Trilinear Higgs coupling determination via single-Higgs differential measurements at the LHC
We study one-loop effects induced by an anomalous Higgs trilinear coupling on total and differential rates for the decay and some of the main single-Higgs production channels at the LHC, namely, VBF, , and . Our results are based on a public code that calculates these effects by simply reweighting samples of Standard-Model-like events for a given production channel. For and production, where differential effects are particularly relevant, we include Standard Model electroweak corrections, which have similar sizes but different kinematic dependences. Finally, we study the sensitivity of future LHC runs to determine the trilinear coupling via inclusive and differential measurements, considering also the case where the Higgs couplings to vector bosons and the top quark is affected by new physics. We find that the constraints on the couplings and the relevance of differential distributions critically depend on the expected experimental and theoretical uncertainties.
CP3-17-37 MCNET-17-18 TUM-HEP-1099/17
Since its discovery in 2012 [1, 2], evidence has been steadily accumulating that the properties of the scalar particle at 125 GeV correspond to those of the Higgs boson predicted by the Standard Model (SM) of elementary particles and interactions. ATLAS and CMS experiments have analysed data from several inverse femtobarns of integrated luminosity at different energies, providing already rather precise measurements of the Higgs couplings to the vector bosons and to the fermions of the third generation [3, 4, 5, 6, 7]. Prospects of the next LHC runs for further improving the precision on the value of these couplings and for measuring also the couplings to the second-generation fermions are very good. On the contrary, the situation and prospects for determining the properties of the scalar potential, i.e., of the Higgs self-couplings at the LHC are less clear and therefore the subject of an intense theoretical and experimental activity.
At low energy, the potential for a scalar particle of mass can be parametrised as a polynomial
where is the vacuum expectation value after electroweak-symmetry-breaking (EWSB). In the SM, renormalisability and gauge invariance dictate that the Higgs potential depends only on two parameters, and ,
where is the Higgs doublet. After EWSB the potential in eq. (2) gives rise to the mass of the physical Higgs boson , which, together with the vacuum expectation value , are related to and via and . As a result, by fixing and , in the SM the Higgs self couplings are completely determined, leading in eq. (1) to with and for , at LO. Thus, with ,
On the other hand, new physics could modify the Higgs potential at low energy, by altering the value of (or in general for the -point Higgs self couplings) without affecting the value of and . This can be realised either directly (e.g. by extending the scalar sector) or indirectly (due to the exchange of new virtual states). In addition, modifications in the self interactions would be induced if the Higgs boson is a composite state.
Since double Higgs production directly depends on the Higgs trilinear coupling at LO, it is the standard process for studying at the LHC. However, the cross section of its main production channel, the gluon fusion, is only about 35 fb at 13 TeV [8, 9, 10], so it is much smaller than single Higgs production cross section, which is about 50 pb . Several phenomenological studies have been performed on the determination of via the relevant experimental signatures emerging from this process: [12, 13, 14, 15, 16, 17], [18, 13],  and [20, 21, 22]. Also [23, 24] and  production processes have beeen considered. Nevertheless, given the complexity of a realistic experimental set-up, the final precision that could be achieved on the determination of is still unclear. On the contrary, it is established that at the LHC perspectives of inferring information on from the triple Higgs production are quite bleak [26, 27], due to the smallness of the corresponding cross section [8, 28] together with a rather weak dependence on this parameter. Even at a future 100 TeV proton–proton collider a considerable amount of integrated luminosity will be necessary in order to obtain rather loose bounds [29, 30, 31].
At the moment the strongest experimental bounds on non-resonant double-Higgs production have been obtained in the CMS analysis of the signature , where cross-sections larger than about 19 times the predicted SM value have been excluded. However, exclusion limits on are found to be strongly dependent on the value of the top-Higgs coupling and they are of the order and for the SM-like case. These new limits, together with the slightly weaker ones from the ATLAS  and CMS  analyses at 13 TeV, improve the results from 8 TeV data ( and ), however, also with a high integrated luminosity (HL) of 3000 fb, a further improvement may not be so tremendous. The most optimistic experimental studies for HL-LHC suggest that it could be possible to exclude values in the range and via the signatures . Additional and complementary strategies for the determination of are thus desirable at the moment.
In ref.  an indirect method of measuring via EW radiative corrections in process was proposed. Recently, the same idea has also been extended to the LHC, by (globally) studying -dependent EW corrections in single Higgs production and decay processes [38, 39, 40, 41]. Moreover, limits on can also be derived by two-loop effects in EW precision observables [42, 43] such as the measurements of and of the oblique parameters. These studies have confirmed that indirect bounds on can be competitive with the direct ones inferred from the di-Higgs production channel. For example, a simple one-parameter fit to the signal strengths measurements at 8 TeV  gives , comparable to the current best constraints from the CMS measurement mentioned above. In both analyses no other deviations for the Higgs couplings are considered. The usefulness of a joint analysis of indirect effects on single-Higgs production and direct effects on di-Higgs production has been discussed and quantified in ref. . As already suggested in ref. [39, 40], the role of differential distributions and their non-flat dependence on has been proved to be crucial in ref. , especially when not only anomalous effects but also modifications of the couplings to the other particles are considered, as expected in a general Effective-Field-Theory (EFT) approach.
The purpose of this work is threefold. First, we present an automated public code for generating events including effects at one loop, thus allowing the study of differential effects in VBF, and production111The impact of differential distributions in gluon-fusion is not studied here as one would need to consider the effects of the trilinear coupling in jet. This involves the calculation of two-loop amplitude with four independent scales, which is not yet feasible. and all the relevant Higgs decays. The code is based on two independent and procedurally different implementations in the MadGraph5_aMC@NLO framework . 222The code can be found at the webpage:
https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/HiggsSelfCoupling. Second, with the help of this code, we extend at the differential level the results of ref. , where all the relevant single Higgs production (, , , ) and decay channels (, , , ) have been analysed and included in a global fit only at the inclusive level. Indeed, in ref.  the usefulness of differential distributions has already been explored but only for the case of and production, providing the results on which the analysis in ref.  rely. Differential information for VBF and production has been presented also in ref. . Here we scrutinise all the relevant distributions that are potentially affected by anomalous effects, presenting for the first time detailed results at the differential level for production, for the decay and also for the process, for which also inclusive results are new. Moreover, for the case of and production, where loop-induced effects are not flat, we repeat the calculation of NLO EW corrections [45, 46, 47, 48, 49, 50] in the SM, which are also not flat, in order to check the robustness of our strategy. As expected, we find that NLO EW corrections are essential for a precise determination of anomalous effects, but also that they do not jeopardise the sensitivity of indirect determination. We use for the calculation the EW extension of the automated MadGraph5_aMC@NLO framework that has already been used and validated in refs. [48, 49, 51, 52, 53, 54]. 333To our knowledge, NLO EW corrections to the process are calculated for the first time here.
Finally we perform a fit for based on the future projections of ATLAS-HL for single-Higgs production and decay at 14 TeV [55, 56]. We consider the effects induced on the fit by additional degrees of freedom, namely anomalous Higgs couplings with the vector bosons and/or the top quark. We investigate the impact on the fit of three different factors: the differential information, the experimental and theoretical uncertainties, and the inclusion of the two aforementioned additional degrees of freedom. We find that in a global fit, including all the possible production and decay channels, two additional degrees of freedom such as those considered here do not preclude the possibility of setting sensible bounds, especially, they have a tiny impact on the upper bound for positive values. On the contrary the role of differential information may be relevant, depending on the assumptions on experimental and theoretical uncertainties.
The structure of the paper is the following. In sec. 2 we briefly repeat and comment the main formulas that are relevant in the study of one-loop induced effects. Differential results for the various processes that have been mentioned before are given in sec. 3. In sec. 4 we present the extension of the analysis of effects including NLO EW corrections and we study the impact on differential distributions, as well as inclusive rates. In sec. 5 we introduce the framework for the fit and discuss the results obtained. Details about the statistical treatment of uncertainties in the fit are reposted in Appendix A.
2 Technical setup
2.1 Self couplings effects in single Higgs production and decays at one loop
Higgs self-couplings can be studied in a model-dependent approach, e.g., choosing a particular UV-complete scenario, or in a model-independent approach, as done in this work. However, there are different ways in which the modifications of trilinear and quartic couplings can be parametrised and they rely on different theoretical assumptions. If new physics is at scales sufficiently higher than the energies where measurements are performed, the SMEFT offers a consistent and model-independent way of organising generic deformations to the Higgs interactions. Moreover, radiative corrections can be consistently performed within this framework. However, it has been shown  that in the case of the trilinear coupling and at the order we are considering, one-loop corrections for single Higgs processes, adding higher dimensional operators that only affect the Higgs self couplings (the operators with ) or directly introducing an anomalous coupling
are two fully equivalent approaches for deforming the SM Higgs potential. In other words, regarding the Higgs self-couplings, differences between an EFT and an anomalous coupling parametrisation will arise only for final states featuring more Higgs bosons and/or at higher loop -level, i.e., with the appearance of higher-point interactions (starting from the quadrilinear).
It is essential to note that the single Higgs production and the decay channels are not sensitive to at one loop. For this reason, although results are delivered in terms of , they can be easily translated in terms of the Wilson coefficient in front of the dimension-6 operator , or those for higher-dimension operators. In the following we briefly repeat and comment the main formulas that have been introduced and discussed at length in ref. ; very minor modifications are present in the notation and in the definition of the corresponding quantities.
The -dependent part of the NLO EW corrections to single Higgs processes are gauge invariant and finite. These contributions can be organised in two categories: a universal part proportional to , which arises from the wave-function renormalisation of external Higgs boson and thus does not depend on the kinematics, and a process-dependent part linear in , which is also sensitive to the kinematics. In presence of modified trilinear coupling, the master formula for the -dependence of a generic observable (total/differential cross section or decay width) can be written as
where is the process- and kinematic-dependent component and stands for the LO prediction including any factorisable higher-order correction. In particular, we assume that QCD corrections do in general factorise, which, for the and VBF case, has been shown to be a correct approach up to NNLO in ref. .444As the weak loops considered here are always characterised by scales of the order of the mass of the heavy particles in the propagators (weak bosons, top quarks and the Higgs) while QCD corrections at threshold are typically dominated by lower scales, factorisation is a reasonable working assumption. Representative diagrams contributing to the for the different processes are depicted in Fig. 1.
The SM component is directly included at fixed NLO via the term appearing in eq. (5). Numerically, the difference between eq. (5) and in ref.  is at sub-permill level and thus negligible. On the other hand, in the limit , and thus goes to the SM case at fixed NLO
This is particularly convenient for the discussion in sec. 4, where we will analyse NLO EW corrections in the SM in conjunction with -induced effects. In conclusion, the relative corrections due to trilinear coupling can be expressed as
which manifestly goes to zero in the limit.
Numerical values of at the inclusive level for the processes considered in this work are reported in Tab. 1. The calculation of for single-top-Higgs production, which appears for the first time here, is non-trivial and discussed in sec. 3.4. The range of validity of eq. (9) has been identified in ref.  as , given the values of and in Tab. 1. As we will see, at the differential level this limit may be too loose since can receive large enhancements (see sec. 3.3). On the other hand, we believe that the constraint identified in ref.  is appropriate for inclusive double Higgs production, but is too strong for the case of single-Higgs production. Indeed the violation of perturbativity for the vertex is kinematic dependent and the condition arises from the configuration with two bosons on-shell and the third one with virtuality slightly larger than . This is the kinematic configuration present above the threshold in double Higgs production, where the bulk of its cross section comes from, but is never present in single Higgs production, since only one Higgs boson can be on-shell in the vertex appearing at one loop.
2.2 Automated codes for the event-by-event calculation of
While is a universal quantity, is process and kinematics dependent. We have employed two independent methods for the computation of for differential cross sections and decay rates. They correspond to the two different codes publicly available, which agree within their numerical accuracy. In the first we parametrise the finite one-loop corrections due to the trilinear Higgs self coupling as form factors that are function of the external momenta. One-loop integrals are computed using the LoopTools package  and the form factors are implemented as effective new vertices in a dedicated UFO model file , which is then used in MadGraph5_aMC@NLO . As a result, parton level events can be generated including effects and any interesting observable analysed. Our current implementation of form factors allows the computation of differential for VBF, and processes. At the order of accuracy of our calculation, F production and all the other decays do not have a kinematic dependence for ; results at the inclusive level are sufficient for any kinematic configuration and thus taken from ref. .
On the other hand, the implementation of form factors for and processes would be quite cumbersome as there are many one-loop integrals that contribute. A different strategy, based on reweighting, has therefore been devised. With this second method, one starts by generating a sample of (unweighted) parton-level events at leading order. These events are then used as input for a code that computes the momentum-dependent weight
where, following the notation of ref. , refers to the tree-level amplitude and to the SM virtual corrections depending on . Then, LO events are reweighted by multiplying the weight of each event by the corresponding . In this way, also with this method it is possible to calculate for any differential distribution.
The required one-loop matrix elements are computed using the capabilities of MadGraph5_aMC@NLO for evaluating loop diagrams [60, 61, 48, 53]. For each process, we use diagram filters in order to select the relevant one-loop matrix elements featuring the Higgs self coupling. We find this method much faster and efficient than the one based on form-factors also for and VBF processes, thus we actually employ it for deriving all the results presented in this work and we suggest the usage of this version of the code. Note, however, that the other method offers at least in principle the possibility of explicitly including NLO QCD corrections on top of -induced effects for VBF and production.
3 Results for differential distributions
Results are presented in this section and have been obtained with the following input parameters,
which are taken from ref. . We use as PDF set the PDF4LHC2015 distributions with the factorisation scale at , where are the masses of the particles in the final state. 555As discussed in ref. , the choice of the factorisation scale has a negligible effects on at inclusive level. The effect is even smaller at differential level. In the following subsections we provide differential results for various relevant observables in VBF, , and production channels and in the decay channel. Each plot has the layout that is described in the following. The upper panel displays the LO distribution (red) and corrections alone (blue), both normalised by their value for the total cross section. In other words, we compare the shape of LO distributions with the shape of the contributions induced by in eq. (5), which is thus independent on the value of . The lower panel display both at differential level (green) and for the total cross section/decay (blue). The latter values are also summarised in Tab. 1 and will be used in the sec. 5 for the representative fit results.
Vector boson fusion is generated by requiring EW production of Higgs plus two jets, which includes also configurations with the vector boson decaying into two jets. We effectively eliminate contributions by applying the following kinematic cuts  on the two final-state jets,
In Fig. 2, we present for representative distributions, namely, , , and . In fact, we have checked that similar effects characterise other observables, which however we do not show. As already noticed in refs. [39, 40] the value of is not particularly large and rather flat for all the distributions shown here; for the total cross section and never exceeds at the differential level. At variance with the case of and considered in the following, loop corrections featuring trilinear Higgs self couplings involve Higgs propagators connecting the final-state Higgs and internal propagators. Thus, no Sommerferld enhancement is present at threshold. In this respect, the interest of VBF for what concerns the indirect determination of is mostly limited to the shift in the total rate, which, even though modest, is anyway relevant. Indeed, VBF is the channel with the second largest cross section and the smallest of the theory uncertainties , as can also be seen in Tab. 4 in Appendix A.
In Figs. 3 and 4 we show the differential for and , respectively. As discussed in refs [39, 40] the main enhancements are present at threshold, where the interaction of the final-state vector and Higgs bosons via a Higgs propagator leads to a Sommerfeld enhancement due to the non-relativistic regime. Indeed the shape of the corrections is quite different from the LO case for and distributions; the former are softer than the latter. For this reason, grows at threshold, where, however, the cross section is rather small. In particular, while in () is 1.19 (1.03)% at the inclusive level, it grows up to, e.g., 2.3(1.8)% for at threshold, with the binning used in Figs. 3(4). Thus, in order to detect anomalous effects, dedicated measurements close to threshold but with enough events, such as the region GeV, would be desirable. For we also show for the rapidity and the difference of the pseudo-rapidity of the and bosons . The latter is particularly interesting because is enhanced w.r.t. the inclusive case in the region corresponding to the largest cross section.
We also looked at possible effects due to the polarisation or, in other words, measurable via the angular distributions of (and ) decay products. We did not see any enhancement or shape dependence for these distributions. Furthermore, one should bear in mind that also the loop-induced process gives a non-negligible amount of the NNLO cross section, order at 13 TeV. This process also has a dependence on , but only at two-loop level, and should exhibit a shape dependence. However, this calculation is not technically feasible yet.
Together with gluon-fusion production, the channel plays a major role in providing information of the top-quark couplings to the Higgs. Its importance can be gauged by simply considering its weight in a global -framework fit  or in the SMEFT framework . The same importance should be ascribed to this process also from the point of view of the sensitivity to : for is the largest among all production channels and with the most significant kinematic dependence . In Fig. 5, we show the most important kinematic distributions in this channel. for total cross section is and can increase up to in distributions. Similarly, with the binning chosen in Fig. 5, for the invariant mass distributions can be as large as 10% close to threshold, even though, once again, in the same region the cross section is suppressed by phase space. The origin of the large phase-space dependence of is again due to Sommerfeld enhancements in the threshold regions that are induced by interactions among the top (anti)quark and the Higgs boson.
Although it is characterised by a rather small cross section at the LHC, single associate production of a Higgs with a single top is a particularly rich and interesting process, especially in searching for observables sensitive to relative phases among the Higgs couplings to fermions and bosons [64, 65, 66, 67]. Naively, one would expect this process to have a sensitivity to the trilinear between that of VBF and ; the process features a top quark in the final state as well as boson(s) in the propagators. The contribution of one-loop diagrams featuring the Higgs self coupling to this process has not been considered in ref.  for two main reasons. The first one was of phenomenological nature: in the SM this process is barely observable at the Run II of the LHC. The second one is of technical nature, as the calculation needs a careful check of EW gauge invariance and UV finiteness, since a few subtleties, which are not present for the other processes discussed in this work, arise. We describe them in the following.
Similar to the case of the decay [38, 39], Goldstone bosons appear in the Feynman diagrams contributing to the LO. Thus, as well interactions are present in one-loop EW corrections. While the former is not modified by effective operators, the latter is indeed modified [38, 39]. The calculation can be consistently performed in two different ways: either directly eliminating Goldstone bosons by employing the unitary gauge, as also done for other quantities in refs. [39, 42], or keeping track of effects in the intermediate calculation steps, as we explain in the following and done in our calculation.
In a generic gauge, the on-shell renormalisation of the EW sector  involves the counterterm for the Goldstone self-energy, which depends on the Higgs tadpole counter term , which in turn depends on the trilinear coupling . Therefore, if we only modify the value of , the Goldstone self-energy counterterm receives a UV-divergent contribution proportional to (), which is not cancelled by any divergence from loop diagrams. Instead, if we consistently take into account the modification of the vertex, loop diagrams featuring a seagull in the propagator are also present; they exactly cancel the UV-divergent contribution proportional to in the Goldstone self-energy counter term, leading to the same result one would obtain in the unitary gauge. Having understood this point, the calculation is straightforward and can be performed automatically in the Feynman gauge.
In our results we include both and channels and we do not apply cuts on the jet, since the result is infrared finite. We find the for the total cross section is about 0.91%. In Fig. 6, we show for kinematic distributions such as , , and . We note that unlike other variables does not decrease monotonically as we move from low to high values. Near threshold displays a quite impressive difference in shape.
The Higgs decay into four fermions is the only Higgs decay channel with nontrivial final state kinematics. Moreover, it is the only one where a priori also can have a shape dependence. Indeed, all the other decays correspond to a process, and since the boson is a scalar, there is not a preferred direction in its reference frame. In the previous study  the for decay was calculated to be 0.83. Although, the full off-shell configuration was taken into account, possible angles between the decay products were not analysed. Using the form factor code mentioned above we calculate for channel. We analysed for many observables involving the four leptons, but we found that it has in general almost no kinematic dependence. As an example, in Fig. 7, we display for leading and sub-leading lepton pair invariant masses. Since the Higgs boson interactions with the final-state fermions are negligible, this result can be extended to all the other decays into four leptons and in general into four fermions.
4 Anomalous trilinear effects and the NLO electroweak corrections
The set of one-loop corrections to single Higgs production and decays involving the trilinear Higgs self coupling is gauge invariant and finite. However, in the SM, performing a perturbative expansion in power of and , other contributions are present at the same order of accuracy. In other words, -induced effects at one-loop level should be considered as a gauge-invariant and finite subset of the complete NLO EW corrections, which also include effects form virtual and photons as well as real emissions contributions. 666In the EW sector of the SM all the interactions are determined by the mass of the fermions, and three additional parameters, which are typically , and or . In general, it is not possible to alter at NLO EW accuracy a derived quantity, such as , without spoiling the renormalisability of the theory. The special case of in single Higgs production at one loop has been discussed in detail in refs.  and .
As shown in the previous section, the possibility of measuring anomalous effects via precise predictions in single Higgs production relies both on the precision of experimental measurements and SM theory predictions. In particular, regarding the theory accuracy, while it is reasonable to assume that QCD corrections in general factorise effects, as explained in sec. 2, this is in general not true for NLO EW corrections. The purpose of this section is to provide a consistent extension of the master formula in eq. (5) that includes also NLO EW corrections and to investigate their impact in the determination of anomalous effects. All the calculations of the NLO EW corrections in the SM, with the exception of F taken from , are performed in a completely automated approach via an extension of the MadGraph5_aMC@NLO framework that has already been used and validated in refs. [48, 49, 51, 52, 53, 54]. Concerning the renormalisation, we use the -scheme, consistently with the input parameters listed in sec. 3.
At the differential level we limit ourselves to the study of the and processes, where the dependence on the kinematics is large. We have also computed the differential EW corrections to the production channel, but we do not report plots here, as its phenomenological relevance will be marginal at 13 TeV LHC for our purposes. The differential case is of particular interest since EW corrections in the SM, due to the Sudakov logarithms, are large in the boosted regime, i.e., exactly in the opposite phase-space region where -induced effects are sizeable, the production threshold, as already discussed before.
The master formula in eq. 5 can be improved including NLO EW correction in the following way
where represents the part of the NLO EW -factor in the SM
that does not depend on , namely 777Here, in order to keep the notation simple, with the symbol we still refer to only the contributions to the Higgs wave-function counterterm. Thus, contains further contributions to the Higgs wave-function counterterm that do not depend on .,
In eq. (14), stands for the observable at LO + NLO EW accuracy. Thus, in the limit , . As can be noted, the term factorises the NLO EW contributions in the SM, while does not. Indeed, in general, EW loop corrections on top of -induced effects need a dedicated two-loop calculation and a full-fledged EFT approach in order to obtain UV-finite results; only the contribution is completely model-independent and factorises the NLO EW corrections in the SM. However, it is worth to note that, assuming factorisation also for contributions, terms of the order would be anyway negligible, since either (Sudakov logarithms in the boosted regime) or (Sommerfeld enhancement in the threshold region) is sizeable, but never both of them at the same time. This will be clear in the differential plots we display in the following.
The EW -factor at the inclusive level can be found for all processes in Tab. 2, while relevant differential results for , and are displayed in Figs. 8, 9 and 10, respectively. In each figure, plots on the left show the distributions, while plots on the right those for the invariant mass of the final state. In the upper plots we display the ratio for different values of , (-10,0,1,2,10). In practice, the case , directly denoted as SM in the plots, corresponds to the differential in the SM. The lower plots display the ratio (solid lines) and the term (dashed lines) for different values of , (-10,0,1,2,10). In practice, the former is our prediction at NLO EW accuracy for the signal strengths that will enter in the fits of the next section888The signal strengths is better defined afterwards in eq. (17). In this section ., the latter is the definition at LO used in the previous works.
First, we comment on the shape of the NLO EW corrections in the upper plots. The general trend in the SM is characterised by large negative Sudakov logarithms in the tails, especially for in production, and positive corrections in the threshold region, especially for the invariant mass distributions and in general . The latter are precisely the effects due to . Thus, changing the value of , the shape of the ratio is highly affected in the threshold region, while it is not deformed in the tail. On the other hand, the change induced by on results in a constant shift in the tail of the distributions. The small bump around and in production is simply due to the threshold in diagrams involving a top-quark loop.
By looking at the upper plots in Figs. 8, 9 and 10 is evident that EW corrections have to be included in order to correctly identify anomalous effects. On the other hand, NLO EW corrections do not largely affect the value of the signal strengths, i.e. the ratio of the BSM and SM prediction. This fact can be seen in the lower plots, where we display (solid lines) and (dashed lines), which indeed corresponds to the aforementioned ratio with or without NLO EW corrections both in the numerator and denominator. 999Actually, the ratio without NLO EW corrections should be , but its difference with , which is used in previous works and more useful for a direct comparison, is negligible. Solid and dashed lines are in general very close, especially for small values of . It is interesting to note that for very small values of a value is leading to corrections that are negative and larger in absolute value than the LO. This is due to the very large (see Fig. 5) and points to the necessity of including higher order -induced effects for large values of and for this specific phase-space region.
For the case of total cross sections, we plot the ratio as a function of in the range [-10, 10] for all the single Higgs production processes, including also . Differences with the corresponding ratios, which have been presented in ref. , are hardly visible and thus we do not show them here.
In conclusion, constraints on from a global fit based on the value of the signal strengths at inclusive  or differential level  will not be affected by NLO EW corrections. On the other hand, in the experimental analyses EW corrections have to be taken into account, especially at the differential level, for the determination of the value of the signal strengths, which is in general important for any BSM study and not peculiar for our case.
5 Constraining through a global fit
In this section we discuss the role of differential distributions in the determination of via single-Higgs production and decays measurements. The aim of this section is threefold. First, we show that the interplay of theory and experimental uncertainties have a large impact in the determination of the constraints on , especially when differential information is exploited. Second, we discuss how bounds on are affected by the presence of additional anomalous couplings in the fit. In particular, we progressively lift the assumption that the Higgs couplings to the top quark and to vector bosons are SM-like. Third, we include the EW effects discussed in the previous section in the fit analysis, providing consistent formulas for repeating the fit in conjunction with additional Higgs anomalous couplings. The reader that is only interested in the results can go directly to sec. 5.2 and skip sec. 5.1, where formulas are given and a few technical details are discussed.
5.1 Combined parameterisation of , and effects
In order to parametrise the Higgs anomalous couplings to the top quark and to the vector bosons we use the coupling modifiers and , respectively (see ref.  for definitions). We are interested in how additional BSM effects entering at LO may alter the determination of and the relevance of differential measurements. The choice of the () kappa framework is driven by simplicity; our main purpose is adding new degrees of freedom in the fit and identify in which configurations the differential information may be particularly relevant. On the other hand, while the cases of new physics entering only via and a very general EFT parametrisation (10 independent parameters) have already been explored , simplified intermediate parameterisations have not been considered yet. These parameterisations, such as the one used here, may be useful to identify relevant scenarios for which the determination of is feasible. 101010 In fact, the analysis carried here is a particular choice of two (linear combinations) of the 10 Wilson coefficients identified in ref. : is related to and to .
First of all we extended the framework and the notation introduced in ref.  in order to take into account in the fit differential information, EW corrections in the production, and and dependence. The experimental inputs entering the fit are the signal strengths, which are defined for any particular combination of production and decay channel as
In eq. (16), the quantities and are the production cross sections ( F, VBF, , , ) and the divided by their SM values, respectively. Assuming on-shell production, the product is the measured rate for the process divided by the corresponding SM prediction. This is valid also for differential distributions involving the reconstructed momentum of the Higgs boson. For simplicity, in the following we will refer with the symbol to both total cross sections or (bins in) differential distributions.
The signal strength productions are given by
where and and the effect of is parameterised via the quantity . In presence of NLO EW corrections is given by
where we have explicitly shown which quantities depend on the specific production process . For differential distributions, differential have to be used. As can be noted, we did not include and effects entering at one loop. As we will see in the results of the fit, we are going to probe deviations at the percent level in and . Thus, and effects are negligible for our purposes. On the other hand, terms of order and may be more important and in fact those from the Higgs-wave-function can be consistently resummed; they are included in eq. (17) via the term. 111111Eq. (18) can be in principle generalised to an EFT framework. In that case, EW corrections can be performed also on top of new physics effects entering at the tree level as well as -induced correction. However, the latter involves non-trivial higher-dimensional corrections. However, unless differently specified, we verified that also the inclusion of the and contributions has a negligible impact in the results presented in the following. It is also important to note that any further or dependence that may be introduced by NLO EW corrections, on top of those already present at LO, is negligible. Indeed, NLO EW corrections are per se at the percent level and their anomalous component would be of the order of few percents of the corrections themselves. Thus, these kinds of effects, which similarly to those of order and can actually be calculated only in an EFT framework, are expected to be of the order and therefore at the permille level or even smaller in our analysis. For this reason, we can safely ignore them.
Similarly, the signal strength for the Higgs decays is given by
NLO EW corrections in Higgs decays are small at inclusive level, therefore we can safely ignore them. The partial decay width in a given channel is given by
where is the total width at LO in the SM. The SM widths in eq. (19)
can be obtained by setting in . In order to ensure that the contribution of the Higgs wave-function renormalisation does not affect the branching ratios, in this case we resummed also the SM part () and factorise it to the dependence, as done in ref.  for the LO analysis.
For the decay channel depends on and
121212The relevant expression is
where in the last step we have assumed that is small, which is indeed true for the decay channels.
5.2 Results: comparison of differential and inclusive information in different scenarios
A first global fit on single Higgs channels has been performed in ref.  using the 8 TeV LHC data, and a similar analysis has been applied to a future LHC scenario (CMS-HL-II) with . Only total cross section information was used and especially the fit included only as a variable. In ref.  a first attempt to use differential rate information provided in ref.  was made by extrapolating the projections on total cross section from ATLAS-HL [55, 70, 56] with .
Since no differential information is available in the measured data at the moment, we focus on the same future scenario at 14 TeV (ATLAS-HL) considered in ref. . However, our results cannot be directly compared with those in ref. , since there are a few differences in the treatment of the inputs from experimental projections. Details are reported in Appendix A, where we also carefully describe the procedure of the fit we performed and the assumptions on the uncertainties. In short, bounds on , and are obtained by maximising a log-likelihood function.
We perform the fit considering two very different scenarios for the uncertainties. In the first scenario (S1), only the statistical uncertainty is included. This crude assumption corresponds to the ideal (and rather unrealistic) situation where theory and experimental systematic uncertainties are negligible. On the other hand, we exploit it for a direct comparison with the second scenario (S2), where both theory and experimental systematic uncertainties are taken into account. At the differential level we performed the combination of the uncertainties via two different approaches that are described in detail in Appendix A. For this reason differential results for this second scenario always appear as bands rather than lines, accounting the uncertainty related to the different assumptions on the systematic and theory errors.
Before performing the global fit, we separately consider the different experimental inputs corresponding to F, VBF, and production131313In this section when we refer to a production mode in fact we mean one of the different -like categories in Tab.3. As can be seen, in any -like category the contribution of the actual process is in general dominant, so we can refer directly to it on the text for simplicity. Only the VBF-like category receives a non negligible contribution from F, which on the other hand has a very similar to VBF. and we restrict to the configuration with only (). We remind that different decay channels are entering for each of the production processes. Results are shown in Fig. 12, where the plot on the left refers to the scenario S1 and the plot on the right to the scenario S2. For the case of and production dashed lines correspond to the fit of differential information; details on the binning are reported in Appendix A.
The different shapes of the curves for values smaller and larger than can be understood from the behaviour of and terms in eqs. 6 and 13. While for both the and terms induce negative contributions in the production signal strengths, for there are large cancellations that suppress the effect of . If we only include statistical uncertainty (S1) the F-like channel provides the best constraints for both for the region and , where also is giving strong constraints, which are not improved by the inclusion of differential information. A similar effect is visible also for ; differential information does not lead to any significant improvement. On the other hand, in the region we see a clear improvement due to differential information for , although bounds from this single production process are not sufficient to set a constraint in the region for .
The plot on the right (S2) shows that including theory and experimental systematic uncertainties makes a difference. The process is giving the strongest constraints in the region and receive improvements from the differential information, with a tiny dependence on the assumption made for the combination of the uncertainties. This difference is induced by the change of the F result moving from the scenario S1 to the scenario S2 rather than by an improvement for . Note that, however, the impact of the differential information for F production is not known and while the exact calculation of the (two-)loop-induced effects from in would be useful, it is currently out of reach. Although constrains from F becomes much weaker in the scenario S2, in the region they are still the strongest. At variance with F, is in general very slightly affected by theory and systematic uncertainties since the dominant error is of statistical origin. Regarding the bounds on from VBF-like and -like channels, they are always worse than those from F and , even when the differential information is used for .
Next, we perform the global fit including all the experimental data as input and taking into account the anomalous couplings and . In Fig. 13 we present bounds after combining all the production channels, under different assumptions: i) only is anomalous, ii) or are anomalous, iii) all three parameters are anomalous. In the presence of anomalous couplings other than , we marginalise over them. The plot on the left refers to the scenario S1, only statistical uncertainties, and the one on the right to the scenario S2, systematic and theory uncertainties included. As we expect, in the scenario S1 the differential information (dashed line) does not noticeably improve any of the constraints, while in the scenario S2 in the region and especially in the region differential information from and leads to a clear improvement of the constraints. What is instead not obvious, especially given the findings of ref. , is the effect induced by anomalous and/or terms to the fit. While constraints in the region are relaxed, although not washed out completely, by the inclusion of one or two new degrees of freedom, in the region they are almost unaltered. In other words, in the scenario S2, bounds in the region are more affected by differential information than by the addition of the or parameters. Moreover, especially in the region , these two parameters alter the constraints more in the unrealistic scenario S1 than S2. We describe in a bit of detail the observed features exploiting the information contained in Fig. 12.
In the scenario S1 for the constraints are strongly affected by the inclusion of and/or since the global fit with only is completely dominated by F in that region. For this process only the total cross section information is used in the fit, so that a flat direction appears, i.e., the fit is dominated by one input 141414Note we have three decay channels for F that are almost fully controlled by , namely and . Indeed, also for the contribution from top-quark loop is known to be much smaller than -loop contribution., which is sufficient for setting constraints on only but not at the same time on and . To resolve this degeneracy more constraining information must be added to the fit. Indeed, the constraints with two parameters ( or ) or three are in the region of the constraints from VBF and in Fig. 12.
The previous argument cannot be applied to the region for the scenario S1, where the bounds in the global fit with only are not completely dominated by F. Indeed the (and in a smaller way the VBF) contribution is not negligible in that region, as can be seen from the left plot of Fig. 12. Moreover, at variance with F production, there is not a large background in production for the experimental signatures involving the Higgs to decay, whose branching ratio has a different and dependence w.r.t. and decays, and for values the impact of decays is more relevant. For this reason and F are sufficient for constraining one, two or three parameters, with negligible difference when parameters other than are marginalised. We explicitly verified this feature.
Moving to the scenario S2, the plot on the right where all uncertainties are included, for the bounds are dominated by channel. For this reason there is a smaller dependence on the number of parameters considered in the fit and a larger sensitivity to the differential information, which is present for the same reason also in the region .
It is clear that the role of the F is essential when the impact of differential information is investigated in the global fit. When F is dominant, since there is no differential dependence, it masks the relevance of differential distributions. On the other hand, when is dominant, the differential information becomes relevant. Above all, one should bear in mind that the impact of on F distributions has not been calculated because of technical reasons; the exact two-loop calculation is beyond the current technology, but could be relevant too. To this purpose, in the following we look at constraints in the and plane with and without the contributions from VBF and F, which hides the impact of the differential information. We consider only the scenario S2, which is more realistic.
In Fig. 14, we provide and contours in plane without (left) and with (right) anomalous , which is anyway marginalised. Upper plots includes all the production channels, whereas in the lower ones only and enter. Analogous plots are provided in Fig. 15 for the , without and with anomalous . First of all, one can note that due to the dependence of the gluon fusion channel and channel, in the upper plots the constraints on in presence of (Fig. 14) are stronger than those in presence of (Fig. 15). 151515 For the same reason, comparing these results with those that would be obtained in the scenario S1, one may also find that after including all uncertainties, the bounds on are enlarged more significantly than those on , since the dominant contribution to bounds on is F and , and the experimental systematic uncertainty and theory uncertainty are much larger than statistical uncertainty for F. Also, in the upper plots, having two independent parameters (left) or marginalising on an additional third one (right) does not lead to qualitatively significant differences. As discussed also before, the impact of the differential information is more important.
If we consider the lower plots the situation is very different. First, constraints with two or three parameters are qualitatively different. Second, the impact of the distributions is much more relevant. In the lower-left plot of Fig. 15 a flat direction is clearly resolved by differential information. The bottom-line is that by changing the number of free parameters and the number of inputs entering in the fit, the relevance of differential distributions and the sensitivity of the -limits on additional parameters can be considerably altered. The range of the lower plots is much larger than in the upper plots, for this reason the exclusion of and/or terms from eq. (17) would lead to visible effects to the 2 contours, anyway without altering the qualitative information.
We have studied one-loop effects for all the relevant single Higgs production modes at the LHC (, , , , ) and decays (, , , ), extending and completing the results presented in ref. . In particular, we have calculated differential results for VBF, , and