Associated Higgs-Bottom Quark Production: Reconciling the 4FS and the 5FS Approach

Associated Higgs-bottom quark production: reconciling the 4FS and the 5FS approach

R. Harlander, M. Kr¨amer, M. Schumacher

6. April 2011 — v0.52

1 Two approaches

The cross section for associated Higgs-bottom quark production, pp → b¯bH + X, can be calculated in two different schemes. As the mass of the bottom quark is large compared to the QCD scale, mb ≫ ΛQCD, bottom quark production is a perturbative process and can be calculated order by order. Thus, in a four-flavour scheme (4FS), where one does not consider b quarks as partons in the proton, the lowest-order QCD production processes are gluon-gluon fusion and quark-antiquark annihilation, gg → b¯bH and qq¯ → b¯bH, respectively. However, the inclusive cross section for gg → b¯bH develops logarithms of the form ln(µF/mb), which arise from the splitting of gluons into nearly collinear b¯b pairs. The large scale µF ≈ MH /4 corresponds to the upper limit of the collinear region up to which factorization is valid [1, 2, 3]. For MH ≫ 4mb the logarithms become large and spoil the convergence of the perturbative series. The ln(µF/mb) terms can be summed to all orders in perturbation theory by introducing bottom parton densities. This defines the so-called five-flavour scheme (5FS). The use of bottom distribution functions is based on the approximation that the outgoing b quarks are at small transverse momentum. In this scheme, the LO process for the inclusive b¯bH cross section is bottom fusion, b¯b → H. If all orders in perturbation theory were taken into account, the four- and five-flavour schemes would be identical, but the way of ordering the perturbative expansion is different. At any finite order, the two schemes include different parts of the all-order result, and the cross section predictions do thus not match exactly. While this leads to an ambiguity in the way the cross section is calculated, it also offers an opportunity to test the importance of various higher-order terms and the reliability of the theoretical prediction. In the next section we briefly summarize some of the features of the two schemes.

2 Discussion

In this section, we collect the main arguments and counter-arguments that have been discussed in the physics community for and against either the 4FS or the 5FS approach. We supplement some of them by our personal comments (in italic type-set).

4FS: • The 4FS includes the full splitting g → b¯b order by order in perturbation theory.

1 • The 4FS starts at LO with the correct kinematics of the process.

• For Higgs masses MH ≫ 4 mb the 4FS calculation misses potentially large logarithmic corrections.

• As the LO process in the 4FS is already a 2 → 3 process, the 4FS prediction is only available at NLO [4, 5].

• The NLO calculation in the 4FS contains contributions that are formally of N3LO in the 5FS.

• 4FS PDFs are rare. Comment: Not true anymore; there are now 4FS pdfs from all major ﬁt groups. Also, it is possible to convert 5FS to 4FS pdfs.

5FS:

• The 5FS resums collinear logarithms of the form ln(MH /mb). • The 5FS calculation is available at NNLO [6].

• The 5FS neglects the kinematical region from large pT,b. Comment: This is not true if NNLO is taken into account in the 5FS.

• The 5FS neglects eﬀects from a ﬁnite bottom quark mass (except those from the Yukawa coupling).

Comment: (a) The leading terms in mb are taken into account from the resum- mation of the ln(MH /mb) terms, see above. (b) All other mb eﬀects are of order 2 2 ∼ −3 mb /MH 10 or higher. This has been conﬁrmed quantitatively in Ref.[7] (com- pare the markers to the dashed blue line in Fig.2).

• Once cuts are applied, or b-jets are tagged, the 5FS is no longer valid. Comment: This is to be discussed in more detail, once we deal with more exclusive cross sections.

• The 5FS calculation neglects contributions where the Higgs couples to a top quark triangle. Comment: (a) Most of these eﬀects are taken into account in the gluon fusion cross section. At NNLO, interference terms between the b¯bh and gluon fusion arise. They are (currently) indeed neglected. (b) The b¯bh process becomes important for inter- mediate to large tan β. This suppresses the top-triangle contributions anyway.

• The PDF uncertainty for the b-densities is under-estimated. Comment: The PDF groups provide uncertainties, and we have to trust them. One source of possibly uncontrolled uncertainty is the matching between the low and the high-Q2 region in the DGLAP evolution. However, since Higgs production happens 2 2 at energies way above the b threshold, these eﬀects should be suppressed by mb /MH as well.

2 σ/pb 2 LHC b-bbar 1.75 MH=100 GeV 1.5 CTEQ6M NNLO 1.25 1 0.75 gg

0.5 bq 0.25

0 bb, q-qbar -0.25 bg

-0.5 -1 10 1 10 µ F/MH

Figure 1: The NNLO cross section for the process b¯b → h + X (pp@14 TeV, MH = 100GeV) (solid line) as a function of the factorization scale µF . The dashed and dotted lines represent the individual sub-processes in the MS scheme. The markers denote the contribution of the gg sub-process, with mass divergences subtracted minimally, and bottom quark mass terms kept in the matrix element calculation. From Ref. [7] (Fig. 140).

• The uncertainty due to renormalization/factorisation scale dependence in the 5FS is unnaturally small. Comment: Since it is a NNLO calculation, we expect a small scale dependence. NNLO and NLO uncertainty bands nicely overlap and we observe a perfectly ﬁne convergence behavior (for µF = MH /4). It is therefore fully reasonable to believe that the NNLO uncertainty band covers higher order eﬀects.

3 Santander matching1

The 4FS and 5FS should both describe the cross section reliably for moderate Higgs masses. With increasing Higgs masses the 5FS should become more reliable, simply be- cause the effect of the collinear logarithms ln(MH /mb) becomes more and more important. This is why it is not surprising that the 5FS and the 4FS begin to differ once larger Higgs masses are considered (see Fig.23 of Ref.[8]). Nevertheless, fully ignoring this difference may make one feel uncomfortable. Therefore, we suggest to combine the two approaches in such a way that they are given variable weight, depending on the value of the Higgs mass.

1This name originates from the fact that the following procedure arose from discussions among the authors at the Higgs Days at Santander 2009.

3 The diﬀerence between the two approaches is formally logarithmic. Therefore, the dependence of their relative importance should be controlled by a logarithmic term. We determine the coeﬃcients such that

(a) the 5FS gets 100% weight in the limit mH → ∞ (b) the 4FS gets 100% weight in the limit where the logarithms are “small”. There is obviously quite some arbitrariness in this statement. We assume here that “small” means ln(MH /mb) = 2. The consequence of this particular choice is that the 4FS and the 5FS both get the same weight for Higgs masses around 100 GeV, consistent with the observed agreement between the 4FS and the 5FS in this region.2

This leads to the following formula 1 σ(b¯b → h + X)= σ4FS + tσ5FS , (1) 1+ t where M t = ln H − 2 . (2) mb

For mb = 4.8GeV and speciﬁc values of MH , this leads to

σ(b¯b → h + X) = 0.49 σ4FS + 0.51 σ5FS ,

MH =100 GeV

σ(b¯b → h + X) = 0.37 σ4FS + 0.63 σ5FS , (3)

MH =200 GeV

σ(b¯b → h + X) = 0.32 σ4FS + 0.68 σ5FS .

MH =300 GeV

We suggest to add the uncertainties quadratically using the same weights:

1/2 (∆σ4FS)2 +(t∆σ5FS)2 ∆σ = . (4) 1+ t

References

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