New features in the JHU generator framework: constraining Higgs boson properties from on-shell and off-shell production
Andrei V. Gritsan, Jeffrey Roskes, Ulascan Sarica, Markus Schulze, Meng Xiao, Yaofu Zhou
HHU-EP-20/02
New features in the JHU generator framework: Constraining Higgs boson propertiesfrom on-shell and off-shell production
Andrei V. Gritsan, Jeffrey Roskes, Ulascan Sarica,
1, 2
Markus Schulze, Meng Xiao,
1, 4 and Yaofu Zhou
1, 5 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA Department of Physics, University of California, Santa Barbara, CA 93106, USA Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, D-12489 Berlin, Germany Zhejiang Institute of Modern Physics, Department of Physics,Zhejiang University, Hangzhou, 310027, P. R. China Department of Physics, Missouri University of Science and Technology, Rolla, MO 65409, USA (Dated: February 21, 2020)We present an extension of the JHUGen and MELA framework, which includes an event gen-erator and library for the matrix element analysis. It enables simulation, optimal discrimination,reweighting techniques, and analysis of a bosonic resonance and the triple and quartic gauge bosoninteractions with the most general anomalous couplings. The new features, which become especiallyrelevant at the current stage of LHC data taking, are the simulation of gluon fusion and vectorboson fusion in the off-shell region, associated ZH production at NLO QCD including the gg initialstate, and the simulation of a second spin-zero resonance. We also quote translations of the anoma-lous coupling measurements into constraints on dimension-six operators of an effective field theory.Some of the new features are illustrated with projections for experimental measurements with thefull LHC and HL-LHC datasets. PACS numbers: 12.60.-i, 13.88.+e, 14.80.Bn
I. INTRODUCTION
We present a coherent framework for the measurement of couplings of the Higgs ( H ) boson and a possible secondspin-zero resonance. Our framework includes a Monte Carlo generator and matrix element techniques for optimalanalysis of the data. We build upon the earlier developed framework of the JHU generator and MELA analysis pack-age [1–4] and extensively use matrix elements provided by MCFM [5–9]. Thanks to the transparent implementationof standard model (SM) processes in MCFM, we extend them to add the most general scalar and gauge couplingsand possible additional states. This allows us to build on the previously studied topics [1–4, 10–58] and present phe-nomenological results in a unified approach. This framework includes many options for production and decay of the H boson. Here we consider gluon fusion (ggH), vector boson fusion (VBF), and associated production with a vectorboson ( V H ) in both on-shell H and off-shell H ∗ production, with decays to two vector bosons. In the off-shell case,interference with background processes is included. Additional heavy particles in the gluon fusion loop and a secondresonance interfering with the SM processes are also considered. In the V H process, we include next-to-leading orderQCD corrections, as well as the gluon fusion process for ZH . The processes with direct sensitivity to fermion Hf ¯ f couplings, such as t ¯ tH , b ¯ bH , tqH , or H → τ + τ − , are discussed in Ref. [4].In an earlier version of our framework, we focused mostly on the Run-I targets and their possible extensions asdocumented in Refs. [1–3]. It was adopted in Run-I analyses using Large Hadron Collider (LHC) data [59–70]. Somenew features in this framework have been reported earlier [41] and have been used for LHC experimental analyses.Most notably, this framework was employed in recent Run-II measurements of the HV V anomalous couplings fromthe first joint analysis of on-shell production and decay [71, 72], from the first joint analysis of on-shell and off-shell H boson production [73], for the first measurement of the CP structure of the Yukawa interaction between the H bosonand top quark [74], in the search for a second resonance in interference with the continuum background [75, 76], andin projections to future on-shell and off-shell H boson measurements at the High Luminosity (HL) LHC [77]. In thispaper, we document, review, and highlight the new features critical for exploring the full Run-II dataset at the LHCand preparing for Run-III and the HL-LHC. We also broaden the theoretical underpinning, allowing interpretation interms of either anomalous couplings or an effective field theory (EFT) framework.Both Run-I and Run-II of the LHC have provided a large amount of data on H boson properties and its interactionswith other SM particles, as analyzed by the ATLAS and CMS experiments. The H boson has been observed in a r X i v : . [ h e p - ph ] J a n (a) HV V (b) Hf ¯ f (c) HV f ¯ f (d) Hf ¯ ff ¯ f FIG. 1: Vertices relevant for
HV V and Hf ¯ f interactions.all accessible production channels, gluon fusion, weak vector boson fusion, V H associated production, and top-quark associated production [78–84], and its production strength is consistent with the SM prediction within theuncertainties [41]. Also its decay channels into gauge bosons (
ZZ, W W, γγ ) have been observed and do not showsignificant deviations within the uncertainties [78–80]. The fermionic interactions have been established for the thirdgeneration quarks ( t, b ) and the τ lepton [81–86], and so far, they are consistent with the SM within the uncertainties.While this picture shows that Nature does not radically deviate from the SM dynamics, it should be noted thatmany generic extensions of the SM predict deviations only below the current precision. Open questions remain, forexample about CP-odd mixtures, the Yukawa coupling hierarchy, and other states involved in electroweak symmetrybreaking. These questions can be addressed in the years to come by fully utilizing the existing and upcoming LHCdata sets. In particular, the study of kinematic tails of distributions involving the H boson is becoming accessiblefor the first time. These signals involve off-shell H boson production and strong interference effects with irreduciblebackgrounds that are subject to the electroweak unitarization mechanism in the SM. This feature turns the kinematictails into particularly sensitive probes of the mechanism of electroweak symmetry breaking and possible extensionsbeyond the SM. Moreover, the study of electroweak production of the H boson (VBF and V H ) is probing
HV V interactions over a large range of momentum transfer, which can expose possible new particles that couple throughloops. Even the direct production of new resonances will first show up as deviations from the expected high-energytail of kinematic distributions. Hence, analyzing these newly accessible features in off-shell H boson production is ofparamount importance to understand electroweak symmetry breaking in the SM and possible extensions involving newparticles. In the following, we review the framework and demonstrate its capabilities through examples of possibleanalyses. The technical details of the framework are described in the manual, which can be downloaded at [87],together with the source code. II. PARAMETERIZATION OF ANOMALOUS INTERACTIONSA. H boson interactions We present our parameterization of anomalous couplings relevant for on-shell and off-shell H boson production anddecay. Following the notation of Refs. [1–3], the HV V scattering amplitude of a spin-zero boson H and two vectorbosons V V with polarization vectors and momenta ε µ , q and ε µ , q , as illustrated in Fig. 1(a), is parameterized by A ( HV V ) = 1 v (cid:26) M V (cid:18) g V V + κ V V q + κ V V q (cid:0) Λ V V (cid:1) + κ V V ( q + q ) (cid:16) Λ V VQ (cid:17) + 2 q · q M V g V V (cid:19) ( ε · ε ) − g V V ( ε · q )( ε · q ) − g V V ε ε ε q q (cid:27) , (1)where M V is the vector boson’s pole mass, v is the SM Higgs field vacuum expectation value, and g V V , , , κ V V , / (Λ V V ) ,and κ V V / (Λ V VQ ) are coupling constants to be measured from data. This parametrization represents the most generalLorentz-invariant form.At tree level in the SM, only the CP-even HZZ and
HW W interactions contribute via g ZZ = g W W = 2. The loop-induced interactions of HZγ , Hγγ , and
Hgg contribute effectively via the CP-even g V V terms and are parametericallysuppressed by α or α s . The CP-violating couplings g V V are generated only at three-loop level in the SM and aretherefore tiny. Beyond the SM, all of these couplings can receive additional contributions, which do not necessarilyhave to be small. For example, the Hgg interaction can be parameterized through a fermion loop, as discussed laterin application to Eq. (37). The fermions in the loop interact with the H boson as illustrated in Fig. 1(b), with thecouplings κ f and ˜ κ f and the amplitude A ( Hf ¯ f ) = − m f v ¯ ψ f ( κ f + i ˜ κ f γ ) ψ f , (2)where ¯ ψ f and ψ f are the Dirac spinors and m f is the fermion mass. One may equivalently choose to express thecouplings through a Lagrangian (up to an unphysical global phase) L hff = − m f v ¯ ψ f ( κ f + i ˜ κ f γ ) ψ f h , (3)which allows a connection to be made between the couplings κ f and ˜ κ f and anomalous operators in an effective fieldtheory. In the SM, the dominant contribution to gluon fusion comes from a top quark loop with ( κ t , ˜ κ t ) = (1 , κ V Vi / (Λ V Vi ) in Eq. (1) are introduced to allow for additional momentum dependence. Below, we alsoshow that these terms can be reinterpreted as the contact interactions shown in Figs. 1(c) and 1(d). By symmetry wehave κ ZZ = κ ZZ , but we do not enforce κ W W = κ W W for W ± bosons. Note that κ γγ = κ γγ = κ gg = κ gg = κ Zγ = 0,while κ γZ = κ Zγ may contribute [66]. The coupling κ V V / (Λ V VQ ) allows for scenarios which violate the gaugesymmetries of the SM.For the Hgg couplings entering the gluon fusion process we also consider the full one-loop dependence instead ofthe effective g gg , couplings in Eq. (1). This feature is important for correctly describing off-shell Higgs productionand additional broad, heavy resonances, where the q -dependence of the interaction cannot be approximated as aconstant coupling. In addition to the closed quark loop with explicit dependence on the bottom and top quarkmasses, we allow for the insertion of fourth generation b (cid:48) and t (cid:48) quarks into the loop.If a gauge boson in Eq. (1) is coupled to a light fermion current, we replace its polarization vectors by ε µi ( q i ) → j µi = e ¯ ψ f (cid:48) γ µ (cid:16) g V f (cid:48) f L ω L + g V f (cid:48) f R ω R (cid:17) ψ f q i − M V + i M V Γ V , (4)where e is the electron electric charge, Γ V is the gauge boson’s width, ω L , R are the left- and right-handed chiralityprojectors, and the g V f (cid:48) f L , R are the corresponding couplings of the gauge boson V to fermions. We also allow forexchanges of additional spin-1 bosons V (cid:48) between the H boson and the fermions. Hence, we add ε µi ( q i ) → j µi − ¯ ψ f (cid:48) γ µ (cid:16) e V (cid:48) f (cid:48) f L ω L + e V (cid:48) f (cid:48) f R ω R (cid:17) ψ f q i − M V (cid:48) + i M V (cid:48) Γ V (cid:48) , (5)with the chirality and flavor dependent couplings e V (cid:48) f (cid:48) f L , R . In this approach, we allow for flavor changing interactions( f (cid:48) (cid:54) = f ) in both the neutral and charged V (cid:48) interactions. In the case where the V (cid:48) boson is very heavy, the limit M V (cid:48) /q i → ∞ yields the contact interaction ε µi ( q i ) → j µi + 1 M V (cid:48) ¯ ψ f (cid:48) γ µ (cid:16) e V (cid:48) f (cid:48) f L ω L + e V (cid:48) f (cid:48) f R ω R (cid:17) ψ f (6)in Fig. 1(c-d). We note that these contact terms and new V (cid:48) states are not the primary interest in this studybecause their existence would become evident in resonance searches and in electroweak measurements, without theneed for H boson production. Moreover, the HZf ¯ f contact terms are equivalent to the already constrained κ ZZ , and κ Zγ terms [31, 33] if coupling flavor universality is assumed. Under the approximation that the Z boson has anarrow width, this correspondence, given in Eq. (28), only involves real couplings. For example, in the limit whereΓ Z (cid:28) M Z , a nonzero κ ZZ / (Λ ZZ ) in Eq. (1) is equivalent to shifting g ZZ → g ZZ + 2 κ ZZ ( M Z (cid:14) Λ ZZ ) and activatinga contact interaction g ZZ (cid:48) = κ ZZ ( M Z (cid:48) (cid:14) Λ ZZ ) , e Z (cid:48) f (cid:48) fλ = e g Zf (cid:48) fλ .The parameterization of the amplitude in Eq. (1) can be related to a fundamental Lagrange density function. Here,we closely follow the so-called Higgs basis of Ref. [41], which is based on an effective field theory expansion up todimension six. The relevant SU(3) × SU(2) × U(1) invariant Lagrangian for H boson interactions with gauge bosons(in the mass eigenstate parameterization) reads L hvv = hv (cid:20) (1 + δc z ) ( g + g (cid:48) ) v Z µ Z µ + c zz g + g (cid:48) Z µν Z µν + c z (cid:3) g Z µ ∂ ν Z µν + ˜ c zz g + g (cid:48) Z µν ˜ Z µν + (1 + δc w ) g v W + µ W − µ + c ww g W + µν W − µν + c w (cid:3) g (cid:0) W − µ ∂ ν W + µν + h .c. (cid:1) + ˜ c ww g W + µν ˜ W − µν + c zγ e (cid:112) g + g (cid:48) Z µν A µν + ˜ c zγ e (cid:112) g + g (cid:48) Z µν ˜ A µν + c γ (cid:3) gg (cid:48) Z µ ∂ ν A µν + c γγ e A µν A µν + ˜ c γγ e A µν ˜ A µν + c gg g s G aµν G aµν + ˜ c gg g s G aµν ˜ G aµν (cid:21) , (7)in accordance with Eq. (II.2.20) in Ref. [41] . The fields and real-valued couplings, as well as the correspondingdimension-six operators, are defined in Ref. [41]; for example, g + g (cid:48) = e / ( s w c w ) , e = 4 πα and g s = 4 πα s . Wenote that when restricting the discussion to the dimension-six effective field theory (see Eq. (II.2.38) in Ref. [41]),Eq. (7) is parameterized by ten real degrees of freedom, so not all of the coefficients are independent. For example, thecoefficients δc w , c ww , ˜ c ww , c w (cid:3) , and c γ (cid:3) can be expressed through linear combinations of the other couplings. Theredundancy was introduced intentionally in Ref. [41] for easier connections to observable quantities in Higgs physics.The generality of our amplitude parameterization allows us to uniquely represent each EFT coefficient in Eq. (7)by an anomalous coupling in Eq. (1). Limiting our couplings to real-valued numbers, we find δc z = 12 g ZZ − , c zz = − s w c w e g ZZ , c z (cid:3) = M Z s w e κ ZZ (Λ ZZ ) , ˜ c zz = − s w c w e g ZZ ,δc w = 12 g W W − , c ww = − s w e g W W , c w (cid:3) = M W s w e κ W W (Λ W W ) , ˜ c ww = − s w e g W W ,c zγ = − s w c w e g Zγ , ˜ c zγ = − s w c w e g Zγ , c γ (cid:3) = s w c w e M Z (Λ Zγ ) κ Zγ ,c γγ = − e g γγ , ˜ c γγ = − e g γγ , c gg = − g s g gg , ˜ c gg = − g s g gg . (8)The Lagrangian for SM HV V interactions is retained by setting δc z = δc w = 0 and all other c i = 0. Hence, only theCP-even HZZ and
HW W interactions remain at tree level.Not every anomalous coupling in Eq. (1) has a corresponding term in the EFT Lagrangian of Eq. (7). Forexample, the gauge invariance violating term κ V V / (Λ V VQ ) has no correspondence because L hvv is gauge invariantby construction. Similarly, charge symmetry in L hvv enforces κ W W = κ W W , which does not necessarily have tobe true in our amplitude setting. For a unique comparison at the level of dimension-six interactions, the abovementioned dependencies amongst EFT coefficients also have to be enforced in the amplitude parameterization ofEq. (7). We quote these relations later in Section II C. Correspondences to other EFT bases are obviously possible.As an illustration, we quote relationships of the CP violating couplings to the Warsaw basis [88] in Appendix A.The dimension-six Lagrangian for HV f ¯ f contact interactions (cfg. Eq. (II.2.24) in Ref. [41]) reads L hvff = 2 e hv (cid:40) W + µ √ s w (cid:16) ¯ u L γ µ δg hW q L d L + ¯ u R γ µ δg hW q R d R + ¯ ν L γ µ δg hW (cid:96) L e L (cid:17) (9)+ W − µ √ s w (cid:16) ¯ d L γ µ δg hW q L u L + ¯ d R γ µ δg hW q R u R + ¯ e L γ µ δg hW (cid:96) L ν L (cid:17) + Z µ s w c w (cid:18) (cid:88) f = u,d,e,ν ¯ f L γ µ δg hZf L f L + (cid:88) f = u,d,e ¯ f R γ µ δg hZf R f R (cid:19)(cid:41) . We note that the so-called Higgs basis is based on a set of Lagrangians for Higgs physics that do not contain the whole SM. Hence, itis not a complete operator basis in the strict mathematical sense. In this work, however, all contributions have direct relations to theWarsaw basis, which fulfills the requirements of a complete basis. (a)
V V V (b)
V V V V
FIG. 2: Gauge boson self interactions related to the
HV V vertices.It contributes to the amplitude shown in Fig. 1(c). A relationship to our framework with anomalous couplings canbe obtained in the limit M V (cid:48) (cid:14) q i → ∞ . It is given by δg hW fλ = M W M W (cid:48) √ s w e g W W e W (cid:48) f (cid:48) fλ , δg hZfλ = M Z M Z (cid:48) s w c w e g ZZ e Z (cid:48) ffλ , (10)where λ = L , R. Similar to the above, the coefficients δg hV fλ are not independent couplings and can be expressedthrough other coefficients of the dimension-six effective field theory. B. Gauge boson self-interactions
In studying off-shell H boson production, some Feynman diagrams not involving an H boson also contribute. Inparticular, the processes gg → W W and qq (cid:48) → qq (cid:48) + W W/ZZ/Zγ ∗ /γ ∗ γ ∗ ( → f ) involve diagrams with triple andquartic gauge boson self couplings, shown in Fig. 2, instead of an H boson vertex. Since there is an intricate interplaybetween gauge boson self couplings and H boson gauge couplings (which guarantees unitarity of the cross section athigh energies), we also consider gauge boson self couplings in our study. Their parameterization reads A ( V W + W − ) = ( − e ) d V W W (cid:26) ( ε V · ε + )( q V · ε − ) + ( ε + · ε − )( q V · ε V ) + ( ε V · ε − )( q V · ε + ) + d V ε ε V ε + ε − p (cid:27) , (11) A ( V V W + W − ) = (+ e ) d V V W W (cid:26) ( ε · ε + )( ε · ε − ) + ( ε · ε − )( ε · ε + ) − ε · ε )( ε + · ε − ) (cid:27) , (12)where q Vij = d Vi p i − d Vj p j is the relative momentum transfer. We fix d γW W = 1 and d ZW W = c w /s w per conventionand allow all other couplings to vary. In the SM, their values are d V = d V = d V = 1 , d V = 0 , (13) d γγW W = 1 , d ZγW W = c w s w , d ZZW W = c w s w , d W W W W = 12 s w . Extensions of the gauge sector of the SM lead to modifications of the above couplings. For example, the CP-violatingterm d V in Eq. (13) can be non-zero. The relevant contributions of the dimension-six Lagrangian for the triple andquartic gauge boson self-interactions are (see Eqs. (3.12, 3.14, 3.15) in Ref. [89]) L tgc = i e (cid:0) W + µν W − µ − W − µν W + µ (cid:1) A ν + i e (cid:104) (1 + δκ γ ) A µν W + µ W − ν + ˜ κ γ ˜ A µν W + µ W − ν (cid:105) + i e c w s w (cid:104) (1 + δg ,z ) (cid:0) W + µν W − µ − W − µν W + µ (cid:1) Z ν + (1 + δκ z ) Z µν W + µ W − ν + ˜ κ z ˜ Z µν W + µ W − ν (cid:105) , (14) L qgc = e ( W + µ A µ W − ν A ν − W + µ W − µ A ν A ν ) + e s w (1 + 2 c w δg ,z )( W + µ W + µ W − ν W − ν − W + µ W − µ W + ν W − ν )+ e c w s w (1 + 2 δg ,z )( W + µ Z µ W − ν Z ν − W + µ W − µ Z ν Z ν )+ e c w s w (1 + δg ,z )( W + µ Z µ W − ν A ν + W + µ A µ W − ν Z ν − W + µ W − µ Z ν A ν ) . (15)The anomalous coefficients in Eqs. (14,15) are related to couplings in Eqs. (11,12) by δκ γ = 12 ( d γ − , ˜ κ γ = 12 d γ , δκ z = 12 (cid:0) d Z − (cid:1) , ˜ κ z = 12 d Z ,δg ,z = d Z − d Z − (cid:18) s w c w d ZZW W − (cid:19) = s w c w d ZγW W − . (16)Similar to the case of L hvv , not all coefficients in Eqs. (14,15) are independent in the effective field theory framework,and we discuss their dependence in the next subsection. Moreover, additional anomalous triple and quartic contri-butions, the λ γ,Z , ˜ λ γ,Z terms in Ref. [41, 89], can arise. These additional terms are unrelated to any of the H bosoncontributions, and therefore, we do not consider them here. C. Coupling relations
In the previous subsections we related our anomalous couplings to the effective field theory coefficients of theso-called Higgs basis [41]. As mentioned above, not all of the EFT coefficients are independent when limiting the dis-cussion to dimension-six interactions . The linear relations for the dependent coefficients can be found in Ref. [41] andthey translate into relations amongst our anomalous couplings. Enforcing these relations allows a unique comparisonbetween the two frameworks, based on a minimal set of degrees of freedom. We find for the HV V interactions g W W = g ZZ + ∆ M W M W , (17) g W W = c w g ZZ + s w g γγ + 2 s w c w g Zγ , (18) g W W = c w g ZZ + s w g γγ + 2 s w c w g Zγ , (19) κ W W (Λ W W ) ( c w − s w ) = κ ZZ (Λ ZZ ) + 2 s w g γγ − g ZZ M Z + 2 s w c w ( c w − s w ) g Zγ M Z , (20) κ Zγ (Λ Zγ ) ( c w − s w ) = 2 s w c w (cid:18) κ ZZ (Λ ZZ ) + g γγ − g ZZ M Z (cid:19) + 2( c w − s w ) g Zγ M Z . (21)The term ∆ M W in Eq. (17) induces a shift in the W boson mass. Given that M W is experimentally measured tohigh precision one can assume ∆ M W ≈
0. The couplings e V (cid:48) f (cid:48) fλ for HV f ¯ f contact interactions in Eq. (6) are equalto the corresponding V ¯ f f couplings g V f (cid:48) fλ in the SM. Therefore, one can often neglect them as they are stronglyconstrained by electroweak precision measurements. The gauge boson self couplings in Eqs. (11-13) are determinedby HV V couplings in Eq. (1) through d γ = 1 + ( g γγ − g ZZ ) c w + g Zγ (cid:18) c w s w − s w c w (cid:19) , (22) d γ = ( g γγ − g ZZ ) c w + g Zγ (cid:18) c w s w − s w c w (cid:19) , (23) d Z = 1 − s w c w c w − s w (cid:0) g γγ − g ZZ (cid:1) − s w c w g Zγ − M Z c w − s w ) κ ZZ (Λ ZZ ) , (24) d Z = d Z = 1 − s w c w − s w (cid:0) g γγ − g ZZ (cid:1) − s w c w g Zγ − M Z c w − s w ) κ ZZ (Λ ZZ ) , (25) d Z = − s w c w d γ , (26) d ZZW W = c w s w (cid:0) d Z − (cid:1) , d ZγW W = c w s w d Z . (27) It should be noted that contributions of dimension-eight can invalidate the relations. See the comments in Section II.2.1.d of Ref. [41].
D. Correspondence to a Pseudo Observable framework
Here we briefly quote relations between our parameterization and the so-called Pseudo Observable framework [31].Similar to our work, the Pseudo Observables are derived from on-shell amplitudes. For the H → ZZ/Zγ ∗ /γ ∗ γ ∗ → (cid:96) amplitude we find the relations κ ZZ = 12 g ZZ + M Z − i M Z Γ Z (Λ ZZ ) κ ZZ , ε ZZ = g ZZ , ε CP ZZ = g ZZ ,ε γγ = g γγ , ε CP γγ = g γγ , ε Zγ = − g Zγ , ε CP Zγ = − g Zγ ,ε Zf λ = M Z − i M Z Γ Z ZZ ) κ ZZ eg Zffλ − M Z − i M Z Γ Z Zγ ) κ Zγ eQ f , (28)for the couplings given in Eqs. (9–11) and Eqs. (20–21) of Ref. [31]. Similarly, the relations for the H → W + W − → (cid:96) ν amplitude read κ W W = 12 g W W + M W − i M W Γ W W W ) ( κ W W + κ W W ) , ε W W = g W W , ε CP W W = g W W ,ε ∗ W (cid:96) λ = M W − i M W Γ W W W ) κ W W eg W (cid:96)νλ , ε
W (cid:96) (cid:48) λ = M W − i M W Γ W W W ) κ W W eg W (cid:96) (cid:48) ν (cid:48) λ . (29)Note that the imaginary terms in these relations are proportional to Γ V /M V , so that in the limit Γ V (cid:28) M V , realcouplings in one framework translate to real couplings in the other. The g V f (cid:48) fλ are the chiral couplings of fermions togauge bosons in Eq. (4). Similar to the effective field theory framework, the κ V V / (Λ V VQ ) term in Eq. (1) does nothave a counter piece in the Pseudo Observable framework. For all other couplings, there is a unique correspondenceto our parameterization in Eq. (4). Gauge boson self couplings can also be incorporated in the Pseudo Observableframework (see Refs. [33, 90]), but we do not explicitly quote the relations to our framework here. E. Unitarization
The above interactions describe all possible dynamics involving the H boson as appearing in gluon fusion gg → H ,vector boson fusion V V → H , associated production V → V H , and its decays to bosons and fermions. For on-shell H boson production and decay, the typical range of invariant masses is O (100 GeV). However, in associated andoff-shell production of the H boson, there is no kinematic limit on q V i or q H other than the energy of the collidingbeams. When anomalous couplings with q -dependence are involved, this sometimes leads to cross sections growingwith energy, which leads to unphysical growth at high energies. Obviously, these violations are unphysical and anartifact of the lacking knowledge of a UV-complete theory. Therefore, one should dismiss regions of phase spacewhere a violation of unitarity happens. To mend this issue, we allow the option of specifying smooth cut-off scalesΛ V ,i , Λ V ,i , Λ H,i for anomalous contributions with the form factor scalingΛ V ,i Λ V ,i Λ H,i (Λ V ,i + | q V | )(Λ V ,i + | q V | )(Λ H,i + | ( q V + q V ) | ) . (30)Studies of experimental data should include tests of different form-factor scales when there is no direct bound on the q -ranges. An alternative approach is to limit the q -range in experimental analysis by restricting the data sample,using, for example, a requirement on the transverse momentum p T of the reconstructed particles. The experimentalsensitivity of both approaches is equivalent and no additional tools are required for the latter approach. However,such restrictions of the data sample lead to statistical fluctuations and therefore noisy results. They are also difficultexperimentally since each new restriction requires re-analysis of the data, rather than simply a change in the signalmodel. Moreover, while p T of the particles and q of the intermediate vector bosons are correlated, this correlation isnot 100%. Therefore, it is not possible to have a fully consistent analysis in all channels using this approach. Finally,we note that other unitarization prescriptions have been presented in Refs. [55, 91]. III. PARAMETERIZATION OF CROSS SECTIONS
In this Section, we discuss the relationship between the coupling constants and the cross section of a processinvolving the H boson. We denote the coupling constants as a n , which could stand for g n , c n , or κ n as used inSection II. The cross section of a process i → H → f can be expressed asd σ ( i → H → f )d s ∝ (cid:16)(cid:80) α ( i ) jk a j a k (cid:17) (cid:16)(cid:80) α ( f ) lm a l a m (cid:17) ( s − M H ) + M H Γ . (31)where (cid:16)(cid:80) α ( i ) jk a j a k (cid:17) describes the production for a particular initial state i and (cid:16)(cid:80) α ( f ) lm a l a m (cid:17) describes the decayfor a particular final state f . Here we assume real coupling constants a n , though these formulas can also be extendedto complex couplings. The coefficients α ( i ) jk and α ( f ) lm evolve with s and may be functions of kinematic observables.These coefficients can be obtained from simulation, as we discuss in Section IV. In this Section, we discuss integratedcross sections, and for this reason we deal with α ( i ) jk and α ( f ) lm as constants that have already been integrated over thekinematics. We will come back to the kinematic dependence in Section V.In the narrow-width approximation for on-shell production, we integrate Eq. (31) over s in the relevant range, ∼ M H Γ tot around the central value of M H , to obtain the cross section for the process of interest σ ( i → H → f ) ∝ (cid:16)(cid:80) α ( i ) jk a j a k (cid:17) (cid:16)(cid:80) α ( f ) lm a l a m (cid:17) Γ tot . (32)One can express the total width as Γ tot = Γ known + Γ other , (33)where Γ known represents decays to known particles and Γ other represents other unknown final states, either invisibleor undetected in experiment.Without direct constraints on Γ other , if results are to be interpreted in terms of couplings via the narrow-widthapproximation in Eq. (32), assumptions must be made on Γ other . However, in the case of the ZZ and W W final states,there is an interplay between the massive vector boson or the H boson going off-shell, resulting in a sizable off-shell H ∗ production [92] with ( s − M H ) (cid:29) M H Γ tot in Eq. (31). The resulting cross section in this region s > (2 M W ) is independent of the width. It should be noted that Eq. (31) represents only the signal part of the off-shell processwith the H boson propagator. The full process involves background and its interference with the signal [92, 93],as we illustrate in Section VII. Nonetheless, the lack of width dependence in the off-shell region is the basis forthe measurement of the H boson’s total width Γ tot [93], provided that the evolution of Eq. (31) with s is known.Therefore, a joint analysis of the on-shell and off-shell regions provides a simultaneous measurement of Γ tot and ofthe cross sections corresponding to each coupling a n in a process i → H ( ∗ ) → f , as illustrated in Refs. [67, 73]. Ina combination of multiple processes, the measurement can be further interpreted as constraints on Γ other and thecouplings, following Eqs. (32) and (33), and with the help of the identityΓ known = (cid:88) f Γ f = Γ SMtot × (cid:88) f (cid:32) Γ SM f Γ SMtot × Γ f Γ SM f (cid:33) = (cid:88) f Γ SM f (cid:88) lm α ( f ) lm a l a m . (34)The coefficients α ( f ) lm describe couplings to the known states and are normalized in such a way that R f ( a n ) = (cid:16)(cid:80) α ( f ) lm a l a m (cid:17) = 1 in the SM, and otherwise R f ( a n ) = Γ f / Γ SM f .In the following, we proceed to discuss the on-shell part of the measurements using the narrow-width approximation.In Table I, we summarize all the coefficients and functions R f needed to calculate Γ known in Eq. (34). These expressionswith explicit coefficients α ( f ) lm help us to illustrate the relationship between the coupling constants introduced inSection II and experimental cross-section measurements. We will also use these expressions in Section VI in applicationto particular measurements. For almost all calculations, we use the JHUGen framework implementation discussed inSection IV. The only exceptions are R γγ and R Zγ , which are calculated using HDECAY [94, 95]. The calculationsare performed at LO in QCD and EW, with the MS-mass for the top quark m t = 162 . m b = 4 .
18 GeV, and QCD scale µ = M H / H → q ¯ q , where we generically use q = b, c, τ, µ to denote either quarks or leptons, in thelimit m q (cid:28) M H we obtain R qq = κ q + ˜ κ q . (35)For the gluon final state H → gg , we allow for top and bottom quark contributions through the couplings fromEq. (2). In addition, we introduce a new heavy quark Q with mass m Q (cid:29) M H and couplings to the H boson κ Q andTABLE I: Partial widths Γ f of the dominant H → f decay modes in the SM in the narrow-width approximation [41]and their modifications with anomalous couplings at M H = 125 GeV, where Γ SMtot = 4 . × − GeV. Final stateswith Γ SM f < Γ SM µµ are neglected. H → f channel Γ SM f / Γ SMtot Γ f / Γ SM f Eq. H → b ¯ b κ b + ˜ κ b ) Eq. (35) H → W + W − R WW ( a n ) Eq. (38) H → gg R gg ( a n ) Eq. (36) H → τ + τ − κ τ + ˜ κ τ ) Eq. (35) H → c ¯ c κ c + ˜ κ c ) Eq. (35) H → ZZ/Zγ ∗ /γ ∗ γ ∗ R ZZ/Zγ ∗ /γ ∗ γ ∗ ( a n ) Eq. (39) H → γγ R γγ ( a n ) Eq. (40) H → Zγ R Zγ ( a n ) Eq. (41) H → µ + µ − κ µ + ˜ κ µ ) Eq. (35) ˜ κ Q . The result is R gg = 1 . κ t + 0 . κ b − . κ t κ b + 2 . κ t + 0 . κ b − . κ t ˜ κ b (36)+ 1 . κ Q + 2 . κ Q κ t − . κ Q κ b + 2 . κ Q + 4 . κ Q ˜ κ t − . κ Q ˜ κ b . The κ Q and ˜ κ Q couplings are connected to the g gg and g gg point-like interactions introduced in Eq. (1) through g gg = − α s κ Q / (6 π ) , g gg = − α s ˜ κ Q / (4 π ) (37)in the limit where m Q (cid:29) M H . The function R gg also describes the scaling of the gluon fusion cross section withanomalous coupling contributions. Setting κ q = κ t = κ b and ˜ κ q = ˜ κ t = ˜ κ b , we find the ratio σ (˜ κ q = 1) /σ ( κ q =1) = 2 .
38, which differs from the ratio for a very heavy quark σ (˜ κ Q = 1) /σ ( κ Q = 1) = (3 / = 2 .
25 due to finitequark mass effects. The latter ratio follows from the observation σ ( g gg = 1) = σ ( g gg = 1). In experiment, it is hardto distinguish the point-like interactions g gg and g gg , or equivalently κ Q and ˜ κ Q , from the SM-fermion loops. Inthe H → gg decay, there is no kinematic difference. In the gluon fusion production, there are effects in the tails ofdistributions, such as the transverse momentum, or in the off-shell region, as we discuss in Section VII. However, inSection VI these effects are negligible and we do not distinguish the g gg and g gg couplings from the SM-fermion loops.For the H → W W → four-fermion final state, we set Λ W W = 100 GeV in Eq. (1) in order to keep all numericalcoefficients of similar order, and rely on the κ W W = κ W W relationship to obtain R W W = (cid:18) g W W (cid:19) + 0 . (cid:0) κ W W (cid:1) + 0 . (cid:0) g W W (cid:1) + 0 . (cid:0) g W W (cid:1) (38)+ 0 . (cid:18) g W W (cid:19) κ W W + 0 . (cid:18) g W W (cid:19) g W W + 0 . κ W W g W W . For the H → ZZ/Zγ ∗ /γ ∗ γ ∗ → four-fermion final state, we set Λ Zγ = Λ ZZ = 100 GeV in Eq. (1) and rely on the κ Zγ and κ ZZ = κ ZZ parameters to express R ZZ/Zγ ∗ /γ ∗ γ ∗ = (cid:18) g ZZ (cid:19) + 0 . (cid:0) κ ZZ (cid:1) + 0 . (cid:0) g ZZ (cid:1) + 0 . (cid:0) g ZZ (cid:1) (39)+ 0 . (cid:18) g ZZ (cid:19) κ ZZ + 0 . (cid:18) g ZZ (cid:19) g ZZ + 0 . κ ZZ g ZZ + 0 . (cid:16) κ Zγ (cid:17) + 0 . (cid:18) g ZZ (cid:19) κ Zγ + 0 . κ ZZ κ Zγ + 0 . g ZZ κ Zγ . We set g Zγ = g Zγ = g γγ = g γγ = 0 in Eq. (39). These four couplings require a coherent treatment of the q cutofffor the virtual photon and are left for a dedicated analysis. We note that some final states in the H → W W and
ZZ/Zγ ∗ /γ ∗ γ ∗ → four-fermion decays may interfere, but their fraction and phase-space overlap are very small. Wetherefore neglect this effect.0For the H → γγ and Zγ final states, we include the W boson and the top and bottom quarks in the loops andobtain R γγ = 1 . (cid:18) g W W (cid:19) + 0 . κ t − . (cid:18) g W W (cid:19) κ t + 0 . κ b − . κ t κ b (40)+ 0 . (cid:18) g W W (cid:19) κ b + 0 . κ t + 0 . κ b − . κ t ˜ κ b .R Zγ = 1 . (cid:18) g W W (cid:19) + 0 . κ t − . (cid:18) g W W (cid:19) κ t + 0 . κ b − . κ t κ b (41)+ 0 . (cid:18) g W W (cid:19) κ b + 0 . κ t + 0 . κ b − . κ t ˜ κ b . The point-like interactions g γγ and g γγ or g Zγ and g Zγ could be considered in Eqs. (40) and (41). However, followingthe approach in Eq. (39), these are left to a dedicated analysis. Within the SM EFT theory approach, a fully generalstudy is available in Ref. [96]. We do not consider higher-order corrections, such as terms involving κ W W , , g W W , or g W W , in Eqs. (40) and (41). We also neglect the H → γ ∗ γ contribution.To conclude the discussion of the cross sections, we note that the relative contribution of an individual coupling a n , either to production (cid:16)(cid:80) α ( i ) jk a j a k (cid:17) or to decay (cid:16)(cid:80) α ( f ) lm a l a m (cid:17) , can be parameterized as an effective cross-sectionfraction f ( i,f ) an = α ( i,f ) nn a n (cid:80) m α ( i,f ) mm a m × sign (cid:18) a n a (cid:19) , (42)where the sign of the a n coupling relative to the dominant SM contribution a is incorporated into the f an definition.In the denominator of Eq. (42), the sum runs over all couplings contributing to the i → H or H → f process. Byconvention, the interference contributions are not included in the effective fraction definition in Eq. (42) so that thisparameter can be more easily interpreted.We adopt the definition of f an used by the LHC experiments [66, 69, 97] for HW W , HZZ , HZγ , and
Hγγ anomalous couplings in the H → ZZ/Zγ ∗ /γ ∗ γ ∗ → e µ process, with the HW W couplings related through Eqs. (17)–(20); f ggCP in the ggH process for the effective Hgg couplings [3]; and f qq CP for processes involving Hq ¯ q fermion couplings,such as H → q ¯ q , with α mm = 1 in Eq. (42). The latter convention for f tt CP is extended to the Ht ¯ t couplings as well,despite the fact that Eq. (35) is not valid for the heavy top quark [4]. It is also easy to invert Eq. (42) to relate thecross section fractions to coupling ratios via a n a m = (cid:115) | f an | α mm | f am | α nn × sign ( f an f am ) , (43)where we omit the process index for either i → H or H → f . Because (cid:80) n | f an | = 1, only all but one of theparameters are independent. We choose to use the f an corresponding to anomalous couplings as our independent setof parameters, leaving for example f a = (cid:16) − (cid:80) n (cid:54) =1 | f an | (cid:17) as a dependent one.There are several advantages in using the f an parameters in Eq. (42) in analyzing a given process on the LHC. Firstof all, the f an and signal strength µ i → f = σ i → f /σ i → f SM form a complete and minimal set of measurable parametersdescribing the process i → H → f . Measuring directly in terms of couplings introduces degeneracy in Eq. (32),because, for example, the production couplings can be scaled up and the decay couplings down without changing theresult. A similar interplay occurs between the couplings appearing in the numerator and the denominator of Eq. (32).Second, the f an parameters are independent of Γ tot , which is absorbed into µ i → f . In contrast, the direct couplingmeasurement a n depends on the assumptions in Eq. (33), including Γ other . Third, f an has the same meaning in allproduction and all decay channels of the H boson. For example, the f an measurement in VBF production is invariantwith respect to the H → f decay channel used. This can be seen from Eq. (32), where (cid:16)(cid:80) α ( f ) lm a l a m (cid:17) / Γ tot can beabsorbed into the µ i → f parameter. Fourth, f an is a ratio of observable cross sections, and therefore it is invariant The situation when production and decay cannot be decoupled in analysis of the data due to the same couplings appearing in bothprocesses, such as in i → V V → H → V V → f , is discussed in detail in Section VI. (a) Signal (b) Interfering background (c) Non-interfering backgroundGluonfusionVectorbosonfusion FIG. 3: Sample diagrams for signal, interfering background and non-interfering background in the processes pp → (cid:96) (gluon fusion) and pp → (cid:96)jj (weak vector boson fusion).with respect to the a n coupling scale convention. For example, the f an value is identical for either the c n or g n couplings related in Eq. (8). Fifth, in the experimental measurements of f an most systematic uncertainties cancel inthe ratios, making it a clean measurement to report. Sixth, the f an are convenient parameters for presenting results astheir full range is bounded between − f an have an intuitive interpretation, as their values indicate the fractional contribution to the measurable cross section,while there is no convention-invariant interpretation of the coupling measurements. In the end, the measurements inindividual processes can be combined, and at that point their interpretation in terms of couplings becomes natural.However, this becomes feasible only when the number of measurements is at least equal to, or preferably exceeds, thenumber of couplings. IV. JHUGEN/MELA FRAMEWORK
The JHUGen (or JHU generator) and MELA (or Matrix Element Likelihood Approach) framework is designedfor the study of a generic bosonic resonance decaying into SM particles. JHUGen is a stand-alone event generatorthat generates either weighted events into pre-defined histograms or unweighted events into a Les Houches Events(LHE) file. A subsequent parton shower simulation as well as full detector simulation can be added using otherprograms compatible with the LHE format. The MELA package is a library of probability distributions based onfirst-principle matrix elements. It can be used for Monte Carlo re-weighting techniques and the construction ofkinematic discriminants for an optimal analysis. The packages are based on developments reported in this work andRefs. [1–4]. It can be freely downloaded at [87]. The package has been employed in the Run-I and Run-II analyses ofLHC data for the H boson property measurements [59–76].Our framework supports a wide range of production processes for spin-zero, spin-one, and spin-two resonances andtheir decays into SM particles. All interaction vertices can have the most general Lorentz-invariant structure withCP-conserving or CP-violating degrees of freedom. We put a special emphasis on spin-zero resonances H , for whichwe allow production through gluon fusion, associated production with one or two jets, associated production with aweak vector boson ( Z/γ ∗ H, W H, γH ), weak vector boson fusion (
V V jj → Hjj ), and production in association withheavy flavor quarks, such as t ¯ tH , tH and b ¯ bH at the LHC. The supported decay modes include H → ZZ / Zγ ∗ / γ ∗ γ ∗ → f , H → W W → f , H → Zγ / γ ∗ γ → f γ , H → γγ , H → τ τ , and generally H → f ¯ f , with the mostgeneral Lorentz-invariant coupling structures. Spin correlations are fully included, as are interference effects fromidentical particles.To extend the capabilities of our framework, JHUGen also allows interfacing the decay of a spin-zero particle afterits production has been simulated by other MC programs (or by JHUGen itself) through the LHE file format. As anexample, this allows production of a spin-zero H boson through NLO QCD accuracy with POWHEG [98] and furtherdecay with the JHUGen. Higher-order QCD contributions are discussed in Ref. [4] for the t ¯ tH process and belowfor the ZH process. Another interface with the MCFM Monte Carlo generator [5–9] allows accessing backgroundprocesses and off-shell H ∗ boson production, including interference with the continuum.In the following we briefly outline new key features in our JHUGen/MELA framework that become available withthis publication. In the subsequent Sections, we apply these new features and demonstrate how they can be usedfor LHC physics analyses. In the simulation, the values of s w , M W , Γ W , m Z , and Γ Z are parameters configurableindependently, and in this paper we set s w = 0 . M W = 80 .
399 GeV, Γ W = 2 .
085 GeV, M Z = 91 . Z = 2 . (a) q ¯ q LO (b) q ¯ q NLO QCD (c) gg LO box (d) gg LO triange
FIG. 4: ZH sample diagrams for leading order q ¯ q and gg initial states, including higher order contributions. A. Off-shell simulation of the H boson in gluon fusion and a second scalar resonance
We extend our previous calculation of gg → H → V V → f by allowing m (cid:96) to be far off the H resonance masspeak. In these regions of phase space the irredicible background from q ¯ q/gg → V V → f continuum productionbecomes significant and, in the case of the gg initial state, interferes with the H production amplitudes, as illustratedin Fig. 3. The MCFM generator [7] contains the SM amplitudes for this process at LO. Our add-on extends theMCFM code and incorporates the most general anomalous couplings in the H boson amplitude. We allow twopossible parameterizations of the CP-even and CP-odd degrees of freedom: the point-like Hgg couplings g gg , g gg andthe full one-loop amplitude with heavy quark flavors, using the Yukawa-type couplings κ q , ˜ κ q . Additional hypotheticalfourth-generation quarks with anomalous HQ ¯ Q couplings can be included as well. For the study of a second H -likeresonance X with mass m X and width Γ X , we allow for the same set of couplings and decay modes. B. Off-shell simulation of the H boson in electroweak production and a second scalar resonance
Similar to the gluon fusion process, we extend our previous calculation of vector boson fusion qq → qq + H ( → V V → f ) and associated production qq → V + H ( → V V → f ), and allow the full kinematic range for m f . TheSM implementation in MCFM [8] includes the s - and t -channel H boson amplitudes, the continuum backgroundamplitudes, and their interference, as illustrated in Fig. 3. We supplement the necessary contributions for the mostgeneral anomalous coupling structure. In particular, this affects the H boson amplitudes but also the triple andquartic gauge boson couplings. We also add amplitudes for the intermediate states ZZ/Zγ ∗ /γ ∗ γ ∗ in place of ZZ in both decay and production with the most general anomalous coupling structure, which are not present in theoriginal MCFM implementation. It is interesting to note that the off-shell VBF process qq → qq + H ( → f ) includescontributions of the q ¯ q → V H ( → f ) process for the case of hadronic decays of the V boson. As in the case ofgluon fusion, we also allow the study of a second H -like resonance X with mass m X , width Γ X , and the same set ofcouplings and decay modes. C. Higher-order contributions to VH production
We calculate the NLO QCD corrections to the associated H boson production process q ¯ q → V H where V = Z, W, γ ,shown in Fig. 4. We use standard techniques and implement the results in JHUGen, relying on the
COLLIER [101]loop integral library. This improves the physics simulation of previous studies at LO and allows demonstrating therobustness of previous matrix element method studies. We also calculate the loop-induced gluon fusion contribution gg → ZH , which is parameterically of next-to-next-to-leading order but receives an enhancement from the largegluon flux, making it numerically relevant for studies at NLO precision. In contrast to the q ¯ q → V H process whichis sensitive to
HV V couplings, the gg → ZH process is additionally sensitive to the Yukawa-type Hq ¯ q couplings. Inboth cases we allow for the most general CP-even and CP-odd couplings. Strong destructive interference betweentriangle and box amplitudes in the SM leads to interesting physics effects that enhance sensitivity to anomalous Ht ¯ t couplings, as we demonstrate in Section VIII. D. Multidimensional likelihoods and machine learning
We extend the multivariate maximum likelihood fitting framework to describe the data in an optimal way andprovide the multi-parameter results in both the EFT and the generic approaches. The main challenge in this analysisis the fast growth of both the number of observable dimensions and the number of contributing components in thelikelihood description of a single process with the increasing number of parameters of interest. We present a practical3FIG. 5: Illustrations of an H boson production and decay in three topologies: (1) boson fusion and decay V V → H → V V → f ; (2) boson fusion with associated jets q q → q q ( V V → H → V V ); and (3) associatedproduction q q → V → V ( H → V V ). Five angles fully characterize the orientation of the production or decaychain and are defined in suitable frames [1, 3].approach to accommodate both challenges, while keeping the approach generic enough for further extensions. Thisapproach relies on the MC simulation, reweighting tools, and optimal observables constructed from matrix elementcalculations. We extend the matrix element approach by incorporating the machine learning procedure to account forparton shower and detector effects when these effects become sizable. Some of these techniques are illustrated withexamples below. V. LHC EVENT KINEMATICS AND THE MATRIX ELEMENT TECHNIQUE
Kinematic distributions of particles produced in association with the H boson or in its decay are sensitive to thequantum numbers and anomalous couplings of the H boson. In the 1 → H → V V → f decay,six observables Ω decay = { θ , θ , Φ , m , m , m f } fully characterize kinematics of the decay products, while two otherangles Ω prod = { θ ∗ , Φ } orient the decay frame with respect to the production axis, as described in Ref. [1] andshown in Fig. 5. The Ω prod angles are random for the production of a spin-zero particle, but provide non-trivialinformation to distinguish signal from either background or alternative spin hypotheses. A similar set of observablescan be defined in a production process. For example, the observables Ω assoc = { θ assoc1 , θ assoc2 , Φ assoc , q , assoc1 , q , assoc2 } characterize VH and weak or strong boson fusion (VBF or ggH) in association with two hadronic jets, as illustratedin Fig. 5 and described further in Ref. [3]. Similar kinematic diagrams defining observables for the t ¯ tH , tqH , and H → τ τ processes are discussed elsewhere [4].In the 2 → H boson production and its subsequent decay to a four-fermion final state,such as VBF, 13 kinematic observables are defined, which include angles and the invariant masses of intermediatestates. There is also the overall boost of the six-body system, which depends on QCD effects. We decouple this boostfrom these considerations. Only a reduced set of observables is available when there are no associated particles inproduction or when the decay chain has less than four particles in the final state.Kinematic distributions with anomalous couplings of the H boson have been shown previously in Refs. [1–3] forboth decay and associated production. Here, we emphasize kinematics in associated production with two jets, shownin Fig. 6. There are distinct features depending on the gg , γγ , Zγ , ZZ , and W W fusion, which is reflected in theassociated jet kinematics. Note that for the production processes we define q i for each vector boson, where q i < (cid:112) − q i . In this case, θ assoc i angles, usually defined in the rest frame of the vectorbosons, are calculated in the H frame instead. We would like to stress that a consistent treatment of all contributionswith γγ , Zγ , ZZ , and W W intermediate states in weak boson fusion is critical in a study of anomalous couplings.While for the SM H boson one could often neglect photon intermediate states when couplings to the Z and W bosons dominate, one generally cannot neglect them when comparing to other contributions generated by higher-dimension operators. In reference to the EFT operators discussed in Section II, the Higgs basis becomes the natural4one to disentangle the Hγγ , HZγ , HZZ , and
HW W operators from the experimentally observed kinematics ofevents. This is visible, for example, in the q VBF1 , distributions corresponding to the pseudoscalar operators, wherethe photon intermediate states lead to a much softer spectrum compared to W and Z . The advantage of the Higgsbasis for experimental analysis becomes especially evident when considering off-shell effects, because there is no off-shell enhancement with the intermediate γ states. Once experimental results are obtained in the Higgs basis, themeasurements can be translated to any other basis. - - - - - VBF1 ,2 q cos ZZWW g Z gg gg - - - VBF F ZZWW g Z gg gg VBF1 ,2 q ZZWW g Z gg gg [GeV] - - - - - VBF1 ,2 q cos ZZWW g Z gg gg - - - VBF F ZZWW g Z gg gg ZZWW g Z gg ggVBF1 ,2 q [GeV] FIG. 6: Distributions of observables in vector boson fusion jet associated production: { θ VBF1 , , Φ VBF , (cid:113) − q , VBF1 , } ,comparing gg , γγ , Zγ , ZZ , and W W fusion for the SM couplings (top) and pseudoscalar couplings (bottom). A looseselection, ∆ R JJ > . p JT >
15 GeV, is applied, consistently for all processes, to avoid divergences in processeswith photons and gluons. All distributions are normalized to unit area.With up to 13 observables Ω sensitive to the Higgs boson anomalous couplings, it is a challenging task to performan optimal analysis in a multidimensional space of observables, creating the likelihood function depending on morethan a dozen parameters in Eq. (7). Full detector simulation and data control regions in LHC data analyses maylimit the number of available events and, as a result, the level of detail in the likelihood. Therefore, it is importantto develop methods that are close to optimal under the practical constraints of the available data and simulation. Inthe rest of this Section, we discuss some of the experimental applications of the tools developed in our framework,which target these tasks in the study of the H boson kinematics.Analysis of experimental observables typically requires the construction of a likelihood function, which is maximizedwith respect to parameters of interest. The complexity of the likelihood function grows quickly both with the numberof observables and with the number of parameters, and the two typically increase simultaneously. Examples of suchlikelihood construction will be discussed in Section VI. Typically, the likelihood function will be parameterized withtemplates (histograms) of observables, using either simulated MC samples or control regions in the data. The challengein this approach is to keep the number of bins of observables to a practical limit, typically several bins for severalobservables, due to statistical limitations in the available data and simulation. Similar practical limitations appear inthe number of parameters of interest, which will be discussed later.The information content in the kinematic observables is different, and one could pick some of the most informative5kinematic observables of interest. The difficulty of this approach is illustrated in Fig. 6 where all five observables (notethat θ , and q , each represent two independent observables) provide important information and it is hard to picka reduced set without substantial loss of information. Another approach is to create new observables optimal for theproblem of interest, and in the next subsections we illustrate optimal observables based on both the matrix elementand the machine learning techniques. Nonetheless, it is not possible to have a prior best set of observables universallygood for all measurements and at the same time limited in the number of dimensions for practical reasons. We notethat alternative methods may try to avoid creation of templates and parameterize the multi-dimensional likelihoodfunction directly with certain approximations. We illustrated some of these methods in Refs. [1, 3] and a broaderreview may be found in Ref. [102]. However, the complexity of those methods also provides practical limitations ontheir application. We present some of the practical approaches in Section VI.One popular example of the reduced set of bins of observables adopted for the study of the H boson kinematics is theso-called Simplified Template Cross Section approach (STXS) [41, 103]. The main focus at this stage [103] is on thethree dominant H boson production processes, namely gluon fusion, VBF, and V H . These main production processesare subdivided into bins based on transverse momentum or mass of various objects, for example the H boson andassociated jets. At future stages, the available information may be subdivided further. This approach became a strongframework for collaborative work of both theorists and experimentalists, as information from all LHC experimentsand theoretical calculations can be combined and shared in an efficient way. Nonetheless, as we illustrate below, thisapproach is still limited in its application for two important reasons. First, the STXS measurements are based on theanalysis of SM-like kinematics. The measurement strategy may not be appropriate for interpretations appearing withnew tensor structures or new virtual particles (such as γ ∗ in place of Z ∗ ) unless a full detector simulation of sucheffects is performed. Additionally, the binning of STXS may not be optimal for all the measurements of interest. A. Matrix element technique
The matrix element likelihood approach (MELA) [1–4] was designed to extract all essential information from thecomplex kinematics of both production and decay of the H boson and retain it in the minimal set of observables.Two types of discriminants were defined for either the production or the decay process, and here we generalize it forany sequential process of both production and decay: D alt ( Ω ) = P sig ( Ω ) P sig ( Ω ) + P alt ( Ω ) , (44) D int ( Ω ) = P int ( Ω )2 (cid:112) P sig ( Ω ) × P alt ( Ω ) , (45)where P sig , P alt , and P int represent the probability distribution for a signal model of interest, an alternative model tobe rejected (either background, a different production process of the H boson, or an alternative anomalous couplingof the H boson), and the interference contribution, which may in general be positive or negative. The probabilitiesare obtained from the matrix elements squared, calculated by the MELA library described in Section IV, and donot generally need to be normalized. The denominator in Eq. (45) is chosen to reduce correlation between thediscriminants, but this choice is equivalent to that of Ref. [3]. The above definition leads to the convenient arrangement0 ≤ D alt ≤ − ≤ D int ≤ P for the two hypotheses provides optimal discrimination power. However, for a continuous set of hypotheses with anarbitrary quantum-mechanical mixture several discriminants are required for an optimal measurement of their relativecontributions. There are three interference discriminants when anomalous couplings appear both in production and indecay. Let us conside only real g and g couplings in Eq. (1), which appear once in production and once in decay, asshown in Eq. (32). The total amplitude squared would have five terms proportional to ( g /g ) m with m = 0 , , , , P ( Ω ; g , g ) ∝ (cid:88) m =0 ( g /g ) m × P m ( Ω ) , (46)where we absorb g and the width into the overall normalization. Equation (44) corresponds to the ratio of the m = 4 and m = 0 terms. Three other ratios give rise to interference discriminants. The four discriminants may6be re-arranged into two discriminants of the form in Eq. (44) and two of the form in Eq. (45), in each case oneobservable defined purely for the production process and the other for the decay process. One could apply theNeyman-Pearson lemma to each pair of points in the parameter space of ( g , g ), but this would require a continuous,and therefore infinite, set of probability ratios. However, equivalent information is contained in a linear combinationof only four probability ratios, which can be treated as four independent observables. Above the 2 m V threshold,there are also interference discriminants appearing due to interference between the off-shell tail of the signal processand the background. A subset of equivalent optimal observables was also introduced independently in earlier work ondifferent topics [105–107].The number of discriminants in Eqs. (44, 45) is still limited if we consider just one anomalous coupling. Nonethe-less, this number grows quickly as we consider multiple anomalous couplings, especially the number of interferencediscriminants. A subset of these discriminants may contain most of the information, depending on the situation. Forthe near-term LHC measurements, the D alt using full production and decay information and D int using productioninformation from correlation of associated particles provide the most optimal information. In the very long term, thelowest powers of m may provide most of the discriminating power when testing data for tiny anomalous contributions,because effects may be most visible in interference. Therefore, the MELA approach still allows us to select a limitedset of the most optimal discriminants, as we illustrate with the practical applications below.Detector resolution effects may come into play in experimental analyses and affect the calculations of the probabil-ities in Eqs. (44, 45). These can be parameterized with transfer functions. However, in most practical applications,the “raw” matrix element probabilities can be used, and the resulting performance degradation is small when thedistributions of the angular and mass observables are broad compared to the resolution. One notable exception isthe invariant mass of relatively narrow resonances, such as the H boson mass in H → ZZ → (cid:96) or the V boson massin associated production V H . In such a case, the signal and background probabilities P sig and P bkg can incorporatethe empirical invariant mass parameterization giving rise to the D bkg discriminant for optimal background rejection.We find such an approach computationally effective without any visible loss of performance. Nonetheless, below wealso introduce machine learning to enhance the matrix element technique. This approach incorporates matrix elementknowledge combined with the parton shower and detector effects. B. Application of the matrix element technique to the boson fusion processes
Let us illustrate the power of the matrix element technique in application to both weak boson fusion (VBF in thefollowing) and strong boson fusion (ggH in the following), where at higher orders in QCD the ggH process may include gg, qg, and qq initial states. This illustration is similar to our earlier study in Ref. [3], but we would like to expandthis illustration in several directions. The boson fusion process is particularly important in the off-shell region, whichis a new feature of this work. However, most kinematic considerations apply equally to both the on-shell and off-shellregions. In the weak boson fusion, for illustration purposes we consider equal strength of W W and ZZ fusion, with g ZZ = g W W and g ZZ = g W W in Eq. (1) and vary the relative contribution of the CP-even and CP-odd amplitudes,with the f VBF g parameter representing their relative cross section fraction. The relative strength of W W and ZZ fusion is fixed in this study because the two processes are essentially indistinguishable in their observed kinematics,as shown in Fig. 6. In strong boson fusion, the parameter f ggHg represents a similar relative cross section fraction ofthe pseudoscalar coupling component.Figure 7 shows the D − and D CP discriminants, calculated according to Eqs. (44) and (45), for the VBF process,to distinguish between the SM hypothesis g ZZ = g W W = 2, the alternative hypothesis g ZZ = g W W (cid:54) = 0, and theinterference between these two contributions. Figure 8 shows the same type of discriminants defined and shown forthe ggH process, enhanced with the events in the VBF-like topology using the requirement m JJ >
300 GeV forillustration. In both cases, information from the H boson and the two associated jets, as illustrated in Fig. 5, is usedin the discriminant calculation. The m JJ requirement is based on the following observation. Among the initial statesin the ggH process, we could have gg, qg, and qq parton pairs. The events with the qq initial state carry most of theinformation for CP measurements and have the topology most similar to the VBF process, which is also known tohave a large di-jet invariant mass. In Section VI, we will use this feature when developing the analysis techniques,but with the matrix-element technique applied to isolate the VBF-like topology. In both the VBF and ggH cases, theazimuthal angle difference between the two jets ∆Φ JJ is also shown for comparison [12]. It is similar to the Φ VBF angle defined in Fig. 5 and shown in Fig. 6, but differs somewhat because it is calculated in a different frame.The ∆Φ JJ angle is defined as follows. The direction of the two jets is represented by the vectors (cid:126)j , in thelaboratory frame, and (cid:126)j T , are the transverse components in the xy plane. If we label j as the jet going in the − z direction (or less forward) and j as the jet going in the + z direction (or more forward), then ∆Φ JJ is the azimuthal7 VBF0- D + - + = VBFg4 f 0.5 - = VBFg4 f - - - - - VBFCP D + - + = VBFg4 f 0.5 - = VBFg4 f - - - JJ FD + - + = VBFg4 f 0.5 - = VBFg4 f FIG. 7: Two discriminants defined in Eq. (44) (left) and Eq. (45) (middle) for the measurement of the CP-sensitiveparameter f VBF g in VBF production. Also shown is the ∆Φ JJ observable (right). The values of f VBF g = ± . ggH0- D + - + = ggHg4 f 0.5 - = ggHg4 f - - - - - ggHCP D + - + = ggHg4 f 0.5 - = ggHg4 f - - - JJ FD + - + = ggHg4 f 0.5 - = ggHg4 f FIG. 8: Two discriminants defined in Eq. (44) (left) and Eq. (45) (middle) for the measurement of the CP-sensitiveparameter f ggH g in ggH production. Also shown is the ∆Φ JJ observable (right). The values of f ggH g = ± . m JJ >
300 GeV is applied to enhance theVBF-like topology of events.angle difference between the first and the second jets, or φ − φ . In vector notation,∆Φ JJ = (ˆ j T × ˆ j T ) · ˆ z | (ˆ j T × ˆ j T ) · ˆ z | · ( (cid:126)j − (cid:126)j ) · ˆ z | ( (cid:126)j − (cid:126)j ) · ˆ z | · cos − (cid:16) ˆ j T · ˆ j T (cid:17) , (47)where the angle between (cid:126)j T and (cid:126)j T defines ∆Φ JJ and the two ratios provide the sign convention. This definition isinvariant under the exchange of the two jets and the choice of the positive z axis direction.The information content of the observables can be illustrated with the Receiver Operating Characteristic (ROC)curve, which is a graphical plot that illustrates the diagnostic ability of a binary classifier system as its discriminationthreshold is varied. Figure 9 (left) shows the ROC curves illustrating discrimination between scalar and pseudoscalarmodels in the VBF process using the D − and ∆Φ JJ observables. The optimal observable D − , which incorporates allkinematic and dynamic information, has the clear advantage. Figure 9 (right) shows the same comparison in the ggHprocess. The gain in using the optimal observable in the ggH process is not as large as in VBF because of the smallerdifferences in dynamics of the scalar and pseudoscalar models, as both are generated by higher-dimension operatorswith the same powers of q i in Eq. (1). While the D − observable incorporates all kinematic and dynamic information,the truly CP-sensitive observable D CP does not rely on dynamics. It provides optimal separation between the modelswith maximal mixing of the CP-even and CP-odd contributions and opposite phases. We illustrate this in Fig. 9(middle) with a ROC curve for discrimination between the f g = ± . C. Matrix element technique with machine learning
The discriminants calculated with the matrix elements directly, as discussed in Section V A, are powerful tools inthe analysis of experimental data. Most importantly, they provide scientific insight into the problem under study.Nonetheless, there could be practical considerations limiting their application in certain cases. For example, eventswith partial reconstruction would require integration over unobserved degrees of freedom. Substantial detector effectsor incorrect particle assignment in reconstructed events may lead to poor experimental resolution and would requiremodeling with transfer functions. All of these effects can be taken into account, but may make calculations ineffi-cient or impractical. Here we provide a practical prescription for overcoming these complications with the help ofmachine learning, while still retaining the functionality of the optimal matrix-element approach. We achieve this byconstructing the training samples and the observables used according to the matrix-element approach.Machine learning is a popular approach to data analysis, especially with the growing computational power ofcomputers. The problem of differentiating between two models, as in Eq. (44), becomes a trivial task with supervisedlearning, where two samples of events with the signal and alternative models are provided as input for training. Onekey aspect where the matrix element approach provides the insight is the set of input observables Ω . As long asthe complete set of observables, sufficient for the matrix element calculations, is provided to the machine learningalgorithm, the outcome of proper training is guaranteed to be a discriminant optimal for this task, equivalent to thatin Eq. (44). We illustrate this with such a discriminant D ML0 − in Fig. 9 (left) in application to the VBF process, usingthe Boosted Decision Tree implementation from Ref. [108].Application of the machine learning approach to the discriminant in Eq. (45) is less obvious, because it requiresknowledge of quantum mechanics to isolate the interference component. Nonetheless, we provide a prescription forobtaining such a discriminant. A discriminant trained to differentiate the models with maximal quantum-mechanicalmixing of the signal and alternative contributions with opposite phases becomes a machine-learning equivalent to thatin Eq. (45), following the discussion in Section V B. The complete kinematic information Ω of the event should beprovided to training. We illustrate this approach with such a discriminant D MLCP in Fig. 9 (middle) in application tothe VBF process. There is a small degradation in performance of the D MLCP discriminant with respect to the matrixelement calculation, but this is attributed to the more challenging task of training in this case and should be recoveredin the limit of perfect training.To summarize, the matrix element technique, expressed in Eqs. (44) and (45), can be expanded with the helpof machine learning with two important ingredients: (1) the complete set of matrix-element input observables Ω ,or equivalent, has to be used, and (2) the machine learning process should be based on the carefully preparedsamples according to the models discussed above. The machine learning approach is still based on the matrix elementcalculations, as the training samples are generated based on the same matrix elements as the discriminants in Eqs. (44)and (45). VI. APPLICATION TO ON-SHELL H(125) BOSON PRODUCTION
We start by investigating the on-shell production and decay of the H boson with its coupling to either weak orstrong vector bosons in the VBF and ggH processes. There has already been extensive study of the HV V couplings,and the current challenge is in the measurement of multiple possible anomalous contributions. On the other hand,there have been limited studies of the anomalous
Hgg couplings, due to lower statistical precision at this time. Thelatter could be interpreted as both an effective coupling to gluons, or as a coupling to quarks in the gluon fusion loop.Prospects of both
HV V and
Hgg studies with either 3000 f b − (HL-LHC) or 300 f b − (full LHC) are presentedbelow. Let us first discuss some general features in analysis of LHC data.For the HV V studies, we will use the example of the H → V V → (cid:96) decay and VBF, V H , or ggH production.Equation (1) defines several anomalous couplings, which we generically denote as g V Vi . All of these processes includethe interference of several
V V intermediate states, such as
V V = ZZ, Zγ, γγ, W W . In the analysis of the data (MCsimulation in our case), a likelihood fit is performed [109, 110]. The probability density function for a given signalprocess, before proper normalization, is defined for the two possible numbers of couplings N in the product: N = 4 : P (cid:16) x ; (cid:126)f (cid:17) ∝ K (cid:88) k,l,m,n =1 k ≤ l ≤ m ≤ n P klmn ( x ) (cid:113) | f gk · f gl · f gm · f gn | sign( f gk · f gl · f gm · f gn ) , (48) N = 2 : P (cid:16) x ; (cid:126)f (cid:17) ∝ K (cid:88) k,l =1 k ≤ l P kl ( x ) (cid:113) | f gk · f gl | sign( f gk · f gl ) , (49)9 ) + P(0 ) - P ( VBF0- D ML0- D JJ FD + = VBFg4 fP . - = VB F g4 f P VBFCP D MLCP D JJ FD ) + P(0 ) - P ( ggH0- D JJ FD FIG. 9: Left: a ROC curve showing the separation power between the scalar (SM-like 0 + ) and pseudoscalar (0 − )models in the VBF process using the D − and ∆Φ JJ observables. The diagonal dashed line shows the hypotheticalno-separation scenario. The points represent the efficiency of selecting each model as the threshold of selection isvaried. Right: same as the left plot, but for the ggH process, with a requirement m JJ >
300 GeV applied to enhancethe VBF-like topology of events. Middle: a ROC curve showing the separation power between the f VBF g = +0 . f VBF g = − . D CP and ∆Φ JJ observables. Also shown on the left andmiddle plots are the ROC curves representing performance of the optimal observables obtained with machine learningtechniques.where x are the observables, but not necessarily the complete set Ω , and f gn are K terms corresponding to thecross-section fractions of the couplings, defined in Eq. (42). Equations (48) and (49) are obtained from Eq. (32) andusing Eq. (43), where the width and f g are absorbed into the overall normalization. In the case of the electroweakprocess, the HV V coupling appears on both the production and the decay sides. As a result, the amplitude squaredhas a product of N = 4 couplings. In the gluon fusion production, on the other hand, the electroweak HV V couplingsappear only in decay, and therefore N = 2. Similarly, if one considers the Hgg coupling on production, N = 2.There are ( N + K − / ( N !( K − K = 5 in our analysis of four anomalous HV V couplings. Therefore, in the case of electroweak production( N = 4, K = 5), we have to deal with 70 terms. If we were to consider K = 13 independent couplings in Eq. (1), wewould formally have to deal with 1820 terms describing production and decay (the actual number would be somewhatsmaller because not all terms contribute to a given decay mode). While such analysis of 1820 terms is in principlefeasible, at the current stage it is not practical. In the case of gluon fusion, there are 15 terms for HV V couplings( N = 2, K = 5) and 3 terms for Hgg couplings ( N = 2, K = 2). If both sets of anomalous couplings are consideredsimultaneously, the total number of terms is the product of these, that is 45.In the simplified analysis of LHC data, using simulation of pp collisions at 13 TeV, we adopt the following approach.We take the analysis of the H → V V → (cid:96) channel as the most interesting for illustration, because both productionand decay information can be used. All production modes of the H boson are included in this study and are generatedwith the JHU generator as discussed in Section IV. The JHU generator framework is also used to generate gluon fusionand electroweak background production of the V V → (cid:96) final states. The dominant q ¯ q → V V → (cid:96) backgroundprocess is generated with POWHEG [111] and scaled to cover for other possible background contributions not modeledotherwise [112]. All events are passed through Pythia 8 [113] for parton shower simulation. The detector effects aremodeled with ad-hoc acceptance selection, and the lepton and hadronic jet momenta are smeared to achieve realisticresolution effects. Going beyond the H → (cid:96) channel, inclusion of the H → γγ , H → τ τ , and H → bb channels mightincrease the dataset by about an order of magnitude, but only for analysis of the production information. In addition,analysis of the H → W W → (cid:96) ν decay may bring some information on the decay side, but not exceeding that fromthe H → (cid:96) case. While we focus on the H → (cid:96) channel, we comment on improvements which will be achieved witha combination of the above channels. A. HVV anomalous couplings
In order to illustrate the power of the matrix element techniques and the analysis tools discussed above, let usconsider the
HV V coupling of the H boson to two weak vector bosons using the H → (cid:96) decay, with vector boson0fusion, associated production with the vector bosons W and Z , or inclusive production, and using both on-shell andoff-shell production. Some of these techniques have already been applied in analyses of LHC data [71–73]. However,the rich kinematics in production and decay of the H boson represents particular challenges in analysis.There are 13 independent HV V anomalous couplings in Eq. (1). An optimal simultaneous measurement of all thesecouplings, or even a sizable subset, represents a practical challenge in data analysis and, as far as we know, has notbeen attempted experimentally yet. Here we stress that an optimal measurement means that the precision of anygiven parameter measurement is not degraded when comparing a multi-parameter approach with all other couplingsconstrained and an optimal single-parameter measurement discussed below. Several approaches have been adopted.In one approach, a small number of couplings, typically two or at most three, is considered. One of these is theSM-like coupling and the other could be parameterized with the cross-section fraction f gi defined above. While thisapproach is optimal for each parameter measurement, the problem with this approach is that correlations betweenmeasurements of different anomalous couplings are not considered.Another recently adopted approach is the STXS measurement, where cross sections of several H boson productionprocesses are measured in several bins based on kinematics of the event. While this approach is attractive due to itsapplicability to a number of various use cases, the problems with this approach are that observables are not necessarilyoptimal for any given measurement, and that the kinematics of events are assumed to follow the SM when measuringthe cross section in each bin. For a correct measurement, a full detector simulation of each coupling scenario isneeded, because the kinematics of associated particles and decay products would affect the measurement in eachbin. The STXS approach based on SM-only kinematics does not include these effects. The latter effect is especiallyimportant because neglecting it may lead to biases in the measurements. In the following, we illustrate the strengthsand weaknesses of each approach, and propose a practical method based on the matrix element approach.First, we would like to note that it is difficult to perform an unambiguous measurement of all 13 independent HV V anomalous couplings in Eq. (1) in a given process. For example, while all these couplings contribute to theVBF production, kinematics of
W W and ZZ fusion are essentially identical, as shown in Fig. 6. The measurementbecomes feasible when the W W and ZZ couplings are related. We adopt two examples of this relationship. Inone case, we simply set g W Wi = g ZZi , which could be interpreted as relationships in Eqs. (17–20) under the c w = 1condition. Such results could be re-interpreted for a different relationship of the couplings. In the second case, weadopt the relationships in Eqs. (17–21) without any conditions. With such a simplification, we are still left with nineparameters in the first case and eight parameters in the second case. To simplify the analysis further, we reduce thenumber of free parameters by setting g γγ = g γγ = g Zγ = g Zγ = 0. While we do expect to observe non-zero values of g γγ and g Zγ even in the SM, constraints on all four couplings are possible from decays H → γγ and Zγ with on-shellphotons. We leave the exercise to include all couplings in an optimal analysis to future studies. In addition, we keepthe g gg and g gg couplings as two free parameters as well. While the dedicated studies of these couplings are presentedin Section VI B, kinematics of the ggH process may affect measurements in the VBF process.As a reference, we take the STXS stage-1.1 binning as applied by the CMS experiment [41, 112]. In this approach,seven event categories are defined, which are optimal for separating the VBF topology with two associated jets; two V H categories, with leptonic and hadronic decay of the V , respectively; the VBF topology with one associated jet; two t ¯ tH categories, with leptonic and fully hadronic top decay, respectively; and the untagged category, which includesthe rest of the events. We call it stage-0 categorization. Each category of events is further split into sub-categories tomatch the requirements on the transverse momenta and invariant masses, as defined in the STXS stage-1.1 binning.In total, there are 22 categories defined [112]. While the above STXS stage-1.1 categorization provides fine binningfor capturing some kinematic features in production of the H boson, it does not keep any information from decay, ithas no information sensitive to CP violation, and more generally, it is not guaranteed to be optimal for measuringany of the parameters of our interest.Since we target the optimal analysis of four anomalous couplings expressed through f g , f g , f Λ1 , and f Zγ Λ1 , webuild the analysis in the following way. Instead of STXS stage-1.1 binning, we start from the seven categories definedin stage-0 for isolating different event topologies. Since in this analysis we do not target fermion couplings , thetwo t ¯ tH categories are merged with the untagged category. There are four discriminants relevant for this analysis,as defined by Eq. (44): D g , D g , D Λ1 , and D Zγ Λ1 . In addition, two interference discriminants, D CP and D int , aredefined by Eq. (45) for the g and g couplings, respectively. The two other interference discriminants are found toprovide little additional information due to large correlations with the discriminants defined in Eq. (44). The fullavailable information is used in calculating the discriminants in the following way. In the untagged category, the H → V V → (cid:96) information is used. In addition, the transverse momentum of the H boson is included, because it is There is an additional factor of ( −
1) in the definition of f Λ1 and f Zγ Λ1 following the convention in experimental measurements [73]. For a study of fermion couplings with this technique, see Ref. [4]. - - - - -
68% CL95% CL - f g4
68% CL95% CL l n L D - VV → H → 4ℓ, L = 3000 (300) fb -1 g , g , Λ , Λ unconstrained MELA STXS (stage-1)
MELA (decay) - - - - - - f g2
68% CL95% CL
68% CL95% CL l n L D - VV → H → 4ℓ, L = 3000 (300) fb -1 g , g , Λ , Λ unconstrained MELA STXS ( stage - ) MELA (decay) - - - - -
68% CL95% CL - f L
68% CL95% CL l n L D - VV → H → 4ℓ, L = 3000 (300) fb -1 g , g , g , Λ unconstrained MELA STXS (stage-1) MELA (decay) - - - - - - g L f
68% CL95% CL
68% CL95% CL l n L D - MELA STXS (stage-1) MELA (decay)
VV → H → 4ℓ, L = 3000 (300) fb -1 g , g , g , Λ unconstrained FIG. 10: Expected constraints from a simultaneous fit of f g , f g , f Λ1 , and f Zγ Λ1 using associated production and H → (cid:96) decay with 3000 (300) fb − data. Three analysis scenarios are shown: using MELA observables with productionand decay (or decay only) information, and using STXS binning. The dashed horizontal lines show the 68 and 95%CL regions.sensitive to production. In both the VBF and V H topologies with two associated jets, both production and decayinformation are used, except for the two interference discriminants, where production information is chosen becauseit dominates. In the leptonic
V H category and the VBF topology with one associated jet, where information is ingeneral missing, the transverse momentum of the H boson is used, with finer binning than in the untagged category.In the end, for each event in a category j a set of observables x is defined.To parameterize the 70 terms in Eq. (48) or the 15 terms in Eq. (49), we rely on samples generated with JHUGen.However, it is not necessary to generate 70 or 15 separate samples. Instead, we generate a few samples that adequatelycover the phase space and re-weight those samples using the MELA package to parameterize the other terms. Topopulate the probability distributions, we use a simulation of unweighted events with detector modeling, and smallstatistical fluctuations are inevitable. A critical step in the process is to ensure that even with these statisticalfluctuations, the probability density function P , defined in Eqs. (48) and (49), remains positive for all possible valuesof (cid:126)f . We detect negative probability by minimizing P , which is a polynomial in (cid:112) | f gi | · sign( f gi ). In the case ofEq. (48), where the polynomial is quartic, we use the Hom4PS program [114–116] to accomplish this minimization.If negative probability is possible, we modify P klmn or P kl using the cutting planes algorithm [117], using the Gurobiprogram [118] in each iteration of the procedure, until P is always positive. We find that only small modifications to P klmn or P kl are needed.In Fig. 10 we show the expected constraints on the four parameters of interest f g , f g , f Λ1 , and f Zγ Λ1 , using bothassociated production and H → (cid:96) decay with 3000 fb − (or 300 fb − ) of data at a single LHC experiment. Theconstraints on each parameter are shown with the other parameters describing the HV V and
Hgg couplings profiled,including f ggCP and the signal strength parameters µ V and µ f . The µ V and µ f parameters correspond to production2 - - - c d l n L D - MELASTXS (stage-1) c z , c zz , c~ zz unconstrained VV → H → 4ℓ , L = 3000 fb -1
95% CL68% CL - c l n L D - d c z , c z , c~ zz STXS ( stage - ) unconstrained MELA
VV → H → 4ℓ , L = 3000 fb -1
95% CL68% CL - c l n L D - MELA c zz , c~ zz STXS ( stage - ) unconstrained VV → H → 4ℓ , L = 3000 fb -1 d c z ,
95% CL68% CL - c~ l n L D - MELASTXS ( stage - ) unconstrained d c z , c z , c zz VV → H → 4ℓ , L = 3000 fb -1
95% CL68% CL
FIG. 11: Expected constraints from a simultaneous fit of (from left to right) δc z , c zz , c z (cid:3) , and ˜ c zz using associatedproduction and H → (cid:96) decay with 3000 fb − data. The EFT coupling constraints are the result of re-interpretationfrom the signal strength and f gi measurements discussed in text. The constraints on each parameter are shown withthe other parameters describing the HV V and
Hgg couplings profiled. Two analysis scenarios are shown: usingMELA observables and using STXS binning. The dashed horizontal lines show the 68 and 95% CL regions.strength of electroweak and other processes, respectively. Therefore, there are a total of seven free parametersdescribing
HV V and
Hgg couplings. The MC scenario has been generated with the SM expectation. The productioninformation dominates in all constraints. However, as discussed in Section II E, this is due to unbounded growthof anomalous couplings with q . Since this behavior cannot continue forever, it is still interesting to look at thedecay-only constraints, which do not rely on the q -dependent growth of the amplitude. Therefore, in Fig. 10 bothkinds of constraints are shown for illustration of the two limiting cases. We point out that form factor scaling, suchas introduced in Eq. (30), can be used for continuous study of this effect.In addition, a comparison is made to the approach where instead of the optimal discriminants, the STXS stage-1.1bins are used as observables, while using full simulation of all processes otherwise. There is a significant difference inexpected precision. The most striking effect is the lack of constraints from decay information, but there is a loss inprecision using production information in STXS as well. It is interesting to point that there is still weak decay-relatedinformation in the categories used in the STXS approach, because interference between identical leptons producesdifferent rates of 2 e µ events compared to 4 e and 4 µ , depending on the couplings.Since f gi measurements involve ratios of couplings, most systematic uncertainties that would otherwise affect thecross section measurements cancel in the ratio. Therefore, the f gi measurements are still expected to be statisticslimited with 3000 fb − of data. For this reason, the expected results can be easily reinterpreted for another scenarioof integrated luminosity, as for example the expectation with 300 fb − shown in parentheses. However, when the f gi measurements are re-interpreted in terms of couplings (as we illustrate below), both the signal strength and the f gi results need to be combined. This leads to sizable systematic uncertainties affecting the couplings. In the following,we assign 5% theoretical and 5% experimental uncertainties on the measurements of the signal strength, which is theratio of the measured and expected cross sections.We also perform a fit with three cross-section fraction parameters f g , f g , and f Λ1 with the EFT relationshipamong couplings following Eqs. (17–21). The conclusions of this study are similar to those presented above. We re-interpret these results as constraints on the δc z , c zz , c z (cid:3) , and ˜ c zz couplings, defined in the EFT parameterization inthe Higgs basis. This fit requires reinterpreting the process cross section and the three fractions in terms of couplings,and one has to take dependence of the width on the couplings into account, following Eq. (32). We assume thatΓ other = 0 and express the width using Eq. (34). The values of κ f = κ t = κ b = κ τ = κ µ and ˜ κ f = ˜ κ t = ˜ κ b = ˜ κ τ = ˜ κ µ are left unconstrained independently for the CP-even and CP-odd fermion couplings. The resulting one-dimensionalconstraints are shown in Fig. 11 and two-dimensional contours with the other parameters profiled are shown in Fig. 12.In Fig. 11 it is evident again that analysis based on the optimal discriminants provides the best constraints on thecouplings of interest. B. Hgg anomalous couplings
The gluon fusion process in association with two jets allows analysis of kinematic distributions for the measurementof potential anomalous contributions to the gluon fusion loop. Resolving the loop effects is a separate task, whichwe do not attempt to perform in this work. However, we point out that unless the particles in the loop are light,their mass does not significantly affect the kinematics of the H boson and associated jets. The main effect is on the H boson’s transverse momentum [41], where heavy particles in the loop may enhance the tail of the distribution at3 - - - c d - zz c l n L D - LHC , L = 3000 fb -1 VV → H → 4ℓ
68% CL95% CLSM - - - c d - z c l n L D - LHC , L = 3000 fb -1 VV → H → 4ℓ
68% CLSM95% CL - - - c d - zz c ~ l n L D - LHC , L = 3000 fb -1 VV → H → 4ℓ
68% CL95% CLSM - c - z c l n L D - LHC , L = 3000 fb -1 VV → H → 4ℓ
68% CL95% CLSM - c - zz c ~ l n L D - LHC , L = 3000 fb -1 VV → H → 4ℓ
SM68% CL95% CL - c - zz c ~ l n L D - LHC , L = 3000 fb -1 VV → H → 4ℓ
95% CL68% CLSM
FIG. 12: Expected two-dimensional constraints from a simultaneous fit of δc z , c zz , c z (cid:3) , and ˜ c zz as shown in Fig. 11 forthe MELA observables. The constraints on each parameter are shown with the other parameters describing the HV V and
Hgg couplings profiled. Top-left: ( δc z , c zz ); top-middle: ( δc z , c z (cid:3) ); top-right: ( δc z , ˜ c zz ); bottom-left: ( c zz , c z (cid:3) );bottom-middle: ( c zz , ˜ c zz ); bottom-right: ( c z (cid:3) , ˜ c zz ). p T >
200 GeV, but will not significantly affect the bulk of the distribution relevant for our study, at p T <
200 GeV.Our analysis of the CP properties of this interaction depends primarily on the angular kinematics of the associatedjets and H boson, as discussed in Section V B. Therefore, in the rest of this work we treat the gluon fusion processwithout resolving the loop contribution, allowing for any particles to contribute, either from SM or beyond. The onlyobservable difference in this analysis is between the CP-even and CP-odd couplings, which can be parameterized asthe overall strength of the H boson’s coupling to gluons and the fraction of the CP-odd contribution f ggCP defined inEq. (42).The analysis strategy follows the approach discussed in application to the HV V measurements in the previoussection, with the difference being the two-jet category optimized for the measurement of the gluon fusion process. Inaddition to a discriminant optimal for signal over background separation, the events are described by three observables.The D observables follows Eq. (44), with the VBF and gluon fusion matrix elements used to isolate the VBFtopology. The D ggH0 − and D ggHCP observables follow Eq. (44) and Eq. (45) for separating the SM-like coupling and CP-odd coupling, but with one modification to the process definition. Only the quark-initiated process defines the matrixelement in these two formulas, because only such a VBF-like topology of the gluon fusion process carries relevant CPinformation. This is illustrated in Fig. 13, where the left plot shows that the D ggH0 − discriminant starts to separatethe two couplings at higher values of D , which correspond to more VBF-like topology. The right plot shows thatonly at higher values of D can one observe the separation. Only a small fraction of the total gluon fusion eventsend up in that region. This illustrates the challenge of the CP analysis in the gluon fusion process. The D ggHCP leadsto forward-backward asymmetry in the distribution of events in the case of CP violation, when both CP-odd andCP-even amplitudes contribute.A projection of f ggCP sensitivity with 3000 and 300 fb − at an LHC experiment is performed. The overall normal-ization of the gluon fusion production rate in the VBF-like topology is provided by the untagged events and eventswith two associated jets in a non-VBF topology. The electroweak VBF process is a background to the f ggCP measure-4 -50 46 -6 -4 -1 1 -1 1 3 -13 -2 15 -5100 -67 4 -7 -2 0 -1 6 1 18 28 50-20 8 9 -3 -1 0 0 2 20 73 7 55-100 -41 -28 0 -7 -3 -1 2 2 21 14 52 33-100 -61 -25 -19 -5 -2 -2 5 3 11 31 51 77-79 -44 -45 -14 -1 -0 2 3 4 12 15 32 89-86 -57 -42 -19 -9 -3 -1 5 8 12 23 47 83-73 -54 -30 -17 -12 -5 -0 1 6 21 30 54 76-81 -52 -36 -20 -11 -6 -6 12 11 17 26 53 74-81 -56 -42 -33 -20 -7 -3 11 17 31 36 56 83 ggH0- D j e t D - - D i ff. [ % ] ggH D <0.5 et D >0.5 ( x 20 ) et D < e t D >0.5 ( x 20 ) et D ++-- FIG. 13: Left: The 2D distribution of the difference between the scalar and pseudoscalar populations of events forthe D and D ggH0 − discriminants, both self-normalized to 1, respectively. Right: D ggH0 − discriminant distributions ofthe scalar and pseudoscalar population events with the requirement on the D discriminant below or above 0.5. CP f (2) (4) (6) (8) gg - D l n L L H C , L = 3000 (300) fb -1 gg → H → 4ℓ , γγ, ττ f κ − f κ∼ − − l n L ∆ - SMbest fit68% CL95% CL L H C , L = 300 fb -1 gg → H → 4ℓ , γγ, ττ FIG. 14: Expected constraints on f ggCP with 3000 (300) fb − (left) and κ f and ˜ κ f couplings in the gluon fusion loopwith 300 fb − (right), using the H → (cid:96), γγ, and τ τ decays. The dashed horizontal lines (left) and contours (right)show the 68 and 95% CL regions.ment in this case, but its kinematics are still distinct enough to keep it separated in the fit on the statistical basis.Keeping its CP properties unconstrained has little effect on the CP analysis in the gluon fusion process. We use the H → (cid:96) analysis to illustrate the sensitivity, but scale the expected constraints with an effective luminosity ratioto account for the relative sensitivity of the H → γγ and τ τ channels based on the typical sensitivity in the VBFtopology [71, 72, 79, 80]. The expected constraints are shown in Fig. 14. With 3000 (300) fb − , one can separateCP-even and CP-odd Hgg couplings with a confidence level of about 9 (3) σ .We re-interpret these expected constraints on f ggCP and the cross section as constraints on the κ f = κ t = κ b and˜ κ f = ˜ κ t = ˜ κ b couplings in the gluon fusion loop, assuming that only the SM top and bottom quarks dominate theloop. There are additional considerations when re-interpreting the signal strength and f ggCP in terms of couplingsfollowing Eq. (32). As in the HV V measurements in Section VI A, we assume Γ other = 0 and express the width Γ tot using Eq. (34). In this approach, the overall signal strength of the VBF and VH processes, proportional to g , remainsunconstrained. Using the f ggCP and gluon fusion cross section constraints expected with 300 fb − of data at LHC withthe H → (cid:96), γγ, and τ τ decays, we show the expected constraints on ( κ f , ˜ κ f ) in Fig. 14 (right). The sign ambiguity( κ f , ˜ κ f ) ↔ ( − κ f , − ˜ κ f ) remains unresolved with experimental data, but the relative sign of κ f and ˜ κ f can be resolved,due to the D ggHCP observable. The measurement of the gluon fusion cross section alone leads to an elliptical constraint5in the 2D likelihood scan, with the eccentricity determined by the ratio σ (˜ κ f = 1) /σ ( κ f = 1) discussed earlier. The f ggCP measurement leads to constraints within the ellipse.Should sizable CP violation effects be hidden in the gluon fusion loop, they can be uncovered with the HL-LHC data sample. Further improvements in the CP constraints on ( κ f , ˜ κ f ) can be obtained by measuring the t ¯ tH process, where even stronger constraints are expected [4]. However, we would like to point out that the ratio σ (˜ κ f = 1) /σ ( κ f = 1) = 0 .
39 in the t ¯ tH process is by a factor of six different from the ggH process. This largedifference will lead to stronger constraints in the combination of the two measurements under assumption of the topquark dominance in the loop because of additional information from the ratio of cross sections. Nonetheless, themeasurement in the gluon fusion is not limited to the top quark Yukawa coupling, but may include other BSM effectsin the loop. Therefore, it is possible for CP effects to show up in the ggH measurement, but not in t ¯ tH . VII. APPLICATION TO OFF-SHELL H(125) BOSON PRODUCTION
We continue by investigating the off-shell production and decay of the H boson with its coupling to either strongor weak vector bosons. There have already been previous studies of the anomalous HV V couplings using these tools,with the most extensive analyses from CMS [67, 73]. Here we document and extend these studies, in particular toanomalous
Hgg couplings and to anomalous couplings in background processes, which do not include the H bosonpropagator, using the EFT relationship. We introduced the off-shell effect in Section III and discussed simulation andanalysis tools in Section IV. The special feature of off-shell production is the strong interference between the signalprocesses, which involve the H boson, and background processes due to the broad invariant-mass distributions of theoff-shell H boson. Analysis of the off-shell region is particularly important to constrain couplings directly, withoutthe complication of the width dependence that appears on-shell in Eq. (32). Equivalently, a joint analysis of theon-shell and off-shell regions leads to constraints on Γ other in Eq. (33). Moreover, the higher q transfer in the off-shelltopology can enhance the effects of anomalous couplings.In this Section, we set Γ tot = 4 .
07 MeV [99] and use the pole mass scheme in the gluon fusion loop calculations with m t = 173 . m b = 4 .
75 GeV [99, 100]. The QCD factorization and renormalization scales are chosen to runas m (cid:96) /
2. In order to include NNLO QCD corrections in the electroweak process, a k factor of 1.12 [99] is applied, seealso discussion of the k factor in application to the V H process in Section VIII. In order to include higher-order QCDcorrections in the gluon fusion process, the LO, NLO, and NNLO signal cross section calculations are performed usingthe MCFM and HNNLO [119–121] programs for a wide range of masses using narrow width approximation. The ratiobetween the NNLO and LO, or between the NLO and LO, values is used as a weight k factor. The NNLO k factorsare applied to simulation as shown below. While this procedure is directly applicable to the signal cross section, itis approximate for background and for signal-background interference. However, the respective NLO calculations areavailable [122–124] for the mass range 150 GeV < m (cid:96) < m t , and Ref. [125] found good agreement between theNLO signal, background, and interference k factors. Any differences set the scale of systematic uncertainties in ourprocedure. A. Off-shell effects due to Hgg anomalous couplings
The off-shell production of the H boson may provide a way to disentangle contributions to the gluon fusion loop fromeither SM-like couplings to the top and bottom quarks, CP-odd couplings of the H boson, or new heavy particles. Weillustrate this in Fig. 15 for CP-even couplings and in Fig. 16 for CP-odd couplings. In order to illustrate the effects,we separate the signal contributions from the top quark, the bottom quark, and an effective point-like interaction,with both CP-even and CP-odd couplings to the H boson. For illustration, the SM values of the HV V couplingsand of the H boson width Γ tot are assumed, but variations of these couplings are considered in Section VII B, and asimultaneous measurement with Γ tot can be considered. The tools allow the modeling of the gluon fusion loop withall possible couplings contributing simultaneously, including interference with the background gg → (cid:96) process. Theeffective point-like interaction is equivalent to heavy t (cid:48) and b (cid:48) quarks in the loop, and this can also be configured inthe JHU generator, with adjustable masses of the these new particles in the loop.While in Section VI it was shown how the CP-even and CP-odd couplings in the gluon fusion loop can be separatedby analyzing of on-shell H boson production in association with two jets, this approach does not allow us to resolvedifferent contributions to the loop. Off-shell production provides a way to separate those contributions. As can be seenin Figs. 15 and 16, the top quark contribution has a distinctive threshold enhancement around 2 m t , with somewhatdifferent behavior of the CP-even and CP-odd components. The heavy particle contribution proceeds without the2 m t threshold, and the light particle contribution is highly suppressed. Therefore, the off-shell spectrum can be usedto resolve the loop effects, such as to differentiate between the top quark and heavy BSM contributions in the loop6or to set limits on the light quark Yukawa couplings or other possible light contributions, similarly to the techniquesusing the on-shell H boson transverse momentum [41, 126]. In all cases, the CP-odd component’s interference withthe background is zero when integrated over the other observables. The actual analysis of the data will benefit fromemploying the full kinematics using the matrix-element approach. B. Off-shell effects due to HVV anomalous couplings
The off-shell production of the H boson also allows testing the anomalous HV V couplings of the H boson to twoelectroweak bosons, V V = W W, ZZ, Zγ, γγ . These couplings appear in the decay H → f in the gluon fusion processand in both production and decay in the electroweak process. The latter includes both VBF and V H production, andin all cases interference with the gluon fusion or electroweak background is included. Examples of such a simulation areshown in Fig. 17. Three anomalous couplings are shown for illustration, g ZZ = g W W , g ZZ = g W W , and κ ZZ , = κ W W , ,which involve interplay of either the H boson or the Z ( W ) boson going off shell. The anomalous couplings of the H boson to the photon are not enhanced off-shell and are not shown here, but can be considered in analysis. Therefore,it is important to stress here that it is natural to use the physical Higgs basis in the EFT analysis of the off-shellregion, since the behavior of the couplings involving the photon is drastically different.Examples of applications of the tools developed here, both simulation and MELA discriminants, can be foundin Refs. [73, 77] where simultaneous analysis of the H boson width and the couplings is performed both with thecurrent LHC data and in projection to the HL-LHC. For example, with 3000 fb of data, a single LHC experimentis expected to constrain Γ tot = 4 . +1 . − . MeV, as shown in Fig. 106 of Ref. [77]. With the current data sample fromthe LHC experiments, the off-shell region significantly improves the anomalous coupling constraints, even with Γ tot profiled [73]. This is evident from the enhancement observed in Fig. 17. The expected gain is not as large at theHL-LHC, as Fig. 39 of Ref. [77] shows, because with access to smaller couplings, the electroweak VBF and
V H production in the on-shell region plays a more important role.It has been pointed out [26, 42] that the gg → (cid:96) process also provides good sensitivity for constraining the topquark electroweak couplings. Similarly, gluon fusion in ZH production is sensitive to the same top quark couplingsand can be used to constrain them [127, 128]. In this work, however, we separate anomalous H boson couplings fromthe rest of the electroweak interactions where this is possible in a consistent way. For the above cases, separating theeffects is certainly possible, because top quark electroweak couplings can also be probed in e.g. pp → t ¯ tZ , which isindependent of the Higgs sector. Moreover, there are no EFT relations between electroweak top quark couplings and H boson couplings. However, the gauge boson self-interactions and the H boson couplings cannot be separated if the m [GeV]
200 400 600 800 - - - -
10 1
JHUGen+
MCFM+
HNNLO H LHC,
13 TeV l d s / d m l [f b / G e V ] gg fi l k t k b k Q [GeV]m
200 400 600 800 - - - - -
10 1 H+bkg+I k t JHUGen+
MCFM+
HNNLO d s / d m l [f b / G e V ] l LHC,
13 TeV bkg gg fi l k b k Q FIG. 15: The invariant mass distribution of four-lepton (4 (cid:96) = 2 e µ ) events produced through gluon fusion at theLHC with a 13 TeV proton collision energy. The different CP-even anomalous Hgg couplings are simulated withJHUGen+MCFM at LO in QCD, and the NNLO k factor is calculated with the HNNLO program, assuming signaland background k factors to be the same as for the SM H boson. Four off-shell scenarios are shown with the couplingschosen to match the SM on-shell gg → H → (cid:96) cross section: SM (solid black), top-quark only (magenta), bottom-quark only (red), and a point-like effective interaction (blue) shown in Eq. (37). The left plot shows the gg → H → (cid:96) process with only signal, while the right plot includes interference with the SM gg → (cid:96) background, which is alsoshown separately in the dotted histogram.7 [GeV]m
200 400 600 800 - - - -
10 1 H d s / d m l [f b / G e V ] l JHUGen+
MCFM+
HNNLO
LHC,
13 TeV gg fi l k t k b k Q ~~~ m [GeV]
200 400 600 800 - - - - -
10 1 d s / d m l [f b / G e V ] l LHC, 13 TeV
JHUGen+
MCFM+
HNNLO
H+bkg+I bkg gg fi l k t k b k Q ~~~ FIG. 16: The four-lepton invariant mass distributions in gluon fusion production as in Fig. 15, but with the threeCP-odd anomalous couplings instead, also chosen to match the SM on-shell gg → H → (cid:96) cross section. [GeV] l m
200 400 600 800 d s / d m l [f b / G e V ] -5 -4 -3 -2 -1
101 H+bkg+Ibkg
JHUGen+
MCFM+
HNNLO
LHC,
13 TeV g g κ ZZZZZZ gg fi l
500 1000 1500 2000 m [GeV] - - - -
10 110 LHC, 13 TeV
JHUGen +MCFM
H+bkg+I bkg l d s /dm4 l [fb/GeV] g g κ ZZZZZZ
EW 4 l +2j FIG. 17: The four-lepton 4 (cid:96) invariant mass distributions in gluon fusion (left, 4 (cid:96) = 2 e µ ) and in associated electroweakproduction with two jets (right, (cid:96) = e, µ, τ ) at the LHC with a 13 TeV proton collision energy. The anomalous HV V couplings are simulated with JHUGen+MCFM at LO in QCD, with the gluon fusion simulation settings and k factorsmatching those for SM in Fig. 15. Three anomalous
HV V couplings are modeled with coupling values chosen tomatch the SM on-shell gg → H → e µ cross section. Interference with the SM gg → (cid:96) (left) and electroweak (right)background is included.EFT relations Eqs. (22–27) are applied in continuum electroweak production. We can account for these relations asdiscussed in the following Section VII C. C. Off-shell effects due to gauge boson self-interactions
In Sections VII A and VII B, only modifications of the H boson couplings to either strong or weak gauge bosons areconsidered, and the background contributions in either gg → f or continuum electroweak production are assumedto be SM-like. However, as discussed in Section II, there is an intricate interplay between gauge boson self-couplingsand H boson gauge couplings. Therefore, under the EFT relationship, the HV V anomalous contribution wouldaffect the triple and quartic gauge boson self-couplings according to Eqs. (22–27). Figure 18 shows examples of the m (cid:96) distributions with anomalous gauge boson self-interactions in the EFT framework. These examples show theCP-conserving anomalous g ZZ and κ ZZ , couplings, which are modified in Eqs. (17–21) for the H boson couplings andin Eqs. (22–27) for the gauge boson self-couplings, keeping all the other anomalous couplings at zero. The size of theanomalous contribution is taken to be similar to the current constraints on anomalous H boson couplings [73] from8
500 1000 1500 2000 - - - - -
10 1
LHC, 13 TeV
JHUGen +MCFM g ZZ =2, g =0.1 (H)g ZZ =2, g = bk g+I)g =2, g =0.1 (bkg) ZZZZ ZZZZ
EW 4 l +2j SM (H+bkg+I) SM (bkg) d s / d m l [f b / G e V ] m l [GeV]
500 1000 1500 2000 - - - - -
10 1
SM ( H+bkg+I) SM (bkg) EW 4 l +2j=2, κ ZZ g ZZ =2, κ g ZZ =2, κ = -10 (bkg) g ZZZZZZ Λ = T eV =- (H) =- (H+bkg+I) LHC, 13 TeV
JHUGen +MCFM m l [GeV] d s / d m l [f b / G e V ] FIG. 18: The four-lepton invariant mass distributions in 4 (cid:96) ( (cid:96) = e, µ, τ ) associated electroweak production withtwo jets at the LHC with a 13 TeV proton collision energy. The anomalous HV V couplings are simulated withJHUGen+MCFM. The black distributions show two SM scenarios: background only (dashed) and the full contributionincluding the H boson (solid). The colored curves show an additional non-zero anomalous contribution from either g ZZ (left) or κ ZZ , (right) in the H boson production component (blue solid), in the background-only component (reddashed), and including all contributions (magenta solid).LHC measurements. It is evident from Fig. 18 that the resonant and nonresonant contributions are of similar size inelectroweak production and that there is a sizable interference between the two. While current analyses of LHC datatypically consider the H boson couplings and gauge boson self-interactions separately [73], the unified framework willallow future joint constraints. VIII. APPLICATION TO THE ZH PROCESS AT NEXT-TO-LEADING ORDER
Production of the H boson in association with an electroweak gauge boson is known for its clean experimentalsignature and its excellent sensitivity to the HV V couplings. During Run-II of the LHC, the experimental precision of ZH analyses [78–82] has reached a level of accuracy that requires theory simulation beyond leading order. Therefore,we account for the dominant perturbative corrections at next-to-leading order QCD in this work and make themavailable in the JHUGen framework.In addition to reducing theoretical uncertainties, the simulation at higher orders also reveals sensitivity to H bosoncouplings that are invisible at the lowest order. In ZH production, the gg → ZH sub-process enters for the first timethrough one-loop diagrams. The box diagram contribution in Fig. 4 yields sensitivity to the Hf ¯ f couplings κ f and˜ κ f in Eq. (2), which are screened in the q ¯ q production process.We must stress that whenever we work with ZH production, γH and γ ∗ H production are equally important. Thisallows us to set constraints on anomalous Hγγ and
HZγ couplings, and we provide this functionality in the JHUGenframework.In Fig. 19 we show the SM m ZH distribution assuming the decays Z → (cid:96) + (cid:96) − and H → b ¯ b at the 14 TeV LHC.Contributions from q ¯ q and gg initial states are shown separately, as are the triangle and box diagram parts ofthe gg partonic process, shown in Fig. 4. The widths of the bands correspond to systematic uncertainties fromvarying the scale by a factor of two around its central value µ = m ZH . Close to the production threshold at m ZH = m Z + m H ≈
220 GeV, the q ¯ q initial state dominates the cross section. Above the 2 m t ≈
345 GeV energy, thetop-quark induced gg initial state becomes much more relevant. In particular, the triangle loop contribution, shownin blue, becomes as large as the q ¯ q contribution, shown in black, and even exceeds it at very high energies. However,as can be seen from the purple band in Fig. 19 the quantum interference between the triangle and box diagrams isstrongly destructive and reduces the overall gg contribution significantly. For this reason, the gg contribution playsonly a marginal role in the SM description of the ZH process.However, in anomalous coupling studies of physics beyond the SM, this strong destructive interference can beperturbed and lead to significantly larger cross sections. Hence, it is a sensitive probe of modifications from the SM.The gg → ZH process has yet another interesting feature. A superficial inspection of the triangle loop contribution inFig. 19 suggests sensitivity to the HV V couplings in Eq. (1). Yet, an explicit calculation shows that the contributions9 − − − −
200 300 400 500 600 700 800 900 1000 d σ d m Z H h f b G e V i m ZH [GeV] q ¯ q → ZH → ‘ − ‘ + b ¯ b , LO q ¯ q → ZH → ‘ − ‘ + b ¯ b , NLObox → ZH → ‘ − ‘ + b ¯ b triangle → ZH → ‘ − ‘ + b ¯ bgg → ZH → ‘ − ‘ + b ¯ b d σ d D gg Z H . . . . . . . . . gg → ZH → ‘ − ‘ + b ¯ bq ¯ q → ZH → ‘ − ‘ + b ¯ b D ggZH σ FIG. 19: Distribution of the m ZH invariant mass (left) and D ggZH discriminant (right) in gg and q ¯ q → ZH → (cid:96) − (cid:96) + b ¯ b processes under the SM hypothesis in 14 TeV LHC collisions, with several components isolated for illustration. Theleft plot shows the differential cross section for the q ¯ q → ZH process at both LO and NLO in QCD. − − − − d σ d m Z H [ f b / G e V ] σ S M σ d σ / d Z H d σ S M / d Z H m ZH [GeV] d σ d c o s θ [ f b ] σ S M σ d σ / d c o s θ d σ S M / d c o s θ -1 -0.8 -0.6 -0.4 -0.2 cos θ d σ d c o s θ [ f b ] σ S M σ d σ / d c o s θ d σ S M / d c o s θ -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cos θ FIG. 20: Kinematic distributions for simulated gg → ZH → (cid:96) − (cid:96) + b ¯ b events at 14 TeV for several scenarios: κ =1 , ˜ κ = 0 (SM, black crosses); κ = − , ˜ κ = 0 (black boxes); κ = 0 , ˜ κ = ± κ = 1 / √ , ˜ κ = ± / √ κ = − / √ , ˜ κ = ± / √ g V V and g V V drop out and yield a zero contribution to the complete squared one-loop amplitude. Moreover,both the triangle and the box diagrams are only sensitive to the axial-vector coupling of the gauge boson to thefermions in the loop. Hence, photons do not couple to the closed fermion loop. The process gg → γH can onlyproceed through an intermediate Z ∗ , which decays into γH via the κ Zγ coupling in Eq. (1). Obviously, gg → W H does not exist because of charge conservation. The absence of sensitivity to anomalous
HV V couplings g V V and g V V does not render the gg → ZH process completely irrelevant to their study. If we assume, for example, that the HZZ interaction involves non-zero g ZZ , which does not contribute to the triangle diagrams, the pattern of destructiveinterference with the box diagram would change. Hence, there is a strong sensitivity to the HV V couplings, which isentangled with possible anomalous values of the Hf ¯ f couplings.The above mentioned special features of the loop-induced gg process motivate separating it from the q ¯ q productionmode. While it is not feasible to isolate one process from the other on an event-by-event basis, it is possible to enhance(or decrease) the relative fraction of the two processes. We use the MELA approach with the D ggZH discriminantcalculated according to Eq. (44) with sig = gg → ZH and alt = q ¯ q → ZH . The distributions of D ggZH for the gg - and q ¯ q -initiated processes are shown in Fig. 19. As an example, in a restricted range of the D ggZH observable which keeps80% of the gg -initiated process, the fraction of this process is enhanced from 7% to 22%. As described in Section V,this approach is superior to selection based on individual kinematic observables, such as selecting higher values of m ZH or p H T [129].0 . . . . . q ¯ q → ZH, Z → ‘ − ‘ + σ L O S M σ L O d σ d m Z H [ f b / G e V ] m ZH [GeV] NLO/LO, SMNLO/LO, g LO/LO, SMNLO,
SMLO,
SMNLO, g LO, g q ¯ q → ZH, Z → ‘ − ‘ + σ L O S M σ L O d σ d c o s θ [ f b ] NLO, SMLO, SMNLO, g LO, g cos θ NLO/LO, SMNLO/LO, g LO/LO, SM . . . . . . . . . . . q ¯ q → ZH, Z → ‘ − ‘ + σ L O d σ d D − NLO, SMLO, SMNLO, g LO, g D − NLO/LO, SMNLO/LO, g LO/LO, SM 0 . . . . . . . . . . . . q ¯ q → ZH, Z → ‘ − ‘ + σ L O d σ d D h + NLO, SMLO, SMNLO, g LO, g D h + NLO/LO, SMNLO/LO, g LO/LO, SM
FIG. 21: Selected kinematic distributions for simulated q ¯ q → ZH → (cid:96) − (cid:96) + b ¯ b events at 14 TeV shown at LO (dotted)and NLO (solid) in QCD. The SM (black) and anomalous coupling model (red) are shown. The anomalous couplingmodel shown is the pseudoscalar model f g = 1 in all plots except for the D h + discriminant distribution, where the f g = 1 model is shown instead. The bottom panels show the k -factor ratios.In the following, we present anomalous coupling results for the gg and q ¯ q processes separately. This serves toillustrate the particular anomalous coupling features of our framework. A full experimental result needs to includeboth processes together. In Fig. 20, we show the effects of several combinations of anomalous Ht ¯ t couplings in the gg → ZH process and compare the shape changes to the SM prediction. We use the general interaction structurefrom Eq. (2) with CP-even ( κ ) and CP-odd (˜ κ ) Yukawa-type couplings and consider the scenarios with a wrong sign Yukawa coupling, pure CP-odd couplings, and mixtures of CP-even and CP-odd couplings. In Fig. 20, we show the ZH invariant mass and the two angles θ and θ . The angles are defined for the V H process in Fig. 5 and Ref. [3], butwe note that it is possible to define the sign of cos θ when θ is the angle between the Z boson and the longitudinaldirection of the overall boost of the V ∗ → ZH system, defined in the V ∗ rest frame. In the case where the V H systemhas finite transverse momentum, we first boost the system in the transverse direction to set the transverse motion tozero. For a practical application of this approach at the LHC, see Ref. [130].In the upper row of Fig. 21 we present the q ¯ q → ZH process and compare the leading order with the next-to-leading QCD prediction. The black and red curves correspond to the SM HZZ coupling and a pure CP-odd couplingfrom g ZZ in Eq. (1), respectively. Shape changes due to the different coupling structure are significant, even on alogarithmic scale. Hence, this process offers strong discrimination power even for small admixtures of g ZZ into theSM-like g ZZ coupling structure. The higher order corrections, shown in the differences between the solid and dottedcurves and in the lower panes, are positive, fairly constant, and O (+10%). In the lower row of Fig. 21 we showthe matrix element discriminants D h + and D − , defined in Eq. (44), for the alternative hypotheses g ZZ = 1 and g ZZ = 1, respectively. Again, the NLO QCD corrections are fairly constant over a wide range. The plots show strongdiscrimination power between each anomalous hypothesis and the SM. The results allow for a more accurate estimateof systematic uncertainties in future analyses, and the flat correction reinforces previous leading order studies.1 IX. APPLICATION TO NEW RESONANCE PRODUCTION
The techniques developed for the study of the H (125) boson would apply to a search for or a study of a newresonance X ( m X ) which may arise in the extensions of the SM, such as any Singlet model or Two Higgs Doubletmodel. For example, if any enhancement or modification of the di-boson spectrum or kinematics in the off-shell regionis observed, one would have to determine the source of this effect. For example, it might come from a modification ofthe H boson couplings in the off-shell region, including anomalous tensor structures; a modification of the continuumproduction, possibly from anomalous self-interactions; or yet another resonance X with a larger mass. This latterscenario is necessary to consider in order to complete the experimental studies.If a new state X is observed, one would need to determine its spin and parity quantum numbers in all accessiblefinal states. The techniques discussed in Section VI would be directly relevant. If the width of the resonance is sizable,interference with background, as discussed in Section VII, will become relevant. Moreover, interference with the off-shell H (125) boson tail would become important as well. All these effects are included in the coherent frameworkof the JHU generator with the modified MCFM matrix element library, and are available in the MELA package forMC re-weighting and optimal discriminant calculations. They have been employed in analyses of Run-II of LHCdata [75, 76].Applications of off-shell H (125) simulation with an additional broad X ( m X ) resonance are shown in Figs. 22,23, and 24. The cross section of the generated resonance X corresponds to the limit obtained by the recent CMSsearch [75], which includes all interference effects of a broad resonance. The most general XV V and
Xgg couplingsdiscussed in application to
HV V and
Hgg in Sections VI and VII are possible. It is interesting to observe that in thescalar case, the interference of X ( m X ) with the H (125) off-shell tail and its interference with the background haveopposite signs and partially cancel each other, but the net effect still remains and alters the distributions. In thecases of anomalous XV V couplings, the size of the interference changes, and in some particular cases, such as the g coupling, even the sign of the interference flips. The point-like Xgg couplings are also tested and shown in Fig. 23,which models the scenario when new heavy states in the gluon fusion loop are responsible for production of the newstate X . - - d s /dm4 l [fb/GeV] m l [GeV] X(450), G =47 GeV H(125), G =0.004 GeV bkg gg fi l X(450)+H(125)+bkg+I
LHC, 13 TeV
JHUGen +MCFM
FIG. 22: Differential cross section of the gluon fusion process gg → ZZ/Zγ ∗ /γ ∗ γ ∗ → (cid:96) as a function of invariantmass m (cid:96) generated with JHUGen+MCFM at LO in QCD. The distribution is shown in the presence of a hypotheticalscalar X (450) resonance with SM-like couplings, m X = 450 GeV, and Γ X = 47 GeV. Several components are eitherisolated or combined as indicated in the legend. Interference (I) of all contributing amplitudes is included. X. SUMMARY
In this paper, we have investigated the Higgs boson interactions at invariant masses both at and well beyondthe mass peak. We considered the weak vector boson fusion process pp → f + jj , gluon fusion pp → f , andassociated production pp → V H . All three processes contribute to both the on-shell and off-shell signal regions.NLO QCD effects, including the gg initial state, are investigated in pp → V H production. Through these processes,we study
HV V interactions in the regime of large momentum transfer, which exposes the unitarization feature inthe Standard Model and is sensitive to the mechanism of electroweak symmetry breaking at high energies. Ourframework allows a general coupling parameterization for the 125 GeV Higgs boson and for a possible second spin-zero resonance. Modifications of the triple and quartic gauge boson couplings are also considered. Deviations from theSM expectation can be parameterized in terms of anomalous couplings, effective field theory operators, and pseudo2
300 400 500 600 700 8000.001 - - g2 =47 GeV, f G X(450), =0.004 GeV G H(125), X(450)+H(125)+bkg+IX(450)+H(125)+bkg+I, point-like d s / d m l [f b / G e V ] m l [GeV] bkg gg fi l LHC, 13 TeV
JHUGen +MCFM
300 400 500 600 700 8000.001 - - g4 =47 GeV, f G X(450), =0.004 GeV G H(125), X(450)+H(125)+bkg+IX(450)+H(125)+bkg+I, point-likebkg gg fi l m l [GeV] d s / d m l [f b / G e V ] LHC, 13 TeV
JHUGen +MCFM
FIG. 23: Same as Fig. 22, but for the anomalous couplings of a new resonance X (450). A scalar resonance with f g = 1 (left) and a pseudoscalar resonance with f g = 1 (right) are considered. Both a top loop and a point-likeinteraction are considered in the gluon fusion production.
200 400 600 800 1000 1200 1400 1600 1800 2000 22000.05 - - · X-H interferenceX-bkg interferenceX(1000), G =100 GeV H(125), G =0.004 GeV bkg EW 4 l +2jX(1000)+H(125)+bkg+I m l [GeV] d s / d m l [f b / G e V ] LHC, 13 TeV
JHUGen +MCFM
200 400 600 800 1000 1200 1400 1600 1800 2000 22000.05 - - · X-H interferenceX-bkg interference=100 GeV G X(1000), =0.004 GeV G H(125), bkg EW 4 l +2j =1 g2 X(1000)+H(125)+bkg+I, f =1 g4 X(1000)+H(125)+bkg+I, f m l [GeV] d s / d m l [f b / G e V ] LHC, 13 TeV
JHUGen +MCFM
FIG. 24: Differential cross section of the electroweak production process qq → qq ( ZZ/Zγ ∗ /γ ∗ γ ∗ → (cid:96) ) as a func-tion of invariant mass m (cid:96) generated with JHUGen+MCFM. The distribution is shown in the presence of a hypo-thetical X (1000) resonance with SM-like couplings (left) and anomalous couplings ( f g = 1 and f g = 1, right), m X = 1000 GeV, and Γ X = 100 GeV. Several components are either isolated or combined as indicated in the legend.Interference (I) of all contributing amplitudes is included.observables. The framework of the JHUGen event generator and MELA library for the matrix element analysis enablesimulation, optimal discrimination, reweighting techniques, and analysis with the most general anomalous couplingsof a bosonic resonance and the triple and quartic gauge boson interactions. The capabilities of the framework havebeen illustrated with projections for measuring the EFT operators with the expected full data samples of the LHCand the High-Luminosity LHC. Acknowledgments : We acknowledge the contributions of CMS collaboration colleagues to the MELA project devel-opment and help with integration and validation of the JHU event generator. We thank Tianran Chen for updatingthe Hom4PS program for this analysis and providing support, we thank Amitabh Basu for suggesting the cuttingplanes algorithm in application to our analysis, we thank Michael Spira and Margarete M¨uhlleitner for help with theHDECAY and C2HDM HDECAY programs, we thank Andrew Gilbert for help with features of the Root program,we thank Giovanni Petrucciani for discussion of the Higgs cross section studies, we thank Jared Feingold for machinelearning studies, we thank Savvas Kyriacou for help with the fit implementation, and we thank Jeffrey Davis for helpin cross section calculations and interface to different coupling conventions in the generator. This research is partiallysupported by the U.S. NSF under grants PHY-1404302 and PHY-1707887, by the Fundamental Research Funds forthe Central Universities (China), and by the U.S. DOE under grant DE-SC0011702. Calculations reported in thispaper were performed on the Maryland Advanced Research Computing Center (MARCC).Note added: A new application of this framework to LHC data appeared in Ref. [131]. We would also like to pointthat the relative sign of the CP-even and CP-odd couplings in Eq. (37) is consistent with Refs. [132–134], while thesign was reversed between Refs. [23, 135]. We adopt the sign convention of the antisymmetric tensor ε = +13consistent with Refs. [132–134] and thank Werner Bernreuther for pointing out the sign ambiguity. Appendix A: Coupling relation to the Warsaw basis
Translations of the EFT operators between the Higgs basis and the Warsaw basis, which is defined in Ref. [88],can be performed with tools such as Rosetta [136]. For example, under the assumption that δc z = δc w in Section II,we have five independent CP-even and three CP-odd electroweak HV V operators, as well as one CP-even and oneCP-odd
Hgg operator in the Higgs basis. The same number of independent H boson operators exists in the Warsawbasis. The relationship between the six CP-even operators is quoted explicitly in Eq. (14) of Ref. [136]. Eliminatingthe assumption δc z = δc w yields one additional degree of freedom: δm in the Higgs basis; ∆ M W in our anomalouscoupling approach in Eq. (17); and a linear combination of three coefficients, called δv in Ref. [136], in the Warsawbasis.We extend the above equivalence to include the CP-odd operators and derive the translation of the four operatorsbetween the Higgs basis and the Warsaw basis as g ZZ = − v Λ (cid:16) s w w φ ˜ B + c w w φ ˜ W + s w c w w φB ˜ W (cid:17) ,g γγ = − v Λ (cid:16) c w w φ ˜ B + s w w φ ˜ W − s w c w w φB ˜ W (cid:17) ,g Zγ = − v Λ (cid:18) s w c w ( w φ ˜ W − w φ ˜ B ) + 12 ( s w − c w ) w φB ˜ W (cid:19) ,g gg = − v Λ w φ ˜ G . (A1)Another set of coefficients is sometimes used and is related to the Warsaw basis through C ϕ ˜ B = − c w Λ e w φ ˜ B ,C ϕ ˜ W = − s w Λ e w φ ˜ W ,C ϕB ˜ W = − s w c w Λ e w φB ˜ W ,C ϕ ˜ G = − w φ ˜ G Λ g s . (A2)We would like to note that it is a question of convenience which operators in the Higgs basis are chosen asindependent ones under the SU(2) × U(1) symmetry. For studies performed in this paper, we find it convenient to pick δc z , c zz , c z (cid:3) , ˜ c zz , c zγ , ˜ c zγ , c γγ , ˜ c γγ , c gg , and ˜ c gg as an independent set of HV V and
Hgg couplings and use Eqs. (17–21)to express the other couplings listed in Eq. (8). In other analyses, another convention may be more convenient. Forinstance, when performing measurements with the
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