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The effective model with a CP violating
Hγγ coupling is given as,Lh=cγcosξγvhFμνFμν+cγsinξγ2vhFμν˜Fμν+cgvhGaμνGaμν,
(1) where F,
Ga denote theγ and gluon field strengths, a = 1, ..., 8 areSU(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=ϵμνσρXσρ ,cγ andcg are the effective couplings in SM to leading order, andξγ∈[0,2π) is a phase that parametrizes CP violation. Whenξγ=0 , this is the SM case; whenξγ≠0 , there must exist CP violation (except forξγ=π ) and new physics beyond SM. This kind of parametrization makes certain that the total signal strength of the Higgs decay into diphoton is equal to the prediction of SM.In SM, to leading order,
cγ is brought by the fermion and W loops, andcg is due to the fermion loops only, which can be expressed ascg=αs16π∑f=t,bF1/2(4m2f/ˆs),
(2) cγ=α8π[F1(4m2W/ˆs)+∑f=t,bNcQ2fF1/2(4m2f/ˆs)],
(3) where
αs(α) are the running QCD (QED) couplings,Nc=3 ,Qf andmf are the electric charge and mass of the fermions, andF1/2(τ)=−2τ[1+(1−τ)f(τ)],
(4) F1(τ)=2+3τ[1+(2−τ)f(τ)],
(5) f(τ)={arcsin2√1/ττ⩾
(6) The helicity amplitudes for
gg\to H\to\gamma\gamma andgg\to \gamma\gamma can be written as [30, 38, 39],\begin{split} {\cal M} = & -{\rm e}^{-{\rm i} h_3\xi_\gamma}\delta^{h_1h_2}\delta^{h_3h_4}\delta^{a b}\frac{M^4_{\gamma\gamma}}{v^2} \frac{4c_gc_\gamma}{M^2_{\gamma\gamma}-M^2_H+iM_H\Gamma_H} \\ & + 4\alpha\alpha_s\delta^{a b}\sum_{f = u, d, c, s, b} Q^2_f {\cal A}_{\rm{box}}^{h_1 h_2 h_3h_4}\;, \end{split}
(7) where a, b are the same as a in Eq. (1), the spinor phases (see their exact formulas in [38, 39] and [16]) are dropped for simplicity,
h_i are the helicities of outgoing gluons and photons,Q_f is the electric charge of the fermion,{\cal A}_{\rm{box}}^{h_1 h_2 h_3h_4} are the reduced 1-loop helicity amplitudes ofgg\to \gamma\gamma mediated by five flavor quarks, while the contribution from the top quark is considerably suppressed [28] and is neglected in our analysis.{\cal A}_{\rm{box}} for non-zero interference is [30, 38, 39]\begin{split} {\cal A}_{\rm{box}}^{++++} =& {\cal A}_{\rm{box}}^{----} = 1, \\ {\cal A}_{\rm{box}}^{++--} =& {\cal A}_{\rm{box}}^{--++} = -1 + z \ln\left (\frac{1 + z}{1 - z}\right) \\&- \frac{1 + z^2}{4}\left[\ln^2 \left(\frac{1 + z}{1-z}\right)+\pi^2\right], \end{split}
(8) where
z = \cos\theta , with\theta the scattering angle of\gamma in the diphoton center-of-mass frame. It may be noted by the careful reader that we use the formulas for{\cal A}_{\rm{box}}^{++++/----} and{\cal A}_{\rm{box}}^{++--/--++} as in [38, 39], while they are exchanged in [30]. This is because the convention we use here is for outgoing gluons, while the helicities have a reversed sign for incoming gluons. It is also worth noting that Eq. (7) is different from Eq. (2) in Ref. [30] because of the{\rm e}^{-{\rm i} h_3\xi_\gamma} factor, which ensures that the Higgs signal strength is not affected by the CP violation factor\xi_\gamma , while the interference strength has a simple\cos\xi_\gamma dependence (see Eqs. (9) and (10)).After considering the interference, the line shape of the smooth background is composed of both the signal and interference line shapes, and can be expressed by
\frac{{\rm d}\sigma_{\rm{sig}}}{{\rm d}M_{\gamma\gamma}} = \frac{G(M_{\gamma\gamma})}{128\pi M_{\gamma\gamma}} \frac{ |c_g c_\gamma|^2 } {(M^2_{\gamma\gamma}-M^2_H)^2+M^2_H\Gamma^2_H}\times \int {\rm d}z,
(9) \begin{split} \frac{{\rm d}\sigma_{\rm{int}}}{{\rm d}M_{\gamma\gamma}} = & \frac{G(M_{\gamma\gamma})}{128\pi M_{\gamma\gamma}} \frac{(M^2_{\gamma\gamma}-M^2_H)\operatorname{Re}\left(c_g c_\gamma \right) +M_H\Gamma_H \operatorname{Im}\left(c_g c_\gamma \right) } {(M^2_{\gamma\gamma}-M^2_H)^2+M^2_H\Gamma^2_H} \\ &\times\int {\rm d}z [{\cal A}_{\rm{box}}^{++++}+{\cal A}_{\rm{box}}^{++--}]\times\cos\xi_\gamma, \end{split}
(10) where
\sigma_{\rm{sig}}, \sigma_{\rm{int}} are the cross-sections of the signal and interference terms, respectively,M_{\gamma\gamma} = \sqrt{\hat{s}} , the integral region z depends on the detector angle coverage, andG(M_{\gamma\gamma}) is the gluon-gluon luminosity function written asG(M_{\gamma\gamma}) = \int^1_{M^2_{\gamma\gamma}/s}\frac{{\rm d}x}{sx}[g(x)g(M^2_{\gamma\gamma}/(sx))]\; .
(11) The interference term consists of two parts: the antisymmetric term (the first term in Eq. (10)), and the symmetric term (the second term in Eq. (10)) around the Higgs boson mass. It is worth noting that to leading order
\operatorname{Im}\left(c^{\rm{SM}}_g c^{\rm{SM}}_\gamma \right) is suppressed bym_b/m_t compared to\operatorname{Re}\left(c^{\rm{SM}}_g c^{\rm{SM}}_\gamma \right ) , because the imaginary parts ofc^{\rm{SM}}_g ,c^{\rm{SM}}_\gamma are mainly from the bottom quark loop, while their real parts are from the top quark or W boson loops. Thus, the symmetric part of the interference term is suppressed to leading order and its contribution to the total cross-section is mainly from the next-to-leading order [28, 34]. In contrast, the antisymmetric term can have a large magnitude aroundM_H .The observable
A_{\rm{int}} extracts the antisymmetric part of the interference by the sign-reversed integral aroundM_H , which is defined asA_{\rm{int}}(\xi_\gamma) = \frac{\int {\rm d}M_{\gamma\gamma} \displaystyle\frac{{\rm d}\sigma_{\rm{int}}}{{\rm d}M_{\gamma\gamma}} \Theta(M_{\gamma\gamma}-M_H)} {\int {\rm d}M_{\gamma\gamma}\displaystyle\frac{{\rm d}\sigma_{\rm{sig}} }{{\rm d}M_{\gamma\gamma}} }\;,
(12) where the region of integration is around the Higgs resonance (e.g.
[121, 131] GeV forM_H = 126 GeV), and the\Theta -function is\Theta (x) \equiv \left\{ {\begin{array}{*{20}{c}} { - 1,}&{x < 0}\\ {1,}&{x > 0} \end{array}} \right..
Therefore, the numerator is the antisymmetric contribution from the interference, and the denomenator is the cross-section from the signal, so that
A_{\rm{int}} is an observable that roughly gives the ratio of the interference to the signal.As
\xi_\gamma = 0 represents the SM case, we can defineA_{\rm{int}}^{\rm{SM}}\equiv A_{\rm{int}}(\xi_\gamma = 0) and rewriteA_{\rm{int}}(\xi_\gamma) simply asA_{\rm{int}}(\xi_\gamma) = A_{\rm{int}}^{\rm{SM}}\times \cos\xi_\gamma \; .
(13) The largest deviation
A_{\rm{int}}(\pi) = -A_{\rm{int}}^{\rm{SM}} occurs when\xi_\gamma = \pi , which represents the inverse CP-evenH\gamma\gamma coupling from new physics without CP violation. It is interesting that this degenerate coupling can only be revealed by the interference effect. -
The numerical results are obtained for proton-proton collisions at
\sqrt{s} = 14 TeV by using the MCFM [40] package, in which the subroutines for helicity amplitudes of Eq. (7) are added. The Higgs boson mass and width are set asM_H = 126 GeV, and\Gamma_H = 4.3 MeV. Each photon is required to havep^{\gamma}_T>20 GeV and|\eta^{\gamma}|<2.5 . Based on the simulation, we studyA^{\rm{SM}}_{\rm{int}} first, and thenA_{\rm{int}} for the CP violation cases. Finally, we estimate the feasibility of obtainingA_{\rm{int}} at the LHC. -
Fig. 1 shows the theoretical line shapes of the signal (a sharp peak shown in the black histogram) and the interference (a peak and dip shown in the red histogram); Fig. 1(a) is the overall plot, Fig. 1(b) and Fig. 1(c) are close-ups. As shown in Fig. 1(a) and Fig. 1(b), the signal has a mass peak that is about four times larger than the interference. The mass peak of the interference is wider and has a much longer tail. The resonance region
[125.9, 126.1] GeV is shown in Fig. 1(c) with a bin width reduced from 100 MeV to 2 MeV. The signal exceeds the interference from the energyM_{\gamma\gamma}\approx M_H-10\times\Gamma_H . After integrating,A_{\rm{int}}^{\rm{SM}} is 36% , as shown in Table 1, which is quite large. As the smearing from the mass resolution (MR) is not considered yet, we denote this case as\sigma_{\rm{MR}} = 0 .\sigma_{\rm{MR}} (GeV)A^{\rm{SM}}_{\rm{int}} denominator (fb)A^{\rm{SM}}_{\rm{int}} numerator (fb)A^{\rm{SM}}_{\rm{int}} (%)0 39.3 14.3 36.3 1.1 39.3 4.0 10.2 1.3 39.3 3.7 9.4 1.5 39.3 3.4 8.6 1.7 39.3 3.1 7.9 1.9 39.3 2.8 7.2 Table 1.
A^{\rm{SM}}_{\rm{int}} values for different mass resolution widths.\sigma_{\rm{MR}}=0 represents the theoretical case before Gaussian smearing.Figure 1. (color online) Diphoton invariant mass
{M_{\gamma \gamma }} distribution of the signal and the interference given by Eq. (9) and (10).{\xi _\gamma } = 0 represents the SM case,{\sigma _{{\rm{MR}}}} = 0 represents the theoretical distribution before Gaussian smearing. (a) is the overall plot, (b) and (c) are close-ups.The invariant mass of the diphoton
M_{\gamma\gamma} has a mass resolution of about1\sim2 GeV in the CMS experiment [41]. For simplicity we include the mass resolution by convoluting the histograms with a Gaussian function with widths\sigma_{\rm{MR}} = 1.1, 1.3, 1.5, 1.7, 1.9 GeV. This convolution procedure is also called Gaussian smearing. Fig. 2 shows the line shapes after Gaussian smearing with\sigma_{\rm{MR}} = 1.5 GeV. The sharp peak of the signal becomes a wide bump (the black histogram), while the peak and dip of the interference are also wider. As they cancel each other nearM_H , the former peak and dip take a moderately antisymmetric shape aroundM_H (the red histogram).A_{\rm{int}}^{\rm{SM}} after Gaussian smearing is thus reduced, and ranges from 10.2% to 7.2% when\sigma_{\rm{MR}} increases from 1.1 to 1.9 GeV, as shown in Table 1. -
Fig. 3 shows the interference line shapes when
\xi_\gamma = 0, \pi, \pi/2 and\sigma_{\rm{MR}} = 1.5 GeV. The blue histogram (\xi_\gamma = \pi , sign-reversed CP-evenH\gamma\gamma coupling) is almost opposite to the red histogram (\xi_\gamma = 0 , SM), and they correspond to the minimum and maximum ofA_{\rm{int}} . The black dashed histogram (\xi_\gamma = \pi/2 , CP-oddH\gamma\gamma coupling) looks like a flat line (actually with some tiny fluctuations from the simulation), and corresponds to zero ofA_{\rm{int}} . Fig. 4 showsA_{\rm{int}} and its absolute statistical error\delta A_{\rm{int}} . The statistical error is estimated using an integrated luminosity of 30 fb−1, and the efficiency of the detector is assumed to be one.\delta A_{\rm{int}} decreases asA_{\rm{int}} becomes smaller. However, the relative statistical error\delta A_{\rm{int}}/A_{\rm{int}} increases quickly and becomes very large asA_{\rm{int}} approaches zero. In SM (\xi_\gamma = 0 in Fig. 4), the relative statistical error\delta A_{\rm{int}}/A_{\rm{int}} is about 18% with the assumption of zero correlation between symmetric and antisymmetric cross-sections. -
In the CMS or ATLAS experiments, the
\gamma\gamma mass spectrum is fitted by a signal function and a background function. To consider the interference effect, the antisymmetric line shape should also be included. That is, instead of a Gaussian function (or a double-sided Crystal Ball function) as the signal in the LHC experiments [41, 42], a Gaussian function (or a double-sided Crystal Ball function) plus an asymmetric function should be used as the modified signal, while the background should be kept the same.To see whether or not the asymmetric line shape could be extracted, we carry out a fit of the modified signal from two background-subtracted data samples. As the background fluctuation would be dealt similarly as in the real experiment, we ignore it here for simplicity. One data sample is from the CMS experiment Ref. [41], from where we take 10 data points with their errors between
[121, 131] GeV in the background-subtracted\gamma\gamma mass spectrum for 35.9 fb−1 integrated luminosity with proton-proton collision energy of 13 TeV (see Fig. 13 in Ref. [14]). The fitting function is given asf(m) = c_1\times f_{\rm{sig}}(m-\delta m)+c_2\times f_{\rm{int}}(m-\delta m),
(14) where
c_1, c_2, \delta m are the fitting parameters, m is the\gamma\gamma invariant mass, the functionsf_{\rm{sig}}(m), f_{\rm{int}}(m) are evaluated from the two histograms in Fig. 2 and they describe the signal and interference. Fig. 5 shows the fit result of the CMS data, in which the crosses represent CMS data with their errors, the red solid line is the combined function, and the black dashed line and the blue dotted line are the signal and interference components, respectively. The black dashed line is almost the same as the red solid line, while the blue dotted line is almost flat. The fitting parameterc_2 for the interference component has a huge uncertainty that is even larger than the central value ofc_1 , which indicates that it is hard to extract the interference component from 35.9 fb-1 of CMS data. For comparison, we simulate a pseudo-data sample from the combined histogram in Fig. 2, which is normalized to about 80 times the amount of CMS data (corresponding to an integrated luminosity of 3000 fb-1), with a bin width of0.5 GeV and Poission fluctuation. The result of the fit is shown in Fig. 6, where the red solid line is shifted from the black dashed line, and the blue dotted line can be clearly distinguished.c_1 andc_2 are fitted asc_1 =0.999\pm 0.002 ,c_2 = 0.947 \pm 0.028 , which are consistent with their SM expected value 1 and deduce to a relative error ofA_{int}\sim 3% according to the error propagation formula. 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. Furthermore, our study shows that the optimal fitting strategy is taking Higgs mass as a free parameter together withc_1 andc_2 . AlthoughM_H has been measured in many channels, its fluctuation is usually too large to get a converged fitting if we take it as a known input value.Figure 5. (color online) Fit of the background-substracted CMS data sample. The crosses represent CMS data from Ref. [41]. The red solid line is the combined function, the black dashed line and the blue dotted line represent the signal and interference components, respectively.
Figure 6. (color online) Fit of the simulated data sample. The crosses represent simulated data from the combined histogram in Fig. 2 normalized to an integrated luminosity of 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.
In contrast, a simulation that also studied the interference effect including the systematic errors has been carried out by the ATLAS collaboration for the HL-LHC with an integrated luminosity of 3000 fb−1 [43]. In that simulation, the mass shift of the Higgs boson caused by the interference effect has been studied with different assumptions for the Higgs width. A pseudo-data was produced by smearing a Breit-Wigner distribution with a model of the detector resolution, and the interference effect was described by the shift of the smeared Breit-Wigner distribution. Based on fitting, the mass shift of the Higgs from the interference effect was estimated to be
\Delta m_H = -54.4 MeV for the SM case, and the systematic error of the mass difference was about 100 MeV. If this result is used to estimate the mass shift effect for the non-SM\xi_\gamma\ne 0 cases, (\xi_\gamma = \pi/2 corresponds to a zero mass shift, and\xi_\gamma = \pi to a reverse mass shift of\Delta m_H = +54.4 MeV, as shown in Fig. 3), then the largest deviation of the mass shift from the SM case is2\times 54.4 MeV (when\xi_\gamma = \pi ), which is almost covered by the systematic error of100 MeV. Therefore, the non-SM\xi_\gamma\ne 0 cases can not be distinguished using this mass shift effect. Nevertheless, it is worth noting that the antisymmetric line shape of the theoretical interference effect is quite different from the shift of two smeared Breit-Wigner distributions in the ATLAS simulation [43], especially in the region far from the Higgs peak, where the antisymmetric line shape of the interference effect has a long flat tail while the Breit-Wigner distribution falls quickly. The authors of the ATLAS study have also noted this difference and have planned to include it in their new search [43]. -
In the above study,
Hgg coupling is assumed to be SM-like. Furthermore, the observableA_{\rm{int}} could also be used to probe CP violation inHgg coupling. In this section, we add one more parameter,\xi_g , to describe CP violation inHgg coupling, and studyA_{\rm{int}} following the same procedure as above.Based on Eq. (1), the parameter
\xi_g to describe CP violation inHgg coupling is added, and the effective Lagrangian is modified as\begin{split} {\cal L}_{\rm h} = & \frac{c_\gamma \cos\xi_\gamma}{v}\; h\, F_{\mu\nu} F^{\mu\nu} + \frac{c_\gamma \sin\xi_\gamma}{2v}\; h\, F_{\mu\nu}{\tilde{F}}^{\mu\nu} \\ &+\frac{c_g \cos\xi_g}{v}\; h\, G^a_{\mu\nu}G^{a\mu\nu} \\&+ \frac{c_g \sin\xi_g}{2v}\; h\, G^a_{\mu\nu}{\tilde{G}}^{a\mu\nu}\; . \end{split}
(15) The helicity amplitude in Eq. (7) and the differential cross-section of the interference in Eq. (10) should be changed correspondingly, and become
\begin{split} {\cal M} =& -{\rm e}^{-{\rm i} h_1 \xi_g}{\rm e}^{-{\rm i} h_3\xi_\gamma}\delta^{h_1h_2}\delta^{h_3h_4}\delta^{a b}\frac{M^4_{\gamma\gamma}}{v^2} \frac{4c_gc_\gamma}{M^2_{\gamma\gamma}-M^2_H+iM_H\Gamma_H} \\ &+ 4\alpha\alpha_s\delta^{a b}\sum_{f = u, d, c, s, b} Q^2_f {\cal A}_{\rm{box}}^{h_1 h_2 h_3h_4}\;, \end{split}
(16) \begin{split} \frac{{\rm d}\sigma_{\rm{int}}}{{\rm d}M_{\gamma\gamma}} \propto& \frac{(M^2_{\gamma\gamma}-M^2_H)\operatorname{Re}\left(c_g c_\gamma \right) +M_H\Gamma_H \operatorname{Im}\left(c_g c_\gamma \right) } {(M^2_{\gamma\gamma}-M^2_H)^2+M^2_H\Gamma^2_H} \\ & \times \int {\rm d}z[\cos(\xi_g+\xi_\gamma){\cal A}_{\rm{box}}^{++++}+\cos(\xi_g-\xi_\gamma){\cal A}_{\rm{box}}^{++--}]\; . \end{split}
(17) Then,
A_{\rm{int}}^{\rm{SM}}\equiv A_{\rm{int}}(\xi_g = 0, \xi_\gamma = 0) and\begin{split} A_{\rm{int}}(\xi_g, \xi_\gamma) =& A_{\rm{int}}^{\rm{SM}}\\&\times \frac{\int {\rm d}z[\cos(\xi_g+\xi_\gamma){\cal A}_{\rm{box}}^{++++}+\cos(\xi_g-\xi_\gamma){\cal A}_{\rm{box}}^{++--}]}{\int {\rm d}z[{\cal A}_{\rm{box}}^{++++}+{\cal A}_{\rm{box}}^{++--}]}\;, \end{split}
(18) where the integral can be calculated numerically once the region of z integration is given. For example, if the pseudorapidity of
\gamma is required to be|\eta^{\gamma}|<2.5 , that is,z\in [-0.985, 0.985] , the integral\int {\rm d}z {\cal A}_{\rm{box}}^{++--}\approx -9 , and Eq. (18) can be simplified asA_{\rm{int}}(\xi_g, \xi_\gamma)\approx A_{\rm{int}}^{\rm{SM}}\times \frac{2\cos(\xi_g+\xi_\gamma)-9\cos(\xi_g-\xi_\gamma)} {-7}\; .
(19) A_{\rm{int}}(\xi_g, \xi_\gamma) thus has the maximum and minimum about 1.6 times that ofA_{\rm{int}}^{\rm{SM}} . If\xi_g = 0 ,A_{\rm{int}}(\xi_g = 0, \xi_\gamma) degenerates toA_{\rm{int}}(\xi_\gamma) in Eq. (13). In contrast, if\xi_\gamma = 0 ,A_{\rm{int}}(\xi_g) = A_{\rm{int}}^{\rm{SM}}\times \cos(\xi_g),
(20) which shows the same dependence as
A_{\rm{int}}(\xi_\gamma) on\xi_\gamma when\xi_g = 0 as in Eq. (13). Hence, a CP violatingHgg coupling can cause similar deviation ofA_{\rm{int}} fromA_{\rm{int}}^{\rm{SM}} as a CP violatingH\gamma\gamma coupling, and a single observedA_{\rm{int}} can not distinguish between them since there are two free parameters for one observable.Fig. 7 shows the interference line shapes for different
\xi_g, \; \xi_\gamma . The red histogram (\xi_g = 0, \; \; \xi_\gamma = 0 ) represents the SM case; the magenta histogram\left(\xi_g = \displaystyle\frac{\pi}{2}, \; \; \xi_\gamma = \displaystyle\frac{\pi}{2}\right) has the largestA_{\rm{int}} ; the cyan histogram\left( \xi_g = \displaystyle\frac{\pi}{2}, \; \; \xi_\gamma = \displaystyle\frac{3\pi}{2} \right) corresponds to the smallestA_{\rm{int}} ; and the black histogram is for the case of\xi_g = 0, \; \; \xi_\gamma = \displaystyle\frac{\pi}{2} withA_{\rm{int}} equal to zero. In the general case where both\xi_g, \; \xi_\gamma are free parameters,A_{\rm{int}}(\xi_g, \; \xi_\gamma) has a wider range of values thanA_{\rm{int}}(\xi_\gamma) , which makes it easier to probe in future experiments.
Probing the CP violating Hγγ coupling using interferometry
- Received Date: 2019-02-15
- Accepted Date: 2019-04-22
- Available Online: 2019-07-01
Abstract: The diphoton invariant mass distribution from the interference between