Anomalous Couplings in Double Higgs Production

The process of gluon-initiated double Higgs production is sensitive to non-linear interactions of the Higgs boson. In the context of the Standard Model, studies of this process focused on the extraction of the Higgs trilinear coupling. In a general parametrization of New Physics effects, however, an even more interesting interaction that can be tested through this channel is the (ttbar hh) coupling. This interaction vanishes in the Standard Model and is a genuine signature of theories in which the Higgs boson emerges from a strongly-interacting sector. In this paper we perform a model-independent estimate of the LHC potential to detect anomalous Higgs couplings in gluon-fusion double Higgs production. We find that while the sensitivity to the trilinear is poor, the perspectives of measuring the new (ttbar hh) coupling are rather promising.


Introduction
Measuring the couplings of the Higgs boson at the LHC is a difficult but important task. It will give crucial information to distinguish among different theoretical scenarios that can lead to a Higgslike particle, and can thus shed light on the mechanism behind electroweak symmetry breaking (EWSB). In the Standard Model (SM), the request of perturbativity and unitarity up to Planckian scales fixes the strength of all the interactions of the Higgs boson in terms of its mass. Sizable modifications of the couplings can arise in weakly-coupled extensions, such as supersymmetry, through the mixing of the Higgs boson with new light states. In this case one expects to produce these new particles directly at the collider. A second compelling possibility is that the EWSB is triggered by new strong dynamics at the TeV scale, and a light Higgs emerges as the pseudo Nambu-Goldstone boson (pNGB) of a larger spontaneously broken symmetry [1]. In this case the modification of couplings is not necessarily accompanied by the presence of new light scalars and the direct manifestation of New Physics can be postponed to TeV energies. In addition to a modified pattern of linear couplings, this scenario predicts new non-linear interactions of the Higgs to the SM fields, which can lead to striking signatures at the collider and are a genuine feature of the underling strong dynamics.
Aim of this paper is to study the effect of anomalous couplings in the process of gluon-initiated double Higgs production at the LHC, gg → hh. Under the reasonable assumption of weak couplings to light fermions, this process proceeds through top quark loops and it is thus sensitive, in the first place, to the Higgs-top couplings. It also receives a contribution from the Higgs trilinear coupling, and for this reason has been studied in detail in the context of the SM [20][21][22][23][24][25][26]. In theories of New Physics, however, a much more interesting coupling that can be probed through this process is the non-linear interaction tthh. The latter is generally present in theories of composite Higgs, like for example the minimal models MCHM4 [27] and MCHM5 [28], and gives a genuine signal of the Higgs strong interactions. As first noticed by the authors of Ref. [29] (see also Ref. [30] for a discussion of the role of the tthh coupling in context of Little Higgs theories), the presence of the new coupling can lead to a dramatic increase of the cross section. For example, enhancements larger than one order of magnitude are possible in the MCHM5, and even for (v/f ) 2 ∼ 0.15, where f is the decay constant of the pNGB Higgs, the total cross section doubles compared to its SM value. Given that the deviations due to Higgs compositeness are usually much milder, gg → hh seems an extremely favored channel which is worth investigating.
In this paper we derive a first quantitative assessment on the detectability of the anomalous coupling tthh in the process gg → hh. We do not consider the rarer process of double-Higgs production via vector boson fusion, which has been investigated at the LHC in previous studies [31][32][33][34], since it is sensitive to the couplings of the Higgs to vector bosons and to the Higgs trilinear coupling. Neglecting such process is a very good approximation, considering that in absence of dedicated kinematic cuts its rate at the LHC is much smaller than the rate of gg → hh. In section 2 we briefly summarize the parametrization of the couplings of a generic Higgs-like scalar which we adopt. In section 3 we analyze the gg → hh process in the presence of modified Higgs interactions. In particular we study the dependence of the cross section on the various couplings and show that there is high sensitivity to the new tthh coupling, much larger than that on the trilinear self-interaction. For our analysis we wrote a dedicated computer code which computes the exact 1-loop matrix element for single and double Higgs production via gluon fusion as a function of the relevant couplings. The code has been implemented as one of the available processes of the event generator ALPGEN [35] and will be made public with its next official release. We then discuss two of the most promising decay channels of the Higgs pair: hh → W W γγ → lνjjγγ in the case in which the Higgs has a suppressed single coupling to the top (fermiophobic limit), and hh → bbγγ in the case in which the linear couplings are SM like. For the latter case, we follow the strategy proposed in Ref. [26] and in section 4 we perform a first collider study to estimate the exclusion and discovery limits on the anomalous tthh coupling. To compute the SM background cross section we use the results of [26] with updated b and γ efficiencies and rejection factors. We collect the results of our collider study in section 4.1. Finally, conclusions are reported in section 5.

General parametrization of the Higgs couplings
In this section we introduce the general parametrization of Higgs couplings that will be used in this paper.
The most general effective Lagrangian that parametrizes the interactions of a Higgs-like scalar at low energy has been discussed in [33] and extended in [11]. Under the assumption of custodial symmetry, the Nambu-Goldstone bosons associated to the electroweak symmetry breaking can be described as the coordinates of the coset SU (2) L × SU (2) R /SU (2) V ∼ SO(4)/SO (3). They can be conveniently parametrized by the 2 × 2 matrix where σ a are the Pauli matrices and v = 246 GeV. At energy scales much below possible new physics states, the effective Lagrangian describing a light Higgs h has the form where q L = (t L , b L ) and a, c t,b , c 2 and d 3 are the numerical coefficients that parametrize the Higgs couplings. The dots stand for terms which are not relevant for double Higgs production via gluonfusion. In particular, we assume that the strength of single interactions of the Higgs to the fermions is not extremely enhanced compared to its SM value, so that the contribution of the light fermions and the bottom quark to double Higgs production can be safely neglected. The coupling to the bottom, c b , is relevant only in the Higgs decay, and we will set for simplicity in the following. We also neglect ggh and gghh local interactions which can be generated by new heavy states at 1-loop level, as for example scalar or fermionic partners of the top quark (see Ref. [36] for a study of the effect of such local interactions).
In this analysis we will freely vary the parameters that appear in the effective Lagrangian. In specific models, however, they can be related to each other. For example, the SM Lagrangian is obtained for A class of theories that we will consider in the following are the composite Higgs models based on the symmetry pattern SO(5)/SO(4) [27,28]. In these models the parameter a is given by fermions in the spinorial (MCHM4 [27]) and fundamental (MCHM5 [28]) representations one gets Equations (5), (6) and (7) account for the value of the Higgs couplings as due to the non-linearities of the chiral Lagrangian. The exchange of new heavy particles can however give further corrections to these expressions. In the following we will neglect these effects since they are parametrically subleading [37], although they can be numerically important when the top or bottom degree of compositeness becomes large [38]. This is especially justified considering that in minimal composite Higgs models with partial compositeness these additional corrections to the couplings do not affect the gg → h rate because they are exactly canceled by the contribution from loops of heavy fermions, as first observed in Refs. [39,37] and explained in Ref. [38]. For double Higgs production we expect this cancellation to occur only in the limit of vanishing momentum of the Higgs external lines. In general, numerically important contributions might come from light top partners (light custodians). In models with partial compositeness, where the dominant contribution to the Higgs potential comes from top loops, the presence of light fermionic resonances is essential to obtain a light Higgs [28,40]. In particular, m h 120 − 130 GeV requires top partners around or below 1 TeV. It would be interesting to analyze in detail their effects on double Higgs production.

Double Higgs production via gluon fusion
In the scenario we are considering, the leading-order contributions to the process gg → hh come from Feynman diagrams containing a top-quark loop. The three relevant diagrams are shown in Fig. 1, and can be computed by using the results of Ref. [21]. We have implemented the automatic computation of the matrix element as one of the processes of the ALPGEN MonteCarlo generator [35]. The code will be made public with the next official release of ALPGEN, and it allows one to compute the total cross section and differential distributions, as well as to generate events for an arbitrary choice of the Higgs couplings c, d 3 , c 2 . The validation of the code has been performed by means of an independent C++ program linked to the QCDLoop [41] and to the LHAPDF routines [42]. All the results reported in the following have been derived by use of the ALPGEN matrix element calculation with CTEQ6l parton distribution functions and renormalization and factorization scales Q = m(hh). The top quark mass has been set to m t = 173 GeV.
The amplitude of each diagram in Fig. 1 is characterized by a different energy scaling at large The cross section depends on the couplings as a quadratic polynomial in the variables c 2 , c 2 and cd 3 , associated respectively to the three diagrams of Fig. 1. It can thus be conveniently expressed by the formula where the value of the (real) coefficients σ, α, β, A 1 , A 2 , A 3 has been extracted by fitting the results of a Montecarlo integration, and is reported in Table 1 Table 1: Coefficients for the fit of eq.(11) of the total LO pp → hh cross section via gluon fusion at the LHC. and 14 TeV. In the above parametrization, the coefficients α and β measure the sensitivity of the cross section on the parameters c 2 and (cd 3 ), relative to c 2 . One can see from Table 1 that the dependence on c 2 is significant while the one on (cd 3 ) is rather mild. This can be tracked back to the additional factor (m 2 h /ŝ) in the amplitude of the triangle diagram which carries the dependence on (cd 3 ), see eq.(9). This factor leads to a suppression at largeŝ and thus, because of the kinematic thresholdŝ > 4m 2 h , to a reduction of the sensitivity on (cd 3 ) of the total cross section. We see in Table 1 that increasing the LHC center-of-mass energy from 8 TeV to 14 TeV increases the LO total cross section (hence the coefficientσ) by a factor ∼ 5.2, while the relative strength among α, β, A 1 , A 2 , A 3 varies by less than ∼ 15%. When the Higgs mass is varied from 120 GeV to 125 GeV,σ decreases by ∼ 5 − 7%, while the other coefficients change by less than ∼ 1 − 3%. The the full one-loop matrix element calculation, while the dot-dashed curve is obtained by taking the limit of infinite top mass. As previously noticed [23], 1 this approximation is reasonably accurate in the case of the total cross section, but completely fails to reproduce the correct m(hh) distribution, as illustrated by the corresponding curve in Fig. 2.
We have seen that modified Higgs couplings, in particular a non-vanishing tthh interaction, can lead to a strong enhancement of the total pp → hh cross section. However, in order to determine the signal yield at the LHC in a given final state one has to take into account also the change in the Higgs decay branching ratios. In our case, these latter depend only on the ratio of the parameters c and a. We will consider two illustrative situations: i) the case in which the branching ratios are similar to the SM ones; ii) the case in which the couplings of one Higgs boson to two fermions are suppressed (fermiophobic limit). The first situation is realized in models where all single Higgs couplings are rescaled by the same factor (as in the MCHM4, see eqs.(5), (6)), or their shift from  with two or three leptons. The γγ branching ratio is also strongly enhanced, so that in this case hh → W W γγ also seems a promising final state. In particular, the request of one lepton from the decay of the W pair should be sufficient to reduce the background and lead to a clean signature at the LHC.  [44]. Figure 5 shows the final yield per fb −1 predicted in the MCHM5 in the two final states hh → γγbb and hh → W W γγ → lνqqγγ as a function of ξ for m h = 120 GeV.
The rate has been computed using the cross section for gg → hh at LO in α s (i.e. no K-factor is included) given by eq.(11) and  and hh → W W γγ → lνqqγγ (dashed curves) as a function of ξ. The thick (thin) curves correspond to the LHC with 14 TeV (8 TeV) center-of-mass energy. The Higgs mass is set to m h = 120 GeV. The rate has been computed using the cross section for gg → hh at LO in α s (i.e. no K-factor is included) given by eq.(11) and Table 1. channel is γγbb, whose rate can be significantly enhanced compared to the SM expectation. in hh → 4W → l ± l ± νν4q (two same-sign leptons) and hh → 4W → 3l3νqq. These high rates suggest that it might be possible to distinguish the hh → 4W signal over the SM background even at 8 TeV.
The results discussed in this section are rather encouraging, and show that double Higgs produc-tion can be an important process to extract or constrain the tthh interaction and, more in general, to probe the Higgs compositeness. However, a more robust assessment of the LHC sensitivity in this sense requires a dedicated analysis of each of the relevant final states and a careful estimate of the background. In the next section we will focus on the γγbb channel and use the studies of Ref. [26] to get a first determination of the precision which can be obtained on c 2 at the 14 TeV LHC.

Analysis of the bbγγ channel
The analysis of the gg → hh → bbγγ process performed in Ref. [26] aimed at measuring the Higgs trilinear coupling, and assumed SM values for the other couplings. In this section we make use of the results of [26] to estimate the LHC sensitivity on the tthh non-linear interaction.
We assume a center-of-mass energy √ s = 14 TeV and set m h = 120 GeV. Given the value of the signal rate and the large SM background, an analysis of bbγγ at 8 TeV seems rather challenging and for this reason it will not be considered here. We expect our results at 14 TeV to be representative of what the LHC sensitivity will be in its future high-energy phase, even if the actual value of c.o.m. energy turns out to be different (on the prospects of 14 and 13 TeV see for example [45]).
The analysis is performed at the parton level: signal events are generated by means of our implementation of double Higgs production via gluon fusion in ALPGEN, while the computation of the background processes is taken from [26]. We take the NLO QCD corrections to gg → hh into account by multiplying the LO cross section by a factor K = 2 [22], 3 while we neglect the smaller contribution of the vector-boson-fusion process to double Higgs production. To ensure an effective suppression of the background, we select events with two photons and two b-jets. Two photon tags are necessary to suppress the huge QCD background. On the other hand, the optimal number of b-tags depends on the strength of the signal and on the collider energy. In Ref. [26] only one b-tag was required for the analysis at the 14 TeV LHC. This choice was motivated by the necessity of preserving the small SM signal as much as possible, at the expense of having to cope with a larger background. In our case the signal cross section is much larger than the SM one in a sizable part of the parameter space. Furthermore, recent studies have shown that a larger b-tagging efficiency than that assumed in Ref. [26] is possible at the LHC [46], maintaining an acceptable rejection rate on jets. In particular, we adopt the following conservative estimates for the efficiencies ( ) and fake rates (r) for b-tagging [46] and photon reconstruction [47]: 4 This allows us to require two b-tags, thus obtaining a stronger suppression of the background at the price of an affordable reduction of the signal. We find that including the additional category of events with one b-tag in the analysis does not sensibly improve our results.
Following [26], we impose the set of kinematic cuts which have a high selection efficiency on the signal and ensure that the bb and γγ invariant masses are reconstructed in the given windows around the Higgs mass. To take into account the detector resolution, we have assumed a 79% efficiency for the reconstruction of the bb pair and a 79% efficiency for the reconstruction of the γγ pair in the signal, as done in Ref. [26]. After the above cuts, the most important irreducible backgrounds come from the bbγγ continuum, and potentially by single Higgs production in association with two b-quarks (h(→ γγ)bb) or two photons (h(→ bb)γγ).
The reducible backgrounds are QCD processes (ccγγ, bbγj, ccγj, bbjj, ccjj, γγjj, γjjj, jjjj) or single-Higgs production processes (hjj and hjγ) where some of the jets fake one or more b-quarks or photons. While the QCD backgrounds are universal and as such are not affected by New Physics, those coming from single Higgs production depend on the value of the modified Higgs couplings.
In the SM, all single Higgs processes are much smaller than the double Higgs signal after the cuts of eq.(13), and can be safely neglected [26]. This approximation is still valid in our context, and for this reason we will not include these backgrounds in our analysis. The list of relevant processes and their cross sections after the cuts of eq. (13) (without including b and photon reconstruction efficiencies) is reported in Table 2.
A further suppression of the background can be obtained by exploiting the particular topology of the signal, where the two Higgs bosons are produced back to back in the center-of-mass frame.   (13) and (14) are taken from [26], and include an additional rescaling factor 1.3 introduced to take into account the increase that can possibly come from NLO corrections. The last line reports the value of the cross sections after the inclusion of the b-jet and photon tagging efficiencies of eq. (12).
We thus select events where the γγ pair has a small opening angle, while the minimal angular separation between a b-jet and a photon is large. We require [26]: In most of the parameter space these cuts imply a moderate reduction of the signal (20% − 40%), while the total background is suppressed by one order of magnitude. The corresponding background cross sections are reported in the second line of Table 2.
Finally, the reducible backgrounds are drastically suppressed once the efficiencies for reconstructing two photons and two b-jets of eq.(12) are included. The resulting final cross sections are shown in the last line of Table 2. The corresponding total background cross section is r b = 5.5 ab.
After performing the kinematic cuts of eqs. (13), (14) and including the efficiencies for the reconstruction of the γγ and bb pairs, the tagging efficiencies (12), and the K-factor, the signal rate (r s ≡ σ(pp → hh) × BR(hh → γγbb)) at m h = 120 GeV is well approximated by the formula For the SM case (c = d 3 = 1, c 2 = 0) we find r s = 4.9 ab. Notice that, compared to the fit of eq.(11) at 14 TeV (see Table 1), the cuts have further weakened the dependence of the cross section on d 3 , since their efficiency is smaller for events with low m(hh) invariant mass.

Results
Using the signal and background rates derived above, we can estimate the sensitivity of the 14 TeV LHC on pp → hh → bbγγ for m h = 120 GeV. Since large luminosities are typically needed to distinguish the signal from SM background, we expect that by the time the analysis of double Higgs production is performed, the Higgs branching ratios to γγ and bb and the linear couplings a, c are known with good accuracy. Double Higgs production can thus be used to extract (or set limits on) the couplings c 2 and d 3 . Figure 6 shows the luminosity required to discover the signal as a function of c, c 2 and d 3 , assuming that the branching ratio BR(hh → γγbb) has the value predicted in the SM. 5 The plots on the left and on the right show the luminosity contours respectively in the plane (c 2 , d 3 ) for c = 1, and in the plane (c, c 2 ) for d 3 = c. As expected, the sensitivity on c and c 2 is stronger than that on the Higgs trilinear coupling d 3 . In particular, while a discovery in the SM would require at least 1200 fb −1 , we find that much lower luminosities are sufficient even for moderately small values of c 2 . Figure 7 shows the corresponding discovery luminosity in the composite Higgs models MCHM4 and MCHM5 as a function of ξ. We find that values of ξ as small as 0.15 can be probed with 300 fb −1 of integrated luminosity. Compared to other processes like double Higgs production via vector boson fusion [33], these results show that gg → hh can be extremely powerful to study the non-linear couplings of a composite Higgs and thus probe its strong interactions.
Once a discovery is established, one can measure the couplings c 2 and d 3 by using the value of c and of the Higgs branching ratios determined in single-Higgs processes. The left plot of Fig. 8 shows the region of 68% probability in the plane (c 2 , d 3 ) with 300, 600 and 1200 fb −1 (light, medium and dark blue regions) obtained by injecting the SM signal (c = d 3 = 1, c 2 = 0) and assuming that the coupling c and the branching fraction BR(hh → γγbb) have been determined with a 20% accuracy. 6 In this case the precision on c 2 is poor even with 1200 fb −1 , while d 3 is basically unconstrained. A more precise determination of c 2 can be obtained if its value is non-vanishing.
The right plot of Fig. 8 shows the case in which the injected signal is that of the MCHM5 with ξ = 0.3, corresponding to (c = d 3 = 0.48, c 2 = −0.6). It assumes that the branching fraction BR(hh → γγbb) and the coupling c = 0.48 predicted by this model have been measured with 20% accuracy in single-Higgs processes. We find that with 300 fb −1 the coupling c 2 can be determined, up to a discrete ambiguity, with a precision of ∼ 20 − 30%. 7 On the other hand, even in this case d 3 remains largely unconstrained with our analysis. Finally, Fig. 9 shows how precisely the parameter ξ can be determined in the MCHM5 through gg → hh → γγbb by making use only of the value of the decay branching ratios determined in single-Higgs processes (that is: without fixing c to its measured value in the fit). As before, we assume that the branching fraction BR(hh → γγbb) the expected signal yield is sufficiently large to establish a discovery (see Fig. 7).

Conclusions
The discovery of a light Higgs-like scalar at the LHC will mark a first important step forward in which the signal rate is too small to make a discovery, (see Fig. 7). In particular, for ξ → 0.5 the h → bb decay rate vanishes (the Higgs becomes fermiophobic), and the signal cannot be distinguished from the SM background. The curves are derived by assuming that the branching fraction BR(hh → γγbb) is determined with 20% accuracy from single-Higgs production.
In this paper we have performed a first model-independent study of double Higgs production via gluon fusion, gg → hh, and we have shown that its cross section is greatly enhanced by the non- not necessarily at largeŝ). Furthermore, none of the diagrams which contribute to tt → hh in the SM grows with the energy, so that the cancellation between the box and the triangle diagram in gg → hh should be rather seen as a numerical accident, not driven by the unitarity of its subprocess.
The strong enhancement of the gg → hh cross section makes this process quite powerful to measure or constrain the strength of the tthh interaction, which we have denoted as c 2 . This should be compared with the much weaker sensitivity on the Higgs trilinear interaction, d 3 , which makes the extraction of this latter coupling extremely challenging at the LHC. In particular, the weaker dependence on d 3 follows from an extra suppressing factor (m 2 h /ŝ) carried by the triangle diagram with Higgs exchange, which thus contributes mainly at threshold. In order to assess the LHC precision on c 2 we have made use of the results of Refs. [23][24][25][26], where double Higgs production via gluon fusion was studied in the context of the SM and several Higgs decay channels were investigated. In a generic scenario of New Physics, what is the best final state largely depends on the value of the Higgs decay branching ratios. In particular, enhanced branching ratios can combine with the increase in the double Higgs production cross section and lead to dramatic effects at the LHC. For example, in the (fermiophobic) limit in which the linear couplings of the Higgs to the SM fermions are suppressed (c → 0), the h → W W and h → γγ branching ratios can be sensibly enhanced compared to their SM values. In this case the final states hh → W W W W and hh → W W γγ seem to be extremely promising, and might be visible even at the 8 TeV LHC. At this energy and with L = 20 fb −1 , for example, we find that for m h = 120 GeV the MCHM5 at ξ = 0.5 predicts ∼ 15 signal events in hh → W W γγ → lνqqγγ (see Fig. 5).
If the Higgs decay branching ratios do not differ much from their SM values and the Higgs boson is light, the most powerful final state should be hh → bbγγ, as suggested by the analysis of Ref. [26]. Even in this case, the signal rate can be significantly enhanced compared to the SM prediction if c 2 is not too small. At 14 TeV, for example, the signal rate predicted in the MCHM5 with ξ = 0.1 is larger than the SM one by more than a factor two. This sensitivity on small values of ξ = (v/f ) 2 shows that double Higgs production via gluon fusion is an extremely powerful process to probe the Higgs compositeness at the LHC. In order to estimate the LHC sensitivity on c 2 in a model-independent way, we have followed the strategy proposed in Ref. [26] and performed a Montecarlo study of pp → hh → bbγγ. We have computed the SM background cross section by using the results reported in [26] and rescaling them to take account of updated b and γ efficiencies and rejection factors. The results that we obtained are quite encouraging. With L = 300 fb −1 the 14 TeV LHC can probe values c 2 −0.2 and c 2 0.8 if c, d 3 ∼ 1 (see Fig. 6). In the case of the MCHM5, an integrated luminosity of 300 fb −1 is sufficient to discover a signal with ξ 0.15 (see Fig. 7). In general, once the signal can be statistically distinguished from the background and a discovery is made, the value of c 2 can be extracted with good accuracy. For example, we find that by injecting a signal with c = d 3 = 0.48 and c 2 = −0.6 (as predicted in the MCHM5 for ξ = 0.3), the coupling c 2 can be measured, up to a discrete ambiguity, with a precision of ∼ 20 − 30% (see Fig. 8).
Our partonic analysis of pp → hh → bbγγ should be considered as a first estimate of the LHC potentiality, although we expect it to be robust and moderately conservative. For example, we have followed a cut-based strategy to reduce the background, although a realistic analysis will certainly make use of shape variables and extract the background from data, similarly to what has been done by the ATLAS and CMS collaborations in single Higgs searches. Also notice that we did not make use of the information on the total invariant mass distribution, dσ/dm(bbγγ), which was instead used in Ref. [26] to further increase the signal significance. Finally, a more precise assessment of the LHC sensitivity will require full inclusion of showering and hadronization effects, as well as a detector simulation. for given number of observed events N and luminosity L. We denote with π(r) the prior distribution and with L(N |rL) the likelihood function, which we take to be a Poisson distribution For the plots of Figs. 6, 7 we use a prior distribution which is flat for r > 0 and vanishing otherwise, and normalize the probability so that  Fig. 7. We define a point in this space to be 'discoverable' at a certain luminosity L if the probability of having a total number of events smaller than or equal to r b L is below 1%, and if the number of observed events (r s + r b )L is 5 or larger. The discovery luminosity is thus defined to be the smallest value of L which satisfies these two conditions.
The prior distribution is assumed to be flat over the plane (c 2 , d 3 ) and the posterior probability is normalized so that dc 2 dd 3 p L (c 2 , d 3 |N ) = 1.
We choose a flat prior for 0 ≤ ξ ≤ 1 (vanishing otherwise) and normalize the probability so that 1 0 dξ p L (ξ|N ) = 1. For each value ξ th the number of observed events has been set to the total expected value N = (r s (ξ th ) + r b )L.