Constraints on anomalous HVV couplings from the production of Higgs bosons decaying to τ lepton pairs

A study is presented of anomalous HVV interactions of the Higgs boson, including its CP properties. The study uses Higgs boson candidates produced mainly in vector boson fusion and gluon fusion that subsequently decay to a pair of τ leptons. The data were recorded by the CMS experiment at the LHC in 2016 at a center-of-mass energy of 13 TeV and correspond to an integrated luminosity of 35 . 9 fb − 1 . A matrix element technique is employed for the analysis of anomalous interactions. The results are combined with those from the H → 4 l decay channel presented earlier, yielding the most stringent constraints on anomalous Higgs boson couplings to electroweak vector bosons expressed as effective cross section fractions and phases: the CP -violating parameter f a 3 cos ð ϕ a 3 Þ ¼ ð 0 . 00 (cid:2) 0 . 27 Þ × 10 − 3 and the CP -conserving parameters f a 2 cos ð ϕ a 2 Þ ¼ ð 0 . 08 þ 1 . 04 − 0 . 21 Þ × 10 − 3 , f Λ 1 cos ð ϕ Λ 1 Þ ¼ ð 0 . 00 þ 0 . 53 − 0 . 09 Þ × 10 − 3 , and f Z γ Λ 1 cos ð ϕ Z γ Λ 1 Þ ¼ ð 0 . 0 þ 1 . 1 − 1 . 3 Þ × 10 − 3 . The current dataset does not allow for precise constraints on CP properties in the gluon fusion process. The results are consistent with standard model expectations.


I. INTRODUCTION
The Higgs boson (H) discovered in 2012 at the CERN LHC [1-3] has thus far been found to have properties consistent with expectations from the standard model (SM) [4][5][6][7][8][9][10]. In particular, its spin-parity quantum numbers are consistent with J PC ¼ 0 þþ according to measurements performed by the CMS [11][12][13][14][15][16][17] and ATLAS [18][19][20][21][22][23] experiments. It is still to be determined whether small anomalous couplings contribute to the HVV or Hff interactions, where V stands for vector bosons and f stands for fermions. Because nonzero spin assignments of the H boson have been excluded [13,19], we focus on the analysis of couplings of a spin-0 H boson. Previous studies of anomalous HVV couplings were performed by both the CMS and ATLAS experiments using either decay-only information [11][12][13]18,19,21], including associated production information [15][16][17]20,22,23], or including off-shell H boson production [14,17]. In this paper, we report a study of HVV couplings using information from production of the H boson decaying to τ leptons. These results are combined with the previous CMS measurements using both associated production and decay information in the H → 4l channel [17], resulting in stringent constraints on anomalous H boson couplings. Here and in the following, l denotes an electron or muon.
The H → ττ decay has been observed by the CMS experiment, with over five standard deviation significance [24]. The H → ττ sample can be used to study the quantum numbers of the H boson and its anomalous couplings to SM particles, including its CP properties. The dominant production mechanisms of the H boson considered in this paper are shown at leading order in QCD in Fig. 1. Anomalous HWW, HZZ, HZγ, Hγγ, and Hgg couplings affect the correlations between the H boson, the beam line direction, and the two jets in vector boson fusion (VBF), in associated production with a vector boson decaying hadronically (VH, where V ¼ W; Z), or gluon fusion production with an additional two jets. The gluon fusion production with two additional jets appears at higher order in QCD with an example of gluons appearing in place of the vector bosons shown in the VBF diagram in the middle of Fig. 1. A study of anomalous Htt couplings in associated production with top quarks, ttH or tqH, and anomalous Hττ couplings in the decay of the H boson are also possible using ττ events [25]. However, more data are needed to reach sensitivity to such anomalous effects, and it has been confirmed that these anomalous couplings would not affect the measurements presented in this paper.
To increase the sensitivity to anomalous couplings in the H boson production, the matrix element likelihood approach (MELA) [2,[26][27][28][29] is utilized to form optimal observables. The analysis is optimized for VBF production and is not additionally optimized for VH or gluon fusion production. However, all three production mechanisms are included in the analysis, using a general anomalous coupling parametrization. The H → ττ channel has advantages over other H boson decay channels because of the relatively high significance of the signal events in the VBF channel [24]. Three mutually exclusive categories of events are reconstructed in the analysis: the VBF category targets events with two associated jets in the VBF event topology, the boosted category contains events with one jet or more jets if the event is not in the VBF category, and the 0-jet category targets H boson events produced via gluon fusion without associated jets. The simultaneous analysis of all three categories of events is necessary to boost the sensitivity to anomalous HVV couplings from events with partial kinematic information reconstructed in the non-VBF categories and to normalize the relative contribution of different production mechanisms.
The analysis utilizes the same data, event selection, and categorization as Ref. [24] and is described in Sec. III. The phenomenological model and Monte Carlo (MC) simulation are described in Sec. IV. The matrix element techniques used to extract the kinematic information are discussed in Sec. V. The implementation of the likelihood fit using kinematic information in the events is presented in Sec. VI. The results are presented and discussed in Secs. VII and VIII, before conclusions are drawn in Sec. IX.

II. CMS DETECTOR
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the solenoid volume, there are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter, each composed of a barrel and two end cap sections. Forward calorimeters extend the pseudorapidity, η, coverage provided by the barrel and end cap detectors. Muons are detected in gas-ionization chambers embedded in the steel flux-return yoke outside the solenoid.
Events of interest are selected using a two-tiered trigger system [30]. The first level (L1), composed of custom hardware processors, uses information from the calorimeters and muon detectors to select events at a rate of around 100 kHz within a time interval of less than 4 μs. The second level, known as the high-level trigger, consists of a farm of processors running a version of the full event reconstruction software optimized for fast processing and reduces the event rate to about 1 kHz before data storage.
A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [31].
The data samples used in this analysis correspond to an integrated luminosity of 35.9 fb −1 collected in Run 2 of the LHC during 2016 at a center-of-mass energy of 13 TeV.

III. EVENT RECONSTRUCTION AND SELECTION
The analysis uses the same dataset, event reconstruction, and selection criteria as those used in the analysis leading to the observation of the H boson decay to a pair of τ leptons [24].

A. Event reconstruction
The reconstruction of observed and simulated events relies on the particle-flow (PF) algorithm [32], which combines the information from the CMS subdetectors to identify and reconstruct particles emerging from pp collisions. Combinations of these PF candidates are used to reconstruct higher-level objects such as jets, τ candidates, or missing transverse momentum, ⃗p miss T . The reconstructed vertex with the largest value of summed physics object p 2 t is taken to be the primary pp interaction vertex, where p T is the transverse momentum. The physics objects are the objects constructed by a jet finding algorithm [33,34] applied to all charged tracks associated with the vertex and the corresponding associated missing transverse momentum.
Electrons are identified with a multivariate discriminant combining several quantities describing the track quality, the shape of the energy deposits in the ECAL, and the FIG. 1. Examples of leading-order Feynman diagrams for H boson production via the gluon fusion (left), vector boson fusion (middle), and associated production with a vector boson (right). The HWW and HZZ couplings may appear at tree level, as the SM predicts. Additionally, HWW, HZZ, HZγ, Hγγ, and Hgg couplings may be generated by loops of SM or unknown particles, as indicated in the left diagram but not shown explicitly in the middle and right diagrams. visible decay products of the τ leptons. The observable m ττ is reconstructed using the SVFIT [41] algorithm, which combines the ⃗p miss T and its uncertainty with the 4-vectors of both τ candidates to calculate a more accurate estimate of the mass of the parent boson. The estimate of the 4-momentum of the H boson provided by SVFIT is used to calculate the kinematic observables discussed in Sec. V.

B. Event selection and categorization
Selected events are classified according to four decay channels, eμ, eτ h , μτ h , and τ h τ h . The resulting event samples are made mutually exclusive by discarding events that have additional loosely identified and isolated electrons or muons.
The largest irreducible source of background is Drell-Yan production of Z → ττ, while the dominant background sources with jets misidentified as leptons are QCD multijet and W þ jets. Other contributing background sources are tt, single top, Z → ll, and diboson production.
The two leptons assigned to the H boson decay are required to have opposite charges. The trigger requirements, geometrical acceptances, and transverse momentum criteria are summarized in Table I. The p T thresholds in the lepton selections are optimized to increase the sensitivity to the H → ττ signal, while also satisfying the trigger requirements. The pseudorapidity requirements are driven by reconstruction and trigger requirements.
In the lτ h channels, the large W þ jets background is reduced by requiring the transverse mass, m T , to be less than 50 GeV. The transverse mass is defined as follows, TABLE I. Kinematic selection criteria for the four decay channels. For the trigger threshold requirements, the numbers indicate the trigger thresholds in GeV. The lepton selection criteria include the transverse momentum threshold, pseudorapidity range, as well as isolation criteria.

Trigger requirement
Lepton selection where p l T is the transverse momentum of the electron or muon and Δϕ is the azimuthal angle between the lepton direction and the ⃗p miss T direction. In the eμ channel, the tt background is reduced by requiring p ζ − 0.85p vis ζ > −35 GeV or −10 GeV depending on the category, where p ζ is the component of ⃗p miss T along the bisector of the transverse momenta of the two leptons and p vis ζ is the sum of the components of the lepton transverse momenta along the same direction [42]. In addition, events with a b-tagged jet are discarded to further suppress the tt background in this channel.
In the same way as in Ref.
[24], the event samples are split into three mutually exclusive production categories: (i) 0-jet category: This category targets H boson events produced via gluon fusion. Events containing no jets with p T > 30 GeV are selected. Simulations indicate that about 98% of signal events in the 0-jet category arise from the gluon fusion production mechanism. (ii) VBF category: This category targets H boson events produced via the VBF process. Events are selected with exactly (at least) two jets with p T > 30 GeV in the eμ (eτ h , μτ h , and τ h τ h ) channels. In the μτ h , eτ h , and eμ channels, the two leading jets are required to have an invariant mass, m JJ , larger than 300 GeV. The vector sum of the ⃗p miss T and the ⃗p T of the visible decay products of the tau leptons, defined as ⃗p ττ T , is required to have a magnitude greater than 50 (100) GeV in the lτ h (τ h τ h ) channels. In addition, the p T threshold on the τ h candidate is raised to 40 GeV in the μτ h channel, and the two leading jets in the τ h τ h channel must be separated in pseudorapidity by jΔηj > 2.5. Depending on the decay channel, up to 57% of the signal events in the VBF category is produced via VBF. This fraction increases with m JJ . Gluon fusion production makes 40%-50% of the total signal, while the VH contribution is less than 3%. (iii) Boosted category: This category contains all the events that do not enter one of the previous categories, namely events with one jet and events with several jets that fail the requirements of the VBF category. It targets events with a H boson produced in gluon fusion and recoiling against an initial state radiation jet. It contains gluon fusion events produced in association with one or more jets (78%-80% of the signal events), VBF events in which one of the jets has escaped detection or events with low m JJ (11%-13%), as well as H boson events produced in association with a W or a Z boson decaying hadronically (4%-8%).
In addition to these three signal regions for each channel, a series of control regions targeting different background processes are included in the maximum likelihood fit used to extract the results of the analysis. The normalization of the W þ jets background in the eτ h and μτ h channels is estimated from simulations and adjusted to data using control regions obtained by applying all selection criteria, with the exception that m T is required to be greater than 80 GeV instead of less than 50 GeV. An uncertainty on the extrapolation from the control region to the signal region is determined in the same way as described in Ref. [24]. The normalization of the QCD multijet background in the eτ h and μτ h channels is estimated from events where the electron or the muon has the same charge as the τ h candidate. The contributions from Drell-Yan, tt, diboson, and W þ jets processes are subtracted. The factor to extrapolate from the same-sign to the opposite-sign region is determined by comparing the yield of the QCD multijet background for events with l candidates passing inverted isolation criteria, in the same-sign and opposite-sign regions. It is constrained by adding the opposite-sign region, where the l candidates pass inverted isolation criteria, to the global fit.
In the τ h τ h channel, the QCD multijet background is estimated from events where the τ h candidates pass relaxed isolation conditions, and the extrapolation factor is derived from events where the τ h candidates have charges of the same sign. The events selected with opposite-sign τ h candidates passing relaxed isolation requirements form a control region included in the global fit. Finally, the normalization of the tt background is adjusted using a control region defined similarly to the eμ signal region, except that the p ζ requirement is inverted and the events are required to contain at least one jet.

IV. PHENOMENOLOGY OF ANOMALOUS COUPLINGS AND SIMULATION
We follow the formalism used in the study of anomalous couplings in earlier analyses by CMS [11][12][13][14][15][16][17]. The theoretical approach is described in [43][44][45][46][47][48][49][50][51]. Anomalous interactions of a spin-0 H boson with two spin-1 gauge bosons VV, such as WW, ZZ, Zγ, γγ, and gg, are parametrized by a scattering amplitude that includes three tensor structures with expansion of coefficients up to ðq 2 =Λ 2 Þ where q i , ϵ Vi , and m V1 are the 4-momentum, polarization vector, and pole mass of the gauge boson, indexed by i ¼ 1, 2. The gauge boson's field strength tensor and dual field strength tensor are f ðiÞμν ¼ ϵ μ Vi q ν i − ϵ ν Vi q μ i and f ðiÞ μν ¼ 1 2 ϵ μνρσ f ðiÞρσ . The coupling coefficients a VV i , which multiply the three tensor structures, and κ VV i =ðΛ VV 1 Þ 2 , which multiply the next term in the q 2 expansion for the first tensor structure, are to be determined from data, where Λ 1 is the scale of beyond the SM (BSM) physics.
In Eq.
(2), the only nonzero SM contributions at tree level are a WW 1 and a ZZ 1 , which are assumed to be equal under custodial symmetry. All other ZZ and WW couplings are considered anomalous contributions, which are either due to BSM physics or small contributions arising in the SM due to loop effects and are not accessible with the current precision. As the event kinematics of the H boson production in WW fusion and in ZZ fusion are very similar, they are analyzed together assuming a WW ¼ a 3 . The a 3 coupling corresponds to the CP-odd amplitude, and its interference with a CP-even amplitude would result in CP violation.
It is convenient to measure the effective cross section ratios f ai rather than the anomalous couplings a i themselves, as most uncertainties cancel in the ratio. Moreover, the effective fractions are conveniently bounded between 0 and 1, independent of the coupling convention. The effective fractional cross sections f ai and phases ϕ ai are defined as follows, where σ i is the cross section for the process corresponding to a i ¼ 1 and all other couplings are set to zero. Since the production cross sections depend on the parton distribution functions (PDFs), the definition with respect to the decay process is more convenient. The cross section ratios defined in the H → 2e2μ decay analysis [12] are adopted. Their values are σ 1 =σ 3 ¼ 6.53, σ 1 =σ 2 ¼ 2.77, ðσ 1 =σ Λ1 Þ × TeV 4 ¼ 1.47 × 10 4 , and ðσ 1 =σ Zγ Λ1 Þ × TeV 4 ¼ 5.80 × 10 3 , as calculated using the JHUGEN7.0.2 event generator [26][27][28][29]. The ellipsis (…) in Eq. (3) indicates any other contribution not listed explicitly. Under the assumption that the couplings in Eq. (2) are constant and real, the above formulation is equivalent to an effective Lagrangian notation. Therefore, in this paper, the real coupling constants are tested, which means only ϕ ai ¼ 0 or π are allowed. The constraints are set on the product f ai cosðϕ ai Þ, which ranges from −1 to þ1.
Anomalous effects in the H → ττ decay and ttH production are described by the Hff couplings of the H boson to fermions, with generally two couplings κ f andκ f , CP-even and CP-odd, respectively. Similarly, if the gluon coupling Hgg is dominated by the top quark loop, it can be described with the κ t andκ t parameters. However, since other heavy states may contribute to the loop, we consider the effective Hgg coupling using the more general parametrization given in Eq.
(2) instead of explicitly including the quark loop. In particular, the effective cross section fraction in gluon fusion becomes where the cross sections σ ggH 2 ¼ σ ggH 3 drop out from the equation following the coupling convention in Eq. (2).
Experimentally observable effects resulting from the above anomalous couplings are discussed in the next section. In this paper, anomalous HWW, HZZ, and HZγ couplings are considered in VBF and VH production, and anomalous Hgg couplings are considered in gluon fusion. Since CP-violating effects in electroweak (VBF and VH) and gluon fusion production modify the same kinematic distributions, both CP-sensitive parameters, f a3 and f ggH a3 , are left unconstrained simultaneously. It has been checked that CP violation in H → ττ decays would not affect these measurements. Under the assumption that the couplings are constant and real, the above formulation is equivalent to an effective Lagrangian notation. Therefore, in this paper, the real coupling constants are tested, and results are presented for the product of f ai and cosðϕ ai Þ, the latter being the sign of the real ratio of couplings a i =a 1 .
Following the formalism discussed in this section, simulated samples of H boson events produced via anomalous HVV couplings (VBF, VH, gluon fusion in association with two jets) are generated using JHUGEN. The associated production in gluon fusion with two jets is affected by anomalous interactions, while the kinematics of the production with zero or one jet are not affected. The latter events are generated with POWHEG2.0 [52][53][54][55], which is used for yield normalization of events selected with two jets and for the description of event distributions in categories of events where the correlation of the two jets is not important. For the kinematics relevant to this analysis in VBF and VH production, the effects that appear at nextto-leading order (NLO) in QCD are well approximated by the leading-order (LO) QCD matrix elements used in JHUGEN, combined with parton showering. The JHUGEN samples produced with the SM couplings are compared with the equivalent samples generated by the POWHEG event generator at NLO QCD, with parton showering applied in both cases, and the kinematic distributions are found to agree.
The PYTHIA8.212 [56] event generator is used to model the H boson decay to τ leptons and the decays of the τ leptons. Both scalar and pseudoscalar H → ττ decays and their interference have been modeled to confirm that the analysis does not depend on the decay model. The default samples are generated with the scalar hypothesis in decay. The PDFs used in the generators are NNPDF30 [57], with their precision matching that of the matrix elements. All MC samples are further processed through a dedicated simulation of the CMS detector based on GEANT4 [58].
To simulate processes with anomalous H boson couplings, for each type of anomalous coupling, we generate events with both the pure anomalous term and its interference with the SM contribution in the production HVV interaction. This allows extraction of the various coupling components and their interference. The MELA package, based on JHUGEN matrix elements, permits the application of weights to events in any sample to model any other HVV or Hff couplings with the same production mechanism. Reweighting enables one to increase the effective simulated event count by using all samples at once to describe any model, even if it has not been simulated. The MELA package also allows calculation of optimal discriminants for further analysis, as discussed in Sec. V.
Simulated samples for the modeling of background processes and of the H boson signal processes with SM couplings are the same as those used for the observation of the H boson decay to a pair of τ leptons [24]. All the corrections applied to samples are the same as in Ref. [24]. The MG5_aMC@NLO [59] generator is used for Z þ jets and W þ jets processes. They are simulated at LO with the MLM jet matching and merging [60]. The MG5_aMC@NLO generator is also used for diboson production simulated at NLO with the FxFx jet matching and merging [61], whereas POWHEG versions 2.0 and 1.0 are used for tt and single top quark production, respectively. The generators are interfaced with PYTHIA to model the parton showering and fragmentation. The PYTHIA parameters affecting the description of the underlying event are set to the CUETP8M1 tune [62].

V. DISCRIMINANT DISTRIBUTIONS
The full kinematic information for both production and decay of the H boson can be extracted from each event. This paper focuses on the production process, illustrated in Fig. 2. The techniques discussed below are similar to those used in earlier analyses by CMS, such as in Ref. [17].
Sensitivity to quantum numbers and anomalous couplings of the H boson is provided by the angular correlations between the two jets, the H boson, and the beam line direction in VBF, VH, or gluon fusion production with an additional two jets. A set of observables could be defined in VBF or VH production, such as ⃗ Ω ¼ fθ 1 ; θ 2 ; Φ; θ Ã ; Φ 1 ; q 2 1 ; q 2 2 g for the VBF or VH process with the angles illustrated in Fig. 2 and the q 2 1 and q 2 2 discussed in reference to Eq. (2), as described in detail in Ref. [28]. It is, however, a challenging task to perform an optimal analysis in a multidimensional space of observables. The MELA is designed to reduce the number of observables to the minimum while retaining all essential information for the purpose of a particular measurement. In this analysis, the background suppression is still provided by the observables defined in Ref. [24].
When the H boson and two associated jets are reconstructed, two types of discriminants can be used to optimally search for anomalous couplings. These two discriminants rely only on signal matrix elements and are well defined. One can apply the Neyman-Pearson lemma [63] to prove that the two discriminants constitute a minimal and complete set of optimal observables [28,29] for the measurement of the f ai parameter. One type of discriminant is designed to separate the process with anomalous couplings, denoted as BSM, from the SM signal process, where P is the probability for the signal VBF production process (either SM or BSM), calculated using the matrix element MELA package and is normalized so that the matrix elements give the same cross sections for either f ai ¼ 0 or 1 in the relevant phase space of each process. Such a normalization leads to an optimal population of events in the range between 0 and 1. The discriminants are denoted as D 0− , D 0hþ , D Λ1 , or D Zγ Λ1 , depending on the targeted anomalous coupling a 3 , a 2 , Λ 1 , or Λ Zγ 1 , respectively.
The second type of discriminant targets the contribution from interference between the SM and BSM processes, where P int SM−BSM is the probability distribution for interference of SM and BSM signals in VBF production. This discriminant is used only for the CP-odd amplitude analysis with f a3 and is denoted D CP in the rest of the paper. In the cases of f Λ1 and f Zγ Λ1 , the interference discriminants do not carry additional information because of their high correlation with the D Λ1 and D Zγ Λ1 discriminants. The f a2 interference discriminant is not used in this analysis either, as it only becomes important for measurements of smaller couplings than presently tested and because of the limited number of events available for background parametrization.
Kinematic distributions of associated particles in gluon fusion are also sensitive to the quantum numbers of the H boson and to anomalous Hgg couplings. A set of observables, ⃗ Ω, identical to those from the VBF process also describes this process. In this analysis, the focus is on the VBF-enhanced phase space in which the selection efficiency for the gluon fusion process is relatively small. Furthermore, the observables defined in Eqs. (5) and (6) for the VBF process are found to provide smaller separation between CP-even and CP-odd H boson couplings for gluon fusion production than MELA discriminants that would be dedicated to the gluon fusion process. Nonetheless, both parameters sensitive to CP violation, f a3 and f ggH a3 , are included in a simultaneous fit using the observables optimized for the VBF process to avoid any possible bias in the measurement of f a3 .
While the correlations between the two jets, the H boson, and the beam line provide primary information about CP violation and anomalous couplings in electroweak production (VBF and VH), even events with reduced kinematic information can facilitate this analysis. For example, in cases where both jets lie outside of the detector acceptance, the p T distribution of the H boson is different for SM and BSM production. This leads to different event populations across the three categories and to a different p T distribution of the H boson in the boosted category. For example, the fraction of signal events is much smaller in the 0-jet category, and the p T distribution is significantly harder in the boosted category for pseudoscalar H boson production than it is for the SM case. These effects are illustrated in Figs. 3-5. The same effects are, however, negligible in gluon fusion production, where both scalar and pseudoscalar Hgg couplings are generated by higher-dimension operators, which correspond to the a gg 2 and a gg 3 terms in Eq. (2).

FIG. 2. Illustrations of H boson production in qq
The H → ττ decay is shown without further illustrating the τ decay chain. Angles and invariant masses fully characterize the orientation of the production and two-body decay chain and are defined in suitable rest frames of the V and H bosons, except in the VBF case, where only the H boson rest frame is used [26,28].
Other observables, such as ΔΦ JJ [43], defined as the azimuthal difference between the two associated jets, have been suggested for the study of CP effects. While they do provide sensitivity to CP measurements, they are not as sensitive as the discriminant variables for VBF production used in this analysis. Nonetheless, as an alternative to the optimal VBF analysis with the MELA discriminants, we also performed a cross-check analysis where the ΔΦ JJ observable is used instead. It was verified that the expected precision on f a3 is indeed lower than in the optimal VBF analysis. On the other hand, the sensitivity of the ΔΦ JJ observable to the f ggH a3 parameter is better than that of the VBF discriminants, and it is close to but not as good as the optimal MELA observables targeting the gluon fusion topology in association with two jets. Both results are discussed in Sec. VII.

VI. ANALYSIS IMPLEMENTATION
Five anomalous HVV coupling parameters defined in Sec. IV are studied: f a3 , f a2 , f Λ1 , f Zγ Λ1 , and f ggH a3 describing anomalous couplings in VBF, VH, and gluon fusion production. The CP-sensitive parameters f a3 and f ggH a3 are studied jointly, while all other parameters are examined independently. Anomalous H boson couplings in other production mechanisms and in the H → ττ decay do not affect these measurements, as the distributions studied here are insensitive to such effects.
The data, represented by a set of observables ⃗ x, are used to set constraints on anomalous coupling parameters. In the case of the CP study, the coupling parameters are f a3 and ϕ a3 . We also consider the scalar anomalous couplings described by f a2 and ϕ a2 , f Λ1 and ϕ Λ1 , and f Zγ Λ1 and ϕ Zγ Λ1 .
Since only real couplings are considered, we fit for the products f a3 cosðϕ a3 Þ with cosðϕ a3

A. Observable distributions
Each event is described by its category k and the corresponding observables ⃗ x. In the 0-jet and boosted categories, which are dominated by the gluon fusion production mechanism, the observables are identical to those used in Ref.
[24], namely ⃗ x ¼ fm vis ; Mg in the eτ h and μτ h 0-jet categories, ⃗ x ¼ fm vis ; p μ T g in the eμ0-jet category, ⃗ x ¼ fm ττ g in the 0-jet τ h τ h category, and ⃗ x ¼ fm ττ ; p H T g in the boosted categories, where M is the τ h decay mode, p μ T is the transverse momentum of the muon, and p H T is the transverse momentum of the H boson. There are no dedicated observables sensitive to anomalous couplings in these categories, as it is not possible to construct them in the absence of a correlated jet pair. Nonetheless, distributions of events in the above observables and categories still differ between signal models with variation of anomalous couplings.
In Figs. 3 and 4, the distributions of m vis and m ττ are displayed for selected events in the 0-jet category, and the transverse momentum distribution of the H boson is shown for the boosted category. Anomalous couplings would result in higher transverse momentum of the H boson and, unlike SM production, would cause the events to preferentially populate the boosted category instead of the one with no jets in the final state. The observable m ττ is used in the τ h τ h decay channel and m vis in other channels in the 0-jet category. Two observables are used in the likelihood fit in the boosted category, m ττ and p H T .  Observed The contributions from BSM and SM yields in the boosted category are different in the τ h τ h and lτ h channels because of different trigger conditions and classification requirements. In Fig. 3, the contribution from the eμ channel is omitted because of its low sensitivity and different binning in the fit. The normalization of the predicted background distributions corresponds to the result of the likelihood fit described in Sec. VI B. In all production modes in Figs. 3 and 4, the H → ττ process is normalized to its best-fit signal strength and couplings and is shown as an open overlaid histogram. The background components labeled in the figures as "others" include events from diboson and single top quark production, as well as H boson decays to W boson pairs. The uncertainty band accounts for all sources of uncertainty. The SM prediction for the VBF H → ττ signal, multiplied by a factor 5000 (300) in Fig. 3 (4), is shown as a red open overlaid histogram. The black open overlaid histogram represents a BSM hypothesis for the VBF H → ττ signal, normalized to 5000 (300) times the predicted SM cross section in Fig. 3 (4).
In Figs. 5-9, the discriminant distributions in the VBF category are displayed. In the VBF category, either three or four observables are used in the likelihood fit: ⃗ x ¼ fm JJ ; m ττ ; D 0− ; D CP g are used to determine the f a3 parameter, ⃗ x ¼ fm JJ ; m ττ ; D 0hþ g are used to determine the f a2 parameter, ⃗ x ¼ fm JJ ; m ττ ; D Λ1 g are used to determine the f Λ1 parameter, and ⃗x ¼ fm JJ ;m ττ ;D Zγ Λ1 g are used to determine the f Zγ Λ1 parameter, as defined in Eqs. (5) and (6). In order to keep the background and signal templates sufficiently populated, a smaller number of bins is chosen for m JJ and m ττ compared to Ref. [24]. It was found that four bins in D 0− , D 0hþ , D Λ1 , and D Zγ Λ1 are sufficient for close-to-optimal performance. At the same time, we adopt two bins in D CP with D CP < 0 and D CP > 0. This choice does not lead to the need for additional bins in the templates, because all distributions except the CP-violating interference component are symmetric in D CP , and this symmetry is enforced in the templates. A forwardbackward asymmetry in D CP would be a clear indication of CP-sensitive effects and is present only in the signal interference template.

B. Likelihood parametrization
We perform an unbinned extended maximum likelihood fit [64] to the events split into several categories according to the three production topologies and four tau-lepton pair final states using the RooFit toolkit [65,66]. The probability density functions for signal P j;k sig ð⃗ xÞ and background P j;k bkg ð⃗ xÞ are binned templates and are defined for each production mechanism j in each category k. Each event is characterized by the discrete category k and up to four observables ⃗ x, depending on the category. For the VBF, VH, or gluon fusion production mechanisms, the signal probability density function is defined as where T j;k ai is the template probability of a pure anomalous coupling a i term and T j;k a1;ai describes the interference between the anomalous coupling and SM term a 1 , or SM term a ggH 2 in the case of gluon fusion. Here, f ai stands for either f a3 , f a2 , f Λ1 , f Zγ Λ1 , or f ggH a3 . Each term in Eq. (7) is extracted from a dedicated simulation.  in the VBF category. All four decay channels, eμ, eτ h , μτ h , and τ h τ h , are summed. The BSM hypothesis depends on the variable shown; it corresponds to f a3 cosðϕ a3 Þ ¼ 1 for the D 0− (upper left) distribution, the maximal mixing in VBF production ("BSM mix," corresponding to f a3 cosðϕ a3 Þ ¼ 0.013) for the D CP distribution (upper right), f a2 cosðϕ a2 Þ ¼ 1 for the D 0hþ distribution (middle left), f Λ1 cosðϕ Λ1 Þ ¼ 1 for the D Λ1 distribution (middle right), and f Zγ Λ1 cosðϕ Zγ Λ1 Þ ¼ 1 for the D Zγ Λ1 distribution (lower). The expected D CP distribution is always symmetric, unless a CP-violating effect is present in the signal.    The signal strength parameters μ V and μ f are introduced as two parameters of interest. They scale the yields in the VBF þ VH and gluon fusion production processes, respectively. They are defined such that for f ai ¼ 0 they are equal to the ratio of the measured to the expected cross sections for the full process, including the H → ττ decay. The likelihood is maximized with respect to the anomalous coupling f ai cosðϕ ai Þ and yield (μ V , μ f ) parameters and with respect to the nuisance parameters, which include the constrained parameters describing the systematic uncertainties. The f a3 cosðϕ a3 Þ and f ggH a3 cosðϕ ggH a3 Þ parameters are tested simultaneously, while all other f ai cosðϕ ai Þ parameters are tested independently. All parameters except the anomalous coupling parameter of interest f ai cosðϕ ai Þ are profiled. The confidence level (C.L.) intervals are determined from profile likelihood scans of the respective parameters. The allowed 68 and 95% C.L. intervals are defined using the profile likelihood function, −2Δ ln L ¼ 1.00 and 3.84, respectively, for which exact coverage is expected in the asymptotic limit [67]. Approximate coverage has been tested with generated samples.

C. Systematic uncertainties
A log-normal probability density function is assumed for the nuisance parameters that affect the event yields of the various background and signal contributions, whereas systematic uncertainties that affect the distributions are represented by nuisance parameters of which the variation results in a continuous perturbation of the spectrum [68] and which are assumed to have a Gaussian probability density function. The systematic uncertainties are identical to those detailed in Ref. [24]. They are summarized in the following.
The rate uncertainties in the identification, isolation, and trigger efficiencies of electrons and muons amount to 2%. For τ h , the uncertainty in the identification is 5% per τ h candidate, and the uncertainty related to the trigger amounts to an additional 5% per τ h candidate [39]. In the 0-jet category, where one of the dimensions of the two-dimensional fit is the reconstructed τ h decay mode, the relative reconstruction efficiency in a given τ h reconstructed decay mode has an uncertainty of 3% [24]. For muons and electrons misreconstructed as τ h candidates, the τ h identification leads to rate uncertainties of 25% and 12%, respectively [39]. This leads to the corresponding uncertainty in the rates of the Z → μμ and Z → ee backgrounds misidentified as the μτ h and eτ h final states, respectively. The requirement that there are no b-tagged jets in eμ decay channel events results in a rate uncertainty as large as 5% in the tt background [69].
The uncertainties in the energy scales of electrons and τ h leptons amount to 1.0%-2.5% and 1.2% [24,39], while the effect of the uncertainty in the muon energy scale is negligible. This uncertainty increases to 3.0% and 1.5%, respectively, for electrons and muons misidentified as τ h candidates [24]. For events where quark-or gluon-initiated jets are misidentified as τ h candidates, a linear uncertainty that increases by 20% per 100 GeV in transverse momentum of the τ h and amounts to 20% for a τ h with p T of 100 GeV is taken into account [24]. This uncertainty affects simulated events with jets misidentified as τ h candidates, from various processes like the Drell-Yan, tt, diboson, and W þ jets productions. Uncertainties in the jet and p miss T energy scales are determined event by event [70], and propagated to the observables used in the analysis.
The uncertainty in the integrated luminosity is 2.5% [71]. Per bin uncertainties in the template probability parametrization related to the finite number of simulated events, or to the limited number of events in data control regions, are also taken into account [68].
The rate and acceptance uncertainties for the signal processes related to the theoretical calculations are due to uncertainties in the PDFs, variations of the renormalization and factorization scales, and uncertainties in the modeling of parton showers. The magnitude of the rate uncertainty depends on the production process and on the event category. In particular, the inclusive uncertainty related to the PDFs amounts to 2.1% for the VBF production mode [72], while the corresponding uncertainty for the variation of the renormalization and factorization scales is 0.4% [72]. The acceptance uncertainties related to the particular selection criteria used in this analysis are less than 1% for all production modes. The theoretical uncertainty in the branching fraction of the H boson to τ leptons is 2.1% [72].
An overall rate uncertainty of 3%-10% affects the Z → ττ background, depending on the category, as estimated from a control region enriched in Z → μμ events. In the VBF category, this process is also affected by a shape uncertainty that depends on m JJ and ΔΦ JJ and can reach a magnitude of 20%. In addition to the uncertainties related to the W þ jets control regions in the eτ h and μτ h final states, the W þ jets background is affected by a rate uncertainty ranging between 5% and 10% to account for the extrapolation of the constraints from the high-m T to the low-m T regions. In the eμ and τ h τ h final states, the rate uncertainties in the W þ jets background yields are 20% and 4%, respectively.
The uncertainty in the QCD multijet background yield in the eμ decay channel ranges from 10% to 20%, depending on the category. In the eτ h and μτ h decay channels, uncertainties derived from the control regions are considered for the QCD multijet background, together with an additional 20% uncertainty that accounts for the extrapolation from the relaxed-isolation control region to the isolated signal region. In the τ h τ h decay channel, the uncertainty in the QCD multijet background yield is a combination of the uncertainties obtained from fitting the dedicated control regions with τ h candidates passing relaxed isolation criteria, of the extrapolation to the signal region ranging from 3% to 15%, and of residual differences between prediction and data in signal-free regions with various loose isolation criteria.
The uncertainty from the fit in the tt control region results in an uncertainty of about 5% on the tt cross section in the signal region. The combined systematic uncertainty in the background yield arising from diboson and single top quark production processes is taken to be 5% [73,74].
The additional D 0− , D 0hþ , D Λ1 , and D Zγ Λ1 observables do not change the procedure for estimating the systematic uncertainty, as any mismodeling due to detector effects is estimated with the same procedure as for any other distribution. None of the systematic uncertainties   introduces asymmetry in the D CP distributions which remain symmetric, except for the antisymmetric signal interference contribution.

VII. RESULTS
The four sets of f ai and ϕ ai parameters describing anomalous HVV couplings, as defined in Eqs. (2) and (3), are tested against the data according to the probability density defined in Eq. (7). The results of the likelihood scans are shown in Fig. 10 and listed in Table II. In each fit, the values of the other anomalous coupling parameters are set to zero. In the case of the CP fit, the f a3 parameter is measured simultaneously with f ggH a3 , as defined in Eq. (4). All other parameters, including the signal strength parameters μ V and μ f , are profiled. The results are presented for the product of f ai and cosðϕ ai Þ, the latter being the sign of the real a i =a 1 ratio of couplings. In this approach, the f ai parameter is constrained to be in the physical range f ai ≥ 0. Therefore, in the SM, it is likely for the best-fit value to be at the physical boundary f ai ¼ 0 for both signs of the a i =a 1 ratio.   The constraints on f ai cosðϕ ai Þ appear relatively tight compared to similar constraints utilizing the H boson decay information, e.g., in Ref. [17]. This is because the cross section in VBF and VH production increases quickly with f ai . The definition of f ai in Eq. (3) uses the cross section ratios defined in the H → 2e2μ decay as the common convention across various measurements. Because the cross section increases with respect to f ai at different rates for production and decay, relatively small values of f ai correspond to a substantial anomalous contribution to the production cross section. This leads to the plateau in the −2 lnðL=L max Þ distributions for larger values of f ai cosðϕ ai Þ in Fig. 10. If we had used the cross section ratios for VBF production in the f ai definition in Eq. (3), the appearance of the plateau and the narrow exclusion range would change. For example, the 68% C.L. upper constraint on f a3 cosðϕ a3 Þ < 0.00093 is dominated by the VBF production information. If we were to use the VBF cross section ratio σ VBF The observed maximum value of −2 lnðL=L max Þ is somewhat different from expectation and between the four analyses, mostly due to statistical fluctuations in the distribution of events across the dedicated discriminants and other observables, leading to different significances of the observed signal driven by VBF and VH production. In particular, the best-fit values for ðμ V ; μ f Þ in the four analyses, under the assumption that f ai ¼ 0, are ð0. 55  This results in a somewhat lower yield of VBF and VH events observed in the first two cases, leading to lower confidence levels in constraints on f a3 cosðϕ a3 Þ and f a2 cosðϕ a2 Þ.
In the f a3 analysis, a simultaneous measurement of f a3 and f ggH a3 is performed. These are the parameters sensitive to CP in the VBF and gluon fusion processes, respectively. Both the observed and expected exclusions from the null hypothesis for any BSM gluon fusion scenario with either MELA or the ΔΦ JJ observable are below one standard deviation.

VIII. COMBINATION OF RESULTS WITH OTHER CHANNELS
The precision of the coupling measurements can be improved by combining the results in the H → ττ channel, presented here, with those of other H boson decay channels. A combination is possible only with those channels where anomalous couplings in the VH, VBF, and gluon fusion processes are taken into account in the fit in a consistent way. If it is not done, the kinematics of the associated jets and of the H boson would not be modeled correctly for BSM values of the f ai or f ggH a3 parameters.
In the example of the CP fit, in the stand-alone fit with the H → ττ channel, the parameters of interest are f a3 cosðϕ a3 Þ, f ggH a3 cosðϕ ggH a3 Þ, μ Hττ V , and μ Hττ f . When reporting one parameter, all other parameters are profiled. In a combined fit of the H → ττ and H → VV channels, such as in Ref. [17], in principle there are four signal strength parameters in the two channels (μ Hττ V , μ Hττ f , μ HVV V , μ HVV f ). However, this can be reduced to three parameters because the ratio between the VBF þ VH and gluon fusion cross sections is expected to be the same in each of the two channels, that is μ Hττ V =μ Hττ . Therefore, the three signal strength parameters are chosen as μ V , μ f , and η τ , where the last one is the relative strength of the H boson coupling to the τ leptons. We should note that, as discussed earlier, the HWW couplings are analyzed together with the HZZ couplings assuming a ZZ i ¼ a WW i . The results can be reinterpreted for a different assumption of the a ZZ i =a WW i ratio [17]. In the combined likelihood fit, all common systematic uncertainties are correlated between the channels, both theoretical uncertainties, such as those due to the PDFs, and experimental uncertainties, such as jet energy calibration. The results using the H → ττ decay are combined with those presented in Ref.
[17] using the on-shell H → 4l decay. The latter employs results from Run 1 (from 2011 and 2012) and Run 2 (from 2015, 2016, and 2017) with data corresponding to integrated luminosities of 5.1, 19.7, and 80.2 fb −1 at center-of-mass energies 7, 8, and 13 TeV, respectively. In this analysis, information about HVV anomalous couplings both in VBF þ VH production and in H → VV → 4l decay is used. In all cases, the signal strength parameters are profiled, and the parameters common to the two analyses are correlated. The combined 68% C.L. and 95% C.L. intervals are presented in Table III, and the likelihood scans are shown in Fig. 11. While the constraints at large values of f ai are predominantly driven by the decay information in the H → VV analysis, the constraints in the narrow range of f ai near 0 are dominated by the production information where the H → ττ channel (2), as shown in Table IV.

IX. CONCLUSIONS
A study is presented of anomalous HVV interactions of the H boson with vector bosons V, including CP violation, using its associated production with two hadronic jets in vector boson fusion, in the VH process, and in gluon fusion, and subsequently decaying to a pair of τ leptons. Constraints on the CP-violating parameter f a3 cosðϕ a3 Þ and on the CP-conserving parameters f a2 cosðϕ a2 Þ, f Λ1 cosðϕ Λ1 Þ, and f Zγ Λ1 cosðϕ Zγ Λ1 Þ, defined in Eqs. (2) and (3), are set using matrix element techniques. The observed and expected limits on the parameters are summarized in Table II. The 68% confidence level constraints are generally tighter than those from previous measurements using either production or decay information. Further constraints are obtained in the combination of the H → ττ and H → 4l decay [17] channels and are summarized in Table III. This combination places the most stringent constraints on anomalous H boson couplings: f a3 cosðϕ a3 Þ ¼ ð0.00 AE 0.27Þ × 10 −3 , f a2 cosðϕ a2 Þ ¼ ð0.08 þ1.04 −0.21 Þ × 10 −3 , f Λ1 cosðϕ Λ1 Þ ¼ ð0.00 þ0.53 −0.09 Þ × 10 −3 , and f Zγ Λ1 cosðϕ Zγ Λ1 Þ ¼ ð0.0 þ1.1 −1.3 Þ × 10 −3 . A simultaneous measurement of f a3 cosðϕ a3 Þ and f ggH a3 cosðϕ ggH a3 Þ parameters is performed, where the latter parameter, defined in Eqs. (2) and (4), is sensitive to CP-violation effects in the gluon fusion process. The current dataset does not allow for precise constraints on CP properties in the gluon fusion process. The results are consistent with expectations for the standard model H boson.

ACKNOWLEDGMENTS
We thank Markus Schulze for optimizing the JHUGEN Monte Carlo simulation program and matrix element library for this analysis. We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMBWF and FWF (  IV. Summary of the allowed 95% C.L. intervals for the anomalous HVV couplings using the results in Table III. The coupling ratios are assumed to be real and include the factor cosðϕ Λ1 Þ or cosðϕ Zγ Λ1 Þ ¼ AE1.  research Grants No. 123842, No. 123959, No. 124845, No. 124850, No. 125105, No. 128713, No. 128786, and No. 129058