Interference effects on Higgs mass measurement in $e^+e^-\to H(\gamma\gamma) Z$ at CEPC

A high luminosity Circular Electron Positron Collider (CEPC) as a Higgs Factory will be helpful to the precision measurement of the Higgs mass. The signal-background interference effect is carefully studied for the Higgs diphoton decay mode in the associated Z boson production at the future $e^+e^-$ colliders at energy $246 {\rm GeV}$. The mass shifts go up from about $20 {\rm MeV}$ to $50 {\rm MeV}$ for the experimental mass resolution ranging from $0.8 {\rm GeV}$ to $2 {\rm GeV}$.


Introduction
The ATLAS and CMS collaborations at the Large Hadron Collider (LHC) announced the amazing discovery of a new particle with a mass of around 125GeV in July 2012 [1,2] and it's properties are well compatible with the Standard Model (SM) Higgs boson [3] but leave the room for new physics. One of next tasks is the precision measurement, such as the mass, spin, couplings, and decay patterns, to determine the nature for Higgs boson. The future e + e − colliders such as the International Linear Collider (ILC), a linear particle accelerator, Triple-Large Electron-Positron Collider (TLEP), and the Circular Electron Positron Collider (CEPC), proposed by Chinese high energy physics community in 2012, will play an important role on this task.
For future Higgs factories, the e + e − → Z * → Z + H(γγ) process will be an excellent channel for precision measurement. The diphoton decay channel is a rare decay mode but provides a clear signature for the Higgs boson. At the LHC, the γγ mode for Higgs production involves a huge background including the dominant reducible jet and the irreducible contributions from the continuum. The background is significantly suppressed at lepton colliders to measure the Higgs properties [4]. The Higgs-bremsstrahlung process, e + e − → ZH, is the most important process for Higgs production when the center-of-mass energy is less than 500GeV. With the leading-order calculations, the production cross section of a 125GeV Higgs reaches the maximum value when the center-of-mass energy is around 246GeV.
The diphoton decay rate has been calculated up-to the complete three-loop level [5] and a four-loop estimation is also considered [6]. The contributions at threeloop and four-loop levels can be neglected in comparison of the one-loop decay rate. For the two-loop level, the QCD and electroweak corrections are nearly completely cancelled in the numerical calculations for Higgs with mass of 125GeV. The electroweak radiative correction for e + e − → ZH was calculated [7][8][9] and the contribution is only below 5% of the tree-level cross section for Higgs with mass of 125GeV [9]. For the background from the continuum, the next-to-leading order electroweak corrections have been considered for the e + e − → Zγγ process in the SM by Y. Zhang et. al in the recent work [10]. A correction of 2.32% is observed as the center-of-mass energy increased to 250GeV. Though the investigation of several typical distributions for the final photons, they also found a dramatical separation in background and in the signal process, which indicated that the background can be significantly suppressed to study the Higgs signal by taking appropriate kinematic cut.
The interference effects for the Higgs mass in the diphoton decay mode at the hadron colliders have been discussed based on the theoretical aspects. The signalcontinuum interference for diphoton final states at LHC was first studied by L. J. Dixon and M. S. Siu in 2003 [11]. According to S. P. Martin, the mass shift from the interference effect is 150MeV or more [12], but the effect becomes smaller on final states containing one extra jet [13]. The interference was also evaluated to the nextto-leading order level in recent works [14,15]. The interference effect of other final states at hadron colliders was also considered in Refs. [16][17][18][19][20][21][22].
This work will focuse on the interference effect of the Higgs mass through diphoton decay mode in Higgsbremsstrahlung process at the future e + e − colliders CEPC (Several discussions at CEPC refer to Refs. [23][24][25][26]). Recently, this interference effect with fixed polarisation at the initial state has been considered in Ref. [27], and a mass shift in range of O(100MeV) is found. In this work, the interference effect but with unpolarized initial state will be revisited. Following the method used in Refs. [11][12][13][14][15], with the narrow-width approximation, the pure signal and interference cross sections for the production can be expressed as:

Calculations and analysis
where R and I represent the real and imaginary parts of the interference amplitude (A e + e − →ZH A H→γγ A * cont ), respectively, and A * cont is the continuum amplitude. The real part is odd in the vicinity of Higgs mass because of the factor m 2 γγ − m 2 H and the total contribution to the decay width is negligible. However, as stated in Ref. [12], a sharp peak and a dip exist near the M H in the diphoton distribution, and the effect slightly moves the peak position. As mentioned in the introduction, the next-to-leading order electroweak corrections to the continuum part only contribute less than 5% to the treelevel cross section. Therefore, only the tree-level contributions for the continuum part is considered in our calculation. For the amplitude of Higgs boson coupled with two photons, we also apply the result at one-loop level [11][12][13][14]. For the input parameters, the resonance mass and width of M H = 125.6GeV, Γ H = 4.2MeV, the fine structure constant α = 1/137, and the running fermion masses m t = 168.2GeV, m b = 2.78GeV, m c = 0.72GeV, m τ = 1.744GeV are adopted, respectively. The signal cross section would reach the maximum at round 245 − 246GeV. Here we take the centerof-mass energy of 246GeV in the following calculations. Fig.2 illustrates the pure signal for diphoton production from Higgs decay and the continuum cross sections.  Three type of cuts on the scattering angle are implemented, which are |cosθ cut γ | = 0.8, |cosθ cut γ | = 0.9 and |cosθ cut γ | = 0.95, respectively. Notably, the experimental cut for the scattering angle may be larger than these values. However, the cross section from ISR, i.e. continuum contribution, is sensitive to the choice of this kind of cuts because its behavior depends on 1/(1 − cosθ cut γ ). That indicates the background contributions sharply in-crease compared with the signals when larger angle cuts are chosen, making the interference effect to the mass measurement insensitive to a larger cut. Another cut on the final photon energy is taken to 20GeV. For the above cut selection, the signal process has sharp peak at the range of 50 − 70fb, and the continuum cross sections only reaches 0.5 − 1fb with the diphoton invariant mass in the range of 120 − 130GeV.
The real-part cross sections of the interference as a function of the diphoton invariant mass are shown on the left panel of Fig.3, whereas the signal with and without the interference effect are shown on the right panel of Fig.3. sig., |cosθ γ |<0.95 sig.+int., |cosθ γ |<0.95 sig., |cosθ γ |<0.9 sig.+int., |cosθ γ |<0.9 sig., |cosθ γ |<0.8 sig.+int., |cosθ γ |<0.8 The imaginary part effects are observed to be even but negligible negative values with their maximum values are less than 2% of that of real part cross sections, and we neglect then in the analysis.
To study the effect of the interference to the Higgs mass measurement, in Ref. [12], convolution integrals with a Gaussian function were created to the cross section to simulate the smearing effect of the Higgs mass due to finite experimental resolution. In Fig.4, the re-sults are plotted with the Gaussian width as σ M R = 0.8, 1.0, 1.5, and 2.0GeV. Compared with the signals without smearing effects shown in Fig.3, the peak slightly moves toward the larger mass direction when the interference effects are taken into account. The behavior of the right-side shift is consistent with that shown in [27]. Similar effects are also observed in the previous studies of hadron colliders [11][12][13][14][15] (The left-side or right-side shift effect might occurs for different sub-process).   The strategy stated by S. P. Martin in Ref. [13] is applied to estimate the mass shift and a least-square fit to the line shape of mass shifts as a function of the Gaussian width (σ M R ) is performed. The results are shown as Fig.5.  Similar to the case of Higgs production in the associated one jet, the mass shifts linearly increased with increasing mass resolution width σ M R [12]. The mass shifts increased from about 20MeV to 50MeV, corresponding to the range of the mass solution width from 0.8GeV to 2.0GeV. Form the figure, the two lines corresponding to the final photon scattering angle cut |cosθ cut γ | = 0.9 and |cosθ cut γ | = 0.95, respectively, are very close to each other referring to the gap with the line "|cosθ cut γ | = 0.8". This result implies that the shifts from the interference effect are not sensitive to a larger angle cut, as mentioned in the above analysis.

Summary
Therefore, in this work, followed by the previous works regarding hadron colliders [11][12][13][14][15], the signalbackground interference effect of the Higgs mass through diphoton decay mode in the associated Z boson production at the future e + e − colliders at energy 240 ∼ 250GeV was considered. Different cut conditions for the final photon scattering angle and different smearing width to simulate the experiments were also considered. Similar right-side shifts to the Higgs mass in the spectrum were also observed, consistent with the statement in Ref. [27]. Considering the smearing Gaussian width σ M R (which simulated to the experimental mass resolution) ranging from 0.8GeV to 2GeV, the corresponding mass shifts increased from about 20MeV to 50MeV. These results will be beneficial in studying the precision measurement of the Higgs mass.