Probing the Charm Yukawa Coupling in Higgs + Charm Production

We propose a new method to determine the coupling of the Higgs boson to charm quarks, via Higgs production in association with a charm-tagged jet: $pp\to hc$. As a first estimate, we find that at the LHC with 3000 fb$^{-1}$ it should be possible to derive a constraint of order one, relative to the SM value of the charm Yukawa coupling. As a byproduct of this analysis, we present an estimate of the exclusive $pp \to hD^{(*)}$ electroweak cross section. Within the SM, the latter turns out to be not accessible at the LHC even in the high-luminosity phase.


I. INTRODUCTION
While the Yukawa couplings of the heavy thirdgeneration fermions to the Higgs boson can be measured at the LHC with a O(10 %) accuracy, see e.g. Ref. [1], constraining the diagonal Yukawa couplings of the second (first) generation quarks at a level close to the Standard Model (SM) expectation is very challenging. An interesting possibility, especially for the second generation, is trying to indirectly access these couplings via the radiative decays h → M + γ(Z) [2][3][4][5], where M is a quarkonium state. 1 As pointed out in Ref. [8], the exclusive h → MV decays (V = γ, Z, W ) may indeed be accessible at the SM level at the LHC and represent a precious source of information on physics beyond the SM. In the specific case of the charm Yukawa coupling (Y c ), it should be possible to obtain bounds 2-3 times larger than the SM value in the high-luminosity (HL) phase of the LHC [9]. These constraints are driven mainly by the direct search for h → cc and, to a smaller extent, also by the indirect sensitivity via h → J/Ψγ.
In this paper, we propose a new method to measure Y c by means of Higgs production in association with a charm-tagged jet. A particular advantage of this method, compared to the search for h → cc, lies in the fact that we probe Y c in production -via the interaction with a charm quark from the abundant gc initial state-allowing to reconstruct the Higgs from its clean decay modes (h → γγ or h → W W ). This procedure strongly reduces the problem of the non-Higgs background, compared to h → cc. Moreover, requiring a single c-tagged jet in the final state allows to adopt high-purity (and low-efficiency) ctag algorithms in order to reduce background (mainly * ilaria.brivio@uam.es † florian.goertz@cern.ch ‡ isidori@physik.uzh.ch 1 For indirect bounds on first generation Yukawa couplings see Ref. [6,7]. from b-quark jets), compared to the case of two c-tagged jets (as in h → cc).
Compared to the indirect sensitivity to Y c in h → J/Ψγ, our new method has the advantage of being sensitive to Y c at the tree-level and being based on a process that, after charm-and Higgs-tagging efficiencies, yields O(1000) signal events at the HL-LHC. For comparison, we recall that B(h → J/Ψγ → µ + µ − γ) ∼ 10 −7 , corresponding to O(10) signal events in pp collisions at 14 TeV with 3000 fb −1 . The main limiting factor of our approach is the theoretical uncertainty on σ(pp → hc), as a function of Y c . This error could be reduced in the future by means of higher-order QCD calculations of the ratio σ(pp → hc)/σ(pp → hb) as a function of Y c and Y b .
In principle, the production of the Higgs boson in association with a charm jet (or a charm hadron) can also proceed via electroweak interactions, with the charm being produced by a real or virtual W boson. To complement this analysis, and previous studies of exclusive hadronic Higgs decays [2,3,5,8], we present here the first estimate of the electroweak production of the Higgs boson in association with a single D or D * meson (qq → hD ( * ) ). These processes are insensitive to the charm Yukawa coupling and could have represented a potential background for the extraction of Y c . We have analyzed them in generic extensions of the SM, along the lines of Ref. [8]. We find that, within the SM, the exclusive electroweak production should not be visible at the LHC, even in the high-luminosity phase. Moreover, we find that these process are not competitive with the corresponding exclusive Higgs decays (h → MV ) as far as generic new physics (NP) searches are concerned.
The paper is organized as follows. In Sect. II we introduce the setup to describe Higgs physics with modified Yukawa couplings. The QCD-Yukawa pp → hc process and the corresponding extraction of Y c is discussed in Sect. III. The exclusive electroweak pp → hD ( * ) process is analyzed in Sect. IV. The results are summarized in the Conclusions.

II. SETUP
Within the SM the couplings of the physical Higgs boson to the fermions are completely determined in terms of fermion masses. However, in the presence of NP, a misalignment between quark-mass and Yukawa matrices is possible. This can be parametrized in a modelindependent way by adding the D = 6 operators to the SM Lagrangian. Here, Φ denotes the Higgs doublet, parametrized in unitary gauge as where v corresponds to the vacuum ex- where the Yukawa matrixŶ After performing a rotation to the mass basiŝ with U d L = U u L V CKM , we finally arrive at the couplings of the physical quarks to the Higgs boson Here, we concentrate on possible experimental constraints on the diagonal entry Y c ≡ (Y u ) 22 . For convenience, we parametrize the deviations from the SM pre- which we assume to be real for simplicity. 2

III. THE QCD-YUKAWA pp → hc PROCESS
We consider the production of a Higgs boson in association with a charm-quark jet. At the LHC, the main partonic process inducing this final state is gc → hc and the corresponding Feynman diagrams are presented in Figure 1. The charm Yukawa coupling, depicted as a black dot, enters in the first two graphs, that yield a contribution to the amplitude of O(g s Y c ). The t−channel diagram turns out to be largely dominant. The third diagram is formally of higher order in α s but is enhanced by the top-quark Yukawa coupling. Here the crossed vertex corresponds to the effective ggh interaction obtained by integrating out the top quark. This diagram yields the contribution to the amplitude that survives in the limit κ c → 0 (see Table I).
The challenge of the proposed process is to tag the charm-quark jet, as in h → cc. However, as anticipated, it offers some interesting virtues compared to h → cc. In particular, it allows us to fully reconstruct the Higgs boson in a clean decay channel such as h → γγ or h → W W , and it requires only a single charm tag. The main drawback is that the process does not vanish in the limit Y c → 0 (contrary to h → cc) requiring a good theoretical control on the cross section as a function of Y c . While a full analysis, including the optimization of the event selection, is beyond the scope of this article, here we just want to examine the potential of the channel by deriving the expected number of signal and background events, based on reasonable efficiency assumptions.
We have calculated the cross section of pp → hc at leading order in QCD (including the effective ggh as discussed above) at the LHC with 14 TeV center-of-mass energy for various values of κ c , employing MadGraph5 [10], with a tailored model file and CTEQ6L1 parton distribution functions. Using m c (m Z ) = 0.63 GeV and m h = 125 GeV, for κ c = 1 (i.e., the SM) we obtain a cross section of σ(pp → hc) = 166.1 fb, employing the default cuts of p T (j) > 20 GeV, η(j) < 5, ∆R(j 1 , j 2 ) > 0.4 for all processes considered here. In the following, we focus on the h → γγ decay channel, with a branching fraction of B(h → γγ) = 0.0023. This leads to S 0 = 2292 events at the HL-LHC with 3000 fb −1 , taking into account also the pp → hc process. Assuming a charm-tagging efficiency of c = 0.4 (see e.g. Ref. [9]), we finally end up with S = c S 0 = 917 signal events. The different number of events obtained by varying κ c are reported in Table I.
The main backgrounds to the process studied here are pp → hg, with the gluon mis-identified as a charm quark, as well as pp → hb, with the bottom quark being mis-tagged. In the first case, we treat separately the case pp → hcc, where only one charm-quark jet is reconstructed and the case where the gluon produces a light quark jet. The backgrounds feature σ(pp → hg) = 12.25 pb, σ(pp → hb) = 203 fb, as well as σ(pp → hcc) = 55 fb. We employ a conservative assumptions for the jet reconstruction efficiency of 1 − miss = 95%, as well as g → c and b → c mis-tag rates of g→c = 1% and b→c = 30%. With these figures we obtain B = 1705 background events at 3000 fb −1 , leading to N (κ c = 1) = S(κ c = 1) + B = 2622 total events. We then assume a statistical error on the total number of events ( √ N ) and a theoretical (relative) error on the signal events of 20%. The latter is deduced by the recent next-to-leading order (NLO) analysis of the Higgs production in association with bottom quarks [11]. Finally, statistical and theoretical error are added in quadrature. 3 In the following, we want to examine the expected constraints that can be set on κ c from the process under consideration. To this purpose, we assume the SM to be true and calculate how many standard deviations ∆N (κ c ) away a prediction N (κ c ) is from N (κ c = 1), which is the expected outcome of the experiment. The values of κ c that lead to a discrepancy of more than n standard deviations are then expected to be excluded at n σ. We plot the corresponding p-value, p(κ c ), in Figure 2 approximating the Poisson distribution of the number of events by a Gaussian. The 1σ and 2σ equivalents are depicted by the solid and dashed lines, respectively. A conservative estimate for the expected 1-σ (95% CL) constraint on κ c is thus obtained as |κ c | < 2.5 (3.9), (5) which lies in the ballpark of the results quoted in [9], where the latter combines ATLAS and CMS to arrive at 2 × 3000 fb −1 of integrated luminosity. On the other hand, an improved prediction of the SM cross section σ(pp → hc), leading to δ th = 10%, would strengthen our expected 1-σ (95% CL) limit to |κ c | < 1.9 (2.6), (6) approaching the SM value of Y c . We note that optimized cuts can still increase S/B and in particular lead to an enhanced sensitivity on κ c . As the statistics at 3000 fb −1 is large enough, there are good prospects to still improve the bounds. A corresponding detailed investigation, including detector simulation, is beyond the purpose of this letter and can be performed best by the experimental community.
We further stress that the dominant source of uncertainty, at present, is the theoretical error on σ(pp → hc). We have indeed checked that the result does not change significantly worsening the g → c and b → c mis-tag rates to 5% and 40%, respectively. As far as the reliability (and possible reduction) on the theoretical error is concerned, a promising possibility would be a dedicated calculation of σ(pp → hc)/σ(pp → hb) at NLO (or NNLO), as a function of Y c /Y b , supplemented by measurements of this ratio and σ(pp → hb) with a combination of normal and inverted b vs. c tags.

IV. THE ELECTROWEAK pp → hM PROCESS
As anticipated in the introduction, the production of the Higgs boson in association with charm can proceed also via electroweak interactions, starting form an initial charm-less qq state (ud → hW ( * ) → hcs). The case of an on-shell W producing a charm jet can be discriminated from the QCD-Yukawa process by means of appropriate cuts on the jet momentum. Less obvious is the discrimination in the case of a virtual W * producing a low-momentum c-jet, or even a single charmed hadron. In the following we estimate in detail the specific case of the single meson production: pp → hM, with M being a charmed meson or a charmonium state.
The leading partonic amplitude within the SM is shown in Fig. 3. Following Refs. [8,12], we parameterize the quark currents appearing in the initial and final state with arbitrary vector and axial couplings: The matrix element of the current that generates the meson in the final state assumes one of the following structures, depending on the spin of M: where f M is the meson's decay constant, and g M encodes the dependence on the coupling to the relevant gauge boson (g P = g A,ij , g V = g V,ij for a q j q i meson). 4 With this notation, the SM expression for the partonic cross section for the case of a pseudoscalar meson reads where V = W ± , Z, and we have suppressed the indices of g A,V for simplicity. The vector case has the same functional form with P → V, up to tiny O m 2 V /m 2 V corrections. In the above expression, q 2 denotes the total momentum of the initial state in the partonic process and Convoluting the cross sections with the appropriate PDF in the region 130 ≤ q 2 ≤ 1 TeV, and assuming an integrated luminosity of 3000 fb −1 we obtain the expected number of events for each channel at the HL-LHC. The results, summarized in Table II, show that these processes will not be observable at the SM level, and certainly do not represent a dangerous background for the QCD-Yukawa process discussed in Sect. III. Given the smallness of the SM signal, it is worth to investigate if these cross sections can be significantly altered beyond the SM. This can be done generalizing the approach of Refs. [8,12]. The leading (helicityconserving) transition amplitude can be decomposed in full generality as are obtained rescaling bin-by-bin the cross section distribution of Drell-Yan processes provided by MadGraph 5 [10]. The computation of Method (b) is performed via numerical convolution of the analytic cross section with the PDF of the MSTW 2008 libraries [13]. Both account only for SM contributions.
To a good accuracy the quark current is conserved (q µ J µ q = 0), and the tensor T µν can be decomposed in terms of only four Lorentz structures. Using the same notation as in Ref. [8]: where q µ is the total momentum of the quark pair in the initial state, and p µ is the meson momentum (p 2 = m 2 M ). With these notations the partonic cross section reads where, similarly to the SM case, σ(qq → hV) has the same functional form up to tiny O m 2 V /m 2 V corrections. Neglecting the latter terms, we obtain where f SM 1 (q 2 ) ∝ 1/(v(q 2 −m 2 V )) and we disregard potential changes to the fermionic currents. Deviation from the SM are thus induced by possible non-pole-terms (i.e. contact terms) in the form factor f 1 (q 2 ). Within a generic effective-field theory (EFT) approach to Higgs physics (both linear and non-linear EFT), contact terms in f 1 (q 2 ) are generated by dimension-six operators. However, their effect would show-up exactly in the same functional form either in the on-shell associated production (pp → V h) or in h → V M decays, that share the same current structure [8,12]. Since the latter processes can be measured (or at least bounded) to a better accuracy, we conclude that σ(pp → hM) is not a very sensitive probe of generic extensions of the SM.

V. CONCLUSIONS
In this letter, we proposed a new strategy for the measurement of the Yukawa coupling of the charm quark: the measurement of the production cross section of the Higgs boson in association with a charm jet. A first estimate showed that Y c could be determined at a level approaching the SM value in this channel, which offers virtues and drawbacks quite different with respect to the h → cc search. A fully realistic analysis was beyond the scope of the present paper. A more realistic evaluation of the efficiencies is likely to decrease the number of signal events S compared to our naive estimate; however, as we have discussed, the sensitivity on Y c could even increase with properly designed b and c tag strategies aimed to measure the background from data and to reduce the theoretical error on the normalization of the cross-section. This first analysis therefore calls for more detailed studies both on the theory and on the experimental side.