Dark matter, Z^{\prime}, vector-like quark at the LHC and b\to s μμ anomaly
DDark matter, Z (cid:48) , vector-like quark at the LHC and b → sµµ anomaly Wei Chao , Hongxin Wang , Lei Wang , Yang Zhang Center for Advanced Quantum Studies, Department of Physics,Beijing Normal University, Beijing, 100875, P. R. China Department of Physics, Yantai University, Yantai 264005, P. R. China School of Physics and Microelectronics,Zhengzhou University, ZhengZhou 450001, P. R. China
Abstract
In this paper, combining the b → sµ + µ − anomaly and dark matter observables, we study thecapability of LHC to test dark matter, Z (cid:48) , and vector-like quark. We focus on a local U (1) L µ − L τ model with a vector-like SU (2) L doublet quark Q and a complex singlet scalar whose lightestcomponent X I is a candidate of dark matter. After imposing relevant constraints, we find that the b → sµ + µ − anomaly and the relic abundance of dark matter favor m X I <
350 GeV and m Z (cid:48) < m Q < m X R < m X I ). The current searchesfor jets and missing transverse momentum at the LHC sizably reduce the mass ranges of thevector-like quark, and m Q is required to be larger than 1.7 TeV. Finally, we discuss the possibilityof probing these new particles at the high luminosity LHC via the QCD process pp → D ¯ D or pp → U ¯ U followed by the decay D → s ( b ) Z (cid:48) X I or U → u ( c ) Z (cid:48) X I and then Z (cid:48) → µ + µ − . Taking abenchmark point of m Q =1.93 TeV, m Z (cid:48) = 170 GeV, and m X I = 145 GeV, we perform a detailedMonte Carlo simulation, and find that such benchmark point can be accessible at the 14 TeV LHCwith an integrated luminosity 3000 fb − . a r X i v : . [ h e p - ph ] F e b . INTRODUCTION At present, there are several interesting excesses in B -physics measurements involvingthe transition b → s(cid:96) + (cid:96) − ( (cid:96) = µ, e ), R K ( ∗ ) ≡ B → K ( ∗ ) µ + µ − B → K ( ∗ ) e + e − . (1)The LHCb results for the R K ratio in one q bin [1, 2] and the R K ∗ ratio in two q bins [3]were found to lie significantly below one: R K = 0 . +0 . − . (stat) +0 . − . (syst) , q ∈ [1 ,
6] GeV ,R K ∗ = 0 . +0 . − . (stat) ± . , q ∈ [0 . , .
1] GeV ,R K ∗ = 0 . +0 . − . (stat) ± . , q ∈ [1 . , .
0] GeV . (2)Belle announced its measurement of R K ∗ [4] R K ∗ = . +0 . − . ± . , . ≤ q ≤ . , . +0 . − . ± . , . ≤ q ≤ . , . +0 . − . ± . , . ≤ q ≤ . , . +0 . − . ± . , . ≤ q ≤ . , . +0 . − . ± . , . ≤ q . (3)The global fits to the experimental data show the new physics (NP) model can explainthe anomalies of R ( K ) and R ( K ∗ ) by contributing to C µ . With C µ,NP =0, the best fit valuefor C µ,NP is − . ± .
16 [5].A U (1) L µ − L τ gauge boson couples only to µ ( τ ) but not to electron [6], and this type of U (1) L µ − L τ model has also been modified from its minimal version to explain b → sµ + µ − anomaly [7–23]. In Ref. [23], in addition to the U (1) L µ − L τ gauge boson Z (cid:48) and a complexsinglet S breaking U (1) L µ − L τ symmetry, a vector-like SU (2) L doublet quark Q and a complexsinglet X are introduced to produce the Z (cid:48) bs coupling large enough to explain the anomaliesof R ( K ( ∗ ) ). As the lightest component of X , X I is a candidate of dark matter (DM). Inthis paper, we will combine the b → sµ + µ − anomaly and the experimental data of DM, andstudy the capability of LHC to test dark matter, Z (cid:48) , and vector-like quark.2 ABLE I: The quantum numbers of the vector-like quark Q ≡ ( U, D ), the scalars X and S underthe gauge group SU (3) C × SU (2) L × U (1) Y × U (1) L µ − L τ .SU(3) c SU(2) L U(1) Y U(1) B − L Q +1 / − q x X q x S − q x Our work is organized as follows. In Sec. II we recapitulate the model. In Sec. III weconsider the relevant theoretical constraints and b → s flavor observables, and explain the b → sµ + µ − anomaly. In Sec. IV, we discuss the DM observables. In Sec. V, we use thecurrent searches at the LHC to constrain the parameter space, and analyze the possibilityof probing the new particles at the high luminosity LHC. Finally, we give our conclusion inSec. VI. II. THE MODEL
In addition to the U (1) L µ − L τ gauge boson Z (cid:48) , the model predicts a complex singlet S ,a complex singlet X , and a SU (2) L doublet quark Q . Their quantum numbers under thegauge group SU (3) C × SU (2) L × U (1) Y × U (1) L µ − L τ are shown in Table I.The Lagrangian which remains invariant under the SU (3) C × SU (2) L × U (1) Y × U (1) L µ − L τ symmetry is given by L = L SM − Z (cid:48) µν Z (cid:48) µν + g Z (cid:48) Z (cid:48) µ (¯ µγ µ µ + ¯ ν µ L γ µ ν µ L − ¯ τ γ µ τ − ¯ ν τ L γ µ ν τ L ) − V + ¯ Q ( i (cid:54) D − M Q ) Q + ( D µ X † )( D µ X ) + ( D µ S † )( D µ S ) − (cid:88) i =1 ( λ i ¯ q iL QX + h.c. ) . (4)Where we ignore the kinetic mixing term of gauge bosons of U (1) L µ − L τ and U (1) Y . q iL denotes the SM left-handed quark doublet with i = 1 , ,
3, and D µ is the covariant derivative.The field strength tensor Z (cid:48) µν = ∂ µ Z (cid:48) ν − ∂ ν Z (cid:48) µ , and g Z (cid:48) is the gauge coupling constant of the3 (1) L µ − L τ group. The scalar potential V is given by V = − µ h ( H † H ) − µ S ( S † S ) + m X ( X † X ) + (cid:2) µX S + h . c . (cid:3) + λ H ( H † H ) + λ S ( S † S ) + λ X ( X † X ) + λ SX ( S † S )( X † X )+ λ HS ( H † H )( S † S ) + λ HX ( H † H )( X † X ) . (5)The SM Higgs doublet H , the singlet filed S and X is expressed by H = G +1 √ ( h + v h + iG ) , S = 1 √ h + v S + iω ) , X = 1 √ X R + iX I ) , (6)Where v h = 246 GeV and v S are respectively vacuum expectation values (VeVs) of H and S , and the X field has no VeV. The mass parameters µ h and µ S in the potential of Eq. (5)are determined by the potential minimization conditions, µ h = λ H v h + 12 λ HS v S ,µ S = λ S v S + 12 λ HS v h . (7)After S acquires the VeV, the µ term makes the complex scalar X split into two real scalarfields X R , X I , and their masses are given by m X R = m X + 12 λ HX v H + 12 λ SX v S + √ µv S m X I = m X + 12 λ HX v H + 12 λ SX v S − √ µv S . (8)The discrete Z symmetry of the scalar potential in Eq. (5) makes the lightest component X to be as a candidate of DM, which we assume is X I .The two physical CP-even states h and S are from the mixing of h and h by the followingrelation, h h = cos θ sin θ − sin θ cos θ hS , (9)where θ is the mixing angle. The two CP-even Higgses mediate the DM interactions, L ( X I X I , h, S ) = − (cid:104) λ HX v H c θ − ( λ SX v S − √ µ ) s θ (cid:105) hX I − (cid:104) λ HX v H s θ + ( λ SX v S − √ µ ) c θ (cid:105) SX I . (10)4n this paper, in order to suppress the stringent constraints from the DM direct detectionand indirect detection experiments, we simply assume the hX I X I coupling is absent, namelytaking θ = 0 and λ HX = 0. For θ = 0, we obtain the following expressions, λ HS = 0 , λ H = m h v h , λ S = m S v S . (11)After S gets VEV, the U (1) L µ − L τ gauge boson Z (cid:48) obtains a mass, m (cid:48) Z = 2 g (cid:48) Z | q x | v S . (12)The complex singlet X mediates the new Yukawa interactions of the vector-like quarksand the SM left-handed quark,∆ L Yukawa = − √ (cid:88) i =1 , , (cid:0) λ u i ¯ u iL U + λ d i ¯ d Li D (cid:1) ( X R + iX I ) + h.c., (13)where we assume that the down-type quarks are already in the mass basis, and rotate theinteraction eigenstates of up-type quarks to the mass eigenstates via the CKM matrix V .Thus, λ u i ≡ (cid:80) j V ij λ j and λ d i ≡ λ i with u i = u, c, t and d i = d, s, b . We will simply set λ = 0 to remove the constraints related to the first generation quarks. As a result, λ u ismuch smaller than λ c and λ t due to the suppression of the factors of V us and V ub . III. b → sµ + µ − ANOMALY
We apply the upper bound of g Z (cid:48) /m Z (cid:48) ≤ (550 GeV) − from the neutrino trident process[24], and require g Z (cid:48) q x ≤ Z (cid:48) couplings. The tree-levelstability of the potential of Eq. (5) requires λ H ≥ , λ S ≥ , λ X ≥ ,λ HS ≥ − (cid:112) λ H λ S , λ HX ≥ − (cid:112) λ H λ X , λ SX ≥ − (cid:112) λ S λ X , (cid:113) λ HS + 2 (cid:112) λ H λ S (cid:113) λ HX + 2 (cid:112) λ H λ X (cid:113) λ SX + 2 (cid:112) λ S λ X +2 (cid:112) λ H λ S λ X + λ HS (cid:112) λ X + λ HX (cid:112) λ S + λ SX (cid:112) λ H ≥ . (14)We scan over the other parameters in the following ranges:60GeV ≤ m X I ≤ , ≤ m X R ≤ , ≤ m Q ≤ , ≤ m Z (cid:48) ≤ , ≤ m S ≤ , . ≤ λ bs ( ≡ λ b λ s ) < . λ b,s ≤ . (15)5e consider four relevant b → s flavor observables, R K ( ∗ ) , ∆ m s , B → X s γ , and R ννK ( ∗ ) ,which are introduced in detail in Ref. [23]. Here we give the expressions for calculating thefour observables briefly. A. Numerical calculations I. R K ( ∗ ) anomalies The model does not contain the tree-level Z (cid:48) - b - s flavor-changing coupling, but producesthe Z (cid:48) - b - s coupling via the one-loop involving the vector-like quarks, X R and X I . The b → sµ + µ − transition operator O µ is generated by Z (cid:48) -exchanging penguin diagrams. Thecorresponding Wilson coefficient C µ,NP is given by [23], C µ, NP9 = − √ q x G F m Z (cid:48) α Z (cid:48) α em λ s λ ∗ b V ∗ ts V tb (cid:34)
12 ( k (cid:48) ( x I ) + k (cid:48) ( x R )) − k ( x I , x R ) (cid:35) , (16)where x R,I = m X R,I /m Q , k ( x ) = x log xx − , k ( x , x ) = k ( x ) − k ( x ) x − x . (17)The prime on the k functions denotes a derivative with respect to the argument. A largemass splitting between m X R and m X I can enhance the absolute value of C µ,NP which canexplain R K ( ∗ ) anomaly. II. ∆ m s for B s − ¯ B s mixing, B → X s γ , and R ννK ( ∗ ) The model gives the new contributions to B s − ¯ B s mixing via the box diagrams involvingthe vector-like quarks, X R and X I , which can be written in the form H ∆ B =2 ,NPeff = C NP (¯ sγ µ P L b )(¯ sγ µ P L b ) . (18)Where C NP is given as [23] C NP1 = ( λ s λ ∗ b ) π M D k (1 , x R , x I ) , (19)where k (1 , x R , x I ) = k (1 , x I ) − k ( x R , x I )1 − x R . (20)At the 2 σ confidence level, the measurement of the mass difference in the B s − ¯ B s systemgives a constraint on the value of C NP [13], − . × − ≤ C NP1 ≤ . × − (GeV − ) . (21)6he model gives the new contributions to B → X s γ via the one-loop diagram involvingthe vector-like quarks, X R and X I . The Wilson coefficients C γ, g is corrected [23], C NP7 γ = √ λ s λ ∗ b V ∗ ts V tb G F M D ( J ( x I ) + J ( x R )) ,C NP8 g = − √ λ s λ ∗ b V ∗ ts V tb G F M D ( J ( x I ) + J ( x R )) , (22)where J ( x ) = 1 − x + 3 x + 2 x − x log x − x ) . (23)The experimental measurement of the inclusive branching fraction of B → X s γ is (3 . ± . × − [25], and the SM prediction is (3 . ± . × − [26]. The explanation ofexperimental values at 2 σ level requires − . × − ≤ C NP7 γ + 0 . C NP8 g ≤ . × − . (24)The model gives the additional contributions to B → K ( ∗ ) ν ¯ ν via the diagrams which areobtained by replacing the external muon lines of the b → sµ + µ − diagrams with the neutrinolines. The current experimental bounds are R ν ¯ νK < . , R ν ¯ νK ∗ < . , (at 90% C.L.) . (25)with R ν ¯ νK ( ∗ ) = B ( B → K ( ∗ ) ν ¯ ν ) exp B ( B → K ( ∗ ) ν ¯ ν ) SM . (26)In the model, the prediction value of R ννK ( ∗ ) is [23] R ν ¯ νK ( ∗ ) = (cid:80) i =1 (cid:12)(cid:12)(cid:12) C SM L + C ii, NP L (cid:12)(cid:12)(cid:12) | C SM L | = 1 + 2 (cid:12)(cid:12)(cid:12) C , NP L (cid:12)(cid:12)(cid:12) | C SM L | , (27)with C SM L ≈ − . C ,NPL = 0, and C ,NPL = − C ,NPL = − √ q x G F m Z (cid:48) α Z (cid:48) α em λ s λ ∗ b V ∗ ts V tb (cid:34)
12 ( k (cid:48) ( x I ) + k (cid:48) ( x R )) − k ( x I , x R ) (cid:35) . (28)7 .10.1250.150.1750.20.2250.250.2750.3 1000 1200 1400 1600 1800 2000 m Q (GeV) λ b s FIG. 1: The surviving samples projected on the planes of m Q versus λ bs . All the samples acco-modate the R K ( ∗ ) anomaly, and the bullets (green) and circles (red) are respectively allowed andexcluded by the ∆ m s . B. Results and discussions
After imposing the constraints mentioned above, we use the model to explain the R K ( ∗ ) anomalies. The bounds of B → X s γ and R ννK ( ∗ ) are almost satisfied in the whole parameterspace being consistent with R K ( ∗ ) . However, there is a strong correlation between ∆ m s and R K ( ∗ ) , as shown in the Eq. (16) and Eq. (19). Fig. 1 shows that R K ( ∗ ) are explained in thewhole region of 1000 GeV ≤ m Q ≤ ≤ λ bs ≤ m s imposesan upper bound on λ bs , which increases with m Q . Due to the constraints of ∆ m s , the R K ( ∗ ) anomaly can be only explained in the region of λ bs ≤ b → s flavor observables, the neutrino trident process, and thetheoretical constraints, the samples explaining the R K ( ∗ ) anomaly are projected on the Fig.2. The left panel shows that the parameters g Z (cid:48) q X and m Z (cid:48) are imposed strong constraints.Due to the constraints of the neutrino trident process, the region with small m Z (cid:48) and large g Z (cid:48) q X is empty. To accomodate the R K ( ∗ ) anomaly, m Z (cid:48) is required to increase with g Z (cid:48) q X .Since we take g Z (cid:48) q x ≤ Z (cid:48) couplings, m Z (cid:48) >
600 GeVis excluded. Similarly, g Z (cid:48) q x ≤ . m Z (cid:48) is taken as100 GeV.The right panel of Fig. 2 shows that m X I is required to increase with m X R since a sizable8
00 200 300 400 500 600 m Z ′ (GeV) g Z ′ q X m X I (GeV) m X R (GeV) m X I ( G e V ) m Q (GeV) FIG. 2: All the samples accomodate the R K ( ∗ ) anomaly, and satisfy the relevant b → s flavorobservables, the neutrino trident process, and the theoretical constraints. mass splitting between m X R and m X I is favoured to explain the R K ( ∗ ) anomaly. Because wechoose m X R ≤ m X I is required to be smaller than 900 GeV. Similarly, m X R ≤ m X I is taken as 60 GeV. IV. DARK MATTER
In the chosen parameter space, the DM can annihilation into Z (cid:48) Z (cid:48) , SS , and the SMquarks. The corresponding Feynman diagrams are shown in the Fig. 3. The X I X I → q ¯ q processes proceed through the D ( U )-exchanging t-channel diagrams. For 1 TeV ≤ m Q ≤ λ b <
1, and λ s <
1, the annihilation cross sections are very small, and their con-tributions to the relic density can be ignored. The X I X I → SS processes proceed throughthe S -exchanging s-channel diagram and the diagram of the quartic coupling X I X I SS . The X I X I → Z (cid:48) Z (cid:48) proceed through the S -exchanging s-channel diagram, the X R -exchangingt-channel diagram, and the diagram of the quartic coupling X I X I Z (cid:48) Z (cid:48) .We use micrOMEGAs [27] to calculate the relic density and the spin-independent DM-nucleon cross section. The model file is generated by FeynRules [28]. The Planck collabora-tion reported the relic density of cold DM in the universe, Ω c h = 0 . ± . X I X I → Z (cid:48) Z (cid:48) from the diagram of Fig. 3(c) only9 Z ′ Z ′ X R X I X I Z ′ Z ′ ( a ) ( b ) SX I X I Z ′ Z ′ X I X I ( c ) SSX I X I ( d ) ( f )( e ) SSX I X I X I X I Q q ¯ q FIG. 3: The Feynman diagrams for X I X I → Z (cid:48) Z (cid:48) , SS, q ¯ q . depends on three parameters g Z (cid:48) q X , m Z (cid:48) , m X I . Since the R K ( ∗ ) anomaly imposes a lowerbound on g Z (cid:48) q X , for m Z (cid:48) < m X I the annihilation cross sections of X I X I → Z (cid:48) Z (cid:48) are muchlarger than the value producing the correct relic density. Similarly, for m S < m X I theannihilation cross sections of X I X I → SS are too large to obtain the correct relic density.Therefore, we need to use the effects of forbidden channel to produce the relic density, namelythat m Z (cid:48) or m S is appropriately larger than m X I . In the calculation of the thermal averagedcross section, the kinetic energy of the DM is nonnegligible in the early universe. When themass difference is not too large and the DMs move fast, the center of mass energy exceedstwice m Z (cid:48) or m S . Therefore, the process X I X I → Z (cid:48) Z (cid:48) ( SS ) can occur in the early universewhen m X I has appropriate mass difference from m Z (cid:48) ( m S ). In addition, the temperatureat the present time is much lower than the freeze-out temperature, and the velocity of DMis much smaller than that in the early universe. The channel X I X I → Z (cid:48) Z (cid:48) ( SS ) arekinematically forbidden at the present time, therefore the experimental constraints of theindirect detection of DM can be naturally satisfied.After imposing the constraints of ”pre-DM” (denoting the R K ( ∗ ) anomaly, the relevant b → s flavor observables, the neutrino trident process, and the theoretical constraints), we10
00 150 200 250 300 350 m X I (GeV) m Z ′ − m X I ( G e V ) m S − m X I m X I (GeV) m S − m X I ( G e V ) m Z ′ − m X I FIG. 4: The surviving samples satisfying the DM relic density and the constraints of ”pre-DM”. find some samples which can achieve the correct DM relic density. The surviving samplesare project on the Fig. 4. From the left panel, we find that the relic density favors m X I <
350 GeV, and most of the surviving samples lie in the region of m Z (cid:48) − m X I <
60 GeV.For a large m S , the annihilation cross section of X I X I → Z (cid:48) Z (cid:48) from the diagram of Fig.3(a) is suppressed. Therefore, a small value of m Z (cid:48) − m X I is required to enhance the crosssection. For a large value of m Z (cid:48) − m X I , the X I X I → Z (cid:48) Z (cid:48) channel is still forbidden in theearly universe, and does not contribute to the relic density. For such case, the X I X I → SS channel will play the dominant contribution to the relic density. As shown in the rightpanel, for a large value of m Z (cid:48) − m X I , a small value of m S − m X I is required to open the X I X I → SS channel in the early universe.Exchanging an initial state X I and a final state quark of Fig. 3(f), we can obtain theFeynman diagrams which give the contributions to the cross section of the DM scatteringoff the nuclei. In the chosen parameter space, we find that the bounds of the XENON1Tfail to exclude the parameter space achieving the correct relic density [30].11 . THE DARK MATTER, Z (cid:48) , AND VECTOR-LIKE QUARK AT THE LHCA. The current constraints from the direct searches at the LHC At the LHC, the vector-like quarks D and U are produced in pairs via the QCD processes, pp → D ¯ D, U ¯ U . (29)In the chosen parameter space, the D and U have following decay modes, D → X I d i , X R d i , U → X I u i , X R u i (30)with X R → X I Z (cid:48) → X I µ + µ − , X I τ + τ − , X I ν µ ¯ ν µ , X I ν τ ¯ ν τ . (31)Since the R K ( ∗ ) anomaly and the DM relic density favor X R to be much larger than X I , D and U will mainly decay into X I s , X I b , and X I u i . In this paper, the coupling of X I and d quark is taken as zero.In order to restrict the productions of the above processes at the LHC for our model,we perform simulations for the samples using MG5 aMC-2.7.3 [31] with
PYTHIA8 [32] and
Delphes-3.2.0 [33], and adopt the constraints from all the analysis for the 13 TeV LHC inversion
CheckMATE 2.0.28 [34]. For the excluded samples, the most sensitive experimentalanalysis is the ATLAS search for the squarks and gluinos in final states containing jets andmissing transverse momentum at 13 TeV LHC with 139 fb − integrated luminosity data[35]. The final states E missT + jets are just the main signal of the D ¯ D and U ¯ U in the model.In Fig. 5, all the samples satisfy the constraints of ”pre-DM” and the DM observables.The current direct searches at the LHC exclude m Q < λ s , some sampleswith m Q around 1.8 TeV can be also excluded. With an increase of m Q , the productioncross sections of pp → D ¯ D, U ¯ U are suppressed by the phase space, and the direct searchesat the LHC can be satisfied. B. The searches for the new particles at the high luminosity LHC
Since the vector-like quark U and D are charged under the U (1) L µ − L τ , the gauge boson Z (cid:48) has the tree-level couplings to the vector-like quarks. Therefore, the model provides a12
000 1200 1400 1600 1800 2000 m Q (GeV) λ s λ b m Q (GeV) λ b λ s m Q (GeV) m X I ( G e V ) λ s m Q (GeV) m Z ′ ( G e V ) λ s FIG. 5: All the samples satisfy the constraints of ”pre-DM” and the DM observables. The squaresand bullets are respectively excluded and allowed by the current direct searches at the LHC. novel approach of searching for Z (cid:48) , the vector-like quark, and DM. Z (cid:48) is produced via theQCD process pp → D ¯ D or pp → U ¯ U followed by the decay D → s ( b ) X R → s ( b ) Z (cid:48) X I or U → u ( c ) X R → u ( c ) Z (cid:48) X I , and then decays into µ + µ − .We pick a benchmark point which accomodates the b → sµ + µ − anomaly, and satisfiesthe constraints of ”pre-DM”, the DM observables, and the current searches at the LHC.Several key input and output parameters are shown in Table II.Now we perform detailed simulations on the signal and backgrounds at the 14 TeV LHCwith high luminosity. We choose the signal to contain opposite sign di-muon ( µ + µ − ), missingtransverse momentum E missT , and multijet ( ≥ b -jet. Themajor SM irreducible background processes to this signal are t ¯ t , W W + jets, ZZ + jets, and13 Z (cid:48) (GeV) m X I (GeV) m X R (GeV) m Q (GeV) Br ( D → X I b ) Br ( D → X I s ) Br ( D → X R b ) Br ( D → X R s )170 145 1309 1930 0.63 0.14 0.19 0.04 TABLE II: Several key input and output parameters for the benchmark point.
W Z + jets.We identify the muon candidates by requiring them to have p T >
15 GeV and | η | < . R = 0 . p T >
20 GeV and | η | < .
5. We assume an average b -tagging efficiency of 80% for real b -jets.In order to suppress the contributions from the SM process, we apply the ”stransverse”mass, m T [37–39], defined as m T = min q T (cid:2) max (cid:0) m T ( p (cid:96) T , q T ) , m T ( p (cid:96) T , p missT − q T ) (cid:1)(cid:3) (32)where p (cid:96) T and p (cid:96) T are the transverse momenta of the di-muon. q T is a transverse vectorthat minimizes the larger of the two transverse masses m T , m T ( p T , q T ) = (cid:112) p T q T − p T · q T ) . (33)Fig. 6 shows the distributions of some kinematical variables at the LHC with √ s = 14TeV for the signal and the background t ¯ t . The other processes are not shown since they aresubdominant. According to the distribution differences between the signal and backgrounds,we can improve the ratio of signal to backgrounds by making some kinematical cuts. Weimpose the following cuts P j T >
290 GeV , P j T >
60 GeV , P b T >
60 GeV , ∆ R µ + µ − < . , < M µ + µ − < ,E missT > , m T > , H b(cid:96)T > . (34)Where P j T and P j T denote the transverse momentum of the hardest and the second hardestjets which include b -jet, and P b T denotes the transverse momentum of the hardest b -jet.∆ R = (cid:112) (∆ φ ) + (∆ η ) is the particle separation with ∆ φ and ∆ η being the separation inthe azimuthal angle and rapidity respectively. M µ + µ − is the invariant mass of µ + and µ − ,and H b(cid:96)T is scalar sum of transverse momentums of all the b -jets, µ ± . Since µ + and µ − of14
50 500 750 1000 1250 1500 1750 2000 P j T (GeV) E v e n t S ( s c a l e d t o o n e ) signalt ̄t 0 1 2 3 4 5 6 R μ ̄ μ − E v e n t S ( s c a l e d t o o n e ) signalt ̄t 0 500 1000 1500 2000 2500 3000 H bℓT (GeV) E v e n t S ( s c a l e d t o o n e ) signalt ̄t200 400 600 800 1000 1200 1400 m T2 (GeV) E v e n t S ( s c a l e d t o o n e ) signalt ̄t 100 200 300 400 500 600 700 ℓ μ ̄ μ − (GeV) E v e n t S ( s c a l e d t o o n e ) signalt ̄t 200 400 600 800 1000 1200 1400 μ missT (GeV) E v e n t S ( s c a l e d t o o n e ) signalt ̄t FIG. 6: The signal and the t ¯ t background distributions of P j ,T , ∆ R µ + µ − , H b(cid:96)T , m T , M µ + µ − , and E missT at the 14 TeV LHC, after requiring an opposite sign di-muon and multijet ( ≥ b -jet. the signal are from the decay of Z (cid:48) with a mass of 170 GeV, M µ + µ − appears a peak at 170GeV, and ∆ R µ + µ − favors a small value. The jets, X I and µ ± of the signal are the decayproducts of the vector-like quark with a mass of 1930 GeV, and such heavy mass leads thatthese products tend to have large transverse momentums. The distributions of m T for t ¯ t and W W +jets backgrounds peak before m W . In addition, the DM X I has a mass of 145GeV, therefore the signal events tend to have a large E missT .We compute the significance as S = n s √ n s + n b , where n s and n b are the normalized signaland background event yields, respectively. After making the kinematical cuts of Eq. (34), n b is drastically reduced, and dominated over by n s . For example, n s ∼
33 and n s + n b ∼ − at the 14 TeV LHC. Fig. 7 shows that for the benchmark point, thesignificance can reach 2 σ and 5.6 σ at the 14 TeV LHC with an integrated luminosity 400fb − and 3000 fb − . 15 Luminosity (fb -1 ) S FIG. 7: The significance versus the integrated luminosity of the 14 TeV LHC for the benchmarkpoint.
VI. CONCLUSION
In this paper we study the capability of LHC to test DM, Z (cid:48) , and vector-like quarkin a local U (1) L µ − L τ model in light of the b → sµ + µ − anomaly and the DM observables.We take m Q < m X R < b → sµ + µ − anomaly andthe DM observables favor m X I <
350 GeV and m Z (cid:48) <
450 GeV after imposing relevantconstraints from theory and b → s flavor observables. The current searches for jets andmissing transverse momentum at the 13 TeV LHC with 139 fb − integrated luminosity dataexclude m Q < pp → D ¯ D or pp → U ¯ U followed by thedecay D → s ( b ) Z (cid:48) X I or U → u ( c ) Z (cid:48) X I and then Z (cid:48) → µ + µ − . Taking a benchmark pointof m Q =1.93 TeV m Z (cid:48) = 170 GeV, and m X I = 145 GeV, we perform a detailed Monte Carlosimulation, and find that such benchmark point can be accessible at the 14 TeV LHC withan integrated luminosity 3000 fb − . 16 cknowledgment We thank Biaofeng Hou for the helpful discussions. This work was supported by theNational Natural Science Foundation of China under grant 11975013, 11775025, and by theNatural Science Foundation of Shandong province (ZR2017JL002 and ZR2017MA004). [1] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. (2014) 151601.[2] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. (2019) 191801.[3] R. Aaij et al. [LHCb Collaboration], JHEP (2017) 055.[4] M. Prim (for the Belle Collaboration), arXiv:1904.02440.[5] A. Datta, J. Kumar, D. London, Phys. Lett. B (2019) 134858.[6] X. G. He, G. C. Joshi, H. Lew and R. R. Volkas, Phys. Rev. D (1991) 22–24.[7] A. Crivellin, G. D’Ambrosio and J. Heeck, Phys. Rev. Lett. (2015) 151801.[8] W. Altmannshofer, S. Gori, S. Profumo and F. S. Queiroz, JHEP (2016) 106.[9] C.-H. Chen and T. Nomura, Phys. Lett. B (2018) 420–427.[10] S. Baek, Phys. Lett. B (2018) 376–382.[11] W. Altmannshofer, S. Gori, M. Pospelov and I. Yavin, Phys. Rev. D (2014) 095033.[12] W. Altmannshofer and I. Yavin, Phys. Rev. D (2015) 075022.[13] P. Arnan, L. Hofer, F. Mescia and A. Crivellin, JHEP (2017) 043.[14] S. Singirala, S. Sahoo and R. Mohanta, Exploring dark matter, Phys. Rev. D (2019)035042.[15] P. T. P. Hutauruk, T. Nomura, H. Okada and Y. Orikasa, Phys. Rev. D (2019) 055041.[16] A. Biswas, A. Shaw, JHEP (2019) 165.[17] Z.-L. Han, R. Ding, S.-J. Lin, B. Zhu, Eur. Phys. Jour. C (2019) 1007.[18] A. S. Joshipura, N. Mahajan, K. M. Patel, JHEP (2020) 001.[19] L. Bian, H. M. Lee, C. B. Park, Eur. Phys. Jour. C (2019) 54-58.[21] P. Ko, T. Nomura and H. Okada, Phys. Rev. D (2017) 111701.[22] D. Liu, J. Liu, C. E. M. Wagner, X.-P. Wang, JHEP (2018) 150.[23] S. Baek, JHEP (2019) 104.
24] W. Altmannshofer, S. Gori, M. Pospelov and I. Yavin, Phys. Rev. Lett. (2015) 221801.[27] G. Belanger, F. Boudjema, A. Pukhov, A. Semenov, Comput. Phys. Commun.
594 (2016).[30] E. Aprile et al. [XENON Collaboration], arXiv:1805.12562.[31] J. Alwall et al. , JHEP , (2014) 079.[32] P. Torrielli and S. Frixione, JHEP , (2010) 110.[33] J. de Favereau et al. [DELPHES 3 Collaboration], JHEP , (2014) 057.[34] D. Dercks, N. Desai, J. S. Kim, K. Rolbiecki, J. Tattersall and T. Weber, Comput. Phys.Commun. , (2017) 383.[35] ATLAS Collaboration, ATLAS-CONF-2019-040.[36] M. Cacciari, G. P. Salam, G. Soyez, JHEP (2008) 063.[37] C. Lester and D. Summers, Phys. Lett. B
99 (1999).[38] A. Barr, C. Lester and P. Stephens, J. Phys. G
063 (2008).063 (2008).