-
Typically, there are six types of scalar LQs [35], which carry a conserved quantum number
F≡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.SU(3)C×SU(2)L×U(1)Y representationlabel F SU(3)C×SU(2)L×U(1)Y representationlabel (ˉ3,3,1/3) S3 −2 (3,1,2/3) TL,R (3,2,7/6) R2 0 (3,1,−1/3) BL,R (3,2,1/6) ˜R2 0 (3,2,7/6) (X,T)L,R (ˉ3,1,4/3) ˜S1 −2 (3,2,1/6) (T,B)L,R (3,2,−5/6) (B,Y)L,R (ˉ3,1,1/3) S1 −2 (3,3,2/3) (X,T,B)L,R (ˉ3,1,−2/3) ˉS1 −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 "
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 modelsR2+BL,R/(B,Y)L,R andS1+BL,R/(B,Y)L,R are almost the same as those in the minimalR2 andS1 models. There are nine modelsR2+TL,R/(X,T)L,R/(T,B)L,R/(T,B,Y)L,R andS1+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 modelsR2/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 .Model Chiral enhancement R2 mt/mμ S1 mt/mμ R2+BL,R/(B,Y)L,R mt/mμ S1+BL,R/(B,Y)L,R mt/mμ R2+TL,R/(X,T)L,R/(T,B)L,R/(T,B,Y)L,R mt/mμ,mT/mμ S1+TL,R/(X,T)L,R/(T,B)L,R/(X,T,B)L,R/(T,B,Y)L,R mt/mμ,mT/mμ R2+(X,T,B)L,R mt/mμ,mT/mμ,mb/mμ,mB/mμ S3+(X,T,B)L,R mt/mμ,mT/mμ,mb/mμ,mB/mμ ˜R2+(B,Y)L,R mb/mμ,mB/mμ ˜S1+(B,Y)L,R mb/mμ,mB/mμ 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¯QLidjRϕ and(¯BL,¯YL)djR˜ϕ and the mass term−MB(ˉBB+ˉYY) . Here, we define the SM Higgs doublet˜ϕ≡iσ2ϕ∗ withσa(a=1,2,3) to be the Pauli matrices. TheQiL anddiR (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 betweendi 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]→[cbLsbL−sbLcbL][bLBL],[bRBR]→[cbRsbR−sbRcbR][bRBR].
(1) Here,
sbL,R andcbL,R are abbreviations ofsinθbL,R andcosθbL,R , respectively. In fact,θbL can be correlated withθbR through the relationtanθbL=mbtanθbR/mB [42]. Here,mb andmB represent the physical b and B quark masses, respectively. Additionally, the mass of the Y quark ismY=MB= √m2B(cbR)2+m2b(sbR)2 . Then, we can choosemB andθbR as the new input parameters. After the transformations in Eq. (1), we obtain the following mass eigenstate Higgs Yukawa interactions:LYukawaH⊃−mbv(cbR)2hˉbb−mBv(sbR)2hˉBB−mbvsbRcbRh(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 andUY(1) gauge fields asWaμ andBμ . Then, the electroweak covariant derivativeDμ is defined as∂μ−igWaμσa/2−ig′YqBμ for a doublet and∂μ−ig′YqBμ for a singlet, in whichYq is theUY(1) charge of the quark field acted byDμ . 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 asL⊃g√2W+μ(¯tLγμbL+ˉBγμY)+h.c..
(3) The Z gauge interactions can be written as
L⊃gcW[(−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:
LgaugeBY⊃g√2W+μ[cbL¯tLγμbL+sbL¯tLγμBL+cbL¯BLγμYL−sbL¯bLγμYL+cbR¯BRγμYR−sbR¯bRγμYR]+h.c..
(5) We also have the mass eigenstate Z gauge interactions:
LgaugeBY⊃gcWZμ[(cbL)2−(sbL)22(¯BLγμBL−¯bLγμbL)−sbLcbL(¯bLγμBL+¯BLγμbL)+(sbR)22¯bRγμbR+(cbR)22¯BRγμBR−sbRcbR2(¯bRγμBR+¯BRγμbR)+s2W3(ˉbγμb+ˉBγμB)+(−12+4s2W3)ˉYγμY].
(6) -
Let us denote the SM lepton fields as
LiL andeiR . The˜R2 can be parametrized as[˜R2/32,˜R−1/32]T , where the superscript labels the electric charge. Then, the˜R2 and˜S1 can induce the followingF=0 andF=2 type gauge eigenstate LQ Yukawa interactions:L˜R2+(B,Y)L,R⊃xi¯eiR(˜R2)†(BLYL)+yij¯LiLϵ(˜R2)∗djR+h.c.,
(7) and
L˜S1+(B,Y)L,R⊃xij¯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,R⊃y˜R2μBLˉμω−B(˜R2/32)∗+y˜R2μbRˉμω+b(˜R2/32)∗+y˜R2μBLˉμω−Y(˜R−1/32)∗−y˜R2μbR¯νLω+b(˜R−1/32)∗+h.c.,
(9) and
L˜S1+(B,Y)L,R⊃y˜S1μbLˉμω−bC(˜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,R⊃ˉμ(−y˜R2μBLsbLω−+y˜R2μbRcbRω+)b(˜R2/32)∗+ˉμ(y˜R2μBLcbLω−+y˜R2μbRsbRω+)B(˜R2/32)∗+y˜R2μBLˉμω−Y(˜R−1/32)∗−y˜R2μbR¯νLω+(cbRb+sbRB)(˜R−1/32)∗+h.c.,
(11) and
L˜S1+(B,Y)L,R⊃ˉμ(y˜S1μbLcbRω−−y˜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(g−2)μ , which can be described by the functionsfq(x) andfS(x) . Then, we use the functionsfq,SLL(x) andfq,SLR(x) to label the parts without and with chiral enhancements. Starting from thefq,SLL(x) andfq,SLR(x) given in our previous paper [41], let us define the following integrals:f˜R2μYLL(x)≡4fqLL(x)−fSLL(x)=3+2x−7x2+2x3+2x(4−x)logx4(1−x)4,f˜R2μbLL(x)≡fqLL(x)+2fSLL(x)=x[5−4x−x2+(2+4x)logx]4(1−x)4,f˜R2μbLR(x)≡fqLR(x)+2fSLR(x)=−5−4x−x2+(2+4x)logx4(1−x)3,f˜S1μbLL(x)≡−fqLL(x)+4fSLL(x)=−2−7x+2x2+3x3+2x(1−4x)logx4(1−x)4,f˜S1μbLR(x)≡−fqLR(x)+4fSLR(x)=−1+4x−5x2−(2−8x)logx4(1−x)3.
(13) For the
˜R2+(B,Y)L,R model, there are b, B, and Y quark contributions to the(g−2)μ . The complete expression is calculated asΔa˜R2+BYμ=m2μ8π2{|y˜R2μBL|2m2˜R−1/32f˜R2μYLL(m2Ym2˜R−1/32)+|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˜R−1/32≡m˜R2 . Compared with the bottom partner chirally enhanced contribution (i.e., thef˜R2μbLR(m2Bm2˜R2/32) related term), the non-chirally enhanced parts are suppressed by the factormμ/(mBsbR)∼1/(104sbR) and the bottom quark chirally enhanced part is suppressed by the factor(mbsbL)/(mBsbR)∼(m2b/m2B) . For the interesting values ofsbR atO(0.01∼0.1) ,Δa˜R2+BYμ is dominated by the bottom partner chirally enhanced contribution. Then, the above expression can be approximated asΔa˜R2+BYμ≈mμmB4π2m2˜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(g−2)μ . The complete expression is calculated asΔa˜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+BYμ≈mμmB4π2m2˜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=1.1664×10−5 GeV−2 ,mW=80.377 GeV ,mZ=91.1876 GeV , andmh=125.25GeV [43]. Thev,g,θW are defined byGF=1/(√2v2),g=2mW/v, cosθW≡cW=mW/mZ . There are also new parametersmB ,mLQ ,θbR , and the LQ Yukawa couplingsyLQμqL,R . The VLQ mass can be constrained from the direct search, which is required to be above1.5 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(sbR)4 orm2b(sbR)2/m2B [48, 49], which leads to a less constrainedθbR . The weak isospin third component ofBR is positive; thus, the mixing with the bottom quark enhances the right-handedZbb coupling. As a result, theAbFB deviation [50, 51] can be compensated, which leads to looser constraints onθbR . Conservatively, we can choose the mixing anglesbR to be smaller than0.1 [42]. The LQ mass can also be constrained from the direct search, which is required to be above1.7 TeV assumingBr(LQ→bμ)=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 andmLQ<mB . For the scenario ofmLQ>mB , we adopt mass parameters ofmB= 1.5 TeV andmLQ=2 TeV . For the scenario ofmLQ<mB , we adopt mass parameters ofmB= 2.5 TeV andmLQ=2 TeV . In Table 3, we give the approximate numerical expressions ofΔaμ in the˜R2/˜S1+(B,Y)L,R models. We also show the allowed ranges forsbR=0.1 andsbR=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 whenmB=m˜R2 , which causes the|Re[y˜R2μBL(y˜R2μbR)∗]| to be under the stress of perturbative unitarity. If there is a large hierarchy betweenmB andm˜R2 , the allowed|Re[y˜R2μBL(y˜R2μbR)∗]| can be smaller. Additionally,Re[y˜R2μBL(y˜R2μbR)∗] should be positive (negative) whenmB<m˜R2 (mB>m˜R2 ). In the˜S1+(B,Y)L,R model,f˜S1μbLR(x) is always negative, which requiresRe[y˜S1μBR(y˜S1μbL)∗]<0 .Model (mB,mLQ)/TeV Δaμ×107 sbR Re[y˜R2μBL(y˜R2μbR)∗] orRe[y˜S1μBR(y˜S1μbL)∗] 1σ 2σ ˜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) 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(g−2)μ at1σ and2σ confidence levels (CLs).We can also choose benchmark points of
sbR and LQ Yukawa couplings to constrain themB andmLQ . Fig. 1 presents the(g−2)μ allowed regions in the plane ofmB−mLQ . As shown,mB<m˜R2 andmB>m˜R2 are favored in the left and middle plots, respectively. This can be understood from the asymptotic behavioursf˜R2μbLR(x)∼−log(x)/2>0 forx→0 andf˜R2μbLR(x)∼−1/(4x)<0 forx→∞ . To produce a positiveΔaμ ,mB<m˜R2 andmB>m˜R2 are favored forRe[y˜R2μBL(y˜R2μbR)∗]>0 andRe[y˜R2μBL(y˜R2μbR)∗]<0 , respectively. Thef˜S1μbLR(x) has asymptotic behavioursf˜S1μbLR(x)∼log(x)/2<0 forx→0 andf˜S1μbLR(x)∼−5/(4x)<0 forx→∞ . To produce a positiveΔaμ ,Re[y˜S1μBR(y˜S1μbL)∗]<0 is favored. Furthermore, the allowed regions in the plane ofmB−mLQ are sensitive to the choice ofyLQμqL,R . Generally, a larger|yLQμqL,R| corresponds to a largermB andmLQ . -
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 ofmLQ>mB , there are new LQ decay channels. When searching for the LQ˜R2/32 , we propose theμjbZ andμjbh signatures. When searching for the LQ˜R−1/32 , we propose theμjbW signatures. When searching for the LQ˜S1 , we propose theμjbZ ,μjbh ,⧸ETjbW signatures. For the scenario ofmLQ<mB , there are new VLQ decay channels. When searching for the VLQ B, we propose theμ+μ−jb signatures. When searching for the VLQ Y, we propose theμ⧸ETjb signatures. It seems that such decay channels have not been searched for by the experimental collaborations.Model Scenario LQ decay VLQ decay new signatures ˜R2+(B,Y)L,R mLQ>mB ˜R2/32→μ+b,μ+B B→bZ,bh ˜R2/32→μjbZ,μjbh ˜R−1/32→μ+Y,νLb Y→bW− ˜R−1/32→μjbW mLQ<mB ˜R2/32→μ+b B→bZ,bh,μ−˜R2/32 B→μ+μ−jb ˜R−1/32→νLb Y→bW−,μ−˜R−1/32 Y→μ⧸ETjb ˜S1+(B,Y)L,R mLQ>mB ˜S1→μ+ˉb,μ+ˉB,νLˉY B→bZ,bh ˜S1→μjbZ ,μjbh ,⧸ETjbW Y→bW− mLQ<mB ˜S1→μ+ˉb B→bZ,bh,μ+(˜S1)∗ B→μ+μ−jb Y→bW−,νL(˜S1)∗ Y→μ⧸ETjb 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:
Γ(˜R2/32→μ+B)Γ(˜R2/32→μ+b)∼|y˜R2μBL|2|y˜R2μbR|2,Γ(˜S1→μ+ˉB)Γ(˜S1→μ+ˉb)∼|y˜S1μBR|2|y˜S1μbL|2,Γ(B→μ−˜R2/32)Γ(B→bh)∼v2|y˜R2μBL|2m2B(sbR)2,Γ(B→μ+(˜S1)∗)Γ(B→bh)∼v2|y˜S1μBR|2m2B(sbR)2.
(18) In Fig. 2, we show the contour plots of above four ratios under the consideration of
(g−2)μ constraints. In these plots, we include the full contributions. We find that the new LQ decay channels can become important for larger|y˜R2μBL| in the˜R2+(B,Y)L,R model and|y˜S1μBR| in the˜S1+(B,Y)L,R model. As for the VLQ decay, the importance of new decay channels depends significantly onsbR . ForsbR=0.1 andmB=2.5 TeV , the new VLQ decay channels are less significant. For smallersbR , the new VLQ decay channels can play an important role.Figure 2. (color online) Contour plots of
log10Γ(˜R2/32→μ+B)Γ(˜R2/32→μ+b) (upper left),log10Γ(˜S1→μ+ˉB)Γ(˜S1→μ+ˉb) (lower left),log10Γ(B→μ−˜R2/32)Γ(B→bh) (upper right), andlog10Γ(B→μ+(˜S1)∗)Γ(B→bh) (lower right), where the colored regions are allowed by the(g−2)μ at2σ CL. For the LQ decay, we choosemB=1.5 TeV andmLQ=2 TeV . For the VLQ decay, we choosemB=2.5 TeV andmLQ=2 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∼0.01 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∼2 fb and∼0.01 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 modesΥ→μ+μ−,νˉνγ . If we consider a more complex flavour structure (e.g., turn on theLQμs interaction), this can affect theB→Kμ+μ− channel. Here, we will not study this detailed phenomenology. -
When the
˜R2 masses are degenerate, there are no gauge boson decay channels such as˜R2/32→˜R−1/32W+ . For the˜R2/32 toμ+b andμ+B decay channels, the widths are calculated asΓ(˜R2/32→μ+b)=m˜R216π√(1−m2μ+m2bm2˜R2)2−4m2μm2bm4˜R2×{(1−m2μ+m2bm2˜R2)[|y˜R2μBL|2(sbL)2+|y˜R2μbR|2(cbR)2]+4mμmbm2˜R2sbLcbRRe[y˜R2μBL(y˜R2μbR)∗]},Γ(˜R2/32→μ+B)=m˜R216π√(1−m2μ+m2Bm2˜R2)2−4m2μm2Bm4˜R2×{(1−m2μ+m2Bm2˜R2)[|y˜R2μBL|2(cbL)2+|y˜R2μbR|2(sbR)2]−4mμmBm2˜R2cbLsbRRe[y˜R2μBL(y˜R2μbR)∗]}. For the
˜R−1/32 toμ+Y,νLb,νLB decay channels, the widths are calculated asΓ(˜R−1/32→μ+Y)=m˜R216π√(1−m2μ+m2Ym2˜R2)2−4m2μm2Ym4˜R2(1−m2μ+m2Ym2˜R2)|y˜R2μBL|2,
Γ(˜R−1/32→νLb)=m˜R216π(1−m2bm2˜R2)2|y˜R2μbR|2(cbR)2,Γ(˜R−1/32→νLB)=m˜R216π(1−m2Bm2˜R2)2|y˜R2μbR|2(sbR)2.
Considering
mμ,mb≪mB andθbL,R≪1 , we have the following approximations:Γ(˜R2/32→μ+B)≈Γ(˜R−1/32→μ+Y)≈m˜R216π(1−m2Bm2˜R2)2|y˜R2μBL|2,Γ(˜R2/32→μ+b)≈Γ(˜R−1/32→νLb)≈m˜R216π|y˜R2μbR|2.
For the
˜S1 toμ+ˉb,μ+ˉB,νLˉY decay channels, the widths are calculated asΓ(˜S1→μ+ˉb)=m˜S116π√(1−m2μ+m2bm2˜S1)2−4m2μm2bm4˜S1×{(1−m2μ+m2bm2˜S1)[|y˜S1μBR|2(sbL)2+|y˜S1μbL|2(cbR)2]+4mμmbm2˜S1sbLcbRRe[y˜S1μBR(y˜S1μbL)∗]},Γ(˜S1→μ+ˉB)=m˜S116π√(1−m2μ+m2Bm2˜S1)2−4m2μm2Bm4˜S1×{(1−m2μ+m2Bm2˜S1)[|y˜S1μBR|2(cbL)2+|y˜S1μbL|2(sbR)2]−4mμmBm2˜S1cbLsbRRe[y˜S1μBR(y˜S1μbL)∗]},Γ(˜S1→νLˉY)=m˜S116π(1−m2Ym2˜S1)2|y˜S1μBR|2.
Considering
mμ,mb≪mB andθbL,R≪1 , we have the following approximations:Γ(˜S1→μ+ˉB)≈Γ(˜S1→νLˉY)≈m˜S116π(1−m2Bm2˜S1)2|y˜S1μBR|2,Γ(˜S1→μ+ˉb)≈m˜S116π|y˜S1μbL|2.
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If
mB∼TeV andθbR≪0.1 , we havemB−mY≈mB(sbR)2/2≲5 GeV , which leads to the kinematic prohibition of some decay channels. For theY→bW− decay channel, the width is calculated asΓ(Y→bW−)=g264πmY√(1−m2b+m2Wm2Y)2−4m2bm2Wm4Y⋅{[(sbL)2+(sbR)2](m2Y−m2b)2+m2W(m2Y+m2b)−2m4Wm2W−12mYmbsbLsbR}.
For the
B→bZ, bh, tW− decay channels, the widths are calculated asΓ(B→bh)=mB32π√(1−m2b+m2hm2B)2−4m2bm2hm4B[(1+m2b−m2hm2B)m2b+m2Bv2+4m2bv2](sbR)2(cbR)2,Γ(B→bZ)=g232πc2WmB√(1−m2b+m2Zm2B)2−4m2bm2Zm4B×{[(sbLcbL)2+(sbRcbR)24](m2B−m2b)2+m2Z(m2B+m2b)−2m4Zm2Z−6mBmbsbLsbRcbLcbR},Γ(B→tW−)=g2(sbL)264π√(1−m2t+m2Wm2B)2−4m2tm2Wm4B(m2B−m2t)2+m2W(m2B+m2t)−2m4Wm2WmB.
Considering
mb, mt, mZ, mW≪mB andθbL,R≪1 , we have the following approximations:Γ(B→bZ)≈Γ(B→bh)≈12Γ(Y→bW−)≈m3B32πv2(sbR)2,Γ(B→tW−)≈m2bmB16πv2(sbR)2.
In the
˜R2+(B,Y)L,R model, the VLQ can also decay into the˜R2 final state. For theY→μ−˜R−1/32 decay channel, the width is calculated asΓ(Y→μ−˜R−1/32)=mY32π√(1−m2μ+m2˜R2m2Y)2−4m2μm2˜R2m4Y(1+m2μ−m2˜R2m2Y)|y˜R2μBL|2.
For the
B→μ−˜R2/32, νL˜R−1/32 decay channels, the widths are calculated asΓ(B→μ−˜R2/32)=mB32π√(1−m2μ+m2˜R2m2B)2−4m2μm2˜R2m4B×{(1+m2μ−m2˜R2m2B)[|y˜R2μBL|2(cbL)2+|y˜R2μbR|2(sbR)2]+4mμm˜R2m2BcbLsbRRe[y˜R2μBL(y˜R2μbR)∗]},Γ(B→νL˜R−1/32)=mB32π(1−m2˜R2m2B)2|y˜R2μbR|2(sbR)2.
Considering
mμ≪mB andθbL,R≪1 , we have the following approximations:Γ(B→μ−˜R2/32)≈Γ(Y→μ−˜R−1/32)≈mB32π(1−m2˜R2m2B)2|y˜R2μBL|2.
In the
˜S1+(B,Y)L,R model, the VLQ can also decay into the˜S1 final state. For theY→νL(˜S1)∗ decay channel, the width is calculated asΓ(Y→νL(˜S1)∗)=mY32π(1−m2˜S1m2Y)2|y˜S1μBR|2.
For the
B→μ+(˜S1)∗ decay channel, the width is calculated asΓ(B→μ+(˜S1)∗)=mB32π√(1−m2μ+m2˜S1m2B)2−4m2μm2˜S1m4B×{(1+m2μ−m2˜S1m2B)[|y˜S1μBR|2(cbL)2+|y˜S1μbL|2(sbR)2]+4mμmBcbLsbRRe[y˜S1μBR(y˜S1μbL)∗]}.
Considering
mμ≪mB andθbL,R≪1 , we have the following approximations:Γ(B→μ+(˜S1)∗)≈Γ(Y→νL(˜S1)∗)≈mB32π(1−m2˜S1m2B)2|y˜S1μBR|2.
Scalar leptoquark and vector-like quark extended models as the explanation of the muon g–2 anomaly: bottom partner chiral enhancement case
- Received Date: 2023-02-14
- Available Online: 2023-07-15
Abstract: Leptoquark (LQ) models are well motivated solutions to the