A Three-loop Neutrino Model with Leptoquark Triplet Scalars
aa r X i v : . [ h e p - ph ] M a r KIAS-P17002
A Three-loop Neutrino Model with Leptoquark Triplet Scalars
Kingman Cheung,
1, 2, 3, ∗ Takaaki Nomura, † and Hiroshi Okada ‡ Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 300 Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan Division of Quantum Phases and Devices, School of Physics,Konkuk University, Seoul 143-701, Republic of Korea School of Physics, KIAS, Seoul 130-722, Korea (Dated: November 6, 2018)
Abstract
We propose a three-loop neutrino mass model with a few leptoquark scalars in SU (2) L -tripletform, through which we can explain the anomaly of B → K ( ∗ ) µ + µ − , a sizable muon g − Keywords: ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION Recently, there was an 2 . σ anomaly in lepton-universality violation in the ratio R K ≡ B ( B → Kµµ ) /B ( B → Kee ) = 0 . +0 . − . ± .
036 by the LHCb Collaboration [1]. Inaddition, sizable deviations were observed in angular distributions of B → K ∗ µµ [2]. Theresults can be interpreted by a large negative contribution to the Wilson coefficient C of thesemileptonic operator O , and also contributions to other Wilson coefficients, in particularto C ′ [3–6].The discrepancy between the theoretical prediction and experimental value on the muonanomalous magnetic dipole moment has been a long-standing problem, which stands at 3 . σ level with the deviation from the SM prediction at [7].∆ a µ = a exp µ − a SM µ = 288(63)(49) × − . If one insists on fulfilling the muon g − σ − σ of the experimental value in anymodels, it puts a strong constraint on the parameter space. For example, it requires arelatively light spectrum in the supersymmetric particles in the MSSM in order to bring theprediction to be within 1 σ − σ of the experimental value. A number of leptoquark modelshave been proposed to solve the B → K ( ∗ ) µµ anomaly, but however it is very hard to satisfysimultaneously the muon g −
2: see for example Ref. [8].In this work, we propose a three-loop neutrino mass model with a few leptoquark scalars in SU (2) L -triplet form. We attempt to use the model to explain the anomaly of B ( ∗ ) → Kµ + µ − ,to achieve a sizable muon g −
2, and to provide a bosonic dark matter candidate, and at thesame time satisfying all the constraints from lepton flavor violations. The concrete modelis based on the SM symmetry and a Z symmetry as SU (3) C × SU (2) L × U (1) Y × Z . Themodel consists of the SM fields, 3 additional leptoquark triplet fields ∆ a , , , and one colorlessdoublet scalar field η . These fields are assigned different Z parities and hypercharges insuch a way that each of the Yukawa-type couplings contributes to either neutrino mass, B → K ( ∗ ) µµ anomaly, muon g −
2, or the dark matter interactions. In this way, althoughthe model contains more parameter, it can however explain all the above anomalies. Theachievements of the model can be summarized in the following.1. The neutrino mass pattern and oscillation can be accommodated with the Yukawa2 uarks Leptons Vector Fermions Q aL i u aR i d aR i L L i e R i L ′ i SU (3) C SU (2) L U (1) Y
16 23 − − − − Z + + + + + − TABLE I: Field contents of fermions and their charge assignments under SU (3) C × SU (2) L × U (1) Y × Z , where the superscript (subscript) index a ( i ) = 1 − coupling terms f, g, h in three-loop diagrams .2. The Yukawa coupling term f can give useful contributions to the Wilson coefficients C , in such a way that it can explain successfully the B → K ( ∗ ) µµ anomaly.3. The muon g − r . Withsome adjustment of the parameters a level of 10 − is possible.4. It provides a dark matter (DM) candidate η R , the real part of the neutral componentof the η field with correct relic density.5. The model can satisfy all the existing constraints from the lepton-flavor violations(LFVs), meson mixings, and rare B decays.This paper is organized as follows. In Sec. II, we describe the neutrino mass matrix andthe solution to the anomaly in b → sµ ¯ µ . In Sec. III, we discuss various constraints of themodel, including lepton-flavor violations, FCNC’s, oblique parameters, and dark matter. InSec. IV, we present the numerical analysis and allowed parameter space, followed by thediscussion on collider phenomenology. Sec. IV is devoted for conclusions and discussion. II. THE MODEL
In this section, we describe the model setup, derive the formulas for the active neutrinomass matrix, and calculate the contributions to b → sµ ¯ µ . See refs. [9–11] for representative three loop neutrino mass models η ∆ a ∆ a ∆ a SU (3) C ¯ ¯ SU (2) L U (1) Y
12 12 23 13 13 Z + − − − + TABLE II: Field contents of bosons and their charge assignments under SU (3) C × SU (2) L × U (1) Y × Z , where the superscript index a = 1 − A. Model setup
We show all the field contents and their charge assignments in Table I for the fermionicsector and in Table II for the bosonic sector. Under this framework, the relevant part ofthe renormalizable Lagrangian and Higgs potential related to the neutrino masses are givenby −L = y ℓ i ¯ L L i Φ e R i + f ij ¯ L L i ∆ † ( iσ ) Q cL j + g ij ¯ L ′ R i ∆ † Q L j + h ij ¯ L ′ L i ∆ † Q cL j + r ij ¯ L ′ L i ηe R j + M i ¯ L ′ L i L ′ R i − λ η † ∆ ∆ Φ ∗ − λ ′ η † ∆ ∆ ∗ Φ − λ ( η † Φ) + h . c ., (II.1)where we have defined L ′ ≡ [ N, E ] T , σ is the second Pauli matrix and we have abbreviatedthe trivial terms for the Higgs potential. The scalar fields can be parameterized asΦ = v + φ √ , η = η + η R + iη I √ , ∆ = δ (1)2 / √ δ (1)5 / δ (1) − / − δ (1)2 / √ , ∆ = δ (2)1 / √ δ (2)4 / δ (2) − / − δ (2)1 / √ , ∆ = δ (3)1 / √ δ (3)4 / δ (3) − / − δ (3)1 / √ , (II.2)where the subscript next to the each field represents the electric charge of the field, v = 246GeV, and Φ is written in the form after the Goldstone fields are aboserbed as the longitudinalcomponents of W and Z bosons. Notice here that each of the components of ∆ and η is The same contents of the field are found in the systematic analysis in the last part of Table 3 of ref. [12].
4n mass eigenstate, since there are no mixing terms that are assured by the Z and U (1) Y symmetries. On the other hand, components of ∆ and ∆ can mix via Φ ∗ Φ ∗ ∆ ∆ term.In the following analysis, we ignore such mixing effects assuming the relevant coupling issmall. Oblique parameters : Each of the mass components among ∆ i is strongly restricted by theoblique parameters. In order to evade such a strong constraint, we simply assume that eachof the components should be of the same mass [13]. Thus, we define m ∆ i as the mass forthe components of ∆ i . On the other hand, each component of η cannot have the same mass,because the neutrino mass is proportional to the mass difference between the components of η , as you shall see later. Hence, we consider the oblique parameter constraints on η , whichare characterized by ∆ T and ∆ S . Their formulae are given by [14]∆ T = F [ η ± , η I ] + F [ η ± , η R ] − F [ η I , η R ]32 π α em v , ∆ S = 12 π Z x (1 − x ) ln " xm η R + (1 − x ) m η I m η ± , (II.3)where α em ≈ /
137 is the fine structure constant, and F [ a, b ] = m a + m b − m a m b m a − m b ln (cid:20) m a m b (cid:21) , m a = m b . (II.4)The experimental bounds are given by [7](0 . − . ≤ ∆ S ≤ (0 .
05 + 0 . , (0 . − . ≤ ∆ T ≤ (0 .
08 + 0 . . (II.5)We consider these constraints in the numerical analysis. Active neutrino mass matrix : The neutrino mass matrix is induced at three-loop level asshown in Fig. 1, and its formula is generally given by M ν ij = M dν ij + M uν ij + tr., [ M uν = 2 M dν ( d → u, δ ( i )1 / → δ ( i )2 / )] , (II.6) M dν ab = 3 λ λ ′ ( m R − m I ) v √ π ) M max X ( a,b,c )=1 f ia g Tab M b h ∗ bc f Tcj F III [ r ∆ , r ∆ , r ∆ , r b , r R , r I , r d c , r d a ] , (II.7)where we used the shorthand notation m R/I ≡ m η R/I , and define M Max ≡ Max[ M b , m ∆ i , m R , m I ], r f ≡ m f /M , and the three-loop function F III is given in theAppendix. Here we adopt an assumption M max = m ∆ , and require 1TeV . m ∆ i (which5 IG. 1: Neutrino mass matrix at the three-loop level, where we have two kind of diagrams thatare running up-quarks and down quarks inside the loop. suggests x d , x u ≈ M ν ij ≈ λ λ ′ ( m R − m I ) v √ π ) m (cid:2) f g T ( M F
III ) h ∗ f T (cid:3) ij + tr., (II.8)where we have abbreviated the symbol of summation and the argument of F III . Thenwe derive the Yukawa coupling in terms of the experimental values and the parameters byintroducing an arbitrary anti-symmetric matrix with complex values A [15], that is A T + A =0, as follows: g = 12 R − ( h † ) − (cid:2) f − V MNS D ν V TMNS ( f T ) − + A (cid:3) T , (II.9)or h = 12 R ∗− ( g † ) − (cid:2) f − V MNS D ν V TMNS ( f T ) − + A (cid:3) ∗ , (II.10)where we shall adopt the former formula in the numerical analysis below, and we define D ν ≡ V TMNS M ν V MNS and parametrize as R = 3 λ λ ′ ( m R − m I ) v M F
III √ π ) m , A ≡ a a − a a − a − a . (II.11)Here we assume one massless neutrino (with normal ordering) in the numerical analysisbelow. On the term f : The new physics contributions to account for the B → K ( ∗ ) µµ anomaly [2]can be interpreted as the shifts in the Wilson coefficients C , . In our model, the relevant6ilson coefficients can be calculated as follows [13]:( C ) µµ = − ( C ) µµ = − C SM f bµ f sµ m , C SM ≡ V tb V ∗ ts G F α em √ π , (II.12)where m ∆ ≡ m δ (3)4 / , G F ≈ . × − GeV − is the Fermi constant. We can then comparethem to the best-fit values of C , from a global analysis based on the LHCb data in Ref. [3]as C = − C : − . . (II.13)Here we also have to work within the − . . C . − .
35 in order to satisfy the the LHCbmeasurement of R K = B ( B + → K + µ + µ − ) /B ( B + → K + e + e − ) = 0 . +0 . − . ± . . σ deviation from the SM prediction [13]. Notice here that various constraintsarising from the f term include B d/s → ℓ + ℓ − , ℓ i → ℓ j γ , which were given in Refs. [13]and [8], and we consider these constraints in the current numerical analysis. Although themuon g − is also induced from this term, the typical order is − ∼ − with a negativesign [13]. Thus, we neglect this contribution to the muon g − .On the terms g and h : The main constraint on g and h comes from B ( b → sγ ). Thepartial decay width for b → sγ is given byΓ( b → sγ ) ≈ α em m b π ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g † a g a F bsγ [ δ (1)2 / , a ] − h † a h a (cid:18) F bsγ [ a, δ (2)1 / ] + F bsγ [ δ (2)1 / , a ] (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (II.14) F bsγ [ a, b ] = 2 m a + 3 m a m b − m a m b + m b + 12 m a m b ln[ m b /m a ]12( m a − m b ) , (II.15)then the branching ratio and its experimental bound [16] are given by B ( b → sγ ) ≈ Γ( b → sγ )Γ tot. . . × − , (II.16)where Γ tot. ≈ . × − GeV is the total decay width of the bottom quark. In ournumerical analysis, we consider this constraint only for the g and h terms. On the term r : This term is very important in our model because it can induce a largecontribution to the muon g − η R to be the DM candidate. First of all, let us consider the LFVs processes, ℓ a → ℓ b γ , via one-loop diagrams. The branching ratio is given by B ( ℓ a → ℓ b γ ) = 48 π C ab α em G m a ( | ( a R ) ab | + | ( a L ) ab | ) , (II.17)7here m a ( b ) is the mass for the charged-lepton eigenstate, C ab ≈ (1 , . , . a, b ) = (2 , , (3 , , (3 , a L ( R ) is simply given by( a L ( R ) ) ab ≈ X i =1 r † bi r ia m a (4 π ) (cid:18) F L ( R ) lfv [ N i , η ± ] − F L ( R ) lfv [ η I , E i ] − F L ( R ) lfv [ η R , E i ] (cid:19) , (II.18) F Llfv [ a, b ] = m a + 2 m a m b + 2 m b ( m a + m b ) ln (cid:16) m b m a + m b (cid:17) m a , F Rlfv [ a, b ] = m a + m b ln (cid:16) m b m a + m b (cid:17) m a , (II.19)where the mass of E ( N ) a is defined by M E ( N ) a . Current experimental upper bounds aregiven by [17, 18] B ( µ → eγ ) ≤ . × − , B ( τ → µγ ) ≤ . × − , B ( τ → eγ ) ≤ . × − . (II.20) Muon g −
2: The muon anomalous magnetic moment is simply given by ∆ a µ ≈ − m µ [ a L + a R ] in Eq. (II.19). Experimentally, it has been measured with a high precision, and itsdeviation from the SM prediction is ∆ a µ = O (10 − ) [19]. It would be worth mentioning anew contribution to the leptonic decay of the Z boson. In our case, the Z boson can decayinto a pair of charged leptons with a correction at one-loop level, and it is proportional to theYukawa couplings related to the muon g −
2. Therefore it can be enhanced due to the largeYukawa couplings. However, we have checked that this mode is within the experimentalbound: B ( Z → ℓ ¯ ℓ ) . × − . 8 − ¯ Q mixing : The forms of K − ¯ K , B d − ¯ B d , and D − ¯ D mixings are, respectively,given by∆ m K ≈ π ) X i,j =1 " g i g † i g † j g j F Kbox [ N i , N j , δ (1)1 / ] + F Kbox [ E i , E j , δ (1)2 / ]4 ! + f † i f i f j f † j (cid:18) F Kbox [ ℓ i , ℓ j , δ (3)4 / ] + F Kbox [ ν i , ν j , δ / ]4 (cid:19) + h † i h i h j h † j F Kbox [ E i , E j , δ (2)4 / ] + F Kbox [ N i , N j , δ (2)1 / ]4 ! . . × − [GeV] , (II.21)∆ m B d ≈ π ) X i,j =1 " g i g † i g † j g j F Bbox [ N i , N j , δ (1)1 / ] + F Bbox [ E i , E j , δ (1)2 / ]4 ! + f † i f i f j f † j (cid:18) F Bbox [ ℓ i , ℓ j , δ (3)4 / ] + F Bbox [ ν i , ν j , δ / ]4 (cid:19) + h † i h i h j h † j F Bbox [ E i , E j , δ (2)4 / ] + F Bbox [ N i , N j , δ (2)1 / ]4 ! . . × − [GeV] , (II.22)∆ m D ≈ π ) X i,j =1 " g i g † i g † j g j F Dbox [ E i , E j , δ (1)5 / ] + F Dbox [ N i , N j , δ (1)2 / ]4 ! + f † i f i f j f † j (cid:18) F Dbox [ ℓ i , ℓ j , δ (3)4 / ] + F Dbox [ ν i , ν j , δ / ]4 (cid:19) + h † i h i h j h † j F Dbox [ N i , N j , δ (2)2 / ] + F Dbox [ E i , E j , δ (2)1 / ]4 ! . . × − [GeV] , (II.23) F Qbox ( x, y, z ) = 5 m Q f Q (cid:18) m Q m q + m q (cid:19) Z δ (1 − a − b − c − d ) dadbdcdd [ am x + bm y + ( c + d ) m z ] , (II.24)where ( q , q ) are respectively ( d, s ) for K , ( b, d ) for B , and ( u, c ) for D . Each of thelast inequalities in Eqs.(II.21 – II.23) represents the upper bound on the correspondingexperimental value [7]. Here we used f K ≈ .
156 GeV, f B ≈ .
191 GeV, m K ≈ .
498 GeV,and m B ≈ .
280 GeV. Dark Matter : Here we identify η R as the DM candidate, and denote its mass by m R ≡ M X . Direct detection : We have a Higgs portal contribution to the DM-nucleon scattering process, Since we assume that one of the neutrino masses to be zero with normal ordering that leads to the firstcolumn in g to be almost zero, i.e., ( g ) , , ≈
0, and so these constraints can easily be evaded. σ N ≈ . × − × (cid:18) ( λ + λ + 2 λ ) vM X (cid:19) [cm] , (II.25)where λ and λ are quartic couplings proportional to ( η † η )(Φ † Φ) and ( η † Φ)(Φ † η ), respec-tively. The current experimental minimal bound is σ N . × − cm at M X ≈
50 GeV.Once we apply this bound on our model, we obtain λ + λ + 2 λ . × − . Hence weassume that all the Higgs couplings are small enough to satisfy the constraint, and we ne-glect DM annihilation modes via Higgs portal in estimating the relic density below. Noticehere that photon and Z boson fields transform as V µ → − V µ under charge-parity ( CP )conjugation, while X is CP -even. Thus X − X − γ ( Z ) couplings are not allowed becausethey violates the CP invariance. Relic density : We consider parameter region in which the DM annihilation cross section is d -wave dominant and the dark matter particles annihilate into a pair of charged-leptons, viathe process η R η R → ℓ i ¯ ℓ j with an E a exchange. Notice that there exist annihilation modessuch as η R η R → W + W − / Z arising from the kinetic term. These modes require a DMmass heavier than at least 500 GeV [22] in order to obtain the correct relic density wherecoannihilation processes should be included. This case is, however, not in favor of explainingthe muon g − Thus we assume that M X . GeV and η R η R → W + W − / Z processes are not kinematically allowed. Then the relic density is simply given byΩ h ≈ . × x f √ g ∗ M P d eff [GeV] , d eff ≈ X ( i,j,a )=1 | r ia r † a,j | M X π ( M E a + M X ) , (II.26)where g ∗ ≈ M P ≈ . × , x f ≈
25. In our numerical analysis below, we use thecurrent experimental range for the relic density: 0 . ≤ Ω h ≤ .
13 [23]. Here we impose the condition m Z / ≈ . m R + m I to forbid the invisible decay of Z boson inour numerical analysis, although the invisible decay of SM Higgs is automatically suppressed in the limitof zero couplings in the Higgs potential. II. NUMERICAL ANALYSIS
As a first step, we perform the numerical analysis on the r term since this term isindependent of the other parameters. We prepare
25 million random sampling points forthe relevant input parameters as follows: M X ∈ [1 , , m I ≈ m η ± ∈ [1 . × M X ,
200 ]GeV ,M ∈ [1 . × M X ,
330 ]GeV , M ∈ [ M ,
600 ]GeV , M ∈ [ M ,
700 ]GeV ,r ′ ∈ [ − , ln(4 π )] , (III.1)where we define M i ≡ M E i = M N i ( i = 1 − r ij ≡ ( ± × r ′ ij and the lower mass bound1 . × M X is expected to forbid the coannihilation processes. Under such parameter ranges,we have found 246 allowed points, which are shown in Fig. 2 satisfying all the constraintsincluding LFVs, oblique parameters, and invisible decay of Z boson, as discussed before.The left panel shows the allowed region to be25[GeV] . M X . , . m η ± . , (III.2)where the lower bound of η ± comes from the LEP experiment [24, 25]. The right panelshows the correlation of r versus ∆ a µ , where r is the most relevant parameter to obtainthe sizable muon g − r is possible to achieve the range 10 − ≤ ∆ a µ ≤ × − , but we need a large r for therange 2 × − < ∆ a µ ≤ − × − . In the next step, we attempt to find the parameter space points that can also solve the B → K ( ∗ ) µµ anomaly by contributing to the C , and at the same time satisfy all theconstraints of the LFVs and FCNCs. Since the number of parameters is getting more andmore, we are content with a benchmark point that we obtained in the first step. We preparethe benchmark point for the masses to fix the three-loop neutrino function F III as follows: M X ≈ .
26 [GeV] , m η I ≈ m η ± ≈ .
57 [GeV] , (III.3) M ≈
277 [GeV] , M ≈
296 [GeV] , M ≈
401 [GeV] , (III.4) m ∆ ≈ , m ∆ ≈ . , m ∆ ≈ . , (III.5) In addition to r , a little bit larger r is also needed. This is technically difficult to obtain the whole numerical values, due to its complicated structure. - £ D a Μ £ ´ - £ W h £
30 40 50 60 70 8080100120140160 M Χ @ GeV D m Η + @ G e V D £ W h £ - - D a Μ ´ r FIG. 2: Scattering plots of the allowed parameter space sets to satisfy LFVs, oblique parameters,and invisible decay of Z boson, in the plane of M X - m η ± in the left panel; and in the plane of∆ a µ - r in the right panel, where r is the most relevant parameter to obtain the sizable muon g − where the above first two lines are provided by the first step so that Ω h ≈ .
120 and∆ a µ ≈ . × − are obtained with r ≈ − .
6. The values in the last line are simply takento evade the collider bound. To satisfy bound on the direct detection search, λ + λ +2 λ . .
002 is needed, where experimental upper bound is σ I . . × − cm , while λ ≈ . m η I − M X = 2 λ v . Therefore a little fine tuning is needed among λ , λ , λ .With this benchmark point, we have the following values: F III [ x ] ≈ − . , F III [ x ] ≈ − . , F III [ x ] ≈ − . . (III.6)Also, we fix − λ λ ′ v / ≈ ] for simplicity. Then, we prepare randomsampling points for the following relevant input parameters:[ a , a , a ] ∈ [ − − i, i ] × [10 − , , f ′ ∈ [ − , ln(4 π )] , h ′ ∈ [ − , ln(4 π )] , (III.7)where we define f ( h ) ij ≡ ( ± × f ′ ( h ′ ) ij . In these parameter ranges, we have found allowed points shown in Fig. 3, which satisfy all the constraints as discussed before. If wefocus on the best-fit value of C , these figures show that the Yukawa couplings f j ( j = 1 − | f | . . , | f | . . , | f | . . , (III.8)where these coupling may affect the collider physics.12 - - - - - f C - - - - - - f C - - - - - - f C FIG. 3: Scattering plots of the allowed parameter space sets in the plane of ( f j )( j = 1 − C ,where the horizontal line is the best fit value of C = − .
68. While the thin red region [ − . , − . σ range. Notice here that we have taken the range C ∈ [ − . , − .
50] to satisfy R K at 1 σ confidential level [3]. Collider Phenomenology : There are two types of new particles in this model other than theSM particles: leptoquarks ∆ a , , and a set of scalar bosons η ± , resulting from an isodoubletscalar field.Note that ∆ a , are assigned with Z = − a are assigned with Z = +1. All∆ , , can be pair produced at hadronic colliders and being searched at the LHC. The directsearch bound is roughly 1 TeV [26]. On the other hand, only ∆ a can participate in the 4-fermion contact interaction, because of the Z parity. The bound from the 4-fermion contactinteraction was worked out in Ref. [8, 13] that the bound is currently inferior to the directsearch bound of about 1 TeV. Therefore, we shall use 1 TeV as the current bound on the∆ , , bosons.The isodoublet field η gives rise to a pair of charged bosons η ± , a scalar η R , and apseudoscalar η I . The charged bosons η ± can run in the triangular loop vertex of Hγγ .Nevertheless, we can suppress such effects by choosing the λ term in Eq. (II.1) very small.As we have explained above such a term is small to avoid the conflict of the direct detection13ound. Although the interaction between the η field and the SM Higgs field is suppressedfor the above reasons, the η can still interact through the kinetic term as it has SU (2) L and U (1) Y interactions. We expect some typical interactions with the gauge bosons: Zη + η − , Zη R η I , W + η − η R , etc . An interesting signature would be Drell-Yan type production of η ± η R via a virtual W . The η ± decays into multi-leptons and η R via the virtual L ′ fields. Therefore, the final stateconsists of multi-charged-leptons and missing energies. Similarly, in the process pp → Z ∗ → η R η I , the η I would decay into η R eventually with a number of very soft leptons, which maynot be detectable. Therefore, the best would be the one produced via virtual W . IV. CONCLUSIONS
We have investigated a three-loop neutrino mass model with some leptoquark scalars with SU (2) L -triplet, in which we have explained the anomaly in B → K ( ∗ ) µ + µ − , sizable muon g −
2, bosonic dark matter, satisfying all the constraints such as LFVs, FCNCs, invisibledecay, and so on. Then we have performed the global numerical analysis and shown theallowed region, in which we have found restricted parameter space, e.g., . M X . , . m η ± . , | f | . . , | f | . . , | f | . . . We find that ∼ O (100) GeV inert doublet scalar is preferred to obtain sizable muon g − pp → W ± → η ± η R which provides signals of multi-leptons plusmissing transverse momentum. Acknowledgments
This work was supported by the Ministry of Science and Technology of Taiwan underGrants No. MOST-105-2112-M-007-028-MY3.14 ppendix A: Loop function
Here we show the explicit form of three-loop function F III , which is given by F III = Z [ dx i ] δ ( P i =1 x i − x − x Z [ dy i ] δ ( P i =1 y i − y ) x x − x − x + 1] Z [ dz i ] δ ( P i =1 z i − z [ y ∆ + z r δ (2)13 + z r d a ] , ∆ = x − x [(1 + y ) x x − x − x + 1] (cid:16) x r δ (1)13 + x r δ (2)13 + x r b + x r η R + x r η I (cid:17) − x x A y [(1 + y ) x x − x − x + 1] (cid:16) y r δ (3)13 + y r d c (cid:17) , where A ≡ x / ( x − dx i ] ≡ R dx R − x dx R − x − x dx R − x − x − x dx , [ dy i ] ≡ R dy R − y dy , and [ dz i ] ≡ R dz R − z dz . [1] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. , 151601 (2014) [arXiv:1406.6482[hep-ex]].[2] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. , 191801 (2013) [arXiv:1308.1707[hep-ex]].[3] S. Descotes-Genon, L. Hofer, J. Matias and J. Virto, JHEP , 092 (2016) [arXiv:1510.04239[hep-ph]].[4] G. Hiller and M. Schmaltz, Phys. Rev. D , 054014 (2014) [arXiv:1408.1627 [hep-ph]].[5] G. Hiller, D. Loose and K. Schonwald, JHEP , 027 (2016) [arXiv:1609.08895 [hep-ph]].[6] S. Descotes-Genon, J. Matias and J. Virto, Phys. Rev. D , 074002 (2013) [arXiv:1307.5683[hep-ph]].[7] K.A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014) and 2015 update.[8] K. Cheung, T. Nomura and H. Okada, arXiv:1610.02322 [hep-ph].[9] L. M. Krauss, S. Nasri and M. Trodden, Phys. Rev. D , 085002 (2003)[arXiv:hep-ph/0210389].[10] M. Aoki, S. Kanemura and O. Seto, Phys. Rev. Lett. , 051805 (2009) [arXiv:0807.0361].[11] M. Gustafsson, J. M. No and M. A. Rivera, Phys. Rev. Lett. , 211802 (2013)arXiv:1212.4806 [hep-ph].[12] C. S. Chen, K. L. McDonald and S. Nasri, Phys. Lett. B , 388 (2014) [arXiv:1404.6033[hep-ph]].
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