Soft-gluon resummation for pseudoscalar Higgs boson production at hadron colliders
aa r X i v : . [ h e p - ph ] N ov Soft-gluon resummation for pseudoscalar Higgs boson production athadron colliders.
Daniel de Florian ∗ and Jos´e Zurita † Departamento de F´ısica, FCEYN, Universidad de Buenos Aires,(1428) Pabell´on 1 Ciudad Universitaria, Capital Federal, Argentina
Abstract
We compute the threshold-resummed cross section for pseudo-scalar MSSM Higgs boson production by gluonfusion at hadron colliders. The calculation is performed at next-to-next-to leading logarithmic accuracy. Wepresent results for both the LHC and Tevatron Run II. We analyze the factorization and renormalization scaledependence of the results, finding that after performing the resummation the corresponding cross section can becomputed with an accuracy better than 10%.October 2007 ∗ email address: defl[email protected] † email address: [email protected] Introduction
The hunt for the Standard Model (SM) Higgs boson is clearly one of the biggest physics goals at the LHC. Itssearch is not only a great challenge from the experimental point of view, it also requires a huge theoretical effortin order to provide very precise predictions both for signal and background. Besides the SM, there are manyother options for New Physics (NP). Among the possible scenarios, the MSSM is a promising one. It provides asolution for the hierarchy problem and also introduces a good dark matter candidate, the lightest supersymmetricparticle. The Higgs boson sector of this model consists in two complex Higgs doublets. After the EWSB, threedegrees of freedom are absorbed by the electroweak massive bosons and the remaining five give rise to the usualSM scalar Higgs boson ( h ), a heavier neutral one ( H ), two charged scalars ( H ± ) and a pseudoscalar neutral Higgs( A ). In the MSSM, the tree level masses depend upon two parameters, which can be selected to be the mass of thepseudoscalar Higgs ( m A ) and the ratio of the vacuum expectation values of the two doublets tan β = v /v [1, 2].These parameters are constrained by LEP and Tevatron experiments, with bounds given by m A >
100 GeV andwith the interval 0 . < tan β < . Theoretical frame
According to the mass factorization theorem, the inclusive cross section for the production of the pseudoscalarHiggs boson by the collision of hadrons h and h may be written as σ ( s, M A ) = X a,b Z dx dx f a/h ( x , µ F ) f b/h ( x , µ F ) Z dz δ (cid:18) z − τ A x x (cid:19) · ˆ σ ab (ˆ s, M A )( z ; α S ( µ R ) , M A /µ R ; M A /µ F ) , (1)where M A is the pseudoscalar Higgs boson mass, τ A = M A /s , and µ F and µ R are the factorization and renormal-ization scales, respectively. ˆ σ ab denote the partonic cross section for the process a + b → A + X , computable inperturbative QCD. The parton densities of the colliding hadrons are denoted by f a/h ( x, µ F ), where the subscript a labels the parton type. We use parton densities as defined in the MS factorization scheme. For practical purposes,one works with the hard coefficient functions G ab instead of the partonic cross sections, the first one given byˆ σ ab = σ (0) z G ab ( z ) , (2)where σ (0) is the Born level contribution. The incoming massless partons a, b do not couple to the pseudoscalarHiggs boson directly. In hadron collisions the main production mechanism is through heavy quark triangle loopsand therefore, the total cross section also depends on the top ( M t ) and bottom ( M b ) quark masses.The NLO coefficients G (1) ab have been exactly computed in Ref. [5], where it was also observed that the NLO Higgsboson cross section can be well approximated in the low tan β regime by considering its limit M t ≫ M A [18]. Hence,along this paper we will work within the large- M t approximation: we consider the case of a single heavy quark(the top), and N f = 5 light-quark flavors, neglecting all the contributions to G ( n ) ab that vanish when M A /M t → M t and M b is included in σ (0) in order to improve the accuracy of thecalculation. The large- M t approximation allows the use of the effective-Lagrangian approach [6, 19, 20], thatshrinks the top quark triangle loop into an effective point-like vertex, considerably simplifying the evaluation ofthe Feynman diagrams involved in the process.The heavy top mass limit is reliable as long as tan β ≪ M t /M b [21], because of the quark-Higgs couplings.Therefore, any calculation relying in the M t → ∞ approximation is valid for low values of tan β . Moreover,we assume that the lighter squarks are much heavier than the top quark, thus their contribution can be safelyneglected.The coefficient function G ab ( z ) is dominated by soft terms in the limit z →
1. Therefore, our main objective isto study the effect of soft-gluon contributions to all perturbative orders. This task requires to work in the Mellin(or N -moment) space [22, 23]. We thus introduce our notation in the N -space.We consider the Mellin transform σ N ( M A ) of the hadronic cross section σ ( s, M A ). The N -moments with respectto τ A = M A /s at fixed M A are customarily defined as follows: σ N ( M A ) ≡ Z dτ A τ N − A σ ( s, M A ) . (3)In N -moment space, Eq. (1) takes a simple factorized form σ N ( M A ) = X a,b f a/h , N ( µ F ) f b/h , N ( µ F ) ˆ σ ab, N ( α S ( µ R ) , M A /µ R ; M A /µ F ) , (4)where f i/h, N and ˆ σ ij, N represent the Mellin transforms of the parton distributions f i/h and of the partonic crosssections ˆ σ ij , respectively.In order to perform the threshold resummation, we first note that the threshold region z → N → ∞ in Mellin space. The dominant contribution in this limit is due to the large logarithmicterms α nS ln m N . Being the only channel open at the Born level, the gg → A subprocess is the unique partoniccontribution that can give rise to the large threshold logarithms. The formalism to systematically perform soft-gluon resummation for hadronic processes was developed in Refs. [22, 23]. In the case of Higgs boson production,2ne has G gg, N = α S ( + ∞ X n =1 α nS n X m =0 G ( n,m ) A ln m N ) + O (1 /N ) = G (res) gg, N + O (1 /N ) , (5)where the dominant contributions in the large- N limit may be reorganized in the following all-order resummationformula: G (res) gg, N ( α S ( µ R ) , M A /µ R ; M A /µ F ) = α S ( µ R ) C gg ( α S ( µ R ) , M A /µ R ; M A /µ F ) · exp {G h ( α S ( µ R ) , ln N ; M A /µ R , M A /µ F ) } . (6)The function C gg ( α S ) contains all the contributions that are constant in the large- N limit. They are originatedfrom the hard virtual contributions and non-logarithmic soft corrections, and can be computed as a power seriesexpansion in α S . The large logarithmic terms α nS ln m N (with 1 ≤ m ≤ n ), which are due to soft-gluon radiation,are included in the exponential factor exp G h . It can be expanded as G h (cid:18) α S ( µ R ) , ln N ; M A µ R , M H µ F (cid:19) = ln N g (1) h ( λ ) + g (2) h ( λ, M A /µ R ; M A /µ F )+ α S ( µ R ) g (3) h ( λ, M A /µ R ; M A /µ F ) + O ( α S ( α S ln N ) k ) (7)where λ = 0¯ α S ( µ R ) ln N and 0¯ is the first coefficient of the QCD β -function.Due to the universality of the soft-gluon emission, the G factor is independent on the type of Higgs bosonproduced in the final state. Hence, the coefficients of the expansion are the same for both h and A . The expressionsfor these coefficients can be found in Ref. [17]. The difference in the resummed expansion between the scalarand pseudoscalar Higgs only shows up in the C gg factor which, being partially originated on the hard-virtualcontribution, obviously depends on the Higgs type under consideration. For the sake of brevity, we only write thedifferences between the A and the h terms, which read∆ C (1) gg = C (1) gg,A − C (1) gg,h = 12∆ C (2) gg = C (2) gg,A − C (2) gg,h = 1939144 + 3 γ E + π − N f − ( 198 + N f M A M t + ( 38 − γ E − N f ) ln M A µ F + ( − − N f ) ln M A µ R , (8)where N f is the number of different light quark flavors.Going back to Eq. (7), the term ln N g (1) h resums all the leading logarithmic (LL) contributions α nS ln n +1 N , g (2) h contains the next-to-leading logarithmic (NLL) terms α nS ln n N , α S g (3) h collects the next-to-next-to-leading logarithmic (NNLL) terms α n +1 S ln n N , and so forth. In this context, the product α S ln N is formally considered asbeing of order unity. Therefore, the ratio of two successive terms in the expansion (7) is formally of O ( α S ), whichmakes the resummed logarithmic expansion in Eq. (7) as systematic as the usual fixed-order expansion in powersof α S .The leading collinear contributions can also be included in the soft-gluon resummation formula by performing[17, 19] the following shift C (1) gg → C (1) gg + 6 ln NN (9)that correctly resums all the terms of the type ( α nS ln n − N ) /N that appear in G ( n ) gg .When attempting for the resummation, one is interested in taking some advantage of the full fixed-order crosssection calculation as well. It is therefore customary [17] to perform a matching between both approaches, whichcan be schematically written as σ matched = σ res + σ f.o − σ res | f.o (10)where σ res corresponds to the result obtained using Eq. (6), σ f.o is the fixed-order cross section and σ res | f.o represents the expansion of the resummed result at the same order in α S as the fixed-order result. This improved matched cross section is our final result for the process. In consequence, throughout the paper we shall refer to3he different orders of the matched cross sections directly as LL, NLL and NNLL. The accuracy of the matching isassured by the order at which the C gg coefficient is computed, being obtained by a direct comparison between theresummed and the fixed-order calculation. Therefore, in σ f.o − σ res | f.o only the hard terms, which are stronglysuppressed in the N → ∞ limit, survive.Recently, the function g (4) h was presented [24, 25]. In principle, it allows to perform the resummation up toNNNLL accuracy. However, the full matched calculation at NNNLL, can only be done if the fixed-order NNNLOresult † were available. Nowadays, only the soft contribution was derived [25]. We have decided to look at theeffect of including the g (4) h term and setting C (3) gg = 0. This inclusion leads to a really slight modification of the fullNNLL result thus validating the convergence of the resummed series expansion, and allowing us to safely neglectthe g (4) h function along this work.Finally, once the expression of the cross section has been computed in N -space, the physical result can beobtained by Mellin inversion. In order to avoid the Landau singularities explicitly present in the exponential factorin Eq. (7) we use the Minimal Prescription , as described in Ref. [26].
We have developed the program THIGRES, a FORTRAN code to compute the resummed (fixed-order) crosssection up to NNLL (NNLO) accuracy, both for scalar and pseudoscalar Higgs boson in the heavy M t limit. Theimprovement over previous calculations lies in the fact that the partonic cross sections up to NNLO are directlywritten in Mellin space. The Mellin coefficients of the hard functions G ( n ) ab were presented in [27]. We haverecomputed these coefficients in an analytical way, by performing the Mellin transform of the results presented inRefs [11–13,15,16], finding some non-negligible differences in the gg and q ¯ q channels at NNLO with respect to [27].The essential ingredients for Mellin transformation can be found in [28]. We have taken advantage of the ANCONTFortran code [29] which provides most of the required special functions in N -space.One essential missing element to tackle the calculation in Mellin space are the PDFs. The available partondistributions are always given in the x space. In order to transform them to N -space, we first perform a fit ofthe densities at the needed scale using a functional form that allows for a simple analytical Mellin transform. Wefind that a linear combination of Eulerian functions x ( α ) (1 − x ) β , which in Mellin space give rise to beta functions B ( α + N, β −
1) are enough to reproduce all the features of the usual parton distributions [30]. After performingsome clever sampling, the result of the fit allows to compute N -moments analytically with an accuracy better than0.5%. Having both the necessary fixed-order and resummed coefficients and the PDFs in Mellin space, one is ableto perform the calculation in the most efficient way with a considerable reduction in the required computer timeand a gain in precision.We have worked with the MRST set of PDFs; using the 2001 LO [32] and the 2002 NLO and NNLO [33],although the code THIGRES allows the use of other PDFs. For the presentation of our results we use M t = 176GeV and M b = 4 .
75 GeV. Therefore, the cross section for the pseudoscalar Higgs boson is reliable only if tan β < ‡ .In order to check the validity range the approximation in tan β , we will start studying the dependence ofthe results upon this parameter. As we are neglecting the finite top mass effects in the partonic cross sections,tan β only appears in the Born term σ . In Fig. 1 we plot σ as a function of tan β , showing the correspondingvariation of the Born cross section, for a M A = 115 GeV boson at the LHC, according to whether one includestop and/or bottom mass effects. It is clear that the use of the infinite top mass limit in the Born cross sectionis not reliable. At least the M t dependence must be included in σ . The solid (top + bottom) and dashed (onlytop) curves are similar around tan β = 1; for this particular value, the pseudoscalar Higgs coupling to up anddown type quarks is the same as for the scalar one. The inclusion of the bottom mass can be neglected until oneenters the region where tan β becomes close to p M t /M b § . Near this point, both curves start to separate fromeach other. It is rather noticeable that for values of tan β ≥
10 the bottom effects are completely dominant, andtherefore any calculation relying on the infinite top mass approximation can not be trusted for those values. This † At least, the full soft-virtual contributions are necessary to compute the coefficient C (3) gg . ‡ We have explicitly checked the accuracy of the infinite top mass approximation by comparing our NLO results with the exact NLOones provided by the FORTRAN code HIGLU [31]. Within this bounds the accuracy is always better than 10 percent. § At this particular value, the A -top coupling is equal to the A -bottom coupling. tan β dependence of the Higgs production Born cross section at the LHC for M A = 115 GeV, in theinfinite top mass limit (dots), including top mass effects (dashes) and both top and bottom mass effects (solid) .behaviour is certainly expected, since the couplings of the pseudoscalar Higgs boson to the up (down) type quarksare suppressed (enhanced) by a factor tan β .It is very important to know the dependence of the cross section upon the renormalization and factorizationscales, as a way to estimate the size of the higher order terms not yet included in the perturbative expansions andtherefore evaluate the uncertainties on the theoretical calculations. This dependence is shown in Fig. 2, consideringa M A = 150 GeV Higgs boson at the LHC, using tan β = 3, for the fixed-order LO, NLO and NNLO results. InFigure 2: Scale dependence of the Higgs production cross section at the LHC for M A = 150 GeV, tan β = 3 , at LO(dots), NLO (dashes) and NNLO (solid). this figure, both scales were varied from M A / M A in three different ways. In the plot on the left, the varyingscales were chosen to be equal ( µ F = µ R = χM A ). In the plot on the center the factorization scale was changedand the renormalization scale was kept fixed ( µ R = M A , µ F = χM A ), while in the one on the right, the oppositevariation was performed ( µ F = M A , µ R = χM A ). As expected from the running of α S , the cross section typicallydecreases when µ R increases. This effect is clearly noticeable in the right-side plot, and, more moderate in theleft-side plot. The graph on the center shows how the variation of µ F leads to and opposite behavior, ie: the crosssection increases with the growing of µ F . This can be explained by the following fact: at the LHC the cross sectionis mainly sensitive to partons with momentum fraction x ∼ − . In this x range, the scaling violation of the partondensities is moderately positive and therefore one observes an artificially reduced factorization scale dependence.In the left-side plot one sees a partial compensation of the two effects, although the µ R variation clearly dominates.5igure 3: Scale dependence of the Higgs production cross section at the LHC for M A = 150 GeV, tan β = 3 , at LL(dots), NLL (dashes) and NNLL (solid). Another interesting feature of this plot is that it shows how sizeable are the higher order corrections. The changewhen going from LO to NLO is quite large, while the inclusion of the NNLO corrections has a moderate impact.We can consider this fact as a hint for the convergence of the perturbative series. Fig. 3 shows the same as Fig. 2,for the resummed cross section. One sees that the plot on the right keeps the typical dependence on µ R , due to therunning of the coupling constant. Nevertheless, the behavior for fixed µ R has changed, especially if we compare thehigher orders (NLO vs NLL and NNLO vs NNLL). The rather flat result of Fig. 2 is now replaced by a considerablyhigher variation, due to the inclusion of both soft and collinear higher order terms. Therefore, in the left-side plot,the scale variations are fairly compensated, and particularly the NNLL result exhibits a tenuous dependence onthe combined scales. Fig. 2 shows that the scale dependence is very slightly reduced when going from LO to NLO(as was already mentioned), and considerably reduced when going from NLO to NNLO. The implementation ofresummation effects leads to a further reduction of the scale dependence.In Fig. 4 we present the fixed-order scale dependence for M A = 150 GeV with tan β = 5 now at the Tevatron.In this Figure, many of the overall features that appeared in the LHC plots are present. The right plot still showsFigure 4: Scale dependence of the Higgs production cross section at the Tevatron for M A =150 GeV, tan β = 5 , atLO (dots), NLO (dashes) and NNLO (solid). the same dependence, again because of the running of α S . In the middle plot, in direct contrast with the resultfrom Fig. 2, the cross sections increases with µ F . At the Tevatron, the partons with roughly x ∼ . Scale dependence of the Higgs production cross section at the Tevatron for M A = 150 GeV, tan β = 5 ,at LL (dots), NLL (dashes) and NNLL (solid). the rather flat scale dependence of the NNLL result in the left plot. It is quite remarkable that the LO and LLcurves look very much alike. The effects of threshold resummation become important only when going to higher(NLL and NNLL) orders. As an overall feature of the scale dependence graphs, it is important to stress the factthat the resummed cross section is clearly more stable against scale variations than the fixed-order result.The importance of higher-orders effects in commonly presented through the introduction of K-factors, which aredefined as the ratio of the cross section at a given order over the LO result. As it was mentioned before, within thelarge- M t approximation, the higher order cross sections are proportional to σ , which is the only term that dependsupon tan β . Therefore, in the infinite top mass limit, the K-factors are fully independent on that parameter. InFig. 6 we present the K factor at LHC including its scale dependence. The bands were obtained by independentlyvarying the scales in the region 0 . M A ≤ µ F , µ R ≤ M A , with the constraint 0 . < µ F /µ R <
2. The LO result thatrenormalizes the K factor was computed with µ F = µ R = M A . The NLO K factor is around 2, accounting for anFigure 6: Fixed-order and resummed K factors for Higgs production at LHC. increase in the cross sections of approximately the same amount as the LO result itself. The NNLO K factor showsa rather more moderate impact. The inclusion of soft-gluon effects at NLL and NNLL accuracy slightly increasesthe cross section on top of the fixed-order contributions and show a larger overlap between the corresponding bands,indicating a better convergence for the resummed series. It becomes clear the reduction of the scale dependence7or the higher orders, as the bands become thinner as the order grows. We also notice an increase of the K factorswith M A , consistent with the fact that the soft-gluon contributions become more dominant as the process getscloser to the hadronic threshold. Once the resummation is performed, the uncertainty due to scale variation canbe estimated to the order of 10 percent.Finally, Fig. 7 shows the K-factor Higgs mass dependence at the Tevatron. Here we can see the same overallFigure 7: Fixed-order and resummed K factors for Higgs production at Tevatron. features as presented in Fig. 6. As a major difference, we note that the K factors are considerably bigger for theTevatron. This is due to the fact that the Tevatron center of mass energy is closer to the hadronic threshold, whichis the kinematical region where soft-gluon effects become relevant.
We have presented the resummed cross section for pseudoscalar Higgs production by gluon fusion at NNLL accuracyin hadronic colliders, presenting the most relevant results for the LHC and Tevatron.Comparing with the fixed-order calculation, the numerical impact of the resummation effects was found to berather moderate. This slight variation when going from NNLO to NNLL is providing a hint for the (hopefully)faster convergence of the perturbative series. Probably the most striking feature of the resummed result is theconsiderably reduction of the scale dependence. It allows to make theoretical predictions with a precision of about10 %, which are accurate enough for discovery of a Higgs boson at the LHC and Tevatron. Moreover, our resultspresents a probe to supersymmetry, as can also be useful to directly test the MSSM and/or another supersymmetricextension of the Standard Model in the low tan β regime.The Fortran code THIGRES, which computes total cross sections for both scalar and pseudoscalar Higgs bosonup to NNLO (or NNLL), is provided upon request from the authors. . Acknowledgements.
This work has been partially supported by ANPCYT, UBACyT and CONICET.
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