Các bổ chính bức xạ điện yếu của giản đồ Feynman một vòng cho quá trình e⁺e⁻ → ZH với chùm tia tới phân cực tại ILC

 TRƯỜNG ĐẠI HỌC SƯ PHẠM TP HỒ CHÍ MINH 
TẠP CHÍ KHOA HỌC 
HO CHI MINH CITY UNIVERSITY OF EDUCATION
JOURNAL OF SCIENCE
ISSN: 
1859-3100 
KHOA HỌC TỰ NHIÊN VÀ CÔNG NGHỆ 
Tập 15, Số 3 (2018): 24-35 
NATURAL SCIENCES AND TECHNOLOGY
Vol. 15, No. 3 (2018): 24-35 
 Email: tapchikhoahoc@hcmue.edu.vn; Website:  
24 
FULL ङ(ࢻ) ELECTROWEAK RADIATIVE CORRECTIONS 
TO ࢋାࢋି → ࢆࡴ WITH BEAM POLARIZATIONS AT THE ILC 
Phan Hong Khiem*, Pham Nguyen Hoang Thinh 
University of Science Ho Chi Minh City 
Received: 18/12/2017; Revised: 16/01/2018; Accepted: 26/3/2018 
ABSTRACT 
We present full ࣩ(ߙ) electroweak radiative corrections to ݁ା݁ି → ܼܪ with the initial beam 
polarizations at the International Linear Collider (ILC). The calculation is checked numerically by 
using three consistency tests that are ultraviolet finiteness, infrared finiteness, and gauge 
parameter independence. In phenomenological results, we study the impact of the electroweak 
corrections to total cross section as well as its distributions. In addition, we discuss the possibility 
of searching for an additional Higgs in arbitrary beyond the Standard Model (BSM) through ZH 
production at the ILC. 
Keywords: Higgs physics at future colliders, numerical method for particle physics, one – 
loop electroweak corrections, physics beyond the Standard Model. 
TÓM TẮT 
Các bổ chính bức xạ điện yếu của giản đồ Feynman một vòng cho quá trình ࢋାࢋି → ࢆࡴ 
với chùm tia tới phân cực tại ILC 
Chúng tôi trình bày các bổ chính bức xạ điện yếu của giản đồ Feynman một vòng cho quá 
trình ݁ା݁ି → ܼܪ với chùm tia tới phân cực tại máy gia tốc tuyến tính quốc tế (ILC). Kết quả tính 
toán được kiểm tra số bằng ba phép kiểm tra: Hữu hạn tử ngoại, hữu hạn hồng ngoại và tính độc 
lập với các tham số gauge. Trong phần kết quả hiện tượng luận, chúng tôi nghiên cứu về sự ảnh 
hưởng của các bổ chính điện yếu đối với tiết diện tán xạ và các phân bố tiết diện tán xạ. Hơn nữa, 
chúng tôi cũng thảo luận về khả năng tìm ra một hạt Higgs (khác với hạt Higgs trong mô hình 
chuẩn) trong số những mô hình mở rộng của mô hình chuẩn (BSM) thông qua quá trình ݁ା݁ି →
ܼܪ tại ILC. 
Từ khóa: vật lí Higgs tại máy gia tốc tương lai, phương pháp giải số trong vật lí hạt, bổ 
chính điện yếu của giản đồ Feynman một vòng, vật lí trong các mô hình mở rộng của mô hình 
chuẩn. 
1. Introduction 
The discovery of the Standard Model-like Higgs boson at the Large Hadron Collider 
(LHC) in 2012 [1], [2] has opened up a new era in particle physics which focuses on 
precision measurement of the Standard Model (SM) as well as search for physics beyond 
the Standard Model. In particular, one of the main targets of future colliders such as the 
* Email: phkhiem@hcmus.edu.vn 
TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM Phan Hong Khiem et al. 
25 
LHC at high luminosities [3], [4], the ILC [5], is to measure the properties of the Higgs 
boson. These measurements will be performed at high precision, e.g. the Higgs boson’s 
couplings will be probed at the precision of 1% or better for a statistically significant 
measurement [5]. This level of precision can be archived at the clean environment of 
lepton colliders (the ILC as a typical example) rather than hadron colliders. In order to 
match the high precision data in near future, higher-order corrections to Higgs productions 
at the ILC are necessary. 
The ILC is a proposed eାeିcollider including the initial beam polarizations with 
center of-mass energy ൫√ݏ൯ in range of 250 GeV to 500 GeV. The energy can be also 
expanded up to 1 TeV. The main Higgs production channels at the ILC are Higgsstrahlung 
(ZH) and WW-, ZZ- fusions. With 250 GeV ≤ √ݏ ≤ 500 GeV, the Higgsstrahlung process 
is the dominant channel. For the process ݁ା݁ି → ܼܪ, the advantage of the recoil mass 
technique [6] can be applied to extract the ZH event which is independent of the Higgs 
decay channels. Hence, the cross section for this process and its relevant distributions can 
be measured to few sub-percent accuracy. 
Full one-loop electroweak radiative corrections have been computed in Refs. [7] - 
[9]. In above calculations, the authors have provided the results for polarized leptons as 
well as polarized Z-boson. However, the detailed numerical investigation for polarizations 
of eା, eି at the ILC, e.g. two beam polarizations which are (ܲ݁ି,ܲ݁ା) = (−80%, +30%) 
and (+80%,−30%) have not been presented yet. Recently, mixed electroweak-QCD 
corrections to this process have been considered in Ref. [10]. The paper has only presented 
the results for unpolarized beams of eା , eି. 
In view of the importance of the process eାeି → ZH, we perform the computation 
again in order to cross-check the previous results, update the physical predictions by using 
the modern input parameters, and include the initial beam polarizations at the ILC. 
Moreover, in this paper we develop a model-independent way introducing an additional 
Higgs boson to the SM. The coupling of the extra Higgs to ZZ which follows the sum rules 
for Higgs bosons [11]. We then discuss the possibility to probe BSM through ZH 
production at the ILC. 
Our paper is organized as follows: In the next section, we present the calculation in 
detail. First, the GRACE-LOOP is described briefly. One then performs the numerical 
checks for the calculation. We next show the physical results for the process eାeି → ZH 
with non - polarized beams at the ILC in more detail. In section III, search for the 
additional Higgs boson at the ILC is discussed. Finally, conclusions and prospects are 
devoted in section IV. 
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26 
2. The calculations 
In this section, we explain the computation for full one-loop radiative corrections to 
process eାeି → ZH in detail. The GRACE program at one-loop [12] used for this 
computation is described in next subsection. 
2.1. GRACE at one loop 
GRACE-LOOP is a generic program for the automatic calculation of scattering 
processes at one-loop electroweak corrections in High Energy Physics. With the 
complexity of the automatic calculation, the internal consistency checks for the 
computation are necessary. For this purpose, the program has implemented non-linear 
gauge fixing terms in the Lagrangian which will be described in the next paragraphs. In 
GRACE-LOOP, the renormalization has been carried out with the on-shell condition 
(follows Kyoto scheme) as reported in Ref. [12]. This program has been checked carefully 
with many of 2 → 2-body electroweak processes in Ref. [12]. The GRACE-LOOP has also 
been used to calculate 2 → 3-body processes such as eାeି → ܼܼܪ, eାeି → ݐݐ̅ܪ, eାeି →
ߥߥ̅ܪ. Moreover, the 2 → 4-body process as eାeି → ߥఓνതஜܪܪ has been performed by using 
GRACE-LOOP. Recently, full one-loop electroweak radiative corrections to two important 
processes which are eାeି → ݐݐ̅ߛ, eାeିߛ have been computed successfully with the help of 
the program. 
Full one-loop electroweak corrections to a process in the GRACE program are 
computed as follows. First, we edit a file (it is called in.prc) in which the users declare the 
model (Standard Model in this case), the names of the incoming and outgoing particles, 
and kinematic configurations for the phase space integration. In the intermediate stage, 
symbolic manipulation FORM [13] handles all Dirac and tensor algebra in d-dimensions, 
decomposes the scattering amplitude into coefficients of tensor one-loop integrals and 
writes the formulas in terms of FORTRAN subroutines on a diagram by diagram basis. 
The generated FORTRAN code will be combined with libraries which contain the routines 
that reduce the tensor one-loop integrals into scalar one-loop functions. These scalar 
functions will be numerically evaluated by one of the FF [14] or LoopTools [15] packages. 
The ultraviolet divergences (UV-divergences) are regulated by dimensional regularization 
and the infrared divergences (IR-divergences) is regulated by giving the photon an 
infinitesimal mass λ. Eventually all FORTRAN routines are linked with the GRACE 
libraries which include the kinematic libraries and the Monte Carlo integration program 
BASES [16]. The resulting executable program can finally calculate cross-sections and 
generate events. Ref [12] describes the method used by the GRACE-LOOP to reduce the 
tensor one-loop five- and six-point functions into one-loop four-point functions. 
As mentioned before, the GRACE-LOOP allows the use of non-linear gauge fixing 
conditions [12] which are defined as follows 
TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM Phan Hong Khiem et al. 
27 
	ℒ࣡ℱ = − 1ξௐ ቚ൫∂ஜ − ݅݁α෥ܣஜ − ݅݃ܿௐβ෨ܼஜ൯ܹஜା + ξௐ ݃2 ൫ν + δ෨ܪ + ݅κଷ෦χଷ൯χାቚଶ 
−
12ξ௓ ൬∂ஜܼ + ξ௓ ݃2ܿௐ (ν + ε෤ܪ)χଷ൰ଶ − 12ξ஺ ൫∂ஜܣ൯ଶ. (1) 
We work in the ܴஞ-type gauges with condition ξௐ = ξ௓ = ξ஺ = 1 (with so-called the 
’t Hooft Feynman gauge), there is no contribution of the longitudinal term in the gauge 
propagator. This choice not only has the advantage of making the expressions much 
simpler, but also avoids unnecessary large cancellations, high tensor ranks in the one-loop 
integrals and extra powers of momenta in the denominators which cannot be handled by 
the FF package. 
Recently, we have used our one-loop integral program which has been reported in 
Ref. [17]. The polarizations for initial beam have been also included in this program [18]. 
Both new features are used for the calculations in this report. 
2.2. ݁ା݁ି → ܼܪ with unpolarized beams 
The full set of Feynman diagrams with the nonlinear gauge fixing, as described in the 
previous section, consists of 4 tree diagrams and 341 one-loop diagrams. This includes the 
counterterm diagrams. In Fig. 1, we show some selected diagrams. 
Figure 1. Typical Feynman diagrams for the reaction ݁ା݁ି → ܼܪ generated 
by the GRACE-Loop system 
We use the following input parameters for the calculation: The fine structure 
constant in the Thomson limit is ߙିଵ = 	137.0359895. The mass of the Z boson is taken 
ܯ௓ = 91.1876 GeV and its decay width is Γ௓ = 2.35ܩܸ݁. The mass of the Higgs boson is 
ܯு = 126 GeV. In the on-shell renormalization scheme, the mass of W boson is treated as 
an input parameter. Because of the limited accuracy of the measured value for ܯௐ, we 
hence take the value that is derived from the electroweak radiative corrections to the muon 
TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM Tập 15, Số 3 (2018): 24-35 
28 
decay width (∆r) [12] with ܩఓ = 1.16639 × 	10 − 5	ܩܸ݁ିଶ. As a result, ܯௐ is a function 
of ܯு. The resulting ܯௐ = 80.370 GeV is corresponding to ∆r = 2.49%. Finally, for the 
lepton masses we take ݉௘ = 0.51099891 MeV, ݉ఓ = 105.658367 MeV and ݉ఛ= 1776.82 
MeV. The quark masses are ݉௨ = 63 MeV, ݉ௗ = 63 MeV, ݉௖ = 1.5 GeV, ݉௦ = 94 MeV, 
݉௧ = 173.5 GeV, and ݉௕ = 4.7 GeV. 
The full ࣩ(α) electroweak cross section considers the tree graphs and the full 
one-loop virtual corrections as well as the soft and hard bremsstrahlung contributions. 
In general, the total cross section in full one-loop electroweak radiative corrections is 
given by 
σࣩ(஑)௓ு = න݀ σ்௓ு + න݀σ௏௓ு(ܥ௎௏ , {α෥,β෨ ,δ෨ , ε෤, ̃ߢ}, λ)	 
	+න݀ σ்௓ுߜ௦௢௙௧൫ߣ ≤ ܧఊ௦ < ݇௖൯ + න݀ σு௓ு൫ܧஓ௦ ≥ ݇௖൯. (2) 
In this formula, σ்௓ு is the tree-level cross section, σ௏௓ு is the cross section due to the 
interference between the one-loop and the tree diagrams. The contribution must be 
independent of the UV-cutoff parameter (ܥ௎௏) and the nonlinear gauge parameters 
(α෥,β෨,δ෨ , ε෤, ̃ߢ). Because of the way we regularize the IR divergences, σ௏௓ு depends on the 
photon mass λ. This λ dependence must cancel against the soft-photon contribution, which 
is the third term in Eq. (2). The soft-photon part can be factorized into a soft factor, which 
is calculated explicitly in Ref [12], and the cross section from the tree diagrams. 
In Tables 1, 2 and 3 in this section, we present the numerical results for the checks of 
UV finiteness, gauge invariance, and the IR finiteness at one random point in phase space, 
evaluated with double precision. The results are stable over a range of 14 digits. 
Finally, we consider the contribution of the hard photon bremsstrahlung, σு
௓ுஓೄ(௞೎). 
This part is the process eାeି → ZHγୗ with an added hard bremsstrahlung photon. The 
process is generated by the tree-level version of the GRACE [12]. By taking this part into 
the total cross section, the final results must be independent of the soft-photon cutoff 
energy ݇௖ . Table 4 shows the numerical result of the check of ݇௖ - stability. Changing ݇௖ 
from 0.0001 GeV to 0.1 GeV, the results are consistent to an accuracy better than 0.04% 
(this accuracy is better than that in each Monte Carlo integration). 
Table 1. Test of ܥ௎௏ independence of the amplitude. In this table, we take the nonlinear gauge 
parameters to be (0,0,0,0,0), ߣ = 10ିଵ଻ GeV and we use 1 TeV for the center-of-mass energy 
ܥ௎௏ 2ℛℯ(ℳ்∗ℳ௅) 0 −8.6563074319085317. 10ିଶ 10ଶ −8.6563074319085359 · 10ିଶ 10ଷ −8.6563074319085234 · 10ିଶ 
TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM Phan Hong Khiem et al. 
29 
Table 2. Test of the IR finiteness of the amplitude. In this table we take the nonlinear gauge 
parameters to be (0,0,0,0,0), ܥ௎௏= 0 and the center-of-mass energy is 1 TeV. 
λ[ܩܸ݁] 2ℛℯ(ℳ்∗ℳ௅)+ soft contribution 10ିଵହ −4.3320229357755305 ⋅ 10ିଷ 10ିଵ଻ −4.3320229357753596 ⋅ 10ିଷ 10ିଶ଴ −4.3320229357753995 ⋅ 10ିଷ 
Table 3. Gauge invariance of the amplitude. In this table, we set ܥ௎௏= 0, 
the fictitious photon mass is 10ଵ଻ GeV and a 1 TeV center-of-mass energy 
൫ࢻ෥,ࢼ෩, ࢾ෩, ࢿ෤,ࣄ෥൯ ૛जऩ(गࢀ∗गࡸ) + soft contribution (0,0,0,0,0) −8.6563074319085317 ⋅ 10ିଶ (1, 2, 3, 4, 5) −8.6563074319085234 ⋅ 10ିଶ (10, 20, 30, 40, 50) −8.6563074319075561 ⋅ 10ିଶ 
Table 4. Test of the ݇௖-stability of the result. We choose the photon mass to be 10ିଵ଻ GeV 
and the center-of-mass energy is 1 TeV. The second column presents the hard photon 
cross-section and the third column presents the soft photon cross-section. The final column 
is the sum of both 
݇௖ [GeV] ߪௌ × 10ିଶ [pb] ߪு × 10ିଶ [pb] ߪௌାு × 10ିଶ [pb] 10ିହ 3.291191± 0.002435 2.933921± 0.002614 6.225112 10ିସ 3.647297± 0.002698 2.579148± 0.002259 6.226445 10ିଷ 4.003403± 0.002961 2.220851± 0.001956 6.224254 10ିଶ 4.359510± 0.003225 1.864859± 0.001564 6.224369 10ିଵ 4.715616± 0.003488 1.507799± 0.001270 6.223415 
Having verified the stability of the results, we proceed to generate the physical 
results of the process. Hereafter, we use λ = 10ିଵ଻GeV, ܥ௎௏ = 0, ݇௖ = 10ିଶ GeV, and 
(ߙ෤ ,ߚ෨,ߜሚ, ߝ̃, ̃ߢ) = (0,0,0,0,0). We defined the percentage of full electroweak radiative 
corrections as follows: 
δாௐ[%] = σࣩ(஑)௓ு − σ்௓ுσ்௓ு × 100%. (3) 
TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM Tập 15, Số 3 (2018): 24-35 
30 
The ܭாௐ factor is also shown in the physical results. It is defined as 
ܭாௐ = σࣩ(஑)௓ுσ்௓ு − 1. (4) 
In Fig. 2 (left Figure), we present the total cross section and full electroweak 
corrections as a function of center-of-mass energy. The energy varies from 220 GeV to 
1000 GeV. The cross section has a peak around √ݏ ≈ 250ܩܸ݁(≈ ܯு + ܯ௓). It then 
decreases when √ݏ > 250 GeV. On the right corner of this Figure, the percentage of full 
radiative corrections to the total cross section is shown as a function of √ݏ. We observe 
that the corrections are from ≈ −40% to ≈ 20% which are corresponding to 220 GeV ≤ √ݏ 
≤ 1000 GeV. In the low energy region, QED corrections are dominant. While the weak 
corrections are the large contribution at higher-energy region. It is well-known that the 
weak corrections in the high-energy region are attributed to the enhancement contribution 
of the single Sudakov logarithm. Its contribution can be estimated as follows: 
ߜௐ ≈
ߙ(ܯ௓ଶ)
ߨ ݏ݅݊ଶ ߠௐ
݈݊ ቆ
ݏ
ܯ௓
ଶቇ ≈ ࣩ(10%)	at	√ݏ = 1000ܩܸ݁. (5) 
It is clear that the corrections make a sizable contribution to the total cross section 
and cannot be ignored for the high-precision program at the ILC. 
In Fig. 2 (right Figure), the angular distribution of Z boson is generated at √ݏ = 250 
GeV. In this Figure, the ܭாௐ given in Eq. (4) indicates the electroweak corrections to the 
differential cross section. One finds that the corrections are about ≈	−8%. Again, this 
contribution should be taken into account at the high precision program of the ILC. 
Figure 2. The total cross-section and its distribution 
TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM Phan Hong Khiem et al. 
31 
2.3. ݁ା݁ି → ܼܪ with polarized beams 
The phenomenological results for the reaction eାeି → ZH including the beam 
polarizations are shown in this subsection. In GRACE program, the polarizations of 
electron and positron are implemented by introducing projection operator [18] as follows 
෍ ݑ௘ష(݌)ݑത௘ష(௣)
௦ୀଵ,ଶ
1	 + 	 ߣ௘షߛହ2 ൫p/	 + 	݉൯, (6)	 
෍ ݑ௘శ(݌)ݑത௘శ(௣)
௦ୀଵ,ଶ
1	 − 	ߣ௘శߛହ2 ൫p/	 − 	݉൯. (7) 
Where ߣ௘ష = ±1(ߣ௘శ = ±1)	are LR for electron (and positron). In this article, we 
are interested in computing full one-loop electroweak radiative corrections to ZH 
production with two options of the initial beam polarizations which are ( ௘ܲష ,ܲ௘శ)	= 
(−80%, +30%) and ( ௘ܲష,ܲ௘శ)	= (+80%, −30%). In general, we use GRACE to generate 
the following processes 
݁௅
ି݁ோ
ା(݁ோି݁௅ା) → ܼܪ, (8) 
݁௅
ି݁௅
ା(݁ோି݁ோା) → ܼܪ. (9) 
We know that two latter processes give small cross sections in comparison with the 
former reactions. Having the cross sections ߪ௅ோ, ߪோ௅, ߪ௅௅ and ߪோோ 	for this reaction, we then 
evaluate the cross section at general polarization ( ௘ܲష ,ܲ௘శ)	 for electron and positron. It is 
given by 
ߪ( ௘ܲష,ܲ௘శ) = (1 + ௘ܲష)(1 + ܲ௘శ)ߪோோ + (1 − ௘ܲష)(1 − ܲ௘శ)ߪ௅௅ 
	+(1 − ௘ܲష)(1 + ܲ௘శ)ߪ௅ோ + (1 + ௘ܲష)(1− ܲ௘శ)ߪோ௅ . (10) 
Of course, after generating all related process by GRACE, we are going to perform 
the numerical checks as the previous case. The independence of the squared amplitude on 
the ܥ௎௏, gauge parameters and λ have been tested. One also confirms that the results are 
stable over a range of 14 digits. The ݇௖	stability also verified at the cross section level. We 
obtain that the results are consistent to an accuracy better than 0.02% (this accuracy is 
better than that in each Monte Carlo integration). After passing the numerical checks, we 
set the ܥ௎௏, gauge parameters, λ and ݇௖ back to the default values which have been shown 
in previous subsection. 
We are going to discuss on the physical results for this reaction at the ILC. In Fig. 
(3), the cross sections at the polarizations for electron, positron which are (−80%, +30%) 
(left panel) and (+80%, −30%) (right panel) are shown as a function of center-of-mass 
energy. Again, we vary the energy from 220 GeV to 1000 GeV. We observe the peaks 
where the cross sections are maximum, around 250 GeV for both cases. The cross sections 
in the polarization of (−80%, +30%) case are larger than those of (+80%, −30%). In these 
Figures, the ܭாௐ are presented for the full electroweak corrections to both cases. For the 
case of (−80%, +30%), one obtains the corrections which vary from ≈ −45% to ≈ 10%. 
TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM Tập 15, Số 3 (2018): 24-35 
32 
For another case of (+80%, −30%), we find that the corrections change from ≈ −20% to ≈ 
40%. The corrections in both cases are significant contributions. They play important role 
at the ILC. The latter case of beam polarizations obtains the larger corrections in high 
energy regions than former case. They come from the weak correction of vector W boson 
which couples only to left handed electron. 
We concern the differential cross sections which are functions of cosine of Z boson’s 
angle at 250 GeV, as shown in Fig. (4). The left (right) panel is shown for ( ௘ܲష,ܲ௘శ)	= 
(−80%, +30%) and (+80%, −30%) respectively. In the first case, the corrections to the 
differential cross sections are about ≈ −15%. For the second case, one obtains that the 
corrections are ≈ 7.5%. In both cases, the corrections are important for future analysis at 
the ILC. 
Figure 3. The total cross-section and full electroweak corrections. 
(−80%, +30%) (+80%, −30%) 
(−80%, +30%) (+80%, −30%) 
Figure 4. The angular distributions. 
TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM Phan Hong Khiem et al. 
33 
3. Search for additional higgs at the ILC 
In this section, we are going to discuss on a method to search indirectly the 
additional Higgs in arbitrary physics beyond the Standard Model. In this model, besides 
the SM-like Higgs boson (ℎ), we assume that there is an additional Higgs boson (H). We 
note that the coupling of Standard Model Higgs boson and the additional Higgs boson to 
ZZ are λ௛௓௓ , ߣு௓௓ respectively. Following the sum rules for Higgs boson [11], these 
couplings satisfy the below condition 
ߣ௛௓௓
ଶ
൫ߣ௛௓௓
SM ൯ଶ + ߣு௓௓ଶ൫ߣ௛௓௓SM ൯ଶ = 1, (11) 
With ߣ௛௓௓SM = ௚ெೋ஼ೢ , ܥௐ = ெೈெೋ . We know that in the SM ߣு௓௓ = 0. In this analysis, we 
vary 0.5 ≤ ߣ௛௓௓ ≤ 0.95. It means that the coupling of the additional Higgs boson to ZZ is in 
the range of 0.31 ≤ ߣு௓௓ ≤ 0.87. In Fig. (5), the total cross-sections generated with varying 
0.5 ≤ ߣ௛௓௓ ≤ 0.95 are shown as a function of center-of-mass energy. √ݏ changes from 220 
GeV to 1 TeV. The left and right figures present the cross sections for polarizations of 
initial beam which are ( ௘ܲష,ܲ௘శ)	 = (−80%, +30%) and (+80%, −30%) respectively. In 
these figures, the red line is for Standard Model case. The blue region is the cross section 
which the coupling ߣ௛௓௓ are from 0.5 to 0.9. While the green area shows the cross section 
corresponding to ߣ௛௓௓ ∈ [0.9,0.95]. From these analyses, we find that the BSMs effects 
through this reaction could be tested clearly at the ILC. 
(−80%, +30%) (+80%, −30%) 
Figure 5. The total cross-section and full electroweak corrections 
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34 
4. Conclusions 
In this article, full ࣩ(ߙ)	electroweak radiative corrections to the process eାeି → ZH 
at the ILC have been computed successfully. The non-polarized and polarized cases for 
initial beams have been computed. The radiative corrections are order of 10% 
contributions to the total cross section as well as its distributions. The corrections are 
significant contributions and they must be taken into account at the ILC. 
We have been also discussed on the model-independent way to probe the additional 
Higgs in physics of beyond the SM through ZH production at the ILC. With high 
luminosity program at the ILC, thanks to the full one-loop radiative corrections to this 
production, we could probe the BSM’s effects and may discriminate many of BSMs at the 
ILC. In future work, we will apply this method for specific BSMs. 
 Conflict of Interest: Authors have no conflict of interest to declare. 
 Acknowledgment: This research is funded by Vietnam National Foundation for Science 
and Technology Development (NAFOSTED) under grant number 103.01-2016.33. 
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