A revisit to scalar dark matter with radiative corrections

Extended Higgs sectors have been studied extensively in context of dark matter phenomenology in tandem with other aspects. In this study, we compute radiative corrections to the dark matter-Higgs portal coupling, which is in fact a common feature of all scalar dark matter models irrespective of the hypercharge of the multiplet from which the dark matter candidate emerges. We select the popular inert doublet model (IDM) as a prototype in order to demonstrate the impact of the next-to-leading order corrections, thereby probing the plausibility of extending the allowed parameter space through quantum effects. Given that the tree level portal coupling is a prima facie free parameter, the percentage change from loop effects can be large. This modifies the dark matter phenomenology at a quantitative level. It also encourages one to include loop corrections to all other interactions that are deemed relevant in this context.


Introduction
The successful discovery of the Higgs boson in 2012 by the CMS [1] and ATLAS [2] collaborations completed the search for the last missing piece in the Standard Model (SM) of particle physics. However, SM is unable to answer certain fundamental observations, viz., the existence of dark matter (DM), massive neutrinos, the excess of baryons over anti-baryons, three generations of leptons etc. Besides, there are certain theoretical issues like the hierarchy problem, which the SM fails to answer. We are thus led to consider physics beyond the standard model (BSM) in order to explain such observations. After the discovery of the Higgs boson, the theoretical and experimental community have spent all their resources in studying the couplings and the CP nature of this discovered boson. The coupling strengths of the Higgs boson to other SM particles conform with their SM expectations within 1σ. A purely CP -odd scenario is also shown to be disfavoured by experiments. The invisible branching ratio of an SM-like Higgs boson is also constrained by experiments and global fits to ∼ 20% at 95% CL [3][4][5][6]. The high-luminosity run of the LHC (HL-LHC) has the potential to constrain all the Higgs couplings to an even greater precision [7,8]. Besides, it also promises to shed light on the cubic and quartic couplings of the Higgs boson through its pair production.
On the other hand, the existence of dark matter in the universe has been repeatedly verified by astrophysical and cosmological observations ranging from galactic to cosmological scales. Apart from the fact that dark matter interacts gravitationally, the only quantitative JHEP05(2019)150 aspect that we know about it is its relic abundance [9], Ω c h 2 = 0.1199 ± 0.0027. However, its nature is still unknown and we expect it to be some electrically neutral particle with no colour quantum number. Amidst the various propositions, the Weakly Interacting Massive Particle (WIMP) stands out as one of the most attractive candidates by attributing to its simplicity and predictability. The observed relic abundance can be explained by the thermal freeze-out mechanism of the WIMP, when its mass is around the electroweak scale. An extension of the SM with a WIMP can help us understand better the origin of the electroweak symmetry breaking (EWSB). The predicted interactions of the WIMP with the SM particles greatly motivate the experimental community to search for this elusive particle at collider experiments, direct dark matter detection experiments at underground laboratories and from indirect detections from cosmological and astrophysical observations.
The lightest supersymmetric particle (LSP) has been considered as the most attractive candidate for cold dark matter due to the fact that the supersymmetric (SUSY) theories alleviate most of the aforementioned limitations faced by the SM. Unfortunately however, SUSY models are gradually getting severely constrained because of the lack of any evidence for superpartners. Lack of any conclusive signatures of WIMPs have gradually pushed the celebrated WIMP scenarios to the corner. In the present study we take recourse to one of the simplest models, the inert Higgs doublet model (IDM) [10,11] which has an inbuilt WIMP candidate. The IDM is possibly the simplest limit of a general two Higgs doublet model, where the additional doublet consisting of complex scalar fields only couples to the SM Higgs and gauge bosons and not to the fermions. However, the most interesting aspect of this model is that the additional doublet is odd under a Z 2 symmetry rendering the occurrence of an even number of inert particles at any interaction vertex. It has also been shown that the neutral scalar or pseudoscalar in the additional doublet can be considered as WIMPs and hence as a viable cold dark matter candidate in the universe [12,13]. In this model, obtaining the correct relic abundance does not require a fine tuning but only requires adjusting its couplings or through co-annihilation with another particle [14]. Several experiments like LUX [15], SuperCDMS [16], Fermi-LAT [17,18], AMS-02 [19,20] and very recently XENON 1T [21][22][23], have tested the dark matter scenario in the context of WIMP searches. These direct dark matter searches have now constrained the mass of the dark matter candidate in the IDM to around half of the mass of the SM Higgs boson (125 GeV) or above ∼ 500 GeV [24][25][26]. The resonance mass around ∼ 62 GeV might still not be completely excluded in the future [27] by direct detection experiments like LZ [28]. Outside these ranges, the dark matter candidate can only contribute to a fraction of the total thermal relic density. Because of its simplicity and richness, the IDM has been exhaustively studied in astrophysical and cosmological studies [29][30][31][32][33][34] and studies pertaining to collider physics [35][36][37][38][39][40][41][42]. There have been several studies in the context of the LHC which considers the Drell-Yan production of HA or H + H − [43][44][45] and then further decays of H → AZ ( * ) , H ± → AW ±( * ) to yield a final state of dileptons or dijets + / E T . Here, H and A are respectively the additional neutral scalar and pseudoscalar and H ± is the charged scalar, in this doublet. Depending on the H − H − h coupling, a monojet signature can also be looked for at the LHC by a pair production of these scalars, if H is the dark matter candidate. To give a broad picture, IDM connects the Higgs to JHEP05(2019)150 dark matter by acting as a portal between the visible and the invisible sector. The onshell corrections to hV V, hf f and hhh couplings have been obtained in refs. [46,47] after considering constraints from perturbative unitarity, vacuum stability and relic abundance. The corrections have been shown to be substantial and can be 100% in the regime of light dark matter masses. These calculations could be of immense importance when the experiments start to constrain the cubic and quartic higgs self-couplings more precisely. Moreover, it has been shown in ref. [48] that the electroweak corrections to the direct detection cross-sections in the IDM can be substantial.
In the present work, we revisit the electroweak correction of the H − H − h vertex and show its effects on the relic abundance calculation and the direct detection cross-section. We are guided by the principle that in order to ascertain the importance of any model for dark matter searches, one needs to look at the higher order corrections which might lead to a significant shift in the parameter space under the constraints from relic density, direct detection cross-section, oblique corrections and collider limits. IDM is one of the simplest models to test our claim and through this we show the importance of precision measurements in the dark matter sector.
We organise the paper as follows. In section 2, we briefly sketch the inert doublet model and its important aspects. We outline the constraints coming from perturbativity, vacuum stability etc. in section 3. We then discuss our renormalisation procedure in section 4 but leave all the details in appendices A and B. In section 5, we discuss the numerical results and finally we summarise and conclude in section 6.

A brief review of the inert doublet model
In addition to the SM fields, the inert doublet model employs an additional scalar doublet, Φ 2 . Moreover, the framework is endowed with a global Z 2 symmetry under which Φ 2 has a negative charge whereas the SM fields have a positive charge. The most general renormalisable scalar potential involving two doublets is then given by [47] where all parameters are real, and Φ 1 is the SM Higgs doublet. Noting that the Z 2 symmetry prevents Φ 2 from picking a vacuum expectation value (vev ), enables us to parametrise the doublets directly in terms of the physical scalars as . (2. 2) The charge of Φ 2 under the SU(2) L × U(1) Y gauge group is 2, 1 2 irrespective of the values chosen for λ 4 and λ 5 . Therefore, it indeed has interactions with the gauge bosons of the ΦΦV V and ΦΦV forms, where, Φ = H, A, H + and V = W ± , Z. A ΦV V vertex, JHEP05(2019)150 however, is disallowed by the Z 2 symmetry. The masses are calculated to be h v 2 is determined using m h = 125 GeV. We choose h to be the SM-like Higgs with mass ∼ 125 GeV. It is easily seen that H, A, H ± are rendered stable by the Z 2 symmetry and thus consequently, H and A are potential candidates for DM. While a detailed account of DM phenomenology for the IDM can be found in [35][36][37][38][39][40][41][42], a few statements are still in order. Relic abundance in the PLANCK ballpark is achieved in the two mass regions (a) 50 m DM 80 and (b) m DM 500 GeV. In region (a), annihilation dominantly proceeds through the exchange of an s-channel h. The sub-dominant contribution to the relic density comes from the t-channel processes to vector boson final states mediated by A and H ± . On the other hand, one must have m H m A m H ± in order to generate Ω DM h 2 0.1 in region (b). Co-annihilation thus becomes inevitable in this case.
In this study, H is chosen to be the DM candidate. A crucial observation that emerges is, in region (a), Ω DM h 2 is highly sensitive to the value of the H − H − h trilinear coupling which is −λ L v at LO with λ L = λ 3 + λ 4 + λ 5 . One therefore expects the region (a) to be naturally more sensitive to the aforementioned radiative effect. This motivates one to review the entire phenomenology by incorporating radiative corrections to, if not all parameters, to the H − H − h portal interaction nonetheless. In addition, beyond the leading order, the parameters that do not participate in the tree level phenomenology of region (a) (such as λ 2 and masses of the CP-odd and charged scalars), will now have their respective roles in the ensuing quantum effects.

Constraints
Our goal is to take a recourse to the DM phenomenology in the IDM after carrying out one-loop corrections to the H − H − h coupling (the Feynman diagrams for the three point function are shown in figure 1). In the process, we obey various constraints stemming from both theory and experiments. On the theoretical side, perturbativity, unitarity and vacuum stability can appreciably constrain an extended Higgs sector as in the IDM. From perturbativity, we impose the constraints |λ i | ≤ 4π, for i = 1, 2, . . . 5. The 2 → 2 matrix element corresponding to the scattering of the longitudinal components of the gauge bosons can be mapped to a corresponding matrix for the scattering of the goldstone bosons [49][50][51][52]. The theory respects unitarity if the absolute value of each eigenvalue of the aforementioned amplitude matrix does not exceed 8π.
The tree level potential remains positive definite along various directions in the field space if the following conditions are met [47], Here, φ is used to denote h, H, A, H + , G 0 , G + or a subset that preserves the Z 2 symmetry (even number of particles from Φ 2 ) in each vertex.
On the experimental side, the bounds on the oblique parameters are taken into account. The most prominent of these is the constraint on the T parameter which puts restriction on the mass splitting between the Z 2 odd scalars. The contribution coming from the inert scalars can be expressed as [54] follows.
We use the NNLO global electroweak fit results obtained by the Gfitter group [55], In the absence of any mixing between h and the Z 2 odd scalars, the tree level couplings of h with the fermions and gauge bosons remain unaltered with respect to their SM values. This implies that the production cross sections of h at the LHC in case of the IDM do not change w.r.t. the corresponding SM values. And in case of an unaltered h-production cross section, the signal strength in the diphoton channel becomes R γγ = the h → γγ amplitude [56]. That is, where G F and α denote respectively the Fermi constant and the QED fine-structure constant. The loop functions are listed below.
and A S (x) are the respective amplitudes for the spin-1 2 , spin-1 and spin-0 particles in the loop and x = m 2 h /4m 2 f /V /S . Therefore, ensuring µ γγ to lie within the experimental uncertainties, implies that the analysis respects the latest signal strength at 13 TeV from ATLAS [57] and CMS [58][59][60], viz., Upon using the standard combination of signal strengths and uncertainties, 1 we obtain µ γγ = 1.04 ± 0.1. Furthermore, we also require the invisible branching ratio of the SM-like Higgs to be BR(h → inv) < 0.15. Lastly, the LEP exclusion limits have been imposed as m A 100 GeV [61] and m H + 90 GeV [36,62]. The IDM has been one of the most popular models that has garnered attention amidst astrophysicists and particle physicists alike, owing to its simplicity and predictive power. Apart from the modification of the diphoton partial width with respect to the SM expectation, the constraint on the T -parameter and the bound from the invisible decay of the SM-like Higgs boson, there are a multitude of search channels which have be used to constrain the IDM parameter space. In refs. [36,61,62] it has been shown that the points satisfying the intersection of the following conditions where µi and σi are the individual signal strengths and their 1-σ uncertainties respectively.

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are excluded by the LEP II data as they would lead to a di-lepton/di-jet signature along with missing energy. Reference [36] showed this in the context of a reinterpretation of the second neutralino. This study has been studied for low values of m H and m A with a significant mass gap in the context of the LHC Run I data [63]. Moreover, ref. [64] studied the process e + e − → H + H − in the context of LEP II data and obtained a bound of m H ± > 70 GeV. In refs. [42,65], the authors study multifarious channels at the LHC in order to impose constraints on the IDM parameter space. Both these studies first look into the constraints ensuing from the SM-like Higgs mass, present limit on the SM-like Higgs width and the Higgs signal strengths. Recast of searches in mono-jet (HH+ jet and HA+ jet, gg, gq and qq initiated), mono-Z (qq initiated), mono-Higgs (gg initiated with HHh and qq initiated with HAh) and vector boson fusion (HH+ jets) were performed in refs. [42,65]. The mono-jet processes having the highest cross-section amongst the rest were considered for the 8 TeV and the projected 13 TeV scenarios in ref. [42]. The projected LHC 13 TeV study in ref. [42] allows small values of λ L between m H ∈ [50, 80] GeV. Besides, from the same study, m H ∈ [55, 80] GeV is allowed when m A > 100 GeV from the monojet requirements. Similar bounds are presented in the m A − m H ± plane. As has been pointed out in ref. [66], the Galactic Centre Excess [67][68][69] (GCE) best-fit data favours small values of λ L which is the driving coupling for the mono-X like searches. Thus, LHC has a very low impact on constraining the parameter space of the IDM from such searches. However, there are other searches which are independent of the size of λ L and depend on the masses of the particles and the splitting. In ref. [63], the chargino and neutralino pair production processes have been studied in details. In particular, they

Outline of renormalisation
In this section, we present an outline of the renormalisation procedure adopted in [47]. The absence of mixing between h and the inert scalars simplifies the machinery to some extent compared to a general two-Higgs doublet model. The IDM scalar sector can be conveniently described using {m h , v, µ 2 , m H , m A , m H + , λ 2 , T h } as independent parameters.
Here T h denotes the tadpole parameter for h. The necessary counterterms are generated by shifting these parameters about their renormalised values as follows.
It is to be noted that we do not need to compute the shift in T h in our case. Moreover, the wave-function renormalisation is invoked as where φ = h, H, A, H + . A more detailed treatment of the renormalisation scheme can be found in [47,70]. In the on-shell (OS) scheme, the mass and field shifts can be expressed in terms of the 1PI amplitudes as follows: These 1PIs are detailed in A and B. The quantity of central importance in this study is the renormalised H − H − h form factor which replaces its tree level counterpart in dark matter calculations. We denote it by Γ ren HHh and decompose it as Γ ren HHh (p 2 1 , p 2 2 , p 2 ) = Γ tree HHh + Γ 1PI HHh (p 2 1 , p 2 2 , p 2 ) + δΓ HHh . Here, p 1 , p 2 and p = p 1 + p 2 refer respectively to the momenta of the two annihilating H and the h. On the right hand side of the equation, the first term refers to the tree level form factor. The second and the third terms respectively denote the unrenormalised 1PI amplitude at the one-loop level; and the corresponding counterterm. It is necessary to express the tree level form factor in terms of the independent parameters in order to generate the corresponding counterterm.
Leading to and Upon considering all these counterterms, we get an effective correction of the form The counterterms for the independent parameters are thus fixed. The quantity directly entering into our analysis is Γ ren HHh (p 2 1 , p 2 2 , p 2 ). The expressions for the various two-and three-point 1PI amplitudes are relegated to appendices A and B.
In passing, we remark that an alternate way of fixing δµ 2 2 is to assume that the SM gauge symmetry is unbroken, compute directly the one-loop correction to µ 2 2 itself and finally define δµ 2 2 to be the UV-divergent part of the same. This particular way of fixing this counterterm is therefore similar to what is done in the MS scheme. Hence, a dependence on the renormalisation scale (say µ) is expected.

Numerical results
The quantity directly entering our analysis is Γ ren HHh (p 2 1 , p 2 2 , p 2 ). Moreover, since dark matter particles annihilate manifestly in an on-shell fashion, we take p 2 1 = p 2 2 = m 2 H . Cold dark matter particles are non-relativistic, and this allows to write p 2 = 4m 2 H . 4,5 This allows us to treat the quantity λ L v + Γ ren HHh (m 2 H , m 2 H , 4m 2 H ) as an "effective" coupling in the present analysis.
Coming to the analysis, we choose the following values chosen for the SM parameters, m h = 125.0 GeV, m t = 173.2 GeV, m b = 4.7 GeV, m W = 80.3 GeV and m Z = 91.2 GeV. 6 The model points are sampled randomly through a scan of the parameter space within the ranges specified below. , (4.8) and in the last term of (4.9), we set m 2 H = m 2 h /4. 4 We must note that even upon considering p 2 off-resonance will yield the same result because the vertex correction will be a function f (p 2 − m 2 h ) which will partially cancel with the propagator, ∼ 1 p 2 −m 2 h , and the correction will be a "constant" one. 5 The consequences of p 2 = 4m 2 H too are examined in the following section. 6 Our mass choices lie in the 1-σ uncertainty range of the measured Higgs and top masses from the LHC Run I and Run II data [71][72][73][74]. We avoid choosing µ 2 2 < 0 in order to prevent the inert doublet from picking a vev, alongside obeying the vacuum stability, perturbativity and unitarity constraints described in section 3. Since the DM self-interaction λ 2 cannot be directly constrained, we choose λ 2 = 0.1, 1 and 5.0 in our scans. We also remind the readers that the upper bound on the invisible branching fraction of h remains an important constraint whenever m H < m h /2. The aforementioned constraint on the Higgs invisible branching ratio leads to |λ L v + Γ ren HHh | < 0.05 for m H = 55 GeV for instance, and a tighter bound for a lower mass. We focus on the regions 50 GeV < m H < 80 GeV and m H > 500 GeV (see section 2) to illustrate our results.

50 GeV < m H < 80 GeV
We propose the following benchmarks in table 1 that are combined with λ 2 = 0.1, 1, 5 discussed before. In addition to the constraints discussed in the previous section, such choices are also guided by a somewhat conservative requirement of |λ 3,4,5 | < 2. Besides in table 1, we also show the constraints ensuing from the T -parameter and from R γγ . All the points abide by the T -parameter constraints mentioned in section 3. As for R γγ , BP1a, BP1b, BP1c and BP1d are allowed within 1-σ uncertainty of the combined signal strength mentioned above. For the remaining four benchmark points, the agreement is within 1.3-σ.   Figure 7 shows the change in the spin-independent dark matter-nucleon scattering rate under radiative corrections. 7 The rate being proportional to |λ L v+Γ ren Ref. [48] examines one-loop corrections to the direct detection process. However, they seem to take into account only the amplitudes involving the gauge couplings. This is somewhat complimentarity to our approach where we focus on correcting the scalar portal coupling, and, this does include the gauge bosons running in the loops. Near mH = 57 GeV for BP1a, the direct detection cross section increases by a factor of ∼ 4 when λ2 = 5 is taken. And this is comparable to a ∼ 2-3 fold enhancement reported in the aforementioned study. Both these numbers surely lead us to the correct ballpark nonetheless. That  it increases upon increasing the effective portal coupling through loop effects and vice versa. An understanding of the aforementioned features governing the strength of the renormalised one-loop form factor therefore suffices to predict the loop-corrected direct detection rates. We plot the direct detection rates for BP1a, BP1b, BP4a and BP4b in figure 7 near the m H < m h /2 mass point. One naturally witnesses a higher direct detection cross section upon incorporating loop corrections in case of BP1a and BP1b. However, it still stays within the XENON 1T bound. On the other hand, the latter two benchmarks are characterised by λ L < 0 and hence this opens up the possibility of the cancellation between the tree level and the 1PI amplitudes. The cancellation is obviously maximum for λ 2 = 5 and therefore the corresponding curves leads to the lowest direct detection rates as can be read from figure 7. Such plots for the other benchmark points are not shown for brevity. The aforementioned discussion applies to the case where the DM annihilates with an exactly zero relative velocity. This is a reasonable approximation to the actual scenario where such non-relativistic annihilations are indeed at play. However, for the sake of completeness, it is useful to demonstrate the effect of a non-zero velocity on the NLO said, the most accurate figure will only emerge when the calculation is not restricted to a particular set of diagrams.   Figure 6. The same as in figure 5 for BP3a, BP3b, BP4a and BP4b.
found to be either positive or negative depending on the value of λ 2 (see figure 9). It is approximately −42.96% for λ 2 = 5 and 26.67% for λ 2 = 0.1 near m H = 580 GeV. Though these numbers look sizable, the relic density changes only slightly w.r.t. its tree level value.
In fact, such a small change is almost indistinguishable from the PLANCK error band and this result remains qualitatively similar for other parameter points. This result does not come as a surprise because the dominant fraction of the relic in this mass region is generated by annihilation and co-annihilation processes of the type φφ −→ V V where gauge interactions are at play. A more complete picture of loop corrections to the thermal relic is therefore expected to emerge only after loop corrections are incorporated in the gauge interactions. In case of direct detection, the corresponding amplitude features the H − H − h coupling and therefore, loop correction to this coupling is expected to match a full radiative correction to a reasonable degree. It is seen that the deviations in relic density and direct detection rates, w.r.t. the tree level, increase as one considers higher values of the inert scalar masses, even if the mass-splitting is kept fixed. This is due to the fact that the parameters λ 3 , λ 4 and λ 5 can individually grow in magnitude in the process, thereby increasing the appropriate one-loop form factors. We remind the readers that one should also include one-loop corrections to the coannihilation processes for a more accurate analysis. The crucial co-annihilation mediating HZA, HW ∓ H ± and H ± H ∓ Z/γ vertices shall also receive potentially large corrections from the extended Higgs sector. In addition, a complete NLO analysis of the DM-nucleon scattering rates requires going beyond computing loop corrections to the portal coupling only. To give an example, one-loop triangle graphs with A and Z in the internal lines will be encountered. It is also customary to examine the dependence of the results on the renormalisation scheme chosen. A more exhaustive radiative treatment of the present scenario is currently under preparation.

Summary and outlook
In this study, we have evaluated one-loop radiative corrections to the dark matter-Higgs portal interaction in the context of the inert doublet model (IDM). Canonical constraints from vacuum stability, perturbative unitarity and LHC data have been taken into account. The motivation behind this work was to obtain a measure of deviation from the leading order results, given that an additional doublet furnishes more bosonic degrees of freedom that can participate in a next-to-leading order analysis. The present renormalisation scheme is based on demanding an unchanged h → HH decay width, upon adding the 1PI amplitudes and the counterterms. We have restricted our numerical analysis to a set of representative and somewhat conservative benchmark points that encompass the salient features. For the dark matter lighter than m W , the inert doublet cannot be fully decoupled even if the other inert scalars are taken to be heavy. This non-decoupling effect induces sizeable loop corrections in the observable quantities and this effect is more prominent for m H < m h /2. In this region, the radiatively corrected interaction can grow or diminish in magnitude depending on whether the tree level coupling respectively carries a positive or negative sign. On the other hand, the m H > 500 GeV region witnesses comparatively small radiative shifts to the relic abundance. This can be safely attributed to the fact that the portal interaction plays only a subdominant role in this mass region. And therefore, a complete picture can only emerge if one incorporates one-loop effects to all the relevant JHEP05(2019)150 interactions. Moreover, the radiative corrections will also affect the search prospects of a dark matter particle at the colliders, a promising channel to probe being the monojet + / E T final state. A more exhaustive study on the impact of one-loop corrections to all the relevant interactions in this scenario is presently underway. In all, scalar dark matter models have served as popular frameworks to study interactions of dark matter with the visible world. Through the present study, we have tried to highlight the fact that these models can be made more accurate once they are augmented with relevant radiative corrections. +(−2p 1 .p 2 −p 2 1 )C 0 (p 2 1 , p 2 2 , q 2 , m 2 A , m 2 Z , m 2 A ) − λ 3 2 g 2 p 2 1 C 21 +p 2 2 C 22 +2p 1 .p 2 C 23 +4C 24 +2p 1 .p 2 C 11 +2p 2 2 C 12 +(−2p 1 .p 2 −p 2 1 )C 0 (p 2 1 , p 2 2 , q 2 , m 2 H + , m 2 W , m 2 H + ) − (g 2 +g 2 ) 2 8 v p 2 1 C 21 +p 2 2 C 22 +2p 1 .p 2 C 23 +4C 24 +(3p 2 1 −p 1 .p 2 )C 11 +(3p 1 .p 2 −p 2 2 )C 12 +(2p 2 1 −2p 1 .p 2 )C 0 (p 2 1 , p 2 2 , q 2 , m 2 Z , m 2 A , m 2 Z ) − g 4 4 v p 2 1 C 21 +p 2 2 C 22 +2p 1 .p 2 C 23 +4C 24 +(3p 2 1 −p 1 .p 2 )C 11 +(3p 1 .p 2 −p 2 2 )C 12 +(2p 2 1 −2p 1 .p 2 )C 0 (p 2 1 , p 2 2 , q 2 , m 2 W , m 2 H + , m 2 W ) . (B.1)

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