ISR effects on loop corrections of a top pair-production at the ILC
Precise predictions for an cross section are presented at an energy region from 400 GeV to 800 GeV. Cross sections are estimated including the beam-polarization effects with full , and also with effects of the initial-state photon emission. A resummation technique is used for the initial-state photon emission up to two-loop order. A pure-weak correction is defined as the full electroweak corrections without the initial-state photonic corrections. As a result, it is obtained that the total cross section of a top quark pair-production receives the pure-weak corrections of 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 correction over the one-loop ones at the center of mass energy of GeV.
The standard theory of particle physics are established finally by a discovery of the Higgs boson[1, 2] in 2012. A current target of particle physics is searching for a more fundamental theory beyond the standard theory (BST). A keystone along this direction must be the Higgs boson and top quark. Since the top quark is the heaviest fermion with a mass at the electroweak symmetry-breaking scale, it is naturally expected to have a special role in the BST.
The international linear collider (ILC), which is an electron-positron colliding experiment with centre of mass (CM) energies above 250 GeV, is proposed and intensively discussed as a future project of high-energy physics. One of main goals of ILC experiments is a precise measurement of top quark properties. Detailed Monte Carlo studies have shown that the ILC would be able to measure most of the standard model parameters to within sub-percent levels. Because of the improvement of the experimental accuracy of the ILC, theoretical predictions are required be given with new level of precision. In particular, a radiative correction due to the electroweak interaction (including spin polarizations) is mandatory for such requirements. Before the discovery of the top quark, a full electroweak radiative correction was conducted for an process at a lower energy, and was then obtained independently for higher energies[6, 7]. The same correction fo a process has also been reported. Recently, full electroweak radiative corrections for the process using a narrow-width approximation for the top quarks including the spin-polarization effects are reported by authors including those of a present report. In this report, the precise estimation of the initial-state photon-radiations is discussed in detail
2 Calculation Method
2.1 GRACE system
For precise cross-section calculations of target process, a GRACE-Loop system is used in this study. The GRACE system is an automatic system to calculate cross sections of scattering processes at one-loop level for the standard theory and the minimum SUSY model. The GRACE system has treated electroweak processes with two, three or four particles in the final state are calculated[12, 13, 14, 15]. The renormalization of the electroweak interaction is carried out using on-shell scheme[16, 17]. Infrared divergences are regulated using fictitious photon-mass. The symbolic manipulation package FORM is used to handle all Dirac and tensor algebra in -dimensions. For loop integrations, all tensor one-loop integrals are reduced to scalar integrals using our own formalism, then performed integrations using packages FF or LoopTools. Phase-space integrations are done using an adaptive Monte Carlo integration package BASES[21, 22]. For numerical calculations, we use a quartic precision for floating variables.
In the GRACE system, while using -gauge in linear gauge-fixing terms, the non-linear gauge fixing Lagrangian[23, 10] is employing for the sake of the system checking. Before calculating cross sections, we performed numerical tests to confirm that the amplitudes are independent of all redundant parameters around digits at several randomly chosen phase points. In addition to above checks, soft-photon cut-off independence was examined: cross sections at the one-loop level, results must be independent from a head-photon cut-off parameter . We confirmed that, while varying a parameter from GeV to GeV, the results of numerical phase-space integrations are consistent each other within the statistical errors.
2.2 Structure function method
The effect of the initial photon emission can be factorized when a total energy of emitted photons are small enough compared with a beam energy. The calculations under a such approximation is referred as the “soft photon approximation(SPA)”. Under the SPA, the corrected cross sections with the initial state photon radiation(ISR), , can be obtained from the tree cross sections using a structure function as follows:
where is the CM energy square and is a energy fraction of emitted photons. The structure function can be calculated using the perturbative method with the SPA. Concrete formulae of the structure function are known up to two loop order. A further improvement of the cross section estimation is possible using the “exponentiation method”. Initial state photon emissions under the SPA, a probability to emit each photon should be independent each other. Thus, the probability to emitt any number of photons can be calculated as
where is a probability to emit -photons. A factor appeared because of a number of identical particles (photons) in the final state. This is nothing more than the Taylor expansion of the exponential function. Therefore, the effect of the multiple photon emissions can be estimated by putting the one photon emission probability in an argument of the exponential function. This technique is referred as the exponentiation or the resummation method.
When the resummation method is applied to the cross section calculations at loop level, the corrected cross sections can not be expressed simply like the formula (1), because the same loop corrections are included in both of the structure function and the loop amplitudes. To avoid a double counting of the same effects both in the structure function and the loop amplitudes, one have to rearrange terms of corrections as follows: The total cross section at one-loop (fixed) order without the resummation, , can be expressed as
where and are the cross sections from the loop and (hard) real-emission corrections, respectively. A factorized function is obtained form the real-radiation diagrams using the SPA. Here the hard photon is defined as the photon whose energy is greater than the threshold energy . Final results must be independent of the value of . The cross section with fixed order correction (3) can be improved using the resummation method as follows:
where shows the improved cross section using the structure function method (1) with the resummation, and is a correction factor from the initial state photon-loop diagrams, which is also included in the structure function .
3 Results and discussions
3.1 Input parameters
The input parameters used in this report are listed in Table 1. The masses of the light quarks (i.e., other than the top quark) and boson are chosen to be consistent with low-energy experiments. Other particle masses are taken from recent measurements. The weak mixing angle is obtained using the on-shell condition because of our renormalization scheme. The fine-structure constant is taken from the low-energy limit of Thomson scattering, again because of our renormalization scheme.
Because the initial state photonic corrections are independent of the beam polarization, all cross section calculations in this report are performed with left (right) polarization of an electron (positron), respectively.
|-quark mass||GeV||-quark mass||GeV|
|-quark mass||GeV||-quark mass||GeV|
|-quark mass||GeV||-quark mass||GeV|
|-boson mass||GeV||-boson mass||GeV|
|Higgs mass||126 GeV|
3.2 Electroweak radiative corrections
At first, let us look at the results of the fixed order correction without a resummantion method shown in Fig. 1. The NLO calculations near the top-quark production threshold shows negative corrections about at the CM energy of GeV. This large correction is a so-called “coulomb correction”, which is caused by that produced particles are moving slowly and have enough time duration to interact each other. The corrections becomes very small around the CM energy of GeV and increases at the high energy region as to at the CM energy of GeV.
Among several types of the radiative correction, e.g., the initial and final state photon radiations, the vertex and box correction, and so on, one can see the initial state photonic corrections give a largest contribution at the high energy region. As shown in Fig. 2, the cross sections at tree level including the ISR correction (a gray line in the figure) are almost the same as the full order electroweak correction (an orange line in the figure) at the CM energy above GeV. This means that the main contribution of the higher order corrections is caused by the initial state photonic corrections. On the other hand, other corrections form the loop diagrams also give a large correction near the threshold region.
After subtracting a (rather trivial) ISR correction from the total corrections, one can discuss the “pure” weak correction in the full electroweak radiative corrections. While the NLO correction degree is defined as
the weak correction degree is defined as
In the definition of , the trivial initial state photonic corrections are subtracted from the full electroweak radiative corrections. Thus, we can understand shows a fraction of the pure-weak correction in the full electroweak radiative corrections. The behaviors of and are shown in Fig. 3 with respect to the CM energies. One can see the pure-weak correction becomes smaller and smaller at a high energy region, and arrive at almost zero at the CM energy of GeV. On the other hand, at the CM energy of GeV, the pure-weak corrections gives correction over the trivial ISR corrections.
3.3 Photonic correction at two-loop order
As mentioned above, the structure function are known up to two-loop order. All of above results are obtained using formulae including two-loop contributions. As also mentioned in previous subsection, a main contribution of the radiative corrections comes from the initial state photonic correction at the high energy region. Therefore, the ISR correction is important among many terms of the radiative corrections. If the two-loop contribution has a large fraction over the one-loop ones, one need to consider even higher-loop corrections for the future experiments. The fraction of two-loop contribution over the one-loop one is defined as
where and show the ISR corrected cross sections by the structure function with one-loop diagrams only and with both one- and two-loop diagrams, respectively. One can see in Fig. 4 that the two-loop contribution becomes larger according to increase the CM energy. It is still about at the CM energy of GeV and reaches about at the CM energy of GeV.
We calculated the precise cross sections of an process at an energy region from 400 GeV to 800 GeV. Especially, the initial-state photon emissions are discussed in details. A resummation technique is applied for the initial-state photon emission up to two-loop order. We found that the total cross section of a top quark pair-production at a center of mass energy of 500 GeV receives the pure-weak corrections of over the trivial ISR corrections. Among the ISR contributions, two-loop diagrams gives correction with respect to the one-loop ones at the CM energy of GeV.
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