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 $(g-2)_{\mu}$ anomaly. In the minimal LQ models, only specific representations can lead to the chiral enhancements. For the scalar LQs, the $R_2$ and $S_1$ can lead to the top quark chiral enhancement. For the vector LQs, the $V_2$ 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 work, 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 enhancement, which originate from the bottom and bottom partner mixing. Then, we propose new LQ and VLQ search channels under the constraints of $(g-2)_{\mu}$.

For the mediators with mass above TeV, the chiral enhancements are required, which can show up when left-handed and right-handed muon couples to a heavy fermion simultaneously. In the new lepton extended models [10][11][12][13], the chiral enhancements originate from the heavy lepton mass. Besides, the LQ models can be the alternative choice [14][15][16][17][18][19][20], 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 lepton sector and quark sector. In our previous paper [21], we investigated the scalar LQ and VLQ 1 extended models with top and top partner chiral enhancements. In this work, we will study the scalar LQ and VLQ extended models, which can produce the bottom partner chiral enhancements. This paper is complementary to our paper [21]. 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) µ and perform the numerical analysis in Sec. III. In Sec. IV, we discuss the possible collider phenomenology. Finally, we make the summary and conclusions in Sec. V.

II. MODEL SETUP
Typically speaking, there are six type of scalar LQs [15], which carry a conserved quantum number F ≡ 3B + L. Here, the B and L are the baryon and lepton numbers. As to the VLQs, there are seven typical representations [22]. In Tab. I, we list their representations and labels. For the six type of scalar LQs and seven type of VLQs, there can be totally 42 combinations, which are named as "LQ + VLQ" for convenience. While, only 17 of them can lead to the chiral enhancements. In Tab. II, 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 quark have already been investigated in our paper [21]. Here, we will study the pure bottom partner chirally enhanced models 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, m b /mµ, mB/mµ S3 + (X, T, B)L,R mt/mµ, mT /mµ, m b /mµ, mB/mµ In the above, the s b L,R and c b L,R are abbreviations of sin θ b L,R and cos θ b L,R . In fact, the θ b L can be correlated with θ b R through the relation tan θ b L = m b tan θ b R /m B [22]. Here, the m b and m B label the physical b and B quark masses. Besides, the mass of Y quark is m Then, we can choose m B and θ b R as the new input parameters. After the transformations in Eq. (1), we obtain the following mass eigenstate Higgs Yukawa interactions: Note that the Y quark does not interact with Higgs at tree level.

II.2. The VLQ gauge interactions
Now, let us label the SU (2) L and U Y (1) gauge fields as W a µ and B µ . Then, the electroweak covariant derivative D µ is defined as charge of the quark field acted by D µ . Thus, the related gauge interactions can be written as After the EWSB, the W gauge interactions can be written as The Z gauge interactions can be written as After the rotations in Eq. (1), we have the mass eigenstate W gauge interactions: We also have the mass eigenstate Z gauge interactions: ] T , where the superscript labels the electric charge. Then, the R 2 and S 1 can induce the following F = 0 and F = 2 type gauge eigenstate LQ Yukawa interactions: and After the EWSB, they can be parametrized as and In the above, we define the chiral operators ω ± as (1 ± γ 5 )/2. After the rotations in Eq. (1), we have the mass eigenstate interactions: and L S1+(B,Y ) L,R ⊃μ(y S1µb III. CONTRIBUTIONS TO THE (g − 2)µ

III.1. Analytic results of the contributions
For the LQµq interaction, there are quark-photon and LQ-photon vertex mediated contributions to the (g−2) µ , which can be described by the functions f q (x) and f S (x). Then, we use the functions f q,S LL (x) and f q,S LR (x) to label the parts without and with chiral enhancements. Starting from the f q,S LL (x) and f q,S LR (x) given in our paper [21], let us define the following integrals: At tree level, we have m R 2/3 is dominated by the bottom partner chirally enhanced contribution. Then, the above expression can be approximated as ).
For the S 1 +(B, Y ) L,R model, there are b and B quark contributions to the (g−2) µ . The complete expression is calculated as ) . (16) Similarly, it can be approximated as ).

III.2. Numerical analysis
The input parameters are chosen as m µ = 105.66MeV, R , and the LQ Yukawa couplings y LQµq L,R . The VLQ mass can be constrained from the direct search, which is required to be above 1.5TeV [24][25][26][27]. 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 28,29], thus it leads to less constrained θ b R . The weak isospin third component of B R is positive, then the mixing with bottom quark enhances the right-handed Zbb coupling. As a result, the A b F B deviation [30,31] can be compensated, which also leads to looser constraints on θ b R . Conservatively speaking, we can choose the mixing angle s b R to be less than 0.1 [22]. The LQ mass can also be constrained from the direct search, which is required to be above 1.7TeV assuming Br(LQ → bµ) = 1 [32,33].
We can choose benchmark points of m B , m LQ , s b R to constrain the LQ Yukawa couplings. Here, we consider two scenarios m LQ > m B and m LQ < m B . For the scenario m LQ > m B , we adopt the mass parameters to be m B = 1.5TeV and m LQ = 2TeV. For the scenario m LQ < m B , we adopt the mass parameters to be m B = 2.5TeV and m LQ = 2TeV. In Tab. III, we give the approximate numerical expressions of the ∆a µ in the R 2 / S 1 + (B, Y ) L,R models. Besides, we also show the allowed ranges for s b R = 0.1 and s b R = 0.05. Of course, these behaviours can be understood from the Eqs. (15) and (17)

IV. LQ AND VLQ PHENOMENOLOGY AT HADRON COLLIDERS
In Tab. IV, we list the main LQ and VLQ decay channels 2 . The decay formulae of LQ and VLQ are given in App. A and B. For the scenario m LQ > m B , there are new LQ decay channels. When searching for the LQ R 2/3 2 , we propose the µj b Z and µj b h signatures. When searching for the LQ R −1/3 2 , we propose the µj b W signatures. When searching for the LQ S 1 , we propose the µj b Z, µj b h, / E T j b W signatures. For the scenario m LQ < m B , there are new VLQ decay channels. When searching for the VLQ B, we propose the µ + µ − j b signatures. When searching for the VLQ Y , we propose the µ / E T j b signatures. It seems that such decay channels have not been searched by the experimental collaborations.

Model
Scenario LQ decay VLQ decay new signatures To estimate the effects of new decay channels, we will compare the ratios of new partial decay widths to the tradition ones. Because of gauge symmetry, the different partial decay widths can be correlated. Then, we choose the following four ratios: In Fig. 2, we show the contour plots of above four ratios under the consideration of (g − 2) µ constraints. In these plots, we have included the full contributions. We find that the new LQ decay channels can become important for larger |y R2µB   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 [34] to generate the model files and compute the cross sections with MadGraph5 − aMC@NLO [35]. For the 2 TeV scale LQ pair production [36][37][38], the cross section can be ∼ 0.01fb at 13 TeV LHC. For the 1.5 TeV and 2.5 TeV scale VLQ pair production [39][40][41], the cross section can be ∼ 2fb and ∼ 0.01fb at 13 TeV LHC. For the single LQ and VLQ production channels, they depend on the electroweak couplings [22,42,43]. 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 TeV and 100 TeV, can be necessary. Besides the collider direct search, there can be some indirect footprints, for example, B physics related decay modes Υ → µ + µ − , ννγ. If we consider more complex flavour structure (say, turn on the LQµs interaction), it can also affect the B → Kµ + µ − channel. Here, we will not study these detailed phenomenology.

V. SUMMARY AND CONCLUSIONS
In this paper, we study the scalar LQ and VLQ extended models to explain the (g − 2) µ anomaly. Then, we find two new models Note added: In paper [44], the authors study the model with R 2 , S 3 , and (B, Y ) L,R . In their work, they do not consider the bottom and B quark mixing, and the chiral enhancements are produced through the R 2 and S 3 mixing. In paper [45], the authors explain the (g − 2) µ and B physics anomalies in the S 1 + (B, Y ) L,R model. to µ + b and µ + B decay channels, the widths are calculated as For the R −1/3 2 to µ + Y, ν L b, ν L B decay channels, the widths are calculated as

R2
)|y R2µB L | 2 , Considering m µ , m b ≪ m B and θ b L,R ≪ 1, we have the following approximations: For the S 1 to µ +b , µ +B , ν LȲ decay channels, the widths are calculated as )[|y S1µB Considering m µ , m b ≪ m B and θ b L,R ≪ 1, we have the following approximations: 16π |y S1µb , which leads to the kinematic prohibition of some decay channels. For the Y → bW − decay channel, the width is calculated as For the B → bZ, bh, tW − decay channels, the widths are calculated as Considering m b , m t , m Z , m W ≪ m B and θ b L,R ≪ 1, we have the following approximations: In the R 2 + (B, Y ) L,R model, the VLQ can also decay into R 2 final state. For the Y → µ − R −1/3 2 decay channel, the width is calculated as For the B → µ − R Considering m µ ≪ m B and θ b L,R ≪ 1, we have the following approximations: In the S 1 + (B, Y ) L,R model, the VLQ can also decay into S 1 final state. For the Y → ν L ( S 1 ) * decay channel, the width is calculated as For the B → µ + ( S 1 ) * decay channel, the width is calculated as Considering m µ ≪ m B and θ b L,R ≪ 1, we have the following approximations: