Processing math: 26%

Scalar leptoquark and vector-like quark extended models as the explanation of the muon g–2 anomaly: bottom partner chiral enhancement case

  • Leptoquark (LQ) models are well motivated solutions to the (g2)μ anomaly. In the minimal LQ models, only specific representations can lead to chiral enhancements. For the scalar LQs, R2 and S1 can lead to the top quark chiral enhancement. For the vector LQs, V2 and U1 can lead to the bottom quark chiral enhancement. When we consider the LQ and vector-like quark (VLQ) simultaneously, there can be more scenarios. In our previous study, we considered the scalar LQ and VLQ extended models with up-type quark chiral enhancement. Here, we study the scalar LQ and VLQ extended models with down-type quark chiral enhancement. We find two new models with B quark chiral enhancements, which originate from the bottom and bottom partner mixing. Then, we propose new LQ and VLQ search channels under the constraints of (g2)μ.
  • The (g2)μ anomaly is a longstanding puzzle in the standard model (SM) of elementary particle physics. It was first announced by the BNL E821 experiment [1]. Last year, the FNAL muon g2 experiment revealed increased deviation from the SM prediction [2]. When combining the BNL and FNAL data, the averaged result is aExpμ=116592061(41)×1011. Compared to the SM prediction aSMμ=116591810(43)×1011 [323], the deviation is ΔaμaExpμaSMμ=(251±59)×1011, which shows a 4.2σ discrepancy. Many new physics models were proposed to explain the anomaly [2429].

    For the mediators with mass above the TeV level, the chiral enhancements are required, which can appear when left-handed and right-handed muons couple to a heavy fermion simultaneously. In the new lepton extended models [3033], the chiral enhancements originate from the large lepton mass. The LQ models are an alternative choice [3440], in which the chiral enhancements originate from the large quark mass. For the minimal LQ models, there are scalar LQs R2/S1 with top quark chiral enhancement and vector LQs V2/U1 with bottom quark chiral enhancement. The LQ can connect the lepton sector and quark sector. On the other hand, the VLQ naturally occurs in many new physics models and is free of quantum anomaly. It can mix with SM quarks and provide new source of CP violation. Hence, the LQ and VLQ extended models can lead to interesting flavour physics in both the lepton sector and quark sector. In our previous study [41], we investigated the scalar LQ and VLQ 1 extended models with top and top partner chiral enhancements. In this study, we investigate the scalar LQ and VLQ extended models, which can produce the bottom partner chiral enhancements. This paper is complementary to our previous paper [41]. Moreover, the top partner and bottom partner lead to different collider signatures.

    In Sec. II, we introduce the models and show the related interactions. Then, we derive the new physics contributions to (g2)μ and perform the numerical analysis in Sec. III. In Sec. IV, we discuss the possible collider phenomenology. Finally, we present a summary and conclusions in Sec. V.

    Typically, there are six types of scalar LQs [35], which carry a conserved quantum number F3B+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.

    Table 1

    Table 1.  Scalar LQ (left) and VLQ (right) representations.
    SU(3)C×SU(2)L×U(1)Y representationlabelFSU(3)C×SU(2)L×U(1)Y representationlabel
    (ˉ3,3,1/3)S32(3,1,2/3)TL,R
    (3,2,7/6)R20(3,1,1/3)BL,R
    (3,2,1/6)˜R20(3,2,7/6)(X,T)L,R
    (ˉ3,1,4/3)˜S12(3,2,1/6)(T,B)L,R
    (3,2,5/6)(B,Y)L,R
    (ˉ3,1,1/3)S12(3,3,2/3)(X,T,B)L,R
    (ˉ3,1,2/3)ˉS12(3,3,1/3)(T,B,Y)L,R
    DownLoad: CSV
    Show Table

    For the six types of scalar LQs and seven types of VLQs, there can be a total of 42 combinations, which are named "LQ+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 models R2+BL,R/(B,Y)L,R and S1+BL,R/(B,Y)L,R are almost the same as those in the minimal R2 and S1 models. There are nine models R2+TL,R/(X,T)L,R/(T,B)L,R/(T,B,Y)L,R and S1+TL,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 models R2/S3+(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 ˜R2/˜S1+(B,Y)L,R.

    Table 2

    Table 2.  Chiral enhancements in the minimal LQ and LQ+VLQ models.
    ModelChiral enhancement
    R2mt/mμ
    S1mt/mμ
    R2+BL,R/(B,Y)L,Rmt/mμ
    S1+BL,R/(B,Y)L,Rmt/mμ
    R2+TL,R/(X,T)L,R/(T,B)L,R/(T,B,Y)L,Rmt/mμ,mT/mμ
    S1+TL,R/(X,T)L,R/(T,B)L,R/(X,T,B)L,R/(T,B,Y)L,Rmt/mμ,mT/mμ
    R2+(X,T,B)L,Rmt/mμ,mT/mμ,mb/mμ,mB/mμ
    S3+(X,T,B)L,Rmt/mμ,mT/mμ,mb/mμ,mB/mμ
    ˜R2+(B,Y)L,Rmb/mμ,mB/mμ
    ˜S1+(B,Y)L,Rmb/mμ,mB/mμ
    DownLoad: CSV
    Show Table

    Let us start with the (B,Y)L,R related Higgs Yukawa interactions. In the gauge eigenstates, there are two interactions ¯QLidjRϕ and (¯BL,¯YL)djR˜ϕ and the mass term MB(ˉBB+ˉYY). Here, we define the SM Higgs doublet ˜ϕiσ2ϕ with σa(a=1,2,3) to be the Pauli matrices. The QiL and diR (i=1,2,3) represent the SM quark fields. We can parametrize ϕ as [0,(v+h)/2]T in the unitary gauge. After the electroweak symmetry breaking (EWSB), there are mixings between di 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:

    [bLBL][cbLsbLsbLcbL][bLBL],[bRBR][cbRsbRsbRcbR][bRBR].

    (1)

    Here, sbL,R and cbL,R are abbreviations of sinθbL,R and cosθbL,R, respectively. In fact, θbL can be correlated with θbR through the relation tanθbL=mbtanθbR/mB [42]. Here, mb and mB represent the physical b and B quark masses, respectively. Additionally, the mass of the Y quark is mY=MB= m2B(cbR)2+m2b(sbR)2. Then, we can choose mB and θbR as the new input parameters. After the transformations in Eq. (1), we obtain the following mass eigenstate Higgs Yukawa interactions:

    LYukawaHmbv(cbR)2hˉbbmBv(sbR)2hˉBBmbvsbRcbRh(ˉbLBR+ˉBRbL)mBvsbRcbRh(ˉBLbR+ˉbRBL).

    (2)

    Note that the Y quark does not interact with Higgs at the tree level.

    Now, let us label the SU(2)L and UY(1) gauge fields as Waμ and Bμ. Then, the electroweak covariant derivative Dμ is defined as μigWaμσa/2igYqBμ for a doublet and μigYqBμ for a singlet, in which Yq is the UY(1) charge of the quark field acted by Dμ. Thus, the related gauge interactions can be written as ¯QLiiDμγμQiL+¯dRiiDμγμdiR+(¯B,¯Y)iDμγμ(B,Y)T. After the EWSB, the W gauge interactions can be written as

    Lg2W+μ(¯tLγμbL+ˉBγμY)+h.c..

    (3)

    The Z gauge interactions can be written as

    LgcW[(12+13s2W)¯bLγμbL+13s2W¯bRγμbR+(12+13s2W)ˉBγμB+(12+43s2W)ˉYγμY]Zμ.

    (4)

    After the rotations in Eq. (1), we have the mass eigenstate W gauge interactions:

    LgaugeBYg2W+μ[cbL¯tLγμbL+sbL¯tLγμBL+cbL¯BLγμYLsbL¯bLγμYL+cbR¯BRγμYRsbR¯bRγμYR]+h.c..

    (5)

    We also have the mass eigenstate Z gauge interactions:

    LgaugeBYgcWZμ[(cbL)2(sbL)22(¯BLγμBL¯bLγμbL)sbLcbL(¯bLγμBL+¯BLγμbL)+(sbR)22¯bRγμbR+(cbR)22¯BRγμBRsbRcbR2(¯bRγμBR+¯BRγμbR)+s2W3(ˉbγμb+ˉBγμB)+(12+4s2W3)ˉYγμY].

    (6)

    Let us denote the SM lepton fields as LiL and eiR. The ˜R2 can be parametrized as [˜R2/32,˜R1/32]T, where the superscript labels the electric charge. Then, the ˜R2 and ˜S1 can induce the following F=0 and F=2 type gauge eigenstate LQ Yukawa interactions:

    L˜R2+(B,Y)L,Rxi¯eiR(˜R2)(BLYL)+yij¯LiLϵ(˜R2)djR+h.c.,

    (7)

    and

    L˜S1+(B,Y)L,Rxij¯eiR(djR)C(˜S1)+yi¯LiLϵ(BLYL)C(˜S1)+h.c..

    (8)

    After the EWSB, they can be parametrized as

    L˜R2+(B,Y)L,Ry˜R2μBLˉμωB(˜R2/32)+y˜R2μbRˉμω+b(˜R2/32)+y˜R2μBLˉμωY(˜R1/32)y˜R2μbR¯νLω+b(˜R1/32)+h.c.,

    (9)

    and

    L˜S1+(B,Y)L,Ry˜S1μbLˉμωbC(˜S1)+y˜S1μBRˉμω+BC(˜S1)y˜S1μBR¯νLω+YC(˜S1)+h.c..

    (10)

    In the above, we define the chiral operators ω± as (1±γ5)/2. After the rotations in Eq. (1), we have the mass eigenstate interactions:

    L˜R2+(B,Y)L,Rˉμ(y˜R2μBLsbLω+y˜R2μbRcbRω+)b(˜R2/32)+ˉμ(y˜R2μBLcbLω+y˜R2μbRsbRω+)B(˜R2/32)+y˜R2μBLˉμωY(˜R1/32)y˜R2μbR¯νLω+(cbRb+sbRB)(˜R1/32)+h.c.,

    (11)

    and

    L˜S1+(B,Y)L,Rˉμ(y˜S1μbLcbRωy˜S1μBRsbLω+)bC(˜S1)+ˉμ(y˜S1μbLsbRω+y˜S1μBRcbLω+)BC(˜S1)y˜S1μBR¯νLω+YC(˜S1)+h.c..

    (12)

    For the LQμq interaction, there are quark-photon and LQ-photon vertex mediated contributions to (g2)μ, which can be described by the functions fq(x) and fS(x). Then, we use the functions fq,SLL(x) and fq,SLR(x) to label the parts without and with chiral enhancements. Starting from the fq,SLL(x) and fq,SLR(x) given in our previous paper [41], let us define the following integrals:

    f˜R2μYLL(x)4fqLL(x)fSLL(x)=3+2x7x2+2x3+2x(4x)logx4(1x)4,f˜R2μbLL(x)fqLL(x)+2fSLL(x)=x[54xx2+(2+4x)logx]4(1x)4,f˜R2μbLR(x)fqLR(x)+2fSLR(x)=54xx2+(2+4x)logx4(1x)3,f˜S1μbLL(x)fqLL(x)+4fSLL(x)=27x+2x2+3x3+2x(14x)logx4(1x)4,f˜S1μbLR(x)fqLR(x)+4fSLR(x)=1+4x5x2(28x)logx4(1x)3.

    (13)

    For the ˜R2+(B,Y)L,R model, there are b, B, and Y quark contributions to the (g2)μ. The complete expression is calculated as

    Δa˜R2+BYμ=m2μ8π2{|y˜R2μBL|2m2˜R1/32f˜R2μYLL(m2Ym2˜R1/32)+|y˜R2μBL|2(sbL)2+|y˜R2μbR|2(cbR)2m2˜R2/32f˜R2μbLL(m2bm2˜R2/32)2mbmμsbLcbRm2˜R2/32Re[y˜R2μBL(y˜R2μbR)]f˜R2μbLR(m2bm2˜R2/32)+|y˜R2μBL|2(cbL)2+|y˜R2μbR|2(sbR)2m2˜R2/32f˜R2μbLL(m2Bm2˜R2/32)+2mBmμcbLsbRm2˜R2/32Re[y˜R2μBL(y˜R2μbR)]f˜R2μbLR(m2Bm2˜R2/32)}.

    (14)

    At the tree level, we have m˜R2/32=m˜R1/32m˜R2. Compared with the bottom partner chirally enhanced contribution (i.e., the f˜R2μbLR(m2Bm2˜R2/32) related term), the non-chirally enhanced parts are suppressed by the factor mμ/(mBsbR)1/(104sbR) and the bottom quark chirally enhanced part is suppressed by the factor (mbsbL)/(mBsbR)(m2b/m2B). For the interesting values of sbR at O(0.010.1), Δa˜R2+BYμ is dominated by the bottom partner chirally enhanced contribution. Then, the above expression can be approximated as

    Δa˜R2+BYμmμmB4π2m2˜R2sbRRe[y˜R2μBL(y˜R2μbR)]f˜R2μbLR(m2Bm2˜R2).

    (15)

    For the ˜S1+(B,Y)L,R model, there are b and B quark contributions to (g2)μ. The complete expression is calculated as

    Δa˜S1+BYμ=m2μ8π2{|y˜S1μBR|2(sbL)2+|y˜S1μbL|2(cbR)2m2˜S1f˜S1μbLL(m2bm2˜S1)2mbmμsbLcbRm2˜S1Re[y˜S1μBR(y˜S1μbL)]f˜S1μbLR(m2bm2˜S1)+|y˜S1μBR|2(cbL)2+|y˜S1μbL|2(sbR)2m2˜S1f˜S1μbLL(m2Bm2˜S1)+2mBmμcbLsbRm2˜S1Re[y˜S1μBR(y˜S1μbL)]f˜S1μbLR(m2Bm2˜S1)}.

    (16)

    Similarly, it can be approximated as

    Δa˜S1+BYμmμmB4π2m2˜S1sbRRe[y˜S1μBR(y˜S1μbL)]f˜S1μbLR(m2Bm2˜S1).

    (17)

    The input parameters are chosen as mμ=105.66 MeV, mb=4.2 GeV, mt=172.5 GeV, GF=1.1664×105 GeV2, mW=80.377 GeV, mZ=91.1876 GeV, and mh=125.25GeV [43]. The v,g,θW are defined by GF=1/(2v2),g=2mW/v, cosθWcW=mW/mZ. There are also new parameters mB, mLQ, θbR, and the LQ Yukawa couplings yLQμqL,R. The VLQ mass can be constrained from the direct search, which is required to be above 1.5 TeV [4447]. The mixing angle is mainly bounded by the electro-weak precision observables (EWPOs). The VLQ contributions to T parameter are suppressed by the factor (sbR)4 or m2b(sbR)2/m2B [48, 49], which leads to a less constrained θbR. The weak isospin third component of BR is positive; thus, the mixing with the bottom quark enhances the right-handed Zbb coupling. As a result, the AbFB deviation [50, 51] can be compensated, which leads to looser constraints on θbR. Conservatively, we can choose the mixing angle sbR to be smaller than 0.1 [42]. The LQ mass can also be constrained from the direct search, which is required to be above 1.7 TeV assuming Br(LQbμ)=1 [52, 53].

    We can choose benchmark points of mB,mLQ,sbR to constrain the LQ Yukawa couplings. Here, we consider two scenarios: mLQ>mB and mLQ<mB. For the scenario of mLQ>mB, we adopt mass parameters of mB= 1.5 TeV and mLQ=2 TeV. For the scenario of mLQ<mB, we adopt mass parameters of mB= 2.5 TeV and mLQ=2 TeV. In Table 3, we give the approximate numerical expressions of Δaμ in the ˜R2/˜S1+(B,Y)L,R models. We also show the allowed ranges for sbR=0.1 and sbR=0.05. Of course, these behaviours can be understood from Eqs. (15) and (17). In the ˜R2+(B,Y)L,R model, f˜R2μbLR(x) vanishes when mB=m˜R2, which causes the |Re[y˜R2μBL(y˜R2μbR)]| to be under the stress of perturbative unitarity. If there is a large hierarchy between mB and m˜R2, the allowed |Re[y˜R2μBL(y˜R2μbR)]| can be smaller. Additionally, Re[y˜R2μBL(y˜R2μbR)] should be positive (negative) when mB<m˜R2 (mB>m˜R2). In the ˜S1+(B,Y)L,R model, f˜S1μbLR(x) is always negative, which requires Re[y˜S1μBR(y˜S1μbL)]<0.

    Table 3

    Table 3.  In the third column, we show the leading order numerical expressions of the Δaμ. In the fifth and sixth columns, we show the ranges allowed by (g2)μ at 1σ and 2σ confidence levels (CLs).
    Model(mB,mLQ)/TeVΔaμ×107sbRRe[y˜R2μBL(y˜R2μbR)] or Re[y˜S1μBR(y˜S1μbL)]
    1σ2σ
    ˜R2+(B,Y)L,R(1.5,2)0.35sbRRe[y˜R2μBL(y˜R2μbR)]0.1(0.55,0.88)(0.38,1.05)
    0.05(1.1,1.77)(0.76,2.11)
    (2.5,2)0.224sbRRe[y˜R2μBL(y˜R2μbR)]0.1(1.38,0.86)(1.65,0.59)
    0.05(2.76,1.71)(3.29,1.19)
    ˜S1+(B,Y)L,R(1.5,2)6.88sbRRe[y˜S1μBR(y˜S1μbL)]0.1(0.045,0.028)(0.054,0.019)
    0.05(0.09,0.056)(0.11,0.039)
    (2.5,2)6.37sbRRe[y˜S1μBR(y˜S1μbL)]0.1(0.049,0.03)(0.058,0.021)
    0.05(0.097,0.06)(0.12,0.042)
    DownLoad: CSV
    Show Table

    We can also choose benchmark points of sbR and LQ Yukawa couplings to constrain the mB and mLQ. Fig. 1 presents the (g2)μ allowed regions in the plane of mBmLQ. As shown, mB<m˜R2 and mB>m˜R2 are favored in the left and middle plots, respectively. This can be understood from the asymptotic behaviours f˜R2μbLR(x)log(x)/2>0 for x0 and f˜R2μbLR(x)1/(4x)<0 for x. To produce a positive Δaμ, mB<m˜R2 and mB>m˜R2 are favored for Re[y˜R2μBL(y˜R2μbR)]>0 and Re[y˜R2μBL(y˜R2μbR)]<0, respectively. The f˜S1μbLR(x) has asymptotic behaviours f˜S1μbLR(x)log(x)/2<0 for x0 and f˜S1μbLR(x)5/(4x)<0 for x. To produce a positive Δaμ, Re[y˜S1μBR(y˜S1μbL)]<0 is favored. Furthermore, the allowed regions in the plane of mBmLQ are sensitive to the choice of yLQμqL,R. Generally, a larger |yLQμqL,R| corresponds to a larger mB and mLQ.

    Figure 1

    Figure 1.  (color online) (g2)μ allowed regions at 1σ (green) and 2σ (yellow) CLs with sbR=0.1. The parameters are chosen as y˜R2μBL=y˜R2μbR=0.8 in the ˜R2+(B,Y) model (left), y˜R2μBL=y˜R2μbR=0.8 in the ˜R2+(B,Y) model (middle), and y˜S1μbL=y˜S1μBR=0.2 in the ˜S1+(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 of mLQ>mB, there are new LQ decay channels. When searching for the LQ ˜R2/32, we propose the μjbZ and μjbh signatures. When searching for the LQ ˜R1/32, we propose the μjbW signatures. When searching for the LQ ˜S1, we propose the μjbZ, μjbh, ETjbW signatures. For the scenario of m_{ \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.

    Table 4

    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.
    ModelScenarioLQ decayVLQ decaynew 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
    DownLoad: CSV
    Show Table

    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 on s_R^b . For s_R^b=0.1 and m_B=2.5 ~\mathrm{TeV}, the new VLQ decay channels are less significant. For smaller s_R^b , the new VLQ decay channels can play an important role.

    Figure 2

    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} at 2\sigma CL. For the LQ decay, we choose m_B=1.5 ~\mathrm{TeV} and m_{ \mathrm{LQ}}=2 ~\mathrm{TeV}. For the VLQ decay, we choose m_B=2.5 ~\mathrm{TeV} and m_{ \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 [5658], 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 [5961], 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 the B\rightarrow K\mu^+\mu^- channel. Here, we will not study this detailed phenomenology.

    In this study, we investigate the scalar LQ and VLQ extended models to explain the (g-2)_{\mu} anomaly. Then, we find two new models \tilde{R}_2/\tilde{S}_1+(B,Y)_{L,R} , which can lead to the B quark chiral enhancements because of the bottom and bottom partner mixing. In the numerical analysis, we consider two scenarios: m_{ \mathrm{LQ}}>m_B and m_{ \mathrm{LQ}}<m_B . After considering the experimental constraints, we choose relative light masses, which are adopted to be (m_B,m_{ \mathrm{LQ}})=(1.5 ~\mathrm{TeV},2 ~\mathrm{TeV}) for the first scenario and (m_B,m_{ \mathrm{LQ}})=(2.5 ~\mathrm{TeV},2~ \mathrm{TeV}) for the second scenario. In the \tilde{R}_2+(B,Y)_{L,R} model, the | \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]| is bounded to be \mathcal{O}(1) , because f_{LR}^{\tilde{R}_2\mu b}(x) vanishes accidentally as m_B= m_{\tilde{R}_2} . Meanwhile, we can expect smaller | \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]| for largely splitted m_{\tilde{R}_2} and m_B . In the \tilde{S}_1+(B,Y)_{L,R} model, the \mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast] is bounded to the range (-0.06, -0.02) at a 2\sigma CL if s_R^b=0.1 .

    Under the constraints from (g-2)_{\mu} , we propose new LQ and VLQ search channels. In the scenario of m_{ \mathrm{LQ}}>m_B , there are new LQ decay channels: \tilde{R}_2^{2/3}\rightarrow \mu^+B , \tilde{R}_2^{-1/3}\rightarrow \mu^+Y , and \tilde{S}_1\rightarrow \mu^+\bar{B},\nu_L\bar{Y} . For larger y_L^{\tilde{R}_2\mu B} and y_R^{\tilde{S}_1\mu B} , it is important to take into account these decay channels. In the scenario of m_{ \mathrm{LQ}}<m_B , there are new VLQ decay channels: B\rightarrow \mu^-\tilde{R}_2^{2/3},\mu^+(\tilde{S}_1)^{\ast} and Y\rightarrow \mu^-\tilde{R}_2^{-1/3},~ \nu_L(\tilde{S}_1)^{\ast}. For s_R^b=0.1 , these channels are negligible compared with the traditional B\rightarrow bZ,~bh and Y\rightarrow bW^- channels. For smaller s_R^b , these new VLQ decay channels can also become important.

    Note added: In a prevoius study [64], the authors examined the model with \tilde{R}_2 , S_3 , and (B,Y)_{L,R} . In this work, they did not consider the bottom and B quark mixing, and the chiral enhancements were produced through the \tilde{R}_2 and S_3 mixing. In [65], the authors explained the (g-2)_{\mu} and B physics anomalies in the S_1+(B,Y)_{L,R} model.

    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 have m_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 the Y\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 the Y\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 the Y\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}

    1The terminology VLQ should not be confused with the vector leptoquark in some bibliographies.

    2The \begin{document}$ \tilde{R}_2^{-1/3}\rightarrow \nu_LB $\end{document} and \begin{document}$ B\rightarrow \nu_L\tilde{R}_2^{-1/3} $\end{document} decay channels are suppressed by the factor \begin{document}$ (s_R^b)^2 $\end{document}. The \begin{document}$ B\rightarrow tW^- $\end{document} decay channel is suppressed by the factor \begin{document}$ m_b^2/m_B^2 $\end{document}.

    [1] G. W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D 73, 072003 (2006), arXiv:hep-ex/0602035 doi: 10.1103/PhysRevD.73.072003
    [2] B. Abi et al. (Muon g-2 Collaboration), Phys. Rev. Lett. 126, 141801 (2021), arXiv:2104.03281[hep-ex doi: 10.1103/PhysRevLett.126.141801
    [3] T. Aoyama, M. Hayakawa, T. Kinoshita et al., Phys. Rev. Lett. 109, 111808 (2012), arXiv:1205.5370[hep-ph doi: 10.1103/PhysRevLett.109.111808
    [4] T. Aoyama, T. Kinoshita, and M. Nio, Atoms 7, 28 (2019) doi: 10.3390/atoms7010028
    [5] A. Czarnecki, W. J. Marciano, and A. Vainshtein, Phys. Rev. D67, 073006 (2003), [Erratum: Phys. Rev. D 73, 119901 (2006)], arXiv: hep-ph/0212229
    [6] C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim, Phys. Rev. D 88, 053005 (2013), arXiv:1306.5546[hep-ph doi: 10.1103/PhysRevD.88.053005
    [7] M. Davier, A. Hoecker, B. Malaescu et al., Eur. Phys. J. C 77, 827 (2017), arXiv:1706.09436[hep-ph doi: 10.1140/epjc/s10052-017-5161-6
    [8] A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D 97, 114025 (2018), arXiv:1802.02995[hep-ph doi: 10.1103/PhysRevD.97.114025
    [9] G. Colangelo, M. Hoferichter, and P. Stoffer, JHEP 02, 006 (2019), arXiv:1810.00007[hep-ph doi: 10.1007/JHEP02%282019%29006
    [10] M. Hoferichter, B.-L. Hoid, and B. Kubis, JHEP 08, 137 (2019), arXiv:1907.01556[hep-ph doi: 10.1007/JHEP08(2019)137
    [11] M. Davier, A. Hoecker, B. Malaescu et al., Eur. Phys. J. C 80, 241 (2020), [Erratum: Eur. Phys. J. C 80, 410 (2020)], arXiv: 1908.00921[hep-ph]
    [12] A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D 101, 014029 (2020), arXiv:1911.00367[hep-ph doi: 10.1103/PhysRevD.101.014029
    [13] A. Kurz, T. Liu, P. Marquard et al., Phys. Lett. B 734, 144 (2014), arXiv:1403.6400[hep-ph doi: 10.1016/j.physletb.2014.05.043
    [14] K. Melnikov and A. Vainshtein, Phys. Rev. D 70, 113006 (2004), arXiv:hep-ph/0312226 doi: 10.1103/PhysRevD.70.113006
    [15] P. Masjuan and P. Sanchez-Puertas, Phys. Rev. D 95, 054026 (2017), arXiv:1701.05829[hep-ph doi: 10.1103/PhysRevD.95.054026
    [16] G. Colangelo, M. Hoferichter, M. Procura et al., JHEP 04, 161 (2017), arXiv:1702.07347[hep-ph doi: 10.1007/JHEP04(2017)161
    [17] M. Hoferichter, B.-L. Hoid, B. Kubis et al., JHEP 10, 141 (2018), arXiv:1808.04823[hep-ph doi: 10.1007/JHEP10(2018)141
    [18] A. Gérardin, H. B. Meyer, and A. Nyffeler, Phys. Rev. D 100, 034520 (2019), arXiv:1903.09471[hep-lat doi: 10.1103/PhysRevD.100.034520
    [19] J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, Phys. Lett. B 798, 134994 (2019), arXiv:1908.03331[hep-ph doi: 10.1016/j.physletb.2019.134994
    [20] G. Colangelo, F. Hagelstein, M. Hoferichter et al., JHEP 03, 101 (2020), arXiv:1910.13432[hep-ph doi: 10.1007/JHEP03(2020)101
    [21] T. Blum, N. Christ, M. Hayakawa et al., Phys. Rev. Lett. 124, 132002 (2020), arXiv:1911.08123[hep-lat doi: 10.1103/PhysRevLett.124.132002
    [22] G. Colangelo, M. Hoferichter, A. Nyffeler et al., Phys. Lett. B 735, 90 (2014), arXiv:1403.7512[hep-ph doi: 10.1016/j.physletb.2014.06.012
    [23] T. Aoyama et al., Phys. Rept. 887, 1 (2020), arXiv:2006.04822[hep-ph doi: 10.1016/j.physrep.2020.07.006
    [24] A. Czarnecki and W. J. Marciano, Phys. Rev. D 64, 013014 (2001), arXiv:hep-ph/0102122 doi: 10.1103/PhysRevD.64.013014
    [25] F. Jegerlehner and A. Nyffeler, Phys. Rept. 477, 1 (2009), arXiv:0902.3360[hep-ph doi: 10.1016/j.physrep.2009.04.003
    [26] A. Freitas, J. Lykken, S. Kell, and S. Westhoff, JHEP 05, 145 [Erratum: JHEP 09, 155 (2014)], arXiv: 1402.7065[hep-ph]
    [27] F. S. Queiroz and W. Shepherd, Phys. Rev. D 89, 095024 (2014), arXiv:1403.2309[hep-ph doi: 10.1103/PhysRevD.89.095024
    [28] M. Lindner, M. Platscher, and F. S. Queiroz, Phys. Rept. 731, 1 (2018), arXiv:1610.06587[hep-ph doi: 10.1016/j.physrep.2017.12.001
    [29] P. Athron, C. Balázs, D. H. Jacob et al., JHEP 09, 080 (2021), arXiv:2104.03691[hep-ph doi: 10.1007/JHEP09(2021)080
    [30] K. Kannike, M. Raidal, D. M. Straub, and A. Strumia, JHEP 02, 106, [Erratum: JHEP 10, 136 (2012)], arXiv: 1111.2551[hep-ph]
    [31] M. Frank and I. Saha, Phys. Rev. D 102, 115034 (2020), arXiv:2008.11909[hep-ph doi: 10.1103/PhysRevD.102.115034
    [32] R. Dermisek, K. Hermanek, and N. McGinnis, Phys. Rev. D 104, 055033 (2021), arXiv:2103.05645[hep-ph doi: 10.1103/PhysRevD.104.055033
    [33] A. Crivellin and M. Hoferichter, JHEP 07, 135 (2021), [Erratum: JHEP 10, 030 (2022)], arXiv: 2104.03202[hep-ph]
    [34] K.-m. Cheung, Phys. Rev. D 64, 033001 (2001), arXiv:hep-ph/0102238 doi: 10.1103/PhysRevD.64.033001
    [35] I. Doršner, S. Fajfer, A. Greljo et al., Phys. Rept. 641, 1 (2016), arXiv:1603.04993[hep-ph doi: 10.1016/j.physrep.2016.06.001
    [36] E. Coluccio Leskow, G. D’Ambrosio, A. Crivellin et al., Phys. Rev. D 95, 055018 (2017), arXiv:1612.06858[hep-ph doi: 10.1103/PhysRevD.95.055018
    [37] I. Doršner, S. Fajfer, and O. Sumensari, JHEP 06, 089 (2020), arXiv:1910.03877[hep-ph doi: 10.1007/JHEP06(2020)089
    [38] A. Crivellin, D. Mueller, and F. Saturnino, Phys. Rev. Lett. 127, 021801 (2021), arXiv:2008.02643[hep-ph doi: 10.1103/PhysRevLett.127.021801
    [39] I. Doršner, S. Fajfer, and S. Saad, Phys. Rev. D 102, 075007 (2020), arXiv:2006.11624[hep-ph doi: 10.1103/PhysRevD.102.075007
    [40] D. Zhang, JHEP 07, 069 (2021), arXiv:2105.08670[hep-ph doi: 10.1007/JHEP07(2021)069
    [41] S.-P. He, Phys. Rev. D 105, 035017 (2022), [Erratum: Phys.Rev.D 106, 039901 (2022)], arXiv: 2112.13490[hep-ph]
    [42] J. A. Aguilar-Saavedra, R. Benbrik, S. Heinemeyer, and M. Pérez-Victoria, Phys. Rev. D 88, 094010 (2013), arXiv:1306.0572[hep-ph doi: 10.1103/PhysRevD.88.094010
    [43] R. L. Workman (Particle Data Group), PTEP 2022, 083C01 (2022) doi: 10.1093/ptep/ptac097
    [44] A. M. Sirunyan et al. (CMS Collaboration), Phys. Rev. D 102, 112004 (2020), arXiv:2008.09835[hep-ex doi: 10.1103/PhysRevD.102.112004
    [45] A. M. Sirunyan et al. (CMS), Phys. Lett. B 772, 634 (2017), arXiv:1701.08328[hep-ex doi: 10.1016/j.physletb.2017.07.022
    [46] M. Aaboud et al. (ATLAS), Phys. Rev. Lett. 121, 211801 (2018), arXiv:1808.02343[hep-ex doi: 10.1103/PhysRevLett.121.211801
    [47] M. Aaboud et al. (ATLAS), JHEP 05, 164 (2019), arXiv:1812.07343[hep-ex doi: 10.1007/JHEP10(2017)141
    [48] C.-Y. Chen, S. Dawson, and E. Furlan, Phys. Rev. D 96, 015006 (2017), arXiv:1703.06134[hep-ph doi: 10.1103/PhysRevD.96.015006
    [49] J. Cao, L. Meng, L. Shang et al., Phys. Rev. D 106, 055042 (2022), arXiv:2204.09477[hep-ph doi: 10.1103/PhysRevD.106.055042
    [50] S. Schael et al. (ALEPH, DELPHI, L3, OPAL, SLD Collaboration, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavour Group), Phys. Rept. 427, 257 (2006), arXiv:hep-ex/0509008 doi: 10.1088/0305-4470/39/5/L02
    [51] M. Baak, J. C′uth, J. Haller et al. (Gfitter Group), Eur. Phys. J. C 74, 3046 (2014), arXiv:1407.3792[hep-ph doi: 10.1140/epjc/s10052-014-3046-5
    [52] A. M. Sirunyan et al. (CMS), Phys. Rev. D 99, 032014 (2019), arXiv:1808.05082[hep-ex doi: 10.1103/PhysRevD.99.032014
    [53] G. Aad et al. (ATLAS), JHEP 10, 112 (2020), arXiv:2006.05872[hep-ex doi: 10.1007/JHEP10(2020)112
    [54] A. Alloul, N. D. Christensen, C. Degrande et al., Comput. Phys. Commun. 185, 2250 (2014), arXiv:1310.1921[hep-ph doi: 10.1016/j.cpc.2014.04.012
    [55] J. Alwall, R. Frederix, S. Frixione et al., JHEP 07, 079 (2014), arXiv:1405.0301[hep-ph doi: 10.1007/JHEP07(2014)079
    [56] J. Blumlein, E. Boos, and A. Kryukov, Z. Phys. C 76, 137 (1997), arXiv:hep-ph/9610408 doi: 10.1007/s002880050538
    [57] B. Diaz, M. Schmaltz, and Y.-M. Zhong, JHEP 10, 097 (2017), arXiv:1706.05033[hep-ph doi: 10.1007/JHEP10(2017)097
    [58] I. Doršner and A. Greljo, JHEP 05, 126 (2018), arXiv:1801.07641[hep-ph doi: 10.1007/JHEP05(2018)126
    [59] J. A. Aguilar-Saavedra, JHEP 11, 030 (2009), arXiv:0907.3155[hep-ph doi: 10.1088/1126-6708/2009/11/030
    [60] O. Matsedonskyi, G. Panico, and A. Wulzer, JHEP 12, 097 (2014), arXiv:1409.0100[hep-ph doi: 10.1007/JHEP12(2014)097
    [61] B. Fuks and H.-S. Shao, Eur. Phys. J. C 77, 135 (2017), arXiv:1610.04622[hep-ph doi: 10.1140/epjc/s10052-017-4686-z
    [62] L. Buonocore, U. Haisch, P. Nason et al., Phys. Rev. Lett. 125, 231804 (2020), arXiv:2005.06475[hep-ph doi: 10.1103/PhysRevLett.125.231804
    [63] L. Buonocore, A. Greljo, P. Krack et al., JHEP 11, 129 (2022), arXiv:2209.02599[hep-ph doi: 10.1007/JHEP11(2022)129
    [64] T. A. Chowdhury and S. Saad, Phys. Rev. D 106, 055017 (2022), arXiv:2205.03917[hep-ph doi: 10.1103/PhysRevD.106.055017
    [65] I. Bigaran, J. Gargalionis, and R. R. Volkas, JHEP 10, 106 (2019), arXiv:1906.01870[hep-ph doi: 10.1007/JHEP10(2019)106
  • [1] G. W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D 73, 072003 (2006), arXiv:hep-ex/0602035 doi: 10.1103/PhysRevD.73.072003
    [2] B. Abi et al. (Muon g-2 Collaboration), Phys. Rev. Lett. 126, 141801 (2021), arXiv:2104.03281[hep-ex doi: 10.1103/PhysRevLett.126.141801
    [3] T. Aoyama, M. Hayakawa, T. Kinoshita et al., Phys. Rev. Lett. 109, 111808 (2012), arXiv:1205.5370[hep-ph doi: 10.1103/PhysRevLett.109.111808
    [4] T. Aoyama, T. Kinoshita, and M. Nio, Atoms 7, 28 (2019) doi: 10.3390/atoms7010028
    [5] A. Czarnecki, W. J. Marciano, and A. Vainshtein, Phys. Rev. D67, 073006 (2003), [Erratum: Phys. Rev. D 73, 119901 (2006)], arXiv: hep-ph/0212229
    [6] C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim, Phys. Rev. D 88, 053005 (2013), arXiv:1306.5546[hep-ph doi: 10.1103/PhysRevD.88.053005
    [7] M. Davier, A. Hoecker, B. Malaescu et al., Eur. Phys. J. C 77, 827 (2017), arXiv:1706.09436[hep-ph doi: 10.1140/epjc/s10052-017-5161-6
    [8] A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D 97, 114025 (2018), arXiv:1802.02995[hep-ph doi: 10.1103/PhysRevD.97.114025
    [9] G. Colangelo, M. Hoferichter, and P. Stoffer, JHEP 02, 006 (2019), arXiv:1810.00007[hep-ph doi: 10.1007/JHEP02%282019%29006
    [10] M. Hoferichter, B.-L. Hoid, and B. Kubis, JHEP 08, 137 (2019), arXiv:1907.01556[hep-ph doi: 10.1007/JHEP08(2019)137
    [11] M. Davier, A. Hoecker, B. Malaescu et al., Eur. Phys. J. C 80, 241 (2020), [Erratum: Eur. Phys. J. C 80, 410 (2020)], arXiv: 1908.00921[hep-ph]
    [12] A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D 101, 014029 (2020), arXiv:1911.00367[hep-ph doi: 10.1103/PhysRevD.101.014029
    [13] A. Kurz, T. Liu, P. Marquard et al., Phys. Lett. B 734, 144 (2014), arXiv:1403.6400[hep-ph doi: 10.1016/j.physletb.2014.05.043
    [14] K. Melnikov and A. Vainshtein, Phys. Rev. D 70, 113006 (2004), arXiv:hep-ph/0312226 doi: 10.1103/PhysRevD.70.113006
    [15] P. Masjuan and P. Sanchez-Puertas, Phys. Rev. D 95, 054026 (2017), arXiv:1701.05829[hep-ph doi: 10.1103/PhysRevD.95.054026
    [16] G. Colangelo, M. Hoferichter, M. Procura et al., JHEP 04, 161 (2017), arXiv:1702.07347[hep-ph doi: 10.1007/JHEP04(2017)161
    [17] M. Hoferichter, B.-L. Hoid, B. Kubis et al., JHEP 10, 141 (2018), arXiv:1808.04823[hep-ph doi: 10.1007/JHEP10(2018)141
    [18] A. Gérardin, H. B. Meyer, and A. Nyffeler, Phys. Rev. D 100, 034520 (2019), arXiv:1903.09471[hep-lat doi: 10.1103/PhysRevD.100.034520
    [19] J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, Phys. Lett. B 798, 134994 (2019), arXiv:1908.03331[hep-ph doi: 10.1016/j.physletb.2019.134994
    [20] G. Colangelo, F. Hagelstein, M. Hoferichter et al., JHEP 03, 101 (2020), arXiv:1910.13432[hep-ph doi: 10.1007/JHEP03(2020)101
    [21] T. Blum, N. Christ, M. Hayakawa et al., Phys. Rev. Lett. 124, 132002 (2020), arXiv:1911.08123[hep-lat doi: 10.1103/PhysRevLett.124.132002
    [22] G. Colangelo, M. Hoferichter, A. Nyffeler et al., Phys. Lett. B 735, 90 (2014), arXiv:1403.7512[hep-ph doi: 10.1016/j.physletb.2014.06.012
    [23] T. Aoyama et al., Phys. Rept. 887, 1 (2020), arXiv:2006.04822[hep-ph doi: 10.1016/j.physrep.2020.07.006
    [24] A. Czarnecki and W. J. Marciano, Phys. Rev. D 64, 013014 (2001), arXiv:hep-ph/0102122 doi: 10.1103/PhysRevD.64.013014
    [25] F. Jegerlehner and A. Nyffeler, Phys. Rept. 477, 1 (2009), arXiv:0902.3360[hep-ph doi: 10.1016/j.physrep.2009.04.003
    [26] A. Freitas, J. Lykken, S. Kell, and S. Westhoff, JHEP 05, 145 [Erratum: JHEP 09, 155 (2014)], arXiv: 1402.7065[hep-ph]
    [27] F. S. Queiroz and W. Shepherd, Phys. Rev. D 89, 095024 (2014), arXiv:1403.2309[hep-ph doi: 10.1103/PhysRevD.89.095024
    [28] M. Lindner, M. Platscher, and F. S. Queiroz, Phys. Rept. 731, 1 (2018), arXiv:1610.06587[hep-ph doi: 10.1016/j.physrep.2017.12.001
    [29] P. Athron, C. Balázs, D. H. Jacob et al., JHEP 09, 080 (2021), arXiv:2104.03691[hep-ph doi: 10.1007/JHEP09(2021)080
    [30] K. Kannike, M. Raidal, D. M. Straub, and A. Strumia, JHEP 02, 106, [Erratum: JHEP 10, 136 (2012)], arXiv: 1111.2551[hep-ph]
    [31] M. Frank and I. Saha, Phys. Rev. D 102, 115034 (2020), arXiv:2008.11909[hep-ph doi: 10.1103/PhysRevD.102.115034
    [32] R. Dermisek, K. Hermanek, and N. McGinnis, Phys. Rev. D 104, 055033 (2021), arXiv:2103.05645[hep-ph doi: 10.1103/PhysRevD.104.055033
    [33] A. Crivellin and M. Hoferichter, JHEP 07, 135 (2021), [Erratum: JHEP 10, 030 (2022)], arXiv: 2104.03202[hep-ph]
    [34] K.-m. Cheung, Phys. Rev. D 64, 033001 (2001), arXiv:hep-ph/0102238 doi: 10.1103/PhysRevD.64.033001
    [35] I. Doršner, S. Fajfer, A. Greljo et al., Phys. Rept. 641, 1 (2016), arXiv:1603.04993[hep-ph doi: 10.1016/j.physrep.2016.06.001
    [36] E. Coluccio Leskow, G. D’Ambrosio, A. Crivellin et al., Phys. Rev. D 95, 055018 (2017), arXiv:1612.06858[hep-ph doi: 10.1103/PhysRevD.95.055018
    [37] I. Doršner, S. Fajfer, and O. Sumensari, JHEP 06, 089 (2020), arXiv:1910.03877[hep-ph doi: 10.1007/JHEP06(2020)089
    [38] A. Crivellin, D. Mueller, and F. Saturnino, Phys. Rev. Lett. 127, 021801 (2021), arXiv:2008.02643[hep-ph doi: 10.1103/PhysRevLett.127.021801
    [39] I. Doršner, S. Fajfer, and S. Saad, Phys. Rev. D 102, 075007 (2020), arXiv:2006.11624[hep-ph doi: 10.1103/PhysRevD.102.075007
    [40] D. Zhang, JHEP 07, 069 (2021), arXiv:2105.08670[hep-ph doi: 10.1007/JHEP07(2021)069
    [41] S.-P. He, Phys. Rev. D 105, 035017 (2022), [Erratum: Phys.Rev.D 106, 039901 (2022)], arXiv: 2112.13490[hep-ph]
    [42] J. A. Aguilar-Saavedra, R. Benbrik, S. Heinemeyer, and M. Pérez-Victoria, Phys. Rev. D 88, 094010 (2013), arXiv:1306.0572[hep-ph doi: 10.1103/PhysRevD.88.094010
    [43] R. L. Workman (Particle Data Group), PTEP 2022, 083C01 (2022) doi: 10.1093/ptep/ptac097
    [44] A. M. Sirunyan et al. (CMS Collaboration), Phys. Rev. D 102, 112004 (2020), arXiv:2008.09835[hep-ex doi: 10.1103/PhysRevD.102.112004
    [45] A. M. Sirunyan et al. (CMS), Phys. Lett. B 772, 634 (2017), arXiv:1701.08328[hep-ex doi: 10.1016/j.physletb.2017.07.022
    [46] M. Aaboud et al. (ATLAS), Phys. Rev. Lett. 121, 211801 (2018), arXiv:1808.02343[hep-ex doi: 10.1103/PhysRevLett.121.211801
    [47] M. Aaboud et al. (ATLAS), JHEP 05, 164 (2019), arXiv:1812.07343[hep-ex doi: 10.1007/JHEP10(2017)141
    [48] C.-Y. Chen, S. Dawson, and E. Furlan, Phys. Rev. D 96, 015006 (2017), arXiv:1703.06134[hep-ph doi: 10.1103/PhysRevD.96.015006
    [49] J. Cao, L. Meng, L. Shang et al., Phys. Rev. D 106, 055042 (2022), arXiv:2204.09477[hep-ph doi: 10.1103/PhysRevD.106.055042
    [50] S. Schael et al. (ALEPH, DELPHI, L3, OPAL, SLD Collaboration, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavour Group), Phys. Rept. 427, 257 (2006), arXiv:hep-ex/0509008 doi: 10.1088/0305-4470/39/5/L02
    [51] M. Baak, J. C′uth, J. Haller et al. (Gfitter Group), Eur. Phys. J. C 74, 3046 (2014), arXiv:1407.3792[hep-ph doi: 10.1140/epjc/s10052-014-3046-5
    [52] A. M. Sirunyan et al. (CMS), Phys. Rev. D 99, 032014 (2019), arXiv:1808.05082[hep-ex doi: 10.1103/PhysRevD.99.032014
    [53] G. Aad et al. (ATLAS), JHEP 10, 112 (2020), arXiv:2006.05872[hep-ex doi: 10.1007/JHEP10(2020)112
    [54] A. Alloul, N. D. Christensen, C. Degrande et al., Comput. Phys. Commun. 185, 2250 (2014), arXiv:1310.1921[hep-ph doi: 10.1016/j.cpc.2014.04.012
    [55] J. Alwall, R. Frederix, S. Frixione et al., JHEP 07, 079 (2014), arXiv:1405.0301[hep-ph doi: 10.1007/JHEP07(2014)079
    [56] J. Blumlein, E. Boos, and A. Kryukov, Z. Phys. C 76, 137 (1997), arXiv:hep-ph/9610408 doi: 10.1007/s002880050538
    [57] B. Diaz, M. Schmaltz, and Y.-M. Zhong, JHEP 10, 097 (2017), arXiv:1706.05033[hep-ph doi: 10.1007/JHEP10(2017)097
    [58] I. Doršner and A. Greljo, JHEP 05, 126 (2018), arXiv:1801.07641[hep-ph doi: 10.1007/JHEP05(2018)126
    [59] J. A. Aguilar-Saavedra, JHEP 11, 030 (2009), arXiv:0907.3155[hep-ph doi: 10.1088/1126-6708/2009/11/030
    [60] O. Matsedonskyi, G. Panico, and A. Wulzer, JHEP 12, 097 (2014), arXiv:1409.0100[hep-ph doi: 10.1007/JHEP12(2014)097
    [61] B. Fuks and H.-S. Shao, Eur. Phys. J. C 77, 135 (2017), arXiv:1610.04622[hep-ph doi: 10.1140/epjc/s10052-017-4686-z
    [62] L. Buonocore, U. Haisch, P. Nason et al., Phys. Rev. Lett. 125, 231804 (2020), arXiv:2005.06475[hep-ph doi: 10.1103/PhysRevLett.125.231804
    [63] L. Buonocore, A. Greljo, P. Krack et al., JHEP 11, 129 (2022), arXiv:2209.02599[hep-ph doi: 10.1007/JHEP11(2022)129
    [64] T. A. Chowdhury and S. Saad, Phys. Rev. D 106, 055017 (2022), arXiv:2205.03917[hep-ph doi: 10.1103/PhysRevD.106.055017
    [65] I. Bigaran, J. Gargalionis, and R. R. Volkas, JHEP 10, 106 (2019), arXiv:1906.01870[hep-ph doi: 10.1007/JHEP10(2019)106
  • 加载中

Figures(2) / Tables(4)

Get Citation
Shi-Ping He. Scalar leptoquark and vector-like quark extended models as the explanation of the muon g-2 anomaly: bottom partner chiral enhancement case[J]. Chinese Physics C. doi: 10.1088/1674-1137/accc1d
Shi-Ping He. Scalar leptoquark and vector-like quark extended models as the explanation of the muon g-2 anomaly: bottom partner chiral enhancement case[J]. Chinese Physics C.  doi: 10.1088/1674-1137/accc1d shu
Milestone
Received: 2023-02-14
Article Metric

Article Views(6172)
PDF Downloads(49)
Cited by(0)
Policy on re-use
To reuse of Open Access content published by CPC, for content published under the terms of the Creative Commons Attribution 3.0 license (“CC CY”), the users don’t need to request permission to copy, distribute and display the final published version of the article and to create derivative works, subject to appropriate attribution.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

Scalar leptoquark and vector-like quark extended models as the explanation of the muon g–2 anomaly: bottom partner chiral enhancement case

  • Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea

Abstract: Leptoquark (LQ) models are well motivated solutions to the (g-2)_{\mu} anomaly. In the minimal LQ models, only specific representations can lead to chiral enhancements. For the scalar LQs, R_2 and S_1 can lead to the top quark chiral enhancement. For the vector LQs, V_2 and U_1 can lead to the bottom quark chiral enhancement. When we consider the LQ and vector-like quark (VLQ) simultaneously, there can be more scenarios. In our previous study, we considered the scalar LQ and VLQ extended models with up-type quark chiral enhancement. Here, we study the scalar LQ and VLQ extended models with down-type quark chiral enhancement. We find two new models with B quark chiral enhancements, which originate from the bottom and bottom partner mixing. Then, we propose new LQ and VLQ search channels under the constraints of (g-2)_{\mu} .

    HTML

    I.   INTRODUCTION
    • The (g-2)_{\mu} anomaly is a longstanding puzzle in the standard model (SM) of elementary particle physics. It was first announced by the BNL E821 experiment [1]. Last year, the FNAL muon g-2 experiment revealed increased deviation from the SM prediction [2]. When combining the BNL and FNAL data, the averaged result is a_{\mu}^{ \mathrm{Exp}}=116592061(41)\times10^{-11} . Compared to the SM prediction a_{\mu}^{ \mathrm{SM}}=116591810(43)\times10^{-11} [323], the deviation is \Delta a_{\mu}\equiv a_{\mu}^{ \mathrm{Exp}}-a_{\mu}^{ \mathrm{SM}}=(251\pm59)\times10^{-11} , which shows a 4.2\sigma discrepancy. Many new physics models were proposed to explain the anomaly [2429].

      For the mediators with mass above the TeV level, the chiral enhancements are required, which can appear when left-handed and right-handed muons couple to a heavy fermion simultaneously. In the new lepton extended models [3033], the chiral enhancements originate from the large lepton mass. The LQ models are an alternative choice [3440], in which the chiral enhancements originate from the large quark mass. For the minimal LQ models, there are scalar LQs R_2/S_1 with top quark chiral enhancement and vector LQs V_2/U_1 with bottom quark chiral enhancement. The LQ can connect the lepton sector and quark sector. On the other hand, the VLQ naturally occurs in many new physics models and is free of quantum anomaly. It can mix with SM quarks and provide new source of CP violation. Hence, the LQ and VLQ extended models can lead to interesting flavour physics in both the lepton sector and quark sector. In our previous study [41], we investigated the scalar LQ and VLQ 1 extended models with top and top partner chiral enhancements. In this study, we investigate the scalar LQ and VLQ extended models, which can produce the bottom partner chiral enhancements. This paper is complementary to our previous paper [41]. Moreover, the top partner and bottom partner lead to different collider signatures.

      In Sec. II, we introduce the models and show the related interactions. Then, we derive the new physics contributions to (g-2)_{\mu} and perform the numerical analysis in Sec. III. In Sec. IV, we discuss the possible collider phenomenology. Finally, we present a summary and conclusions in Sec. V.

    II.   MODEL SETUP
    • 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 representationlabelFS 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 models R_2+B_{L,R}/(B,Y)_{L,R} and S_1+B_{L,R}/(B,Y)_{L,R} are almost the same as those in the minimal R_2 and S_1 models. There are nine models R_2+T_{L,R}/(X,T)_{L,R}/(T,B)_{L,R}/(T,B,Y)_{L,R} and S_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 models R_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}.

      ModelChiral 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.

    • A.   VLQ Yukawa interactions with Higgs

    • 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. The Q_L^i and d_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 between d^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 and c_{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 and m_B represent the physical b and B quark masses, respectively. Additionally, the mass of the Y quark is m_Y=M_B= \sqrt{m_B^2(c_R^b)^2+m_b^2(s_R^b)^2}. Then, we can choose m_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.

    • B.   VLQ gauge interactions

    • Now, let us label the S U(2)_L and U_Y(1) gauge fields as W_{\mu}^a and B_{\mu} . Then, the electroweak covariant derivative D_{\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 which Y_q is the U_Y(1) charge of the quark field acted by D_{\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)
    • C.   VLQ Yukawa interactions with LQ

    • Let us denote the SM lepton fields as L_L^i and e_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 following F=0 and F=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)
    III.   CONTRIBUTIONS TO \boldsymbol{(g-2)_{\mu}}

      A.   Analytical results of contributions

    • 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 functions f^q(x) and f^S(x) . Then, we use the functions f_{LL}^{q,S}(x) and f_{LR}^{q,S}(x) to label the parts without and with chiral enhancements. Starting from the f_{LL}^{q,S}(x) and f_{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., the f_{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 factor m_{\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 of s_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)
    • B.   Numerical analysis

    • 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}, and m_h=125.25 \mathrm{GeV} [43]. The v,\;g,\;\theta_W are defined by G_F=1/(\sqrt{2}v^2), g=2m_W/v,~\cos\theta_W\equiv c_W=m_W/m_Z. There are also new parameters m_B , m_{ \mathrm{LQ}} , \theta_R^b , and the LQ Yukawa couplings y_{L,R}^{ \mathrm{LQ}\mu q} . The VLQ mass can be constrained from the direct search, which is required to be above 1.5~ \mathrm{TeV} [4447]. 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 or m_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 of B_R is positive; thus, the mixing with the bottom quark enhances the right-handed Zbb coupling. As a result, the A_{FB}^b deviation [50, 51] can be compensated, which leads to looser constraints on \theta_R^b . Conservatively, we can choose the mixing angle s_R^b to be smaller than 0.1 [42]. The LQ mass can also be constrained from the direct search, which is required to be above 1.7 ~\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 and m_{ \mathrm{LQ}}<m_B . For the scenario of m_{ \mathrm{LQ}}>m_B , we adopt mass parameters of m_B= 1.5 TeV and m_{ \mathrm{LQ}}=2 ~\mathrm{TeV}. For the scenario of m_{ \mathrm{LQ}}<m_B , we adopt mass parameters of m_B= 2.5 TeV and m_{ \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 for s_R^b=0.1 and s_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 when m_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 between m_B and m_{\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) when m_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} at 1\sigma and 2\sigma confidence levels (CLs).

      We can also choose benchmark points of s_R^b and LQ Yukawa couplings to constrain the m_B and m_{ \mathrm{LQ}} . Fig. 1 presents the (g-2)_{\mu} allowed regions in the plane of m_B-m_{ \mathrm{LQ}} . As shown, m_B<m_{\tilde{R}_2} and m_B>m_{\tilde{R}_2} are favored in the left and middle plots, respectively. This can be understood from the asymptotic behaviours f_{LR}^{\tilde{R}_2\mu b}(x)\sim -\log(x)/2 > 0 for x\rightarrow0 and f_{LR}^{\tilde{R}_2\mu b}(x) \sim-1/(4x) < 0 for x\rightarrow \infty . To produce a positive \Delta a_{\mu} , m_B<m_{\tilde{R}_2} and m_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. The f_{LR}^{\tilde{S}_1\mu b}(x) has asymptotic behaviours f_{LR}^{\tilde{S}_1\mu b}(x)\sim \log(x)/2 < 0 for x\rightarrow0 and f_{LR}^{\tilde{S}_1\mu b}(x)\sim-5/(4x) < 0 for x\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 of m_B-m_{ \mathrm{LQ}} are sensitive to the choice of y_{L,R}^{ \mathrm{LQ}\mu q} . Generally, a larger |y_{L,R}^{ \mathrm{LQ}\mu q}| corresponds to a larger m_B and m_{ \mathrm{LQ}} .

      Figure 1.  (color online) (g-2)_{\mu} allowed regions at 1\sigma (green) and 2\sigma (yellow) CLs with s_R^b=0.1 . The parameters are chosen as y_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), and y_L^{\tilde{S}_1\mu b}=-y_R^{\tilde{S}_1\mu B}=0.2 in the \tilde{S}_1+(B,Y) model (right).

    IV.   LQ AND VLQ PHENOMENOLOGY AT HADRON COLLIDERS
    • 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 of m_{ \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 of m_{ \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.

      ModelScenarioLQ decayVLQ decaynew 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 on s_R^b . For s_R^b=0.1 and m_B=2.5 ~\mathrm{TeV}, the new VLQ decay channels are less significant. For smaller s_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} at 2\sigma CL. For the LQ decay, we choose m_B=1.5 ~\mathrm{TeV} and m_{ \mathrm{LQ}}=2 ~\mathrm{TeV}. For the VLQ decay, we choose m_B=2.5 ~\mathrm{TeV} and m_{ \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 [5658], 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 [5961], 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 the B\rightarrow K\mu^+\mu^- channel. Here, we will not study this detailed phenomenology.

    V.   SUMMARY AND CONCLUSIONS
    • In this study, we investigate the scalar LQ and VLQ extended models to explain the (g-2)_{\mu} anomaly. Then, we find two new models \tilde{R}_2/\tilde{S}_1+(B,Y)_{L,R} , which can lead to the B quark chiral enhancements because of the bottom and bottom partner mixing. In the numerical analysis, we consider two scenarios: m_{ \mathrm{LQ}}>m_B and m_{ \mathrm{LQ}}<m_B . After considering the experimental constraints, we choose relative light masses, which are adopted to be (m_B,m_{ \mathrm{LQ}})=(1.5 ~\mathrm{TeV},2 ~\mathrm{TeV}) for the first scenario and (m_B,m_{ \mathrm{LQ}})=(2.5 ~\mathrm{TeV},2~ \mathrm{TeV}) for the second scenario. In the \tilde{R}_2+(B,Y)_{L,R} model, the | \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]| is bounded to be \mathcal{O}(1) , because f_{LR}^{\tilde{R}_2\mu b}(x) vanishes accidentally as m_B= m_{\tilde{R}_2} . Meanwhile, we can expect smaller | \mathrm{Re}[y_L^{\tilde{R}_2\mu B}(y_R^{\tilde{R}_2\mu b})^\ast]| for largely splitted m_{\tilde{R}_2} and m_B . In the \tilde{S}_1+(B,Y)_{L,R} model, the \mathrm{Re}[y_R^{\tilde{S}_1\mu B}(y_L^{\tilde{S}_1\mu b})^\ast] is bounded to the range (-0.06, -0.02) at a 2\sigma CL if s_R^b=0.1 .

      Under the constraints from (g-2)_{\mu} , we propose new LQ and VLQ search channels. In the scenario of m_{ \mathrm{LQ}}>m_B , there are new LQ decay channels: \tilde{R}_2^{2/3}\rightarrow \mu^+B , \tilde{R}_2^{-1/3}\rightarrow \mu^+Y , and \tilde{S}_1\rightarrow \mu^+\bar{B},\nu_L\bar{Y} . For larger y_L^{\tilde{R}_2\mu B} and y_R^{\tilde{S}_1\mu B} , it is important to take into account these decay channels. In the scenario of m_{ \mathrm{LQ}}<m_B , there are new VLQ decay channels: B\rightarrow \mu^-\tilde{R}_2^{2/3},\mu^+(\tilde{S}_1)^{\ast} and Y\rightarrow \mu^-\tilde{R}_2^{-1/3},~ \nu_L(\tilde{S}_1)^{\ast}. For s_R^b=0.1 , these channels are negligible compared with the traditional B\rightarrow bZ,~bh and Y\rightarrow bW^- channels. For smaller s_R^b , these new VLQ decay channels can also become important.

      Note added: In a prevoius study [64], the authors examined the model with \tilde{R}_2 , S_3 , and (B,Y)_{L,R} . In this work, they did not consider the bottom and B quark mixing, and the chiral enhancements were produced through the \tilde{R}_2 and S_3 mixing. In [65], the authors explained the (g-2)_{\mu} and B physics anomalies in the S_1+(B,Y)_{L,R} model.

    APPENDIX A: LQ DECAY WIDTH FORMULAE
    • 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}

    APPENDIX B: VLQ DECAY WIDTH FORMULAE
    • If m_B\sim \mathrm{TeV} and \theta_R^b\ll0.1 , we have m_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 the Y\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 the Y\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 the Y\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}

Reference (65)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return