Lineshape of the Higgs boson in future lepton colliders

The effect of the photon emission (bremsstrahlung) in the cross section of the process of direct production of the Higgs boson in the future high luminosity electron and muon colliders is calculated. It was found that cross section at the top of the Higgs boson resonance peak is reduced by factor 0.347 for the electron collider and 0.548 for the muon collider. Machine spread of the centre of the mass energy of 4.2MeV (equal to the Higgs width) would reduce peak cross section further, by factor 0.170 and 0.256 (QED and energy spread) for electron and muon beams respectively. Possible uncertainties in the resummed QED calculations are discussed. Numerical results for the lineshape cross section including QED and many values of the machine energy spread are provided.


QED initial state radiation formulas
Born cross section for the Higgs particle production in e ± collider is given by the (relativistic [8]) Breit-Wigner (B-W) formula [5] σ B (s) = 4πB ee Γ 2 where M H = 125.7 GeV and Γ H = 4.2M eV according to refs. [9,10]. The branching ratio of e + e − → H is B ee = 5.3 · 10 −9 . The electron and muon branching ratios are related by the factor m 2 µ /m 2 e , thus the above result can be obtained from B µµ = 2.19 · 10 −4 , see also refs. [9,10]. There are variants of the B-W formulas with the s-dependent width, but for the narrow resonance like Higgs they differ negligibly from the above, see more discussion in the Appendix A.

(2.4)
Here, α is the QED coupling constant, m e the electron mass and C is the Euler-Mascheroni constant. In case of the muon beams m e in γ is replaced by m µ and B ee by B µµ . Zero spin nature of the Higgs boson instead of spin one of Z counts negligibly in the QED ISR effects, simply because deformation of the resonance curve is mainly due to soft photons. The constant A which is responsible for the above spin difference is of order α π 1/400 without any logarithmic enhancement. It is influencing mainly an overall normalization -hence at the precision level we are aiming at, it can be safely set to zero 3 On the other hand soft photon exponentiation/resummation in eq. (2.2) is critical and mandatory. The formula of eq. (2.2) comes from standard diagrammatic perturbative QED calculations including Yennie-Frautschi-Suura (YFS) exponentiation, see ref. [5] and was originally introduced for the purpose of the algebraic validation of the Monte Carlo program YFS2 of ref. [11]. Later on it was discussed and used in many papers, see for instance refs. [5,[12][13][14][15] and the references therein.
Three variants (for I = (a), (b) or (c)) of the ISR formula in eq. (2.2) correspond to the increasing sophistication (perturbative order) of the non-soft collinear radiative corrections. Change of the type of ISR formula will be used to estimate uncertainty due to unknown/neglected QED higher orders.

Machine energy spread
In real accelerator experiments the beam is not monoenergetic, i.e., centre-of-mass energy E = √ s has spread δ around the centre value E 0 = √ s 0 of the beam energy. The distribution of E is usually well approximated by the following Gaussian distribution In case of no QED effects, the Born cross section (2.1) gets simply convoluted with the energy spectrum of eq. (3.1): Once QED bremsstrahlung is switched on, the following double convolution provides realistic experimental cross section: Because of rapid decrease of the Gaussian distribution for large arguments, the energy integration range will be restricted to E − 10δ ≤ E ≤ E + 10δ without any loss of the calculation reliability. The numerical integrations in one and two dimensions require a little bit of care, because of strongly singular integrands. The adaptive integration library functions of ROOT library [16] were used. All results were also cross-checked using FOAM adaptive Monte-Carlo simulator/integrator of [17][18][19]

Numerical results for electron-positron colliders
The Born cross sections and results from three variants of the ISR formula of eq. (2.2) are presented in Fig. 1. The case σ (b) and σ (c) are almost indistinguishable. In addition,  Table 2: Cross section of the direct Higgs production process in the electro-positron collider including effect of ISR type (c) and the machine energy spread. No energy spread is in σ (1) , while σ (2) and σ (3) include ISR and energy spread δ = 4.2 M eV and δ = 8.0 M eV according to eq. (3.3). Reference Born cross section σ B is also included and the ratios with respect to Born are also provided. The integration error is below 0.0005f b.
Tab. 1 presents the same results for the energy at the peak E = M H and near the peak E = M H ± Γ H with the 4-digit precision. The QED uncertainty from unaccounted QED higher orders can be estimated from Tab. 1 looking into differences between ISR type I = (c) and I = (b). It is below 0.3% and is compatible with the estimate of neglected A in eq. (2.2) (it is also comparable to the numerical integration error). From now on we shall use ISR formula for I = (c) only.
The introductory exercise on the machine energy spread is shown in Fig. 2 where Higgs production cross section in the electron-positron collider is plotted without energy spread (Born) and for several values of the machine energy spread δ = 4.2, 8, 15, 30, 100MeV. The QED ISR effect is not yet included.
On the other hand Fig. 3 shows our most interesting result, that is Higgs production cross section in the electron-positron collider including both QED ISR effect according to eq. In Fig. 4 we plot again the same Higgs production cross section in the electron-positron collider including both QED ISR effect and machine energy spread in the narrower energy range and for the energy spread parameter δ = 4.2MeV which is ambitiously aimed at the FCCee collider, and also for more realistic δ = 8.0MeV. The ratio of cross section with respect to Born is also shown in the lower plot there.
The same results as in Fig. 4 are also shown in Tab. 2 for three values of the energy, at the resonance peak E = M H and near the peak E = M H ± Γ H , with the 4-digit precision. As we see there, the combined reduction of the Higgs production process at the resonance peak due to QED effect is 0.347 and goes down to 0.170 for the machine energy spread equal to Higgs width ∼ 4.2MeV.
The suppression factor of the peak cross section due to QED ISR can be also quite well reproduced, and in this way cross checked, with very simple approximate calculation presented in Appendix B. This approximation works even better in case of the weaker ISR effect for muon beams.
The above results can be used as an input for further studies of the practical observability of the Higgs resonant cross section in the future electron-positron colliders.
In the above numerical exercises we have used discrete values of the machine energy spread δ. In Fig. 5 we also show the continuous dependence on δ of the Higgs production  Table 3: Cross section of the direct Higgs production process in the µ + µ − collider including effect of ISR type (c) and the machine energy spread. The same values of the energy spread δ an energy E are used as in Tab.2. The Monte Carlo integration error is below 0.03pb. process at the resonance peak, E = M H , divided by the Born cross section, both for QED ISR switched on and off. More precisely, we are plotting there the following two ratios for ISR off and on respectively. One could easily expand the above numerical exercises to more values in the 2dimensional space of the energy E and machine energy spread δ, but we think that numerical results collected above are complete enough and can serve as a reliable starting point for all studies of the Higgs resonance observability in the future electron-positron colliders.
It is worth to pointing out that because of the normalization of convolution as a function of δ at E = M H to unity we can construct the following quantity that is close unity when δ/Γ H is not too big. In Fig. 7 the following quantity is plotted. If R V (E, δ) 1 then double convolution can be replaced by the following handy approximation As seen in Fig. 7 the above approximation for E = M H works reasonably, within 10% for δ ≤ Γ H , but its validity deteriorates significantly for δ > Γ H and in such a case numerical double convolution is unavoidable.

Numerical results for muon colliders
The calculations of the previous section can be easily extended to the case of muon collider, in the so-called Higgs factories (for overview see, e.g., [20]), by means of replacing m e with m µ and B ee with B µµ in the equations of section 2 (see also [6] or [7]). This kind of colliders will produce clean sample of Higgs boson without much background and therefore would allow to measure Higgs mass and its properties very precisely. Our analysis is vital in view of MAP (Muon Accelerator Program) [20] at Fermilab and the new ideas of producing muon monochromatic beams, that would allow to limit the major obstacle, the machine energy spread. Thanks to much higher branching ratio for the Higgs to muon pair, the production cross section for the µ + µ − → H is much higher. Also, QED effects for muon collider are roughly factor 2 weaker, simply because for muon beams at the Higgs peek γ = 0.0611, as compared to γ = 0.1106 for electron beams.
Numerical results for muon collider are presented in Fig. 6 and Tab. 3, and correspond to Higgs production cross section dependence on energy E and machine energy spread δ shown in Fig. 4 and in Tab. 2. As seen in this table, reduction factor of the cross section at the resonance peak due to QED ISR is now only 0.548. For the machine energy spread δ = 8MeV it deteriorates down to 0.162 (which is accidentally comparable to electron collider case with δ = 4MeV) and for δ = 4.2MeV it is equal 0.279 5 .

Summary
It was analysed numerically what is the influence of the QED ISR and machine energy spread on the the resonant Higgs boson production cross section, the so called Higgs line-shape, in the process e + e − → H and µ + µ − → H.
It was found that for electron collider the QED ISR reduces by itself peak cross section by factor 0.338. The QED higher order uncertainty of this results was estimated to be below 0.3%. The proper double convolution of the QED radiative spectrum with the machine energy spread was performed and doubly cross-checked. For instance machine energy spread of the same size as Higgs width (δ 4MeV) reduces Higgs production cross section further down to 0.172σ B . These results are compatible (albeit slightly different) from these in the preliminary analysis shown at the recent FCCee workshop, see [4]. It forms a solid basis for any analysis of the observability of the Higgs resonant cross section in the future e + e − colliders.
The same analysis was also repeated for the muon colliders, where the reduction factor due to QED ISR at the peak position was found to be 0.548 and, for instance, it gets reduced further down to 0.162 for the energy beam spread being twice the Higgs width.
(3) Figure 6: Line-shape of the direct Higgs production process in the µ + µ − collider including effect of ISR type and the machine energy spread. The same input and notation as in Fig. 4 [MeV] δ