Bottom-induced contributions to Higgs plus jet at next-to-next-to-leading order
PPrepared for submission to JHEP
Bottom-induced contributions to Higgs plus jet atnext-to-next-to-leading order
Roberto Mondini and Ciaran Williams
Department of Physics,University at Buffalo, The State University of New York, Buffalo 14260, USA
E-mail: [email protected] , [email protected] Abstract:
We present a next-to-next-to-leading order (NNLO) QCD calculation of thebottom-induced contributions to the production of a Higgs boson plus a jet, i.e. the process pp → H + j to O ( y b α s ) . We work in the five-flavor scheme (5FS) in which the bottom quarkmass is retained only in the coupling to the Higgs boson. Our calculation uses N -jettinessslicing to regulate infrared divergences, allowing for fully-differential predictions for colliderobservables. After extensively validating the methodology, we present results for the 13 TeVLHC. Our NNLO predictions show a marked improvement in the overall renormalizationand factorization scale dependence, the latter of which proves to be particularly troublesomefor 5FS calculations at lower orders. In addition, using the same methodology we presenta NNLO computation of bb → H . Our results are implemented into MCFM. a r X i v : . [ h e p - ph ] F e b ontents bb → H at NNLO 83.2 bb → H + j at NLO 93.3 bb → H + j at NNLO 11 Since its discovery nearly a decade ago [1, 2], the Higgs boson has become an establishedpart of the particle physics landscape, and a significant amount of research effort has beendevoted to the greater understanding of its properties. After the initial establishment of itsintrinsic properties such as its CP and spin [3, 4], the focus has shifted to obtaining precisionmeasurements of the couplings of the Higgs boson to the other particles of the StandardModel (SM) as well as to itself. The Higgs boson self-coupling is a particularly pressingmeasurement to obtain, since it will allow for a more detailed study of the electroweak sym-metry breaking potential in the Standard Model. Although the Higgs self-coupling is fullypredicted from known parameters in the Standard Model, it is often sensitive to extensionsof the SM (referred to as BSM) that change the nature of the electroweak potential. Sim-ilarly, it is also of vital importance to constrain the couplings of the Higgs boson to otherparticles in the Standard Model, namely the W and Z bosons and massive fermions. TheLHC has made significant progress over the last decade [5, 6], and will continue to improveupon existing results over the forthcoming Run III. Further in the future, measurementswith sub-percentage uncertainties will require a collider with upgraded capabilities, forwhich serious planning is now underway [7, 8]. In all of these endeavors, precision predic-tions for differential distributions in the Standard Model are critical in order to avoid thesituation in which theoretical uncertainties become the dominant source of error.– 1 –f the Higgs interactions, one of the most fascinating to study is the coupling betweenthe Higgs boson and third-generation fermions, the top and bottom quarks and tau leptons.Among these, the bottom quark is unique, as there are two ways to gleam insights into itscoupling: through Higgs boson production or its direct decay to the quarks themselves.Given the large hierarchy in mass, the top quark dominates Higgs boson production atthe LHC through the gluon-fusion mechanism and therefore probing the bottom Yukawacoupling through Higgs production is challenging. On the other hand, the SM Higgs bosoncopiously decays to bottom quarks with a branching fraction of around 50%, and thereforethe bottom Yukawa coupling dominates the Higgs decay width, propagating to all other(on-shell) measurements of the Higgs boson at the LHC. Determining the bottom-Higgscoupling as precisely as possible is thus an essential requirement of the future experimentalhigh-energy physics program. For example, in extensions of the SM, extended Higgs sectorstypically modify the coupling of the 125-GeV Higgs boson to up- and down-type fermions,and can lead to enhanced production cross sections [9–12].Given the immense interest, there have been many theoretical studies of processesinvolving the Higgs boson and the bottom quark at hadron colliders. When making pre-dictions at the LHC, one must first decide how to handle the mass of the bottom quark.Since the bottom quark is heavier than the proton, a natural choice is to keep the mass ofthe bottom quark in the calculation and exclude it from initial-state contributions. Thisscheme is known as the four-flavor scheme (4FS), due to the number of active initial-stateflavors in the proton. The leading-order production mechanism at the LHC in the 4FS isthus the process gg → bbH (plus a sub-dominant qq -initiated contribution). The 4FS hasthe advantage that no approximations are made in regards to the kinematics, which is par-ticularly helpful in relation to final-state bottom-quark tagging, since single b -tagged jetscan be isolated without theoretical issues, i.e. no jet cuts are required even though the finalstate contains two partons. However, a major drawback of the 4FS scheme arises from theoccurrence of large logarithms of the form α s log( m b /m H ) . When computing cross sectionsas perturbation series directly in α s , these logarithms induce large corrections and spoil theconvergence of the series. One way to ameliorate this problem is to resum the logarithmswhen possible. Initial-state collinear logarithms can be resummed into the parton distribu-tion functions (PDFs). This introduces the five-flavor scheme (5FS) in which the bottomquark contributes to the initial-state PDFs, and the mass of the quark is neglected in theremaining kinematics. At leading order, Higgs boson production in the 5FS thus proceedsdirectly through bottom-quark fusion bb → H . It is worth noting that, since final-statecollinear splittings g → bb are not resummed, care must be taken in the 5FS when applying b -tagging requirements and comparisons are made to experimental data (i.e. one should tryto remove jets arising from gluon splitting from the experimental analyses).Given this theoretical richness, there have been many detailed calculations, phenomeno-logical studies, and comparisons of the 5FS and 4FS at various orders in α s , matching toparton-showers, and resummed predictions. We refer the interested reader to refs. [13–32]for further details. A particularly impressive calculation is the recent achievement of next-to-next-to-next-to-leading order (N3LO) accuracy for the total cross section in the 5FS [33],which has been subsequently matched to the NLO 4FS result [34].– 2 – further intricacy relating the 5FS to the 4FS comes from the dependence on thefactorization scale through the bottom-quark PDF in the 5FS. It was noted in the earliestcalculations of bottom-quark fusion at NLO in the 5FS [13, 14] that the higher-order cor-rections were large and resulted in cross sections that could have differences of an order ofmagnitude from the 4FS LO result. Detailed analysis in ref. [16] (following the argumentsof ref. [35]) illustrated that the choice of a central factorization scale of m H was too high forthe process, and that a scale of around m H / was more appropriate to ensure the reliabilityof collinear factorization. Predictions made with a central factorization scale choice in thisregion showed much better perturbative convergence, and a broader compatibility with the4FS result. Nevertheless, a strong dependence on the unphysical factorization scale shouldbe seen as a negative feature of the 5FS when used at LO and NLO. Subsequent NNLOand N3LO predictions for bottom-quark fusion [21, 22, 33] show a significant reduction ofthis problem and motivate our computation of H + j in the 5FS at NNLO.On the Higgs boson decay side the theoretical situation is also under good control.Again, there are two scheme possibilities which can be considered as the decay versionsof those discussed above. A commonly-used approximation is to retain the mass of thebottom quark only in the coupling to the Higgs boson (the decay equivalent of the 5FS),and there have been many detailed theoretical studies of this process (see e.g. [36–38]),which is now known up to O ( α s ) inclusively [39] and at N3LO differentially [40]. Increasingthe complexity of the calculation, one can include the mass of the bottom quarks fully (the4FS equivalent), and in this setup recent calculations have pushed the accuracy to NNLOfor fully-differential predictions [41, 42].The continuing maturation of the experimental analyses at the LHC has had a twofoldimpact on Higgs boson studies. Firstly, the increased statistics and precision have allowedfor an extensive range of Higgs boson observables to be studied, including Higgs-plus-multiple-jet production (see e.g. [43, 44]). Secondly, comparison of data to theory has high-lighted the need for increased precision on the theoretical front, emphasizing the importanceof NNLO predictions in QCD. Over the last few years significant progress has been made,resulting in several independent calculations of Higgs-plus-jet at NNLO in the effective fieldtheory (EFT) in which the top quark is integrated out [45–51]. Impressively, in ref. [52] theaccompanying jet was integrated out of the calculation, allowing for a computation of the pp → H + X differential cross section at N3LO accuracy.Our aim in this paper is to provide a similar level of theoretical accuracy for the bottomquark-initiated contribution to H + j as it is present for the dominant EFT productionmechanism. In order to do so, we will work in the 5FS and treat the bottom quark asmassless everywhere except in the coupling between the Higgs boson and the bottom quarks.The different calculations of H + j at NNLO available in the literature used a variety ofinfrared-regulating techniques. For brevity we do not describe them all in detail here, butfocus on the pieces pertinent to this paper. One of the initial calculations of H + j [47] useda non-local slicing procedure (based on the event shape N -jettiness). The results obtainedin this paper seemed to be in conflict with those obtained in other studies based upon localsubtraction techniques. A detailed study in ref. [51] showed that the two methodologies doindeed produce the same results, but that power corrections arising from the approximate– 3 – igure 1 . Top row: representative Feynman diagrams in the 5FS which contribute to the process pp → H + j at NNLO, regardless of b -tagging requirements. Bottom row: representative Feynmandiagrams in the 5FS which contribute to the process pp → H + j at NNLO, but fail b -taggingrequirements. form of the factorization formula used in the slicing procedure must be handled carefully.In this paper we will use the same methodology and follow the same techniques as shown inref. [51] to control and estimate the remaining power corrections. In ref. [51] the gg channelwas shown to have the worst power corrections for EFT H + j production. Thankfullyfor our calculation, this channel does not appear at leading order and we therefore expectpower corrections to be easier to control.Our paper proceeds as follows. In section 2 we present a brief overview of the technicaldetails of our calculation, while section 3 discusses its validation. We present results for the13 TeV LHC in section 4, and finally we draw our conclusions in section 5. The primary focus of this paper is the calculation of the NNLO QCD corrections to thebottom-induced contributions to Higgs plus one jet at the LHC. Given its phenomenologicalrelevance and role as a check of our calculation, we will also present results for the bottomquark fusion process at NNLO (i.e. bottom-induced contributions to Higgs plus zero jets).As discussed in the introduction, the most important theoretical choice when consid-ering bottom-quark processes at hadron colliders is how to treat m b , i.e. whether to workin the 5-flavor (5FS) or 4-flavor (4FS) scheme. In this paper we work in the five-flavorscheme, which will allow us to extend the computation to NNLO accuracy. Representa-tive Feynman diagrams relevant for our calculation at this order are shown in fig. 1. Werecall that in the 5FS the bottom quark mass is taken to zero and bottom quarks havea non-zero contribution to the PDFs. At first glance, the 5FS scheme may appear not tobe useful for computing H + b related processes, since by setting the bottom quark massto zero the bottom Yukawa coupling should also be taken to zero. In order to circumventthis problem, we work in the mixed-renormalization scheme, in which the bottom quarkYukawa is taken in the MS scheme, and the bottom quark mass, used in propagators and– 4 –n the relativistic kinematics, is taken in the on-shell scheme. This scheme allows one totake the limit m OS b → while keeping the Yukawa coupling non-zero. This scheme has twoadvantages in QCD calculations. Firstly, it allows for a robust definition of the 5FS for H + b amplitudes. Secondly, by evolving the scale in the running Yukawa coupling to µ R (i.e. m H ), one avoids large logarithms which arise in the OS scheme at higher orders, andas a result the perturbative corrections are under better control. Downsides of the mixedscheme include breaking the relationship between the OS mass and the MS mass [53, 54]and an inability to consistently renormalize higher-order corrections in the electroweak cou-pling [31]. Nevertheless, the reduction in sensitivity to collinear initial-state logarithms (atthe cost of a strong dependence on the factorization scale at LO), and the ability to pursuehigher-order corrections, renders the 5FS along with the mixed renormalization scheme avery useful theoretical construct for LHC computations. For the bottom-induced H + j process at NNLO, three phase-space topologies contribute(see fig. 1), corresponding to the double-virtual, real-virtual, and double-real corrections tothe underlying LO topology. UV and IR singularities are present at this order and must beappropriately renormalized and regulated. We describe the calculation of the various UV-renormalized matrix elements for each phase-space configuration in ref. [55] for the decay H → bbj at NNLO. This leaves the discussion of the IR regulation, which is different fromthat described in ref. [55] due to the LHC kinematics.In order to regulate the IR divergences present at this order we use the N -jettinessslicing approach [56, 57]. This method has become an established technique for evaluatingNNLO cross sections involving final-state jets at the LHC [51, 57–59], and we provide abrief overview in this section. The central idea is to separate the (differential) cross sectionof a process into two pieces, σ NNLO = σ ( τ N ≤ τ cut n j ) + σ ( τ N > τ cut n j ) , (2.1)where the variable τ N is the N -jettiness variable [60]. For our 1-jet example, this variableis defined as τ = (cid:88) m min i p m · k i P i , (2.2)where { p m } is the set of all partonic momenta in an event, while { k i } are the momenta ofthe two incoming beams and the hardest jet present in the event (after clustering). Thequantity P i is a somewhat arbitrary choice of hard scale, and in our calculation we take P i = 2 E i (known as the geometric measure [61, 62]). The above-cut term σ ( τ N > τ cut n j ) hassufficiently large value of the N -jettiness variable to have at most one unresolved parton, andtherefore corresponds to a NLO computation of the cross section with an additional partonpresent. The below-cut term σ ( τ N ≤ τ cut n j ) contains all of the double-unresolved limits atNNLO. However, in the limit τ cut1 → the cross section can be approximated using the– 5 –ollowing factorization theorem from Soft-Collinear Effective Field Theory (SCET): σ ( τ ≤ τ cut n j ) = (cid:90) τ cut nj dτ S ⊗ n j (cid:89) i =1 J i ⊗ (cid:89) a =1 , B a ⊗ H + F ( τ cut n j ) , (2.3)where in our case n j = 1 . The above equation is valid up to power corrections (denoted bythe F ( τ cut n j ) term), which vanish in the limit τ cut n j → . At NLO the leading power correctionsare well described by the form τ cut n j log( τ cut n j /Q ) , and at NNLO the leading power correctionshave the form τ cut n j log ( τ cut n j /Q ) (where in both cases Q is the hard scale associated withthe process). The general terms that enter the SCET factorization theorem are the soft( S ), jet ( J ), and beam ( B ) functions, for which calculations accurate to O ( α s ) needed forour calculation can be found in refs. [63–68].There are several alternative choices [51, 59] one can make when applying the jettiness-slicing method. Firstly, one can choose whether to work with a fixed version of τ cut1 , in whichall events are compared to a given energy scale, or with a dynamical definition, in whichthe final-state kinematics (of the clustered system) generates different τ cut1 values for eachphase-space point. Typically, for 1-jet NNLO processes it is more prudent to use the latteroption. Since power corrections are sensitive to the overall hardness of the system throughthe expansion parameter τ cut1 /Q , very energetic jets have suppressed power corrections. Byusing a fixed τ cut1 , the calculation for these terms includes points that are very soft andcollinear (relative to the hard scale), resulting in large Monte Carlo uncertainties and codeinstabilities. On the other hand, using a dynamic τ cut1 ensures a more relaxed τ cut1 for moreenergetic jets, reducing this problem and producing more stable results, without increasingthe impact of unwanted power corrections.In order to obtain the remaining process-specific hard function ( H ) appearing in eq. (2.3),we use our double-virtual calculation for the decay amplitude H → bbg presented inref. [55] . The result for LHC kinematics is obtained by performing the relevant crossing,moving the desired final-state partons to the initial state. In practical terms, this involvestaking the appropriate analytic continuation of the various harmonic polylogarithms thatappear in the virtual amplitudes as described in section 4 of ref. [70]. After crossing therelevant final-state partons to the initial state, we have checked that our results have thecorrect factorization properties in the relevant soft and collinear limits [71, 72], findingexcellent agreement. The Standard Model does not allow for the consideration of the impact of a single fermiongeneration in isolation. For the purposes of this calculation, in order to completely specifyour theoretical framework we must also address the role of the top quark in the computation.This is because at O ( α s ) the cross section becomes sensitive to the presence of the topinduced production. One must therefore specify whether one works in the effective fieldtheory or full Standard Model. Precision calculations in the full Standard Model are made See also ref. [69]. – 6 –onsiderably more difficult by the presence of the additional mass scale and are currentlyknown to NLO accuracy for H + j [73]. On the other hand, NNLO predictions are availablein the EFT [45–51]. Typically, in EFT calculations the top mass effects are included via arescaling of the cross section by those computed in the full theory at lower orders.For H + j in the 5FS at O ( α s ) accuracy there are two contributions, the pure bottom-induced and the LO top-induced piece, and since the top-induced contribution is leadingorder one could easily consistently work in the full SM or the EFT. However, it is knownthat the higher corrections to the EFT pieces are large, and therefore including only the LOpiece makes little phenomenological sense. Our strategy in this paper is to ignore the top-induced pieces altogether and focus only on the technical aspects of the NNLO calculation ofthe bottom-induced contribution. In order to obtain reliable phenomenological predictionsat “NNLO”, one would therefore wish to combine the O ( α s y b ) pieces with the O ( α s ) EFTresults. To avoid having the bottom-induced component be entirely overwhelmed by theEFT piece, one would also wish to apply b -tagging requirements (and consider other sourcesof Higgs plus heavy flavor [31] arising from VBF and V H processes). We postpone suchdetailed phenomenology study to a future publication. Additionally, we note that whenworking in the EFT the bottom Yukawa coefficient is matched to that of the full SM asfollows, y EFT b = y SMb (cid:18) (cid:16) α s π (cid:17) ∆ (2) F + O ( α s ) (cid:19) , (2.4)where ∆ (2) F = (cid:18) −
13 log µ m t (cid:19) . (2.5)This means that, when working in the context of the EFT, we should include a termproportional to ∆ (2) F multiplying our LO predictions. In this paper we remain agnostic tothe exact implementation of the top quark and therefore choose to present results in termsof the unmatched y SMb . The impact of adjusting the coefficient to y EFT b is a rather small(sub-percentage) effect and does not affect the conclusions presented in this paper.Finally, we note that there are interference terms between the top quark (or EFT)initiated contributions to H + j and the bottom-induced contributions. This interferencerequires a helicity flip in order to be non-zero, inducing an overall scaling of the form y MS b m OS b (since the helicity flip is a kinematic mass). As a result, the interference vanishesin the 5FS. However, there is an ambiguity in the mixed-renormalization scheme whichrenders this argument not quite complete, since one can relate the OS mass to the MS mass changing the scaling to y MS b m MS b ∝ ( y MS b ) and then take the limit m OS b → toapproach the 5FS. At “LO” in the interference, O ( α s y b y t ) , such a procedure is well-definedsince the interchange of the mass schemes is trivial. However, at higher orders this procedureis much more delicate due to the presence of IR logarithms in m OS b , and rich UV structure.Very recently, this limit was studied in the context of extracting a sensible result at NLOin the interference for Higgs-plus-charm production [74] (where the even larger hierarchybetween y c and y t makes these terms more important). In addition, these pieces were– 7 –tudied in ref. [75] for the decay of H → bb and H → cc at O ( α s ) in the “4FS” in whichthe mass was fully retained. In keeping our focus on the technical aspects of the NNLOcomputation, and being agnostic regarding the top quark implementation, in this paper wetake the first limit, in which the interference is set to zero. However, when pursuing a fullLHC phenomenology study and given the size of y t , we advocate including the term usingthe limit extraction in which the helicity flip mass is coverted into the MS mass prior totaking the limit. We leave this to a more detailed future study. The calculation described in the previous section has been implemented into the MonteCarlo code MCFM [76–79]. We make extensive use of the code’s ability to handle processesinvolving a final-state jet at NNLO, and particularly important for this paper are theMCFM developments outlined in refs. [51, 59]. This section details the various checks wehave performed on our computation (in addition to the analytic soft and collinear checkspreviously mentioned). bb → H at NNLO We validate our calculation of bb → H at NNLO by comparing our results to those knownin the literature. This process has been well studied and public codes are available for thethe computation of cross sections at NNLO accuracy. We use the SusHi framework [22, 80],which can compute a variety of Higgs production cross sections at NNLO accuracy in theSM and its supersymmetric extensions. In this comparison we run MCFM with parametersset to match the default implementation in
SusHi . We use the MMHT14 [81] PDF sets,(taking the NNLO set for all predictions) and the following setup for our comparison: √ s = 13 TeV, µ R / µ F = m H .As discussed in previous sections, the choice of a factorization scale around the Higgs bosonmass is somewhat of a problem for phenomenology, since the perturbation theory is subjectto large corrections. However, in this case the large corrections act in our favor whenattempting to validate our implementation, since the larger NNLO coefficient allows usto separate scales associated with the pure coefficient, power-suppressed corrections, andnumerical Monte Carlo uncertainties. With the parameter choices listed above, the values ofthe bottom quark mass used in the Yukawa couplings are m MS b ( µ R ) = { . , . , . } GeV at LO, NLO, and NNLO respectively. By matching the MCFM parameters to thesevalues we observe excellent agreement at NLO: the result from
Sushi is 701.97 fb, whileMCFM gives 701.95 fb. We then proceed to compare our result for the NNLO coefficientin fig. 2. Here we use the τ parameter in the laboratory (unboosted) frame and presentresults over the range . ≤ τ cut0 ≤ . GeV. As is by now well known in the literature,the leading power corrections at NNLO can be described parametrically as follows, δ NNLO ( τ cut0 ) = δ NNLO0 + c (cid:18) τ cut0 Q (cid:19) log (cid:18) τ cut0 Q (cid:19) + . . . , (3.1)– 8 – ���� τ ���� μ � = � � = μ � / � δ ��������� =- ������ �� �� → � [ ���� ] �� - � ����� ����� ������������������������������������� τ ���� δ �� � � ( τ � � � � ) / δ � � � � � �� � � Figure 2 . Comparison of the NNLO coefficient obtained with MCFM and the equivalent predictionfrom
SusHi . Shown are the results obtained for a selection of choices of the slicing parameter τ cut0 and a two-parameter fit to the power corrections. The shaded band corresponds to the uncertaintyon the extracted τ cut0 → limit from the fit. where the ellipses indicate sub-leading contributions of the form τ log τ etc., and Q is a hardscale associated with the process (e.g. m H ). The residual power corrections in our resultsare clearly well described by this parametric form, and by fitting our results accordingly weare able to simultaneously extract the coefficient in the limit τ cut0 → and parametrize theresidual impact of power corrections present in calculations with non-zero τ cut0 . By fittingour results in this way we determine δ NNLO0 = − . ± .
13 fb , (3.2)which is in excellent agreement with the coefficient obtained from Sushi , δ NNLOSushi = − . fb. Our results clearly show that for τ cut0 in the region − GeV the residual powercorrections are significantly less than 1% of the NNLO coefficient, and subsequently per-mille level relative to the total physical prediction. In addition to the detailed comparisondescribed above, we have performed a similar fit to our calculation with the canonical scalechoice of µ R = m H = 4 µ F , obtaining δ NNLO0 = 18 . ± . fb, which is again in excellentagreement with the result obtained from SusHi , 18.52 fb. bb → H + j at NLO Before studying the slicing dependence of the main result of our paper, the bottom Yukawacontributions to H + j at NNLO, we study the process at NLO. The primary area of interestis to study the different options for the definition of τ cut1 and their associated asymptoticregions of validity. In order to test the various ingredients of our calculation, we begin bycomputing cross sections for H + j at NLO using the different IR-regulating prescriptionsdescribed in the previous sections. For these comparisons we use the following setup: √ s = 13 TeV, µ R = 4 µ F = m H – 9 – jT > GeV, | η j | < . ,with jets clustered using the anti- k T algorithm with R = 0 . . Additionally, we will brieflystudy the power corrections with the more central jet requirement of | η j | < . . No cuts areimplemented on the Higgs boson. We use the MMHT14 [81] PDF sets and for simplicity weuse the NNLO PDF sets for all predictions in this section. Consequently, α s and m MS b ( µ R ) are evaluated using the three-loop running (implemented into MCFM using the results of RunDec [82]). We take as an input m MS b ( m b ) = 4 . GeV, such that m MS b ( m H ) = 2 . GeV. With our central scale choice and the parameters described above, the LO cross sectionis 92.61 fb.Next, we turn our attention to validating the NLO cross section, which we have com-puted using dipole subtraction [83] and the jettiness-slicing approach. As part of the val-idation of the dipole calculation we have checked the (in)dependence of our result on theunphysical α parameters [84], which control the amount of non-singular phase space utilizedin the dipole subtractions. Using the dipole method and the parameter choices above, thecorresponding NLO cross section is 144.98 fb. We now consider the various implementa-tions of the jettiness-slicing method. Our results are presented in fig. 3, where the panelson the left side show the ratio of the cross section obtained using a fixed value for τ cut1 tothe dipole result, for both the boosted and traditional definitions. The data points on thefigure show the results obtained with the full phase-space cuts described above as well as afit to the data of the form δ NLO τ = δ NLO0 + c (cid:18) τ cut1 Q (cid:19) log (cid:18) τ cut1 Q (cid:19) . (3.3)In order to quantify the impact of forward radiation on the power corrections we additionallyshow a fit to similar results obtained with a tighter jet requirement | η j | < . , althoughfor readability we suppress the Monte Carlo output. The difference between the solid anddashed lines on the figure is therefore indicative of the sensitivity of the power-suppressedterms to the presence of forward jets. The panels on the right side of fig. 3 show the samecross section ratios, computed using a dynamic version of τ cut1 , which we define as τ cut1 = (cid:15) (cid:113) m H + ( p HT ) . (3.4)As before, results are evaluated in both the laboratory and boosted frames. The corre-sponding fit for this setup is as follows, δ NLO (cid:15) = δ NLO0 + c (cid:15) log (cid:15). (3.5)We observe the same pattern for the impact of the power corrections as reported in ref. [51].By evaluating in the boosted frame, the size of the power corrections is significantly re-duced [85], especially when the phase space includes contributions from regions in whichthe jet has larger pseudo-rapidity ( | η j | > . . Using the dynamic version of τ cut1 also resultsin smaller power corrections, and in particular the boosted-dynamic definition is the leastsensitive to power corrections. We therefore employ the boosted-dynamic version of theslicing in our subsequent studies at NNLO.– 10 – ���� τ ���� | η � | < ��� [ ����� ]| η � | < ��� [ ������ ( �� ���� )] σ ����� (| η � | < ��� ) = ������ �� σ ����� (| η � | < ��� ) = ������ �� ����������������������������������� σ � � � ( τ � � � � ) / σ � � � � � ◼ ◼ ◼ ◼ ◼ ◼ ◼ ����� τ ���� ( ������� ) | η � | < ��� [ ����� ]| η � | < ��� [ ������ ( �� ���� )] σ ����� (| η � | < ��� ) = ������ �� σ ����� (| η � | < ��� ) = ������ �� ����� ����� ����� ����� ����� ����������������������������������� τ ���� ( ��� ) σ � � � ( τ � � � � ) / σ � � � � � ������� τ ���� | η � | < ��� [ ����� ]| η � | < ��� [ ������ ( �� ���� )] τ ���� = ϵ � � � + � �� � ����������������������������������� σ � � � ( τ � � � � ) / σ � � � � � ◼ ◼ ◼ ◼ ◼ ������� τ ���� ( ������� ) | η � | < ��� [ ����� ]| η � | < ��� [ ������ ( �� ���� )] τ ���� = ϵ � � � + � �� � �� × �� - � �� × �� - � �� × �� - � �� × �� - � �� × �� - � �� × �� - � ������������������������������ ϵ σ � � � ( τ � � � � ) / σ � � � � � Figure 3 . A comparison of different definitions of the jettiness-slicing parameter for the NLOpredictions of H + j . The figures on the left use a fixed definition of τ cut1 , while those on the rightuse a dynamic version. The upper panels show the result in the laboratory frame whereas the lowerpanels evaluate the cut in the rest frame of the H + j system. bb → H + j at NNLO In this section we discuss the validation of our primary result, the NNLO predictions forthe bottom-induced contributions to H + j . We begin by presenting a check of the H + 2 j NLO result, which forms the above-cut piece of our NNLO prediction. As before, we checkthe α (in)dependence of the calculation, for which results are presented in fig. 4. Thesepredictions were obtained using the NNLO CT14 PDF set [86], µ R = µ F = m H , and thetwo-loop running of the bottom Yukawa coefficient. By comparing results at sub per-millelevel accuracy we are able to rigorously test the cancellation of IR singularities at one loopand the subsequent cancellation of dipole-related terms from the real-virtual and double-realcontributions at NNLO. Following the notation of ref. [51], we define (cid:15) ab = σ ( α ab = 1) − σ ( α ab = 10 − ) σ ( α ab = 1) , (3.6)where the indices a and b correspond to either initial- ( I ) or final- ( F ) state dipoles. Fig. 4illustrates that our results are insensitive to the choice of the α parameter at the level of (cid:15) ∼ . for the qg , qg , and qq channels. We have studied these channels in greaterdetail since they receive contributions from both the ggbb amplitudes and the four-quarkamplitudes, and therefore have the most intricate IR structure. Results are also shown for gg and qq fluxes, which we have constrained to the (still stringent) level of (cid:15) ∼ . , (cid:15) ∼ . ,or better. We are therefore confident that the cancellation between the unintegrated andintegrated dipoles has been correctly implemented in our NLO H + 2 j calculation.Finally, we arrive at the main result of this section, namely the validation of the τ cut1 dependence of our result for the NNLO coefficient. We return to our previous setup used in– 11 – ■ ■ ■ α �� α �� α �� α �� �� ���� σ ( α = � )= ����� �� ■ �� ● �� � � � � � � - ����� - �������������������� ϵ �� ■ ■ ■ ■ α �� α �� α �� α �� �� ��� σ ( α = � )= ���� �� ■ �� ● �� � � � � � � - ����� - �������������������� ϵ �� α �� α �� α �� α �� �� ���� σ ( α = � )= ����� �� - ����� - �������������������� ϵ �� α �� α �� α �� α �� �� ���� σ ( α = � )= ����� �� - ����� - �������������������� ϵ �� Figure 4 . The independence on the α parameter for a selection of partonic configurations forbottom-induced H + 2 j production. The shaded band indicates the uncertainty on the α = 1 prediction. The most intricate qg and qq channels, which receive contributions from both bbggH and four-quark amplitudes, have been computed with greater Monte Carlo statistics. the validation of H +0 j at NNLO and H + j at NLO, namely we use MMHT 14 PDF sets atNNLO accuracy. We use the canonical scale choices of µ R = m H = 4 µ F and the three-looprunning for the bottom quark Yukawa coupling and α s , such that m MS b ( m H ) = 2 . GeV.Our results are presented in fig. 5. We study the dependence of the NNLO coefficient onthe dynamic version of τ cut1 , which we recall is defined as τ cut1 = (cid:15) (cid:113) m H + ( p HT ) . (3.7)The results of ref. [51] for H + j in gluon fusion, and our preceding study for this processat NLO, clearly demonstrate that this choice results in the smallest power corrections,particularly when the associated jet is not required to be central (as in our case). Theresults in fig. 5 span the range × − ≤ (cid:15) ≤ × − , which is approximately equivalentto a fixed τ cut1 in the range . − . GeV (setting p HT = 30 GeV, which correspondsto the minimum jet transverse momentum). As expected, our results are well described bythe following approximation for the power corrections, δ NNLO (cid:15) = δ NNLO0 + c (cid:15) log (cid:15) , (3.8)where δ NNLO0 represents the physical correction obtained in the limit (cid:15) → . We find δ NNLO0 = 8 . ± .
35 fb (3.9)– 12 – ������ τ ���� ( ������� ) τ ���� = ϵ � � � + � �� � δ ����� + � � ϵ ��� ( ϵ ) � [ ��� ] μ � = � � = � μ � �� - � �� - � �� - � ��������� ϵ δ ϵ �� � � [ � � ] Figure 5 . The τ cut1 dependence of the NNLO coefficient for H + j production at the 13 TeV LHC.Results are presented for the dynamic version of τ cut1 , evaluated in the rest frame of the H + j system. Also shown is a fit to the results parametrizing the residual (cid:15) ( τ cut1 ) power corrections.The dashed line corresponds to the limit (cid:15) → of the fit, and the shaded band represents fittinguncertainties on the asymptotic limit. and, when added to the NLO cross section, we obtain σ NNLO H + j ( µ R = 4 µ F = m H ) = 153 . ± .
35 (fit) fb , (3.10)which means that we are able to control the remaining unknown power corrections to thelevel of 0.2% on the NNLO cross section. For the remainder of this paper we will use theboosted dynamic τ cut1 with (cid:15) set to × − . From our preceding study we can estimatethat for this value the remaining power corrections should be at the level of a few percenton the NNLO coefficient, and hence around the per-mille level on the full physical NNLOprediction. Such a level of accuracy should be more than adequate for the phenomenologypresented in the next section. In this section we present our results for the NNLO predictions for H + j at the 13 TeVLHC. We use the CT14 PDF sets [86], matched to the appropriate order in perturbationtheory (the running of α S and m MS b ( µ R ) therefore occurs at the next perturbative order).We use the same fiducial cuts as in section 3, namely we cluster jets using the anti- k T algorithm with R = 0 . and require them to have p jT > GeV and | η j | < . . We begin by investigating the factorization and renormalization scale dependence of thetotal cross section for H + 1 j at NNLO accuracy. Additionally, we also investigate the cross– 13 – ��� / σ �� σ ���� / σ ��� ��� ��� ���������������������� μ � / � � � � �� � �� → � + � � = �� ��� μ � = � � ��������� ��� ��� ����������������������� μ � / � � σ �� _ → � + � � [ � � ] �� → � + � + � [ ( � � � )] � = �� ��� μ � = � � ��������� σ ��� / σ �� σ ���� / σ ��� ��� ��� ���������������������� μ � / � � � � �� � ��� ��� ������������� μ � / � � σ �� → � + � + � � [ � � ] Figure 6 . Main figures: dependence of the LO, NLO, and NNLO cross sections on the factorizationscale (left: H + 0 j , right: H + 1 j ). Insets: ratio of the NNLO prediction to the NLO (red) andratio of the NLO prediction to LO (blue, dashed). section for H + 0 j production at the same order. Although higher-order predictions arenow available [33], it is nevertheless interesting to compare the two predictions with andwithout the additional jet requirement at NNLO.Our results for H + 0 j and H + 1 j are shown in fig. 6. We set a central renormalizationscale of µ R = m H and vary the factorization scale in the range m H / ≤ µ F ≤ m H . Asis well known, the factorization scale dependence for the H + 0 j cross section is dramaticat LO and NLO, while at NNLO the behavior is somewhat improved (and even more soat N3LO [33]). The maximum value around µ F = m H / adds weight to the historicalargument of using µ F = m H / as the central scale choice in NLO predictions [16].The presence of initial-state gluons and a final-state jet conspire to decrease the de-pendence of the H + 1 j cross section on the factorization scale when compared to theequivalent H + 0 j result. However, it is clear that the H + j cross section still bears astriking dependence on the factorization scale. At LO, across the range studied the crosssection increases by a factor of 7, while at NLO the increase is a factor of two. The NNLOresults from our calculation lead to a substantial improvement. By including second-ordercorrections, the overall increase in the cross section over the range of µ F drops to a factorof 1.36. Indeed, the vast majority of this increase occurs at lower scale choices, while atlarger scales m H ≤ µ F ≤ m H the NNLO cross section changes only by around 6% (com-pared to a change of 36% at NLO over the same range of µ F ). It is therefore clear thatNNLO accuracy is mandated for a robust estimate of rates, free from large uncertaintiesinduced by the unphysical dependence on the factorization scale. As the dependence on µ F significantly drops for µ F ≥ m H / , we will choose a central scale choice that reflects thisin our subsequent predictions. – 14 – ��� / σ �� σ ���� / σ ��� ��� ��� � � ������������� � � �� � �� → � + � � = �� ��� μ � = � � / � ��������� ��� ��� � � ����������������������� μ � / � � σ �� _ → � + � � [ � � ] �� → � + � + � [ ( � � � )] � = �� ��� μ � = � � / � ��������� σ ��� / σ �� σ ���� / σ ��� ��� ��� � � ������������� � � �� � ��� ��� � � ���������������� μ � / � � σ �� → � + � + � � [ � � ] Figure 7 . Main figures: dependence of the LO, NLO, and NNLO cross sections on the renormal-ization scale (left: H + 0 j , right: H + 1 j ). Insets: ratio of the NNLO prediction to the NLO (red)and ratio of the NLO prediction to LO (blue, dashed). In fig. 7 we turn our attention to the renormalization scale dependence of the NNLO H + 0 j and H + 1 j cross sections. For the bb → H process we make the customarychoice µ F = m H / and, motivated by the results discussed in the preceding paragraphs,we choose µ F = m H / for the H + 1 j predictions. We present the dependence of the crosssections over the range m H / ≤ µ R ≤ m H . For the bb → H process, at leading orderthe only dependence of the cross section on µ R arises from the evolution of m MS b . Higher-order corrections induce a rather mild dependence through α s , and the NNLO predictionis already rather insensitive to the renormalization scale. For the H + j cross section thesituation is rather different, since here the LO result depends on both α s and m MS b . Onthe smaller end of the considered range µ R < m H / , the perturbation theory becomesrather unreliable, with large corrections at each subsequent order. However, in the region m H ≤ µ R ≤ m H the perturbation theory becomes well-behaved. There is also a significantreduction in the residual scale uncertainty at NNLO: σ pp → H + j ( µ R = m H ) /σ pp → H + j ( µ R =8 m H ) is ∼ . at NLO, but reduces to ∼ . at NNLO.The results of this subsection indicate that predictions obtained using central scalechoices comparable to ( µ R , µ F ) = ( m H , m H / should demonstrate convergent behaviorin the perturbative expansion, with a reasonably small residual dependence for excursionsfrom this central choice. We therefore choose these values for the differential predictionspresented in the next section. We turn our attention to differential distributions, focusing exclusively on our new results forthe H +1 j process. Our parameter choices are the same as those in the previous section andthe central scale choice is taken to be ( µ R , µ F ) = ( m H , m H / . In order to assess the impact– 15 – � → � + � + � [ ( � � � )] μ � = � � = � μ � ��������� � ��� ����� ������������� ����������� � σ �� → � + � + � / � � � [ � � ] ��������� ��������������������� � � �� � / � � ������� ��������������� � � �� � / � � � ����� ( μ � � μ � ) ��� [ �� % ���� ] - � - � � � ���������������������� � � � � �� � / �� � � �� → � + � + � [ ( � � � )] μ � = � � = � μ � ��������� � ��� ����� ������������� ����������� � σ �� → � + � + � / � � � [ � � ] ��������� ��������������������� � � �� � / � � ������� ��������������� � � �� � / � � � ����� ( μ � � μ � ) ��� [ �� % ���� ] - � - � � � ���������������������� � � � � �� � / �� � � Figure 8 . Distributions of the rapidity of the Higgs boson ( y H , left) and leading jet ( y j , right)at LO, NLO, and NNLO accuracy. The top panel shows the distribution, while the lower panelspresent ratios to the LO, NLO, and NNLO central predictions. In addition to the six-point scalevariation, the lower panel presents estimates of PDF uncertainties at 68% C.L. (the other panelsonly include the scale variation). of the residual dependence on the renormalization and factorization scales we vary thesechoices using a six-point variation. Specifically, we compute our predictions with µ R and µ F varied by factors of two, i.e. we compute distributions with ( µ R /m H , µ F / (2 m H )) = ( α, β ) where ( α, β ) ∈ { (1 , , (1 , , (1 , / , (1 / , , (2 , , (2 , , (1 / , / } (without being in-creased and decreased in opposite directions). For each bin in the distribution the largestdeviations from the central value are taken as upper and lower estimates of the scale vari-ation.Fig. 8 shows the results for the rapidity distribution of the Higgs boson ( y H ) andleading jet ( y j ) at LO, NLO, and NNLO accuracy. By comparing the two distributionsit is clear that the Higgs boson is produced with a more central distribution than the jet,which has a broader distribution (and hence more forward jets). This can be traced backto the underlying kinematics, since the Higgs boson is a scalar particle and therefore isproduced more isotropically, while the leading jet favours the collinear (forward) regionin which the quark-gluon splitting is enhanced. The pattern of higher-order corrections is– 16 – � → � + � + � [ ( � � � )] μ � = � � = � μ � ��������� � ��� ����� ������������� ���������������� � σ �� → � + � + � / �� � � [ � � / � � � ] ��������� ������������ � � �� � / � � ������� ������������������ � � �� � / � � � ����� ( μ � � μ � ) ��� [ �� % ���� ] � ��� ��� ��� ��������������������� � �� [ ��� ] � � �� � / �� � � �� → � + � + � [ ( � � � )] μ � = � � = � μ � ��������� � ��� ����� ������������� ���������������� � σ �� → � + � + � / �� � � [ � � / � � � ] ��������� ������������ � � �� � / � � ������� ������������������ � � �� � / � � � ����� ( μ � � μ � ) ��� [ �� % ���� ] � ��� ��� ��� ��������������������� � �� [ ��� ] � � �� � / �� � � Figure 9 . Distributions of the transverse momentum of the Higgs boson ( p HT , left) and leadingjet ( p jT , right) at LO, NLO, and NNLO accuracy. The top panel shows the distribution, while thelower panels present ratios to the LO, NLO, and NNLO central predictions. In addition to thesix-point scale variation, the lower panel presents estimates of PDF uncertainties at 68% C.L. (theother panels only include the scale variation). broadly similar for both distributions, with a significant shape change from LO to NLO anda much smaller change from NLO to NNLO. We observe that the scale variation decreasesfrom around ± % at NLO to around ± at NNLO. In the central region | y X | < . ( X = H, j ), the corrections are relatively flat, whereas in the larger rapidity regions theybecome more sizable. We note that some care should be taken in this region, since itcould be prone to larger power corrections in the N -jettiness slicing method. However, weestimate that remaining power corrections enter at around the percent level in the tailsof the two distributions, which should not substantially change the interpretation of theplots. In the lower panel we additionally include an estimate of the uncertainties due tothe PDF extractions, obtained at 68% C.L. using LHAPDF [87]. We see that at NNLO theuncertainties from the PDFs and the six-point scale variation are of the same order ( ∼ ).For very forward Higgs bosons the PDF uncertainties become very large, but there is verylittle cross section in this region.Fig. 9 presents the transverse momentum distribution of the Higgs boson and the lead-ing jet. For the transverse momentum of the Higgs boson the softest bin p HT < GeV– 17 –orresponds to an observable one perturbative order lower than the rest of the calcula-tion (since there exists no → underlying topology) and this is reflected in the largerNNLO/NLO ratio and overall scale variation of this bin. Focusing on the change from NLOto NNLO, we see that the ratios are rather dynamic (while remaining within the uncer-tainty band of the NLO calculations), especially for the transverse momentum of the Higgsboson. In the region p HT < m H / there is a decrease in the prediction of around − ,while in the region around p HT ∼ m H the prediction increases at NNLO by around − before asymptotically approaching a smaller ratio (around ) at larger transverse momen-tum. The leading-jet transverse momentum distribution is different: the main impact ofthe NNLO corrections is to soften the spectrum, especially at high p T . Finally, for bothdistributions the PDF uncertainties are again comparable in size to those obtained using asix-point scale variation. We have presented a NNLO calculation of the bottom-induced contributions to pp → H + j in the 5FS. Our results have been implemented into MCFM and, as a useful by-product andcross-check of our setup, we have also implemented Higgs boson production through bottom-quark fusion bb → H at the same order. Analytic results for the various Higgs-plus-partonamplitudes needed in this paper have been obtained from our previous study of the Higgsboson decay to three partons, crossed to the appropriate LHC kinematic configurations. Wehave performed many cross-checks of our calculation. At the analytic level we have checkedthe collinear and soft factorization properties of our two-loop amplitudes, and have carriedout numerical checks of our H + 4 , parton amplitudes. For the H + 2 j NLO computationwe have done extensive testing on the exact cancellation of the integrated and unintegrateddipole subtractions. We have compared our results for bb → H to the public code Sushi ,finding excellent agreement.Higgs plus jet is fast becoming a standard candle process to study the Higgs bosonat the LHC. While the bottom-initiated contributions remain a small piece of the totalcross section, their study is motivated by the strong desire to constrain the bottom Yukawacoupling wherever possible. Unfortunately, bottom-induced processes in the 5FS have astrong dependence on the factorization scale, making a precision computation of these crosssections challenging. The results of this paper show a dramatic stabilization of the crosssection at NNLO, particularly with regard to the overall dependence on the factorizationscale.While our results focused on the process pp → H + j , there are several interestingextensions of this study to pursue. Firstly, in order to target the bottom Yukawa interaction,experimental analyses would need to impose b -tagging requirements. Therefore, adjustingour computation to target pp → H + b at NNLO is a logical next step. In order to do ameaningful phenomenological study for the LHC, the bottom-induced contributions shouldbe compared to the dominant production mechanism through gluon fusion (with a b -taggingrequirement applied). Dealing with the interference term in a sensible limit in the 5FS atthis order is also interesting for a full phenomenology study. The inclusion of the decay of– 18 –he Higgs boson is also a must in order to adequately describe fiducial volumes, for which aparticularly interesting example is when the Higgs boson decays to bottom quarks (whichmust also be matched to NNLO accuracy). With these alterations in hand, it would alsobe interesting to study Higgs-plus-charm production. Finally, in addition to the SM Higgsboson, the calculation described here could also be used in BSM extensions. For instance,the Higgs-bottom quark coupling could be modified as in the MSSM or SMEFT, or themass of the scalar particle itself could be increased, e.g. in dark matter searches where thescalar particle acts as a mediator of putative dark forces (with potential missing energydecays). We leave these exciting and detailed studies to future work. Acknowledgments
We thank Uli Schubert for useful discussions. We are particularly grateful to John Campbellfor many essential discussions regarding the results of ref. [51]. The authors are supported bya National Science Foundation CAREER award number PHY-1652066. Support providedby the Center for Computational Research at the University at Buffalo.
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