# \(\mathcal {O}(\alpha ^2)\) ISR effects with a full electroweak one-loop correction for a top pair-production at the ILC

## Abstract

Precise predictions for an \(e^+e^-\rightarrow t\bar{t}\) cross-section are presented in the energy region from 400 to 800 GeV. Cross-sections are estimated including the beam-polarization effects with full \(\mathcal {O}(\alpha )\), and also with the effects of the initial-state photon emission. A radiator technique is used for the initial-state photon emission up to two-loop orders. In this investigation, a weak correction is defined as the full electroweak corrections without the initial-state photonic corrections. As a result, it is determined that the total cross-section of a top quark pair-production receives the weak corrections of \(+\,4\%\) over the trivial initial state corrections at a center of mass energy of 500 GeV. Among the initial state contributions, a contribution from two-loop diagrams gives less than \(0.11\%\) correction over the one-loop ones at the center of mass energies of from 400 to 800 GeV. In addition, the effect of a running coupling constant is also discussed.

## 1 Introduction

The standard theory of particle physics has been finally established after the discovery of the Higgs boson [1, 2] in 2012. The next major challenge in particle physics is the search for a more fundamental theory beyond the standard model (BSM). In this regard, the role of the Higgs boson and the top quark is considered to be critical. Since the top quark is the heaviest fermion with a mass in the electroweak symmetry-breaking regime, it is naturally expected to have a special role in the BSM.

The international linear collider [3] (ILC), which is an electron-positron colliding experiment with a center of mass (CM) energies above 250 GeV, is proposed and intensively discussed as a future project of high-energy physics. One of the main goals of the ILC experiments is the precise measurement of the properties of the top quark. Detailed Monte Carlo studies have shown that the ILC can measure most of the standard model parameters to within sub-percent levels [4]. Theoretical predictions are required with a new level of precision because of the improvement in the experimental accuracy of the ILC. In particular, a radiative correction due to the electroweak interaction (including spin polarizations) is mandatory based on this requirement. Before the discovery of the top quark, a full electroweak radiative correction was performed for an \(e^-e^+\rightarrow t\bar{t}\) process at a lower energy [5, 6], and was then independently obtained for higher energies [7, 8]. The same correction for a \(e^-e^+\rightarrow t\bar{t}\gamma \) process has also been reported [9]. NLO QCD corrections for on-shell \(t\bar{t}\) and \(t\bar{t}H\) including decays are calculated in Ref. [10]. A detailed study of the electroweak correction for top quark decay is also reported in Refs. [11, 12]. A possible search of the minimum SUSY particles based on loop corrections of the top-quark pair-production at the ILC is reported in [13].

Recently, electroweak radiative corrections for the process \(e^-e^+\rightarrow t \bar{t}\) \(\rightarrow b \bar{b} \mu ^+\mu ^-\nu _{\mu } \bar{\nu }_{\mu }\) including the spin-polarization effects have been reported [14] by the authors of a present report. Unfortunately, a complete electroweak correction for the six-body final-state is impossible based on the current limits of computing power because the number of Feynman diagrams concerned is too large. In Ref. [14], authors used a simple narrow width approximation (NWA) for the top-quark production and decay. A more precise method to treat both production and decay at a one loop-order is the double-pole approximation method. This method was initially developed for a *W*-boson pair production [15, 16], and was subsequently applied to top quark production [17].

Among several sources of radiative correction, it is known that the initial state photonic correction (ISR) accounts for the largest contribution in general, and thus, it is very important for the precise estimation of production cross-sections. Top-quark decay is not treated in this study, because the ISR effect on the total cross-section does not strongly depend on its decay process. In this report, the precise estimation of the effect due to the initial-state photon-radiation is discussed in detail.

## 2 Calculation method

### 2.1 GRACE system

For precise cross-section calculations of the target process, a GRACE-Loop system is used in this study. The GRACE system is an automatic system for calculating the cross-sections of scattering processes at a one-loop level for the standard theory [18] and the minimum SUSY model [19]. The GRACE system has been used to treat electroweak processes consisting of two, three or four particles in the final state [20, 21, 22, 23] at the one-loop order. The renormalization of the electroweak interaction is performed using on-shell scheme [24, 25]. Divergences in the infrared limit are regulated using fictitious photon-mass [25]. The symbolic manipulation package FORM [26] is used to deal with all Dirac and tensor algebra in *n*-dimensions. For loop integrations, all tensor one-loop integrals are reduced to scalar integrals using our own formalism [18], then performed integrations using packages FF [27] or LoopTools [28]. Phase-space integrations are performed using an adaptive Monte Carlo integration package BASES [29, 30]. For numerical calculations, we used a quartic precision for floating variables.

While using \(R_\xi \)-gauge for the linear gauge-fixing terms in the GRACE system, the non-linear gauge fixing Lagrangian [18, 31] is also employed to check the system. Numerical tests were performed to confirm that the amplitudes are independent of all redundant parameters at approximately 20 digits at several randomly chosen phase points before calculating the cross-sections. In addition to the aforementioned checks, the soft-photon cut-off independence was examined: cross-sections at the one-loop level, results must be independent of the head-photon cut-off parameter \(k_c\). We confirmed that, while varying a parameter \(k_c\) from \(10^{-4}\) to \(10^{-1}\) GeV, the results of numerical phase-space integration are consistent with each other within the statistical errors of numerical integrations, which is typically on the order of \(0.1\%\).

### 2.2 Radiator method

*H*(

*x*,

*s*) as follows:

*s*is the CM energy squared and

*x*is the energy fraction of an emitted photon. The structure function can be calculated using the perturbative method with the SPA. Concrete formulae of the structure function are calculated up to two loop orders [25]. A further improvement of the cross-section estimation is possible using the “exponentiation method”. For initial state photon emissions under the SPA, the probability of emitting each photon should be independent of each other. Thus, the probability of emitting any number of photons can be calculated as;

*k*photons. A factor 1 /

*k*! is necessary due to the appearance of

*k*identical particles (photons)in the final state. This is essentially a Taylor expansion of the exponential function. Therefore, the effect of multiple photon emissions can be estimated by making the one-photon emission probability the argument of the exponential function. This technique is referred to as to the exponentiation method. When the exponentiation method is applied to the cross-section calculations at loop level, the corrected cross-sections cannot simply be expressed as the formula (1), because the same loop corrections are included in both the structure function and loop amplitudes. To avoid a double counting of the same corrections in the structure function and loop amplitudes, the terms of corrections have to be rearranged.

*E*is a beam energy and \(s=4E^2\). The threshold energy \(k_c\) is included in both \(\sigma _{Hard}\) and \(\delta _{SPA}\), and the total cross-section must be independent of the value of \(k_c\) after summing up all contributions. At the same time, final results are also independent of the photon mass \(\lambda \) due to the cancellation among \(\lambda \) contributions in \(\sigma _{Loop}\) and \(\delta _{SPA}\).

*H*(

*s*,

*x*). A term \(\widetilde{\sigma }_{Soft}\) includes only the final-state radiation and interference terms between the initial and final state radiations. Instead, \(\widetilde{\sigma }_{ISR}\) gives the improved cross-section including the initial-state radiation using the radiator method [25]. The total cross-section can be calculated using the radiator function as;

*D*(

*x*,

*s*), which corresponds to the square root of the structure function, gives the probability of emission of a photon with an energy fraction of

*x*at the CM energy square

*s*. In this method, electrons and positrons can emit different energies, and thus a finite boost of the CM system can be treated. The radiator function can be obtained as [25].

## 3 Results and discussions

### 3.1 Input parameters

The input parameters used in this report are listed in Table 1. The mass of the light quarks (i.e., other than the top quark) and *W* boson are chosen to be consistent with low-energy experiments [32]. Other particle masses are taken from recent measurements [33]. The weak mixing-angle is given using the on-shell condition \(\sin ^2{\theta _W} = 1- m_W^2/m_Z^2\) because of our renormalization scheme. The fine-structure constant \(\alpha =1/137.0359859\) is obtained from the low-energy limit of Thomson scattering, because of the renormalization scheme used.

Particle masses used in analysis

| \(58.0\times 10^{-3}\) GeV |
| \(58.0\times 10^{-3}\) GeV |

| 1.5 GeV |
| \(92.0\times 10^{-3}\) GeV |

| 173.5 GeV |
| 4.7 GeV |

| 91.187 GeV |
| 80.370 GeV |

Higgs mass | 126 GeV |

### 3.2 Electroweak radiative corrections

#### 3.2.1 total cross-sections

At first, the fixed order correction without using an exponentiation method is investigated. The total cross-sections obtained at leading (tree) and next-to-leading order (NLO) calculations are shown in Fig. 1. These results are the same as those included in our previous report [14] for a simple NLO correction. The consistency between our current and previous results [5, 7, 8] was numerically confirmed after adjustment of the input parameters. The NLO calculations near the top-quark production threshold (at an approximate CM energy of 400 GeV) reveals negative corrections of approximately \(7\%\). The corrections become very small in the vicinity of the CM energy of 500 GeV and increase in the high-energy region to \(2.8\%\) at the CM energy of 800 GeV. Considering several types of radiative corrections, e.g., the initial and final state photon radiation, the vertex and box correction, etc., the initial-state photonic correction gives the largest contribution at the high energy region. As shown in Fig. 1, the cross-sections at the tree level including the ISR correction (a dotted line in the figure) are almost the same as the full order \(\mathcal {O}(\alpha )\) electroweak correction (a dashed line in the figure) at CM energies above 700 GeV. This implies that the main contribution of the higher order corrections is caused by the initial state photonic corrections. On the other hand, other corrections from the loop diagrams also give a large correction near the threshold region. At a CM energy of approximately 500 GeV, these effects are accidentally canceled and result in a small correction of the total cross-section.

#### 3.2.2 Angular distribution

### 3.3 Photonic correction at two-loop order

*H*(

*s*,

*x*) given in (8) include two-loop effects. All of the aforementioned results were obtained using a full formula (8). As mentioned in the previous subsection, a main contribution of the radiative corrections comes from the initial state photonic-correction in the high energy region. Therefore, the ISR correction is one of many important terms of the radiative corrections associated with these energies. If the two-loop contribution has a significant fraction in the full correction, even higher-loop corrections must be considered in future experiments. The fraction of two-loop contribution over the one-loop one is defined as

### 3.4 Running coupling

*qqW*vertices is provided using Eqs. (12) and (13) with \(q^2=m_W^2\).

## 4 Summary

We calculated the precise cross-sections of an \(e^+e^-\rightarrow t\bar{t}\) process in the energy region from 400 to 800 GeV. In particular, the initial-state photon emissions were discussed in detail. An exponentiation technique was applied for the initial-state photon emissions up to two-loop orders. It was determined that the total cross-section of a top quark pair-production at a center of mass energy of 500 GeV receives the weak corrections of \(+\,4\%\) over the trivial ISR corrections. Among the ISR contributions, two-loop diagrams resulted in less than \(0.4\%\) correction with respect to the one-loop ones at the CM energies from 400 to 800 GeV.

The improved Born approximation yielded cross-sections better than \(2\%\) compared with the NLO cross-sections with ISR.

## Notes

### Acknowledgements

I would like to thank prof. T. Kaneko and prof. F. Yuasa for their continuous encouragement and fruitful discussion. We would like to thank Editage for English language editing.

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