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The scalar sector of the GM model [28, 29] consists of the usual complex doublet (
$ \phi^+ $ ,$ \phi^0 $ ), a real triplet ($ \xi^+ $ ,$ \xi^0 $ ,$ \xi^- $ ), and a complex triplet ($ \chi^+ $ ,$ \chi^+ $ ,$ \chi^0 $ ). To make the global SU(2)$ _{\rm L} \times $ SU(2)$ _{\rm R} $ symmetry explicit, the doublet is expressed in the form of a bi-doublet Φ, whereas the triplets are combined to form a bi-triplet X.$ \Phi = \left( {\begin{array}{*{20}{c}} {{\phi ^{0*}}}&{{\phi ^ + }}\\ { - {\phi ^{ + *}}}&{{\phi ^0}} \end{array}} \right),\quad \;\;\;{\kern 1pt} X = \left( {\begin{array}{*{20}{c}} {{\chi ^{0*}}}&{{\xi ^ + }}&{{\chi ^{ + + }}}\\ { - {\chi ^{ + *}}}&{{\xi ^0}}&{{\chi ^ + }}\\ {{\chi ^{ + + *}}}&{ - {\xi ^{ + *}}}&{{\chi ^0}} \end{array}} \right).$
(1) The vevs (vacuum expectation values) are defined by
$ \langle \Phi \rangle = ({ v_{\phi}}/{\sqrt{2}}) I_{2\times2} $ and$ \langle X \rangle = v_{\chi} I_{3 \times 3} $ , where$ I_{2\times2} $ and$ I_{3\times3} $ are the unit matrices. The W and Z boson masses constrain$ v_{\phi}^2 + 8 v_{\chi}^2 \equiv v^2 = \frac{1}{\sqrt{2} G_{\rm F}} \approx (246\; {\rm GeV})^2 $
(2) with
$ G_{\rm F} $ as the Fermi constant.The most general gauge-invariant scalar potential involving these fields that conserves the custodial SU(2) is given by
$ \begin{aligned}[b] V(\Phi,X) =& \frac{\mu_2^2}{2} {\rm{Tr}}(\Phi^\dagger \Phi) + \frac{\mu_3^2}{2} {\rm{Tr}}(X^\dagger X) \\ & + \lambda_1 [{\rm{Tr}}(\Phi^\dagger \Phi)]^2 + \lambda_2 {\rm{Tr}}(\Phi^\dagger \Phi) {\rm{Tr}}(X^\dagger X) \\& + \lambda_3 {\rm{Tr}}(X^\dagger X X^\dagger X) + \lambda_4 [{\rm{Tr}}(X^\dagger X)]^2\\& - \lambda_5 {\rm{Tr}}( \Phi^\dagger \tau^a \Phi \tau^b) {\rm{Tr}}( X^\dagger T^a_1 X T^b_1) \end{aligned} $
$ \begin{aligned}[b]\quad\quad & - M_1 {\rm{Tr}}(\Phi^\dagger \tau^a \Phi \tau^b)(U X U^\dagger)_{ab} \\& - M_2 {\rm{Tr}}(X^\dagger T^a X T^b)(U X U^\dagger)_{ab}. \end{aligned} $
(3) Here, the SU(2) generators for the doublet representation are
$ \tau^a = \sigma^a/2 $ , where$ \sigma^a $ is the Pauli matrices, and the generators for the triplet representation$ T^a_1 $ are$\begin{aligned}[b]& {T^1} = \left( {\begin{array}{*{20}{c}} 0&{\dfrac{1}{{\sqrt 2 }}}&0\\ {\dfrac{1}{{\sqrt 2 }}}&0&{\dfrac{1}{{\sqrt 2 }}}\\ 0&{\dfrac{1}{{\sqrt 2 }}}&0 \end{array}} \right),\\& {T^2} = \left( {\begin{array}{*{20}{c}} 0&{ - \dfrac{\rm i}{{\sqrt 2 }}}&0\\ {\dfrac{\rm i}{{\sqrt 2 }}}&0&{ - \dfrac{\rm i}{{\sqrt 2 }}}\\ 0&{\dfrac{\rm i}{{\sqrt 2 }}}&0 \end{array}} \right),\\& {T^3} = \left( {\begin{array}{*{20}{c}} 1&0&0\\ 0&0&0\\ 0&0&{ - 1} \end{array}} \right). \end{aligned} $
(4) The matrix U is given by [43]
$ U = \left( {\begin{array}{*{20}{c}} {- \dfrac{1}{\sqrt{2}}} & {0} & {\dfrac{1}{\sqrt{2}} }\\ {- \dfrac{\rm i}{\sqrt{2}}} & {0 } & { - \dfrac{\rm i}{\sqrt{2}} }\\ {0 } & { 1 } & { 0} \end{array}} \right). $
(5) The physical fields can be organized by their transformation properties under custodial SU(2) symmetry into two singlets, a triplet, and a fiveplet. The triplet and fiveplet states are given by
$ \begin{aligned}[b] H_3^+ = &- s_H \phi^+ + c_H \frac{\left(\chi^++\xi^+\right)}{\sqrt{2}}, \\ H_3^0 =& - s_H \phi^{0,i} + c_H \chi^{0,i}, \\ H_5^{++} =& \chi^{++}, \\ H_5^+ =& \frac{\left(\chi^+ - \xi^+\right)}{\sqrt{2}}, \\ H_5^0 =& -\sqrt{\frac{2}{3}} \xi^0 + \sqrt{\frac{1}{3}} \chi^{0,r}, \end{aligned} $
(6) where the vevs are parameterized by
$ c_H \equiv \cos\theta_H = \frac{v_{\phi}}{v}, \qquad s_H \equiv \sin\theta_H = \frac{2\sqrt{2}\,v_\chi}{v}, $
(7) and the neutral fields can be decomposed into real and imaginary parts according to
$ \begin{aligned}[b]& \phi^0 \to \frac{v_{\phi}}{\sqrt{2}} + \frac{\phi^{0,r} + {\rm i} \phi^{0,i}}{\sqrt{2}}, \qquad \chi^0 \to v_{\chi} + \frac{\chi^{0,r} + {\rm i} \chi^{0,i}}{\sqrt{2}}, \\& \xi^0 \to v_{\chi} + \xi^0. \end{aligned} $
(8) The masses within each custodial multiplet are degenerate at tree level, and after eliminating
$ \mu_2^2 $ and$ \mu_3^2 $ in favor of the vevs, the masses can be written as①$ \begin{aligned}[b]m_5^2 =& \frac{M_1}{4 v_{\chi}} v_\phi^2 + 12 M_2 v_{\chi} + \frac{3}{2} \lambda_5 v_{\phi}^2 + 8 \lambda_3 v_{\chi}^2, \\ m_3^2 =& \frac{M_1}{4 v_{\chi}} (v_\phi^2 + 8 v_{\chi}^2) + \frac{\lambda_5}{2} (v_{\phi}^2 + 8 v_{\chi}^2). \end{aligned} $
(9) The two singlet mass eigenstates are given by
$ \begin{aligned}[b]h =& \cos \alpha \, \phi^{0,r} - \sin \alpha \, H_1^{0\prime}, \\ H =& \sin \alpha \, \phi^{0,r} + \cos \alpha \, H_1^{0\prime}, \end{aligned} $
(10) where
$ H_1^{0 \prime} = \sqrt{\frac{1}{3}} \xi^0 + \sqrt{\frac{2}{3}} \chi^{0,r}. $
(11) The mixing angle and masses are given by
$ \begin{aligned}[b]\sin 2 \alpha =& \frac{2 {\cal{M}}^2_{12}}{m_H^2 - m_h^2}, \qquad \cos 2 \alpha = \frac{ {\cal{M}}^2_{22} - {\cal{M}}^2_{11} }{m_H^2 - m_h^2}, \\ m^2_{h,H} =& \frac{1}{2} \left[ {\cal{M}}_{11}^2 + {\cal{M}}_{22}^2 \mp \sqrt{\left( {\cal{M}}_{11}^2 - {\cal{M}}_{22}^2 \right)^2 + 4 \left( {\cal{M}}_{12}^2 \right)^2} \right]. \end{aligned} $
(12) The elements of their mass matrix are given by
$ \begin{aligned}[b]{\cal{M}}_{11}^2 =& 8 \lambda_1 v_{\phi}^2, \\ {\cal{M}}_{12}^2 =& \frac{\sqrt{3}}{2} v_{\phi} \left[ - M_1 + 4 \left(2 \lambda_2 - \lambda_5 \right) v_{\chi} \right], \\ {\cal{M}}_{22}^2 =& \frac{M_1 v_{\phi}^2}{4 v_{\chi}} - 6 M_2 v_{\chi} + 8 \left( \lambda_3 + 3 \lambda_4 \right) v_{\chi}^2. \end{aligned} $
(13) We define H to be heavier than h (
$ m_h < m_H $ ), and either h or H can be the observed 125 GeV Higgs boson at the LHC. -
The program package
$ GMCALC $ (version 1.5.0) [44] with Fortran code is employed in this study to calculate the mass spectrum of the Higgs bosons in the GM model, their decaying branching ratios (BR) and total widths, their relevant mixing angles, and the tree-level couplings of the Higgs bosons to other particles. It also includes a routine to generate the datacard "$ param\_card. dat $ " to be used by$ MadGraph5\_aMC@NLO $ [45] with implementation of the corresponding FeynRules [46] model for the cross sections of the GM Higgs bosons. The FeynRules implementation for the GM model includes the automatic calculation of the next-to-leading order QCD corrections. The Universal FeynRules Output (UFO) file of the FeynRules implementation for the GM model, which can be downloaded from [47], is used by$ MadGraph5\_ aMC@NLO $ (version 2.7.2) in this study. The theoretical constraints in this paper include the conditions for tree-level unitarity, the bounded-from-below requirement on the scalar potential, and the absence of deeper custodial symmetry-breaking minima, as detailed in [44]. Indirect constraints from the S parameter and flavor physics$ b \rightarrow s\gamma $ and$ B_{s}^{0} \rightarrow \mu^{+}\mu^{-} $ are also considered. Constraints from the public tools$ HiggsBounds $ -$ 5 $ [48] (version 5.8.0) and$ HiggsSignals $ -$ 2 $ [49] (version 2.5.1) are applied to further compare the predictions of the custodial-singlet mass eigenstate h or H with the LHC Higgs search results of various channels from Run2 at a center-of-mass energy of 13 TeV. These include the cross section limits, signal rate and mass measurements, and results in the form of simplified template cross section measurements.A new low-
$ m_{5} $ benchmark for the GM model, defined for the neutral custodial fiveplet scalar with a mass$ m_{5} $ $ \in $ (50,550) GeV, was proposed in [37] to study the phenomenological behavior of the$ H_{5} $ states and SM-like Higgs, h. In [37],$ \lambda_{3} $ (= −1.5),$ \lambda_{4} $ (= 1.5 =$ -\lambda_{3} $ ), and$ M_{2} $ (= 20 GeV) were fixed as constants, with the lightest custodial-singlet mass eigenstate h as the LHC observed 125 GeV Higgs boson. In addition, the parameters$ \lambda_{2} $ and$ \lambda_{5} $ were fixed as functions of the mass$ m_{5} $ :$ \lambda_{2} $ = 0.08 ($ m_{5} $ /100 GeV), and$ \lambda_{5} $ = −0.32 ($ m_{5} $ /100 GeV)= −4$ \lambda_{2} $ . Only$ m_{5} $ and$ s_{H} $ were permitted to vary in the studied ranges,$ m_{5} $ $ \in $ (65, 550) GeV and$ s_{H} $ $ \in $ (0, 1). After several iterations of the tests, to generate the points efficiently, we employ the following ranges for the six specific parameters:$ \begin{aligned}[b] &0.0 < M_{2} < 50, \quad 0.0 < s_{H} < 0.66, \quad -1.6 < \lambda_{3} < 0.0, \\ &0.0 < \lambda_{4} < 1.6, \quad 0.0 < \lambda_{2} < 0.08 (m_{5}/50\;{\rm{ GeV}}), \\& -0.32 (m_{5}/50\;{\rm{ GeV}}) < \lambda_{5} < 0.0. \end{aligned} $
(14) We find that wider parameters ranges have practically no impact on our conclusions. Similar to [37], the Fermi constant
$ G_{\rm F} $ is set as$ 1.1663787 \times 10^{-5} $ $ {\rm GeV}^{-2} $ . To study more stringent constraint on the GM model using the results of the search for low-mass Higgs bosons in the mass range from 70 to 110$ {\rm GeV} $ in the diphoton channel at the CMS [40] and to study the discovery potential of other interesting decay channels, the initial scan range of the$ H_{5}^{0} $ mass are specified in the range$ m_{5} $ $ \in $ (65,130) GeV. In addition, any of the custodial-singlet mass eigenstates h or H can be the LHC observed Higgs boson; therefore, we should consider the constraints on$ HiggsBounds $ -$ 5 $ and$ HiggsSignals $ -$ 2 $ , as mentioned in the above paragraph.
Search for a lighter neutral custodial fiveplet scalar in the Georgi-Machacek model
- Received Date: 2022-03-11
- Available Online: 2022-08-15
Abstract: Many researches from both theoretical and experimental perspectives have been performed to search for a new Higgs Boson that is lighter than the 125