The μ + μ − collider to sensitivity estimates on the magnetic and electric dipole moments of the tau-lepton
M. Köksal, A. A. Billur, A. Gutiérrez-Rodríguez, M. A. Hernández-Ruíz
aa r X i v : . [ h e p - ph ] N ov The µ + µ − collider to sensitivity estimates on the magnetic andelectric dipole moments of the tau-lepton M. K¨oksal ∗ , A. A. Billur † , A. Guti´errez-Rodr´ıguez ‡ , and M. A. Hern´andez-Ru´ız § Deparment of Optical Engineering, Sivas Cumhuriyet University, 58140, Sivas, Turkey. Deparment of Physics, Sivas Cumhuriyet University, 58140, Sivas, Turkey. Facultad de F´ısica, Universidad Aut´onoma de ZacatecasApartado Postal C-580, 98060 Zacatecas, M´exico. Unidad Acad´emica de Ciencias Qu´ımicas, Universidad Aut´onoma de ZacatecasApartado Postal C-585, 98060 Zacatecas, M´exico. (Dated: November 6, 2018)
Abstract
Using the effective Lagrangian formalism, the anomalous magnetic and electric dipole momentsof the tau-lepton in the µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − process at the future muon colliders atthe √ s = 1 . , µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − process leads to a remarkable improvement inthe existing experimental bounds on the anomalous magnetic and electric dipole moments of thetau-lepton. PACS numbers: 13.40.Em, 14.60.FgKeywords: Electric and Magnetic Moments, Taus, Muon colliders. ∗ [email protected] † [email protected] ‡ alexgu@fisica.uaz.edu.mx § [email protected] . INTRODUCTION A high priority on the physic program for the current and future of High Energy Physics(HEP) is the quest for physics Beyond the Standard Model (BSM). With this motivation a µ + µ − collider at the CERN, is one of the potential candidates for a future energy frontiercolliding machine. The original idea about the possibility of muon colliders was proposedby G.I. Budker [1], Skrinsky and Parkhomchuk [2] and Neuffer [3]. More recently, a col-laboration of different members, has been formed to coordinate studies on specific designs[4–6]. The design of this collider demonstrates a novel high-energy and high-luminosity col-lider type, which will permit exploration of HEP at energy frontiers beyond the reach ofcurrently existing and proposed electron-positron colliders. In addition, the proposed 1.5,3, 6 TeV center-of-mass energies µ + µ − collider in the CERN provides outstanding discoverypotential and can complement the physics program of the Large Hadron Collider (LHC).The reason as well as the advantage for interest in muon colliders is that they are fun-damental leptons with a mass that is a factor of 207 greater than the mass of the electronor positron. As for an electron, the full center-of-mass energy is available in an interaction.But because of the large mass, there is essentially no synchrotron radiation from the muonin comparison to electrons and positrons. Consequently, the machine can be circular andmuch smaller than the current design of linear electron-positron colliders, and the hope isthat the sum of development and construction costs will not be so high as to make therealization unaffordable.A muon collider will accelerate two muon beams in opposite directions around an under-ground ring 6.3 km of circumference. Beams will collide head-on and scientists will studywhat results from the collision to search for dark matter, dark energy, the matter-antimatterasymmetry, supersymmetric particles, signs of extra dimensions and other subatomic phe-nomena. Furthermore, a muon collider has the characteristic that it focuses on a region ofenergy to discover the physical phenomena that the LHC can not reveal on its own. A muoncollider would provide a clear and unobstructed view of the subatomic world. In addition,the beauty of a muon collider is that the collision events are clean.Starting from the feasibility of a muon collider to study new physics, we study theanomalous Magnetic Moment ( τ MM) and Electric Dipole Moment ( τ EDM) of the tau-lepton in the µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − process at the future muon collider at the2 s = 1 . , L =10 , , , , , , , , f b − and systematic uncertainties of δ sys = 0% , , γ ∗ γ ∗ collisions can be examined by EquivalentPhoton Approximation (EPA) [7–9], that is to say, using the Weizsacker-Williams approx-imation (WWA). In EPA, photons emitted from incoming leptons which have very lowvirtuality are scattered at very small angles from the beam pipe and because the emittedquasi-real photons have a low Q virtuality, these are almost real. These processes havebeen observed experimentally at the LEP, Tevatron and LHC [10–16]. In particular, themost stringent experimental limit on the anomalous τ MM and τ EDM is obtained throughthe process e + e − → e + γ ∗ γ ∗ e − → e + τ ¯ τ e − by using multiperipheral collision at the LEP [17].The study for the dipole moments is a very active field with ongoing experiments tomeasure the dipole moments of a variety of physical systems such as Atoms, Molecules,Nuclei and Particles, see e.g. Refs. [18–20] for recent reviews. In addition, the dipolemoments of particles can be probed by analyzing decay and collision processes. This has beendone for the tau-lepton in processes such as e + e − → e + e − τ + τ − by DELPHI Collaboration[17] and e + e − → τ + τ − by BELLE Collaboration [21], respectively, obtaining the followingsbounds: DELPHI : − . < a τ < . ,
95% C.L. , − . < d τ (10 − ecm ) < . ,
95% C.L. , (1)and BELLE : − . < Re ( d τ (10 − ecm )) < . ,
95% C.L. , − . < Im ( d τ (10 − ecm )) < . ,
95% C.L. . (2)A summary of experimental and theoretical bounds on the dipole moments of the τ -leptonare given in Table I of Ref. [22]. See Refs. [23–48] for another bounds on the τ MM and the τ EDM in different context.The paper is organized in the following way. In Section II, are given the gauge-invariantoperators of dimension six. In Section III, we study the total cross-section and the dipolemoments of the tau-lepton through the process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − at the γ ∗ γ ∗ II. ELECTROMAGNETIC CURRENT AND OPERATORS OF DIMENSION SIXA. τ + τ − γ vertex form factors The proposed high-energy and high-luminosity µ + µ − collider offers new opportunitiesfor the improved determination of the fundamental physical parameters of standard heavyleptons. In this sense, compared to the electron or the muon case, the electromagneticproperties of the τ -lepton are largely unexplored. On this topic, a convenient way of studyingits electromagnetic properties on a model-independent way is through the effective tau-photon interaction vertex which is described by four independent form factors. The possibleelectromagnetic properties of the τ -lepton are summarized in the most general expressionconsistent with Lorentz and electromagnetic gauge invariance for the τ + τ − γ vertex betweenon-shell tau-lepton and the photon [23, 49–52] as followsΓ ατ = eF ( q ) γ α + ie m τ F ( q ) σ αµ q µ + e m τ F ( q ) σ αµ q µ γ + eF ( q ) γ ( γ α − q α m τ q ) . (3)The quantities e and m τ are the charge of the electron and the mass of the τ -lepton, re-spectively. σ αµ = i [ γ α , γ µ ] and q = p ′ − p is the four-momentum of the photon. The formfactors F , , , ( q ) have the following interpretations for q = 0: Q τ = F (0) , Electric charge , (4) a τ = F (0) , τ MM , (5) d τ = e m τ F (0) , τ EDM . (6) F ( q ) is the Anapole form factor. B. Gauge-invariant operators of dimension six
In theoretical, experimental and phenomenological searches most of the tau-lepton elec-tromagnetic vertices search involve off-shell tau-leptons. In our study, one of the tau-leptons4s off-shell and measured quantity is not directly a τ and d τ . For this reason deviations of thetau-lepton dipole moments from the SM values are examined in a model independent wayusing the effective Lagrangian formalism. This formalism is defined by high-dimensional op-erators which lead to anomalous τ + τ − γ coupling. For our study, we apply the dimension-sixeffective operators that contribute to the τ MM and τ EDM [53–56]: L eff = 1Λ h C LW Q LW + C LB Q LB + h.c i , (7)where Q LW = ( ¯ ℓ τ σ µν τ R ) σ I ϕW Iµν , (8) Q LB = ( ¯ ℓ τ σ µν τ R ) ϕB µν , (9)in which, B µν is the U (1) Y gauge field strength tensors and W Iµν is the SU (2) L gauge fieldstrength tensors, respectively, while ϕ and ℓ τ are the Higgs and the left-handed SU (2) L doublets which contain τ , and σ I are the Pauli matrices.The corresponding CP even κ and CP odd ˜ κ observables are obtained with the electroweaksymmetry breaking from the effective Lagrangian given by Eq. (7): κ = 2 m τ e √ υ Λ Re h cos θ W C LB − sin θ W C LW i , (10)˜ κ = 2 m τ e √ υ Λ Im h cos θ W C LB − sin θ W C LW i , (11)where υ = 246 GeV is the breaking scale of the electroweak symmetry, Λ is the new physicsscale and sin θ W is the sin of the weak mixing angle.These observables are related to contribution of the anomalous τ MM and τ EDM throughthe following relations: κ = ˜ a τ , (12)˜ κ = 2 m τ e ˜ d τ . (13)5 II. THE CROSS-SECTION OF THE PROCESS µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − In order to study the opportunities of the muon collider as an option to sensitivity es-timates on the τ MM and τ EDM in detail, we focus here only on the cross-section of theprocess µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − . The Feynman diagrams at the tree level are givenin Fig. 1. A. γ ∗ γ ∗ → τ + τ − cross-section The corresponding matrix elements for the subprocess γ ∗ γ ∗ → τ + τ − in terms of theMandelstam invariants ˆ s , ˆ t , ˆ u and from the anomalous parameters κ and ˜ κ are | M | = 16 π Q τ α e m τ (ˆ t − m τ ) (cid:20) κ ( m τ − ˆ t )( m τ + ˆ s − ˆ t ) m τ − m τ − m τ ˆ s + ˆ t (ˆ s + ˆ t )) m τ + 2( m τ − ˆ t )( κ (17 m τ + (22ˆ s − t ) m τ + ˆ t (9ˆ t − s ))+ ˜ κ (17 m τ + 4ˆ s − t )( m τ − ˆ t )) m τ + 12 κ ( κ + ˜ κ )ˆ s ( m τ − m τ ˆ t ) − ( κ + ˜ κ ) ( m τ − ˆ t ) ( m τ − ˆ s − ˆ t ) (cid:21) , (14) | M | = − π Q τ α e m τ (ˆ u − m τ ) (cid:20) κ ( m τ + (ˆ s − t ) m τ + ˆ t (ˆ s + ˆ t )) m τ + 16(7 m τ − (3ˆ s + 4ˆ t ) m τ + ˆ t (ˆ s + ˆ t )) m τ + 2( m τ − ˆ t )( κ ( m τ + (17ˆ s − t ) m τ + 9ˆ t (ˆ s + ˆ t ))+ ˜ κ ( m τ − t )( m τ − ˆ t − ˆ s )) m τ + ( κ + ˜ κ ) ( m τ − ˆ t ) ( m τ − ˆ s − ˆ t ) (cid:21) , (15) M † M + M † M = 16 π Q τ α e m τ (ˆ t − m τ )(ˆ u − m τ ) × (cid:20) − m τ − m τ ˆ s ) + 8 κm τ (6 m τ − m τ (ˆ s + 2ˆ t ) − ˆ s ) + 6ˆ t ) + 6ˆ s ˆ t ) + ( κ (16 m τ − m τ (15ˆ s + 32ˆ t ) + m τ (15ˆ s ) + 14ˆ t ˆ s + 16ˆ t ) ) + ˆ s ˆ t (ˆ s + ˆ t )) + ˜ κ (16 m τ − m τ (15ˆ s + 32ˆ t )+ m τ (5ˆ s ) + 14ˆ t ˆ s + 16ˆ t ) ) + ˆ s ˆ t (ˆ s + ˆ t ))) − κ ˆ s ( κ + ˜ κ )6 ( m τ + m τ (ˆ s − t ) + ˆ t (ˆ s + ˆ t )) − s ( κ + ˜ κ ) ( m τ − tm τ + ˆ t (ˆ s + ˆ t )) (cid:21) . (16)Here, the Mandelstam variables are ˆ s = ( p + p ) = ( p + p ) , ˆ t = ( p − p ) = ( p − p ) ,ˆ u = ( p − p ) = ( p − p ) , while p and p are the four-momenta of the incoming photons, p and p are the momenta of the outgoing tau-lepton, Q τ is the tau-lepton charge and α e is the fine-structure constant.WWA is another possibility for tau pair production, and the quasi-real photons emittedfrom both lepton beams collide with each other and produce the subprocess γ ∗ γ ∗ → τ + τ − .In WWA, the photon spectrum is given by f γ ∗ ( x ) = απE µ { [ 1 − x + x / x ] log ( Q max Q min ) − m µ xQ min (1 − Q min Q max ) − x [1 − x log ( x E µ + Q max x E µ + Q min ) } , (17)where x = E γ /E µ and Q max is maximum virtuality of the photon. In this work, we havetaken into account the maximum virtuality of the photon as Q max = 2 , , GeV . Theminimum value of the Q min is given by Q min = m µ x − x . (18)The reaction γ ∗ γ ∗ → τ + τ − participates as a subprocess in the main process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − , and the total cross-section is given by σ = Z f γ ∗ ( x ) f γ ∗ ( x ) d ˆ σdE dE . (19)We presented results for the dependence of the total cross-section of the pro-cess µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − on κ (˜ κ ). We consider the following cases with Q max = 2 , , GeV : 7 For √ s = 1500 GeV . σ ( κ ) = h (5 . × ; 1 . × ; 1 . × ) κ + (4 . × ; 6 . × ; 8 . × ) κ + (4 . × ; 7 . × ; 8 . × ) κ + (1 .
14; 1 .
70; 2 . κ + (0 .
35; 0 .
52; 0 . i ( pb ) (20) σ (˜ κ ) = h (5 . × ; 1 . × ; 1 . × )˜ κ + (4 . × ; 7 . × ; 8 . × )˜ κ + (0 .
35; 0 .
52; 0 . i ( pb ) . (21) • For √ s = 3000 GeV . σ ( κ ) = h (2 . × ; 4 . × ; 5 . × ) κ + (7 . × ; 1 . × ; 1 . × ) κ + (7 . × ; 1 . × ; 1 . × ) κ + (1 .
65; 2 .
34; 2 . κ + (0 .
51; 0 .
72; 0 . i ( pb ) (22) σ (˜ κ ) = h (2 . × ; 4 . × ; 5 . × )˜ κ + (7 . × ; 1 . × ; 1 . × )˜ κ + (0 .
51; 0 .
72; 0 . i ( pb ) . (23) • For √ s = 6000 GeV . σ ( κ ) = h (9 . × ; 1 . × ; 2 . × ) κ + (1 . × ; 1 . × ; 2 . × ) κ + (1 . × ; 1 . × ; 2 . × ) κ + (2 .
26; 3 .
00; 3 . κ + (0 .
68; 0 .
93; 1 . i ( pb ) (24) σ (˜ κ ) = h (9 . × ; 1 . × ; 2 . × )˜ κ + (1 . × ; 1 . × ; 2 . × )˜ κ + (0 .
68; 0 .
93; 1 . i ( pb ) . (25)These formulas have been obtained with the help of the package CALCHEP [57], whichcan computate the Feynman diagrams, integrate over multiparticle phase space and eventsimulation. Furthermore, we apply the following acceptance cuts for τ + τ − signal at themuon collider: 8 τ, ¯ τt > GeV, (transverse momentum of the final state particles) , | η τ, ¯ τ | < . , (pseudorapidity reduces the contamination from other particlesmisidentified as tau) , ∆ R ( τ, ¯ τ ) > . , (separation of the final state particles) , (26)of course, is fundamental that we apply these cuts to reduce the background and to optimizethe signal sensitivity.From Eqs. (20)-(25), the dependent terms on κ (˜ κ ) are purely anomalous, and the inde-pendent term of κ (˜ κ ) give the cross-section of the SM. B. Sensitivity on the ˜ a τ and ˜ d τ through µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − at the muoncollider A muon collider is an ideal discovery machine in the multi-TeV energy range. In thissubsection, we assess the capabilities of future muon collider to test the existence of the τ MM and τ EDM by means of the process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − at the mode γ ∗ γ ∗ → τ ¯ τ . Specifically, we assume energies from 1.5 to 6 TeV and integrated luminositiesof at least 10 − f b − . The primary motivation for the √ s = 1 . , , T eV center of-mass energies, luminosities L = 10 , , , , , , f b − of such a collider, as wellas of the virtuality of the photon Q max = 2 , , GeV is to optimize the expected signalcross-section and the sensitivity on ˜ a τ and ˜ d τ .The total cross-section production as a function of κ (˜ κ ) has been computed in the previoussection (see Eqs. (20)-(25)), and is displayed in Figs. 2-7. The graphics are for Q max =2 , , GeV with √ s = 1 . , , T eV . These figures clearly show a strong dependence withrespect to κ (˜ κ ), the virtuality of the photon Q max , as well as with the center-of-mass energy √ s . In Figs. 8-13, the graphics are for √ s = 1 . , , T eV with Q max = 2 , , GeV ,respectively. These figures also show a clear and strong dependence with respect to κ (˜ κ ), √ s and Q max . The total cross-section increase of the order of σ = 20 pb at the upper andlower limit of κ (˜ κ ) and tends to the value of the SM when κ (˜ κ ) tends to zero, as indicatedby Eqs. (20)-(25).To estimate the sensitivity on the parameters ˜ a τ and ˜ d τ we consider the acceptance cutsgiven in Eq. (26), take into account the systematic uncertainties δ sys = 0 , ,
5% and we9dopt the statistical method for the χ defined as [22, 42, 58–63] χ = σ SM − σ BSM ( √ s, Q max , ˜ a τ , ˜ d τ ) σ SM q ( δ st ) + ( δ sys ) ! , (27)with σ BSM ( √ s, Q max , ˜ a τ , ˜ d τ ) the total cross-section incorporating contributions from the SMand new physics, δ st = √ N SM is the statistical error and δ sys is the systematic error. Thenumber of events is given by N SM = L int × σ SM , where L int is the integrated luminosity ofthe µ + µ − collider.We now discuss the reach at different center-of-mass energies in the sensitivity esti-mates determination on the τ MM and τ EDM. As we will show below, the sensitivity ofthe electromagnetic properties of the tau-lepton, and in particular its magnetic and elec-tric dipole moments, may be measured competitively in these facilities, using the pro-cess µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − . The sensitivities from the ˜ a τ and ˜ d τ turn out tobe very strong at µ + µ − collider. For this reason we performing a detailed study andwe present Tables that illustrate the sensitivity on ˜ a τ and ˜ d τ for different virtuality ofthe photon Q max , center-of-mass energies √ s , luminosities L , uncertainties systematic δ sys and at 95% C.L. Our results are presented in Tables I-III. Our most significant re-sults on ˜ a τ and ˜ d τ are the following for √ s = 6 T eV , L = 710 f b − , δ sys = 0 and Q max = 2 , , GeV , ˜ a τ = ( − . , . − . , . − . , . | ˜ d τ | = (0 . , . , . × − at 95% C.L.. From these results it is seen thatthe sensitivity improves for large values of Q max . In addition, notice that these results givean improvement of 1-2 orders of magnitude with respect to the results given in Eqs. (1) and(2) obtained by the DELPHI and BELLE Collaborations.Furthermore, in order to following study the opportunities of the muon collider in detail,we focus now in the bounds contours on the ( κ, ˜ κ ) plane depending on integrated luminosityand for √ s = 1 . , , T eV and Q max = 2 GeV in Figs. 17-19. The sensitivity reach of amuon collider is indicated by a blue, yellow and green solid line in each plot. These resultsshow that anomalous couplings κ and ˜ κ can be probed with very good sensitivity in a muoncollider. 10 ABLE I: Model-independent sensitivity estimates for the ˜ a τ magnetic moment and the ˜ d τ electricdipole moment for Q max = 2 , , GeV , √ s = 1 . T eV and L = 10 , , , , f b − at 95%C.L., through the process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − . √ s = 1 . T eV , 95% C.L. L ( f b − ) δ sys ˜ a τ | ˜ d τ ( ecm ) |
10 0% [(-0.00763; -0.00684; -0.00646), (0.00502; 0.00438; 0.00405)] (3 . . . × −
10 3% [(-0.00916; -0.00862; -0.00839), (0.00654; 0.00616; 0.00598)] (4 . . . × −
10 5% [(-0.01066; -0.01020; -0.01000), (0.00804; 0.00773; 0.00758)] (5 . . . × −
20 0% [(-0.00667; -0.00599; -0.00567), (0.00406; 0.00354; 0.00327)] (2 . . . × −
20 3% [(-0.00874; -0.00832; -0.00815), (0.00613; 0.00586; 0.00574)] (4 . . . × −
20 5% [(-0.01043; -0.01004; -0.00987), (0.00781; 0.00757; 0.00745)] (4 . . . × −
50 0% [(-0.00564; -0.00509; -0.00483), (0.00304; 0.00264; 0.00243)] (2 . . . × −
50 3% [(-0.00845; -0.00812; -0.00798), (0.00584; 0.00566; 0.00597)] (3 . . . × −
50 5% [(-0.01028; -0.00994; -0.00979), (0.00766; 0.00747; 0.00737)] (4 . . . × −
100 0% [(-0.00502; -0.00454; -0.00432), (0.00242; 0.00209; 0.00192)] (1 . . . × −
100 3% [(-0.00834; -0.00805; -0.00793), (0.00573; 0.00559; 0.00552)] (3 . . . × −
100 5% [(-0.01023; -0.00991; -0.00976), (0.00761; 0.00744; 0.00735)] (4 . . . × −
110 0% [(-0.00494; -0.00447; -0.00426), (0.00234; 0.00202; 0.00186)] (1 . . . × −
110 3% [(-0.00833; -0.00804; -0.00792), (0.00572; 0.00559; 0.00551)] (3 . . . × −
110 5% [(-0.01022; -0.00990; -0.00976), (0.00760; 0.00744; 0.00735)] (4 . . . × − IV. CONCLUSIONS
We perform a comprehensive study of the sensitivity to both the total cross-section ofthe process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − and on the ˜ a τ magnetic moment and the ˜ d τ electric dipole moment, with respect to the parameters of the future muon collider, √ s and L , as well as of the virtuality of the photon Q max . We consider the most general Lagrangiancoupling of two tau-leptons and the photon (see Eqs. (7)-(13)), which involves both ˜ a τ and11 ABLE II: Model-independent sensitivity estimates for the ˜ a τ magnetic moment and the ˜ d τ electricdipole moment for Q max = 2 , , GeV , √ s = 3 T eV and L = 50 , , , , f b − at 95%C.L., through the process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − . √ s = 3 T eV , 95% C.L. L ( f b − ) δ sys ˜ a τ | ˜ d τ ( ecm ) |
50 0% [(-0.00473; -0.00430; -0.00408), (0.00256; 0.00224; 0.00207)] (1 . . . × −
50 3% [(-0.00756; -0.00728; -0.00714), (0.00540; 0.00523; 0.00515)] (3 . . . × −
50 5% [(-0.00926; -0.00893; -0.00878), (0.00710; 0.00690; 0.00680)] (4 . . . × −
100 0% [(-0.00421; -0.00383; -0.00365), (0.00203; 0.00177; 0.00164)] (1 . . . × −
100 3% [(-0.00750; -0.00723; -0.00710), (0.00533; 0.00518; 0.00511)] (3 . . . × −
100 5% [(-0.00921; -0.00891; -0.00876), (0.00703; 0.00688; 0.00678)] (4 . . . × −
200 0% [(-0.00378; -0.00345; -0.00329), (0.00160; 0.00139; 0.00128)] (1 . . . × −
200 3% [(-0.00746; -0.00720; -0.00708), (0.00530; 0.00516; 0.00509)] (3 . . . × −
200 5% [(-0.00921; -0.00890; -0.00875), (0.00705; 0.00687; 0.00677)] (4 . . . × −
300 0% [(-0.00356; -0.00326; -0.00312), (0.00139; 0.00120; 0.00111)] (1 . . . × −
300 3% [(-0.00745; -0.00720; -0.00707), (0.00528; 0.00515; 0.00508)] (3 . . . × −
300 5% [(-0.00920; -0.00890; -0.00875), (0.00704; 0.00686; 0.00677)] (4 . . . × −
450 0% [(-0.00337; -0.00310; -0.00296), (0.00120; 0.00103; 0.00095)] (1 . . . × −
450 3% [(-0.00744; -0.00719; -0.00707), (0.00528; 0.00515; 0.00508)] (3 . . . × −
450 5% [(-0.00920; -0.00889; -0.00875), (0.00704; 0.00686; 0.00677)] (4 . . . × − ˜ d τ interactions.We found that, for ˜ a τ and ˜ d τ , the best sensitivity constraints come from consider √ s =6 T eV , L = 710 f b − and Q max = 64 GeV and we estimated the sensitivity to be ˜ a τ =( − . , . | ˜ d τ | = 0 . × − at 95% C.L. as is show in Table III. Thiscompares favorably with earlier DELPHI and BELLE studies for τ MM and τ EDM (see Eqs.(1) and (2)), and readily provides leading sensitivity for ˜ a τ and ˜ d τ .We already show through Figs. 2-19 and Tables I-III that a future µ + µ − collider, currently12 ABLE III: Model-independent sensitivity estimates for the ˜ a τ magnetic moment and the ˜ d τ electric dipole moment for Q max = 2 , , GeV , √ s = 6 T eV and L = 50 , , , , f b − at 95% C.L., through the process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − . √ s = 6 T eV , 95% C.L. L ( f b − ) δ sys ˜ a τ | ˜ d τ ( ecm ) |
50 0% [(-0.00410; -0.00371; -0.00353), (0.00220; 0.00196; 0.00303)] (1 . . . × −
50 3% [(-0.00686; -0.00657; -0.00643), (0.00501; 0.00487; 0.00479)] (3 . . . × −
50 5% [(-0.00839; -0.00815; -0.00789), (0.00658; 0.00640; 0.00630)] (4 . . . × −
100 0% [(-0.00365; -0.00332; -0.00315), (0.00174; 0.00155; 0.00145)] (1 . . . × −
100 3% [(-0.00682; -0.00654; -0.00640), (0.00496; 0.00484; 0.00476)] (3 . . . × −
100 5% [(-0.00837; -0.00804; -0.00788), (0.00655; 0.00638; 0.00628)] (4 . . . × −
300 0% [(-0.00310; -0.00282; -0.00269), (0.00119; 0.00105; 0.00098)] (1 . . . × −
300 3% [(-0.00678; -0.00651; -0.00639), (0.00493; 0.00481; 0.00474)] (3 . . . × −
300 5% [(-0.00836; -0.00803; -0.00787), (0.00654; 0.00637; 0.00627)] (4 . . . × −
500 0% [(-0.00290; -0.00264; -0.00252), (0.00098; 0.00087; 0.00081)] (0 . . . × −
500 3% [(-0.00678; -0.00615; -0.00638), (0.00492; 0.00481; 0.00474)] (3 . . . × −
500 5% [(-0.00835; -0.00803; -0.00787), (0.00654; 0.00637; 0.00627)] (4 . . . × −
710 0% [(-0.00278; -0.00253; -0.00242), (0.00086; 0.00076; 0.00071)] (0 . . . × −
710 3% [(-0.00677; -0.00651; -0.00639), (0.00492; 0.00481; 0.00474)] (3 . . . × −
710 5% [(-0.00835; -0.00803; -0.00787), (0.00653; 0.00637; 0.00627)] (4 . . . × − envisioned as a machine for new physics BSM, will have leading sensitivity to probing both˜ a τ and ˜ d τ couplings simultaneously through the process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − .However, it is worth mentioning that significant room remains to be explored in both the˜ a τ and ˜ d τ couplings.In conclusion, our study complement and extend previous ˜ a τ and ˜ d τ sensitivity estimatesmade for various specific collider environments. In general, the muon collider with thehighest integrated luminosity in proposal can reach better sensitivity couplings in the high13nergy region. In addition, this collider with larger center-of-mas energies is able to explorebroader parameter space in the high energy region. Acknowledgements
M. K. and A. A. B. acknowledge that this work is supported by the Scientific ResearchProject Found of Cumhuriyet University under the project number TEKNO-022. A. G. R.and M. A. H. R. acknowledge support from SNI and PROFOCIE (M´exico). [1] G.I. Budker, Accelerators and Colliding Beams, Proc. of the 7th International Accel. Confer-ence (Erevan), 1969.[2] V.V. Parkhomchuk and A.N. Skrinsky, Ionization Cooling: Physics and Applications, Procof the 12th International Conf. on High Energy Accelerators, 1983, eds. F.T. Cole and R.Donaldson.[3] D. Neuffer, Principles and Applications of Muon Cooling,
Particle Accelerators , 75, (1983).[4] R. B. Palmer, Muon Colliders, Review of Accelerator Science and Technology , 137 (2014)and references therein.[5] E. Eichten, Future high energy colliders, in Muon Accelerator Program 2014 Spring Workshop,Fermilab, Batavia, Illinois, U.S.A., May 2731 2014.[6] Charles M. Ankenbrandt, Muzaffer Atac, et al. , [Muon Collider Collaboration], Physical Re-view Special Topics-Accelerators and Beams , 081001 (1999) and references therein.[7] V. M. Budnev, I. F. Ginzburg, G. V. Meledin and V. G. Serbo, Phys. Rep. , 181 (1975).[8] G. Baur, et al. , Phys. Rep. , 359 (2002).[9] K. Piotrzkowski,
Phys. Rev.
D63 , 071502 (2001).[10] A. Abulencia, et al. , [CDF Collaboration],
Phys. Rev. Lett. , 112001 (2007).[11] T. Aaltonen, et al. , [CDF Collaboration], Phys. Rev. Lett. , 222002 (2009).[12] T. Aaltonen, et al. , [CDF Collaboration],
Phys. Rev. Lett. , 242001 (2009).
13] S. Chatrchyan, et al. , [CMS Collaboration],
JHEP , 052 (2012).[14] S. Chatrchyan, et al. , [CMS Collaboration],
JHEP , 080 (2012).[15] V. M. Abazov, et al. , [D0 Collaboration],
Phys. Rev.
D88 , 012005 (2013).[16] S. Chatrchyan, et al. , [CMS Collaboration],
JHEP , 116 (2013).[17] J. Abdallah, et al. , [DELPHI Collaboration], Eur. Phys. J.
C35 , 159 (2004).[18] J. Engel, M. J. Ramsey-Musolf, and U. van Kolck,
Prog. Part. Nucl. Phys. , 21 (2013).[19] N. Yamanaka, et al. , Eur. Phys. J.
A53 , 54 (2017).[20] T. Chupp, P. Fierlinger, M. Ramsey-Musolf, and J. Singh, arXiv:1710.02504 [physics.atom-ph].[21] K. Inami, et al. , [BELLE Collaboration],
Phys. Lett.
B551 , 16 (2003).[22] M. K¨oksal, A. A. Billur, A. Guti´errez-Rodr´ıguez and M. A. Hern´andez-Ru´ız,
Phys. Rev.
D98 ,015017 (2018).[23] S. Eidelman and M. Passera,
Mod. Phys. Lett.
A22 , 159 (2007).[24] W. Bernreuther, A. Brandenburg and P. Overmann,
Phys. Lett.
B391 , 413 (1997), Erratum:
Phys. Lett.
B412 , 425 (1997).[25] E. O. Iltan,
Eur. Phys. J.
C44 , 411 (2005).[26] B. Dutta, R. N. Mohapatra,
Phys. Rev.
D68 , 113008 (2003).[27] E. Iltan,
Phys. Rev.
D64 , 013013 (2001).[28] E. Iltan,
JHEP , 0305 (2003).[29] E. Iltan,
JHEP 0404 , 018 (2004).[30] L. Tabares, O. A. Sampayo,
Phys. Rev.
D65 , 053012 (2002).[31] S. Eidelman, D. Epifanov, M. Fael, L. Mercolli, M. Passera,
JHEP , 140 (2016).[32] M. K¨oksal, arXiv:1809.01963 [hep-ph].[33] M. A. Arroyo-Ure˜na, et al. , Eur. Phys. J.
C77, 227 (2017).[34] M. A. Arroyo-Ure˜na, et al. , Int. J. Mod. Phys. A32, 1750195 (2017).[35] Xin Chen, et al. , arXiv:1803.00501 [hep-ph].[36] Antonio Pich, Prog. Part. Nucl. Phys. 75, 41-85 (2014).[37] S. Atag and E. Gurkanli,
JHEP , 118 (2016).[38] Lucas Taylor,
Nucl. Phys. Proc. Suppl. , 237 (1999).[39] M. Passera, Nucl. Phys. Proc. Suppl. , 213 (2007).[40] M. Passera,
Phys. Rev.
D75 , 013002 (2007).
41] J. Bernabeu, G. A. Gonz´alez-Sprinberg, J. Papavassiliou, J. Vidal,
Nucl. Phys.
B790 , 160(2008).[42] Y. ¨Ozg¨uven, S. C. Inan, A. A. Billur, M. K¨oksal, M. K. Bahar,
Nucl. Phys.
B923 , 475 (2017).[43] A. Guti´errez-Rodr´ıguez, M. A. Hern´andez-Ru´ız and L.N. Luis-Noriega,
Mod. Phys. Lett.
A19 ,2227 (2004).[44] A. Guti´errez-Rodr´ıguez, M. A. Hern´andez-Ru´ız and M. A. P´erez,
Int. J. Mod. Phys.
A22 ,3493 (2007).[45] A. Guti´errez-Rodr´ıguez,
Mod. Phys. Lett.
A25 , 703 (2010).[46] A. Guti´errez-Rodr´ıguez, M. A. Hern´andez-Ru´ız, C. P. Casta˜neda-Almanza,
J. Phys.
G40 ,035001 (2013).[47] A. A. Billur, M. K¨oksal,
Phys. Rev. D89 , 037301 (2014).[48] W. Bernreuther, O. Nachtmann, P. Overmann,
Phys. Rev.
D48 , 78 (1993).[49] J. A. Grifols and A. M´endez,
Phys. Lett.
B255 , 611 (1991); Erratum ibid. B259 , 512 (1991).[50] R. Escribano and E. Mass´o,
Phys. Lett.
B395 , 369 (1997).[51] C. Giunti and A. Studenikin,
Phys. Atom. Nucl. , 2089 (2009).[52] C. Giunti and A. Studenkin, Rev. Mod. Phys. , 531 (2015).[53] W. Buchmuller and D. Wyler, Nucl. Phys.
B268 , 621 (1986).[54] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek,
JHEP , 085 (2010).[55] M. Fael, Electromagnetic dipole moments of fermions , PhD. Thesis, (2014).[56] S. Eidelman, D. Epifanov, M. Fael, L. Mercolli and M. Passera,
JHEP , 140 (2016).[57] A. Belyaev, N. D. Christensen and A. Pukhov,
Comput. Phys. Commun. , 1729 (2013).[58] A. A. Billur, M. K¨oksal and A. Guti´errez-Rodr´ıguez,
Phys. Rev.
D96 , 056007 (2017).[59] M. K¨oksal, A. A. Billur and A. Guti´errez-Rodr´ıguez,
Adv. High Energy Phys. , 6738409(2017).[60] A. A. Billur, M. K¨oksal,
Phys. Rev.
D89 , 037301 (2014).[61] A. Guti´errez-Rodr´ıguez, M. Koksal, A. A. Billur, and M. A. Hern´andez-Ru´ız, arXiv:1712.02439[hep-ph].[62] M. K¨oksal, S. C. Inan,
Adv. High Energy Phys. , 315826 (2014).[63] I. Sahin and M. Koksal,
JHEP , 100 (2011). IG. 1: The Feynman diagrams contributing to the subprocess γ ∗ γ ∗ → τ + τ − . Q max2 =
64 GeV - - σ pb ) FIG. 2: The total cross-sections of the process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − as a function of κ for center-of-mass energy of √ s = 1 . T eV .
64 GeV - - κ σ ( pb ) FIG. 3: Same as in Fig. 2, but for √ s = 3 T eV . - - κ σ ( pb ) FIG. 4: The same as Fig. 2, but for √ s = 6 T eV .
64 GeV - - κ σ ( pb ) FIG. 5: The total cross-sections of the process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − as a function of ˜ κ for center-of-mass energy of √ s = 1 . T eV . Q max2 =
64 GeV - - κ σ ( pb ) FIG. 6: The same as Fig. 5, but for √ s = 3 T eV . - - κ σ ( pb ) FIG. 7: The same as Fig. 5, but for √ s = 6 T eV . - - κ σ ( pb ) FIG. 8: The total cross-sections of the process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − as a function of κ for Q max = 2 GeV . - κ σ ( pb ) FIG. 9: The same as Fig. 8, but for ˜ κ . - - κ σ ( pb ) FIG. 10: The same as Fig. 8, but for Q max = 16 GeV . - κ σ ( pb ) FIG. 11: The same as Fig. 9, but for Q max = 16 GeV . - - κ σ ( pb ) FIG. 12: The same as Fig. 8, but for Q max = 64 GeV . - κ σ ( pb ) FIG. 13: The same as Fig. 9, but for Q max = 64 GeV .FIG. 14: The total cross-sections of the process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − as a function of κ and ˜ κ for center-of-mass energy √ s = 1 . T eV and Q = 2 GeV . IG. 15: The same as Fig. 14, but for √ s = 3 T eV .FIG. 16: The same as Fig. 14, but for √ s = 6 T eV . =
10 fb - L =
50 fb - L =
100 fb - - - - (cid:1)(cid:0) ˜ FIG. 17: Sensitivity contours at the 95%
C.L. in the ( κ − ˜ κ ) plane for the process µ + µ − → µ + γ ∗ γ ∗ µ − → µ + τ ¯ τ µ − for center-of-mass energy √ s = 1 . T eV and Q = 2 GeV . L =
10 fb - L =
100 fb - L = fb - - - - - - - - κ κ ˜ FIG. 18: The same as Fig. 17, but for √ s = 3 T eV . =
10 fb - L =
100 fb - L =
700 fb - - - - - - - - κ κ ˜ FIG. 19: The same as Fig. 17, but for √ s = 6 T eV ..