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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.
  • Nuclear clustering has recently attracted great attention within the nuclear structure studies, especially for nuclei in the expanded nuclear chart away from the β-stability line [17]. The cluster structure is often formed and stabilized at highly excited resonant states in the vicinity of the cluster-separation threshold [8]. This structure can be probed experimentally by nuclear reaction tools combined with sequential cluster-decay. The reconstruction of the resonant state from the decay fragments, by the so-called invariant mass (IM) method, allows to select the cluster states with large partial decay widths. This selection significantly reduces the level density at high excitation energies, being in favor of the quantitative extraction of the physical properties of the resonances. Furthermore, a model-independent determination of the spin in the reconstructed resonant state can be achieved though the sensitive angular correlation method [9, 10]. This is particularly important at high excitation energies, where the differential cross-section method becomes almost futile due to the overlap of many close-by states and the uncertainties in many fitting parameters [9].

    Determination of the spin of each resonant state is of particular importance to form the molecular band, which is required to firmly establish the clustering structure [11]. Thus far, the angular correlation analysis has been the most sensitive and reliable tool to determine the spin of the resonant state [10, 1215]. Nevertheless, the angular correlation plot depends on the selection of the coordinate system, which varies for different types of experiments and conventions of data analysis. Although in most cases the z-coordinate axis is fixed in the beam direction, the definition of the spherical angle axis differs in different coordinate systems. To date, no detailed analysis and comparison of these coordinate systems is provided in the literature, which could lead to misunderstanding and erroneous application of the angular correlation method.

    In this study, we systemically investigate three kinds of coordinate systems that have been frequently applied in the literature. The definitions and features of these systems are outlined and compared to each other. The consistency of these systems is demonstrated by the experimental data analysis for the 10.29-MeV resonant state in 18O. Some suggestions for the application of the angular correlation method are provided in the summary.

    For a sequential decay reaction a(A,Bc+C)b, the composite resonant particle B may decay into e.g., two spin-zero fragments. The angular correlation of the latter is a sensitive probe for the spin of the resonant state in the mother nucleus B [10]. In a spherical coordinate system, with its z-axis pointing along the beam direction (Fig. 1), the correlation function can be parameterized in terms of four angles [10]. Namely, the center-of-mass (c.m.) scattering polar and azimuthal angles, θ and ϕ, respectively, of the resonant particle B; the polar and azimuthal angles, ψ and χ, respectively, of the relative velocity vector vrel of the two fragments (the arrow connecting HI and LI in Fig. 1). Both polar angles, θ and ψ, are with respect to the beam direction. The azimuthal angle ϕ is 0 (or 180) in the horizontal plane defined by the center positions of the detectors placed at opposite sides of the beam (the chamber plane or the detection plane). Another azimuthal angle χ is defined to be 0 (or 180) in the reaction plane, fixed by the beam axis and the reaction product B. Because of the limited detector geometry in a typical experiment, the correlation is often approximately constrained in the chamber plane, as shown in Fig. 1. In this case, the azimuthal angles ϕ and χ remain at 0 or 180, depending on the selected coordinate system, and the angular correlation appears as a function of only two polar angles θ and ψ. This is referred to as the in-plane correlation.

    Figure 1

    Figure 1.  (color online) Schematic diagram of the sequential decay following a transfer reaction a(A,Bc+C)b , depicting the polar angles θ and ψ in the detection plane. The two decay fragments, c and C, are specified as light ion (LI) and heavy ion (HI), respectively, in the figure.

    When the azimuthal angle χ is restricted to 0 (or 180), the most striking feature of the angular correlation in the θ-ψ plane appears, as the ridge structures associated with the spin of the mother nucleus ([9, 10, 16]). At relatively small θ angles, the structure is characterized by the locus ψ=αθ in the double differential cross-section, where α depicts a constant for the slope of the ridge and is almost inversely proportional to the spin of the resonant state B [10, 16]. The correlation is oscillatory along the ψ angle for a fixed θ, and vice versa. Generally, this in-plane correlation structure can be projected onto the one-dimensional spectrum W(θ=0,ψ=ψαθ) . Within the strong absorption model (SAM) [1719], α may be related to the orbital angular moment li of the dominant partial wave in the entrance channel through α=liJJ, with J being the spin of B [9, 16]. li can be evaluated simply from li=r0(A1/3p+A1/3t)2μEc.m. [20], with Ap and At being the mass numbers of the beam and target nucleus, respectively. Here, μ is the reduced mass and Ec.m. depicts the center-of-mass energy. If the resonant nucleus is emitted at angles close to θ=0, the projected correlation function W(ψ) is simply proportional to the square of the Legendre polynomial of order J, namely |PJ(cos(ψ))|2. This method has been frequently applied in the literature ([21] and references therein) and will also be demonstrated in the following section 2.3.

    As indicated above, in the application of the angular correlation plot, it is important to enhance the ridge structure, which corresponds to the spin of the resonant mother nucleus. Consequently, the selection of the coordinate system for the plot is meaningful. For a non-polarized experiment, the reaction process satisfies the axial symmetry around the beam axis. Moreover, the decay process should satisfy the parity (space inversion) symmetry with respect to the c.m. of the resonant mother nucleus. In the two-dimensional correlation plot (plot of the double differential cross-section), depicting functions of the polar angles θ and ψ, it is natural to plot these symmetrical events in the same ridge band while placing the non-symmetric events elsewhere, to enhance the sensitivity to the associated spin. This should be applied even for the simple case of restricted detection around the chamber plane. For instance, in Fig. 2 we schematically illustrate two processes that generate the same polar angles θ and ψ but are not symmetric in terms of the resonance-decay. These two processes should be distinguished from the ridge plot using an appropriate coordinate system, such as the one using positive and negative θ, as defined below. In contrast, the coordinate definition should keep the four symmetric processes in the same ridge band, as schematically portrayed in Fig. 3, such that the ridge structure appears continuously and a simple projection can bring them together to enhance the sensitivity to the associated spin. We introduce, in the following, three kinds of coordinate systems and demonstrate their differences with respect to plot-definition and consistency in extraction of the spin.

    Figure 2

    Figure 2.  (color online) Schematic diagram of the two independent reaction processes with the same polar angles. The direction of the relative velocity vector vrel is guided by the light ion in both cases.

    Figure 3

    Figure 3.  (color online) Schematic diagram of the four symmetric reaction-decay processes in the chamber plane. (a) and (b) are parity-symmetric processes, while (c) and (d) are their axial-symmetric processes, respectively. All processes are identified by the angles θ and ψ defined in various coordinate systems, as described in the text.

    In the first coordinate system (hereinafter refered to as the ψfix coordinate system), the relative velocity vector of the decay products always points to the fixed detector at one side of the beam (where ϕ = 0). This is convenient when two decay particles are identical, such as in 24Mg12C+12C [10], or each detector is designed to be sensitive to only one type of the particles, such as 18O14C+α with 14C always detected at one-side, while α is at the other side of the beam [9]. By definition, θ is positive on the opposite side of the beam and negative on the same side, in comparison with the fixed positive ψ. With this definition, processes (a) and (b) are plotted at one position (negative θ), while (c) and (d) are at another position (positive θ) (see present Fig. 4(A) or Fig. 5 in Ref. [10]).

    Figure 4

    Figure 4.  (color online) Angular correlation spectra for the 10.29-MeV state in 18O in the (A) ψfix, (B) ψion and (C) ψfull coordinate systems, respectively. The projection lines in (A), (B), and (C) (the red-solid lines) correspond to a slope parameter α=liJJ, with li = 13.9 (using r0 = 1.2 fm). (a), (b), and (c) show the projections of (A), (B), and (C) onto the θ=0 axis, respectively. The black dot-dashed line indicates the simulated detector efficiency in each coordinate system. All experimental distributions are compared with a squared Legendre polynomial of the 4th order.

    Figure 5

    Figure 5.  (color online) Excitation energy spectrum for 18O emitting to small θ angles. The state at 10.29 MeV is selected to demonstrate the angular correlation analysis.

    In another more “physical” convention [16] (hereinafter denoted as the ψion coordinate system), the relative velocity vector vrel always points to a certain decay particle (usually the lighter one, LI), corresponding to ϕ = 0. The positive θ remains at the opposite side of the positive ψ. Under this convention, the axial-symmetric processes (a) and (c) will be plotted at the same position, whereas (b) and (d) are plotted at another position, as demonstrated in the present Fig. 4(B) or in Fig. 3 of Ref. [16]. Due to the different detection efficiencies for light and heavy particles, the correlation structure in this coordinate system may differ from that in the ψfix coordinate system.

    Additionally, on the basis of the ψion coordinate system, the polar angle ψ could also be assigned a positive or negative sign, depending on the azimuthal angle χ. First, 0 of the azimuthal angle ϕ or χ is defined by a detector in the chamber plane. Then, a positive ψ means that χ is close to 0, while negative ψ means that χ is close to 180. Hence, four intrinsically equivalent cases in Fig. 3 are plotted at four different positions in the θ-ψ plane. This coordinate system is denoted as ψfull. This convention was used in some previous studies, such as in Ref. [14], and also demonstrated in the present Fig. 4(C). Thus, both θ and ψ range across positive as well as negative scales. Since the experimental detection system may not be exactly symmetric with respective to the beam axis, the double differential cross-section in Fig. 4(C) seems not symmetric neither. It is evident that this wider scale distribution would give more consistent information for the ridge structure, however in the meantime, it would require higher statistics.

    The above-introduced three coordinate systems are equally meaningful, since the non-symmetric process, as shown in Fig. 2(b), does not appear in any of the defined ridge bands. Meanwhile, these systems should be consistent with each other in terms of extracting the spin of the resonant mother nucleus. This consistency is demonstrated below by experimental data analysis.

    Recently, a multi-nucleon transfer and cluster-decay experiment [22] 9Be(13C, 18O*14C + α)α was performed at the HI-13 tandem accelerator facility at the China Institute of Atomic Energy (CIAE) in Beijing. Resonant states in 18O can be reconstructed according to the invariant mass method [1, 5], as shown in Fig. 5 for events with small θ angles. The state at 10.29 MeV is a good candidate for angular correlation analysis, owing to its clear peak identification and relatively large ψ-angle coverage. In Fig. 4, we plot the angular correlation spectrum for the 18O 10.29-MeV state in the above described three coordinate systems (Fig. 4(A-C)). Further, these two dimensional spectra in θψ plane are projected onto the θ=0 axis according to the above described ψ=ψliJJθ relation, as exhibited in Fig. 4(a-c), respectively. The projections are compared with the square of the Legendre polynomial of order 4. Only the periodicity of the distribution matters, whereas the absolute peak amplitudes depend on the detection efficiency. Although the distributions behave slightly differently in the three coordinate systems, the periodicities of the experimental spectra all agree with the Legendre polynomial of order 4, corresponding to a spin-parity of 4+ for the 10.29-MeV state in 18O. This consistency between various coordinate systems indicates the reliability of the angular correlation method in determining the spin of a resonant state.

    Based on the consistency exhibited in Fig. 4 and the symmetry property of the Legendre polynomial, we may plot the projected correlation spectrum as a function of |cos(ψ)| [21], in order to increase the statistics in each bin of the distribution. Furthermore, this plot is independent of the above-defined coordinate systems. Moreover, the excitation energy spectrum can be reconstructed, similarly to that in Fig. 5, for each bin of |cos(ψ)| and the corresponding event number can be extracted for the pure 10.29-MeV peak by subtracting the smooth background beneath the peak. The experimental correlation spectrum is plotted in Fig. 6. The theoretical function composed of a squared Legendre polynomial with a constant background, corrected by the detection efficiency, is used to describe the experimental results. Not only the periodicity, but also the magnitude of the function for a spin-parity of 4+ provide an excellent fit to the experimental data, whereas other options of spin-parity can be excluded. A constant background is nevertheless needed in the theoretical function, since the experimental data include some uncorrelated components stemming from events within the 10.29-MeV peak, but away from the exact θ = 0 axis [10].

    Figure 6

    Figure 6.  (color online) Angular correlation spectrum for 10.29-MeV state in 18O, in comparison to the Legendre polynomials of order 4 (the red dotted line) and order 6 (the blue dot-dashed line). A uniformly distributed background is assumed to account for the uncorrelated component (the long dot-dashed line). Theoretical angular distributions have been corrected for the detection efficiency. The corresponding reduced ˉχ2 indicating the efficiency of the theoretical description is also indicated in the plot.

    The fragment angular correlation in a sequential cluster-decay reaction provides a method to determine the spins of the resonant nucleus independent of models. When the correlation spectrum is restricted to angles close to the detection chamber plane, the ridged pattern can be clearly seen in the two-dimensional plot with respective to the two polar angles θ and ψ. According to the ways to deal with symmetrical events, three coordinate systems for different θ and ψ definitions have been adopted in the literature for various experiments and for the spin analysis. In the present work, we outlined these systems and compared them to each other to clarify their differences and consistencies. The systems are examined by the cluster-decay data for the 10.29-MeV state in 18O, measured in our recent experiment. This study provides a better understanding of the angular correlation function, and demonstrates the possible choices for the best extraction of the spin in a resonant state.

    In the case where a resonant nucleus decays into two spin-zero fragments, the two-dimensional correlation spectrum for small θ angles can be projected onto the θ = 0 axis. This projected spectrum may be described by a squared Legendre polynomial with an order corresponding to the spin of the resonant mother nucleus. Using this method, a spin-parity of 4+ is decisively determined for the 10.29-MeV state in 18O.

    Based on the above investigations, we propose the basic procedure for the application of the angular correlation method. Firstly, the experiment should be designed to have good detection for events at small θ angles and with wide ψ coverage. Secondly, a proper coordinate system should be selected according to the detection and data-distribution characteristics. In principle, the best choice is the ψfull coordinate system, owing to its wider angular range of the correlation spectrum, which may help identify the ridge structure assuming the statistics are sufficient. However, when the statistics are low, the coordinate systems ψfix or ψion, depending on the detection arrangement, are more convenient. Thirdly, before the projection onto the θ=0 axis, the ridge structure should be distinctly observed. Otherwise, any small shift in the projection direction may lead to the wrong extraction of the spin, especially in the case of higher spin, where more oscillations in the projected spectrum are expected. The detector efficiency should likewise be carefully examined, since large variations in the efficiency curve may give rise to some non-physical structures in the projected distribution. Finally, the projected spectrum may be obtained by subtracting the background for each bin of |cos(ψ)|. This experimental spectrum can be compared with the theoretical function (Legendre polynomial corrected by the detection efficiency), and the goodness of fit can be examined quantitatively. However, the correct projection parameter should be fixed before this fitting procedure.

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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
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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 (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.

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    I.   INTRODUCTION
    • 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.

    II.   MODEL SETUP
    • 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.

      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

      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 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.

      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

      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 ¯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.

    • B.   VLQ gauge interactions

    • 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)
    • C.   VLQ Yukawa interactions with LQ

    • 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)
    III.   CONTRIBUTIONS TO (g2)μ

      A.   Analytical results of contributions

    • 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)
    • B.   Numerical analysis

    • 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.

      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)

      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).

      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.  (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).

    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 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 mLQ<mB, there are new VLQ decay channels. When searching for the VLQ B, we propose the μ+μjb signatures. When searching for the VLQ Y, we propose the μETjb signatures. It seems that such decay channels have not been searched for by the experimental collaborations.

      ModelScenarioLQ decayVLQ decaynew signatures
      ˜R2+(B,Y)L,RmLQ>mB˜R2/32μ+b,μ+BBbZ,bh˜R2/32μjbZ,μjbh
      ˜R1/32μ+Y,νLbYbW˜R1/32μjbW
      mLQ<mB˜R2/32μ+bBbZ,bh,μ˜R2/32Bμ+μjb
      ˜R1/32νLbYbW,μ˜R1/32YμETjb
      ˜S1+(B,Y)L,RmLQ>mB˜S1μ+ˉb,μ+ˉB,νLˉYBbZ,bh˜S1μjbZ, μjbh, ETjbW
      YbW
      mLQ<mB˜S1μ+ˉbBbZ,bh,μ+(˜S1)Bμ+μjb
      YbW,νL(˜S1)YμETjb

      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(˜R2/32μ+B)Γ(˜R2/32μ+b)|y˜R2μBL|2|y˜R2μbR|2,Γ(˜S1μ+ˉB)Γ(˜S1μ+ˉb)|y˜S1μBR|2|y˜S1μbL|2,Γ(Bμ˜R2/32)Γ(Bbh)v2|y˜R2μBL|2m2B(sbR)2,Γ(Bμ+(˜S1))Γ(Bbh)v2|y˜S1μBR|2m2B(sbR)2.

      (18)

      In Fig. 2, we show the contour plots of above four ratios under the consideration of (g2)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 on sbR. For sbR=0.1 and mB=2.5 TeV, the new VLQ decay channels are less significant. For smaller sbR, the new VLQ decay channels can play an important role.

      Figure 2.  (color online) Contour plots of log10Γ(˜R2/32μ+B)Γ(˜R2/32μ+b) (upper left), log10Γ(˜S1μ+ˉB)Γ(˜S1μ+ˉb) (lower left), log10Γ(Bμ˜R2/32)Γ(Bbh) (upper right), and log10Γ(Bμ+(˜S1))Γ(Bbh) (lower right), where the colored regions are allowed by the (g2)μ at 2σ CL. For the LQ decay, we choose mB=1.5 TeV and mLQ=2 TeV. For the VLQ decay, we choose mB=2.5 TeV and mLQ=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 MadGraph5aMC@NLO [55]. For the 2 TeV scale LQ pair production [5658], 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 [5961], 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 modes Υμ+μ,νˉνγ. If we consider a more complex flavour structure (e.g., turn on the LQμs interaction), this can affect the BKμ+μ 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 (g2)μ anomaly. Then, we find two new models ˜R2/˜S1+(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: mLQ>mB and mLQ<mB. After considering the experimental constraints, we choose relative light masses, which are adopted to be (mB,mLQ)=(1.5 TeV,2 TeV) for the first scenario and (mB,mLQ)=(2.5 TeV,2 TeV) for the second scenario. In the ˜R2+(B,Y)L,R model, the |Re[y˜R2μBL(y˜R2μbR)]| is bounded to be O(1), because f˜R2μbLR(x) vanishes accidentally as mB=m˜R2. Meanwhile, we can expect smaller |Re[y˜R2μBL(y˜R2μbR)]| for largely splitted m˜R2 and mB. In the ˜S1+(B,Y)L,R model, the Re[y˜S1μBR(y˜S1μbL)] is bounded to the range (0.06,0.02) at a 2σ CL if sbR=0.1.

      Under the constraints from (g2)μ, we propose new LQ and VLQ search channels. In the scenario of mLQ>mB, there are new LQ decay channels: ˜R2/32μ+B, ˜R1/32μ+Y, and ˜S1μ+ˉB,νLˉY. For larger y˜R2μBL and y˜S1μBR, it is important to take into account these decay channels. In the scenario of mLQ<mB, there are new VLQ decay channels: Bμ˜R2/32,μ+(˜S1) and Yμ˜R1/32, νL(˜S1). For sbR=0.1, these channels are negligible compared with the traditional BbZ, bh and YbW channels. For smaller sbR, these new VLQ decay channels can also become important.

      Note added: In a prevoius study [64], the authors examined the model with ˜R2, S3, 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 ˜R2 and S3 mixing. In [65], the authors explained the (g2)μ and B physics anomalies in the S1+(B,Y)L,R model.

    APPENDIX A: LQ DECAY WIDTH FORMULAE
    • When the ˜R2 masses are degenerate, there are no gauge boson decay channels such as ˜R2/32˜R1/32W+. For the ˜R2/32 to μ+b and μ+B decay channels, the widths are calculated as

      Γ(˜R2/32μ+b)=m˜R216π(1m2μ+m2bm2˜R2)24m2μm2bm4˜R2×{(1m2μ+m2bm2˜R2)[|y˜R2μBL|2(sbL)2+|y˜R2μbR|2(cbR)2]+4mμmbm2˜R2sbLcbRRe[y˜R2μBL(y˜R2μbR)]},Γ(˜R2/32μ+B)=m˜R216π(1m2μ+m2Bm2˜R2)24m2μm2Bm4˜R2×{(1m2μ+m2Bm2˜R2)[|y˜R2μBL|2(cbL)2+|y˜R2μbR|2(sbR)2]4mμmBm2˜R2cbLsbRRe[y˜R2μBL(y˜R2μbR)]}.

      For the ˜R1/32 to μ+Y,νLb,νLB decay channels, the widths are calculated as

      Γ(˜R1/32μ+Y)=m˜R216π(1m2μ+m2Ym2˜R2)24m2μm2Ym4˜R2(1m2μ+m2Ym2˜R2)|y˜R2μBL|2,

      Γ(˜R1/32νLb)=m˜R216π(1m2bm2˜R2)2|y˜R2μbR|2(cbR)2,Γ(˜R1/32νLB)=m˜R216π(1m2Bm2˜R2)2|y˜R2μbR|2(sbR)2.

      Considering mμ,mbmB and θbL,R1, we have the following approximations:

      Γ(˜R2/32μ+B)Γ(˜R1/32μ+Y)m˜R216π(1m2Bm2˜R2)2|y˜R2μBL|2,Γ(˜R2/32μ+b)Γ(˜R1/32νLb)m˜R216π|y˜R2μbR|2.

      For the ˜S1 to μ+ˉb,μ+ˉB,νLˉY decay channels, the widths are calculated as

      Γ(˜S1μ+ˉb)=m˜S116π(1m2μ+m2bm2˜S1)24m2μm2bm4˜S1×{(1m2μ+m2bm2˜S1)[|y˜S1μBR|2(sbL)2+|y˜S1μbL|2(cbR)2]+4mμmbm2˜S1sbLcbRRe[y˜S1μBR(y˜S1μbL)]},Γ(˜S1μ+ˉB)=m˜S116π(1m2μ+m2Bm2˜S1)24m2μm2Bm4˜S1×{(1m2μ+m2Bm2˜S1)[|y˜S1μBR|2(cbL)2+|y˜S1μbL|2(sbR)2]4mμmBm2˜S1cbLsbRRe[y˜S1μBR(y˜S1μbL)]},Γ(˜S1νLˉY)=m˜S116π(1m2Ym2˜S1)2|y˜S1μBR|2.

      Considering mμ,mbmB and θbL,R1, we have the following approximations:

      Γ(˜S1μ+ˉB)Γ(˜S1νLˉY)m˜S116π(1m2Bm2˜S1)2|y˜S1μBR|2,Γ(˜S1μ+ˉb)m˜S116π|y˜S1μbL|2.

    APPENDIX B: VLQ DECAY WIDTH FORMULAE
    • If mBTeV and θbR0.1, we have mBmYmB(sbR)2/25 GeV, which leads to the kinematic prohibition of some decay channels. For the YbW decay channel, the width is calculated as

      Γ(YbW)=g264πmY(1m2b+m2Wm2Y)24m2bm2Wm4Y{[(sbL)2+(sbR)2](m2Ym2b)2+m2W(m2Y+m2b)2m4Wm2W12mYmbsbLsbR}.

      For the BbZ, bh, tW decay channels, the widths are calculated as

      Γ(Bbh)=mB32π(1m2b+m2hm2B)24m2bm2hm4B[(1+m2bm2hm2B)m2b+m2Bv2+4m2bv2](sbR)2(cbR)2,Γ(BbZ)=g232πc2WmB(1m2b+m2Zm2B)24m2bm2Zm4B×{[(sbLcbL)2+(sbRcbR)24](m2Bm2b)2+m2Z(m2B+m2b)2m4Zm2Z6mBmbsbLsbRcbLcbR},Γ(BtW)=g2(sbL)264π(1m2t+m2Wm2B)24m2tm2Wm4B(m2Bm2t)2+m2W(m2B+m2t)2m4Wm2WmB.

      Considering mb, mt, mZ, mWmB and θbL,R1, we have the following approximations:

      Γ(BbZ)Γ(Bbh)12Γ(YbW)m3B32πv2(sbR)2,Γ(BtW)m2bmB16πv2(sbR)2.

      In the ˜R2+(B,Y)L,R model, the VLQ can also decay into the ˜R2 final state. For the Yμ˜R1/32 decay channel, the width is calculated as

      Γ(Yμ˜R1/32)=mY32π(1m2μ+m2˜R2m2Y)24m2μm2˜R2m4Y(1+m2μm2˜R2m2Y)|y˜R2μBL|2.

      For the Bμ˜R2/32, νL˜R1/32 decay channels, the widths are calculated as

      Γ(Bμ˜R2/32)=mB32π(1m2μ+m2˜R2m2B)24m2μm2˜R2m4B×{(1+m2μm2˜R2m2B)[|y˜R2μBL|2(cbL)2+|y˜R2μbR|2(sbR)2]+4mμm˜R2m2BcbLsbRRe[y˜R2μBL(y˜R2μbR)]},Γ(BνL˜R1/32)=mB32π(1m2˜R2m2B)2|y˜R2μbR|2(sbR)2.

      Considering mμmB and θbL,R1, we have the following approximations:

      Γ(Bμ˜R2/32)Γ(Yμ˜R1/32)mB32π(1m2˜R2m2B)2|y˜R2μBL|2.

      In the ˜S1+(B,Y)L,R model, the VLQ can also decay into the ˜S1 final state. For the YνL(˜S1) decay channel, the width is calculated as

      Γ(YνL(˜S1))=mY32π(1m2˜S1m2Y)2|y˜S1μBR|2.

      For the Bμ+(˜S1) decay channel, the width is calculated as

      Γ(Bμ+(˜S1))=mB32π(1m2μ+m2˜S1m2B)24m2μm2˜S1m4B×{(1+m2μm2˜S1m2B)[|y˜S1μBR|2(cbL)2+|y˜S1μbL|2(sbR)2]+4mμmBcbLsbRRe[y˜S1μBR(y˜S1μbL)]}.

      Considering mμmB and θbL,R1, we have the following approximations:

      Γ(Bμ+(˜S1))Γ(YνL(˜S1))mB32π(1m2˜S1m2B)2|y˜S1μBR|2.

Reference (65)

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