Probing doubly charged scalar bosons from the doublet at future high-energy colliders
aa r X i v : . [ h e p - ph ] F e b OU-HET-1079
Probing doubly charged scalar bosons from the doubletat future high-energy colliders
Kazuki Enomoto, ∗ Shinya Kanemura, † and Kento Katayama ‡ Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
Abstract
The isospin doublet scalar field with hypercharge 3 / Y = 3 / ∗ [email protected] † [email protected] ‡ k [email protected] . INTRODUCTION In spite of the success of the Standard Model (SM), there are good reasons to regardthe model as an effective theory around the electroweak scale, above which the SM shouldbe replaced by a model of new physics beyond the SM. Although a Higgs particle has beendiscovered at the LHC [1], the structure of the Higgs sector remains unknown. Indeed, thecurrent data from the LHC can be explained in the SM. However, the Higgs sector in the SMcauses the hierarchy problem, which must be solved by introducing new physics beyond theSM. In addition, the SM cannot explain gravity and several phenomena such as tiny neutrinomasses, dark matter, baryon asymmetry of the universe, and so on. Clearly, extension ofthe SM is inevitable to explain these phenomena.In the SM, introduction of a single isospin doublet scalar field is just a hypothesis withoutany theoretical principle. Therefore, there is still a room to consider non-minimal shapes ofthe Higgs sector. When the above mentioned problems of the SM are considered togetherwith such uncertainty of the Higgs sector, it might happen that it would be one of the naturaldirections to think about the possibility of extended Higgs sectors as effective theories ofunknown more fundamental theories beyond the SM. Therefore, there have been quite a fewstudies on models with extended Higgs sectors both theoretically and phenomenologically.Additional isospin-multiplet scalar fields have often been introduced into the Higgs sectorin lots of new physics models such as models of supersymmetric extensions of the SM, thosefor tiny neutrino masses [2–12], dark matter [13–15], CP-violation [16, 17], and the first-orderphase transition [18, 19]. One of the typical properties in such extended Higgs sector is aprediction of existence of charged scalar states. Therefore, theoretical study of these chargedparticles and their phenomenological exploration at experiments are essentially importantto test these models of new physics.There is a class of models with extended Higgs sectors in which doubly charged scalarstates are predicted. They may be classified by the hypercharge of the isospin-multipletscalar field in the Higgs sector; i.e. triplet fields with Y = 1 [3, 4, 8], doublet fields with Y = 3 / Y = 2 [7, 8, 12, 22]. These fields mainly enterinto new physics model motivated to explain tiny neutrino masses, sometimes together withdark matter and baryon asymmetry of the universe [12, 20, 21, 23–25]. The doubly chargedscalars are also introduced in models for other motivations [26, 27]. Collider phenomenology2f these models is important to discriminate the models. There have also been many studiesalong this line [20, 28–37].In this paper, we concentrate on the collider phenomenology of the model with an ad-ditional isodoublet field Φ with Y = 3 / √ s = 14 TeV and the integrated luminosity of L = 3000 fb − [38].Clearly, Φ cannot couple to fermions directly. The component fields are doubly chargedscalar bosons Φ ±± and singly charged ones Φ ± . In order that the lightest one is able todecay into light fermions, we further introduce an additional doublet scalar field φ with thesame hypercharge as of the SM one φ , Y = 1 /
2. Then, Y = 3 / pp → W + ∗ → Φ ++ H − i have been analyzed, where H ± i ( i = 1 ,
2) are mass eigenstates of singly charged scalar states.They have indicated that masses of all the charged states Φ ±± and H ± i may be measurableform this single process by looking at the Jacobian peaks of transverse masses of several com-binations of final states etc. However, they have not done any analysis for backgrounds. Inthis paper, we shall investigate both signal and backgrounds for this process to see whetheror not the signal can dominate the backgrounds after performing kinematical cuts at theHL-LHC.This paper is organized as follows. In Sec. II, we introduce the minimal model withdoubly charged scalar bosons from the doublet which is mentioned above, and give a briefcomment about current constraints on the singly charged scalars from some experiments.In Sec. III, we investigate decays of doubly and singly charged scalars and a production ofdoubly charged scalars at hadron colliders. In Sec. IV, results of numerical evaluations forthe process pp → W + ∗ → Φ ++ H − i are shown. Final states of the process depend on massspectrums of the charged scalars, and we investigate two scenarios with a benchmark value.Conclusions are given In Sec. V. In Appendix A, we show analytic formulae for decay ratesof two-body and three-body decays of the charged scalars. II. MODEL OF THE SCALAR FIELD WITH Y = 3 / We investigate the model whose scalar potential includes three isodoublet scalar fields3 , φ , and Φ [20]. Gauge groups and fermions in the model are same with those in the SM.Quantum numbers of scalar fields are shown in Table I. The hypercharge of two scalars φ and φ is 1 /
2, and that of the other scalar Φ is 3 /
2. In order to forbid the flavor changingneutral current (FCNC) at tree level, we impose the softly broken Z symmetry, where φ and Φ have odd parity and φ has even parity [39]. SU (3) C SU (2) L U (1) Y Z φ / φ / − Φ / − TABLE I. The list of scalar fields in the model
The scalar potential of the model is given by V = V THDM + µ | Φ | + 12 λ Φ | Φ | + X i =1 ρ i | φ i | | Φ | + X i =1 σ i | φ † i Φ | + n κ (Φ † φ )( ˜ φ † φ ) + h . c . o , (1)where V THDM is the scalar potential in the two Higgs doublet model (THDM), and it is givenby V THDM = X i =1 µ i | φ i | + (cid:16) µ φ † φ + h . c . (cid:17) + X i =1 λ i | φ i | + λ | φ | | φ | + λ | φ † φ | + 12 n λ ( φ † φ ) + h . c . o . (2)The Z symmetry is softly broken by the terms of µ φ † φ and its hermitian conjugate.Three coupling constants µ , λ and κ can be complex number generally. After redefinitionof phases of scalar fields, either µ or λ remains as the physical CP-violating parameter.In this paper, we assume that this CP-violating phase is zero and all coupling constants arereal for simplicity.Component fields of the doublet fields are defined as follows. φ i = ω + i √ ( v i + h i + iz i ) , Φ = Φ ++ Φ + , (3)4here i = 1 ,
2. The fields φ and φ obtain the vacuum expectation values (VEVs) v / √ v / √
2, respectively. These VEVs are described by v ≡ p v + v ≃
246 GeV andtan β ≡ v /v . On the other hand, the doublet Φ cannot have a VEV without violatingelectromagnetic charges spontaneously.Mass terms for the neutral scalars h i and z i are generated by V THDM . Thus, mass eigen-states of the neutral scalars are defined in the same way with those in the THDM (See, forexample, Ref. [40]). Mass eigenstates h , H , A , and z are defined as Hh = R ( α ) h h , zA = R ( β ) z z , (4)where α and β (= Tan − ( v /v )) are mixing angles, and R ( θ ) is the two-by-two rotationmatrix for the angle θ , which is given by R ( θ ) = cos θ sin θ − sin θ cos θ . (5)The scalar z is the Nambu-Goldstone (NG) boson, and it is absorbed into the longitudinalcomponent of Z boson. Thus, the physical neutral scalars are h , H , and A . For simplicity,we assume that sin( β − α ) = 1 so that h is the SM-like Higgs boson.On the other hand, the mass eigenstates of singly charged scalars are different from thosein the THDM, because the field Φ ± is mixed with ω ± and ω ± . The singly charged masseigenstates ω ± , H ± , and H ± are defined as ω ± H ± H ± = χ sin χ − sin χ cos χ cos β sin β − sin β cos β
00 0 1 ω ± ω ± Φ ± . (6)The scalar ω ± is the NG boson, and it is absorbed into the longitudinal component of W ± boson. Thus, there are two physical singly charged scalars H ± and H ± . The doubly chargedscalar Φ ±± is mass eigenstate without mixing.The doublet Φ does not have the Yukawa interaction with the SM fermions because of itshypercharge. Therefore, Yukawa interactions in the model is same with those in the THDM.They are divided into four types according to the Z parities of each fermion (Type-I, II, If we consider higher dimensional operators, interactions between Φ and leptons are allowed [32].
5, and Y [41]). In the following, we consider the Type-I Yukawa interaction where all left-handed fermions have even parity, and all right-handed ones have odd-parity. The type-IYukawa interaction is given by L Y ukawa = − X i,j =1 (cid:26) ( Y u ) ij Q iL ˜ φ u jR + ( Y d ) ij Q iL φ d jR + ( Y ℓ ) ij L iL φ ℓ jR (cid:27) + h . c ., (7)where Q iL ( L iL ) is the left-handed quark (lepton) doublet, and u jR , d jR , and ℓ jR are theright-handed up-type quark, down-type quark and charged lepton fields, respectively. TheYukawa interaction of the singly charged scalars are given by − √ v cot β X i,j =1 (cid:26) V u i d j u i (cid:16) m u i P L + m d j P R (cid:17) d j + δ ij m ℓ i ν i P L ℓ i (cid:27)(cid:16) cos χH +1 − sin χH +2 (cid:17) +h . c ., (8)where V u i d j is the ( u i , d j ) element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [16,42], δ ij is the Kroneker delta, and P L ( P R ) is the chirality projection operator for left-handed (right-handed) chirality. In addition, ( u , u , u ) = ( u, c, t ) are the up-type quarks,( d , d , d ) = ( d, s, b ) are the down-type quarks, ( ℓ , ℓ , ℓ ) = ( e, µ, τ ) are the charged leptons,and ( ν , ν , ν ) = ( ν e , ν µ , ν τ ) are the neutrinos. The symbols m u i , m d i , and m ℓ i are the massesfor u i , d i , and ℓ i , respectively. In the following discussions, we neglect non-diagonal termsof the CKM matrix.Finally, we discuss constraints on some parameters in the model from various experiments.If the coupling constant κ in the scalar potential is zero, the model have a new discrete Z symmetry where the doublet Φ is odd and all other fields are even. This Z symmetrystabilizes Φ ±± or Φ ± , and their masses and interactions are strongly constrained. Thus, κ = 0 is preferred, and it means that sin χ = 0. In this paper, we assume that χ = π/ H ± and H ± have Type-I Yukawa interaction, it isexpected that the constraints on H ± and H ± are almost same with those on the chargedHiggs boson in the Type-I THDM and the difference is caused by the factor sin χ or cos χ in Eq. (8). In the case where sin χ = cos χ = 1 / √
2, the constraints are as follows. Fortan β . .
4, the lower bound on the masses of H ± and H ± are given by flavor experiments.This lower bound depends on the value of tan β , and it is about 400 GeV for tan β = 1 [43–45]. In the region that 1 . . tan β . .
7, the lower bound on the mass is given by the searchfor the decay of the top quark into the bottom quark and the singly charged scalar at theLHC Run-I. This lower bound is about 170 GeV [45, 46]. For tan β & .
7, the direct search6t LEP gives the lower bound on the mass. It is about 80 GeV [47]. From Eq. (8), it isobvious that if we think the case where | sin χ | > | cos χ | , ( | sin χ | < | cos χ | ) the constraintson H ± ( H ± ) are relaxed, and those on H ( H ± ) become more stringent. III. PRODUCTION AND DECAYS OF CHARGED SCALAR STATES
In this section, we investigate the decay of the new charged scalars and the productionof the doubly charged scalar at hadron colliders. In the following discussion, we assumethat Φ ±± , H , and A are heavier than H ± and H ± . Then, H ± , cannot decay into Φ ±± , H ,and A . In addition, the masses of H ± , H ± , and Φ ±± are denoted by m H m H , and m Φ ,respectively. A. Decays of charged scalar sates
First, we discuss the decays of the singly charged scalars H ± and H ± . They decay intothe SM fermions via Yukawa interaction in Eq. (8). Since they are lighter than Φ ±± , H ,and A , their decays into Φ ±± W ∓ ( ∗ ) , HW ± ( ∗ ) , and AW ± ( ∗ ) are prohibited. On the otherhand, the decay of the heavier singly charged scalars into the lighter one and Z ( ∗ ) is allowed,and it is generated via the gauge interaction. In the following, we assume that H ± is heavierthan H ± ( m H > m H ). FIG. 1. The branching ratio of H ± . In Fig. 1, the branching ratio for each decay channel of H ± is shown. Since we assume7hat H ± is lighter than H ± , it decays via the Yukawa interaction [41] . In the region where m H .
140 GeV, the decay into cs and that into τ ν are dominant. When we consider a littleheavier H ± , which are in the mass region between 140 GeV and m t + m b ≃
180 GeV, thebranching ratio for H ± , → t ∗ b → W ± bb is dominant [48]. In the mass region m t + m b < m H ,the branching ratio for H ± → tb is almost 100 %. The decays into cs , τ ν , and t ( ∗ ) b are allinduced by the Yukawa interaction. Since we consider the Type-I Yukawa interaction, thedependence on tan β of each decay channel is same. Thus, the branching ratio in Fig. 1hardly depends on the value of tan β . Analytic formulae of decay rates for each decaychannel are shown in Appendix A 1.The singly charged scalar H ± also decays into the SM fermions via the Yukawa interaction.In addition, H ± → H ± Z ( ∗ ) is allowed. In Fig. 2, the branching ratios of H ± in two cases areshown. The left figure of Fig. 2 is for tan β = 10 and ∆ m ( ≡ m H − m H ) = 20 GeV. In thesmall mass region, the decay H ± → H ± Z ∗ is dominant. In the region where m H &
140 GeV,the decay H ± → t ( ∗ ) b becomes dominant, and the branching ratio for H ± → tb is almost100 % for m H &
180 GeV. If we consider smaller tan β , the decays via Yukawa interactionare enhanced because the Yukawa interaction is proportional to cot β . (See Eq. (8).) Thus,he branching ratio for H ± → H ± Z ∗ decreases.The right figure of Fig. 2 is for the case where tan β = 3 and ∆ m = 50 GeV. In thesmall mass region, the branching ratio for H ± → H ± Z ∗ is about 80 %, and those for otherdecay channels are negligible small. However, in the mass region where m H &
180 GeV, H ± → H ± Z ∗ become negligible small, and the branching ratio for H ± → tb is almost 100 %.If we consider larger tan β , the decays via the Yukawa interaction is suppressed, and thebranching ratio for H ± → H ± Z ∗ increases. Thus, the crossing point of the branching ratiofor H ± → tb ( t ∗ b ) and that for H ± → H ± Z ∗ move to the point at heavier m H . Analyticformulae of decay rates for each decay channel are shown in Appendix A 1.Next, we discuss the decay of the doubly charged scalar Φ ±± . The doubly charged scalarΦ ±± does not couple to fermions via Yukawa interaction . Therefore, it decays via the weak In this paper, we neglect the effects of one-loop induced decays H ± i → W ± γ and H i ± → W ± Z [49]. In Ref [48], Type-II Yukawa interaction is investigated, and the condition tan β . H ± , → t ∗ b dominant. In our case (Type-I), this condition is not necessary because all fermionscouple to φ universally. This is different from doubly charged Higgs boson in the triplet model in which dilepton decays of doublycharged Higgs bosons are important signature to test the model [36]. IG. 2. The branching ratio of H ± . In the left figure, we assume that ∆ m ( ≡ m H − m H ) = 20 GeVand tan β = 10. In the right figure, we assume that ∆ m = 50 GeV and tan β = 3 gauge interaction . We consider the following three cases.First, the case where ∆ m ( ≡ m Φ − m H ) <
80 GeV and ∆ m ( ≡ m Φ − m H ) <
80 GeV isconsidered. In this case, Φ ±± cannot decay into the on-shell H ± , , and three-body decays aredominant. In the upper left figure of Fig. 3, the branching ratio of Φ ±± in this case is shown.We assume that tan β = 3, ∆ m <
20 GeV, ∆ m <
10 GeV. In the small mass region,Φ ±± → H ± f f is dominant. With increasing of m Φ , the masses of H ± , also increase becausethe mass differences between them are fixed. Thus, the branching ratio for Φ ±± → W ± f f is dominant in the large mass region. At the point m Φ ≃
260 GeV, the branching ratiofor Φ ±± → W ± f f changes rapidly. It is because that at this point, the decay channelΦ ±± → W ± tb is open. If we consider the large tan β , the decay rates of Φ ±± → W ∓ f f becomes small because this process includes H ±∗ , → f f via Yukawa interaction which isproportional to cot β . However, the decays Φ ±± → H ± , f f are generated via only the gaugeinteraction. Thus, for tan β &
3, the branching ratio for Φ ±± → W ± f f becomes small.Second, the case where ∆ m >
80 GeV and ∆ m <
80 GeV is considered. In this case,Φ ±± → H ± W ± is allowed while Φ ±± → H ± W ± is prohibited. In the upper right figureof Fig. 3, the branching ratio of Φ ±± in this case is shown. We assume that tan β = 3,∆ m <
100 GeV, ∆ m <
50 GeV. In all mass region displayed in the figure, the branchingratio for Φ ±± → H ± W ± are almost 100 %, and those for other channels are at most about0 . m Φ ≃
260 GeV, the branching ratio for Φ ±± → W ± f f changes rapidly. In triplet Higgs models, if the VEV of the triplet field is small enough the main decay mode of the doublycharged Higgs boson is the diboson decay [31]. On the other hand, in our model, such a decay mode doesnot exist at tree level.
9t is because that at this point, the decay channel Φ ±± → W ± tb is open.Third, the case where ∆ m >
80 GeV and ∆ m >
80 GeV is considered. and both ofΦ ±± → H ± , W ± are allowed. In the lower figure of Fig. 3, the branching ratio in this caseis shown. We assume that tan β = 3, ∆ m = 100 GeV, ∆ m = 90 GeV. In all mass regiondisplayed in the figure, the branching ratio does not change because the mass differencesbetween Φ ±± and H ± , are fixed. The branching ratio for Φ ±± → H ± W ± is about 75 %,and that for Φ ±± → H ± W ± is about 25 %. These decays are generated via only the gaugeinteraction. Thus, the branching ratios of them do not depend on tan β , and they aredetermined by only the mass differences between Φ ±± and m H , . FIG. 3. The branching ratios of the decay of Φ ±± . The upper lift (right) afigure is those in thecase that ∆ m ( ≡ m Φ − m H ) = 20 GeV (100 GeV) and ∆ m ( ≡ m Φ − m H ) = 10 GeV (50 GeV).The bottom one corresponds to the case that ∆ m = 100 GeV and ∆ m = 90 GeV. . Production of Φ ±± at hadron colliders We here discuss the production of the doubly charged scalar Φ ±± . In our model, produc-tion processes of charged scalar states are pp → W + ∗ → H + i A ( H ), pp → Z ∗ ( γ ) → H + i H − i , pp → W + ∗ → Φ ++ H − i , and pp → Z ∗ ( γ ) → Φ ++ Φ −− . In the THDM, the first and secondprocesses (the singly charged scalar production) can also occur [50, 51] However, doublycharged scalar bosons are not included in the THDM . In the model with the isospin tripletscalar with Y = 1 [3, 4, 8, 26, 27], all of these production processes can appear. However, themain decay mode of doubly charged scalar is different from our model. In the triplet model,the doubly charged scalar from the triplet mainly decays into dilepton [36] or diboson [31].In our model, on the other hand, Φ ±± mainly decays into the singly charged scalar and W boson.In this paper, we investigate the associated production pp → W + ∗ → Φ ++ H − i ( i = 1 , ±± and H ± i appear inthe Jacobian peaks of transverse masses of several combinations of final states [20]. Pairproductions are also important in searching for Φ ±± and H ± i , however we focus on theassociated production in this paper. The parton-level cross section of the process qq ′ → W + ∗ → Φ ++ H − i ( i = 1 ,
2) is given by σ i = G F m W | V qq ′ | χ i πs ( s − m W ) h m H ± i + ( s − m ±± ) − m H ± i ( s + m ±± ) i / , (9)where s is the square of the center-of-mass energy, G F is the Fermi coupling constant, and V qq ′ is the ( q, q ′ ) element of CKM matrix. In addition, χ i in Eq. (9) is defined as χ = sin χ, χ = cos χ. (10)In Fig. 4, we show the cross section for pp → W + ∗ → Φ ++ H − in the case that √ s =14 TeV and χ = π/
4. The cross section is calculated by using M AD G RAPH A MC@NLO [58]and FeynRules [59]. The black, red, blue lines are those in the case that ∆ m = 0, 50, and100 GeV, respectively. The results in Fig. 4 do not depend on the value of tan β . At theHL-LHC ( √ s = 14 TeV and L = 3000 fb − ), about the 6 × doubly charged scalars are In the THDM, and also in our model with the Y = 3 / gb → tH ± [52], qb → q ′ bH ± [53], bb → W ± H ∓ [54, 55], gg → W ± H ∓ [55, 56], etc. (See also Ref. [57].) In this paper, we do not consider these processes and concentrateonly on the processes pp → W + ∗ → Φ ++ H − i . m Φ = 200 GeV and ∆ m = 50 GeV. If Φ ±± isheavier, the cross section decreases, and about the 300 doubly charged scalars are expectedto be generated at the HL-LHC in the case that m Φ = 800 GeV. The cross section increaseswith increasing of the mass difference ∆ m . Since we assume that χ = π/
4, the cross sectionof the process pp → W + ∗ → Φ ++ H − is same with that in Fig. 4 if m H = m H . If we considerthe case that | sin χ | > | cos χ | ( | cos χ | > | sin χ | ), the cross section of pp → W + ∗ → Φ ++ H − become larger (smaller) than that of pp → W + ∗ → Φ ++ H − even if m H = m H . FIG. 4. The cross section for pp → W + ∗ → Φ ++ H − , where √ s = 14 TeV and χ = π/
4. The black,red, blue lines are those in the case that ∆ m ( ≡ m Φ − m H ) = 0, 50, and 100 GeV, respectively. IV. SIGNAL AND BACKGROUNDS AT HL-LHC
In this section, we investigate the detectability of the process pp → W + ∗ → Φ ++ H − i ( i = 1 ,
2) in two benchmark scenarios. In the first scenario (Scenario-I), the masses of H ± and H ± are set to be 100 GeV and 120 GeV, so that they cannot decay into tb . In this case,their masses are so small that the branching ratio for three body decay H ± , → W ± bb is lessthan 5 % approximately. Thus, their main decay modes are H ± , → cs and H ± , → τ ν . Inthe second scenario (Scenario-II), masses of H ± and H ± are set to be 200 GeV and 250 GeV,and they predominantly decay into tb with the branching ratio to be almost 100 %.12n our analysis below, we assume the collider performance at HL-LHC as follows [38]. √ s = 14 TeV , L = 3000 fb − , (11)where √ s is the center-of-mass energy and L is the integrated luminosity. Furthermore, weuse the following kinematical cuts (basic cuts) for the signal event [58]; p jT >
20 GeV , p ℓT >
10 GeV , | η j | < , | η ℓ | < . , ∆ R jj > . , ∆ R ℓj > . , ∆ R ℓℓ > . , (12)where p jT ( p ℓT ) and η j ( η ℓ ) are the transverse momentum and the pseudo rapidity of jets(charged leptons), respectively, and ∆ R jj , ∆ R ℓj , and ∆ R ℓℓ in Eq. (12) are the angulardistances between two jets, charged leptons and jets, and two charged leptons, respectively. A. Scenario-I
FIG. 5. The Feynman diagram for the signal process in Scenario-I, where q and q ′ are partons. In this scenario, the singly charged scalars decay into cs or τ ν dominantly. (See Figs. 1and 2.) We investigate the process pp → W + ∗ → Φ ++ H − , → τ + ℓ + ννjj ( ℓ = e, µ ). TheFeynman diagram for the process is shown in Fig. 5. In this process, the doubly chargedscalar Φ ++ and one of the singly charged scalars H − , are generated via s-channel W + ∗ .The produced singly charged scalar decays into a pair of jets, and Φ ++ decays into τ + ℓ + νν through the on-shell pair of the singly charged scalar and W + . Thus, in the distribution ofthe transverse mass of τ + ℓ + (cid:0)(cid:0) E T , where E T is the missing transverse energy, we can see the13acobian peak whose endpoint corresponds to m Φ [20] . In the present process, furthermore,in the distribution of the transverse mass of two jets, we can basically see twin Jacobianpeaks at m H and m H [20]. Therefore, by using the distributions of M T ( τ + ℓ + (cid:0)(cid:0) E T ) and M T ( jj ), we can obtain the information on masses of all the charged scalars H ± , H ± , andΦ ±± . This is the characteristic feature of the process in this model. When we consider thedecay of the tau lepton, the transverse mass of the decay products of the tau lepton and ℓ + νν can be used instead of M T ( τ + ℓ + νν ).In the following, we discuss the kinematics of the process at HL-LHC with the numericalevaluation. For input parameters, we take the following benchmark values for Scenario-I; m Φ = 200 GeV , m H = 100 GeV , m H = 120 GeV , tan β = 10 , χ = π . (15)From the LEP data [47], the singly charged scalars are heavier than the lower bound ofthe mass (80 GeV). In addition, we take the large tan β (=10), so that they satisfy theconstraints from flavor experiments [43, 44] and LHC Run-I [45, 46].The final state include the tau lepton, and we consider the case that the tau lepton decaysinto π + ν . In this case, π + flies in the almost same direction of τ + in the Center-of-Mass(CM) frame because of the conservation of the angular momentum [51]. The branchingratio for τ + → π + ν is about 11 % [60], and we assume that the efficiency of tagging thehadronic decay of tau lepton is 60 % [61]. Under the above setup, we carry out the numericalevaluation of the signal events by using M AD G RAPH A MC@NLO [58], FeynRules [59], andTauDecay [62]. As a result, about 600 signal events are expected to be produced at HL-LHC.The distributions of the signal events for M T ( π + ℓ + (cid:0)(cid:0) E T ) and M T ( jj ) are shown in red line inthe left figure of Fig. 6 and in the right one, respectively.Next, we discuss the background events and their reduction. The main backgroundprocess is pp → W + W + jj → τ + ℓ + ννjj . The leading order of this background process is O ( α ) and O ( α α s ). For O ( α ), the vector boson fusion (VBF) and tri-boson production pp → W + W + W − → W + W + jj are important. On the other hand, for O ( α α s ), the mainprocess is t-channel gluon mediated pp → q ∗ q ′∗ → W + W + jj , where q and q ′ are quarks ininternal lines. The number of the total background events under the basic cuts in Eq. (12) In general, the transverse mass M T of n particles is defined as follows. M T = ( E T + E T + · · · + E T n ) + | p T + p T + · · · + p T n | , (13) E T i = | p T i | + m i ( i = 1 , , · · · , n ) , (14)where p T i and m i are the transverse momentum and the mass of i -th particle, respectively.
50 100 150 200 250 300 350 400 M T ( π + l + /E T ) [GeV] E v e n t / b i n backgroundsignal M T ( jj ) [GeV] E v e n t / b i n backgroundsignal FIG. 6. The distribution of the signal and background events for M T ( π + ℓ + (cid:0)(cid:0) E T ) (the left figure)and M T ( jj ) (the right one) We use the basic cut in Eq. (12). The width of the bin in the figuresis 10 GeV. We use the benchmark values in Eq. (15). is shown in Table II. Transverse mass distributions of background events for M T ( π + ℓ + (cid:0)(cid:0) E T )and M T ( jj ) are shown in the blue line in the left figure of Fig. 6 and in the right one,respectively. The number of the background events is larger than that of the signal. Clearly,background reduction has to be performed by additional kinematical cuts.First, we impose the pseudo-rapidity cut for a pair of two jets (∆ η jj ). The ∆ η jj distri-butions of the signal and background processes are shown in the upper left figure in Fig. 7.For the signal events, the distribution has a maximal value at ∆ η jj = 0 as they are gen-erated via the decay of H − or H − . On the other hand, for the VBF background, two jetsfly in the almost opposite directions, and each jet flies almost along the beam axis. Large | ∆ η jj | is then expected to appear [63], so that we can use | ∆ η jj | < . O ( α ) and O ( α α s ) processes because in these background, the distribution are maximal at ∆ η jj = 0.Second, we impose the angular distance cut for a pair of two jets (∆ R jj ). The ∆ R jj distributions of the signal and background processes are shown in the upper right figure inFig. 7. For the signal events, the distribution has a maximal value at ∆ R jj ≃ .
0. On theother hand, for the O ( α α s ) background events, ∆ R jj has a peak at ∆ R jj ∼ π . In addition,in the O ( α ) ones, ∆ R jj has large values between 3 and 6. Therefore, for ∆ R jj <
2, thebackground events are largely reduced while the almost all signal events remains.Third, we impose invariant mass cut for a pair of two jets ( M jj ). The M jj distributions ofthe signal and background processes are shown in the bottom figure in Fig. 7. For the signal15vents, as they are generated via the decay of the singly charged scalars, the distribution hastwin peaks at the masses of H ± and H ± (100 GeV and 120 GeV). On the other hand, forthe background events, the jets are generated via on-shell W or t-channel diagrams. Then,the distribution of the background has a peak at the W boson mass ( ∼
80 GeV). Thus, thekinematical cut 90 GeV < M jj <
180 GeV is so effective to reduce the background events.We note that this reduction can only be possible when we already know some informationon the masses of the singly charged scalars.We summarize three kinematical cuts for the background reduction.(i) | ∆ η jj | < . , (16)(ii) ∆ R jj < , (17)(iii) 90 GeV < M jj <
180 GeV , (18) signal S background B S/ √ S + B Basic cuts(Eq. (12)) 592 3488 9.3Basic cuts (Eq. (12))and ∆ R jj < | ∆ η jj | < . Let us discuss how the backgrounds can be reduced by using the first two kinematicalcuts (i) and (ii), in addition to the basic cuts given in Eq. (12). This corresponds to the casethat we do not use the information on the masses of the singly charged scalars. The resultsare shown in the third column of Table II. In this case, about 88 % of the background eventsare reduced, while about 82 % of the signal events remain. We obtain the significance as S/ √ S + B = 16. The distributions for M T ( π + ℓ + (cid:0)(cid:0) E T ) and M T ( jj ) are shown in Fig. 8. Inthe left figure of Fig. 8, we can see the Jacobian peak of M T ( π + ℓ + (cid:0)(cid:0) E T ). Consequently, the16 − − ∆ η jj E v e n t / b i n O ( α α s ) O ( α ) signal ∆ R jj E v e n t / b i n O ( α α s ) O ( α ) signal M ( jj ) [GeV] E v e n t / b i n O ( α α s ) O ( α ) signal FIG. 7. The distributions of signal and background events for ∆ η jj (the upper left figure), ∆ R jj (the upper right one), and M jj (the bottom one). The red lines are those for the signal events.The blue (yellow) lines are those for the background events of O ( α ) ( O ( α α s )). In the figuresfor ∆ η jj and ∆ R jj , we take the width of bins as 0 .
1. In the figure for M jj , the width of bins is10 GeV. We use the benchmark values in Eq. (15). signal process can be detected at HL-LHC in Scenario-I of Eq. (15). However, the endpointof the signal is unclear due to the background events, so that it would be difficult to preciselydecide the mass of Φ ++ . On the other hand, we can see the twin Jacobian peaks of M T ( jj )in the right figure of Fig. 8. Therefore, we can also obtain information on masses of boththe singly charged scalars. In this way, all the charged scalar states Φ ±± , H ± , and H ± canbe detected and their masses may be obtained to some extent.Furthermore, if we impose all the kinematical cuts (i), (ii), and (iii) with the basic cuts,the backgrounds can be further reduced. The results are shown in the fourth column of17
50 100 150 200 250 300 350 400 M T ( π + l + /E T ) [GeV] E v e n t / b i n backgroundsignal M T ( jj ) [GeV] E v e n t / b i n backgroundsignal FIG. 8. The distribution of the signal and background events for M T ( π + ℓ + (cid:0)(cid:0) E T ) (the left figure)and M T ( jj ) (the right one) We use the basic cuts in Eq. (12), | ∆ η jj | < .
5, and ∆ R jj <
2. Thewidth of bins in the figures is 10 GeV. We use the benchmark values in Eq. (15).
Table II. The number of signal events are same with that in the previous case. On the otherhand, the background reduction is improved, and 98 % of the background events are reduced.The significance is also improved as S/ √ S + B = 20. Distributions for M T ( π + ℓ + (cid:0)(cid:0) E T ) and M T ( jj ) are shown in Fig 9. In the left figure of Fig 9, we can see that there are only fewbackground events around the end point of Jacobian peak M T ( π + ℓ + (cid:0)(cid:0) E T ). Thus, it would beexpected we obtain the more clear information on m Φ than that from the case where only(i) and (ii) are imposed as additional kinematical cuts. We can also clearly see the twinJacobian peaks in the right figure of Fig 9, and a large improvement can be achieved for thedetermination of the masses of both the singly charged scalar states.Before closing Subsection A, we give a comment about the detector resolution. In theprocess, the transverse momenta of jets ( p jT ) are mainly distributed between 0 and 200 GeV,and the typical value of them is about 100 GeV. According to Ref. [64], at the currentATLAS detector, the energy resolution for p jT ≃
100 GeV is about 10 %. In Figs. 6-9, wetake the width of bins as 10 GeV. Therefore, it would be possible that the twin Jacobianpeaks in the distribution for M T ( jj ) overlap each other and they looks like one Jacobianpeak with the unclear endpoint at the ATLAS detector if the mass differences is not largeenough. Then, it would be difficult to obtain the information on both m H and m H from thetransverse momentum distribution. Even in this case, it would be able to obtain the hint forthe masses by investigating the process. In our analysis, we did not consider the background18
50 100 150 200 250 300 350 400 M T ( π + l + /E T ) [GeV] E v e n t / b i n backgroundsignal M T ( jj ) [GeV] E v e n t / b i n backgroundsignal FIG. 9. The distribution of the signal and background events for M T ( π + ℓ + (cid:0)(cid:0) E T ) (the left figure)and M T ( jj ) (the right figure) We use the basic cut in Eq. (12) and all the kinematical cuts inEq. (16). The width of the bin in the figures is 10 GeV. where the Z boson decays into dijet such as qq → Z ∗ → Zh → jjτ τ → jjπ + ν τ ℓ − ν τ ν ℓ , whichcan be expected to be reduced by veto the events of M jj at the Z boson mass and the cutof the transverse mass M T ( π + ℓ + (cid:0)(cid:0) E T ) below 125 GeV. It does not affect the Jacobian peakand the endpoint at the mass of doubly charged scalar boson Φ ±± . B. Scenario-II
In this scenario, the singly charged scalars predominantly decay into tb with the branchingratio almost 100 %. We investigate the signal pp → W + ∗ → Φ ++ H − , → ttbbℓ + ν → bbbbℓ + ℓ ′ + ννjj ( ℓ, ℓ ′ = e, µ ). The Feynman diagram for the process is shown in Fig. 10. Thedecay products of Φ ++ and H ± , are bbℓ + ℓ ′ + νν and bbjj , respectively. Therefore, in thesame way as Scenario-I, we can obtain information on masses of all the charged scalars byinvestigating the transverse distributions of signal and background events for M T ( bbℓ + ℓ ′ + νν )and M T ( bbjj ). However, in the Scenario-II, decay products of both Φ ++ and H − , includea bb pair, and it is necessary to distinguish the origin of the two bb pairs. We suggest thefollowing two methods of the distinction.In the first method, we use the directions of b and b . In the process, Φ ++ and H − , are generated with momenta in the opposite directions, and decay products fly along thedirections of each source particle. The both of two W bosons generated via the decay of Φ ++ decay into charged leptons and neutrinos, while the W boson via the decay of H , decays19 IG. 10. The Feynman diagram for the signal process in Scenario-II, where q and q ′ are partons. into a pair of jets. By using this topology of the process, we can distinguish the origin oftwo bb pairs. The bb pair which flies along the charged leptons ℓ + and ℓ ′ + (and flies alongthe almost opposite direction of a pair of jets) comes from the decay of Φ ++ . The other bb pair is the decay product of H − , .In the second method, we use the transverse momenta of b and b . As shown in theFeynman diagram in Fig. 10, in the decay chain of Φ ++ , b is generated via the decay of thetop quark while b is generated via the decay of the singly charged scalars from the decayof Φ ++ . On the other hand, in the decay chain of H − , , b is generated via the decay of thesingly charged scalars while b is generated via the decay of the anti-top quark. Therefore,when the singly charged scalars are heavy enough to satisfy the inequality, m H , − m t − m b > m t − m W − m b , (19)the typical value of the transverse momentum of b from H − , is larger than that of b fromthe top quark. In the same way, the typical value of transverse momentum of b from H +1 , islarger than that of b from the anti-top quark. Therefore, in this case, we can construct the bb pair which mainly comes from the decay of Φ ++ by selecting b with the smaller transversemomentum and b with the larger transverse momentum. The other bb pair comes from thedecay of H − , . On the contrary, when the singly charged scalars are light enough to satisfythe inequality, m H , − m t − m b < m t − m W − m b , (20)the typical value of the transverse momentum of b ( b ) from H − , ( H +1 , ) is smaller than that20f b ( b ) from the top quark (the anti-top quark). Therefore, in the case where the singlycharged scalar is so light that they satisfy the inequality in Eq. (20), we can construct the bb pair which mainly comes from the decay of Φ ++ by selecting b with the larger transversemomentum and b with the smaller transverse momentum. The other bb pair comes from thedecay of H − , . Finally, when the masses of singly charged scalars are around 250 GeV, theysatisfy the equation, m H , − m t − m b ≃ m t − m W − m b . (21)Then, the typical values of the transverse momenta of two b are similar, and those of two b are also similar. Therefore, we can construct the correct bb pair only partly by using theabove method, and it is not so effective. In this case, the first method explained in theprevious paragraph is needed.In the following, we discuss the signal and the background events at HL-LHC with thenumerical calculation. In the numerical evaluation, we take the following benchmark valuesas Scenario-II. m Φ = 300 GeV , m H = 200 GeV , m H = 250 GeV , tan β = 3 , χ = π . (22)For tan β = 3, the lower bound on the masses of singly charged scalars is about 170 GeVas mentioned in the end of Sec. II. Then, this benchmark values satisfy the experimentalconstraints on singly charged scalars. In addition, we adopt the assumption about thecollider performance at HL-LHC in Eq. (11), and we use the basic kinematical cuts inEq. (12). The final state of the signal includes two bottom quarks and two anti-bottomquarks, and we assume that the efficiency of the b-tagging is 70 % per one bottom or anti-bottom quark [65]. Thus, the total efficiency of the b-tagging in the signal event is about24 %. In the numerical calculation, we use M AD G RAPH A MC@NLO [58], FeynRules [59].As a result, 145 events are expected to appear at HL-LHC as shown in Table III. In thisbenchmark scenario of Eq. (22), H ± is so light that we can use the distinction of the bb pairin the case where m H − m t − m b < m t − m b − m W . Therefore, we can construct the bb pair which mainly comes from the decay of H − by selecting b with the smaller transversemomentum and b with the larger transverse momentum. On the other hand, the mass of H ± is 250 GeV, and it satisfies the equation m H − m t − m b ≃ m t − m b − m W . Therefore,the selection of b and b by their transverse momenta is partly effective in the signal where21 − is produced with Φ ++ via W + ∗ . In Figs. 11, we show the distributions of M T ( b b ℓ + ℓ ′ + (cid:0)(cid:0) E T ) and M T ( b b jj ), where b ( b )is the bottom quark (anti-bottom quark) with the larger transverse momentum and b ( b )is the other. In the left figure of Fig. 11, the endpoint of the Jacobian peak is not so sharpbecause the selection of the bb pairs do not work well in the associated production of Φ ++ and H − . In the right figure of Fig. 11, we can see the twin Jacobian peaks at the massesof the singly charged scalars. However, the number of events around the Jacobian peaks,especially the one due to H ± , are small, and it would be difficult to obtain information onmasses form the distribution for M T ( b b jj ). In order to obtain the clearer information on m H , , we can use the invariant mass of b b jj instead of M T ( b b jj ).In Fig. 12, we show the distributions of signal and backgrounds for the invariant mass of b b jj . The numbers of events at the twin peaks are O (30) and O (10), which are larger thanthaose at the twin Jacobian peaks in the figure for M T ( b b jj ) (the right figure of Fig 11). Signal S Background
B S/ √ S + B Basic cuts(Eq. (12)) 145 40 11TABLE III. Numbers of signal event and background events under the basic cuts in Eq. (12) inScenario II. We assume that the efficiency of b-tagging is 70 %. We use the benchmark values inEq.(22).
Next, we discuss the background events at HL-LHC. We consider the process pp → ttbbW + → bbbbW + W + W − → bbbbℓ + ℓ ′ + ννjj as the background. As a result of the numericalcalculation, 40 events are expected to appear at HL-LHC as shown in Table. III. This isthe same order with the signal events. In Fig. 11, the distributions of M T ( b b ℓ + ℓ ′ + (cid:0)(cid:0) E T ) and M T ( b b jj ) in the background events are shown. We use only the basic cuts in Eq. (12) inthe numerical calculation. Nevertheless, in the both figures of Fig. 11, the number of signalevents around the Jacobian peaks are much larger than thoes of the background events.In Fig. 12, the distribution of the background events for the invariant mass M ( b b jj ) inthe background events are shown. The numbers of signal events around the two peaks are We note that we assume some information on the mass of singly charged scalars to select the kinematicalcuts.
100 200 300 400 500 600 700 800 M T ( b ¯ b l + l ′ + /E T ) [GeV] E v e n t / b i n backgroundsignal M T ( b ¯ b jj ) [GeV] E v e n t / b i n backgroundsignal FIG. 11. The distribution of M T ( b b ℓ + ℓ ′ + (cid:0)(cid:0) E T ) (the left one) and M T ( b b jj ) (the right one) inthe signal and background events under the kinematical cuts in Eq. (12). In the figures, the widthof bins is 10 GeV. We use the benchmark values in Eq.(22). M ( b ¯ b jj ) [GeV] E v e n t / b i n backgroundsignal FIG. 12. The distribution of the invariant mass of b b jj in the signal and background eventsunder the kinematical cuts in Eq. (12). In the figure, the width of bins is 10 GeV. We use thebenchmark values in Eq.(22). much larger than those of the background events.In summary, it would be possible that we obtain information on masses of all thecharged scalars H ± , H ± , and Φ ±± by investigating the transverse mass distribution for M T ( b b ℓ + ℓ ′ + (cid:0)(cid:0) E T ) and M T ( b b jj ) and the invariant mass distribution for M ( b b jj ) at HL-LHC.Before closing Subsection B, we give a comment about the detector resolution. In theprocess of Scenario-II, the typical value of the transverse momenta of jets and bottom quarksis about 100 GeV. As mentioned in the end of the section for Scenario-I, at the ATLAS23etector, the energy resolution for p jT ≃
100 GeV is about 10 % [64]. In Figs. 11 and 12, wetake the width of bins as 10 GeV. Therefore, it would be possible that the twin Jacobianpeaks in the distribution for M T ( jj ) or M ( jj ) overlap each other and they looks like oneJacobian peak with the unclear endpoint at the ATLAS detector if the mass differences isnot large enough. Then, it would be difficult to obtain the information on both m H and m H from the transverse momentum distribution. Even in this case, it would be able toobtain the hint for masses by investigating the process. V. SUMMARY AND CONCLUSION
We have investigated collider signatures of the doubly and singly charged scalar bosonsat the HL-LHC by looking at the transverse mass distribution as well as the invariant massdistribution in the minimal model with the isospin doublet with the hypercharge Y = 3 / pp → W + ∗ → Φ ++ H − , in the following two cases depending on the mass of the scalar bosons with the appropriatekinematical cuts . (1) The main decay mode of the singly charged scalar bosons is the taulepton and missing (as well as charm and strange quarks). (2) That is into a top bottom pair.In the both cases, we have assumed that the doubly charged scalar boson is heavier thanthe singly charged ones. It has been concluded that the scalar doublet field with Y = 3 / ±± and H ± , are too large. ACKNOWLEDGEMENTS
We would like to thank Arindam Das and Kei Yagyu for useful discussions. This workis supported by Japan Society for the Promotion of Science, Grant-in-Aid for ScientificResearch, No. 16H06492, 18F18022, 18F18321 and 20H00160.24 ppendix A: Some formulae for the decays of charged scalars
In this section, we show some analytic formulae for decay rates of the charged scalars H ± , and Φ ±± .
1. Formulae for decays of the singly charged scalars H ± , a. 2-body decays The decay rate for the decay of H ± i ( i = 1 ,
2) into a pair of quarks qq ′ is given byΓ( H ± i → qq ′ ) = 3 m H i π (cid:18) m H i v (cid:19) χ ′ i cot β | V qq ′ | (cid:16) ( r q + r q ′ ) − ( r q + r q ′ ) − r q r q ′ (cid:17) F ( r q , r q ′ ) , (A1)where r q ( r q ′ ) is the ratio of the squared mass of quark q ( q ′ ) to the squared mass of H ± i : r q = m q m H i , r q ′ = m q ′ m H i , (A2)and χ ′ i is defined as follows. χ ′ = cos χ, χ ′ = sin χ. (A3)The function F ( x, y ) in Eq. (A1) is defined as F ( x, y ) = p x − y ) − x + y ) . (A4)The decay rate for the decay of H ± i into a charged lepton ℓ and a neutrino ν ℓ is given byΓ( H ± i → ℓν ℓ ) = m H i π (cid:16) m ℓ v (cid:17) χ ′ i cot β (cid:18) − m ℓ m H i (cid:19) , (A5)where m ℓ is mass of ℓ .In the case that m H i > m H j + m Z ( i, j = 1 , , i = j ), the decay H ± i → H ± j Z is allowed,and its decay rate is given byΓ( H i ± → H ± j Z ) = m H i π (cid:16) m H i v (cid:17) sin χF ( r Z , r j ) ( i = j ) , (A6)where r Z = m W m H i , r j = m H j m H i . (A7)25 . 3-body decays The decay rate for H ± i → t ∗ b → W ± bb is given byΓ( H ± i → t ∗ b → W ± bb ) = 3 m H i π (cid:16) m t v (cid:17) χ ′ i cot β | V tb | Z r W d xx (1 − x ) ( x − r W ) ( x + 2 r W )( x − r t ) + r t r Γ t , (A8)where mass of the bottom quark is neglected, and r W , r t , and r Γ r are defined as follows. r W = m W m H i , r t = m t m H i , r Γ t = Γ t m H i , (A9)where Γ t is the total decay width of the top quark.In the case that m H i > m H j ( i = j ), the decay H ± i → H ± j Z ∗ → H ± j f f , where f is a SMfermion, is allowed. The decay rate is given byΓ( H ± i → H ± j Z ∗ → H ± j f f ) = N fc m H i π (cid:16) m Z v (cid:17) sin χ (cid:0) ( C fV ) + ( C fA ) (cid:1) × Z (1 −√ r j ) d x F ( x, r j ) ( x − r Z ) + r Z r Γ Z , (A10)where N fc is the color degree of freedom of a fermion f , r Z and r j are defined same withthat in Eq. (A7), and r Γ Z is the ratio of the squared decay rate of Z boson to squared massof H ± i : r Γ Z = Γ Z m H i . (A11)In addition, the coeffitient C fV ( C fA ) in Eq. (A10) is the coupling constant of the vector (axialvector) current: L = g L θ W f γ µ (cid:0) C fV + C fA γ (cid:1) f Z µ , (A12)where g L is the gauge coupling constant of the gauge group SU (2) L , and θ W is the Weinbergangle. In Eq. (A10), mass of fermions are neglected.
2. Formulae for decays of the doubly charged scalar Φ ±± a. 2-body decay If m Φ ±∓ > m H i + m W , the decay Φ ±± → H ± i W ± ( i = 1 ,
2) is allowed. The decay rate isgiven by Γ(Φ ±± → H ± i W ± ) = m Φ π (cid:16) m Φ v (cid:17) χ i F ( R W , R i ) , (A13)26here χ i is defined in Eq. (10), the function F ( x, y ) is defined in Eq. (A4), and R i and R W is defined as follows. R W = m W m , R i = m H i m . (A14) b. 3-body decay In the case that where the mass differences between Φ ±± and H ± i is so small that decaysΦ ±± → H ± i W ± are prohibited, three-body decays Φ ±± → H ± i f f ′ , where f and f ′ areSM fermions, are dominant in small m Φ region. (See Fig. 3.) The branching ratio forΦ ±± → H ± i f f ′ is given byΓ(Φ ±± → H ± i f f ′ ) = N fc π χ i Z (1 −√ R i ) d xx F ( x, R i ) ( x − R W ) + R Γ W R W , (A15)where R Γ W is the squared ratio of the decay width of W boson (Γ W ) to m Φ ; R Γ W = Γ W m . (A16)In Eq. (A15), we neglect the masses of f and f ′ .In the large m Φ region, Φ ±± → W ± f f ′ is also important. The decay rate is given byΓ(Φ ±± → W ± f f ′ ) = N fc m Φ π (cid:16) m Φ v (cid:17) sin 2 χ cot β | V ff ′ | × Z (1 −√ R W ) ( √ R f + √ R f ′ ) d x F (cid:16) R f x , R ′ f x (cid:17) F ( x, R W ) G ( x ) , (A17)where the function G ( x ) is defined as follows. G ( x ) = n ( R f + R f ′ )( x − R f − R f ′ ) − R f R f ′ o × ( x − R ) + R R Γ + 1( x − R ) + R R Γ ) . (A18)The symbols R f , R f ′ , R i , and R Γ i ( i = 1 ,
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