Higgs plus jet production in bottom quark annihilation at next-to-leading order
aa r X i v : . [ h e p - ph ] J u l July 2010 — WUB/10-20
Higgs plus jet production in bottom quark annihilation atnext-to-leading order
Robert V. Harlander, Kemal J. Ozeren, Marius Wiesemann
Fachbereich C, Bergische Universit¨at Wuppertal42097 Wuppertal, Germany [email protected]@[email protected]
Abstract
The cross section for Higgs+jet production in bottom quark annihilation is calcu-lated through
NLO QCD . The five-flavour scheme is used to derive this contributionto the Higgs+jet production cross section which becomes numerically important inthe
MSSM for large values of tan β . We present numerical results for a proton colliderwith 14 TeV center-of-mass energy. The NLO matrix elements for d σ/ d p T are thencombined with the total inclusive cross section in order to derive the integrated crosssection with a maximum cut on p T at next-to-next-to-leading order. The Higgs mechanism [1–3] plays a central role in both the Standard Model ( SM ) [4–6]and its supersymmetric extensions [7]. The gauge bosons and quarks acquire massesthrough interactions with Higgs fields. Up to now, the search for the physical Higgs bosonhas been unsuccessful but has led to the exclusion of a certain Higgs mass range [8, 9].In combination with the fits of electro-weak precision data to higher order perturbativecalculations this leads to a rather small range of allowed values for a SM Higgs bosonmass [10].Supersymmetric (
SUSY ) theories require an enlarged Higgs sector. The minimal
SUSY extension of the SM leads to five physical Higgs bosons. Due to the larger number of freeparameters in SUSY , the sensitivity of experimental data to Higgs bosons is weaker thanin the SM [11, 12].The Large Hadron Collider ( LHC ) is expected to find a Higgs boson if it exists. To do this,various production and decay channels must be considered. The relative utility of eachchannel depends strongly on the Higgs mass and couplings. While in the SM gluon fusionis the Higgs production process with the largest cross section by far, in SUSY theories withlarge tan β , Higgs production in association with bottom quarks is dominant (for reviews,see Refs. [13, 14]; detailed studies of the relative importance of gluon fusion and bottomannihilation have been performed in Ref. [15–17]). This is because in this region of the SUSY parameter space the
Hbb coupling is enhanced relative to the SM .Assuming that only four quark flavours and the gluon make up the proton (the so-called“four-flavour scheme” or ), the dominant leading order Feynman diagram for this1 bH (a) bb H (b) Figure 1: Leading order diagrams for associated bbh production in the (a) four and (b)five flavour scheme.process is shown in Fig. 1(a). If the bottom quark was massless, integration over phasespace would lead to divergences arising from the kinematical region where one or bothbottom quarks are collinear to the incoming partons. The bottom quark mass regulatesthese divergences, but they still leave traces in terms of logarithms of the form ln( m b /m H ).Such logarithms lead to large perturbative coefficients, so ideally one would like to resumthem. This can be achieved by considering this process in the five-flavour scheme ( ) [18,19], i.e. by introducing bottom quark PDF s (parton density functions). Now that the b quarks can appear in the initial state, the leading order process is changed to thatin Fig. 1(b). We note that the scheme choice amounts merely to a re-ordering of theperturbative series. Of course, when truncated at a finite order, results obtained in eitherscheme will differ, with the difference being formally of higher order in α s .However, it was found that the difference between the inclusive cross section in the and the differs by roughly a factor of five when evaluated at µ F = µ R = M H , where µ F /µ R is the factorization/renormalization scale. This remains true also at NLO QCD which was calculated for the in Ref. [21, 22], and for the in Ref. [23, 24]. It wasthus proposed in Refs. [22,25–27] that when using the five flavour scheme the appropriatescale choice is m H / is that it neglects the contribution from large- p T bottom quarks atleading order. This is taken into account only at NNLO and higher (note that the LO set ofFeynman diagrams in the is part of the NNLO set in the ). Indeed, the factorizationscale dependence at
NNLO is very flat [28] and seems to confirm the “natural” scale choiceat lower orders of µ F = M H / H +jetprocess instead of the fully inclusive production when searching for the Higgs boson [29].The gg → H +jet cross section is known at LO , including the full top and bottom quarkmass dependence, both in the SM [29–31] and in the MSSM [31].
NLO QCD corrections areknown in the heavy-top limit [32–34]. It is expected that a very good approximation ofthe
MSSM effects can be obtained by simply replacing the corresponding Wilson coefficientof the effective ggh coupling with its
MSSM expression [35–37], at least as long as tan β is not too large. Otherwise, bottom loop effects which are not covered in the heavy-toplimit will be important. Resummation effects for small and large p T of the Higgs bosonhave been treated in Refs. [38–41].As mentioned already above, for large tan β , it is essential to also take bottom quarkannihilation into account. It is well known that the corresponding QCD corrections can be In fact, this has been the default for all modern
PDF sets; see Ref. [20] though. b gb (a) Hg bb (b)
Figure 2: Representative diagrams for each of the two leading order channels.large. In this paper we present them for distributions of the Higgs boson. Since it has beenshown for the inclusive cross section that the dominant
SUSY effects can be approximatedto high accuracy by an effective b ¯ bh coupling [42], our results are directly applicable tothe MSSM by a trivial overall rescaling.In Ref. [43], a related quantity, namely the Higgs production cross section in associationwith a single tagged b quark was studied in the . Tagging a b may be useful formeasuring the b ¯ bh Yukawa coupling y b , for example. In this paper we consider Higgsproduction without necessarily tagging a final state b . That is, we consider the bb initialstate, and its associated sub-channels, as a contribution to the inclusive Higgs+jet crosssection. We present the p T and y distributions, and study the scale dependence of the crosssection and its component channels. These results are valid to NLO in QCD perturbationtheory. They will be presented in Section 2.Using the knowledge of the total inclusive cross section at
NNLO [28], we can then use ourresult for
NLO H +jet production in order to derive the NNLO cross section with uppercuts on p T . This will be described in Section 3. σ ( b ¯ b → h +jet) The generic leading order diagrams to Higgs plus jet production are shown in Fig. 2. At
NLO each of these receives virtual corrections, and in addition we must include the realemission contributions, which induce also other initial states. The full list of processesat
NLO is: bb → gH and bg → bH at one loop, bb → ggH , bb → bbH , bb → qqH , gb → gbH , bq → bqH , qq → bbH , gg → bbH , bb → bbH at tree level, where q denotesany of the light quarks u, d, s, c . It is understood that the charge conjugated processesmust also be included. Formally, the virtual contributions include diagrams where theHiggs boson is radiated off a closed bottom or top quark loop. The former lead to terms ∼ α s y b · m b /M , however, which is neglected throughout our calculation, and in the spiritof Refs. [28, 43], we discard the latter which are proportional to the top Yukawa coupling.They are separately finite and gauge invariant and could simply be added to our results,once the ratio of top and bottom Yukawa coupling is known.The NLO calculation of a process as the one considered here is by now standard. Weapply the dipole subtraction method [44] in order to cancel the infra-red poles betweenthe virtual and the real radiation contributions in the b ¯ b and bg processes. Introducing the α -parameter for restricting the dipole phase space [45,46] not only improves the numericalintegration, but also serves as a welcome check through the requirement of α -independenceof the final result. Furthermore, our result for the virtual corrections agrees with the resultof Ref. [43]. The leading logarithmic behaviour at small p T can be checked numerically3 -4 -3 -2 p T [GeV] d / dp T [pb / GeV] LO
NLO S had = 14 TeV m H = 120 GeV R = F = (p +m ) (a)
30 40 50 60 70 80 90 100 110 120 130 140 K p T p T [GeV] S had = 14 TeV m H = 120 GeV R = F = (p +m ) (b) Figure 3: (a) Higgs transverse momentum distribution at LO (dashed) and NLO (solid);(b) corresponding K-factor.against the resummed expression of Ref. [47]. The most important check, however, is thenumerical comparison to a fully analytic evaluation of the p T distribution to be publishedelsewhere [48].In order to avoid the infra-red divergence at low Higgs transverse momenta p T , we cutcontributions from p T <
30 GeV in this section. For our numerical analysis we use thefollowing set of input parameters. The
PDF s are taken from the
MSTW
QCD coupling is accordingly set to α s ( M Z ) = 0 . LO , and α s ( M Z ) = 0 . NLO . The b ¯ bh coupling, for which we assume the SM expression m b /v ( v = 246 .
22 GeV),is evaluated with the running bottom quark mass m b ( µ R ) defined in the MS scheme,with an input value m b ( m b ) = 4 . M H = 120 GeV.In Fig. 3 and Fig. 4 we show the transverse momentum and rapidity distributions of theHiggs boson, both at LO and NLO , and the corresponding K -factors K p T ≡ (d σ/ d p T ) NLO (d σ/ d p T ) LO , K y ≡ (d σ/ d y ) NLO (d σ/ d y ) LO , (1)where y = 12 ln E + p z E − p z , (2)and E and p z are the energy and longitudinal component of the Higgs boson in the labframe. The choice of the renormalization and factorization scales is given in the plots. Weremark that the numerator/denominator in Eq. (1) is evaluated with NLO / LO PDF s andcouplings. Both for the p T and the y distribution, the dependence of the K factors on p T and y is very similar to what is observed for the gluon fusion channel [32–34]: K p T israther flat over the considered p T interval, while K y drops mildly towards larger values ofthe rapidity. The absolute size of the corrections is significantly smaller than in the gluonfusion case though. Note that the K factors for the distributions cannot be immediatelydeduced from the one for the inclusive cross section for b ¯ b → H + X due to the strong µ F dependence at LO , cf. Ref. [28] and Fig. 6 (b).4 y d / dy ( p T > 30 GeV ) [pb ] LO NLO S had = 14 TeV m H = 120 GeV R = F = 120 GeV (a) y K y S had = 14 TeV m H = 120 GeV R = F = 120 GeV p T > 30 GeV (b) Figure 4: (a) Higgs rapidity distribution at LO (dashed) and NLO (solid); (b) correspond-ing K-factor. Since the distribution is symmetric around y = 0, only positive values of y are shown. ( p T > 30 GeV ) [pb] F m H LO NLO S had = 14 TeV m H = 120 GeV R = 120 GeV (a) R m H LO NLO ( p T > 30 GeV ) [pb] S had = 14 TeV m H = 120 GeV F = 120 GeV (b) ( p T > 30 GeV ) [pb] R m H R m H LO NLO S had = 14 TeV m H = 120 GeV F = 30 GeV (c) ( p T > 30 GeV ) [pb] R m H R m H LO NLO S had = 14 TeV m H = 120 GeV F = 120 GeV|y| < 2 (d) Figure 5: Scale dependence of the integrated cross section for p T >
30 GeV at M H =120 GeV. (a) µ R = M H fixed, µ F varies — (b) µ F = M H fixed, µ R varies — (c) µ F = M H / µ R varies — (d) same as (b), but with cut on | y | < .1 1 100.00.20.40.60.81.01.2 ( p T < 30 GeV ) [pb] F m H LO NLO
NNLO S had = 14 TeV m H = 120 GeV R = 120 GeV (a) tot [pb] F m H LO NLO
NNLO S had = 14 TeV m H = 120 GeV R = 120 GeV (b) R m H LO NLO
NNLO ( p T < 30 GeV ) [pb] S had = 14 TeV m H = 120 GeV F = 30 GeV (c) tot [pb] R m H LO NLO
NNLO S had = 14 TeV m H = 120 GeV F = 30 GeV (d) Figure 6: Scale dependence of the integrated cross section for p T <
30 GeV at M H =120 GeV. (a) µ R = M H fixed, µ F varies — (b) µ F = M H / µ R varies.In Fig. 5 we show the integrated cross section for h +jet production with a minimum p T ofthe Higgs of p T, cut = 30 GeV: σ ( p T > p T, cut ) = Z p T >p T, cut d p T d σ d p T (3)at M H = 120 GeV as a function of (a) the factorization and (b,c) the renormalizationscale. The factorization scale dependence is already quite small at LO and improvesslightly at NLO . The renormalization scale dependence we show for (b) µ F = M H and(c) µ F = M H /
4, respectively. For both choices, the
NLO corrections improve the scaledependence significantly, although µ F = M H seems to lead to a more natural behaviourof the LO and the NLO curves. This behaviour does not change much if we restrict theHiggs rapidity to | y | <
2, as shown in Fig. 5 (d). NNLO cross section with p T cut The knowledge of the total inclusive cross section σ tot at NNLO [28] allows us to use theresults of the current paper to evaluate the inclusive cross section when applying a finite6 tot (NNLO) p T,cut [GeV] ( p T < p T,cut ) [pb] LO NLO NNLO S had = 14 TeV m H = 120 GeV R = 120 GeV F = 30 GeV Figure 7: Inclusive cross section with an upper cut p T, cut on the Higgs transverse momen-tum, see Eq. (4), as a function of p T, cut . The dotted, dashed, and solid curves show the LO , NLO , and the
NNLO result. The arrow on the right indicates the value of the
NNLO result without cuts, σ tot . p T cut: σ ( p T < p T, cut ) = Z p T
p T, cut d p T d σ d p T . (4)Of course, p T, cut must not be too small in order not to be sensitive to the region wherelarge ln( p T /M H ) terms spoil perturbative convergence. Furthermore, if σ ( p T < p T, cut ) isto be evaluated with NNLO accuracy, we have to evaluate both terms on the right side ofEq. (4) with
NNLO PDF s.Fig. 6 (a) shows σ ( p T < p T, cut ) for p T, cut = 30 GeV as a function of the factorization scale µ F , for µ R = M H , varied over the rather large interval µ F ∈ [0 . , M H . The observationsare very similar to the fully inclusive cross section obtained without cuts [28] which isshown for comparison in Fig. 6 (b): Perturbation theory prefers a scale significantly below M H . With this choice, the renormalization scale dependence is very weak around µ R = M H already at NLO , as shown in Fig. 6 (c) and Fig. 6 (d).Finally, Fig. 7 shows the
NNLO cross section σ ( p T < p T, cut ) as a function of p T, cut . Sincethe LO only contributes at p T = 0, it is independent on p T, cut . The NNLO corrections arenegative with respect to
NLO at small p T, cut and change sign at around p T, cut ≈
50 GeV.
In this paper we have presented a first study of higher order differential distributionsfor Higgs production in bottom quark annihilation. The five-flavour scheme was used tocalculate
NLO p T and y distributions of the Higgs boson. Combination with the inclusive7 NLO total cross section allowed us to derive the inclusive cross section with upper cutson the Higgs transverse momentum at
NNLO .Concerning the choice of the factorization scale, we find that it strongly depends on theobservable under consideration: the value µ F = M H / p T = 0 is involved, but is less motivated otherwise.Although we expect that the results will be phenomenologically relevant on their own, ourapproach should be useful also for an extension to a fully differential NNLO
Monte Carloprogram for the process b ¯ b → H + X along the lines of Ref. [51]. Acknowledgments.
We would like to thank M. Kr¨amer for useful comments on themanuscript. This work was supported by
DFG contract HA 2990/3-1 and by
BMBF grant05H09PXE.
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