to be around or less than per mil level.
As increases, the size of the power-suppressed term starts to grow and becomes important when is comparable with . The power suppressed term is properly included in our prediction via the matching procedure, Eq. (5), described in the previous section.
The growth of the power suppressed contribution leads to the idea of profile scales Liu:2013hba (); Stewart:2013faa (); Ligeti:2008ac (); Abbate:2010xh () that smoothly turn off resummation and merge with the fixed-order predictions in the high region where the effective theory breaks down and the power suppressed terms dominate. However, in our present analysis, we focus a fixed of and study boosted production. We expect that for such a small , the singular contributions dominate and must be resummed. Therefore instead of using profile scales, we parametrize the scales using their canonical values in SCET:
as our central scale choice. The imaginary hard scale choice allows a better convergent perturbative series in evaluating the hard function and a resummation of a towers of terms.
We default to
as the central scale for the fixed order calculation. We vary the scales up and down by a factor of two to estimate the scale uncertainty. For the fixed order results, we use the “Stewart-Tackmann-prescription” for the error estimation Stewart:2011cf ():
where and are the uncertainties for total and inclusive cross sections, respectively, obtained by varying the scale up and down by a factor two. For the resummation improved predictions, we use
where is computed by varying all the scales in the effective theory collectively. For , we vary each scale and independently and take the envelope of all the variations.
We now turn to discuss the main results of this work. In presenting our numerical results, we use the (N)NLO MSTW2008 PDF set Martin:2009iq () for (N)NLO and results and set the value and running of the strong coupling constant accordingly. We use for the active number of quark flavors, and choose the scheme as the electroweak input parameters throughout our analysis. We will only discuss and productions at LHC14 in this manuscript. Jets are formed with anti- jet algorithm333Here anti- is selected for illustrative purposes only. There is no theoretical difficulty for us to switch to other type jet algorithms, for instance Cambridge/Aachen, and the conclusions of this work will not be affected. with Butterworth:2008iy () and are vetoed if they have transverse momentum greater than .
We first study the stability of the perturbative predictions with and without the improvement from the jet veto resummation. Fig. 2 illustrates the impact of the resummation on the convergence of the perturbative cross section predictions at LHC. Here we plot the cross section as a function of the lower transverse momentum cut of the boson . It is clear that the resummation of the jet veto logarithms (right panel) accelerates the convergence of the perturbative series compared to the pure fixed-order ones (left panel), as the curve from the prediction almost overlaps with the one, while the NLO prediction is off by a visible amount with respect to the NNLO result.
The same trends can be observed for the predictions of the production at LHC which are depicted in fig. 3. The convergence of the perturbative expansion is greatly improved after the resummation.
In both fig. 2 and fig. 3, the error bands reflect the scale uncertainties estimated using the prescriptions sketched previously in Eq. (12) and Eq. (13) for the pure fixed-order and the resummation improved predictions, respectively. Although the resummed predictions usually have more conservative handles over the perturbative uncertainties due to variation of multiple scales in the resummed cross section, we still observe sizable reductions in the theoretical errors after invoking the resummation.
To see the reduction more clearly, we plot the NNLO and predictions together in fig. 4 for both and productions. The pure fixed-order and the resummed calculations yield central values close to each other, yet the resummed cross section comes along with a reduced scale dependence. For the experimentally interesting region ( GeV), the scale dependence drops from to for () production at LHC, i.e. the theoretical error band shrinks by around , and the veto efficiency for and processes are roughly 47% and 41% respectively.
Recently, significant effort has been devoted to improved understanding of the Higgs cross section at the LHC, either using conventional NNLO calculations Boughezal:2013uia () or by utilizing resummation techniques Banfi:2012yh (); Banfi:2012jm (); Becher:2012qa (); Becher:2013xia (); Stewart:2013faa (); Liu:2012sz (); Liu:2013hba (); Boughezal:2013oha (). In this work, we investigated the associated production process at the LHC. On the experimental side, a jet veto procedure demanding no jets with transverse momentum larger than is used by both CMS and ATLAS to suppress the backgrounds in the boosted Higgs analysis. However, theoretically the small value of often destabilizes the perturbative expansion due to the existence of large logarithmic structure in the series. This makes the perturbative prediction unreliable and results in large theoretical uncertainties on the cross section as the of the vector boson/Higgs increases. For production, when the transverse momentum of the vector boson is required to be larger than , the perturbative predictions suffer from roughly errors for LHC even at the NNLO. The limited power of the theoretical predictions restricts the accuracy that experimentalists can achieve. Also the large K factors in the presence of jet veto when one goes from LO, NLO to NNLO leads to the concern that the missing higher orders corrections may still be large and have sizable contributions to the cross section.
Here we improved the theoretical predictions through the resummation the jet veto logarithms and a series of terms up to accuracy within the SCET framework. We further matched the resummed result onto the NNLO calculation to provide the full resummation improved cross section for production at hadron colliders. The improved results reduce the theoretical uncertainties. In the highly boosted regime, the scale uncertainty drops from to or below from NNLO to for LHC. Meanwhile resumming the jet veto logarithms greatly improves the convergence of the perturbative series in the region of interest experimentally. It can be seen from the fact that the central value remains virtually unchanged when one moves to higher orders in perturbative calculations, proving the reliability of the theoretical predictions. After resummation, the jet veto efficiency is found to be and for and , respectively.
In this manuscript, we highlight the general features of the resummation improved predictions for associated production, and show the power of resummation in reducing scale uncertainties and providing reliable central values. For illustration purpose here, we have chosen simplified cuts from the current experimental analysis without loss of generality. The results can be extrapolated for experimental use. In the future, extensive numerical studies on production with more practical cuts will be pursued in an upcoming paper, based on the scheme in this work.
We thank Frank Petriello for many helpful suggestions and carefully reading the manuscript. We are grateful for discussions and kind supports from Stefan Höche on cross-checking FEWZ. We thank Jike Wang for explaining current experimental status. X.L. would like to thank Yin Cui for all the supports when this work was carried out. Y.L. was supported by the US Department of Energy under contract DE–AC02–76SF00515. X.L. was supported by DE-AC02-06CH11357 and the grants DE-FG02-95ER40896 and DE-FG02-08ER4153. Y.L would like to thank for the hospitality of Northwestern University during the completion of this work. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
Appendix A modification of FEWZ
Here we give a brief introduction of the master formula we use to modify FEWZ for the Higgsstrahlung process. The squared matrix element of the DY process in FEWZ can be schematically written as,
where is the square of the quark current and includes all QCD corrections, and is the square of the lepton current. As long as no observables related to asymmetry in the lepton phase space, we can neglect the axial part of the current, hence
When integrated over the lepton phase space inclusively, we find that,
with as the momenta of the two leptons and as the momentum of the vector boson. In original FEWZ, individual components of are not separately available since they only appear in the product of two currents in the matrix element. For the DY-like part of the production, it can be thought as a gauge boson first produced via the DY process, and subsequently decaying to and , where further decays to two leptons. Because of gauge invariance, we can write the new squared matrix element for Higgsstrahlung process as,
where denotes the square of the new lepton current. If we integrate inclusively over the and phase space, the new lepton current becomes,
in which and are the momenta of and respectively, and . The second piece drops out when multiplied with the quark current, and therefore we arrive at,
However, Eq. (19) does not apply to the case where acceptance cut is placed on leptons, the vector boson or the Higgs. We can use
in combination with Eq. (16) and obtain
The average is performed by integrating over the original lepton phase space inclusively, and arbitrary cuts can be applied on the new vector boson and Higgs, as well the final leptonic final states, as long as no asymmetry related observables are measured. It can be seen that Eq. (19) is easily recovered by substituting the integrated form of into Eq. (21). Because only the product appears in Eq. (21), FEWZ can be modified in a relatively straightforward way.
Appendix B Fixed-order matrix elements
In this appendix, we list all the ingredients needed for a resummation for production in -jet bin. We start with the fixed order matrix elements to
b.1 Hard Function
The spin and color averaged LO matrix elements squared for processes and are proportional to that of LO DY processes,
with , and . Therefore
where and are the masses for and bosons, respectively and and are the widths.
The NNLO hard function for Drell-Yan has been known for a while and can be found for instance in Becher:2007ty (), which gives
Here we have abbreviated .
b.2 Soft Function
The full two-loop soft function can be extracted from Ref. Stewart:2013faa () by suitably replacing the color factors, and is found to have the form
with , and .
The dependence on the jet algorithm starts to enter at two loops through the two-loop anomalous dimension, , which determines the coefficient of the single logarithm of , as well as the non-logarithmic two-loop soft constant, . For the coefficients of the soft non-cusp anomalous dimensions we find
Here, is the clustering correction due to the jet algorithm:
up to corrections whose explicit form can be lifted from Ref. Becher:2013xia ().
The two-loop soft function constant , which is not determined from RGE constraints, is
b.3 Beam Function
When matching onto pdfs, the NNLO beam function has the form
where the matching kernels read
given that and .
The form of the two-loop matching coefficient can be obtained by expanding the resummed beam function and matching onto PDFs, which results in