-
Typically, there are six types of scalar LQs [35], which carry a conserved quantum number
F\equiv 3B+ L . Here, B and L are the baryon and lepton numbers. As for the VLQs, there are seven typical representations [42]. In Table 1, we list their representations and labels.S U(3)_C\times S U(2)_L\times U(1)_Y representationlabel F S U(3)_C\times S U(2)_L\times U(1)_Y representationlabel (\bar{3},3,1/3) S_3 -2 (3,1,2/3) T_{L,R} (3,2,7/6) R_2 0 (3,1,-1/3) B_{L,R} (3,2,1/6) \widetilde{R}_2 0 (3,2,7/6) (X,T)_{L,R} (\bar{3},1,4/3) \widetilde{S}_1 -2 (3,2,1/6) (T,B)_{L,R} (3,2,-5/6) (B,Y)_{L,R} (\bar{3},1,1/3) S_1 -2 (3,3,2/3) (X,T,B)_{L,R} (\bar{3},1,-2/3) \bar{S}_1 -2 (3,3,-1/3) (T,B,Y)_{L,R} Table 1. Scalar LQ (left) and VLQ (right) representations.
For the six types of scalar LQs and seven types of VLQs, there can be a total of 42 combinations, which are named "
\mathrm{LQ}+ \mathrm{VLQ} " for convenience. Only 17 of them can lead to the chiral enhancements. In Table 2, we list these models that feature the chiral enhancements. The contributons in the four modelsR_2+B_{L,R}/(B,Y)_{L,R} andS_1+B_{L,R}/(B,Y)_{L,R} are almost the same as those in the minimalR_2 andS_1 models. There are nine modelsR_2+T_{L,R}/(X,T)_{L,R}/(T,B)_{L,R}/(T,B,Y)_{L,R} andS_1+T_{L,R}/ (X,T)_{L,R}/(T,B)_{L,R}/(X,T,B)_{L,R}/(T,B,Y)_{L,R} , which produce the top and top partner chiral enhancements. For the two modelsR_2/S_3+(X,T,B)_{L,R} , there are top, top partner, bottom, and bottom partner chiral enhancements at the same time. The models including T quarks were investigated in our previous work [41]. Here, we will study the pure bottom partner chirally enhanced models\tilde{R}_2/\tilde{S}_1+ (B,Y)_{L,R} .Model Chiral enhancement R_2 m_t/m_{\mu} S_1 m_t/m_{\mu} R_2+B_{L,R}/(B,Y)_{L,R} m_t/m_{\mu} S_1+B_{L,R}/(B,Y)_{L,R} m_t/m_{\mu} R_2+T_{L,R}/(X,T)_{L,R}/(T,B)_{L,R}/(T,B,Y)_{L,R} m_t/m_{\mu},m_T/m_{\mu} S_1+T_{L,R}/(X,T)_{L,R}/(T,B)_{L,R}/(X,T,B)_{L,R}/(T,B,Y)_{L,R} m_t/m_{\mu},m_T/m_{\mu} R_2+(X,T,B)_{L,R} m_t/m_{\mu},m_T/m_{\mu},m_b/m_{\mu},m_B/m_{\mu} S_3+(X,T,B)_{L,R} m_t/m_{\mu},m_T/m_{\mu},m_b/m_{\mu},m_B/m_{\mu} {\tilde{\bf R}_2+\bf(B,Y)_{L,R}} m_b/m_{\mu},m_B/m_{\mu} {\tilde{\bf S}_1+\bf(B,Y)_{L,R}} m_b/m_{\mu},m_B/m_{\mu} Table 2. Chiral enhancements in the minimal LQ and LQ+VLQ models.
-
Let us start with the
(B,Y)_{L,R} related Higgs Yukawa interactions. In the gauge eigenstates, there are two interactions\overline{Q_L}^id_R^j\phi and(\overline{B_L},\overline{Y_L})d_R^j\tilde{\phi} and the mass term-M_B(\bar{B}B+\bar{Y}Y) . Here, we define the SM Higgs doublet\tilde{\phi}\equiv {\rm i}\sigma^2\phi^{\ast} with\sigma^a(a=1,2,3) to be the Pauli matrices. TheQ_L^i andd_R^i (i=1,2,3 ) represent the SM quark fields. We can parametrize ϕ as[0,(v+h)/{\sqrt{2}}]^T in the unitary gauge. After the electroweak symmetry breaking (EWSB), there are mixings betweend^i and B. For simplicity, we only consider mixing between the third generation and B quark. Thus, we can perform the following transformations to rotate b and B quarks into mass eigenstates:\begin{aligned}[b]& \left[\begin{array}{c}b_L\\B_L\end{array}\right]\rightarrow \left[\begin{array}{cc}c_L^b& s_L^b\\-s_L^b& c_L^b\end{array}\right] \left[\begin{array}{c}b_L\\B_L\end{array}\right],\\& \left[\begin{array}{c}b_R\\B_R\end{array}\right]\rightarrow \left[\begin{array}{cc}c_R^b& s_R^b\\-s_R^b& c_R^b\end{array}\right] \left[\begin{array}{c}b_R\\B_R\end{array}\right]. \end{aligned}
(1) Here,
s_{L,R}^b andc_{L,R}^b are abbreviations of\sin\theta_{L,R}^b and\cos\theta_{L,R}^b , respectively. In fact,\theta_L^b can be correlated with\theta_R^b through the relation\tan\theta_L^b=m_b\tan\theta_R^b/m_B [42]. Here,m_b andm_B represent the physical b and B quark masses, respectively. Additionally, the mass of the Y quark ism_Y=M_B= \sqrt{m_B^2(c_R^b)^2+m_b^2(s_R^b)^2} . Then, we can choosem_B and\theta_R^b as the new input parameters. After the transformations in Eq. (1), we obtain the following mass eigenstate Higgs Yukawa interactions:\begin{aligned}[b]\mathcal{L}_{\rm H}^{\rm Yukawa}&\supset-\frac{m_b}{v}(c_R^b)^2h\bar{b}b-\frac{m_B}{v}(s_R^b)^2h\bar{B}B\\&-\frac{m_b}{v}s_R^bc_R^bh(\bar{b}_LB_R+\bar{B}_Rb_L)-\frac{m_B}{v}s_R^bc_R^bh(\bar{B}_Lb_R+\bar{b}_RB_L). \end{aligned}
(2) Note that the Y quark does not interact with Higgs at the tree level.
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Now, let us label the
S U(2)_L andU_Y(1) gauge fields asW_{\mu}^a andB_{\mu} . Then, the electroweak covariant derivativeD_{\mu} is defined as\partial_{\mu}-{\rm i}gW_{\mu}^a\sigma^a/2-{\rm i}g^{\prime}Y_qB_{\mu} for a doublet and\partial_{\mu}-{\rm i}g^{\prime}Y_qB_{\mu} for a singlet, in whichY_q is theU_Y(1) charge of the quark field acted byD_{\mu} . Thus, the related gauge interactions can be written as\overline{Q_L}^iiD_{\mu}\gamma^{\mu}Q_L^i+ \overline{d_R}^iiD_{\mu}\gamma^{\mu}d_R^i+(\overline{B},\overline{Y})iD_{\mu}\gamma^{\mu}(B,Y)^T . After the EWSB, the W gauge interactions can be written as\mathcal{L}\supset \frac{g}{\sqrt{2}}W_{\mu}^+(\overline{t_L}\gamma^{\mu}b_L+\bar{B}\gamma^{\mu}Y)+\mathrm{h.c.}.
(3) The Z gauge interactions can be written as
\begin{aligned}[b]\mathcal{L}&\supset\frac{g}{c_W}\Bigg[\left(-\frac{1}{2}+\frac{1}{3}s_W^2\right)\overline{b_L}\gamma^{\mu}b_L+\frac{1}{3}s_W^2\overline{b_R}\gamma^{\mu}b_R\\&+\left(\frac{1}{2}+\frac{1}{3}s_W^2\right)\bar{B}\gamma^{\mu}B+\left(-\frac{1}{2}+\frac{4}{3}s_W^2\right)\bar{Y}\gamma^{\mu}Y\Bigg]Z_{\mu}. \end{aligned}
(4) After the rotations in Eq. (1), we have the mass eigenstate W gauge interactions:
\begin{aligned}[b] \mathcal{L}_{\rm BY}^{\rm gauge}&\supset\frac{g}{\sqrt{2}}W_{\mu}^+[c_L^b\overline{t_L}\gamma^{\mu}b_L+s_L^b\overline{t_L}\gamma^{\mu}B_L+c_L^b\overline{B_L}\gamma^{\mu}Y_L\\&-s_L^b\overline{b_L}\gamma^{\mu}Y_L+c_R^b\overline{B_R}\gamma^{\mu}Y_R-s_R^b\overline{b_R}\gamma^{\mu}Y_R]+\mathrm{h.c.}. \end{aligned}
(5) We also have the mass eigenstate Z gauge interactions:
\begin{aligned}[b] \mathcal{L}_{\rm BY}^{\rm gauge}&\supset\frac{g}{c_W}Z_{\mu}\Bigg[\frac{(c_L^b)^2-(s_L^b)^2}{2}(\overline{B_L}\gamma^{\mu}B_L-\overline{b_L}\gamma^{\mu}b_L)\\&-s_L^bc_L^b(\overline{b_L}\gamma^{\mu}B_L+\overline{B_L}\gamma^{\mu}b_L)+\frac{(s_R^b)^2}{2}\overline{b_R}\gamma^{\mu}b_R\\&+\frac{(c_R^b)^2}{2}\overline{B_R}\gamma^{\mu}B_R-\frac{s_R^bc_R^b}{2}(\overline{b_R}\gamma^{\mu}B_R+\overline{B_R}\gamma^{\mu}b_R)\\&+\frac{s_W^2}{3}(\bar{b}\gamma^{\mu}b+\bar{B}\gamma^{\mu}B)+\left(-\frac{1}{2}+\frac{4s_W^2}{3}\right)\bar{Y}\gamma^{\mu}Y\Bigg]. \end{aligned}
(6) -
Let us denote the SM lepton fields as
L_L^i ande_R^i . The\tilde{R}_2 can be parametrized as[\tilde{R}_2^{2/3},\tilde{R}_2^{-1/3}]^T , where the superscript labels the electric charge. Then, the\tilde{R}_2 and\tilde{S}_1 can induce the followingF=0 andF=2 type gauge eigenstate LQ Yukawa interactions:\begin{array}{*{20}{l}} \mathcal{L}_{\tilde{R}_2+(B,Y)_{L,R}}\supset x_i\overline{e_R^i}(\tilde{R}_2)^{\dagger}\left(\begin{array}{c}B_L\\Y_L\end{array}\right)+y_{ij}\overline{L_L^i}\epsilon(\tilde{R}_2)^\ast d_R^j+\mathrm{h.c.}, \end{array}
(7) and
\begin{array}{*{20}{l}} \mathcal{L}_{\tilde{S}_1+(B,Y)_{L,R}}\supset x_{ij}\overline{e_R^i}(d_R^j)^C(\tilde{S}_1)^{\ast}+y_i\overline{L_L^i}\epsilon\left(\begin{array}{c}B_L\\Y_L\end{array}\right)^C(\tilde{S}_1)^\ast+\mathrm{h.c.}. \end{array}
(8) After the EWSB, they can be parametrized as
\begin{aligned}[b] \mathcal{L}_{\tilde{R}_2+(B,Y)_{L,R}}&\supset y_L^{\tilde{R}_2\mu B}\bar{\mu}\; \omega_-B(\tilde{R}_2^{2/3})^\ast+y_R^{\tilde{R}_2\mu b}\bar{\mu}\; \omega_+b(\tilde{R}_2^{2/3})^\ast\\&+y_L^{\tilde{R}_2\mu B}\bar{\mu}\; \omega_-Y(\tilde{R}_2^{-1/3})^\ast-y_R^{\tilde{R}_2\mu b}\overline{\nu_L}\; \omega_+b(\tilde{R}_2^{-1/3})^\ast+\mathrm{h.c.}, \end{aligned}
(9) and
\begin{aligned}[b] \mathcal{L}_{\tilde{S}_1+(B,Y)_{L,R}}&\supset y_L^{\tilde{S}_1\mu b}\bar{\mu}\; \omega_-b^C(\tilde{S}_1)^\ast+y_R^{\tilde{S}_1\mu B}\bar{\mu}\; \omega_+B^C(\tilde{S}_1)^\ast\\&-y_R^{\tilde{S}_1\mu B}\overline{\nu_L}\; \omega_+Y^C(\tilde{S}_1)^\ast+\mathrm{h.c.}. \end{aligned}
(10) In the above, we define the chiral operators
\omega_{\pm} as(1\pm\gamma^5)/2 . After the rotations in Eq. (1), we have the mass eigenstate interactions:\begin{aligned}[b] \mathcal{L}_{\tilde{R}_2+(B,Y)_{L,R}}&\supset \bar{\mu}(-y_L^{\tilde{R}_2\mu B}s_L^b\omega_-+y_R^{\tilde{R}_2\mu b}c_R^b\omega_+)b(\tilde{R}_2^{2/3})^\ast\\&+\bar{\mu}(y_L^{\tilde{R}_2\mu B}c_L^b\omega_-+y_R^{\tilde{R}_2\mu b}s_R^b\omega_+)B(\tilde{R}_2^{2/3})^\ast\\&+y_L^{\tilde{R}_2\mu B}\bar{\mu}\; \omega_-Y(\tilde{R}_2^{-1/3})^\ast\\&-y_R^{\tilde{R}_2\mu b}\overline{\nu_L}\omega_+(c_R^bb+s_R^bB)(\tilde{R}_2^{-1/3})^\ast+\mathrm{h.c.}, \end{aligned}
(11) and
\begin{aligned}[b] \mathcal{L}_{\tilde{S}_1+(B,Y)_{L,R}}&\supset \bar{\mu}(y_L^{\tilde{S}_1\mu b}c_R^b\omega_--y_R^{\tilde{S}_1\mu B}s_L^b\omega_+)b^C(\tilde{S}_1)^\ast\\&+\bar{\mu}(y_L^{\tilde{S}_1\mu b}s_R^b\omega_-+y_R^{\tilde{S}_1\mu B}c_L^b\omega_+)B^C(\tilde{S}_1)^\ast\\&-y_R^{\tilde{S}_1\mu B}\overline{\nu_L}\; \omega_+Y^C(\tilde{S}_1)^\ast+\mathrm{h.c.}. \end{aligned}
(12) -
For the
\mathrm{LQ}\mu q interaction, there are quark-photon and LQ-photon vertex mediated contributions to(g-2)_{\mu} , which can be described by the functionsf^q(x) andf^S(x) . Then, we use the functionsf_{LL}^{q,S}(x) andf_{LR}^{q,S}(x) to label the parts without and with chiral enhancements. Starting from thef_{LL}^{q,S}(x) andf_{LR}^{q,S}(x) given in our previous paper [41], let us define the following integrals:\begin{aligned}[b] f_{LL}^{\tilde{R}_2\mu Y}(x)\equiv& 4f_{LL}^q(x)-f_{LL}^S(x)=\frac{3 + 2x - 7x^2 + 2x^3 + 2x(4 - x)\log x}{4(1-x)^4},\\ f_{LL}^{\tilde{R}_2\mu b}(x)\equiv& f_{LL}^q(x)+2f_{LL}^S(x)=\frac{x[5-4x-x^2+(2+4x)\log x]}{4(1-x)^4},\\ f_{LR}^{\tilde{R}_2\mu b}(x)\equiv& f_{LR}^q(x)+2f_{LR}^S(x)=-\frac{5-4x-x^2+(2+4x)\log x}{4(1-x)^3},\\ f_{LL}^{\tilde{S}_1\mu b}(x)\equiv&-f_{LL}^q(x)+4f_{LL}^S(x)\\=&-\frac{2-7x+2x^2+3x^3+2x(1-4x)\log x}{4(1-x)^4},\\ f_{LR}^{\tilde{S}_1\mu b}(x)\equiv&-f_{LR}^q(x)+4f_{LR}^S(x)=-\frac{1+4x-5x^2-(2-8x)\log x}{4(1-x)^3}. \end{aligned}
(13) For the
\tilde{R}_2+(B,Y)_{L,R} model, there are b, B, and Y quark contributions to the(g-2)_{\mu} . The complete expression is calculated as\begin{aligned}[b]\Delta a_{\mu}^{\tilde{R}_2+BY}=&\frac{m_{\mu}^2}{8\pi^2}\Bigg\{\frac{|y_L^{\tilde{R}_2\mu B}|^2}{m_{\tilde{R}_2^{-1/3}}^2}f_{LL}^{\tilde{R}_2\mu Y}\Bigg(\frac{m_Y^2}{m_{\tilde{R}_2^{-1/3}}^2}\Bigg)\\&+\frac{|y_L^{\tilde{R}_2\mu B}|^2(s_L^b)^2+|y_R^{\tilde{R}_2\mu b}|^2(c_R^b)^2}{m_{\tilde{R}_2^{2/3}}^2}f_{LL}^{\tilde{R}_2\mu b}\Bigg(\frac{m_b^2}{m_{\tilde{R}_2^{2/3}}^2}\Bigg)\\&-\frac{2m_b}{m_{\mu}}\frac{s_L^bc_R^b}{m_{\tilde{R}_2^{2/3}}^2} \mathrm{Re}\Big[y_L^{\tilde{R}_2\mu B}\Big(y_R^{\tilde{R}_2\mu b}\Big)^\ast\Big]f_{LR}^{\tilde{R}_2\mu b}\Bigg(\frac{m_b^2}{m_{\tilde{R}_2^{2/3}}^2}\Bigg)\\&+\frac{|y_L^{\tilde{R}_2\mu B}|^2(c_L^b)^2+|y_R^{\tilde{R}_2\mu b}|^2(s_R^b)^2}{m_{\tilde{R}_2^{2/3}}^2}f_{LL}^{\tilde{R}_2\mu b}\Bigg(\frac{m_B^2}{m_{\tilde{R}_2^{2/3}}^2}\Bigg)\\&+\frac{2m_B}{m_{\mu}}\frac{c_L^bs_R^b}{m_{\tilde{R}_2^{2/3}}^2} \mathrm{Re}\Big[y_L^{\tilde{R}_2\mu B}\Big(y_R^{\tilde{R}_2\mu b}\Big)^\ast\Big]f_{LR}^{\tilde{R}_2\mu b}\Bigg(\frac{m_B^2}{m_{\tilde{R}_2^{2/3}}^2}\Bigg)\Bigg\}. \end{aligned}
(14) At the tree level, we have
m_{\tilde{R}_2^{2/3}}=m_{\tilde{R}_2^{-1/3}}\equiv m_{\tilde{R}_2} . Compared with the bottom partner chirally enhanced contribution (i.e., thef_{LR}^{\tilde{R}_2\mu b}\left(\dfrac{m_B^2}{m_{\tilde{R}_2^{2/3}}^2}\right) related term), the non-chirally enhanced parts are suppressed by the factorm_{\mu}/(m_Bs_R^b)\sim1/(10^4s_R^b) and the bottom quark chirally enhanced part is suppressed by the factor(m_bs_L^b)/(m_Bs_R^b) \sim(m_b^2/m_B^2) . For the interesting values ofs_R^b at\mathcal{O}(0.01\sim0.1) ,\Delta a_{\mu}^{\tilde{R}_2+BY} is dominated by the bottom partner chirally enhanced contribution. Then, the above expression can be approximated as\Delta a_{\mu}^{\tilde{R}_2+BY}\approx\frac{m_{\mu}m_B}{4\pi^2m_{\tilde{R}_2}^2}s_R^b \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]f_{LR}^{\tilde{R}_2\mu b}\left(\frac{m_B^2}{m_{\tilde{R}_2}^2}\right).
(15) For the
\tilde{S}_1+(B,Y)_{L,R} model, there are b and B quark contributions to(g-2)_{\mu} . The complete expression is calculated as\begin{aligned}[b]\\[-8pt]\Delta a_{\mu}^{\tilde{S}_1+BY}=&\frac{m_{\mu}^2}{8\pi^2}\Bigg\{\frac{|y_R^{\tilde{S}_1\mu B}|^2(s_L^b)^2+|y_L^{\tilde{S}_1\mu b}|^2(c_R^b)^2}{m_{\tilde{S}_1}^2}f_{LL}^{\tilde{S}_1\mu b}\Bigg(\frac{m_b^2}{m_{\tilde{S}_1}^2}\Bigg)-\frac{2m_b}{m_{\mu}}\frac{s_L^bc_R^b}{m_{\tilde{S}_1}^2} \mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast]f_{LR}^{\tilde{S}_1\mu b}\Bigg(\frac{m_b^2}{m_{\tilde{S}_1}^2}\Bigg)\\&+\frac{|y_R^{\tilde{S}_1\mu B}|^2(c_L^b)^2+|y_L^{\tilde{S}_1\mu b}|^2(s_R^b)^2}{m_{\tilde{S}_1}^2}f_{LL}^{\tilde{S}_1\mu b}\Bigg(\frac{m_B^2}{m_{\tilde{S}_1}^2}\Bigg)+\frac{2m_B}{m_{\mu}}\frac{c_L^bs_R^b}{m_{\tilde{S}_1}^2} \mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast]f_{LR}^{\tilde{S}_1\mu b}\Bigg(\frac{m_B^2}{m_{\tilde{S}_1}^2}\Bigg)\Bigg\}. \end{aligned} (16) Similarly, it can be approximated as
\Delta a_{\mu}^{\tilde{S}_1+BY}\approx\frac{m_{\mu}m_B}{4\pi^2m_{\tilde{S}_1}^2}s_R^b \mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast]f_{LR}^{\tilde{S}_1\mu b}\left(\frac{m_B^2}{m_{\tilde{S}_1}^2}\right).
(17) -
The input parameters are chosen as
m_\mu=105.66~\mathrm{MeV} ,m_b=4.2 ~\mathrm{GeV} ,m_t=172.5 ~\mathrm{GeV} ,G_F=1.1664\times10^{-5} ~\mathrm{GeV}^{-2} ,m_W=80.377 ~\mathrm{GeV} ,m_Z=91.1876 ~\mathrm{GeV} , andm_h=125.25 \mathrm{GeV} [43]. Thev,\;g,\;\theta_W are defined byG_F=1/(\sqrt{2}v^2), g=2m_W/v,~\cos\theta_W\equiv c_W=m_W/m_Z . There are also new parametersm_B ,m_{ \mathrm{LQ}} ,\theta_R^b , and the LQ Yukawa couplingsy_{L,R}^{ \mathrm{LQ}\mu q} . The VLQ mass can be constrained from the direct search, which is required to be above1.5~ \mathrm{TeV} [44−47]. The mixing angle is mainly bounded by the electro-weak precision observables (EWPOs). The VLQ contributions to T parameter are suppressed by the factor(s_R^b)^4 orm_b^2(s_R^b)^2/m_B^2 [48, 49], which leads to a less constrained\theta_R^b . The weak isospin third component ofB_R is positive; thus, the mixing with the bottom quark enhances the right-handedZbb coupling. As a result, theA_{FB}^b deviation [50, 51] can be compensated, which leads to looser constraints on\theta_R^b . Conservatively, we can choose the mixing angles_R^b to be smaller than0.1 [42]. The LQ mass can also be constrained from the direct search, which is required to be above1.7 ~\mathrm{TeV} assuming\mathrm{Br}(\mathrm{LQ}\rightarrow b\mu)=1 [52, 53].We can choose benchmark points of
m_B,m_{ \mathrm{LQ}},s_R^b to constrain the LQ Yukawa couplings. Here, we consider two scenarios:m_{ \mathrm{LQ}}>m_B andm_{ \mathrm{LQ}}<m_B . For the scenario ofm_{ \mathrm{LQ}}>m_B , we adopt mass parameters ofm_B= 1.5 TeV andm_{ \mathrm{LQ}}=2 ~\mathrm{TeV} . For the scenario ofm_{ \mathrm{LQ}}<m_B , we adopt mass parameters ofm_B= 2.5 TeV andm_{ \mathrm{LQ}}=2 ~\mathrm{TeV} . In Table 3, we give the approximate numerical expressions of\Delta a_{\mu} in the\tilde{R}_2/\tilde{S}_1+(B,Y)_{L,R} models. We also show the allowed ranges fors_R^b=0.1 ands_R^b=0.05 . Of course, these behaviours can be understood from Eqs. (15) and (17). In the\tilde{R}_2+(B,Y)_{L,R} model,f_{LR}^{\tilde{R}_2\mu b}(x) vanishes whenm_B= m_{\tilde{R}_2} , which causes the| \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]| to be under the stress of perturbative unitarity. If there is a large hierarchy betweenm_B andm_{\tilde{R}_2} , the allowed| \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]| can be smaller. Additionally,\mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast] should be positive (negative) whenm_B<m_{\tilde{R}_2} (m_B>m_{\tilde{R}_2} ). In the\tilde{S}_1+(B,Y)_{L,R} model,f_{LR}^{\tilde{S}_1\mu b}(x) is always negative, which requires\mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast]<0 .Model (m_B,m_{ \mathrm{LQ}})/ \mathrm{TeV} \Delta a_{\mu}\times10^7 s_R^b \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast] or\mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast] 1\sigma 2\sigma \tilde{R}_2+(B,Y)_{L,R} (1.5,2) 0.35s_R^b \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast] 0.1 (0.55,0.88) (0.38,1.05) 0.05 (1.1,1.77) (0.76,2.11) (2.5,2) -0.224s_R^b \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast] 0.1 (-1.38,-0.86) (-1.65,-0.59) 0.05 (-2.76,-1.71) (-3.29,-1.19) \tilde{S}_1+(B,Y)_{L,R} (1.5,2) -6.88s_R^b \mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast] 0.1 (-0.045,-0.028) (-0.054,-0.019) 0.05 (-0.09,-0.056) (-0.11,-0.039) (2.5,2) -6.37s_R^b \mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast] 0.1 (-0.049,-0.03) (-0.058,-0.021) 0.05 (-0.097,-0.06) (-0.12,-0.042) Table 3. In the third column, we show the leading order numerical expressions of the
\Delta a_{\mu} . In the fifth and sixth columns, we show the ranges allowed by(g-2)_{\mu} at1\sigma and2\sigma confidence levels (CLs).We can also choose benchmark points of
s_R^b and LQ Yukawa couplings to constrain them_B andm_{ \mathrm{LQ}} . Fig. 1 presents the(g-2)_{\mu} allowed regions in the plane ofm_B-m_{ \mathrm{LQ}} . As shown,m_B<m_{\tilde{R}_2} andm_B>m_{\tilde{R}_2} are favored in the left and middle plots, respectively. This can be understood from the asymptotic behavioursf_{LR}^{\tilde{R}_2\mu b}(x)\sim -\log(x)/2 > 0 forx\rightarrow0 andf_{LR}^{\tilde{R}_2\mu b}(x) \sim-1/(4x) < 0 forx\rightarrow \infty . To produce a positive\Delta a_{\mu} ,m_B<m_{\tilde{R}_2} andm_B>m_{\tilde{R}_2} are favored for\mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]>0 and\mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]<0 , respectively. Thef_{LR}^{\tilde{S}_1\mu b}(x) has asymptotic behavioursf_{LR}^{\tilde{S}_1\mu b}(x)\sim \log(x)/2 < 0 forx\rightarrow0 andf_{LR}^{\tilde{S}_1\mu b}(x)\sim-5/(4x) < 0 forx\rightarrow \infty . To produce a positive\Delta a_{\mu} ,\mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast]<0 is favored. Furthermore, the allowed regions in the plane ofm_B-m_{ \mathrm{LQ}} are sensitive to the choice ofy_{L,R}^{ \mathrm{LQ}\mu q} . Generally, a larger|y_{L,R}^{ \mathrm{LQ}\mu q}| corresponds to a largerm_B andm_{ \mathrm{LQ}} .Figure 1. (color online)
(g-2)_{\mu} allowed regions at1\sigma (green) and2\sigma (yellow) CLs withs_R^b=0.1 . The parameters are chosen asy_L^{\tilde{R}_2\mu B}=y_R^{\tilde{R}_2\mu b}=0.8 in the\tilde{R}_2+(B,Y) model (left),y_L^{\widetilde{R}_2\mu B}=-y_R^{\tilde{R}_2\mu b}=0.8 in the\tilde{R}_2+(B,Y) model (middle), andy_L^{\tilde{S}_1\mu b}=-y_R^{\tilde{S}_1\mu B}=0.2 in the\tilde{S}_1+(B,Y) model (right). -
In Table 4, we list the main LQ and VLQ decay channels
2 . The decay formulae of LQ and VLQ are given in Appendices A and B, respectively. For the scenario ofm_{ \mathrm{LQ}}>m_B , there are new LQ decay channels. When searching for the LQ\tilde{R}_2^{2/3} , we propose the\mu j_bZ and\mu j_bh signatures. When searching for the LQ\tilde{R}_2^{-1/3} , we propose the\mu j_bW signatures. When searching for the LQ\tilde{S}_1 , we propose the\mu j_bZ ,\mu j_bh ,\not {E_T}j_bW signatures. For the scenario ofm_{ \mathrm{LQ}}<m_B , there are new VLQ decay channels. When searching for the VLQ B, we propose the\mu^+\mu^-j_b signatures. When searching for the VLQ Y, we propose the\mu\; \not {E_T}j_b signatures. It seems that such decay channels have not been searched for by the experimental collaborations.Model Scenario LQ decay VLQ decay new signatures \tilde{R}_2+(B,Y)_{L,R} m_{ \mathrm{LQ}}>m_B \tilde{R}_2^{2/3}\rightarrow \mu^+b,\mu^+B B\rightarrow bZ,bh \tilde{R}_2^{2/3}\rightarrow \mu j_bZ,\mu j_bh \tilde{R}_2^{-1/3}\rightarrow \mu^+Y,\nu_Lb Y\rightarrow bW^- \tilde{R}_2^{-1/3}\rightarrow \mu j_bW m_{ \mathrm{LQ}}<m_B \tilde{R}_2^{2/3}\rightarrow \mu^+b B\rightarrow bZ,bh,\mu^-\tilde{R}_2^{2/3} B\rightarrow \mu^+\mu^-j_b \tilde{R}_2^{-1/3}\rightarrow \nu_Lb Y\rightarrow bW^-,\mu^-\tilde{R}_2^{-1/3} Y\rightarrow \mu \not {E_T}j_b \tilde{S}_1+(B,Y)_{L,R} m_{ \mathrm{LQ}}>m_B \tilde{S}_1\rightarrow \mu^+\bar{b},\mu^+\bar{B},\nu_L\bar{Y} B\rightarrow bZ,bh \tilde{S}_1\rightarrow \mu j_bZ ,\mu j_bh ,\not {E_T}j_bW Y\rightarrow bW^- m_{ \mathrm{LQ}}<m_B \tilde{S}_1\rightarrow \mu^+\bar{b} B\rightarrow bZ,bh,\mu^+(\tilde{S}_1)^{\ast} B\rightarrow \mu^+\mu^-j_b Y\rightarrow bW^-,\nu_L(\tilde{S}_1)^{\ast} Y\rightarrow \mu \not {E_T}j_b Table 4. In the third column, we show the main LQ decay channels. In the fourth column, we show the main VLQ decay channels. In the fifth column, we show the new LQ or VLQ signatures.
To estimate the effects of new decay channels, we will compare the ratios of new partial decay widths with the traditional ones. Because of gauge symmetry, the different partial decay widths can be correlated. Then, we choose the following four ratios:
\begin{aligned}[b]&\frac{\Gamma(\tilde{R}_2^{2/3}\rightarrow \mu^+B)}{\Gamma(\tilde{R}_2^{2/3}\rightarrow \mu^+b)}\sim\frac{|y_L^{\tilde{R}_2\mu B}|^2}{|y_R^{\tilde{R}_2\mu b}|^2},\quad\frac{\Gamma(\tilde{S}_1\rightarrow \mu^+\bar{B})}{\Gamma(\tilde{S}_1\rightarrow \mu^+\bar{b})}\sim\frac{|y_R^{\tilde{S}_1\mu B}|^2}{|y_L^{\tilde{S}_1\mu b}|^2},\\&\frac{\Gamma(B\rightarrow \mu^-\tilde{R}_2^{2/3})}{\Gamma(B\rightarrow bh)}\sim\frac{v^2|y_L^{\tilde{R}_2\mu B}|^2}{m_B^2(s_R^b)^2},\quad\frac{\Gamma(B\rightarrow \mu^+(\tilde{S}_1)^{\ast})}{\Gamma(B\rightarrow bh)}\sim\frac{v^2|y_R^{\tilde{S}_1\mu B}|^2}{m_B^2(s_R^b)^2}. \end{aligned}
(18) In Fig. 2, we show the contour plots of above four ratios under the consideration of
(g-2)_{\mu} constraints. In these plots, we include the full contributions. We find that the new LQ decay channels can become important for larger|y_L^{\tilde{R}_2\mu B}| in the\tilde{R}_2+(B,Y)_{L,R} model and|y_R^{\tilde{S}_1\mu B}| in the\tilde{S}_1+(B,Y)_{L,R} model. As for the VLQ decay, the importance of new decay channels depends significantly ons_R^b . Fors_R^b=0.1 andm_B=2.5 ~\mathrm{TeV} , the new VLQ decay channels are less significant. For smallers_R^b , the new VLQ decay channels can play an important role.Figure 2. (color online) Contour plots of
\log_{10}\frac{\Gamma(\tilde{R}_2^{2/3}\rightarrow \mu^+B)}{\Gamma(\tilde{R}_2^{2/3}\rightarrow \mu^+b)} (upper left),\log_{10}\frac{\Gamma(\tilde{S}_1\rightarrow \mu^+\bar{B})}{\Gamma(\tilde{S}_1\rightarrow \mu^+\bar{b})} (lower left),\log_{10}\frac{\Gamma(B\rightarrow \mu^-\tilde{R}_2^{2/3})}{\Gamma(B\rightarrow bh)} (upper right), and\log_{10}\frac{\Gamma(B\rightarrow \mu^+(\tilde{S}_1)^{\ast})}{\Gamma(B\rightarrow bh)} (lower right), where the colored regions are allowed by the(g-2)_{\mu} at2\sigma CL. For the LQ decay, we choosem_B=1.5 ~\mathrm{TeV} andm_{ \mathrm{LQ}}=2 ~\mathrm{TeV} . For the VLQ decay, we choosem_B=2.5 ~\mathrm{TeV} andm_{ \mathrm{LQ}}=2 ~\mathrm{TeV} .For the LQ and VLQ production at hadron colliders, there are pair and single production channels, which are very sensitive to the LQ and VLQ masses. We can adopt the FeynRules [54] to generate the model files and compute the cross sections with MadGraph5
{}_{-} aMC@NLO [55]. For the 2 TeV scale LQ pair production [56−58], the cross section can be\sim0.01 ~\mathrm{fb} at the 13 TeV LHC. For the 1.5 TeV and 2.5 TeV scale VLQ pair production [59−61], the cross section can be\sim2~ \mathrm{fb} and\sim0.01~ \mathrm{fb} at the 13 TeV LHC. For the single LQ and VLQ production channels, they depend on the electroweak couplings [42, 62, 63]. In the parameter space of large LQ Yukawa couplings, the single LQ production can be important, which may give some constraints at HL-LHC. To generate enough events, higher energy hadron colliders, for example, 27 and 100 TeV, can be necessary. In addition to the collider direct search, there can be indirect footprints, for example, B physics related decay modes\Upsilon\rightarrow \mu^+\mu^-,\nu\bar{\nu}\gamma . If we consider a more complex flavour structure (e.g., turn on the\mathrm{LQ}\mu s interaction), this can affect theB\rightarrow K\mu^+\mu^- channel. Here, we will not study this detailed phenomenology. -
When the
\tilde{R}_2 masses are degenerate, there are no gauge boson decay channels such as\tilde{R}_2^{2/3}\rightarrow \tilde{R}_2^{-1/3}W^+ . For the\tilde{R}_2^{2/3} to\mu^+b and\mu^+B decay channels, the widths are calculated as\begin{aligned}[b]\\[-5pt]\Gamma(\tilde{R}_2^{2/3}\rightarrow \mu^+b)=&\frac{m_{\tilde{R}_2}}{16\pi}\sqrt{\left(1-\frac{m_{\mu}^2+m_b^2}{m_{\tilde{R}_2}^2}\right)^2-\frac{4m_{\mu}^2m_b^2}{m_{\tilde{R}_2}^4}}\;\\& \times\Bigg\{\Bigg(1-\frac{m_{\mu}^2+m_b^2}{m_{\tilde{R}_2}^2}\Bigg)[|y_L^{\tilde{R}_2\mu B}|^2(s_L^b)^2+|y_R^{\tilde{R}_2\mu b}|^2(c_R^b)^2]+\frac{4m_{\mu}m_b}{m_{\tilde{R}_2}^2}s_L^bc_R^b \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]\Bigg\},\\\Gamma(\tilde{R}_2^{2/3}\rightarrow \mu^+B)=&\frac{m_{\tilde{R}_2}}{16\pi}\sqrt{\Bigg(1-\frac{m_{\mu}^2+m_B^2}{m_{\tilde{R}_2}^2}\Bigg)^2-\frac{4m_{\mu}^2m_B^2}{m_{\tilde{R}_2}^4}}\; \\&\times\Bigg\{\Bigg(1-\frac{m_{\mu}^2+m_B^2}{m_{\tilde{R}_2}^2}\Bigg)[|y_L^{\tilde{R}_2\mu B}|^2(c_L^b)^2+|y_R^{\tilde{R}_2\mu b}|^2(s_R^b)^2]-\frac{4m_{\mu}m_B}{m_{\tilde{R}_2}^2}c_L^bs_R^b \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]\Bigg\}. \end{aligned}\tag{A1} For the
\tilde{R}_2^{-1/3} to\mu^+Y,\nu_Lb,\nu_LB decay channels, the widths are calculated as\begin{aligned}[b]\Gamma(\tilde{R}_2^{-1/3}\rightarrow \mu^+Y)=\frac{m_{\tilde{R}_2}}{16\pi}\sqrt{\left(1-\frac{m_{\mu}^2+m_Y^2}{m_{\tilde{R}_2}^2}\right)^2-\frac{4m_{\mu}^2m_Y^2}{m_{\tilde{R}_2}^4}}\left(1-\frac{m_{\mu}^2+m_Y^2}{m_{\tilde{R}_2}^2}\right)|y_L^{\tilde{R}_2\mu B}|^2, \end{aligned}
\begin{aligned}[b]&\Gamma(\tilde{R}_2^{-1/3}\rightarrow \nu_Lb)=\frac{m_{\tilde{R}_2}}{16\pi}\left(1-\frac{m_b^2}{m_{\tilde{R}_2}^2}\right)^2|y_R^{\tilde{R}_2\mu b}|^2(c_R^b)^2,\\&\Gamma(\tilde{R}_2^{-1/3}\rightarrow \nu_LB)=\frac{m_{\tilde{R}_2}}{16\pi}\left(1-\frac{m_B^2}{m_{\tilde{R}_2}^2}\right)^2|y_R^{\tilde{R}_2\mu b}|^2(s_R^b)^2. \end{aligned}\tag{A2}
Considering
m_{\mu},m_b\ll m_B and\theta_{L,R}^b\ll1 , we have the following approximations:\begin{aligned}[b]&\Gamma(\tilde{R}_2^{2/3}\rightarrow \mu^+B)\approx\Gamma(\tilde{R}_2^{-1/3}\rightarrow \mu^+Y)\approx\frac{m_{\tilde{R}_2}}{16\pi}\Bigg(1-\frac{m_B^2}{m_{\tilde{R}_2}^2}\Bigg)^2|y_L^{\tilde{R}_2\mu B}|^2,\\&\Gamma(\tilde{R}_2^{2/3}\rightarrow \mu^+b)\approx\Gamma(\tilde{R}_2^{-1/3}\rightarrow \nu_Lb)\approx\frac{m_{\tilde{R}_2}}{16\pi}|y_R^{\tilde{R}_2\mu b}|^2. \end{aligned}\tag{A3}
For the
\tilde{S}_1 to\mu^+\bar{b},\mu^+\bar{B},\nu_L\bar{Y} decay channels, the widths are calculated as\begin{aligned}[b]\Gamma(\tilde{S}_1\rightarrow \mu^+\bar{b})=&\frac{m_{\tilde{S}_1}}{16\pi}\sqrt{\Bigg(1-\frac{m_{\mu}^2+m_b^2}{m_{\tilde{S}_1}^2}\Bigg)^2-\frac{4m_{\mu}^2m_b^2}{m_{\tilde{S}_1}^4}}\; \\&\times\Bigg\{\Bigg(1-\frac{m_{\mu}^2+m_b^2}{m_{\tilde{S}_1}^2}\Bigg)[|y_R^{\tilde{S}_1\mu B}|^2(s_L^b)^2+|y_L^{\tilde{S}_1\mu b}|^2(c_R^b)^2]+\frac{4m_{\mu}m_b}{m_{\tilde{S}_1}^2}s_L^bc_R^b \mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast]\Bigg\},\\\Gamma(\tilde{S}_1\rightarrow \mu^+\bar{B})=&\frac{m_{\tilde{S}_1}}{16\pi}\sqrt{\Bigg(1-\frac{m_{\mu}^2+m_B^2}{m_{\tilde{S}_1}^2}\Bigg)^2-\frac{4m_{\mu}^2m_B^2}{m_{\tilde{S}_1}^4}}\; \\&\times\Big\{(1-\frac{m_{\mu}^2+m_B^2}{m_{\tilde{S}_1}^2})[|y_R^{\tilde{S}_1\mu B}|^2(c_L^b)^2+|y_L^{\tilde{S}_1\mu b}|^2(s_R^b)^2]-\frac{4m_{\mu}m_B}{m_{\tilde{S}_1}^2}c_L^bs_R^b \mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast]\Bigg\},\\\Gamma(\tilde{S}_1\rightarrow \nu_L\bar{Y})=&\frac{m_{\tilde{S}_1}}{16\pi}\Bigg(1-\frac{m_Y^2}{m_{\tilde{S}_1}^2}\Bigg)^2|y_R^{\tilde{S}_1\mu B}|^2. \end{aligned}\tag{A4}
Considering
m_{\mu},m_b\ll m_B and\theta_{L,R}^b\ll1 , we have the following approximations:\begin{aligned}[b]&\Gamma(\tilde{S}_1\rightarrow \mu^+\bar{B})\approx\Gamma(\tilde{S}_1\rightarrow \nu_L\bar{Y})\approx\frac{m_{\tilde{S}_1}}{16\pi}\Bigg(1-\frac{m_B^2}{m_{\tilde{S}_1}^2}\Bigg)^2|y_R^{\tilde{S}_1\mu B}|^2,\\&\Gamma(\tilde{S}_1\rightarrow \mu^+\bar{b})\approx\frac{m_{\tilde{S}_1}}{16\pi}|y_L^{\tilde{S}_1\mu b}|^2. \end{aligned} \tag{A5}
-
If
m_B\sim \mathrm{TeV} and\theta_R^b\ll0.1 , we havem_B-m_Y\approx m_B(s_R^b)^2/2\lesssim5 ~\mathrm{GeV} , which leads to the kinematic prohibition of some decay channels. For theY\rightarrow bW^- decay channel, the width is calculated as\Gamma(Y\rightarrow bW^-)=\frac{g^2}{64\pi m_Y}\sqrt{\Bigg(1-\frac{m_b^2+m_W^2}{m_Y^2}\Bigg)^2-\frac{4m_b^2m_W^2}{m_Y^4}}\cdot\Bigg\{[(s_L^b)^2+(s_R^b)^2]\frac{(m_Y^2-m_b^2)^2+m_W^2(m_Y^2+m_b^2)-2m_W^4}{m_W^2}-12m_Ym_bs_L^bs_R^b\Bigg\}. \tag{B1}
For the
B\rightarrow bZ,~bh,~tW^- decay channels, the widths are calculated as\begin{aligned}[b]\Gamma(B\rightarrow bh)=&\frac{m_B}{32\pi}\sqrt{\Bigg(1-\frac{m_b^2+m_h^2}{m_B^2}\Bigg)^2-\frac{4m_b^2m_h^2}{m_B^4}}\Bigg[\Bigg(1+\frac{m_b^2-m_h^2}{m_B^2}\Bigg)\frac{m_b^2+m_B^2}{v^2}+4\frac{m_b^2}{v^2}\Bigg](s_R^b)^2(c_R^b)^2, \\\Gamma(B\rightarrow bZ)=&\frac{g^2}{32\pi c_W^2m_B}\sqrt{\Bigg(1-\frac{m_b^2+m_Z^2}{m_B^2}\Bigg)^2-\frac{4m_b^2m_Z^2}{m_B^4}}\\&\times\Bigg\{\Bigg[(s_L^bc_L^b)^2+\frac{(s_R^bc_R^b)^2}{4}\Bigg]\frac{(m_B^2-m_b^2)^2+m_Z^2(m_B^2+m_b^2)-2m_Z^4}{m_Z^2}-6m_Bm_bs_L^bs_R^bc_L^bc_R^b\Bigg\},\\\Gamma(B\rightarrow tW^-)=&\frac{g^2(s_L^b)^2}{64\pi}\sqrt{\Bigg(1-\frac{m_t^2+m_W^2}{m_B^2}\Bigg)^2-\frac{4m_t^2m_W^2}{m_B^4}}\frac{(m_B^2-m_t^2)^2+m_W^2(m_B^2+m_t^2)-2m_W^4}{m_W^2m_B}. \end{aligned}\tag{B2}
Considering
m_b,~m_t,~m_Z,~m_W\ll m_B and\theta_{L,R}^b\ll1 , we have the following approximations:\begin{aligned}[b]&\Gamma(B\rightarrow bZ)\approx\Gamma(B\rightarrow bh)\approx\frac{1}{2}\Gamma(Y\rightarrow bW^-)\approx\frac{m_B^3}{32\pi v^2}(s_R^b)^2,\\&\Gamma(B\rightarrow tW^-)\approx\frac{m_b^2m_B}{16\pi v^2}(s_R^b)^2. \end{aligned}\tag{B3}
In the
\tilde{R}_2+(B,Y)_{L,R} model, the VLQ can also decay into the\tilde{R}_2 final state. For theY\rightarrow \mu^-\tilde{R}_2^{-1/3} decay channel, the width is calculated as\Gamma(Y\rightarrow \mu^-\tilde{R}_2^{-1/3})=\frac{m_Y}{32\pi}\sqrt{\Bigg(1-\frac{m_{\mu}^2+m_{\tilde{R}_2}^2}{m_Y^2}\Bigg)^2-\frac{4m_{\mu}^2m_{\tilde{R}_2}^2}{m_Y^4}}\Bigg(1+\frac{m_{\mu}^2-m_{\tilde{R}_2}^2}{m_Y^2}\Bigg)|y_L^{\tilde{R}_2\mu B}|^2. \tag{B4}
For the
B\rightarrow \mu^-\tilde{R}_2^{2/3},~\nu_L\tilde{R}_2^{-1/3} decay channels, the widths are calculated as\begin{aligned}[b]\Gamma(B\rightarrow \mu^-\tilde{R}_2^{2/3})=&\frac{m_B}{32\pi}\sqrt{\Bigg(1-\frac{m_{\mu}^2+m_{\tilde{R}_2}^2}{m_B^2}\Bigg)^2-\frac{4m_{\mu}^2m_{\tilde{R}_2}^2}{m_B^4}}\; \\&\times\Bigg\{\Bigg(1+\frac{m_{\mu}^2-m_{\tilde{R}_2}^2}{m_B^2}\Bigg)[|y_L^{\tilde{R}_2\mu B}|^2(c_L^b)^2+|y_R^{\tilde{R}_2\mu b}|^2(s_R^b)^2]+\frac{4m_{\mu}m_{\tilde{R}_2}}{m_B^2}c_L^bs_R^b \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]\Big\},\\\Gamma(B\rightarrow \nu_L\tilde{R}_2^{-1/3})=&\frac{m_B}{32\pi}\Bigg(1-\frac{m_{\tilde{R}_2}^2}{m_B^2}\Bigg)^2|y_R^{\tilde{R}_2\mu b}|^2(s_R^b)^2. \end{aligned}\tag{B5}
Considering
m_{\mu}\ll m_B and\theta_{L,R}^b\ll1 , we have the following approximations:\Gamma(B\rightarrow \mu^-\tilde{R}_2^{2/3})\approx\Gamma(Y\rightarrow \mu^-\tilde{R}_2^{-1/3})\approx\frac{m_B}{32\pi}\Bigg(1-\frac{m_{\tilde{R}_2}^2}{m_B^2}\Bigg)^2|y_L^{\tilde{R}_2\mu B}|^2. \tag{B6}
In the
\tilde{S}_1+(B,Y)_{L,R} model, the VLQ can also decay into the\tilde{S}_1 final state. For theY\rightarrow \nu_L(\tilde{S}_1)^{\ast} decay channel, the width is calculated as\Gamma(Y\rightarrow \nu_L(\tilde{S}_1)^{\ast})=\frac{m_Y}{32\pi}\Bigg(1-\frac{m_{\tilde{S}_1}^2}{m_Y^2}\Bigg)^2|y_R^{\tilde{S}_1\mu B}|^2. \tag{B7}
For the
B\rightarrow \mu^+(\tilde{S}_1)^{\ast} decay channel, the width is calculated as\begin{aligned}[b]\Gamma(B\rightarrow \mu^+(\tilde{S}_1)^{\ast})=&\frac{m_B}{32\pi}\sqrt{\Bigg(1-\frac{m_{\mu}^2+m_{\tilde{S}_1}^2}{m_B^2}\Bigg)^2-\frac{4m_{\mu}^2m_{\tilde{S}_1}^2}{m_B^4}}\;\\&\times\Bigg\{\Bigg(1+\frac{m_{\mu}^2-m_{\tilde{S}_1}^2}{m_B^2}\Bigg)[|y_R^{\tilde{S}_1\mu B}|^2(c_L^b)^2+|y_L^{\tilde{S}_1\mu b}|^2(s_R^b)^2]+\frac{4m_{\mu}}{m_B}c_L^bs_R^b \mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast]\Bigg\}. \end{aligned}\tag{B8}
Considering
m_{\mu}\ll m_B and\theta_{L,R}^b\ll1 , we have the following approximations:\Gamma(B\rightarrow \mu^+(\tilde{S}_1)^{\ast})\approx\Gamma(Y\rightarrow \nu_L(\tilde{S}_1)^{\ast})\approx\frac{m_B}{32\pi}\Bigg(1-\frac{m_{\tilde{S}_1}^2}{m_B^2}\Bigg)^2|y_R^{\tilde{S}_1\mu B}|^2. \tag{B9}
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