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A multi-charged particle model with local U(1)μ-τ to explain muon g–2, flavor physics, and possible collider signature

  • We consider a model with multi-charged particles, including vector-like fermions, and a charged scalar under a local U(1)μτ symmetry. We search for an allowed parameter region explaining muon anomalous magnetic moment (muon g2) and bs+ anomalies, satisfying constraints from the lepton flavor violations, Z boson decays, meson anti-meson mixing, and collider experiments. Via numerical analysis, we explore the typical size of the muon g2 and Wilson coefficients to explain the bs+ anomalies in our model when all other experimental constraints are satisfied. Subsequently, we discuss the collider physics of the multicharged vectorlike fermions, considering a number of benchmark points in the allowed parameter space.
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Nilanjana Kumar, Takaaki Nomura and Hiroshi Okada. A multi-charged particle model with local U(1)μ-τ to explain muon g-2, flavor physics, and possible collider signature[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac425a
Nilanjana Kumar, Takaaki Nomura and Hiroshi Okada. A multi-charged particle model with local U(1)μ-τ to explain muon g-2, flavor physics, and possible collider signature[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac425a shu
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A multi-charged particle model with local U(1)μ-τ to explain muon g–2, flavor physics, and possible collider signature

    Corresponding author: Nilanjana Kumar, nilanjana.kumar@gmail.com
    Corresponding author: Takaaki Nomura, nomura@kias.re.kr
    Corresponding author: Hiroshi Okada, hiroshi.okada@apctp.org
  • 1. Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
  • 2. School of Physics, KIAS, Seoul 02455, Korea
  • 3. Asia Pacific Center for Theoretical Physics (APCTP) - Headquarters San 31, Hyoja-dong, Nam-gu, Pohang 790-784, Korea
  • 4. Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea

Abstract: We consider a model with multi-charged particles, including vector-like fermions, and a charged scalar under a local U(1)μτ symmetry. We search for an allowed parameter region explaining muon anomalous magnetic moment (muon g2) and bs+ anomalies, satisfying constraints from the lepton flavor violations, Z boson decays, meson anti-meson mixing, and collider experiments. Via numerical analysis, we explore the typical size of the muon g2 and Wilson coefficients to explain the bs+ anomalies in our model when all other experimental constraints are satisfied. Subsequently, we discuss the collider physics of the multicharged vectorlike fermions, considering a number of benchmark points in the allowed parameter space.

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    I.   INTRODUCTION
    • A muon anomalous magnetic moment (muon g2) is experimentally and theoretically analyzed with high precision, and it is a promising observation, to test/verify novel physics beyond the standard model (SM). Recently, the E989 Run 1 experiment at Fermilab (FNAL) [1] provided new data for muon g2, where the previous measurement at the E821 experiment at Brookhaven National Lab (BNL) two decades ago [2] indicates a deviation from the SM prediction by 3σ. Combining the BNL results, the deviation from the SM prediction [3, 4] is given by

      Δaμ=(25.1±5.9)×1010,

      (1)

      where the deviation reaches 4.2σ with a positive value from the SM prediction. Moreover, a further update of Fermilab E989 and an upcoming J-PARC E34 [5] experiment will provide better results with higher precision. To explain the deviation theoretically, several mechanisms have been proposed over time, such as, gauge contributions [6-8], Yukawa contributions at one-loop level [9], and Barr-Zee contributions [10] at two-loop level. In particular, if muon g2 is related to other phenomenologies such as neutrino mass generations, dark matter, and various flavor physics, the new Yukawa interactions become important, where muon g2 would be explained at a one-loop level via such interactions [9, 11-42] (also refer to recent approaches after new FNAL results [43-78]). In such a case, it is required to simultaneously satisfy several constraints of lepton flavor violations (LFVs), such as ijγ, ijkˉ (i,j,k,=(e,μ,τ)), including lepton flavor conserving (violating) Z boson decays Zˉ, Zνˉν [79]. In particular, the μeγ process presents the most stringent constraint, where the current upper bound on the branching ratio is 4.2×1013 [80], and its future bound will reach the sensitivity at 6×1014 [81]. In addition, Z boson decays will be tested in future experiments such as CEPC [82]. Previously, we analyzed models introducing multi-charged fields (scalars and vector-like leptons) with general U(1)Y hypercharges, to obtain a positive muon g2, and explored the parameter region satisfying several experimental constraints [38]. Another interesting study includes a new U(1) that explains the same case [83].

      Another interesting hint for new physics includes experimental anomalies of semileptonic B-meson decays, deviations in the measurements of the angular observable P5 in the decay of the B meson (BKμ+μ) [84-88], the ratio of branching fractions, RK=BR(B+K+μ+μ)/BR(B+K+e+e) [89-91], and RK=BR(BKμ+μ)/BR(BKe+e) [92]. Various global fits to corresponding Wilson coefficients are also carried out [93-96], thus indicating that the negative contribution to the Wilson coefficient associated with the (ˉsRγμbL)(ˉμγμμ) operator is preferred in explaining the anomalies. We can explain the anomalies by introducing a U(1)μτ gauge symmetry when we include a few extra field contents such as vector-like quarks [97-104].

      Hence, it is worthwhile to consider a model with multi-charged particles– vector-like quarks, vector-like leptons, and charged scalar fields– under a local U(1)μτ framework, where we can combine the ideas in the model discussed in [38] and [97, 102]. The advantages of this approach are as follows: (1) we can constrain the flavor structure of Yukawa couplings associated with extra fermions, to suppress the constraints from lepton flavor violations (LFVs), (2) we have more contributions to muon g2 from one loop diagrams with Z and vector-like leptons, (3) collider signature is controlled by the U(1)μτ charge assignment to provide predictions. Subsequently, we investigate if both muon g2 and B-anomalies can be explained simultaneously by analyzing the correlation among the parameters considering experimental constraints, and then discuss collider physics to demonstrate possible signatures of this scenario.

      In this paper, we discuss the model introducing multi-charged fields (scalars and fermions) under a local U(1)μτ framework, as an extension of the model in [38] and [97, 102]. We investigate contributions to muon g2 from one-loop diagrams, including the new particles such as vector-like lepton, charged scalar, and Z boson. Extra vector-like quarks are introduced and Wilson coefficient is calculated to explain B-anomalies. Constraints from meson anti-meson mixing are discussed in addition to LFV and Z decays. Then, we explore the parameter region accommodating both muon g2 and B-anomalies. We search for the parameters satisfying all the constraints, and from the allowed model parameters, we consider the benchmark points (BP's) for the collider study.

      Because the multi-charged fields can be produced at the Large Hadron Collider (LHC), the signature of the exotic charged particles are also explored. We particularly focus on the LHC signatures of an exotic lepton doublet. Here, the exotic leptons decay via the charged scalar, which in turn produces different collider signatures w.r.t the standard scenario, where exotic leptons (singly charged) decay into SM particles directly (Wν, Z, and H) [105]. We will demonstrate that a small mass difference between the charged scalar and the exotic lepton is naturally favored by the sizable muon (g2). Hence, the collider signature of this particular model will contain very soft muons. We particularly focus on the signature of the two oppositely charged muon and tau pairs at LHC.

      The remainder of this paper is organized as follows. In Sec. II, we present the setup of the model and formulate the Wilson coefficient for B-decay, meson anti-meson mixing, LFV's, muon g2, and Z boson decays. In Sec. III, we perform numerical analysis to identify the allowed region of parameter space. In Sec. IV, we discuss possible extension of the model by introducing the U(1)μτ gauge symmetry and discuss the collider physics signature. We conclude in Sec. V.

    II.   MODEL SETUP AND FORMALISM
    • We consider a model with gauge symmetry GSM×U(1)μτ where GSM is the SM gauge symmetry and U(1)μτ is an extra gauge symmetry. In our setup of the model, we introduce isospin doublet fermions La[ψa,ψa]T(a=13), Qa[q1/3a,q4/3a]T[ua,da]T and a singly-charged boson s+, as presented in Table 1; here x and y for U(1)μτ refer to any real number, and the SM quarks are not charged under U(1)μτ. For vector-like fermions, we introduce three generations to match with the SM. We also introduce three right-handed neutrinos with the U(1)μτ charge. In this paper we do not discuss neutrino mass. Neutrino masses under U(1)μτ can be found e.g. in Refs. [106, 107]. Here, we also introduce a scalar field φ with non-zero VEV to break U(1)μτ spontaneously. The Lagrangian involving the interaction of new particles and SM, including the potential, is given by,

      LLμ LLτ eRμ eRτ νRμ νRτ L Q H s+ φ
      SU(3) 1 1 1 1 1 1 1 3 1 1 1
      SU(2)L 2 2 1 1 1 1 2 2 2 1 1
      U(1)Y 12 12 1 1 0 0 32 56 12 +1 0
      U(1)μτ 1 1 1 1 1 1 1+x x 0 x y

      Table 1.  Charge assignments of fields under SU(2)L×U(1)Y×U(1)μτ for the extended model. We introduce three generations of vector-like fermions L and Q.

      LnY=f2aˉLL2LRas++giaˉQLiQRas++hijˉLcLiLLjs++kij¯νcRieRjs++MQaˉQLaQRa+MψaˉLLaLRa+h.c.=f2a[ˉν2PRψas++ˉ2PRψas+]+gia[ˉuiPRuas+ˉdiPRdas+]+hij[ˉνciPLjs+ˉciPLνjs+]+kij¯νcRieLjs++MQaˉQLaQRa+MψaˉLLaLRa+h.c.,

      (2)

      V=μ2H|H|2+μ2S|s+|2+λH|H|4+λs|s+|4+λHs|H|2|s+|2+μ2φ|φ|2+λφ|φ|4+λHφ|H|2|φ|2+λSφ|s+|2|φ|2,

      (3)

      where (i,j,a)=13 represent generation indices, (iσ2), with σ2 being the second Pauli matrix, and LL[R]a(QL[R]a)PL[R]La(Qa). The SM Yukawa term yiiˉLLieRiH provides masses for charged leptons (miyiiv/2) by developing a nonzero vacuum expectation value (VEV) of H, which is denoted by Hv/2. The exotic fermion mass eigenvalues are, respectively, MQ,Mψ for Q', L'. We expect that the interaction term involving hij plays a role in the s+ decay into SM fields appropriately. However, because this term negatively contributes to the muon g2, we assume the scale of hij is not significantly large. This implies that we do not discuss the LFVs and muon g2 of this term. In addition, note that non-zero components of hij and kij are changed by our choice of parameter x; hence, the decay pattern of s+ depends on x. More concretely, the structure of the third and fourth terms in Eq. (2) depends on the value of x, such that

      hijˉLcLiLLjs+=h{12,13}ˉLcL{e,e}LL{μ,τ}s+for  x={1,1},

      (4)

      kij¯νcRieRjs+=k{12(21),13(31),22,33}¯νcR{e(μ),e(τ),μ,τ}eR{μ(e),τ(e),μ,τ}s+for  x={1,1,2,2},

      (5)

      where we cannot have the Yukawa interaction for x0. Therefore, the decay pattern of s+ is determined by the choice of x, where we consider that our right-handed neutrinos are assumed to be light, such that s+ can decay into states containing them. For x=2, the constraint from the collider experiment is weaker because s± only decays into the third generation of leptons, while we have stronger constraints for x=±1 or 2, because it decays into electrons and/or muons. Hence, we chose x=2 in our numerical analysis. In addition, we do not have an extra term in any choice of y, where x0 and y0.

      In a scalar sector, we assume coupling λHφ is small, such that mixing between φ and H is negligible for simplicity. Under the assumption, the VEV of φ is simply given by vφμ2φ/λφ. After φ develops a VEV, we have massive Z boson, whose mass is given by

      mZ=ygvφ,

      (6)

      where g denotes the gauge coupling associated with U(1)μτ. The mass eigenvalue of s+ is given by

      mS=μ2S+λHs2v2+λSφ2v2φ.

      (7)

      In our numerical analysis, we take mS as a free parameter.

    • A.   M¯M mixing

    • The parameter space of our model get constrained from the neutral meson mixings, where the VLQs appear in the loop. The relevant expressions, as presented in [108], are

      ΔMQmQf2Q3(4π)23a,b=1Re[gkagaigjbgb]Fbox(MQa,MQb,ms),

      (8)

      Fbox(m1,m2,m3)=[dx]3zxm21+ym22+zm23,

      (9)

      where [dx]310dxdydzδ(1xyz), BsˉBs mixing corresponds to (i,j,k,)=(2,3,3,2), BdˉBd mixing corresponds to (i,j,k,)=(1,3,3,1), while KˉK and DˉD correspond to (i,j,k,)=(1,2,2,1). The neutral meson mixing formulas should be lower than the experimental bounds, as given in [108, 109]:

      ΔmK3.48×1015 [GeV],

      (10)

      3.29×1013 [GeV]ΔmBd+ΔmSMBd3.37×1013 [GeV],

      (11)

      1.16×1011 [GeV]ΔmBs+ΔmSMBs1.17×1011 [GeV],

      (12)

      ΔmD6.25×1015 [GeV],

      (13)

      where we have taken the 3σ interval, and mM and fM are the meson mass and the meson decay constant, respectively. The following values of the parameters are used in our analysis: fK0.156 GeV, fBd(Bs)0.191(0.274) GeV [110, 111], fD0.212 GeV, mK0.498 GeV, mBd(Bs)5.280(5.367) GeV, and mD1.865 GeV. The SM contributions are given by [112]:

      2.96×1013 [GeV]ΔmSMBd5.13×1013 [GeV],

      (14)

      1.06×1011 [GeV]ΔmSMBs1.44×1011 [GeV].

      (15)

      Subtracting the SM contributions from the experimental results, the following bounds can be obtained:

      1.85×1013 [GeV]ΔmBd4.05×1014 [GeV],

      (16)

      2.77×1012 [GeV]ΔmBs1.07×1012 [GeV].

      (17)
    • B.   bsiˉj decay

    • In our model, we apply the same mechanism in [97, 102] to generate ΔCμ9 using Z interaction; other mechanisms with U(1)μτ can be found in [103, 104]. We obtain the contribution to ΔCμ9 from diagrams in Fig. 1. Subsequently, we obtain the contribution to ΔCμ,Z9 as in [97, 102]

      Figure 1.  Diagrams that contribute to ΔCμ9.

      ΔCμ,Z9xg2(4π)2m2ZCSM3a=1g3aga2×[dx]2ln(Δ[MQa,mS]Δ[mS,MQa]+M2Qaxm2S+yM2Qa),CSMVtbVtsGFαem2π,Δ[m1,m2]=xm21+ym22,

      (18)

      where [dx]210dxdyδ(1xy) and quark masses are ignored. We can obtain ΔCμ,Z91 satisfying all the experimental constraints as shown in [97, 102] with MQa=O(1) TeV, mS=O(100) GeV and mZ=O(100) GeV, where Z contribution to muon g2 is negligible in this region.

      Here, we simplify the above formula by carrying out an integration

      ΔCμ,Z9ag3aga2xg22(4π)2m2ZCSM.

      (19)

      In addition, we also obtain an Effective Lagrangian to induce bsˉ decay via box diagram, such that

      L[box]=a,bg2aga3f2bfb24(4π)2(ˉsγμPLb)×(ˉ2γμ2ˉ2γμγ52)Fbox(MQa,Mψb,ms),

      (20)

      which corresponds to O9=O10 [93].

      ΔCμ[box]9=ΔCμ[box]10a,bg2aga3f2bfb24(4π)2CSMFbox(MQa,Mψb,ms),

      (21)

      where CSMVtbVtsGFαem2π. In total, we obtain the new physics contribution to the Wilson coefficient, ΔC9, as

      ΔCμ9=ΔCμ,Z9+ΔCμ[box]9.

      (22)

      Furthermore, we should consider the diagrams replacing Z by Z in Fig. 1, which induce flavor universal contributions to C9 and C10 via the Z boson exchange. Calculating the diagrams, we obtain

      ΔC9(Z)ag3aga2g224(4π)2m2Zc2WCSM×(12+43s2W)(12+2s2W),

      (23)

      ΔC10(Z)ag3aga2g228(4π)2m2Zc2WCSM(12+43s2W),

      (24)

      where cW=cosθW, with θW being the Weinberg angle. Because structures of C9,10(Z) are similar to ΔCμ,Z9, we obtain the relationship

      ΔC9(Z)ΔCμ,Z9g22m2Zc2Wm2Zxg212(12+43s2W)(12+2s2W),

      (25)

      ΔC10(Z)ΔCμ,Z9g22m2Zc2Wm2Zxg214(12+43s2W).

      (26)

      Then, the bsμˉμ anomalies can be explained by ΔCμ,Z9=0.97 as the best fit value, [1.12,0.81] at 1σ, and [1.27,0.65] at the 2σ interval [96]. The flavor universal ΔC9(Z) is significantly smaller than ΔCμ,Z9, owing to the suppression factor (1/2+2s2W). For ΔC10(Z), we consider constraint from the Bsμ+μ measurement. Recent LHCb measurement of the branching ratio is given by [113, 114]

      BR(B0sμ+μ)exp=(3.09+0.46+0.150.430.11)×109,

      (27)

      where the first uncertainty is statistical and the second one is systematic. We can estimate the branching ratio in the model, such that [115]

      BR(B0sμ+μ)th=|10.24ΔCμμ10|2BR(B0sμ+μ)SM,

      (28)

      where BR(B0sμ+μ)SM=(3.65±0.23)×109 is the theoretical predication in the SM [116]. In the numerical analysis, we impose that the branching ratio in our model is within the1σ region in Eq. (27). In addition, note that x<0 is preferred because we realize a positive C10 to fit the data.

    • C.   Lepton flavor violations and muon anomalous magnetic moment

    • In our model, we do not have lepton flavor violations from Yukawa coupling fia because only components associated with muon, f2a, are non-zero. Hence, we only focus on the contribution to muon g2 from the Yukawa interactions.

      The muon anomalous magnetic moment (Δaμ): We can estimate the scalar loop contribution to the muon anomalous magnetic moment via (muon g2), which is given by

      ΔaSμmμ(aL+aR)22.

      (29)

      The amplitude aL/R can be expressed as,

      (aL)22(aR)22mμa=13f2afa2(4π)2×[F(Mψa,mS)+2F(mS,Mψa)],

      (30)

      where MψMψ.

      It is worthwhile considering Δaμ via Z, although it would not be required because we already have the contribution via f2a, and the preferred mass range is lighter than that for the B anomalies. The Z boson loop contribution is obtained as [117]

      ΔaZμ=g2m2μ4π210dxx2(1x)x2m2μ+(1x)m2Z.

      (31)

      In summary, muon g2 is given by

      Δaμ=ΔaSμ+ΔaZμ.

      (32)

      The measured value exhibit a 3.3σ deviation from the SM prediction, given by Δaμ=(26.1±8)×1010 [3], which is also a positive value. Note here that the charged scalar contribution using h23 is negligible, as we consider h23 to be small, as discussed below Eq. (3).

    • D.   Flavor-conserving leptonic Z boson decays

    • Here, we consider the Z boson decay into two leptons using the Yukawa terms involving f2a at one-loop level [25]. Because some components of f2a are expected to be large, to obtain a sizable Δaμ value, the experimental bounds on Z boson decays could be an issue at the one-loop level. Note that Z boson decays are modified only when the second generation of leptons are involved due to the U(1)μτ symmetry. This is why we consider the flavor conserving processes of Z boson only.

      First, the relevant Lagrangian is given by

      Lg2cw[ˉγμ(12PL+s2W)+12ˉνγμPLν]Zμ+g2cw[(12PL+s2W)ˉψ+γμψ+(12PL+2s2W)ˉψ++γμψ]Zμ+ig2s2WcW(s+μssμs+)Zμ,

      (33)

      where s(c)Wsin(cos)θW0.23 represents the sine (cosine) of the Weinberg angle. The decay rate of the SM at tree level is then given by

      Γ(Zi+j)SMmZ12πg22c2W(s4Ws2W2+18)δij,

      (34)

      Γ(Zνiˉνj)SMmZ96πg22c2Wδij.

      (35)

      Combining all the diagrams in Fig. 2, the ultraviolet divergence cancels out and only the finite part remains [25] and is given by,

      Figure 2.  Feynman diagrams for Ziˉj (up) and Zνiˉνj (down).

      ΔΓ(Zμμ+)mZ12πg22c2W[|B22|22Re[A22(B)22](s2W2+18)],

      (36)

      ΔΓ(Zνμˉνμ)mZ24πg22c2W[|Bν22|214],

      (37)

      where,

      A22s2W,B2212af2afa2(4π)2G(Mψa,mS),Bν2212+af2afa2(4π)2Gν(Mψa,mS),

      (38)

      G(Mψa,mS)s2W(12+s2w)H1(Mψa,mS)(12+s2w)2H2(mψa,mS)+(12+2s2w)H3(Mψa,mS),

      (39)

      Gν(Mψa,mS)s2W(12+s2w)H1(Mψa,mS)12H2(Mψa,mS)+(12+s2w)H3(Mψa,mS),

      (40)

      H1(m1,m2)=m41m42+4m21m22ln[m2m1]2(m21m22)2,

      (41)

      H2(m1,m2)=m424m21m22+3m414m22(m222m21)ln[m2]4m41ln[m1]4(m21m22)2,

      (42)

      H3(m1,m2)=m21(m21m22+2m22ln[m2m1](m21m22)2).

      (43)

      Notice here that the upper indices of B and G; ,ν, respectively, represent pairs of the muon and muon-neutrino final states. We consider ψ as ψ inside the argument of G, while ψ as ψ inside the argument for Gν. The current bounds on the lepton-flavor-(conserving) changing Z boson decay branching ratios at 95 % CL are given by [79]:

      ΔBR(ZInvisible)i,j=13ΔBR(Zνiˉνj)<±5.5×104,

      (44)

      ΔBR(Zμ±μ)<±6.6×105,

      (45)

      where ΔBR(Zfiˉfj) (i=j) is defined by

      ΔBR(Zfiˉfj)Γ(Zfiˉfj)Γ(Zfiˉfj)SMΓtotZ,

      (46)

      where the total Z decay width ΓtotZ=2.4952±0.0023 GeV [79]. We consider all these constraints in the numerical analysis in the next section.

    • E.   Constraints for Z interaction

    • Here, we discuss the experimental constraints for the gauge interaction associated with Z. The gauge coupling and Z mass are restricted by the neutrino trident process νNνNμ+μ, where N is a nucleon [6, 118]. The approximated bound is given by mZ/g550 GeV for mZ>1 GeV, and we apply the bound in our numerical analysis below.

      The gauge interaction is also constrained by the LHC experiment searching for the signal of ppμ+μZ(μ+μ), as given in [119]. The experimental results put the constraint on a new gauge coupling, in the mass range 5mZ70 GeV. We will compare parameter region, explaining B anomalies with the constraint.

    • F.   Decay of charged scalar

    • Finally, we discuss the decay of charged scalar that provides implication to collider physics when we introduce U(1)μτ symmetry. As discussed below, Eq. (2) charged scalar decays into only the third generation of leptons when we chose x=2. We then chose x=2 to relax the collider constraint from the charged scalar signature. The decay width of s+ for x=2 is given by

      Γs±τ±RνRτk23316πmS,

      (47)

      where we ignored the lepton mass in the final state assuming light right-handed neutrino. In addition, we assume right-handed neutrinos are long-lived and it will be just missing energy at collider experiments. Furthermore, note that the lightest particle among Q, L, and s+ would be stable when there is no interaction associated with hij or kij in other choices of x value.

    III.   NUMERICAL ANALYSIS
    • In this section, we perform a numerical analysis to search for parameter sets that accommodate all the phenomena discussed above. Here, we scan our relevant free parameters {gia,f2a,g,mZ,Mψa,MQa,mS} globally in the following range:

      gia[105,1],f2a[102,1],g[103,1],mZ[10,1000] GeV,Mψ1[100,500] GeV,Mψ2[Mψ1,750] GeV,Mψ3[Mψ2,1000] GeV,MQ1[1000,5000] GeV,MQ2[MQ1,5000] GeV,MQ3[MQ2,5000] GeV,mS[Mψ120,Mψ110] GeV,

      (48)

      where we also chose x=2 for the U(1)μτ charge assignment. Here, we chose Mψ1 and mS values to be nearly degenerated to avoid constraints from the heavy-charged lepton search at collider experiments. We find that bsμˉμ and the neutral meson mixing mainly depends on following Yukawa coupling combinations:

      Cμ9g21g13|f21|2,ΔmKg21g11,ΔmBsg31g12,ΔmBdg31g11,ΔmDg11g12.

      (49)

      Because we would like to increase Cμ9 to as large as possible, while all the meson mixings should be within the experimental ranges, the following hierarchy is preferred

      g11<<g21g31.

      (50)

      Then, we estimate Cμ9 and muon g2 imposing experimental constraints. In Fig. 3 we present the allowed parameter space in terms of mZ and g to explain the bsμˉμ anomalies via ΔCZ9 within the 1σ region of global fit. We also present the parameter region excluded by the LHC measurement searching for ppμˉμZ(μˉμ) process [119]. We determine that the parameter region of mZ50 GeV is excluded by the LHC constraints while heavier Z region can accommodate the B anomalies. For the allowed region, the upper limit of g for fixed mZ is determined by the constraint from the neutrino trident while the lower limit is given by constraint from BR(B0sμ+μ). Consequently, we determine the narrow range of parameter space where the region close to the neutrino trident limit mZ/g>550 is allowed. Note that the maximum |Cμ[box]9| is 0.115 at most, which is out of the 3σ range of experimental results due to the stringent constraint emerging from ΔmBs, because they (ΔCμ[box]9 and ΔmBs) are proportional to the same combination g31g21. If one extends gai to be complex, then one can evade the constraint of ΔmBs and keep large value of |ΔCμ9|. However, in this case, another experimental bound of CP asymmetry ACP arises and it gives more stringent constraint [120]. Therefore, we need the contribution from Z interaction to explain B anomalies.

      Figure 3.  (color online) Allowed points in the parameter space of mZ and g that can explain the anomaly of bsμˉμ, providing ΔC9 within 1σ interval of global fit. We also present the region excluded by the ppμ+μZ(μ+μ) search at the LHC experiment.

      Next, we show muon g2 for allowed parameter sets satisfying all experimental constraints and explaining the B anomalies. In the left(center) plot of Fig. 4, we present the contribution to muon g2 from the scalar (Z) loop as a function of Mψ1(mZ), and the total muon g2 is shown in the right plot of the figure. We find that the contribution from the Z loop can be larger than 2×1010 for mZ600 GeV. Note here that the upper bound up to 600 GeV comes from mZ/g>550 GeV while that above 600 GeV comes from our choice of g<1. The contribution from the scalar loop can be larger than 1010 for mψ1260 GeV. In particular, it can be close to 109 for mψ1100 GeV region. It is thus possible to explain muon g2 within the 2σ level when we add both Z and scalar contributions for light mψ1 region. We also note that the upper bound on f21 is 0.6, which restricts the maximum value of ΔaSμ. Here, this upper bound of f21 originates from the constraints of Z boson decays.

      Figure 4.  (color online) Left : Contribution to muon g2 from scalar loop diagrams. Center : Contribution to muon g2 from the Z loop diagram. Right: Sum of scalar and Z contributions. The regions between solid, dashed, and dotted lines indicate the 1 σ, 2 σ, and 3 σ regions of deviation between the observed value and SM prediction respectively.

    • A.   Collider physics and constraints

    • As discussed in the previous subsections, to get sizable muon g2 that satisfies the flavor constraints together, the mass scale (M) of the exotic lepton doublet is required to be light; to obtain ΔaSμO(1010) we need M300 GeV. Here, we are interested in the production and decay modes of the doubly charged vector like lepton (VLL) given by,

      ppψ++ψ,ψ++(μ+s+)μ+(νll+),ψ(μs)μ(ˉνll).

      Hence, the final state is 1 oppositely charged muon pair + 1 oppositely charged lepton (l) + MET. Because we choose f21=0.5, ψ±± will decay mostly in to muon and a charged scalar. Now the coupling of the charged scalar with the SM lepton and neutrino is defined by k33, as discussed in Sec. II.F, and we consider it to be of the order 0.01 where s+ decays into τ+ˉνRτ with 100 % branching ratio.

      Vector-like leptons and quarks are constrained from the collider physics experiments. The ATLAS Collaboration performed a search for heavy lepton resonances decaying into a Z boson and a lepton in a multi lepton final state at a center-of-mass energy of 8 TeV [121], constraining the singlet VLL model and excluding its mass range of 114 – 176 GeV. For the doublet VLL model, the L3 Collaboration at LEP placed a lower bound of 100 GeV on additional heavy leptons [122]. It has been demonstrated in [105] and [123] that the VLL's in the mass range 120 – 740 GeV are excluded with 95% CL in different multilepton signals. In these analyses, the vectorlike leptons were singly charged and hence it only decays to a SM boson (H, W, Z) and SM leptons. However, in our case, VLLs decay in to a charged scalar and muon specifically, followed by the decay of the off-shell or on-shell charged scalar into a neutrino and another τ lepton. Here, we assume that MψmS and the produced muon is less energetic, which would be missed at detectors by the kinematical cut. Hence, the characteristic of our signal is significantly different from [105] and [123]. Similarly, for vectorlike quarks, the current limit is 1–1.3 TeV [124], but in our model, it decays via the charged scalar, thereby resulting in different final states not searched so far at LHC.

      LEP experiment excludes the charged Higgs masses below 80 GeV [125]. At the LHC, searches for the charged Higgs have been performed through various decay channels, H±cs [126], tb [127] and ντ± [128], and most of these searches exclude m±H<mt. Other searches such as that of [128] give upper limit on the cross section× BR as a function of the charged scalar mass. Notice that the s± only pair produced via theZ/γ propagator in the schannel and the cross section is below the current limit.

      In this analysis, we made our selections differently from [105] and [123]. As a negligible mass difference between the charged Higgs and the VLL naturally implied from the muon (g2), the muon will have a very small pT(10) GeV, but the other two leptons will have a much higher pT. Other two leptons are τ in our case because we chose the U(1)μτ charge, such as charged scalar couples only τ and τ-neutrino. This scenario is still allowed for the VLL mass 300 GeV. There are scenarios [129, 130] when the doubly charged VLLs decay to a W± and lepton(l±), giving a final state of two oppositely charged lepton pair (l±) + MET. In this study we have focused on a more exotic scenario, as proposed by the U(1)μτ extended model, where the charged exotic leptons decays to tau lepton and a neutrino via the charged scalar. Hence, in this study, we select our signal to be 1 oppositely charged muon pair with very small pT + 1 oppositely charged tau pair with a moderate pT + MET, and we keep the mass difference between the charged Higgs and the VLL 10 GeV. The same final state has also been studied for a more general model of vector-like leptons in [131]. One of the advantages of VLL with small mass is that the cross section is large, which can negate the effect of the suppression due to more than one tau tagging. Moreover, in the VLL signatures studied so far by CMS and ATLAS, the assumption was that VLL decays to a W or Z, which is unlikely in our case. Consequently, the W/Z veto can increase the signal efficiency.

      We express the model Lagrangian of Eq. (2) in FeynRules (v2.3.13) [132, 133]. We generate the model file for MadGraph5_aMC@NLO (v2.2.1) [134] using FeynRules. Then, we calculate the production cross section using the NNPDF23LO1 parton distributions [135] with the factorization and renormalization scales at the central m2T scale after thekT-clustering of the event. We have computed the signal cross section of ppψ++ψ, where p=q,ˉq,γ. The cross sections are normalized to the 5-flavor scheme. The inclusion of the photon PDF increases the signal cross section significantly, as the coupling is proportional to the charge of the fermion. We plot the the production cross section in Fig. 5 for 13 TeV, as well as 27 TeV. Production cross section pps+s is much smaller than that of ψ++ψ and mass region M(mS)<150 GeV is still allowed by current experimental constraint [128]. After showering events in PYTHIA [136], events were passed through DELPHES 3 [137] for the detector simulation. In DELPHES, we choose the isolation cut for leptons to be \Delta R_\max} = 0.5, to ensure no hadronic activity inside this isolation cone. While generating the events, we kept the min pT for muons to be 6 GeV, and also follow other trigger requirements for the soft muons following [138]. The tau tagging efficiency is considered to be 0.6, and the misidentification efficiency is 0.01.

      Figure 5.  (color online) The cross section for the pair production process ppψ++ψ as a function of the VLL mass at 13 TeV and 27 TeV.

      The pT distribution of the leading and subleading tau and muon is presented in Fig. 6 for BP1. In Fig. 7 (left), we present the transverse missing energy and HT(l)=ipT(l)i distribution and (right) the ratio MET/meff (meff=ET+HT(l)+HT(j)), which is effective to reduce the QCD-jet backgrounds. Based on these distributions we select a set of simple cuts on different kinematic variables.

      Figure 6.  (color online) The transverse momentum distribution of the leading(1) and the subleading(2) muon (left) and tau pairs (right) in unweighted events of ppΨ++Ψ at 13 TeV p-p collision for BP1.

      Figure 7.  (color online) The transverse missing energy (MET) and sum of all lepton pT distribution are presented in the left. In the right, the distribution is for the ratio of the MET and meff. Events are unweighted and generated by ppΨ++Ψ at 13 TeV p-p collision for BP1.

      Selection 1:

      ● Opposite sign same flavor pair of mu and tau (μ+μ) + (τ+τ),

      pT(μ1)> 6 GeV, pT(μ2)>6 GeV, pT(τ1)> 60 GeV, pT(τ2)>40 GeV,

      |η(μ,τ)|<2.5, ΔR(l,l)>0.3,

      Selection2:

      ● b-jet veto, MET>100 GeV, HT>150 GeV,

      MET/meff>0.5,

      Selection3:

      Z veto with MZ±10 GeV.

      We present the signal cross section after the selections in Table 2 for three BP's. It can be observed that for this multilepton channel, the cross section is well above 1 fb after the selections. The signal does not suffer much from the Z-veto, which is a big advantage for our signal as the Z veto is effective for reducing the backgrounds from Z decays. The b -jet veto and the requirement of the higher ratio of MET and meff will also be effective to reduce the background for these types of signal. For the discussion on the background of this particular channel, refer to [131]. In general, the multilepton channel possesses less background compared to the other processes. After Selection 3, the number of events at 150 fb1 is always more than 150 if background is very small, which makes this channel a good candidate look for new physics at 13 TeV LHC run.

      BP1 BP2 BP3
      k33=0.01 g310.368,MQ=1083 g310.32,MQ=1200 g310.1080,MQ=1201
      g210.166,ms272 g210.2060,ms230 g210.6240,ms304
      g110.0468,M284 g110.0014,M250 g110.0071,M320
      Selection 1 3.44 (9.58) fb 2.87 (11.06) fb 2.67 (9.62)fb
      Selection 2 1.76 (7.31)fb 1.22 (4.88)fb 1.49 (4.36)fb
      Selection 3 1.63 (5.82)fb 1.06 (3.28)fb 1.38 (4.96)fb

      Table 2.  Signal cross section (fb) after the selections at three different benchmark points at 13 TeV and 27 TeV (italic). Masses are in GeV.

    IV.   CONCLUSION
    • We analyzed muon (g2), LFVs, Z decays, ΔCμ9 for B-anomalies, and MˉM mixing in a framework of multi-charged particles, which includes exotic scalars, leptons, and quarks under the local U(1)μτ. Owing to the gauge symmetry, we can suppress the LFV process, which could appear from the Yukawa interactions among the exotic lepton, charged scalar, and SM lepton. Accordingly, we found that the sizable Yukawa couplings are naturally allowed to explain muon g2. First, we formulated phenomenological observables mentioned above in our model and performed numerical analyses to search for allowed parameter sets.

      Via numerical calculations, we determined that our ΔCμ9 can accommodate B-anomalies where the Z boson contribution is dominant. However, the contribution from the box diagram in ΔCμ[box]9 can only reach 0.1 when we impose constraints from the Zνiˉνj invisible decay, Zμˉμ decay, and BsˉBs mixing. This is owing to the stringent constraints from BsˉBs mixing and Zμˉμ, which restrict the relevant Yukawa coupling constants. We demonstrated that the muon g2 in our model is a sum of the contributions from the scalar boson loop and Z loop diagrams. It was inferred that we can explain muon g2 within the 2σ level when we include both of these contributions. Finally, we studied the collider physics focusing on the production of doubly charged leptons, using some benchmark points allowed by the numerical analysis. We verified that the channel with pairs of oppositely charged muon and tau has some unique features that distinguish our model signatures from other vector-like lepton signatures at LHC. The exotic vector-like quarks and the Z will also provide interesting collider phenomenology; however, we reserve it for a future study.

    ACKNOWLEDGMENTS
    • This research was supported by the Korean Local Governments - Gyeongsangbuk-do Province and Pohang City (H.O.). H. O. is sincerely grateful to the KIAS member, and log cabin at POSTECH for providing a comfortable space to come up with this project. N.K. acknowledges the support from the Dr. D. S. Kothari Postdoctoral scheme (201819-PH/18-19/0013). N. K. also acknowledges "(9/27-28 @APCTP HQ) APCTP Mini-Workshop - Recent topics on dark matter, neutrino, and their related phenomenologies" where the problem was proposed and also thanks the hospitality of APCTP, Korea.

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