ATLAS NOTE January 4, 2010

Transcription

1 Draft version x.y ATLAS NOTE January 4, 2 Higgs Cross Sections for Early Data Taking N. Andari a, K. Assamagan b, A.-C. Bourgaux a, M. Campanelli c, G. Carrillo d, M. Escalier a, M. Flechl e, J. Huston f, S. Muanza g, B. Murray h, B. Mellado d, A. Nisati i, J. Qian j, D. Rebuzzi k, M. Schram l, R. Tanaka a, T. Vickey d, M. Warsinsky m, H. Zhang g a LAL-Orsay b Brookhaven National Laboratory c University College London d University of Wisconsin, Madison e University of Uppsala f Michigan State University g CPPM-Marseille h Rutherford Appleton Laboratory i Sezione di Roma I and INFN j University of Michigan k Università di Pavia and INFN, Sezione di Pavia l McGill University m Freiburg University 8 9 This is the abstract Abstract

2 January 4, 2 : 4 DRAFT Gluon-gluon Fusion Process At the LHC, gluon-gluon fusion gg H (Fig. ) is the dominant production process for the Standard Model Higgs boson, over the whole mass spectrum. Since the Higgs boson does not couple to massless particles, this process proceeds at tree level through a triangular loop of heavy quarks, mostly top or bottom quarks, depending on the strong Yukawa coupling of heavy fermions. This process suffers from large QCD corrections and the uncertainty from higher order corrections remains large. The cross section is a slowly converging perturbation expansion of the form σ = σ + σ + σ + = σ ( ). g g t H t H g g Figure : Leading order Feynman diagram for Higgs production via gluon fusion gg H QCD and Electroweak Radiative Corrections At leading order () in QCD perturbation theory, the cross section is proportional to αs 2, α s being the QCD coupling. Its behaviour is shown in Figure 2 as a function of the Higgs boson mass at different center-of-mass energies. At m H 3 GeV, i.e. near the t t threshold where the Hgg amplitude develops an imaginary part, the cross-section presents a bump... Effective Theory An effective theory, where the internal top quark has an infinite mass m top (so called heavy-top limit ), is known to provide a good approximation for the cross section[?]. Recently, it has been shown that the heavy-top limit calculation agrees to the full top mass dependent calculation to better than. % for the Higgs mass range between and 3 GeV/c 2 at [?]. This guarantees the validity of the effective theory for Higgs masses below the t t threshold. Above the t t threshold, however, the infinite top quark mass approximation fails, overestimating the cross section by O( %) for very heavy Higgs. In this mass range, the top quark mass should be trated exactly. The remaining difference between approximated results and exact quark mass calculations is due to the contribution of the bottom quark loop. The cross section receives large correction from higher order QCD diagrams. The QCD processes include real and virtual corrections O(α S ) : gg Hg, gq Hq and q q Hg. The increase of the gg H cross section due to QCD diagrams above is O(8%), almost independent from the Higgs mass, as displayed in Figure 2. Top and bottom quark pole mass are fixed to m t = 72. GeV and m b = 4.7 GeV and both µ R and µ F are set to m H. The two-loop electroweak corrections are small and mostly due to light fermion W contribution in the m H 2m W range (at the level of [% ; 9%]). For m H 2m W they become again very small...2 Cross Sections The complex three-loop QCD corrections to the gg H fusion process have been recently calculated in the limit of a very heavy top quark. In this configuration, the Feynman diagrams contributing

3 January 4, 2 : 4 DRAFT to the process factorize into two pieces: a massive component where the heavy quark has been integrated out and which represents an effective coupling constant multiplying the Hgg vertex, and a massless component involving only gluons and light quarks, which describes the short distance effects and where the finite momenta of the particles have to be taken into account. The calculation effectively reduces then to a two-loop calculation with massless particles. Figure 2 left shows the cross section in comparison to and one, showing explicitly (Fig. 3 right) the nice convergence behaviour of the perturbative series. The value of the cross section including the corrections has still two main uncertainties. The first one comes from the gluon structure function, describing the momentum distribution of a parton in the proton, which has still a large uncertainty in the low-x region NNLL Soft Gluon Resummation To improve further the theoretical predictions for the cross section, one can also resum the soft and collinear gluon radiation parts which in general leads to large logarithmic terms and includes the dominant electroweak radiative corrections (much smaller that the QCD corrections). For gluon-gluon fusion, the resummation of the large logarithms has been performed at the next-to-next-to-leading logarithm (NNLL) accuracy in the heavy top quark mass limit. The resummation relies on the basic factorization theorem for partonic cross section into soft, collinear and hard parts near the phase-space boundary Electroweak Radiative Corrections The total cross section for gluon-gluon fusion can be written as σ(m H ) = [ + δ EW (m H ) ] σ QCD had (m H ), () where δ EW represents the electroweak radiative corrections. They receive contributions from: the exact dependence on the light quark corrections as in [Phys.Lett.B9 (24) 432], improved by the use of complex masses for W and Z; the effect of heavy quark corrections as in Degrassi-Maltoni, only up to GeV. Table?? lists the factor δ EW, for different values of the Higgs boson mass, up to m H = GeV, while Table?? shows the corrections in the mass range from 6 to GeV. In this region the top contribution in not negligible close to 2m t, i.e. between 3 and 4 GeV [Actis, Passarino, Sturm, Uccirati]..2 Available Theoretical Tools There are several programs available for the gluon-gluon fusion calculation, each one with different characteristics. Table summarizes the status of the art..2. Inclusive and Exclusive Calculations HIGLU HIGLU performs the computations of gluon-gluon fusion cross section at and. The program includes the exact contribution of the top and bottom quarks in the loop that generates the

4 January 4, 2 : 4 DRAFT 4 Table : Status of the art for gluon-gluon fusion process: programs and calculation currently available. The first three have been massively used to produce the Table and the plots of the present Note. Order Mode m t m b EW Resum HIGLU [?] inclusive exact exact - - HPro [?] exclusive exact exact - - HggTotal [?] inclusive +F() exact yes no µ = m H /2 De Florian, Grazzini [?] inclusive exact exact yes yes FEHiP [?] exclusive +F() new FEHiP [?] exclusive +F() exact yes - H [?,?] exclusive MCFM [?] exclusive Higgs couplings to gluons. Among the parton distribution functions interfaced to this program by default there are not MSTW28, so a small modification to the source code has been necessary (of which the author is aware). The input parameters have been chosen so that the integration errors never exceed a fraction of percent of the inclusive cross sections. Results are presented in Table?? for the tree level calculation and in Table?? for the QCD calculations, including the scale uncertainties and the PDF uncertainties calculated on the 4 MSTW28 PDF sets (one central pdf and 4 extremal pdfs). HPro The Monte Calro program HPro computes fully differential next-to-leading order QCD cross sections for the gluon-gluon fusion process. The exact dependence of the cross section from the top and bottom quark masses has been implemented in the calculations. The predictions of HPro can be used to correct for the finite quark mass effects the differential cross sections (i.e. the fully differential results of the program FEHiP, see below), as well as HIGLU is adopted to correct the total cross section (in the De Florian, Grazzini computations, for instance)..2.2 Inclusive Calculations HggTotal HggTotal is a program for the gluon-gluon fusion cross section calculations up to. The renormalization and the factorization scales recommended by authors are of M Higgs /2 to effectively take into account the NNLL soft-gluon resummation effect. De Florian and Grazzini De Florian and Grazzini presented in [?] an updated prediction for gluon-gluon fusion Higgs boson production process at hadron colliders, up to in perturbative QCD, including soft gluon resummation up to next-to-next-to leading logatithmic accuracy () and the two-loop EW corrections. Their calculation include the exact treatment of m b up to, basing on the calculation included in HIGLU. For what m t is concerned, the top-quark contribution is considered in the loop, and the calculation up NNLL+ is performed in the large-m t limit, using an effective theory. Then the result is rescaled by the exact m t dependent Born cross section, as an approximation for the top-quark contribution. The authors make use of a not yet public code. Results on,, +NNLL and +NNLL+EW inclusive cross sections, calculated with MSTW28 and our set of input parameters, have been provided us, together with the PDF (68% and 9% CL, with and without the α S uncertainty) and scale uncertainties for

5 January 4, 2 : 4 DRAFT the ++EW cross section. Factorization and renormalization scales have been set equal to the higgs boson masses. Figure 2 and show the results for the total cross section up to ++EW, and the related uncertainties, respectively..2.3 Exclusive Calculations FEHiP FEHiP is a numerical program which computes gluon-gluon fusion fully differential cross sections in hadron collisions up to in perturbative QCD. The authors implement an alternative approach from the dipole sutraction to the problem of real radiation at, where the cancellation of singularities is performed numerically, in a completely automated fashion and no analytic integration are required [?]. The code can handle also the Higgs boson decay into a two photons at approximation in QCD (treating the Higgs in narrow width approximation). Also FEHiP, like De Florian-Grazzini calculations and HggTotal, works in the limit of an infinitely heavy top quark, in which the Higgs boson coupling to gluons is point-like. However, all the results are rescaled by a factor F(m t ) = σ (m t) σ (m t = ). (2) This rescaling, indicated as +F() in Table, accounts for the effects of the finite top mass exactly for cross section, and provides an approximate description of m t dependence at higher orders. This method is precise to (%) at the, but can lead to discrepancies up to (%) for high Higgs boson masses, when the and order are considered. FEHiP uses MRST2 parton distribution functions, provided with the distribution of FEHiP. The use of MSTW28 implies a small modification in the code. A new version of the code is expected to be released soon. It would include (as reported in Table ) the exact treatment of m b and the two-loop EW corrections. H The numerical program H is a parton level Monte Carlo program which computes QCD corrections to the gluon-gluon fusion cross section up to in perturbation theory. It implements the extension to order of the so-called subtraction method proposed in [?], a general algorithm for the handling and cancellation of the infrared singularities. The Higgs boson is treated in the narrow-width approximation, and the calculations are done in the m t limit, without exact treatment of the top- quark mass. In the program, the user can set also the Higgs boson decay into a pair of vector bosons, and apply a set of cuts on the final state leptons (photons) and the associated jets, to study the distribution of the main experimental observables..3 Inclusive Cross Sections.3. Total Cross Sections The total cross sections for gluon-gluon fusion process at center-of-mass energies of 7, and 4 TeV are shown in Figure 2. The calculation is based on the calculations by D. de Florian and M. Grazzini[?]. The cross sections at QCD,, +NNLL correction, +NNLL+Electroweak corrections are also shown. The K-factor of QCD calculation over is roughly 2 %, NNLL correction is about %, where as additional electroweak corrections add % corrections. The total cross sections as a function of centre-of-mass energies for different Higgs mass is shown in Figure 3.

6 January 4, 2 : 4 DRAFT 6 σ [fb] 2 σ [pb] 6 NNLL+EW - 4 TeV NNLL+EW - TeV NNLL+EW - 7 TeV 4 m H =2 GeV m H =2 GeV m H =4 GeV s [TeV] Figure 2: Total cross sections calculated by de Florian program for gg H process as a fuction of Higgs mass at s = 7, and 4 TeV (left). Total cross sections with HggTotal for gg H process as a function of centreof-mass energy for Higgs mass of 2, 2 and 4 GeV (right). σ [fb] NNLL+EW NNLL σ [fb] NNLL+EW NNLL Figure 3: Total cross sections callculated by de Florian program for the gg H process at s = TeV at the order of,, + and +NNLL+EW (left). K-factors based upon de Florian calculation. The cross section ratios of, +NNLL and +NNLL+EW against cross sections are plotted as a function of Higgs mass (left) Comparison between Different Calculations It is very important to compare the cross section results with different approaches at different order of perturvative calculations. The and calculations based upon HIGLU and HggTotal and de Florian calculation are compared in Figure 4. (a) the ratio HggTotal and HIGLU at, where HggTotal does not include the bottom correction which is important at low Higgs mass retion and amounts to - % for Higgs mass of GeV/c 2. For heavy Higgs mass, they agree well within % as heavy-top limit approximation is accurate enough to reproduce the finite top quark mass effect at. (b) the ratio between HggTotal and HIGLU at. Again HggTotal does not contain the bottom correction which amounts to -8 % for Higgs mass of GeV/c 2. For heavy Higgs mass, it is well known that heavy-top limit calculation overestimates the cross section by the order of O( %)[?]. (c) the ratio between de Florian and HIGLU at. De Florian calculation takes into account bottom correction based on HIGLU calculation, thus good agreement between two calculations is

7 January 4, 2 : 4 DRAFT 7 R = σ HggTotal /σ HIGLU - [%] R = σ HggTotal /σ HIGLU - [%] (a) HggTotal vs HIGLU at (b) HggTotal vs HIGLU at R = σ de Florian /σ HIGLU - [%] R = σ HggTotal /σ de Florian - [%] (c) de Florian vs HIGLU at (d) HggTotal vs de Florian at Figure 4: Cross section ratio between different calculations for gg H process at s = TeV observed. For heavy Higgs, howeveer, O( %) discrepancy is observed as de Florian calculation is based on heavy-top limt. (d) the ratio between HggTotal and de Florian at. HggTotal takes into account QCD, bottom correciton and top-bottom interference at and electroweak radiative corrections. The renormalization and the factorization scales have been chosen at M Higgs /2 to effectively take into account the soft-gluon resummation. De Florian calculation takes into account QCD plus NNLL effect due to soft-gluon resummation. Bottom and top-bottom interference effects are considered at as well as the electroweak radiative corrections. The two calculations agree well within 2 % in most of the Higgs mass region except at the top threshold region. One should note that both calculations are based on heavy-top limit, thus both are overestimating the cross section by O( %) for heavy Higgs..4 Theoretical uncertainties.4. PDF uncertainties The uncertainties, on the cross section due to PDF which can be up to 2% for this production process, can be calculated by means of the PDF error sets S i, available for most of PDF schemes. One first

8 January 4, 2 : 4 DRAFT 8 89 evaluates the cross section with the nominal PDF S to obtain the central value σ, then calculates the 9 cross section σ i with the 2N PDF S i PDFs and defines for each σ i value, the deviations σ i ± = σ i ± σ 9 when σ > i < σ. The uncertainties are summed quadratically to calculate σ ± = Σ i (σ i ± ) 2. The cross 92 section, including the error, in then given by σ + σ σ [?]. An another way of computing asymmetric 93 uncertainty is to use the master formula : σ + = N i= N i= [max(σ + i σ,σ i σ ),] 2 (3) σ = [max(σ σ + i,σ σ i ),] 2 (4) 94 An other method for estimating the PDF errors which will be used in the following, is discussed in [?], which leads to symmetric uncertainties. The uncertainty is then σ ± = ( ) 2 i σ + 2. i σi 9 Figure 6 96 shows that the uncertainties from pdfs depends on the chosen pdf. 97 The first period of LHC data taking is expected to increase sensibly the knowledge of the structure 98 function w.r.t HERA and the Tevatron Renormalization and factorization scale dependences The second uncertainty arises from corrections above the. The cross section changes with the renomalization scale µ R (at which one defines the strong coupling constant) and factorization scales µ F (at which one performs the matching between the perturbative calculations of the matrix element and the non-perturbative part which resides in the parton distribution functions) as an effect of un-calculated higher order effects. Starting from a median scale µ, which is considered the natural scale of the process and is expected to absorb the large logarithmic corrections, the current standard convention is to very the two scales, either collectively or independently, within µ /ξ µ R, µ F ξ µ, where ξ is typically 2 for this production process (in some cases it is more prudent to use larger values for ξ, as will be seen in the case of the Higgs production in bottom quark fusion, for instance). The variation of the scales results in a uncertainty band: the narrower the band is, th smaller the higher-order corrections are expected to be. This is by no means a rigorous way to estimate the theoretical uncertainty, it gives indeed an indication of the full uncertainty Finite quark mass effect.4.4 Total theoretical uncertainty The results of an uncertainty analysis are characterized by 2 S i N PDF set of published eigenvector PDF sets along with the central fit.

9 January 4, 2 : 4 DRAFT 9 [%] δ 2 PDF 9% CL PDF+α S 9% CL PDF 68% CL PDF+α S 68% CL [%] δ 2 QCD Scale 68% CL PDF+α S [%] δ 2 PDF 9% CL PDF+α S 9% CL PDF 68% CL PDF+α S 68% CL (a) [%] δ 2 QCD Scale 68% CL PDF+α S (b) [%] δ 2 PDF 9% CL PDF+α S 9% CL PDF 68% CL PDF+α S 68% CL (c) [%] δ 2 QCD Scale 68% CL PDF+α S (d) (e) (f) Figure : Scale and PDF uncertainties calculated by de Florian program for gg H process at s = 7 (top) (middle) and 4 (bottom) TeV. The left plots show the PDF uncrtainties (68% and 9%, with and without the α s contribution), while the right plots show the QCD scale uncertainties in comparison to the 68% PDF + α s uncertainty.

10 January 4, 2 : 4 DRAFT Figure 6: Relative cross-sections and uncertainties from pdf computed with different pdfs. REF : R = σ/σ ξ=..4.3 µ =ξ M R H µ =ξ M F H R = σ/σ ξ=..4.3 µ =ξ M R H µ =M H /ξ F ξ ξ Figure 7: The scale dependence of Higgs cross section via gluon fusion gg H for s = 4 TeV and M H = 2 GeV. Left plot: diagonal scan by changing the renormalization and the factorization scales by the same factor ξ for lowest order (black) and next-leading order (bule) and next-to-next leading order (red) calculations. Right plot: anti-diagonal scan by changing the renormalization scale by a factor ξ, but of the renormalization scale by a factor /ξ. - R = σ run /σ pole (GeV/c ) Figure 8: The bottom mass dependence of Higgs cross section via gluon fusion gg H for s = TeV. The cross sections are calculated with HggTotal with the renormalization and factorizations scales set at the Higgs mass. M Higgs

11 January 4, 2 : 4 DRAFT Differential distributions While integrate cross section allows to know better the expected rate of the Higgs at LHC, increasing the order of the computation may change the topology of the final state. The selection cuts of each channel could be affected by this change in the order of the computation. So one needs to study the dependence of the differential cross section with the order of the computation... H γγ channel 222 The differential distribution of the momentum of the photon is shown on Fig. 9 left. While the curve 223 is kinematically bounded by m H /2, higher order computation goes further. Around m H /2, an instability 224 occurs, that could be absorbed by resummation. While the energy in the center of mass has an importance 22 effect (Fig. 9 right) on the acceptance, the effect of higher order is small. The difference between and 226 order is due to the choice of asymetric cut (p T > 4 GeV, p2 T > 2 GeV ) for the H γγ channel. 227 In conclusion, a global K factor can be used. (γ) [fb/gev] T dσ/dp H γγ m H = GeV acceptance m H = 2 GeV,µ =µ =m F R H cuts: p >(4; 2), η [;.37] U [.2;2.37] T MC@ by Yaquan Fang Pythia Fehip, mstw28, Fehip, mstw28, p (γ) [GeV] T s [TeV] Figure 9: Left : transverse momentum distibutions of the photon computed with H at different orders, for gg H γγ for Higgs mass of GeV/c 2 at s = TeV. Right : acceptance for the process of gg H γγ for Higgs mass of 2 GeV/c 2 at s = TeV, using different computations : MC@, Pythia and FEHiP at and H ZZ ( ) 4l channel The differential distribution of the momentum of the lepton is shown on Fig. 9 left. The effect on the lepton selection efficiency is small. In conclusion, a global K factor can be used.

12 January 4, 2 : 4 DRAFT 2 /σ dσ/dpt - -2 /σ dσ/dpt pt (GeV/c) pt2 (GeV/c) /σ dσ/dpt - -2 /σ dσ/dpt pt3 (GeV/c) pt4 (GeV/c) Lepton Selection Efficiency H (a) Lepton p T distribution,2 3,4 p >2 GeV, p >7 GeV T T η <2. H ZZ* 4l Higgs Mass (GeV) (b) Lepton selection efficiency Figure : : lepton transverse momentum spectrum of gg H ZZ 4leptons for Higgs mass of 3 GeV/c 2 at s = TeV. Right : lepton selection efficiencies for gg H ZZ 4leptons as a function of Higgs mass at s = TeV.

13 January 4, 2 : 4 DRAFT H WW ( ) lνlν channel The effect of the order of the computation is small on the lepton selection efficiency (Fig. left). But due to the large QCD background, a jet veto is used in the analysis. This jet veto effiency is strongly deteriorated (Fig. right) by the order of the computation, and should be taken into consideration for the analysis. Lepton Selection Efficiency H p > GeV, η <2. T H WW* ll Higgs Mass (GeV) Parton Jet Veto Efficiency H Higgs Mass (GeV) Figure : Left : lepton selection efficiency for gg H WW lνlν as a function of Higgs mass at s = TeV. Right : jet veto efficiency for gg H WW lνlν as a function of Higgs mass at s = TeV ?? The production of the Higgs through gluon fusion is sensitive to a fourth generation of quarks. Because the Higgs couples in proportion to the fermion mass, a heavier generation of quarks is not suppressed in the process, as it would be expected for a loop process with a heavier particle in the loop. The Higgs cross section is sensitive to a fourth generation even if the quarks are too heavy for a direct discover at LHC. The mass range is limited by the scale of new physics where the Standard Model breaks down.