Probe CP property in Hγγ coupling through interference

and the cross section of the Higgs signal. We study Aint in Standard Model (SM) and various CP violating Hγγ couplings. The Aint in SM (A SM int ) could reach a value of 10%, which is possible to be detected in HL-LHC experiment. Meanwhile, for CP -violating Hγγ couplings, Aint could range from A int to minus A SM int , which makes it a good observable to study new physics. The Aint with both CP -violating Hγγ and Hgg couplings are also studied and its value range is further extended.

The diphoton invariant mass distribution of interference between gg → H → γγ and gg → γγ is almost antisymmetric around the Higgs mass MH .We propose a new observable Aint to quantify this effect, which is a ratio of a sign-reversed integral around MH ( e.g. ) and the cross section of the Higgs signal.We study Aint in Standard Model (SM) and various CPviolating Hγγ couplings.The Aint in SM (A SM int ) could reach a value of 10%, which is possible to be detected in HL-LHC experiment.Meanwhile, for CP -violating Hγγ couplings, Aint could range from A SM int to minus A SM int , which makes it a good observable to study new physics.The Aint with both CP -violating Hγγ and Hgg couplings are also studied and its value range is further extended.

I. INTRODUCTION
The CP violation, as one of the three Sakharov conditions [1], is necessary when explaining the matterantimatter asymmetry in our universe [2].Its source is considered to have close relation with Higgs dynamics [3].Thus the CP properties of the 125 GeV Higgs boson with spin zero is proposed to be probed in various channels at the Large Hadron Collider (LHC) [4][5][6][7][8][9][10][11][12][13][14][15].Among them, the H → ZZ → 4 channel recently gives a constraint of the CP odd/even mixture in HZZ coupling being less than 40% [4].While other channels have no constraint at current ∼ 30 fb −1 integrated luminosity.Compared to H → ZZ → 4 process, H → γγ process is another golden channel to discover Higgs boson and has a relative clean signature, but it suffers from lacking of CP -odd observable constructed from the self-conjugated diphoton signal kinematic variables.The CP property in Hγγ coupling could also be studied in H → γ * γ * → 4 process [16][17][18], however, it is challenged by the low conversion rate of the off-shell photon decaying into two leptons.In this paper, we study the CP property in Hγγ coupling through the interference between gg → H → γγ and gg → γγ.
This interference has been studied in many papers [19][20][21][22][23][24][25].Compared to the Breit-Wigner lineshape of the Higgs boson's signal, the lineshape of the interference term could be roughly divided into two parts: one is symmetric around M H and the other is antisymmetric around M H .These two kinds of interference lineshapes have different effects: after integrating over a symmetric mass region around M H , the symmetric interference lineshape could reduce the signal Breit-Wigner cross section by ∼ 2% [25]; while the antisymmetric one has no contribution to the total cross section, but could distort the signal lineshape, and shift the resonance mass peak by ∼ 150 MeV [21,24].Besides, a variable A i is proposed [26,27] to quantify the interference effect in a sophisticated way, which defines a sign-reversed integral around M H (e.g.
dM ) in its numerator and a sign-conserved integral around M H (e.g. dM ) in its denominator, with both the integrands being overall lineshape, which is superimposed by the signal lineshape, the symmetric interference lineshape and the antisymmetric interference lineshape.In principal, all three effects from interference, changing signal cross section, shifting resonance mass peak and A i (the ratio of sign-reversed integral and sign-conserved integral), could be used to probe CP violation in Hγγ coupling, but their sensitivities are different.As the symmetric interference lineshape is contributed mainly from the Next-to-Leading order while the antisymmetric one from the leading order [20,25], the effect from antisymmetric interference lineshape has better sensitivity, which means the latter two effects could be more sensitive to CP violation.
Nevertheless, experimentally A i is not trivial, and could be affected a lot by the mass uncertainty of M H [27].The main reason is once M H was deviated a little, the sign-reversed integral in the numerator would get a large extra value from the signal lineshape.To solve this problem, we suggest to separate the antisymmetric interference lineshape from the overall lineshape firstly, then replace the integrand in the numerator with only the antisymmetric interference lineshape.Thus the effect from the mass uncertainty is suppressed.The new modified observable is named as A int and it is used to quantify the interference effect in our analysis.
In this paper, we study to probe the CP property in Hγγ coupling through interference between gg → H → γγ and gg → γγ.The rest of the paper is organized as follows.In Section II, we introduce an effective model with a CP -violating Hγγ coupling, and calculate the interference between gg → H → γγ and gg → γγ.Then we introduce the observable A int , and study its dependence on CP violation.In Section III, we simulate the lineshapes of the signal and the interference, and get the A int in SM and various CP -violation cases.After that, a fitting is performed to show whether or not A int could be extracted in future experiment.In Section IV, we build a general framework with both CP violating Hγγ and Hgg couplings and study the A int in a same proce-dure as above.In Section V, we give a conclusion and discussion.

II. THEORETICAL CALCULATION
The effective model with a CP -violating Hγγ coupling is given as, where F , G a denote the γ and gluon field strengths, a = 1, ..., 8 are SU (3) c adjoint representation indices for the gluons, v = 246 GeV is the electroweak vacuum expectation value, the dual field strength is defined as Xµν = µνσρ X σρ , c γ and c g are effective couplings in SM at leading order, ξ γ ∈ [0, 2π) is a phase that parametrize CP violation.When ξ γ = 0, it is the SM case; when ξ γ = 0, there must exist CP violation (except for ξ γ = π) and new physics beyond SM.
In SM at leading order, c γ is introduced by fermion and W loops and c g is introduced by fermion loops only, which have the expressions as where a, b are the same as a in Eq. ( 1), N c = 3, Q f and m f are electric charge and mass of fermions, and The helicity amplitudes for gg → H → γγ and gg → γγ can be written as [21,28], where the spinor phases are dropped for simplicity, h i s are helicities of outgoing gluons and photons, Q f is the electric charge of fermion, A h1h2h3h4 box are reduced 1-loop helicity amplitudes of gg → γγ mediated by five flavor quarks, while the contribution from top quark is much suppressed [19] and is neglected in our analysis.The A box for non-zero interference are [21,28] A where z = cos θ, with θ being the scattering angle of γ in diphoton center of mass frame.After considering interference, the lineshape over the smooth background is composed of both lineshapes of signal and interference, which can be expressed by where σ sig , σ int are cross sections from signal term and interference term respectively, M γγ = √ ŝ, the integral region of z depends on the detector angle coverage in experiment, G(M γγ ) is gluon-gluon luminosity function written as The interference term consists of two parts: antisymmetric (the first term in Eq. ( 10)) and symmetric (the second term in Eq. ( 10)) parts around Higgs boson's mass.It is worthy to notice that at leading order Im c SM g c SM γ is suppressed by m b /m t compared to Re c SM g c SM γ because the imaginary parts of c SM g , c SM γ are mainly from bottom quark loop while their real parts are from top quark or W boson loop.Thus the symmetric part of the interference term is much suppressed at leading order and its integral value contributed for the total cross section is mainly from Next-to-Leading order [19,25].In contrast, the antisymmetric part could have a larger magnitude around M H .
The observable A int extracts the antisymmetric part of the interference by an sign-reversed integral around M H , which is defined as where the integral region is around Higgs resonance (e.g.[121,131] GeV for M H = 126 GeV), the Θ-function is So the numerator keeps the antisymmetric contribution from the interference, and the denomenator is the cross section from the signal, A int is an observable that roughly indicates the ratio of the interference to the signal.As ξ γ = 0 represents the SM case, we could define A SM int ≡ A int (ξ γ = 0) and rewrite A int (ξ γ ) simply as The largest deviation A int (π) = −A SM int happens when ξ γ = π, which represents an inverse CP-even Hγγ coupling from new physics but without CP violation.It's interesting that this degenerate coupling could only be exhibited by the interference effect.

III. SIMULATION
The simulation is performed for a proton-proton collider with √ s = 14 TeV by using the MCFM [29] package, in which subroutines according to the helicity amplitudes of Eq. ( 7) are added.The Higgs boson's mass and width are set as M H = 126 GeV, and Γ H = 4.3 MeV.Each photon is required to have p γ T > 20 GeV and |η γ | < 2.5.Based on the simulation, we study A SM int firstly and then A int from CP violation cases.After that, we make a fitting to show how the A int is extracted.
A. A SM int Fig. 1 show the theoretical lineshapes of the signal (a sharp peak shown in the black histogram) and the interference (a peak and dig shown in the red histogram), among which Fig. 1a is an overall plot, Fig. 1b and Fig. 1c are close-ups.As shown in Fig. 1a and Fig. 1b, the signal has a mass peak that is about four times of the interference, but the mass peak of the interference is wider and has a much longer tail.The resonance region [125.9,126.1]GeV is further scrutinized in Fig. 1c with bin width changed from 100 MeV to 2 MeV.The value of the signal exceeds that of the interference at the energy point that is about ten times of Higgs boson's width (∼ 43 MeV) below M H .After integrating, the A SM int is 36% as shown in table I, which is quite marvelous.As the smearing by the mass resolution (MR) is not considered yet, we mark it as the σ M R = 0 case.
The invariant mass of the diphoton M γγ has a mass resolution of about 1 ∼ 2 GeV at the LHC experiment [30].For simplicity we include the mass resolution effect by convoluting the histograms with a Gaussian function of width σ M R = 1.1, 1.3, 1.5, 1.7, 1.9 GeV.This convolution procedure is also called Gaussian smearing.Fig. 2 shows the lineshapes after the Gaussian smearing with σ M R = 1.5 GeV.The sharp peak of the signal becomes a wide bump (the black histogram), meanwhile, the peak and dig of the interference are also widened, but they cancel each other a lot near M H and leave a flat bump and down (the red histogram).The A SM int after Gaussian smearing is thus much reduced, which range from 10.2% to 7.2% when σ M R goes from 1.1 to 1.9 GeV as shown in table I.  9) and (10).ξγ = 0 represents the SM case, σMR = 0 represents the theoretical distribution before Gaussian smearing.Among them (a) is an overall plot, (b) and (c) are close-ups.Fig. 3 shows the lineshapes of interference under the ξ γ = 0, π, π/2 cases with σ M R = 1.5 GeV.The blue histogram (ξ γ = π, inverse CP -even Hγγ coupling) is almost opposite to the red histogram (ξ γ = 0, SM), which correspond to the minimum and the maximum of A int values.The black dashed histogram (ξ γ = π/2, CP -odd Hγγ coupling) looks like a flat line, and it corresponds to zero A int value.Fig. 4 shows A int and its absolute statistical error δA int .The statistical error is estimated with an integrated luminosity of 30 fb −1 , and the efficiency of detector is assumed to be one.δA int decrease as A int becomes smaller, however, the relative statistical error δA int /A int increase quickly and becomes very large as A int approaches zero.In SM (ξ γ = 0 in Fig. 4), the relative statistical error δA int /A int is about 18% with an assumption of zero correlation between symmetric and antisymmetric cross-sections.

C. Fitting
In current CMS or ATLAS experiment, the γγ mass spectrum is fitted by a signal function and a background function.To consider the interference effect, the antisymmetric lineshape should also be included.That is, instead of a Gaussian function (or a double-sided Crystal Ball function) as the signal function in current LHC experiment [30,31], a Gaussian function (or a doublesided Crystal Ball function) plus an asymmetric function should be used as the modified signal function, while the background function is kept as same as in the experiment.
To see whether or not the asymmetric lineshape could be extracted, we carry out a modified-signal fitting on two background-subtracted data samples.As the background fluctuation would be dealt similarly as in real experiment, we ignore it here for simplicity.One data sample is from the CMS experiment in Ref. [30], we fetch 10 data points with its errors between [121,131] GeV in the background-subtracted γγ mass spectrum for 35.9 fb −1 luminosity with proton-proton collide energy at 13 TeV (see Fig. 13 in Ref. [30]).The fitting function is described as (14) where c 1 , c 2 , δm are float parameters, m means the value of the γγ invariant mass, the functions f sig (m), f int (m) are evaluated from the two histograms in Fig. 2 and they describe the signal and interference separately.Fig. 5 shows the fitting result on the CMS data, in which the crosses represent CMS data with its error, the red solid line is the combined function, the black dashed line and the blue dotted line represent the signal and interference components respectively.The black dashed line is almost same as the red solid line while the blue dotted line is almost flat, the fitting parameter c 2 for the interference component has a huge uncertainty that even larger than the central value of c 1 , which indicates the interference component is hard to be extracted from the 35.9 fb −1 CMS data.For a comparison, we simulate a pseudodata sample from the combined histogram in Fig. 2, which is normalized to have events of about 80 times the CMS data (corresponding to 3000 fb −1 ), with a binwidth of 0.5 GeV and Poission fluctuation.The fitting result is shown in Fig. 6, where the red solid line has a shift from the black dashed line, the blue dotted line could be distinguishable clearly.The ratio of the fitting parameters c 1 /c 2 is almost equal to one, and each of them have uncertainties less than 3%, which corresponds to a A int value consistent with A SM int .Even though this fitting result looks quite good, it can only reflect that the antisymmetric lineshape could be extracted out when no contamination comes from systematic error.
By contrast, a simulation that also study the interference effect has been carried out with systematic error included by ATLAS collaboration at 3000 fb −1 HL-LHC [32].In that simulation, the mass shift of Higgs boson caused by the interference effect has been studied under different Higgs' width assumptions.A pseudo-data is produced by smearing a Breit-Wigner with the resolution model and the interference effect are described by the shift of smeared Breit-Wigner distributions.Based on fitting, the mass shift of Higgs from the interference effect is estimated to be ∆m H = −54.4MeV for the SM case, and the systematic error on the mass difference is about 100 MeV.If using this result to estimate the mass shift effect for the non-SM ξ γ = 0 cases, that would be, ξ γ = π/2 corresponds to a zero mass shift, and ξ γ = π corresponds to a reverse mass shift of ∆m H = +54.4MeV as shown in Fig. 3. Then the largest deviation of the mass shift from the SM case is 2 × 54.4 MeV (when ξ γ = π), which is almost covered by the systematic error of 100 MeV.There-fore, the non-SM ξ γ = 0 cases could not be distinguished through this mass shift effect.Nevertheless, it is worthing to note that the antisymmetric lineshape of the interference effect by theoretical calculation is quite different from the shift of two smeared Breit-Wigner distributions in ATLAS's simulation [32], especially at the region far from the Higgs' peak, the antisymmetric lineshape of the interference effect has a longer flat tail while the Breit-Wigner distribution falls fast.The authors from ATLAS collaboration has also noticed this difference and planned to add it to their next research [32].

IV. CP VIOLATION IN Hgg COUPLING
In the above study the Hgg coupling is supposed to be SM-like, furthermore, the observable A int could also be used to probe CP violation in Hgg coupling.In this section, we add one more parameter ξ g to describe CP violation in Hgg coupling, and study A int following the same procedure as above.
Based on Eq. ( 1), one more parameter ξ g to describe CP violation in Hgg coupling is added, and the effective Lagrangian is modified as After that, the helicity amplitude in Eq. ( 7) and differential cross section of interference in Eq. ( 10) should be changed correspondingly, which are Then A SM int ≡ A int (ξ g = 0, ξ γ = 0) and where the integral could be calculated numerically once the the integral region of z is given.For example, if the pseudorapidity of γ is required to be |η γ | < 2.5, that is, z ∈ [−0.985, 0.985], the integral dzA ++−− box ≈ −9, and Eq. ( 18) could be simplified as ) A int (ξ g , ξ γ ) thus has a maximum and minimum of about 1.6 times of A SM int .If ξ g = 0, A int (ξ g = 0, ξ γ ) will degenerate to the A int (ξ γ ) in Eq. (13).By constrast, if (GeV) 7. The diphoton invariant mass Mγγ distribution of interference after Gaussian smearing in various ξg, ξγ cases.
which shows the same dependence of A int (ξ γ ) on ξ γ when ξ g = 0 as in Eq. ( 13).So a CP -violating Hgg coupling could cause similar deviation of A int to A SM int as a CP -violating Hγγ coupling, and an single observed A int value could not distinguish them since there are two free parameters for one observable.Fig. 7 shows the lineshapes of interference for different ξ g , ξ γ choices.The red histogram (ξ g = 0, ξ γ = 0) represents the SM case; the magenta histogram (ξ g = π 2 , ξ γ = π 2 ) could get largest A int ; the cyan histogram (ξ g = π 2 , ξ γ = 3π 2 ) corresponds to the smallest A int ; and the black histogram is from ξ g = 0, ξ γ = π 2 case with A int equal to zero.For the general case of both ξ g , ξ γ being free parameters, A int (ξ g , ξ γ ) could have a wider value range than A int (ξ γ ), which makes it easier to be probed in future experiment.

V. CONCLUSION AND DISCUSSION
The diphoton mass distribution from the interference between gg → H → γγ and gg → γγ at leading order is almost antisymmetric around M H and we propose an sign-reversed integral around M H to get its contribution.After dividing this integral value by the cross section of Higgs signal, we get an observable A int .In SM, the theoretical A int value without invariant mass resolution could be large than 30%.After considering mass resolution of ∼ 1.5 GeV, A int is reduced but still could be as large as 10%.The CP violation in Hγγ could change A int from 10% to -10% depending on the CP violation phase ξ γ .In a general framework of both CP -violating Hγγ and Hgg coupling, A int could have a larger value of ∼ ±16%, which makes it easier to be probed.At current experiment, the uncertainties of signal strength in H → γγ process is larger than A int , which could make it difficult to be extracted from the total spectrum.However, the high luminosity and precision in future experiment may let A int be extracted out, and the CP violation be probed to some extent.

FIG. 1 .
FIG.1.The diphoton invariant mass Mγγ distribution of the signal and the interference as in Eq. (9) and(10).ξγ = 0 represents the SM case, σMR = 0 represents the theoretical distribution before Gaussian smearing.Among them (a) is an overall plot, (b) and (c) are close-ups.

FIG. 5 .
FIG.5.A fitting on the background-substracted CMS data sample.The crosses represent CMS data from Ref.[30].The red solid line is the combined function, the black dashed line and the blue dotted line represent the signal and interference components respectively.

FIG. 6 .
FIG.6.A fitting on the simulated data sample.The crosses represent simulated data from the combined histogram in Fig.2normalized to 3000 fb −1 .The red solid line is the combined function, the black dashed line and the blue dotted line represent the signal and interference components respectively.

TABLE I .
The A SM int values with different mass resolution widths.The σMR = 0 represents the theoretical case before Gaussian smearing.