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  3. One-loop contributions to in standard model \(H \to Z*\gamma \to {v_e}{v_e} \gamma \)

One-loop contributions to in standard model \(H \to Z*\gamma \to {v_e}{v_e} \gamma \)

One-loop contributions to the decay process \(H \to Z*\gamma \to {v_e}{v_e} \gamma \) in standard model are performed in the paper One-loop contributions to in standard model \(H \to Z*\gamma \to {v_e}{v_e} \gamma \). The detailed computations are carried out in unitary gauge. In physical results, we present numerical results for partial decay width and its distribution. We find that the partial decay width is given to 0.466 KeV. This result is in the upper bound of the current experimental data

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  1. 70 Khiem Hong Phan, Dung Tri Tran / Tạp chí Khoa học và Công nghệ Đại học Duy Tân 5(48) (2021) 70-78 5(48) (2021) 70-78 One-loop contributions to in standard model Đóng góp tích phân Feynman một vòng cho quá trình phân rã hạt vô hướng Higgs trong mô hình chuẩn Khiem Hong Phana,b*, Dung Tri Tranc Phan Hồng Khiêma,b,*, Trần Trí Dũngc a Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City 700000, Vietnam a Viện Nghiên cứu Khoa học Cơ bản và Ứng dụng, Trường Đại học Duy Tân, Tp. HCM, Việt Nam b Faculty of Natural Sciences, Duy Tan University, Da Nang City 550000, Vietnam b Khoa Khoa học Tự nhiên, Trường Đại học Duy Tân, Đà Nẵng , Việt Nam c University of Science Ho Chi Minh City, 227 Nguyen Van Cu, District 5, HCM City, Vietnam c Đại học Khoa học Tự nhiên Tp. HCM, 227 Nguyễn Văn Cừ, Quận 5, Tp. HCM (Ngày nhận bài: 11/6/2021, ngày phản biện xong: 20/9/2021, ngày chấp nhận đăng: 14/10/2021) Abstract One-loop contributions to the decay process in standard model are performed in this paper. The detailed computations are carried out in unitary gauge. In physical results, we present numerical results for partial decay width and its distribution. We find that the partial decay width is given to KeV. This result is in the upper bound of the current experimental data at the Large Hadron Collider. Keywords: Higgs phenomenology; One-loop corrections; analytic methods for Quantum Field Theory; Dimensional regularization. Tóm tắt Trong bài báo này, chúng tôi tính các đóng góp tích phân Feynman một vòng cho quá trình phân rã hạt vô hướng Higgs trong mô hình Chuẩn. Tính toán chi tiết được xét trong chuẩn unitary. Trong phần kết quả vật lý, chúng tôi trình bày kết quả của bề rộng phân rã và các phân bố của bề rộng phân rã cho quá trình trên. Kết quả bề rộng phân rã nhận được từ tính toán này là $0.466$ KeV và phù hợp với dữ liệu thực nghiệm hiện tại ở máy gia tốc LHC. Từ Khóa: Hiện tượng luận hạt vô hướng Higgs; Bổ chính lượng tử; Phương pháp giải tích cho Lý Thuyết trường lượng tử; Phương pháp chỉnh thứ nguyên. 1. Introduction LHC) as well as future colliders [1, 2, 3] is to After the discovery of the Standard model measure the SM-like Higgs properties. It means like (SM-like) Higgs boson at the Large Hadron that all couplings of the Higgs to gauge bosons Collider (LHC), one of the main targets at the and matter particles are probed as precise as High-Luminosity Large Hadron Collider (HL- possible. From the experimental data, we can * Corresponding Author: Khiem Hong Phan; Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City 700000, Vietnam; Faculty of Natural Sciences, Duy Tan University, Da Nang City 550000, Vietnam Email: phanhongkhiem@duytan.edu.vn
  2. Khiem Hong Phan, Dung Tri Tran / Tạp chí Khoa học và Công nghệ Đại học Duy Tân 5(48) (2021) 70-78 71 verify the Standard model at higher energy decay width and its distribution in detail. The region as well as extract the contributions of partial decay width is to KeV which is in new physics. In all the Higgs decay modes, the the upper bound of the present experimental channel of Higgs boson decay to photon and data at the LHC. Detailed analytical missing energy is great of interest the colliders calculations and physical results for [4, 5, 6, 7, 8, 9] by following reasons: (i) since with are discussed in our forthcoming many new particles which are absent in SM paper. may exchange in the loop diagrams of the Our paper is organized as follows: In section decay process; (ii) new neutral particles rather 2, after describing briefly one-loop tensor than three neutrinos may exist in new physics. reduction method, detailed calculations for one- Subsequently, the decay could provide an loop contributions to are important information for testing Higgs sector explained more. Analytic formulas for one-loop as well as probing to dark matter and form factors and phenomenological results are constraining new physic parameters. presented in this section. Conclusions and In order to analyse new physics, we must outlook are devoted in section . understand fully the Standard model 2. Calculations background. Therefore, precise calculations for are necessary. The relevant In general, one-loop amplitude for the decay Feynman diagrams for this decay channel start is expressed in terms of one- not at tree level but at one-loop level in the loop Feynman tensor integrals which are electroweak interaction. As the above reasons, reduced frequently into scalar functions. In this we carry out one-loop contributions to the computation, we apply the tensor reduction decay . The calculations are method in [10]. The approach will be explained performed in unitary gauge by using briefly in the following paragraphs. Firstly, the dimensional regularization. In definitions for one-loop one-, two-, three-point phenomenological results,we present the partial tensor integrals with rank are given by dd k k µ1 k µ2 · · · k µP Z µ1 µ2 ···µP 2 2−d/2 {A; B; C} = (µ ) . (1) (2π)d {D1 ; D1 D2 ; D1 D2 D3 } Where the inverse Feynman propagators are given as follows: Dj = (k + qj )2 − m2j + iρ, (2) for . In this equation, The explicit reduction formulas for one-loop with are the external momenta and are one-, two- and three-point tensor integrals up to internal masses in the loops. Note that rank involving this process are then shown thanks to momentum as follows [10]: conservation. Space-time dimension is − and parameter is a renormalization scale.
  3. 72 Khiem Hong Phan, Dung Tri Tran / Tạp chí Khoa học và Công nghệ Đại học Duy Tân 5(48) (2021) 70-78 (3) In the above relations, we have utilized the related to the decay in unitary gauge and the short notation [10] relevant Feynman rules for the three- and four- point vertices which we use for this work are . All summarized and devoted in Appendix. The scalar coefficients are so- Ward identity is implied for external photon's called Passarino-Veltman functions (PV) in on-shell condition as where , are [10]. Analytic formulas of the PV functions are momentum and polarization vector of photon well-known and they have been implemented respectively. Besides that Dirac equation's into package LoopTools [12] for numerical conditions for external electron neutrinos computations. and are applied as to simplify more the 2.1 Analytic results calculations. Kinematic invariant variables for The detailed evaluations for the decay this decay process are given in unitary gauge in which only the physical particles appear and with being the Higgs boson ghosts and Goldstone bosons being absent are mass. presented in this subsection. The decay channel Assuming that electron neutrinos are consists of boson and fermion internal massless, so that , we then particles exchanging in one-loop triangle introduce the following Mandelstam variables diagrams (seen Fig. 1 and Fig. 2 respectively). as: The fermion and gauge boson propagators (4) Figure 1: -boson particles exchanging in one-loop triangle diagrams of in unitary gauge.
  4. Khiem Hong Phan, Dung Tri Tran / Tạp chí Khoa học và Công nghệ Đại học Duy Tân 5(48) (2021) 70-78 73 Figure 2: Fermion particles exchanging in one-loop triangle diagrams of in unitary gauge. General one-loop amplitude for the decay with boson internal lines (shown in Fig. 1). channel can be written as follows: The term is to the contributions of triangle Feynman diagrams with fermions exchanging (5) in the loop (seen in Fig. 2). Each component of Eq. (5) is expressed in terms of Lorentz In this formula, presents for the invariant structure as follows: contributions of triangle Feynman diagrams (6) Where we have used left-handed projection of ultraviolet cut-off ( -term) and the mass operator The calculations scale parameter of dimensional regularization are consistently straightforward and we make when taking , we confirm that these full usage of [13, 14] for handling terms are cancelled out after summing all all Dirac traces and contracts in general diagrams. As a result, the total amplitude dimensions. The form factors contributions become consistently finite at are expressed in terms of and are independent of . Taking an example, scalar coefficients - more specifically, in terms one-loop triangle amplitude's contributions of Passarino-Veltman functions at arbitrary related to this decay channel, in particularly, are obtained by summing all diagram consist of the ultraviolet divergences that amplitudes of each type as follows. In the limit appear in the scalar two-points integrals and all scalar Passarino-Veltman their own expressions in terms of in coefficients of the mentioned dimensional regularization. When we take into form factors are expressed in terms of the considerations about difference between two Passarino-Veltman function's basis such as one- 's, the analytical result becomes finite in terms point integrals , two-point integrals and of logarithm functions at . Specifically, we three-point integrals in [10]. In principle, obtain for and as each diagram's contribution can be dependent follows B0 (MH2 , M 2 , M 2 ) − B0 (q12 , M 2 , M 2 ) = p "p # MH4 − 4M 4 MH2 MH4 − 4M 4 MH2 + 2M 4 − MH2 = ln MH2 2M 4 p " p # 2 q12 − 4M 4 q12 2 q12 − 4M 4 q12 + 2M 4 − q12 (7) − ln , q12 2M 4
  5. 74 Khiem Hong Phan, Dung Tri Tran / Tạp chí Khoa học và Công nghệ Đại học Duy Tân 5(48) (2021) 70-78 B0 (P 2 , M 2 , M 2 ) − B0 (0, M 2 , M 2 ) = √ "√ # P 4 − 4M 4 P 2 P 4 − 4M 4 P 2 + 2M 4 − P 2 = 2+ ln . (8) P2 2M 4 Moreover, the three-point integral related to this decay channel is also given by finite result in terms of logarithm functions at as follows: 1 C0 (0, q12 , MH2 , M 2 , M 2 , M 2 ) = 2 × 2(MH − q12 ) ( "p # "p #) M 4 − 4M 4 M 2 + 2M 4 − M 2 q 2 − 4M 4 q + 2M 4 − q H 12 12 12 × ln2 H H − ln2 . 2M 4 2M 4 (9) By performing explicitly -expansion for the space-time dimension in terms of above form factors, we confirm exactly Ward logarithm functions of related masses and identity relation for the -term's Mandelstam variables for above form factors in form factors. Therefore, the analytical results at Eq. (6) are expressed as (W ) MH2 − q12 (W ) F1 = F2 2 α2 = − × 2MH2 MW 3 3 sW (MH2 − q12 )2 (q12 − MZ2 + iΓZ MZ ) ( h i × MH2 MW2 MH2 (q12 − 6MW 2 ) + 12MW 4 + 6MW 2 2 q12 − 2q12 × " p # −MH2 + MH4 − 4MH2 MW 2 2 + 2MW × ln2 2 2MW n +MH2 2MH4 MW 2 − MH4 q12 + 12MH2 MW 4 − 4MH2 MW 2 q12 q h i + q12 2 − 4MW 2 q12 MH2 (q12 − 2MW 2 ) + 2MW 2 (q12 − 6MW 2 ) × " p # 2 2 2 −q12 + q12 − 4MW q12 + 2MW × ln 2 2MW h i 2 −MW MH2 (q12 − 6MW 2 ) + 12MW 4 + 6MW 2 q12 − 2q12 2 × " p # 2 2 2 2 −q12 + q12 − 4MW q12 + 2MW × ln 2 2MW o 2 2 4 2 2 +MH q12 − 12MW q12 + 2MW q12 q h i +q12 MH4 − 4MH2 MW 2 MH2 (2MW 2 − q12 ) + 12MW 4 − 2MW 2 q12 × " p #) −MH2 + MH4 − 4MH2 MW 2 + 2MW 2 × ln 2 2MW (10)
  6. Khiem Hong Phan, Dung Tri Tran / Tạp chí Khoa học và Công nghệ Đại học Duy Tân 5(48) (2021) 70-78 75 and (f ) MH2 − q12 (f ) F1 = F2 2 α2 m2f NCf Qf = (T 3 − 2Qf s2W ) × 2MH2 MW c2W s3W (MH2 − q12 )2 (q12 − MZ2 + iΓZ MZ ) f q −q12 + q12 2 − 4q12 m2f + 2m2f ( " # h × MH2 (4m2f − MH2 + q12 ) ln2 2m2f q −q12 + q12 2 − 4q12 m2f + 2m2f " # q i 2 2 +4 q12 − 4q12 m2f ln − 4M H + 4q12 2m2f q 2 −MH + MH4 − 4m2f MH2 + 2m2f " # +MH2 (MH2 − 4m2f − q12 ) ln2 2m2f q −MH2 + MH4 − 4m2f MH2 + 2m2f " #) q −4q12 MH4 − 4MH2 m2f ln . (11) 2m2f Here is the fine-structure constant, for quarks), the charge (in units of e), and the third component of weak isospin of the and with fermion respectively. is the electroweak mixing angle. In addition The decay width is then calculated as and are the mass, the color follow: multiplicity factor ( for leptons and
  8. 2 dΓ MH2 Eγ3 (MH − 2Eγ )
  10. (W ) X
  11. (f )
  12. =
  13. F1 + F1
  14. (1 + cos2 θ). (12) dEγ d cos θ 128π 3
  15. f
  16. Where is the photon energy and is GeV, GeV and angle of photon and electron neutrino in center- GeV, we gain numerical result for of-mass of neutrino pair. Note that due to the the decay width. Integrating over and − , we get the couplings of the Higgs boson to fermions are proportional to the fermion masses, so we only decay width KeV. This result is in take into considerations about the top quark , the bound of experimental data at the LHC. bottom quark , strange quark charm quark In this Fig. 3, the distribution of decay width and tau lepton masses for the charged fermion as a function of and is shown. We loop diagrams. observe the peak which are corresponding to the -pole that boson decay to neutrinos. The 2.2 Numerical results position of this peak for − is located In this paper, we use the following input at parameters: , MH2 − MZ2 GeV, GeV, Eγ = = 29.3159 GeV. (13) 2MH GeV, GeV, This provides important information for GeV, GeV, testing SM at higher energy regions at LHC.
  17. 76 Khiem Hong Phan, Dung Tri Tran / Tạp chí Khoa học và Công nghệ Đại học Duy Tân 5(48) (2021) 70-78 3. Conclusions Analytic results for one-loop contributions to have presented in this paper. In physical results, we compute the partial decay width and differential decay width with respect to photon's energy. The partial decay width is KeV. This result agrees with upper bound of experimental data at LHC. Acknowledgment This research is funded by Vietnam National Foundation for Science and Technology Figure 3: Differential decay width as a function of Development (NAFOSTED) under the grant . number − Appendix: Feynman rules and Feynman diagrams The fermion , virtual gauge boson propagators and the Feynman rules for the three-point and four-point vertices in unitary gauge are, respectively, given. In the Tables, we use − . The short notation for the standard Lorentz tensors of the gauge boson self couplings − − − with momenta are all inward and − − are introduced for convenience. The notation − from the propagator of the virtual gauge boson is used. Table 1: Feynman rules involving the decay through fermions and bosons loop in the unitary gauge.
  18. Khiem Hong Phan, Dung Tri Tran / Tạp chí Khoa học và Công nghệ Đại học Duy Tân 5(48) (2021) 70-78 77 Table 2: Couplings involving the decay through fermions and bosons loop in unitary gauge. References [6] M. Aaboud et al., Search for invisible Higgs boson [1] A. Liss et al., Physics at a High-Luminosity LHC with decays in vector boson fusion at TeV ATLAS, arXiv:1307.7292. with the ATLAS detector, Phys. Lett. B 793 (2019), 499-519. [2] CMS Collaboration, Projected Performance of an Upgraded CMS Detector at the LHC and HL-LHC: [7] V. S. Ngairangbam, A. Bhardwaj, P. Konar and A. K. Contribution to the Snowmass Process, Nayak, Invisible Higgs search through Vector arXiv:1307.7135. Boson Fusion: A deep learning approach, Eur. Phys. J. C 80 (2020) no.11, 1055. [3] H. Baer, T. Barklow, K. Fujii, Y. Gao, A. Hoang, S. Kanemura, J. List, H. E. Logan, A. Nomerotski and [8] G. Aad et al., Performance of Missing Transverse M. Perelstein, et al. The International Linear Momentum Reconstruction in Proton Proton Collider Technical Design Report - Volume 2: Collisions at 7 TeV with ATLAS, Eur. Phys. J. C 72 Physics, arXiv:1306.6352. (2012), 1844. [4] A. M. Sirunyan et al., Search for invisible decays of a [9] G. Aad et al., Search for invisible decays of the Higgs Higgs boson produced through vector boson fusion boson produced in association with a hadronically in proton-proton collisions at TeV, Phys. decaying vector boson in pp collisions at Lett. B 793 (2019), 520-551. TeV with the ATLAS detector, Eur. Phys. J. C 75 (2015) no.7, 337. [5] M. Aaboud et al., Combination of searches for invisible Higgs boson decays with the ATLAS [10] A. Denner and S. Dittmaier, Reduction schemes for experiment, Phys. Rev. Lett. 122 (2019) no.23, one-loop tensor integrals, Nucl. Phys. B 734 (2006), 231801. 62-115.
  19. 78 Khiem Hong Phan, Dung Tri Tran / Tạp chí Khoa học và Công nghệ Đại học Duy Tân 5(48) (2021) 70-78 [11] A. Denner, Techniques for calculation of [13] H. H. Patel, Package-X: A Mathematica package for electroweak radiative corrections at the one loop the analytic calculation of one-loop integrals, level and results for W physics at LEP-200, Fortsch. Comput. Phys. Commun. 197 (2015), 276-290. Phys. 41 (1993), 307-420. [14] H. H. Patel, Package-X 2.0: A Mathematica package [12] T. Hahn and M. Perez-Victoria, Automatized One- for the analytic calculation of one-loop integrals, Loop Calculations in 4 and D dimensions, Comput. Comput. Phys. Commun. 218 (2017), 66-70. Phys. Commun. 118 (1999), 153-165.
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