Determining the CP Property of ht ¯ t Coupling via a Novel Jet Substructure Observable

Determining the CP property of the Higgs boson is important for a precision test of the Standard Model as well as for the search for new physics. We propose a novel jet substructure observable based on the azimuthal anisotropy in a linearly polarized gluon jet that is produced in association with a Higgs boson at hadron colliders, and demonstrate that it provides a new CP -odd observable for determining the CP property of the Higgs-top interaction. We introduce a factorization formalism to define a polarized gluon jet function with the insertion of an infrared-safe azimuthal observable to capture the linear polarization.

Introduction.-Pinning down the CP nature of the Higgs-top interaction (ht t) is an important program being pursued at the Large Hadron Collider (LHC) [1][2][3][4][5].Any deviation from a Standard-Model-like ht t coupling could indicate new physics as well as provide a potential source for the CP violation as required by the baryogenesis [6].Unlike CP -violating Higgs interactions with vector bosons, which arise from dimension-six operators, CP -violating effects in the ht t coupling could occur via a dimension-four operator, and can be potentially larger.In Eq. ( 1), y t = √ 2m t /v is the Yukuwa coupling of Higgs and top quark in the Standard Model (SM), and (κ, κ) parametrize the CPeven and CP -odd ht t interactions, respectively, which can be reparametrized as (κ, κ) = κ t (cos α, sin α), with α being the CP phase.The SM corresponds to (κ, κ) = (1, 0) or (κ t , α) = (1, 0).
In this Letter, we propose a new CP -odd observable, for probing the ht t interaction, which originates from a linearly polarized gluon in the associated production of a Higgs boson and gluon jet (hg).The essential observation is that a singly polarized gluon can be produced from the hard scattering of unpolarized partons.After its production, the gluon fragments into a jet with some linear polarization that breaks the rotational invariance around the jet direction and orients the jet constituents according to the hippopedal distribution, const.+ ξ 1 cos 2ϕ + ξ 2 sin 2ϕ . ( Here, as will be defined below, ξ 1 and ξ 2 parameterize the two degrees of freedom of the linear polarization and depend on both the kinematics of the hard process and the ht t couplings, κ and κ.The azimuthal angle ϕ is defined in the x-ŷ plane of the coordinate system, shown in Fig. 1 (right), where ẑlab is the beam direction, and k is three-momentum of the gluon jet in the partonic center-of-mass (c.m.) frame.The azimuthally anisotropic jet image in Eq. ( 2) can be measured as a new jet substructure observable and provide sensitivity to the CP phase of the ht t interaction.In particular, we will show that ξ 2 and the associated sin 2ϕ structure are CP -odd.They are more sensitive to a small CP phase α than ξ 1 , including the sign of α.Contrary to building upon a neutral state of charged particles and antiparticles [35], this CP -odd observable is constructed purely out of the kinematic information in the gluon jet.Such CP sensitivity in hg production would not be possible without exploring the gluon jet substructure, which has not been considered previously in the literature.We also note that associated Higgs-top production and indirect measurements via hV or V V production also depend on the hV V couplings and require assumptions on the latter, whereas hg production only depends on the ht t coupling.
Linearly polarized gluon in hg production.-Thepolarization state of the produced gluon is described by a density matrix, represented in the helicity basis as with three polarization degrees of freedom, ξ ≡ (ξ Thus, ξ 1 and ξ 2 are differences between linear polarization degrees along two orthogonal directions.It is readily seen that under CP transformation, (ξ 1 , ξ 2 ) → (ξ 1 , −ξ 2 ) so they are CP -even and CP -odd, respectively.The ambiguity in defining ẑlab in Eq. ( 3) at a pp collider merely implies the change (x, ŷ) → (−x, −ŷ), which does not affect linear polarization states, contrary to transverse spins of fermions [36].
The gluon produced in the hg process has a significant linear polarization.At leading order (LO), both gg fusion and q q annihilation contribute via a top loop, as exemplified in Fig. 1 (left) for the gg channel.Even though the q q channel can also produce a substantially polarized gluon, its contribution to the total cross section is much smaller and will be neglected.Parametrizing the helicity amplitudes g(λ 1 ) g(λ 2 ) → h g(λ 3 ) in the partonic c.m. frame in terms of the gluon's transverse momentum p T , rapidity y g , and azimuthal angle ϕ g , we have with f abc the color factor, and λ i the gluon helicities.The p T and y g sufficiently determine the Higgs energy, T cosh 2 y g , and the partonic c.m. energy √ ŝ = p T cosh y g + E h , with m H being the Higgs mass.A and A are the CP -even and CP -odd helicity amplitudes, respectively, constrained by their CP properties as The gluon density matrix is determined through where the convention of summing over repeated indices is taken, and |M| 2 is the unpolarized squared amplitude, averaged/summed over the spins and colors, with N c,g = 8.Due to their CP properties in Eq. ( 7), A and A individually only contribute to ξ 1 , while it is their interference that contributes to ξ 2 .In terms of the CP phase α, ξ 1 and ξ 2 can be expressed as where we have defined the polarization parameters with the notations Parametrizing ξ 1,2 as in Eq. ( 9) clearly shows that the polarization only depends on the CP phase α, but not on the coupling strength κ t , which only controls the event rate.The helicity polarization ξ 3 is also nonzero as √ ŝ > 2m t , but its value is generally small compared to ξ 1 and ξ 2 , and will not be discussed in this work.
The parameters (∆, ω, β 1 , β 2 ) are functions of p T and y g , as shown in Fig. 2(a) for some benchmark phase-space points.While ∆ describing the relative difference between the CP -even and CP -odd amplitude squares stays relatively flat around −0.4 when p T < 10 TeV, the parameters ω, β 1 , and β 2 , which control the sizes of the polarizations ξ 1 and ξ 2 , vary sizably with p T .Based on their p T dependence, we divide the phase space into three kinematic regions and discuss them in turn.
1. Low-p T region: p T ≲ 100 GeV.Both |ω| and β 1 have large values, whereas β 2 ≃ 0. The linear polarization is thus dominated by ξ 1 , with ξ 2 ≃ 0. The dominance of ω over β 1 further implies that ξ 1 does not depend sensitively on α.Being well below the √ ŝ = 2m t threshold, this region can be well approximated by the infinite-top-mass effective field theory (EFT) [37,38].In Fig. 2(b), the SM predictions for ξ 1 are shown for both the full one-loop calculation and the EFT approximation, where one can see that ξ 1 generally has a large negative value, which means that the produced gluon is dominantly polarized along the x direction in the production plane, cf.Eq. ( 5).Furthermore, it is not dramatically dependent on the gluon rapidity y g .Since the low-p T region contains most of the hg events, it is suitable for testing the linear polarization phenomenon.We expect a significant cos 2ϕ jet anisotropy due to the dominant ξ 1 .The insensitivity to α also enables this region to serve as a calibration region for experimentally measuring the linear polarization, which is important to ensure its viability and to understand the systematic uncertainties of the measurement since such phenomenon has not been observed before.
2. Transition region: 100 GeV ≲ p T ≲ 300 GeV.β 1 and ω rapidly go to 0 and flip their signs, while |β 2 | starts growing to an appreciable value.Hence, the linear polarization is dominated by ξ 2 if α is not too small, as illustrated in Fig. 2(c) for ξ 1 at α = 0, and ξ 2 at α = π/4, which corresponds to a maximal CP mixing.A nonzero α would then lead to linearly polarized gluon jets featuring a sin 2ϕ anisotropy, whose measurement provides a good opportunity for constraining the CPodd coupling.Both ξ 1 and ξ 2 are sensitive to y g , and their magnitudes are larger for gluons at more central rapidity region.Since this region covers the √ ŝ = 2m t threshold, EFT is no longer a good approximation, as indicated in the right half of Fig. 2(b).
3. High-p T region: p T ≳ 300 GeV.Both β 1 and β 2 have appreciable negative magnitude.Their values grow and approach each other as p T increases.Moreover, ω, being smaller than |β 1 |, becomes less important in ξ 1 .Qualitatively, we can interpret this region by taking (ω, ∆) → 0 and (β 1 , β 2 ) → β, which gives (ξ 1 , ξ 2 ) ∼ β(cos 2α, sin 2α).Then the jet anisotropy in Eq. ( 2) can be recast as const.+ β cos 2(ϕ − α) , (10) so that the main axis direction of the jet image gives a direct measure of the CP phase.In fact, it can be shown that as ŝ → ∞, this qualitative simplification becomes exact within one-loop calculation.The quantitative behavior of ξ 1 and ξ 2 in the high-p T region is shown in the right half of Fig. -Around the same time as QCD was developed, it was noted that linearly polarized gluons with nonzero ξ 1 can be produced in hard collision processes [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55], and some non-perturbative arguments were used in favor of oblate gluon jets characterized by a cos 2ϕ distribution.In the presence of a CP -violating interaction as considered in this work, a nonzero ξ 2 polarization is also produced leading to an additional sin 2ϕ structure, which serves as a handle to probe the CP structure.
Here, we introduce the polarized gluon jet in terms of the modern factorization formalism, for the first time.The polarized gluon turns into a jet that imprints its polarization information in the azimuthal distribution of its constituents, which can be projected out by weighting each event by some azimuth-sensitive observable.The azimuthally weighted cross section σ w of the inclusive hg production at a pp collider can be factorized into a hard scattering cross section, as given in Eq. ( 8), multiplied by a polarized gluon jet function, in much the same way as the factorization for an unpolarized jet function [56][57][58] or fragmentation function [59,60].It reads as up to corrections of powers of m J /p T and the jet size R. Here, dσ/dy g dp 2 T = L(s, ŝ) |M| 2 /16πE h √ ŝ is the differential cross section for the on-shell gluon production, where L(s, ŝ) = 1 ŝ/s dx/(xs) f g/p (x, µ F )f g/p (ŝ/xs, µ F ) is the gluon-gluon parton luminosity, with the factorization scale chosen at µ F = p T in the parton distribution function (PDF) f g/p (x, µ F ) of the proton, and we have used the LO kinematics to integrate over the Higgs phase space.
In the partonic c.m. frame, the gluon momentum k defines the jet mass m 2 J = k 2 and direction ẑ as in Eq. ( 3).By defining two lightlike vectors n µ = (1, −ẑ)/ √ 2 and nµ = (1, ẑ)/ √ 2, we can approximate the gluon momentum in the hard part to be on shell by only retaining the large component, p µ g = (k • n)n µ , which then defines the rapidity y g and p T = k • n/( √ 2 cosh y g ).To the leading power of m J /p T , the on-shell gluon carries the polarization ξ and fragments into a jet, described by the polarized jet function dJ(ξ, m 2 J , ϕ)/dϕ, where X denotes the state of the particles within the jet, in accordance with the jet algorithm [57,61], whose momenta are dominantly along n.G µν c is the gluon field strength tensor, and W ab (∞, x; n) is the Wilson line in the adjoint representation from x to ∞ along n, with the color indices a, b, and c summed over.In Eq. ( 12), the gluon polarization states are projected using the on-shell polarization vectors ε µ λ (p g ) with helicity λ = ±1, which are then averaged with the density matrix ρ λλ ′ (ξ).The resultant azimuthal distribution is extracted by inserting the observable where p i,T and ϕ i are, respectively, the transverse momentum and azimuthal angle of the jet constituent i with respect to the x-ŷ plane defined in Eq. ( 3).The ϕ distribution is a new jet substructure observable introduced by the linear polarization.The dependence on ξ 3 would vanish due to parity invariance of O(ϕ, X).
As a result of the p i,T weight, the observable O(ϕ, X) is infrared (IR) safe, and hence the polarized gluon jet function is insensitive to hadronization effects and becomes perturbatively calculable, with a predictable ϕ dependence.However, it was noted long before [43,50] that the gluon polarization information will be greatly washed out by the cancellation between the g → gg and g → q q channels, which was also found recently in a similar situation [62][63][64].It is possible to mitigate these effects by using jet flavor tagging techniques [65][66][67][68][69][70][71][72][73][74][75][76].For example, one may recluster the identified gluon jet into two subjets, and only keep those gluon jets with their two subjets tagged as quarks.At O(α s ), requiring a tagged quark in the gluon jet leaves g → q q as the only diagram, giving the polarized gluon jet function, where the jet algorithm dependence does not come in at this order to the leading power of m J .Eq. ( 14) needs to be multiplied by the tagging efficiency when used in Eq. (11).Although flavor tagging reduces the statistics significantly, it enhances the gluon spin analyzing power from O(1%) to about 50% [50] and will improve the statistical precision.
Before closing this section, we note the difference of the gluon polarization from a quark.While a transversely polarized light (massless) quark can also be produced from hard scattering processes, its transverse spin cannot be conveyed via the perturbative quark jet function due to the chiral symmetry of a massless quark.It is hence related to chiral symmetry breaking and must require the presence of some non-perturbative functions [60,77,78].
Phenomenology.-The gluon jet azimuthal anisotropy in Eq. ( 14) can be experimentally measured by simply constructing the asymmetry observables [79] where i ∈ {1, 2}, F 1 (ϕ) = cos 2ϕ and F 2 (ϕ) = sin 2ϕ.The uncertainties of the asymmetries A 1,2 are dominated by statistical ones, given by 1/ √ N with N being the number of the observed events.Now we provide a simple demonstration of the constraining power of the gluon linear polarization on the CP phase, by confining ourselves to the transition region for both the HL-LHC at 14 TeV and FCC-hh at 100 TeV, with integrated luminosities 3 ab −1 and 20 ab −1 , respectively.
The hg cross section in the transition region is estimated for the Lagrangian [Eq.( 1)] using CT18NNLO PDFs [80] with MG5 aMC@NLO 2.6.7 [81] by first generating the hg events with p T ∈ [100, 300] GeV and |η g | ≤ 2.5 in the lab frame, and then boosting to the partonic c.m. frame with a further cut |y g | ≤ 0.8, which gives κ 2 t (0.57 cos 2 α + 1.3 sin 2 α) pb for the HL-LHC and κ 2 t (13.7 cos 2 α + 30.7 sin 2 α) pb for the FCC-hh.While both κ t and α affect the total production rate and can be constrained by the measurement of the latter, only α determines the polarization.In the following, we take κ t = 1 and consider the constraint on α from the polarization data.
We are interested in final states where the (fat) gluon jet is composed of a pair of quark subjets.While it is possible to also discriminate light quark subjets from gluon subjets, here we only provide a conservative estimate by restricting to the bottom (b) and charm (c) quark tagging as used in experiments [82][83][84][85][86][87][88][89][90][91][92].We estimate the branching fraction f g b b (f gcc ) of g → b b (g → cc) through parton shower simulation using Pythia 8.307 [93], which gives f g b b = 0.013 and f gcc = 0.019 in the selected kinematic region.Following Refs.[86,87], we take b-tagging efficiency ϵ b = 0.7 and c-tagging efficiency ϵ c = 0.3.We consider the diphoton decay channel of the SM Higgs boson and assume a Higgs tagging efficiency ϵ h = 0.002.This then gives about (51 cos 2 α + 115 sin 2 α) reconstructed events at the HL-LHC and (8100 cos 2 α + 18200 sin 2 α) events at the FCC-hh.In Fig. 3, we display the predicted average values of ξ 1,2 in the transition region at the FCC-hh as functions of the CP phase α, together with their uncertainty bands around the SM central values.As expected, it is ξ 2 that constrains small values of α, whereas ξ 1 is too small to have an impact in this region.Assuming the SM scenario with ξ 2 = 0, we can project the constraint |α| ≤ 8.6 • .In this estimate, we have only used the gluon polarization information with Higgs decaying to diphotons.In order to make a significant impact with data from the HL-LHC, one will have to include other Higgs decay channels and light quark flavor tagging in the gluon jets, as well as data from the low-p T and high-p T regions, which shall significantly improve the constraints.To the best of our knowledge, however, none of the event generators currently available in public is capable of correctly generating parton showers for a polarized gluon produced directly from the hard part, stopping us from having a more careful phenomenological study, which we must leave for the future.Nevertheless, the anisotropy substructure of the g → b b splitting studied at ATLAS [94] should render such observables more hopeful and realistic.
Summary.-A precise understanding of the CP property of the Higgs boson is important both to test the SM and to probe new physics.In this Letter, we proposed a novel way of probing the CP structure of the Higgs-top interaction, by measuring the azimuthal anisotropy sub-structure of the gluon jet produced in association with a Higgs boson, which originates from the linear polarization of the final-state gluon.We have introduced a factorization formalism and defined a perturbative polarized gluon jet function with insertion of an IR-safe azimuthal observable.Experimental measurement of the linearly polarized gluon jet will be an important test of the SM and can also serve as a new tool to search for new physics.

FIG. 1 .
FIG. 1. Left: a representative diagram for gg → hg via a box top loop; the others omitted are s-, t-, and u-channel diagrams involving triangular top loops and tri-gluon vertices.Right: the gluon x-ŷ-ẑ frame defined in Eq. (3).

FIG. 3 .
FIG. 3. Constraining power of the FCC-hh gluon polarization data, in the transition region, on the CP phase α. ⟨ξ1,2⟩ are the average values of ξ1,2 in the specified kinematic region.Their statistical uncertainties are indicated by the red and blue bands, respectively, around the SM prediction (with α = 0).The green-hatched region is the α range allowed by the ξ2 measurement.