Transverse-momentum resummation: Higgs boson production at the Tevatron and the LHC
aa r X i v : . [ h e p - ph ] S e p ZU-TH 17/11
Transverse-momentum resummation:Higgs boson production at the Tevatron and the LHC
Daniel de Florian ( a ) , Giancarlo Ferrera ( b,c ) , Massimiliano Grazzini ( d ) ∗ and Damiano Tommasini ( b,d )( a ) Departamento de F´ısica, FCEYN, Universidad de Buenos Aires,(1428) Pabell´on 1 Ciudad Universitaria, Capital Federal, Argentina ( b ) Dipartimento di Fisica e Astronomia, Universit`a di Firenze andINFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Florence, Italy ( c ) Dipartimento di Fisica, Universit`a di Milano andINFN, Sezione di Milano, I-20133 Milan, Italy ( d ) Institut f¨ur Theoretische Physik, Universit¨at Z¨urich, CH-8057 Z¨urich, Switzerland
Abstract
We consider the transverse-momentum ( q T ) distribution of Standard Model Higgsbosons produced by gluon fusion in hadron collisions. At small q T ( q T ≪ m H , m H being the mass of the Higgs boson), we resum the logarithmically-enhanced contributions due to multiple soft-gluon emission to all order in QCDperturbation theory. At intermediate and large values of q T ( q T ∼ < m H ), we consistentlycombine resummation with the known fixed-order results. We use the mostadvanced perturbative information that is available at present: next-to-next-to-leadinglogarithmic resummation combined with the next-to-leading fixed-order calculation.We extend previous results including exactly all the perturbative terms up to order α in our computation and, after integration over q T , we recover the known next-to-next-to-leading order result for the total cross section. We present numerical results at theTevatron and the LHC, together with an estimate of the corresponding uncertainties.Our calculation is implemented in an updated version of the numerical code HqT .September 2011 ∗ On leave of absence from INFN, Sezione di Firenze, Sesto Fiorentino, Florence, Italy.
Introduction
One of the major tasks of the physics program at high-energy hadron colliders, such as the FermilabTevatron and the CERN LHC, is the search for the Higgs boson and the study of its properties.Gluon–gluon fusion, through a heavy-quark (mainly top-quark) loop, is the main productionmechanism of the Standard Model (SM) Higgs boson over the entire range of Higgs boson masses(100 GeV ∼ < m H ∼ < H → W W → l + l − ν ¯ ν , gives the dominant contribution to the Higgs signalin the range of mass 140 GeV ∼ < m H ∼ <
180 GeV. In this mass region, first constraints beyond theLEP lower bound of 114 . < m H <
177 GeV [2].The first results of the ATLAS and CMS collaborations presented at EPS 2011 conference [3], andupdated for Lepton Photon 2011 [4], dramatically extend the excluded region over most of themass range between 145 and 466 GeV.The above exclusion relies on accurate theoretical predictions [5, 6] for the inclusive gg → H cross section, which is now known up to next-to-next-to-leading order (NNLO) [7], with theinclusion of soft-gluon contributions up to next-to-next-to-leading logarithmic accuracy (NNLL)[8], and two-loop electroweak effects [9] † .In this paper we consider the transverse momentum ( q T ) spectrum of the SM Higgs boson H produced by the gluon fusion mechanism. This observable is of direct importance in theexperimental search. A good knowledge of the q T spectrum can help to set up strategies toimprove the statistical significance. When studying the q T distribution of the Higgs boson inQCD perturbation theory it is convenient to define two different regions of q T . In the large- q T region ( q T ∼ m H ), where the transverse momentum is of the order of the Higgs boson mass m H ,perturbative QCD calculations based on the truncation of the perturbative series at a fixed orderin α S are theoretically justified. In this region, the QCD radiative corrections are known up tothe next-to-leading order (NLO) [11, 12, 13] and QCD corrections beyond the NLO are evaluatedin Ref. [14], by implementing threshold resummation at the next-to-leading logarithmic (NLL)level.In the small- q T region ( q T ≪ m H ), where the bulk of the events is produced, the convergenceof the fixed-order expansion is spoiled by the presence of large logarithmic terms, α n S ln m ( m H /q T ).To obtain reliable predictions, these logarithmically-enhanced terms have to be systematicallyresummed to all perturbative orders [15, 16, 17, 18, 19]. It is then important to consistentlymatch the resummed and fixed-order calculations at intermediate values of q T , in order to obtainaccurate QCD predictions for the entire range of transverse momenta.The resummation of the logarithmically enhanced terms is effectively (approximately) per-formed by standard Monte Carlo event generators. In particular, MC@NLO [20] and POWEG[21] combine soft-gluon resummation through the parton shower with the leading order (LO) resultvalid at large q T , thus achieving a result with formal NLO accuracy.The numerical program HqT [18] implements soft-gluon resummation up to NNLL accuracy[22] combined with fixed-order perturbation theory up to NLO in the large- q T region [13]. The † Updated predictions for the inclusive Higgs production cross sections at the LHC are presented in Ref. [10]. q T spectrumof the Monte Carlo event generators used in the analysis and is thus of direct relevance in theHiggs boson search.The program HqT is based on the transverse-momentum resummation formalism described inRefs. [17, 18, 19], which is valid for a generic process in which a high-mass system of non strongly-interacting particles is produced in hadron–hadron collisions. The method has so far been appliedto the production of the SM Higgs boson [18, 23, 24], single vector bosons [25, 26],
W W [27] and ZZ [28] pairs, slepton pairs [29], and Drell-Yan lepton pairs in polarized collisions [30].In this paper we update and extend the phenomenological analysis presented in Ref. [18]. Inparticular, we implement the exact value of the NNLO hard-collinear coefficients H H (2) N computedin Ref. [31, 32], and the recently derived value of the NNLL coefficient A (3) [33].We use the most advanced perturbative information that is available at present: NNLL re-summation at small q T and the fixed-order NLO calculation at large q T . We present numericalresults for Higgs production at the Tevatron Run II and at the LHC and we perform a detailedstudy of the perturbative uncertainties. We also consider the normalized q T spectrum and discussits theoretical uncertainties. Our calculation for the q T spectrum is implemented in the updatedversion of the numerical code HqT , which can be downloaded from [34]. Other phenomenologicalstudies of the Higgs boson q T distribution, which combine resummed and fixed-order perturbativeresults at various levels of theoretical accuracy, can be found in Refs. [35]–[39].The paper is organized as follows. In Sect. 2 we briefly review the resummation formalism ofRefs. [17, 18, 19] and its application to Higgs boson production. In Sect. 3 we present numericalresults for Higgs boson production at the Tevatron and the LHC. In Sect. 4 we summarize ourresults. In this section we briefly recall the main points of the transverse-momentum resummation approachproposed in Refs. [17, 18, 19]. We consider the specific case of a Higgs boson H produced bygluon fusion. As recently pointed out in Ref. [19], the gluon fusion q T -resummation formula hasa different structure than the resummation formula for q ¯ q annihilation. The difference originatesfrom the collinear correlations that are a specific feature of the perturbative evolution of collidinghadron into gluon partonic initial states. These gluon collinear correlations produce, in the small- q T region, coherent spin correlations between the helicity states of the initial-state gluons anddefinite azimuthal-angle correlations between the final-states particles of the observed high-masssystem. Both these kinds of correlations have no analogue for q ¯ q annihilation processes in thesmall- q T region. In the case of Higgs boson production, being H a spin-0 scalar particle, theazimuthal correlations vanishes and only gluon spin correlations are present [19].We consider the inclusive hard-scattering process h ( p ) + h ( p ) → H ( m H , q T ) + X, (1)where h and h are the colliding hadrons with momenta p and p , m H and q T are the Higgs2oson mass and transverse momentum respectively, and X is an arbitrary and undetected finalstate.According to the QCD factorization theorem the corresponding transverse-momentum differ-ential cross section dσ H /dq T can be written as dσ H dq T ( q T , m H , s ) = X a,b Z dx Z dx f a/h ( x , µ F ) f b/h ( x , µ F ) d ˆ σ H,ab dq T ( q T , m H , ˆ s ; α S ( µ R ) , µ R , µ F ) , (2)where f a/h ( x, µ F ) ( a = q, ¯ q, g ) are the parton densities of the colliding hadron h at the factorizationscale µ F , d ˆ σ H,ab /dq T are the perturbative QCD partonic cross sections, s (ˆ s = x x s ) is the squareof the hadronic (partonic) centre–of–mass energy, and µ R is the renormalization scale ‡ .In the region where q T ∼ m H , the QCD perturbative series is controlled by a small expansionparameter, α S ( m H ), and fixed-order calculations are theoretically justified. In this region, theQCD radiative corrections are known up to NLO [11, 12, 13]. In the small- q T region ( q T ≪ m H ),the convergence of the fixed-order perturbative expansion is spoiled by the presence of powers oflarge logarithmic terms, α n S ln m ( m H /q T ) (with 1 ≤ m ≤ n − d ˆ σ H,ab dq T = d ˆ σ (res . ) H,ab dq T + d ˆ σ (fin . ) H,ab dq T . (3)The first term on the right-hand side contains all the logarithmically-enhanced contributions, atsmall q T , and has to be evaluated to all orders in α S . The second term is free of such contributionsand can thus be computed at fixed order in perturbation theory. To correctly take into accountthe kinematic constraints of transverse-momentum conservation, the resummation procedure hasto be carried out in the impact parameter space b . Using the Bessel transformation between theconjugate variables q T and b , the resummed component d ˆ σ (res . ) H,ac can be expressed as d ˆ σ (res . ) H,ac dq T ( q T , m H , ˆ s ; α S ( µ R ) , µ R , µ F ) = Z ∞ db b J ( bq T ) W Hac ( b, m H , ˆ s ; α S ( µ R ) , µ R , µ F ) , (4)where J ( x ) is the 0th-order Bessel function. The resummation structure of W Hac can be organizedin exponential form considering the Mellin N -moments W HN of W H with respect to the variable z = m H / ˆ s at fixed m H § , W HN ( b, m H ; α S ( µ R ) , µ R , µ F ) = H HN (cid:0) m H , α S ( µ R ); m H /µ R , m H /µ F , m H /Q (cid:1) × exp {G N ( α S ( µ R ) , L ; m H /µ R , m H /Q ) } , (5)were we have defined the logarithmic expansion parameter L ≡ ln( Q b /b ), and b = 2 e − γ E ( γ E = 0 . ... is the Euler number). ‡ Throughout the paper we use parton densities f ( x, µ F ) and running coupling α S ( µ R ) as defined in the MSscheme. § For the sake of simplicity we are presenting the resummation formulae only for the specific case of the diagonalterms in the flavour space. In general, the exponential is replaced by an exponential matrix with respect to thepartonic indeces (a detailed discussion of the general case can be found in Ref. [18]). Q ∼ m H , appearing in the right-hand side of Eq. (5), named resummation scale [18],parameterizes the arbitrariness in the resummation procedure. As a matter of fact the argumentof the resummed logarithms can always be rescaled as ln( m H b ) = ln( Q b ) + ln( m H /Q ) (as longas Q ∼ m H and independent of b ). Although W HN (i.e., the product H HN × exp {G N } ) does notdepend on Q when evaluated to all perturbative orders, its explicit dependence on Q appears when W HN is computed by truncation of the resummed expression at some level of logarithmic accuracy(see Eq. (6) below). As in the case of µ R and µ F , variations of Q around m H can thus be used toestimate the uncertainty from yet uncalculated logarithmic corrections at higher orders.The form factor exp {G N } is universal (process independent) ¶ and contains all the terms α n S L m with 1 ≤ m ≤ n , that order-by-order in α S are logarithmically divergent as b → ∞ (or, equiva-lently, q T → exponentiation property, all the logarithmic contributionsto G N with n + 2 ≤ m ≤ n are vanishing. The exponent G N can be systematically expanded as G N ( α S , L ; m H /µ R , m H /Q ) = L g (1) ( α S L ) + g (2) N ( α S L ; m H /µ R , m H /Q )+ α S π g (3) N ( α S L ; m H /µ R , m H /Q ) + O ( α n S L n − ) (6)where the term L g (1) resums the leading logarithmic (LL) contributions α n S L n +1 , the function g (2) N includes the NLL contributions α n S L n [16], g (3) N controls the NNLL terms α n S L n − [22, 33] and soforth. The explicit form of the functions g (1) , g (2) N and g (3) N can be found in Ref. [18].The process dependent function H HN does not depend on the impact parameter b and it includesall the perturbative terms that behave as constants as b → ∞ . It can thus be expanded in powersof α S = α S ( µ R ): H HN ( m H , α S ; m H /µ R , m H /µ F , m H /Q ) = σ (0) H ( α S , m H ) h α S π H H, (1) N ( m H /µ F , m H /Q )+ (cid:16) α S π (cid:17) H H, (2) N ( m H /µ R , m H /µ F , m H /Q ) + O ( α ) i , (7)where σ (0) H ( α S , m H ) is the partonic cross section at the Born level. The first order H H, (1) N [40] andthe second order H H, (2) N [31, 32] coefficients in Eq. (7), for the case of Higgs boson production inthe large- M t approximation, are known.To reduce the impact of unjustified higher-order contributions in the large- q T region, the loga-rithmic variable L in Eq. (5), which diverges for b →
0, is actually replaced by e L ≡ ln ( Q b /b + 1)[18, 23]. The variables L and e L are equivalent when Qb ≫ q T ), but theylead to a different behaviour of the form factor at small values of b . An additional and relevantconsequence of this replacement is that, after inclusion of the finite component (see Eq. (8)), weexactly recover the fixed-order perturbative value of the total cross section upon integration of the q T distribution over q T (i.e., the contribution of the resummed terms vanishes upon integrationover q T ).The finite component of the transverse-momentum cross section dσ (fin . ) H (see Eq. (3)) does notcontain large logarithmic terms in the small- q T region, it can thus be evaluated by truncation of ¶ It only depends on the partonic channel that produces the Born cross section. It is thus usually called quarkor gluon Sudakov form factor. h d ˆ σ (fin . ) H,ab dq T i f . o . = h d ˆ σ H,ab dq T i f . o . − h d ˆ σ (res . ) H,ab dq T i f . o . , (8)where we have introduced the subscript f . o . to denote the perturbative truncation of the variousterms. This matching procedure combines the resummed and the finite component of the partoniccross section by avoiding double-counting in the intermediate q T -region and allows us to achievea prediction with uniform theoretical accuracy over the entire range of transverse momenta.In summary, to carry out the resummation at NLL+LO accuracy, we need the inclusion ofthe functions g (1) , g (2) N , H H, (1) N , in Eqs. (6,7), together with the evaluation of the finite componentat LO (i.e. at O ( α S )) in Eq. (8); the addition of the functions g (3) N and H H, (2) N , together with thefinite component at NLO (i.e. at O ( α )) leads to the NNLL+NLO accuracy k . We point out thatour best theoretical prediction (NNLL+NLO) includes the full NNLO perturbative contributionin the small- q T region plus the NLO correction at large- q T . In particular, the NNLO result for thetotal cross section is exactly recovered upon integration over q T of the differential cross section dσ H /dq T at NNLL+NLO accuracy.Finally we recall that the resummed form factor exp {G N ( α S ( µ R ) , e L ) } has a singular behaviour,related to the presence of the Landau pole in the QCD running coupling, at the values of b where α S ( µ R ) e L ≥ π/β ( β is the first-order coefficient of the QCD β function). To perform the inverseBessel transformation with respect to the impact parameter b a prescription is thus necessary. Wedeal with this singularity by using the regularization prescription of Refs. [42, 43]: the singularityis avoided by deforming the integration contour in the complex b space. q T spectrum of the Higgs boson at the Tevatron andthe LHC In this section we consider Higgs boson production by gluon fusion at the Tevatron ( √ s = 1 . √ s = 7 TeV and 14 TeV). We present our resummed results at NNLL+NLOaccuracy, and we compare them with the NLL+LO results. For the Tevatron we choose m H = 165GeV. For the LHC at √ s = 7 and √ s = 14 TeV we fix m H = 165 GeV and m H = 125 GeV,respectively.The results we present in this section are obtained with an updated version of the numericalcode HqT [34]. The new version of this code was improved with respect to the one used in Ref. [18].The main differences regard the implementation of the second-order coefficients H H, (2) N computedin Ref. [31] (the numerical results in Ref. [18] were obtained by using a reasonable approximationof this coefficient) and the use of the recently derived value of the coefficient A (3) [33] whichcontributes to the NNLL function g (3) N (the results in Ref. [18] were obtained by using the A (3) value from threshold resummation [44]). We have checked the quantitative effect of the exactvalues of H H, (2) and A (3) at the Tevatron and the LHC. We find that the effect is generally small k The evaluation of the second-order coefficient H H, (2) N for complex values of N , necessary to perform the inverseMellin transform, is obtained using the numerical results of Ref. [41]. −
2% at the LHC at 14 TeV, 2 −
3% at the Tevatron, and at the LHC with7 TeV). We also find that the exact values of H H, (2) and A (3) have the same qualitative impact:it makes the q T -spectrum (slightly) harder.The calculation is performed strictly in the large- M t approximation.The hadronic q T cross section at NNLL+NLO (NLL+LO) accuracy is computed by usingNNLO (NLO) parton distributions functions (PDFs) with α S ( µ R ) evaluated at 3-loop (2-loop)order. This choice of the order of the parton densities and α S is fully justified both in the small- q T region (where the calculation of the partonic cross section includes the complete NNLO (NLO)result and is controlled by NNLL (NLL) resummation) and in the intermediate- q T region (wherethe calculation is constrained by the value of the NNLO (NLO) total cross section). Recent sets ofparton densities, which are obtained by analyses of various collaborations, are presented in Refs.[45, 46, 47, 48, 49]. Since the main purpose of our work is the study of the q T distribution upto the NNLL+NLO, we consider here only the PDFs sets of Refs. [46, 47, 48, 49], which provideNNLO parton densities with N f = 5 (effectively) massless quarks. Moreover, to avoid multiplepresentations of similar results, we use the MSTW2008 parton densities unless otherwise stated(the results in Ref. [18] were obtained by using the MRST2004 set [50]).As discussed in Sect. 2, the resummed calculation depends on the factorization and renormal-ization scales and on the resummation scale Q . Our convention to compute factorization and renor-malization scale uncertainties is to consider independent variations of µ F and µ R by a factor of twoaround the central values µ F = µ R = m H (i.e. we consider the range m H / ≤ { µ F , µ R } ≤ m H ),with the constraint 0 . ≤ µ F /µ R ≤
2. Similarly, we follow Ref. [18] and choose Q = m H / m H / < Q < m H .In Fig. 1 (left panels) we present the NLL+LO q T spectrum of a Higgs boson at the Tevatron,and at the LHC with √ s = 7 TeV and √ s = 14 TeV. The NLL+LO result (solid lines) atthe default scales ( µ F = µ R = m H , Q = m H /
2) are compared with the corresponding LO results(dashed lines). The LO finite component of the spectrum (see Eq. (3)) is also shown for comparison(dotted lines). We see that the LO result diverges to + ∞ as q T →
0. The resummation of thesmall- q T logarithms leads to a well-behaved distribution: it vanishes as q T →
0, has a kinematicalpeak, and tends to the corresponding LO result at large values of q T . The finite componentsmoothly vanishes as q T → q T region.The results in the right panels of Fig. 1 are analogous to those in the left panels although sys-tematically at one order higher. The q T spectrum at NNLL+NLO accuracy (solid line) is comparedwith the NLO result (dashed line) and with the NLO finite component of the spectrum (dottedline). The NLO result diverges to −∞ as q T → q T , it has an unphysicalpeak (the top of the peak is above the vertical scale of the plot) that is produced by the numericalcompensation of negative leading and positive subleading logarithmic contributions. In the regionof intermediate values of q T (say, around 50 GeV), the difference between the NNLL+NLO andNLO results gives a sizable contribution with respect to the NLO finite component. This differ-ence is produced by the logarithmic terms (at NNLO and beyond NNLO) that are included inthe resummed calculation at NNLL accuracy. At large values of q T the contribution of the NLOfinite component noticeably increases. This behaviour indicates that the logarithmic terms are nolonger dominant and that the resummed calculation cannot improve upon the predictivity of the6 a) (b)(c) (d)(e) (f) Figure 1:
The q T spectrum of Higgs bosons at the Tevatron and the LHC. Results shown are atNLL+LO (left panels) and NNLL+NLO (right panels) accuracy. Each result is compared to thecorresponding fixed-order result (dashed line) and to the finite component (dotted line) in Eq. (8). q T spectrum increasesat NNLL+NLO accuracy with respect to the NLL+LO accuracy. The height of the peak atNNLL+NLO is larger than at NLL+LO. The NNLO total cross section, which fixes the value ofthe q T integral of our NNLL+NLO result, is larger than the NLO total cross section (by about30% at the Tevatron and 25% at the LHC). This is due to the positive contribution of both theNNLO terms at small q T (the H H, (2) N coefficient of the the H HN function and the g (3) N function inthe Sudakov form factor) and the NLO finite component at intermediate and large values of q T .Comparing Fig. 1(a),1(b) with Fig. 1(c), 1(d) and Fig 1(e), 1(f) we see that the spectrum isharder at the LHC than at the Tevatron. The peak of the NNLL+NLO curve moves from q T ∼ q T ∼
10 GeV at the LHC at √ s = 7 TeV, to q T ∼
12 GeV at the LHCat √ s = 14 TeV.In Fig. 2 we show the scale dependence of the NLL+LO (dashed lines) and NNLL+NLO (solidlines) results. In the left panels we consider variations of the renormalization and factorizationscales. The bands are obtained by varying µ R and µ F as previously described in this section.We note that, in the region of small and intermediate transverse momenta ( q T ∼ <
70 GeV), theNNLL+NLO and NLL+LO bands overlap. This feature, which is not present in the case ofthe fixed-order perturbative results at LO and NLO, confirms the importance of resummation toachieve a stable perturbative prediction. In the region of small and intermediate values of q T ,we observe a sensible reduction of the scale dependence going from NLL+LO to NNLL+NLOaccuracy. At the peak the reduction is from ±
20% to ±
13% at the Tevatron, and from ±
11% to ±
8% ( ±
12% to ± √ s = 7 ( √ s = 14) TeV. Although µ R and µ F are variedindependently, we find that the dependence on µ R dominates at any value of q T .We point out that the q T region where resummed perturbative predictions are definitely signif-icant is a wide region from intermediate to relatively-small (say, close to the peak of the distribu-tion) values of q T . In fact, at very small values of q T (e.g. q T ∼ <
10 GeV) the size of non-perturbativeeffects is expected to be important ∗∗ , while in the high- q T region (e.g. q T ∼ > m H GeV) the resum-mation of the logarithmic terms cannot improve the predictivity of the fixed-order perturbativeexpansion. The inset plots in the figure show the region from intermediate to large values of q T .At large q T , the NLL+LO and NNLL+NLO results deviate from each other, and the deviationincreases as q T increases. As previously stated, this behaviour is not particularly worrying since,in the large- q T region, the resummed results loose their predictivity and should be replaced bycustomary fixed-order results.In the right panels of Fig. 2 we consider resummation scale variations. The bands are obtainedby fixing µ R = µ F = m H and varying Q between m H / m H . Performing variations ofthe resummation scale, we can get further insight on the size of yet uncalculated higher-orderlogarithmic contributions at small and intermediate values of q T . We find that, in the region ofthe peak, at the Tevatron the scale dependence at NNLL+NLO (NLL+LO) is about ±
4% ( ± √ s = 7 TeV the scale dependence at NNLL+NLO (NLL+LO) is about ± ± √ s = 14 it is about ±
3% ( ± ∗∗ See the discussion at the end of this Section. a) (b)(c) (d)(e) (f) Figure 2:
The q T spectrum of Higgs bosons at the Tevatron and the LHC. The bands are obtainedby varying µ F and µ R (left panels) and Q (right panels) as described in the text. T region, at NNLL+NLO accuracy, the factorization and renormalization scale dependence isdefinitely larger than the resummation scale dependence.The integral over q T of the resummed NNLL+NLO (NLL+LO) spectrum is in agreement (forany values of µ R , µ F and Q ) with the value of the corresponding NNLO (NLO) total cross sectionto better than 1%, thus checking the numerical accuracy of the code. We also note that the large- q T region gives a little contribution to the total cross section; therefore, the total cross sectionconstraint mainly acts as a perturbative constraint on the resummed spectrum in the region fromintermediate to small values of q T .In Fig. 3 (left panels) we report our NLL+LO and NNLL+NLO total scale uncertanty bands(the inset plots show the large- q T region). The bands represent our best estimate of the pertur-bative uncertainty, and they are obtained by performing scale variations as follows. We indepen-dently vary µ F , µ R and Q in the ranges m H / ≤ { µ F , µ R } ≤ m H and m H / ≤ Q ≤ m H , withthe constraints 0 . ≤ µ F /µ R ≤ . ≤ Q/µ R ≤
2. The constraint on the ratio µ F /µ R is thesame as used in Fig. 2; it has the purpose of avoiding large logarithmic contributions (powers ofln( µ F /µ R )) that arise from the evolution of the parton densities. Analogously, the constraint onthe ratio Q/µ R avoids large logarithmic contributions (powers of ln( Q /µ R )) in the perturbativeexpansion of the resummed form factor †† exp {G N } (see Eq. (6)). We recall (see e.g. Eq. (19) ofRef. [18]) that the exponent G N of the form factor is obtained by q integration of perturbativefunctions of α S ( q ) over the range b /b ≤ q ≤ Q . To perform the integration with system-atic logarithmic accuracy, the running coupling α S ( q ) is then expressed in terms of α S ( µ R ) (andln( q /µ R )). As a consequence, the renormalization scale µ R should not be too different from theresummation scale Q , which controls the upper bound of the q integration.A more effective way to show the perturbative uncertainties is to consider the fractional differ-ence with respect to a ’reference’ central prediction. We choose the NNLL+NLO result at centralvalue of the scales as ’reference’ result, X C , and we show the ratio ( X − X C ) /X C in Fig. 3 (rightpanels). The label X refers to the NNLL+NLO results including scale variations (solid lines), andto the NLO results including scale variations (dashed lines).We comment on the overall perturbative uncertainty band of our results in Fig. 3 starting fromthe Tevatron. The NNLL +NLO (NLL+LO) uncertainty is about ±
13% ( ± ±
10% ( ± q T = 30 GeV, and becomes ±
18% ( ± q T = 60 GeV. In the region beyond q T ∼
80 GeV the resummed result looses predictivity, andits perturbative uncertainty becomes large.In Fig. 3(b) the scale variation band of the NLO result is compared to the NNLL+NLO band.The NLO band is obtained by varying µ F and µ R as for the NNLL+NLO calculation (the NLOcalculation does not depend on the resummation scale Q ). We see that at large values of q T theNLO and NNLL+NLO bands overlap, and the NLO result has smaller uncertainty. As q T becomessmaller than about 80 GeV, the NNLL+NLO has a smaller uncertainty, and the bands marginallyoverlap. In this region of transverse momenta, the effect of resummation starts to set in. When q T becomes smaller and smaller, the NLO band quickly deviates from the NNLL+NLO band andthe NLO result becomes unreliable.We now consider the perturbative uncertainty at the LHC, √ s = 7 TeV. The NNLL +NLO †† We do not apply additional constraints on the ratio
Q/µ F , since the form factor does not depend on µ F . a) (b)(c) (d)(e) (f) Figure 3:
The q T spectrum of Higgs bosons at the Tevatron and the LHC: NNLL+NLO (solid)and NNL+LO (dashes) uncertainty bands (left panels); NNLL+NLO (solid) and NLO (dashes)uncertainty bands relative to the central NNLL+NLO result (right panels). ±
10% ( ± ±
8% ( ± q T = 30 GeV, and becomes ±
10% ( ± q T = 60 GeV. In the regionbeyond q T ∼
120 GeV the resummed result looses predictivity, and its perturbative uncertaintybecomes large. In Fig. 3(d) we compare the NLO and NNLL+NLO bands. The qualitativefeatures are similar to Fig. 3(b): at large values of q T the NLO and NNLL+NLO scale uncertaintybands overlap, and the NLO result has smaller uncertainty. As q T becomes smaller than about120 GeV, the NNLL+NLO has a smaller uncertainty, but the bands still overlap. In the regionof intermediate transverse momenta ( q T ∼
50 GeV), the bands marginally overlap and the NLOresult underestimates the cross section. When q T becomes smaller, the NLO band quickly deviatesfrom the NNLL+NLO band and the NLO result becomes unreliable.We finally consider the perturbative uncertainty at the LHC when √ s = 14 TeV. The NNLL+NLO (NLL+LO) uncertainty is about ±
9% ( ± ± ± q T = 30 GeV, and moves to ±
12% ( ± q T = 60 GeV. Inthe region beyond q T ∼
150 GeV the resummed result looses predictivity, and its perturbativeuncertainty becomes large. In Fig. 3(f) we compare the NLO and NNLL+NLO scale uncertaintybands. The qualitative features are similar to those of Figs. 3(b), 3(d): at large values of q T the NLO and NNLL+NLO bands overlap and the NLO result has smaller uncertainty. In theregion of intermediate transverse momenta ( q T ∼
50 GeV), the bands marginally overlap and theNLO result underestimates the cross section. When q T becomes smaller, the NLO result becomesunreliable.Comparing Fig. 3(a),3(b) with Fig. 3(c),3(d) and Fig. 3(e),3(f) we see that perturbative un-certainties are larger at the Tevatron than at the LHC. We also note that our NNLL+NLO resultis much more stable at the LHC than at the Tevatron, where its validity is confined to a smallerregion of transverse momenta. This is not completely unexpected. At smaller values of the centerof mass energy, the production of the Higgs boson is accompanied by softer radiation, and thusthe q T spectrum is softer than at the LHC.We conclude this section with a discussion on the uncertainties on the normalized q T spectrum(i.e., 1 /σ × dσ/dq T ). As mentioned in the introduction, the typical procedure of the experimentalcollaborations is to use the information on the total cross section [10] to rescale the best theoreticalpredictions of Monte Carlo event generators, whereas the NNLL+NLO result of our calculation,obtained with the public program HqT , is used to reweight the transverse-momentum spectrum ofthe Higgs boson obtained in the simulation. Such a procedure implies that the important infor-mation provided by the resummed NNLL+NLO spectrum is not its integral, i.e. the total crosssection, but its shape . The sources of uncertainties on the shape of the spectrum are essentially thesame as for the inclusive cross section: the uncertainty from missing higher-order contributions,estimated through scale variations, and PDF uncertainties. One additional uncertainty in the q T spectrum that needs be considered comes from Non-Perturbative (NP) effects.We remind the reader that the quantitative predictions presented in this paper are obtainedin a purely perturbative framework. It is known [15] that the transverse-momentum distributionis affected by NP effects, which become important as q T becomes small. A customary way ofmodelling these effects is to introduce an NP transverse-momentum smearing of the distribution.In the case of resummed calculations in impact parameter space, the NP smearing is implementedby multiplying the b -space perturbative form factor by an NP form factor. The parameterscontrolling this NP form factor are typically obtained through a comparison to data. Since there12s no evidence for the Higgs boson yet, the procedure to fix the NP form factor is somewhatarbitrary. Here we follow the procedure adopted in Ref. [18], and we multiply the resummed formfactor in Eq. (4) by a gaussian smearing S NP = exp {− gb } , where the parameter g is taken in therange ( g = 1 . − .
64 GeV ) suggested by the study of Ref. [37] ‡‡ . The above procedure can giveus some insight on the quantitative impact of these NP effects on the Higgs boson spectrum.In Fig. 4 (left panels) we compare the NNLL+NLO shape uncertainty as coming from scalevariations (solid lines) to the NP effects (dashed lines). The bands are obtained by normaliz-ing each spectrum to unity, and computing the relative difference with respect to the centralnormalized prediction obtained with the MSTW2008 NNLO set (with g = 0). A comparisonof Fig. 4(a),4(c),4(e) to Fig. 3(b),3(d),3(f) shows that the scale uncertainty on the normalizedNNLL+NLO distribution is smaller than the corresponding uncertainty on the NNLL+NLO re-sult. This is not unexpected: a sizeable contribution to the uncertainties shown in Fig. 3 comesactually from uncertainties on the total cross section, which do not contribute in Fig. 4. Inother words, studying uncertainties on the normalized distribution allows us to assess the trueuncertainty in the shape of the resummed q T spectrum.At the Tevatron (Fig. 4(a)) such scale uncertainty ranges from +8% −
3% in the region of thepeak, to +3% −
8% when q T ∼
50 GeV. At larger values of q T the uncertainty of the NNLL+NLOresummed distribution increases consistently with the behaviour observed in Fig. 3(b). The in-clusion of the NP effects makes the distribution harder, the effect ranging from 10% to 20% in thevery small- q T region. For q T ∼ >
10 GeV the impact of NP effects is of the order of about 5% anddecreases as q T increases. At the LHC, √ s = 7 TeV (Fig. 4(c)) the scale uncertainty ranges from+5% −
3% in the region of the peak to +5% −
4% at q T ∼
80 GeV. At the LHC, √ s = 14 TeV(Fig. 4(e)) the shape uncertainty ranges from +5% −
3% in the region of the peak to +8% − q T ∼
100 GeV. The impact of NP effects is similar at √ s = 7 and 14 TeV: it ranges from about10% to 20% in the region below the peak, is about 3 −
4% for q T ∼
20 GeV, and quickly decreasesas q T increases. We conclude that the uncertainty from unknown NP effects is smaller than thescale uncertainty, and is comparable to the latter only in the very small q T region.The impact of PDF uncertainties at 68% CL on the shape of the q T spectrum is studied inFigs. 4(b),4(d),4(f). By evaluating PDF uncertainties with MSTW2008 NNLO PDFs (red bandin Figs. 4(b),4(d),4(f)) we see that the uncertainty is at the ± −
2% level, both at the Tevatronand at the LHC. The use of different PDF sets affects not only the absolute value of the NNLOcross section (see e.g. Ref. [51]) but also the shape of the q T spectrum. The predictions obtainedwith NNPDF 2.1 PDFs are in good agreement with those obtained with the MSTW2008 setand the uncertainty bands overlap over a wide range of transverse momenta. On the contrary,the prediction obtained with the ABKM09 NNLO set is softer and the uncertainty band doesnot overlap with the MSTW2008 band. This behaviour is not completely unexpected: when theHiggs boson is produced at large transverse momenta, larger values of Bjorken x are probed, wherethe ABKM gluon is smaller than MSTW2008 one. The JR09 band shows a good compatibilitywith the MSTW2008 result, at least at the Tevatron and at the LHC for √ s = 7 TeV, wherethe uncertainty is however rather large. At the LHC for √ s = 14 TeV the differences with theMSTW2008 result are more pronounced. ‡‡ We note that the inclusion of this smearing factor does not change the overall normalization, since S NP ( b =0) = 1 a) (b)(c) (d)(e) (f) Figure 4:
Uncertainties in the normalized q T spectrum of the Higgs boson at the Tevatron and theLHC. Left panels: the NNLL+NLO uncertainty bands (solid) computed as in Fig. 3 compared toan estimate of NP effects (dashed). Right panels: PDF uncertainties bands at 68% CL. All resultsare relative to the NNLL+NLO central value computed with MSTW2008 NNLO PDFs. Summary
In this paper we have considered the q T spectrum of Higgs bosons produced in hadron collisions,and we have presented a perturbative QCD study based on transverse-momentum resummationup to NNLL+NLO accuracy.We have followed the formalism developed in Refs. [17, 18, 19], which is valid for the productionof a generic high-mass system of non strongly-interacting particles in hadron collisions. Theformalism combines small- q T resummation at a given logarithmic accuracy with the fixed-ordercalculations. It implements a unitarity constraint that guarantees that the integral over q T of thedifferential cross section coincides with the total cross section at the corresponding fixed-orderaccuracy. This leads to QCD predictions with a controllable and uniform perturbative accuracyover the region from small up to large values of q T . At large values of q T , the resummationformalism is superseded by customary fixed-order calculations.We have considered Higgs bosons produced by gluon fusion in p ¯ p collisions at the Tevatron and pp collisions at LHC energies, and we have presented an update of the phenomenological analysisof Ref. [18]. The calculation now includes the exact value of the NNLO hard-collinear coefficients H H (2) N computed in Ref. [31, 32], and the recently derived value of the NNLL coefficient A (3) [33].We have performed a study of the scale dependence of our results to estimate the correspondingperturbative uncertainty. In a wide region of transverse momenta the size of the scale uncertaintiesis considerably reduced in going from NLL+LO to NNLL+NLO accuracy.Our calculation for the q T spectrum is implemented in the updated version of the numericalcode HqT . We have argued that, given the use that is currently done of our numerical program, theimportant information is in the shape of the q T spectrum. We have thus studied the uncertaintiesof the normalized spectrum, comparing scale and PDF uncertainties, and estimating the impactof NP effects. Acknowledgements.
This work has been supported in part by the European Commission through the ’LHCPhe-noNet’ Initial Training Network PITN-GA-2010-264564.
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