Higgs coupling measurements at the LHC
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Abstract
Due to the absence of tantalising hints for new physics during the LHC’s Run 1, the extension of the Higgs sector by dimensionsix operators will provide the new phenomenological standard for searches of nonresonant extensions of the Standard Model. Using all dominant and subdominant Higgs production mechanisms at the LHC, we compute the constraints on Higgs physicsrelevant dimensionsix operators in a global and correlated fit. We show in how far these constraints can be improved by new Higgs channels becoming accessible at higher energy and luminosity, both through inclusive cross sections as well as through highly sensitive differential distributions. This allows us to discuss the sensitivity to new effects in the Higgs sector that can be reached at the LHC if direct hints for physics beyond the SM remain elusive. We discuss the impact of these constraints on wellmotivated BSM scenarios.
1 Introduction
A question that arises at this stage in the LHC programme is the ultimate extent to which we will be able to probe the presence of such interactions. Or stated differently: what are realistic estimates of Wilson coefficient constraints that we can expect after Run 2 or the highluminosity phase if direct hints for new physics will remain elusive? With a multitude of additional Higgs search channels as well as differential measurements becoming available, the complexity of a fit of the relevant dimensionsix operators becomes immense.
It is the purpose of this work to provide these estimates. Using the Gfitter [30, 31, 32, 33] and Professor [34] frameworks, we construct predictions of fully differential cross sections, evaluated to the correct leadingorder expansion in the dimensionsix extension \(\hbox {d}\sigma =\hbox {d}\sigma ^{\text {SM}} + \hbox {d}\sigma ^{\{O_i\}}/\Lambda ^2\). We derive constraints on the Wilson coefficients in a fit of the dimensionsix operators relevant for the Higgs sector, inputting a multitude of present as well as projections of future LHC Higgs measurements.
This paper is outlined as follows. In Sect. 2 we introduce our approach in more detail. In particular, we discuss the involved Higgs production and decay processes and review our interpolation methods in the dimensionsix operator space, as well as introduce the key elements of our fit procedure. In Sect. 3 we give an overview of the statistical setup used. Our results using LHC Run 1 data are compared to existing and related work in Sect. 4. This sets the stage for the extrapolation to 14 TeV LHC centreofmass energy in Sect. 5, where we detail the assumptions made when extrapolating to higher luminosities. Our results are presented in Sect. 6, where we give estimates of the sensitivity that can be expected at the LHC for the operators considered in this work. An example on how the EFT constraints can be used in the context of a welldefined BSM model is given in Sect. 7. We give a discussion of our results and conclude in Sect. 8.
Throughout this work we will use the socalled strongly interacting light Higgs basis [9] adopting the “bar notation” (this choice is not unique and can be related to other bases [35]), and constrain deviations from the SM with leadingorder electroweak precision. A series of publications have extended the dimensionsix framework to nexttoleading order [36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. The impact of these modified electroweak corrections can in principle be large in phase space regions where SM electroweak corrections are known to be sizable and should be treated on a casebycase basis. However, this is not the main objective of this analysis and we consider higherorder electroweak effects beyond the scope of this work.
2 Framework and assumptions
2.1 Higgs production and decay
2.1.1 Production
For parameter choices close to the SM, including \(\mathcal{{M}}_{d=6}^2\) is typically not an issue and the parameters \(c_i^2\) are often numerically negligible for inclusive observables such as signal strengths. However, to obtain an inclusive measurement, we marginalise over a broad range of energies at the LHC and a positive theoretical cross section might be misleading as momentum dependencies of some dimensionsix operators violate a naive scaling \(c_i E^2/\Lambda ^2 <c_i^2 E^4/\Lambda ^4\) in the tails of momentumdependent distributions, where E denotes the respective and processrelevant energy scale. For this reason, we choose to calculate cross sections to the exact order \({\sim } 1/\Lambda ^2\) and later reject Wilson coefficient choices that lead to a negative differential cross section for integrated bins of a given LHC setting when this part of the phase space is resolved; such negative cross sections signal bigger contributions of the \(d=6\) terms than we expect in the SM, and we cannot justify limiting our analysis to dimensionsix operators if new physics becomes as important as the SM in observable phase space regions. This provides a conservative tool to gauge the validity of our approach, but care has to be taken by interpreting the results when connecting to concrete physics scenarios. In strongly interacting scenarios, it can be shown that the squared \(d=6\) terms are important, for small Wilson coefficients they are negligible. The latter avenue should be kept in mind for our results.
2.1.2 Included production modes and operators
We consider the production modes \(pp\rightarrow H\), \(pp \rightarrow H+{\text {j}}\), \(pp \rightarrow t\bar{t} H\), \(p p \rightarrow WH\), \(p p \rightarrow ZH\) and \(p p\rightarrow H+2{\text {j}}\) (via gluon fusion and weakboson fusion) in a fully differential fashion by including the differential Higgs transverse momentum distributions to setting constraints. As we demonstrate, including energydependent differential information whenever possible, is key to setting most stringent constraints on the dimensionsix extension by including the information of the distributions’ shapes beyond the total cross section, especially when probing blind directions in the signal strength, as shown in Fig. 1a. Note that for the underlying \(2\rightarrow 2\) and \(2\rightarrow 3 \) processes in the regions of detector acceptance, the Higgs transverse momentum is highly correlated with the relevant energy scales that probe the new interactions, Fig. 1b, and therefore is a suitable observable to include in this first step towards a fully differential Higgs fit. Expanding the cross sections to linear order in the Wilson coefficients as done in this work is not a mere technical twist, but allows us to obtain a description of the high\(p_T\) cross sections within our approximations.
The operator \((H^\dagger H)^3\) and offshell Higgs production in the EFT framework [54, 61, 62] deserve additional comments. Dihiggs production is the only process which provides direct sensitivity to \(\bar{c}_6\) [63] and factorises from the global fit, at least at leading order. Hence, the \(\bar{c}_6\) can be separated from the other directions to good approximation. While Higgs pair production process can serve to lift \(y_t\)degeneracies in the dimensionsix extension [64, 65], the sensitivity to \(\bar{c}_6\) is typically small when we marginalise over \(\bar{c}_{u3}\). The latter can be constrained either in \(pp \rightarrow \bar{t} t H\), \(pp\rightarrow ZZ\) in the Higgs offshell regime [54, 61, 62, 66, 67] or \(pp\rightarrow H+{\text {j}}\) [68, 69, 70, 71], however, only the former of these processes provides direct sensitivity to \(\bar{c}_{u3}\) without significant limitations due to marginalisation over the other operator directions. Current recast analyses place individual constraints in the range of \(\bar{c}_{u3}\lesssim 5\) and \(\bar{c}_g\lesssim 10^{3}\) [70] for the 8 TeV data set.
While the expected sensitivity to \(pp\rightarrow HH(+{\text {jets}})\) still remains experimentally vague at this stage in the LHC programme [72, 73], the potential to observe \(pp\rightarrow \bar{t} t H\) is consensus. We therefore do not include \(pp\rightarrow HH\) to our projections and also omit offshell Higgs boson production, since experimental efficiencies during the LHC highluminosity phase will significantly impact the sensitivity in these channels. We leave a more dedicated discussion of these channels to future work [74].
Treelevel sensitivity of the various production mechanisms
Production process  Included sensitivity 

\(pp\rightarrow H\)  \(\bar{c}_g, \bar{c}_{u3}, \bar{c}_H\) 
\(pp \rightarrow H+\text {j}\)  \(\bar{c}_g, \bar{c}_{u3}, \bar{c}_H\) 
\(pp \rightarrow H+2\text {j}\) (gluon fusion)  \(\bar{c}_g, \bar{c}_{u3}, \bar{c}_H\) 
\(pp \rightarrow t\bar{t}H\)  \(\bar{c}_g, \bar{c}_{u3}, \bar{c}_H\) 
\(pp \rightarrow VH\)  \(\bar{c}_{W}, \bar{c}_B, \bar{c}_{HW}, \bar{c}_{HB}, \bar{c}_{\gamma }, \bar{c}_H\) 
\(pp \rightarrow H+2\text {j}\) (weakboson fusion)  \(\bar{c}_{W}, \bar{c}_B, \bar{c}_{HW}, \bar{c}_{HB}, \bar{c}_{\gamma }, \bar{c}_H\) 
2.1.3 Branching ratios
For the branching ratios, we rely on eHdecay to include the correct Higgs branching ratios in the dimensionsix extended Standard Model, which is detailed in [75] and implements a linearisation in the Wilson coefficients. The branching ratios are therefore sensitive to all Wilson coefficients affecting single Higgs physics. An example for the variation of the branching ratios as a function of \(\bar{c}_\gamma \) is shown in Fig. 2.
We sample a broad range of dimensionsix parameter choices and interpolate them using the Professor method detailed in the Appendix A. This also allows us to identify already at this stage a reasonable Wilson coefficient range with a positivedefinite Higgs decay phenomenology that limits the validity (i.e. the positive definiteness) of our narrow width approximation. We find an excellent interpolation of the eHdecay output (independent of the interpolated sample’s size and choice) and we typically obtain per millelevel accuracy of the Higgs partial decay widths and branching ratios, which is precise enough for the limits we can set. Interpolation using Professor is key to performing the fit in the high dimensional space of operators and observables in a very fast and accurate way.
3 Statistical analysis
4 Results for Run 1
In the following we will evaluate the status of the effective Lagrangian Eq. (3) in light of available Run 1 analyses. Similar analyses have been performed by a number of groups; see e.g. [22, 24, 26]. Comparing the above fitprocedure to these results not only allows us to validate the highly nontrivial fitting procedure against other approaches, but also to extend these results by including additional measurements which have become available in the meantime. We include Run 1 experimental analyses using HiggsSignals v1.4 [76, 77], based on HiggsBounds v4.2.1 [78, 79, 80, 81], which calculates \(\chi ^2\) given in Eq. (8) taking into account experimental and theoretical correlations, as well as signal acceptances.
The results are shown in Fig. 3 and are in good agreement with the results obtained in Refs. [24, 28]. Numerical values are given below in Table 5. Small differences can be understood from working under different assumptions (specifically the strict linearisation of dimensionsix effects) as well as including more analyses. It should be noted that our choice of limiting the range of Wilson coefficient values (necessary for the positive definiteness of differential distributions) is necessitated by our extrapolation and inclusion of differential distributions. Consequently, we cannot set a limit on many operators in the light of Run 1 measurements within our approximations. However, the direct comparison to the Figs. 4 and 5 will allow us to see how these can be improved when going to higher centreofmass energy and luminosity. Relaxing these constraints will lead to increased Wilson coefficient intervals for the marginalised scans over the 8 TeV signal strength measurements (for a recent fit without limited coefficient ranges see Ref. [51]).
5 Projections for 14 TeV and the highluminosity phase
 \(H\rightarrow \mathrm {b}\bar{b}\):

We assume a flat btagging efficiency of \(60~\%\), i.e. \(\epsilon _{d,b\bar{b}}=0.36\).
 \(H\rightarrow \gamma \gamma \):

For the identification and reconstruction of isolated photons we assume, respectively, an efficiency of \(85~\%\). Hence, we find \(\epsilon _{d,\gamma \gamma } \simeq 0.72\).
 \(H\rightarrow \tau ^+ \tau ^\):

We consider \(\tau \)decays into hadrons (\(\text {BR}_{\text {had}}=0.648\)) or leptons, i.e. an electron (\(\text {BR}_{e}=0.178\)) or muon (\(\text {BR}_{\mu }=0.174\)). For the reconstruction efficiency of the hadronic \(\tau \) we assume a value of \(50~\%\) and for the electron and muon we use \(95~\%\). Thus, the total reconstruction efficiency is \(\epsilon _{d,\tau \tau } \simeq 0.433\).
 \(H\rightarrow ZZ^* \rightarrow 4l\):

We consider Z decays into electrons and muons only, also taking into account \(\tau \) decays into lighter leptons. For each lepton we assume a reconstruction efficiency of \(95~\%\), which gives a total reconstruction efficiency of \(\epsilon _{d,4l} \simeq 0.815\).
 \(H\rightarrow WW^* \rightarrow 2l 2\nu \):

Only lepton decays into electrons and muons are considered and for each visible lepton we include a \(95~\%\) reconstruction efficiency, i.e. \(\epsilon _{d,2l 2\nu } = 0.9025\)
 \(H\rightarrow Z \gamma \):

Again, we include separately an \(85~\%\) identification and reconstruction efficiency for isolated photons and a \(95~\%\) reconstruction efficiency for each electron and muon. As a result we find \(\epsilon _{d,Z\gamma } \simeq 0.767\).
 \(H\rightarrow \mu ^+ \mu ^\):

Each muon is assumed to have a reconstruction efficiency of \(95~\%\), resulting in \(\epsilon _{d,\mu \mu }=0.9025\).
Relative systematic uncertainties due to background processes for each production and decay channel in \(\%\)
Production process  Decay process  

\(pp\rightarrow H\)  10  \(H\rightarrow b\bar{b}\)  25 
\(pp\rightarrow H+\text {j}\)  30  \(H\rightarrow \gamma \gamma \)  20 
\(pp\rightarrow H+2\text {j}\)  100  \(H\rightarrow \tau ^+ \tau ^\)  15 
\(pp\rightarrow HZ\)  10  \(H\rightarrow 4l\)  20 
\(pp\rightarrow HW\)  50  \(H\rightarrow 2l 2\nu \)  15 
\(pp\rightarrow t\bar{t}H\)  30  \(H\rightarrow Z \gamma \)  150 
\(H\rightarrow \mu ^+ \mu ^\)  150 
Relative systematic uncertainties for each production times decay channel in \(\%\)
\(t\bar{t}H\)  HZ  HW  H incl.  \(H+\text {j}\)  \(H+2\text {j}\)  

\(H\rightarrow b\bar{b}\)  80  25  40  100  100  150 
\(H\rightarrow \gamma \gamma \)  60  70  30  10  10  20 
\(H\rightarrow \tau ^+ \tau ^\)  100  75  75  80  80  30 
\(H\rightarrow 4l\)  70  30  30  20  20  30 
\(H\rightarrow 2l 2\nu \)  70  100  100  20  20  30 
\(H\rightarrow Z \gamma \)  100  100  100  100  100  100 
\(H\rightarrow \mu ^+ \mu ^\)  100  100  100  100  100  100 
Beyond identification and reconstruction efficiencies for production channels and Higgs decays, each channel is plagued by individual experimental systematic uncertainties. We adopt flat systematic uncertainties in \(p_{T,H}\) for the individual channels. The numerical values are based on the results from experimental Run 1 analyses [3, 4, 82, 83, 84, 85, 86, 87, 88, 90, 91, 94, 95, 109, 110, 111, 112], see Table 3. In channels where no measurement has been performed or no information is publicly available, e.g. \(pp\rightarrow H+2\text {j}, H\rightarrow Z \gamma \), we choose a conservative estimate of systematic uncertainties of 100 %. In addition to the uncertainties listed in Table 3, we include a systematic uncertainty of 30 % for the \(H\rightarrow 2l 2\nu \) channel for differential cross sections. This uncertainty is due to the inability of reconstructing the Higgs transverse momentum accurately.
During future runs, experimental uncertainties are likely to improve with the integrated luminosity. Hence for our results at 14 TeV we use the 8 TeV uncertainties as a starting point, as displayed in Tables 2 and 3, and we rescale them by \(L_8/L_{14}\) for a given integrated luminosity at 14 TeV \(L_{14}\). This results in a reduction of statistical and experimental systematic uncertainties by a factor of about 0.3 for \(L_{14}=300~\text {fb}^{1}\) and about 0.1 for \(L_{14}=3000~\text {fb}^{1}\). This simplified procedure has also been adopted by the ATLAS and CMS collaborations to extrapolate the sensitivity of experimental analyses to future runs [113, 114, 115, 116]. We use this extrapolation for ease of comparison and reproducibility.
We only consider measurements with more than five signal events after the application of all efficiencies and a total uncertainty smaller than 100 %. The pseudodata are constructed using the SM hypothesis, i.e. all Wilson coefficients are set to zero. We construct expected signal strength measurements for all accessible production and decay modes. Additionally, differential cross sections as a function of the Higgs transverse momentum are simulated with a bin size of 100 GeV. In \(2\rightarrow 3\) processes like ttH other differential distributions might provide higher sensitivity than \(p_{T,H}\), but at this point we restrict the analysis to include \(p_{T,H}\) distributions only, as these are likely to be provided as unfolded distributions by the experimental collaborations. We leave studies on the sensitivity of additional kinematic variables in a global fit to future work [74].
Theoretical uncertainties for each production and decay channel in \(\%\)
Production process  Decay process  

\(pp\rightarrow H\)  14.7  \(H\rightarrow b\bar{b}\)  6.1 
\(pp\rightarrow H+\text {j}\)  15  \(H\rightarrow \gamma \gamma \)  5.4 
\(pp\rightarrow H+2\text {j}\)  15  \(H\rightarrow \tau ^+ \tau ^\)  2.8 
\(pp\rightarrow HZ\)  5.1  \(H\rightarrow 4l\)  4.8 
\(pp\rightarrow HW\)  3.7  \(H\rightarrow 2l 2\nu \)  4.8 
\(pp\rightarrow t\bar{t}H\)  12  \(H\rightarrow Z \gamma \)  9.4 
\(H\rightarrow \mu ^+ \mu ^\)  2.8 
Constraints at 95 % CL on dimensionsix operator coefficients (first column) from LHC Run 1 data, considering only one operator in the fit (second column) and all operators simultaneously (third column). The results obtained using pseudodata are shown in the last two columns, with signal strengths measurements only (fourth column) and including differential distributions (fifth column). In case no constraints can be derived within the parameter ranges considered in this work, the lower and upper limits are indicated to lie outside this range
Individual  Marginalised (all \(c_i\) free)  

Run 1  Run 1  Pseudodata \(\mu \)  Pseudodata \(\mu \) and \(p_{T,h}\)  
\(\bar{c}_{g}\) \([{\times } 10^{4}]\)  \([0.32, \, 0.35]\)  \([0.64, \, 0.43]\)  \([0.84, \, {>} 1.000]\)  \([0.82, \, {>} 1.000]\) 
\(\bar{c}_{\gamma }\) \([{\times } 10^{4}]\)  \([5.5, \, 3.6]\)  \([7.8, \, 4.3]\)  \([{<} 10.000, \, 7.3]\)  \([{<} 10.000, \, 6.6]\) 
\(\bar{c}_{W}\)  \([{<} 0.01, \, 0.007]\)  \([{<} 0.01, \, {>} 0.01]\)  \([{<} 0.01, \, {>} 0.01]\)  \([{<} 0.01, \, {>} 0.01]\) 
\(\bar{c}_{H}\)  \([{<}0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\) 
\(\bar{c}_{HW}\)  \([0.047, \, 0.014]\)  \([{<} 0.05, \, 0.035]\)  \([{<} 0.05, \, > 0.05]\)  \([0.044, \, {>} 0.05]\) 
\(\bar{c}_{HB}\)  \([{<} 0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\) 
\(\bar{c}_{u3}\)  \([{<} 0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\) 
\(\bar{c}_{d3}\)  \([{<} 0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\)  \([{<} 0.05, \, {>} 0.05]\) 
Theory uncertainties included in the fit are listed in Table 4 and have been obtained by the Higgs cross section working group [101, 102, 103] (see also [117] about their role in Higgs fits). We assume the same size of theory uncertainties for the SM predictions as for calculations using the EFT framework. The theory uncertainties are not scaled with luminosity and retain the values given in Table 4 throughout this work.

the above luminosity scaling of experimental uncertainties,

a clean separation of the measurements of all production and decay channels (no cross talk between channels),

flat experimental systematic uncertainties as a function of \(p_{T,H}\),

flat theory uncertainties as a function of \(p_{T,H}\) as quoted in Table 4, which we assume to be independent of the Wilson coefficients.
6 Predicted constraints
The projected measurements of the Higgs signal strengths and the Higgs transverse momentum (\(p_{T,H}\)) distributions are used to test the sensitivity to the dimensionsix operators that can be obtained with the LHC. In all fits theory uncertainties are included as nuisance parameters with Gaussian constraints. The constraints on individual Wilson coefficients are obtained by a marginalisation over the remaining coefficients and the nuisance parameters related to the theory uncertainties.
In order to test this approach, we first generate pseudodata for 8 TeV following the procedure detailed above. The integrated luminosity is chosen to be \(L_8\), i.e. \(25~\text {fb}^{1}\) per experiment which corresponds to the full Run 1 data. With this setting no luminosity scaling of experimental uncertainties is performed. Besides statistical uncertainties, the generated 8 TeV data have systematic uncertainties corresponding to the values given in Tables 2 and 3. We compare the constraints obtained with these pseudodata with the ones obtained from the Run 1 analysis in Table 5. Similar to the constraints derived in Sect. 4 no reliable constraints at 95 % CL on coefficients other than \(\bar{c}_{g}\) and \(\bar{c}_{\gamma }\) can be derived within the parameter ranges considered in this work. We observe that the constraints using pseudodata are considerably weaker than the ones from the existing Run 1 measurements. This is no surprise, as the simplified approach outlined above cannot reflect the complexity of real analyses, where a number of signal regions are used to disentangle different production modes. This picture does not change when including differential distributions (last column of Table 5) which results in slightly better constraints at 8 TeV compared to the fit with signal strengths only. We note that although the constraints obtained with pseudodata are generally weaker, they are very similar to the ones using current Run 1 experimental data. We therefore trust our method and proceed to derive the expected sensitivity of the LHC.
Predicted constraints at 95 % CL on dimensionsix operator coefficients (first column) for the LHC with 14 TeV with an integrated luminosity of \(300~\text {fb}^{1}\) (LHC300) and \(3000~\text {fb}^{1}\) (LHC3000). In the second and third columns results are given using signal strength measurements only, in the last two columns results including differential \(p_{T,H}\) measurements are shown. In case no constraints can be derived within the parameter ranges considered in this work, the lower and upper limits are indicated to lie outside this range
Signal strengths only  With differential \(p_{T,H}\) measurements  

LHC300  LHC3000  LHC300  LHC3000  
\(\bar{c}_{g}\) \([{\times } 10^{4}]\)  \([0.53, \, 0.66]\)  \([0.49, \, 0.57]\)  \([0.19, \, 0.22]\)  \([0.06, \, 0.07]\) 
\(\bar{c}_{\gamma }\) \([{\times } 10^{4}]\)  \([3.9, \, 3.4]\)  \([2.9, \, 2.7]\)  \([2.5, \, 2.0]\)  \([1.6, \, 1.3]\) 
\(\bar{c}_{W}\)  \([{<} 0.010, \, {>} 0.010]\)  \([{<} 0.010, \, {>} 0.010]\)  \([0.008, \, 0.008]\)  \([0.004, \, 0.004]\) 
\(\bar{c}_{H}\)  \([{<} 0.050, \, {>} 0.050]\)  \([{<} 0.050, \, {>} 0.050]\)  \([{<} 0.050, \, {>} 0.050]\)  \([0.044, \, 0.035]\) 
\(\bar{c}_{HW}\)  \([0.030, \, 0.032]\)  \([0.027, \, 0.028]\)  \([0.007, \, 0.010]\)  \([0.004, \, 0.004]\) 
\(\bar{c}_{HB}\)  \([0.030, \, 0.032]\)  \([0.026, \, 0.027]\)  \([0.008, \, 0.011]\)  \([0.004, \, 0.004]\) 
\(\bar{c}_{u3}\)  \([{<} 0.050, \, {>} 0.050]\)  \([{<} 0.050, \, {>} 0.050]\)  \([{<} 0.050, \, {>} 0.050]\)  \([0.020, \, 0.008]\) 
\(\bar{c}_{d3}\)  \([{<} 0.050, \, {>} 0.050]\)  \([{<} 0.050, \, {>} 0.050]\)  \([{<} 0.050, \, {>} 0.050]\)  \([{<} 0.050, \, {>} 0.050]\) 
In a second step, we include the differential \(p_{T,H}\) measurements from all production modes, except \(pp\rightarrow H\). For the \(pp\rightarrow H\) production mode we include six signal strength measurements (see the Appendix), as no transverse momentum of the Higgs boson is generated on treelevel. This results in \(82+6\) independent measurements included for the fit with \(300~\text {fb}^{1}\) and \(117+6\) for \(3000~\text {fb}^{1}\). In a given production and decay channel, experimental systematic uncertainties are included as correlated uncertainties among bins in \(p_{T,H}\). Comparing the above constraints with those expected from including the differential distributions, Fig. 5, we see a tremendous improvement. The improvement compared to the constraints presented in Fig. 4 is solemnly due to the inclusion of differential distributions, as no new channels are added in this step. We also observe a reduction of the impact of theoretical uncertainties. Twodimensional contours of the expected constraints are shown in Fig. 6 for the scenario with \(3000~\text {fb}^{1}\). The fits using signal strength measurements only (grey) reveal a series of flat directions which cannot be amended by a different operator choice. Several flat directions are resolved with the fit using information from the differential \(p_{T,H}\) measurements. While the improvement on the exact numerical constraints can be somewhat compromised by larger systematic uncertainties, the general feature of lifting flat directions still remains [74]. Even with \(3000~\text {fb}^{1}\) it is not possible to constrain \(\bar{c}_{u3}\) and \(\bar{c}_{g}\) or \(\bar{c}_{HW}\) and \(\bar{c}_{HB}\) simultaneously using signal strength modifiers only. Using information from the differential \(p_{T,H}\) measurements, which are systematically under sufficient control, effectively allows one to constrain all coefficients simultaneously. Elements of studying differential distributions to effective Higgs dimensionsix framework have been investigated with similar findings in the literature [24, 26, 118], but, to our knowledge, Figs. 5 and 6 provide the first consistent fit of all singleHiggs relevant operators in a fully differential fashion, in particular with extrapolations to 14 TeV. The numerical values of the 95 % CL intervals for the different scenarios are given in Table 6.
A series of dimensionsix operators, on which no constraints can be formulated at this stage of the LHC programme or by only including signal strength measurements, can eventually be constrained with enough data and differential distributions. The reason behind this is that differential measurements ipso facto increase the number of (correlated) measurements by number of bins, leading to a highly overconstrained system. Also, since the impact of many operators is most significant in the tails of energydependent distribution, the relative statistical pull is decreased by only considering inclusive quantities.
7 Interpretation of constraints
The whole purpose of interpreting data in terms of an effective field theory is to use this framework as a means of communication between a lowscale measurement at the LHC and a UV model defined at a high scale, out of reach of the LHC. This way, the EFT framework allows us to limit a large class of UV models.
For a welldefined interpretation using effective operators, we assume that the operators, induced by the UV theory, only directly depend on the SM particle and symmetry content, and we also need to assume that the UV theory is weakly coupled to the SM sector. The last condition is necessary to justify the truncation of the effective Lagrangian at dimension six. After establishing limits on Wilson coefficients of the effective theory, as performed in Sects. 4–6, we can now address the implications for a specific UV model.
Two popular ways of addressing the Hierarchy problem are composite Higgs models and supersymmetric theories. Let us quickly investigate in how far these constraints are relevant once we match the EFT expansion to a concrete UV scenario.
8 Discussion, conclusions and outlook
Even though current measurements as performed by ATLAS and CMS show good agreement with the SM hypothesis for the small statistics collected during LHC Run 1, the recently discovered Higgs boson remains one of the best candidates that could be a harbinger of physics beyond the SM. If new physics is heavy enough, modifications to the Higgs boson’s phenomenology from integrating out heavy states can be expressed using effective field theory methods.
In this paper we have constructed a scalable fitting framework, based on adapted versions of Gfitter, Professor, Vbfnlo, and eHdecay and have used an abundant list of available singleHiggs LHC measurements to constrain new physics in the Higgs sector for the results of Run 1. In these fits we have adopted the leadingorder strongly interacting light Higgs basis assuming vanishing treelevel T and S parameters and flavour universality of the new physics sector. Our results represent the latest incarnation of fits at 8 TeV, and update results from the existing literature. The main goal of this work, however, is to provide an estimate of how these constraints will improve when turning to high energy collisions at the LHC with large statistics in light of expected systematics. In this sense our work represents a first step towards an ultimate Higgs sector fit, which is not limited to inclusive measurements, but uses highly sensitive differential distributions throughout.
It is interesting to see that including differential information at the LHC, we can expect the limits on certain operators to become competitive with measurements at a future FCCee [127, 128]. This is not entirely unexpected since the high \(p_{T,H}\) cross sections, especially for hadronic channels, are sensitive probes of BSM physics. A major limiting factor, however, are the involved theoretical uncertainties, especially when moving to differential distributions at large statistics. Obviously, electroweak precision constraints provide a complementary avenue to constrain the presence of higher dimensional operators [24, 51, 129, 130] and are guaranteed to improve the sensitivity. We reserve a dedicated discussion for the future.
Footnotes
Notes
Acknowledgments
We thank A. Falkowski, M. GonzalezAlonso, A. Greljo, D. Marzocca, T. Plehn, M. Trott and T. You for discussions, and T. Stefaniak for help with HiggsSignals. This research was supported in part by the European Commission through the “HiggsTools” Initial Training Network PITNGA2012316704 and by the German Research Foundation (DFG) in the Collaborative Research Centre (SFB) 676 “Particles, Strings and the Early Universe” located in Hamburg.
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