The study of unitarity bounds on the neutral triple gauge couplings in the process e^+e^-\to \ell^+\ell^-γ
aa r X i v : . [ h e p - ph ] F e b The study of unitarity bounds on the neutral triple gaugecouplings in the process e + e − → ℓ + ℓ − γ Qing Fu, Yu-Chen Guo, and Ji-Chong Yang
Department of Physics, Liaoning Normal University, Dalian 116029, China
Abstract
The neutral triple gauge couplings (nTGCs) provide a unique opportunity to probe new physicsbeyond the Standard Model. As an effective theory (EFT), the nTGCs are valid only undera certain energy scale. One of the signatures that an EFT is no longer valid is the violationof unitarity. We study the partial wave unitarity bounds on the coefficients of nTGCs in theprocess e + e − → ℓ + ℓ − γ . The kinematic features and event selection strategy are studied by Monte-Carlo simulation. Based on the statistical significance, the expected constraints in experiments areestimated. The minimal luminosities required for the experiments to reach the unitarity boundsare presented. . INTRODUCTION The extension of gauge interactions has been studied intensively in the searching of newphysics (NP) beyond the Standard Model (SM). In the SM effective field theory (SMEFT)approach [1], there are high-dimensional operators contributing to anomalous tri-linear gaugecouplings (aTGCs) and anomalous quartic gauge couplings (aQGCs). While the SMEFThas mainly been applied with only dimension-6 operators in the phenomenological studies,the importance of dimension-8 operators has been emphasised by many researchers [2]. Ithas been shown that the dimension-8 operators play a very important role from the convexgeometry perspective to the SMEFT space [3]. Besides, there are instances sensitive todimension-8 operators because the dimension-6 operators are absent [4]. The neutral triplegauge couplings (nTGCs) [5–8] are examples of such cases.As an effective theory, the SMEFT is valid under a specific energy scale. The contributionsfrom high dimensional operators typically grow with the energy scale. As a consequence, toprobe the signals of high dimensional operators, one needs a large energy scale where thevalidity of the SMEFT becomes an important issue. In the previous studies, the unitarity [9]is often used to determine whether an effective theory is valid [10]. In a e + e − collider,without uncertainties from the parton distribution functions, the energy scale of a process e + e − → X is just the center mass (c.m.) energy of the collider. Therefore, the constraintson the coefficients of the nTGCs in the sense of unitarity can be obtained straightforwardly.Meanwhile, the constraints at certain luminosity can be obtained by studying the signalsignificance. For a fixed energy scale, there is a minimal luminosity that the constraints setby experiments are tighten than those required by unitarity. Since the SMEFT is only validwithin the constraints set by unitarity, the analysis of the signals of nTGCs are meaninglessif the luminosity is not large enough. In this work, we study the contribution of nTGCsin the process e + e − → ℓ + ℓ − γ which has been studied in Ref. ntgc2. The partial waveunitarity bounds of nTGCs in this process at future e + e − colliders are investigated. Thesignal significance is studied by using Monte-Carlo (MC) simulation, based on which theminimal luminosities required to study the nTGCs at different energy scales are presented.The rest of this letter is organized as follows, in Sec. II we briefly introduce the operatorscontributing to nTGCs; the partial wave unitarity bounds are presented in Sec. III; thenumerical results based on MC simulation is shown in Sec. IV; finally Sec. V is a summary.2 I. DIMENSION-8 OPERATORS CONTRIBUTING TO NTGCS
At dimension-8, there 4 CP-conserving operators contributing to nTGCs, they are [5, 7] L nTGC = sign( c ˜ BW )Λ BW O ˜ BW + sign( c B ˜ W )Λ B ˜ W O B ˜ W + sign( c ˜ W W )Λ W W O ˜ W W + sign( c ˜ BB )Λ BB O ˜ BB , (1)with O ˜ BW = iH + ˜ B µν W µρ { D ρ , D ν } H + h.c., O B ˜ W = iH + B µν ˜ W µρ { D ρ , D ν } H + h.c., O ˜ W W = iH + ˜ W µν W µρ { D ρ , D ν } H + h.c., O ˜ BB = iH + ˜ B µν B µρ { D ρ , D ν } H + h.c., (2)where ˜ B µν ≡ ǫ µναβ B αβ , ˜ W µν ≡ ǫ µναβ W αβ and W µν ≡ W aµν σ a / σ a are Pauli matrices, c X are dimensionless coefficients with sign( c X ) = ± , and the Λ X are related with the cutoffscale as Λ X = Λ / | c X | / . At leading order, the process e + e − → ℓ + ℓ − γ can be affected bythose operators via ZV γ couplings where V is a Z boson or a photon.It has been pointed out that, in the case of ZV γ coupling with Z boson and γ on-shell,there is only one independent operator because the O B ˜ W operator is equivalent to O ˜ BW operator, O ˜ W W and O ˜ BB operators do not contribute [7]. Therefore, in the following, weonly consider the O ˜ BW operator. III. THE PARTIAL WAVE UNITARITY BOUND e + e − Z, γ γZ
FIG. 1. Feynman diagrams of the process e + e − → Zγ induced by nTGCs. When the contributions from nTGCs are taken into account, the cross-section of theprocess e + e − → ℓ + ℓ − γ grow with energy scale, which leads to the violation of unitarity atlarge enough energy. The violation of unitarity indicates that the SMEFT is no longer validto describe the phenomenon perturbatively, therefore partial wave unitarity is often used asa criterion to determine whether the SMEFT is valid. In the case of f ¯ f → V V , where f
3s a fermion, ¯ f is an anti-fermion, V , are vector bosons, the amplitude can be expandedas [11] M ( f σ ¯ f σ → V ,λ V ,λ ) = 16 π X J ( J + 12 ) δ σ , − σ e i ( m − m ) φ d Jm ,m ( θ, φ ) T J , (3)where σ , are helicities of the fermion and the anti-fermion, λ , are helicities of vectorbosons, m = σ − σ , m = λ − λ , d Jm ,m are Winger D functions, φ and θ are azimuthand zenith angles of V and T J are coefficients of the partial wave expansion. The partialwave unitarity bound is then | T J | ≤ e + e − → γγ with photons on-shell, we consider only e + e − → Zγ . The Feynman diagrams of e − e + → Zγ induced by nTGCs are shown in Fig. 1.The helical amplitudes can be obtained as M (cid:0) e −↑ e + ↑ → Z γ ± (cid:1) = e e iφ √ sv ( M Z − s ) (cos( θ ) ∓ c ˜ BW )4 √ M Z c W Λ BW , M (cid:0) e −↑ e + ↑ → Z ± γ ± (cid:1) = ± e e iφ v ( M Z − s ) sin( θ )sign ( c ˜ BW )4 c W Λ BW , M (cid:0) e −↓ e + ↓ → Z γ ± (cid:1) = e e − iφ √ s (2 s W − v ( s − M Z ) (cos( θ ) ± c ˜ BW )8 √ M Z s W c W Λ BW , M (cid:0) e −↓ e + ↓ → Z ± γ ± (cid:1) = ± e e − iφ (2 s W − v ( s − M Z ) sin( θ )sign ( c ˜ BW )8 s W c W Λ BW . (4)With Eqs. (3), (4) and | T J | <
1, the unitarity bounds areΛ + − , BW ≥ (cid:18) e √ sv ( s − M Z )48 √ πM Z c W (cid:19) , Λ + − , ++˜ BW ≥ (cid:18) e v ( s − M Z )48 √ πc W (cid:19) , Λ − + , BW ≥ (cid:18) e √ s (1 − s W ) v ( s − M Z )96 √ πM Z s W c W (cid:19) , Λ − + , ++˜ BW ≥ (cid:18) e (2 s W − v ( M Z − s )96 √ πs W c W (cid:19) , (5)and bounds from e − e + − → Z γ − , e − e + − → Z − γ − , e −− e + → Z γ − and e −− e + → Z − γ − aresame as those from e − e + − → Z γ + , e − e + − → Z + γ + , e −− e + → Z γ + and e −− e + → Z + γ + ,respectively. The unitarity bounds are depicted in Fig. 2.Since there is always s > M Z , the strongest bound is Λ − + , BW . We consider √ s =250 , , , √ s = 5000 GeV which has been investigated for nTGCs [7]. The lower bounds of Λ ˜ BW are listed in Table. I 4 FIG. 2. The lower bounds of Λ ˜ BW from different helical amplitudes. √ s
250 GeV 500 GeV 1000 GeV 3000 GeV 5000 GeVΛ ˜ BW > .
41 GeV > .
38 GeV > .
52 GeV > .
04 GeV > .
19 GeVTABLE I. The constraints on Λ ˜ BW from unitarity bounds. IV. NUMERICAL STUDY
The coefficients of nTGCs can be constraint by experiments. By analysing the signalsignificance, one can estimate the minimal luminosities for the experiments to enter theregion of unitarity bounds where the SMEFT is valid and the constraints make sense. Inthis section, the numerical results obtained by MC simulation are presented.The features of the signals and backgrounds are studied by using the
MadGraph5_aMC@NLO toolkit [17]. The fast detector simulation is applied by using
Delphes [18] with the CEPCdetector card. The basic cuts are set as same as the default settings except for ∆ R ℓℓ which isdefined as p ∆ η ℓℓ + ∆ φ ℓℓ where ∆ η ℓℓ and ∆ φ ℓℓ are the differences between pseudo-rapiditiesand azimuth angles of the leptons, respectively. As will be explained later, in the basic cutswe use ∆ R ℓℓ > .
2. In the MC simulation, we use the coefficients in the ranges listed inTable II.We consider the process e + e − → ℓ + ℓ − γ , the dominant signal of the O ˜ BW operator is from5 s = 250 GeV √ s = 500 GeV √ s = 1 TeV √ s = 3 TeV √ s = 5 TeV[ − , − , − ,
30] [ − ,
2] [ − . , . ˜BW ) / Λ BW (TeV − ) used in the MC simulation. the diagrams depicted in Fig. 1 joint with Z → ℓ + ℓ − . The SM backgrounds are depicted inFig. 3. (a). The signal events are generated with the largest coefficients in Table II. . . .. . .γ γ γγγZ, γe − e − e − e − e + e + e + e + ℓ + ℓ + ℓ + ℓ + ℓ − ℓ − ℓ − ℓ − (a)(b) W FIG. 3. Feynman diagrams which contribute to the process e + e − → ℓ + ℓ − γ We require the particle numbers in the final state to be N ℓ + ≥ N ℓ − ≥ N γ ≥ N ℓ,γ cut, in the following, results are presented after N ℓ,γ cut.To remove the backgrounds without a Z resonance, we require the invariant mass of theleptons (denoted as M ℓℓ ) to be close to M Z . The normalized distributions of M ℓℓ are shownin Fig. 4. (a). In the distributions of the signal, the peaks at M Z are much sharper thanthose in the distributions of the SM backgrounds. Defining ∆ M = | M ℓℓ − M Z | , we cut offthe events with ∆ M >
15 GeV.In the signal events, the photons are emitted from the s-channel diagram. Therefore, in6
50 100 15010 -3 -2 -1 (a) -3 -2 -1 (b) -3 -2 -1 (c) -3 -2 -1 (d) FIG. 4. The normalized distributions of M ℓℓ , | cos θ γ | , | cos θ ℓ | and ∆ R ℓℓ . the c.m. frame of e + e − , the zenith angles of the photons (denoted as θ γ ) are typically smallerthan those emitted from a e ± in the initial state. θ γ has been proposed to discriminate thesignal from the backgrounds in the study of nTGCs in the process e + e − → Zγ . [7]. Thenormalized distributions of | cos( θ γ ) | are shown in Fig. 4. (b). We cut off the events with alarge | cos( θ γ ) | .The polarization of e ± beam has been studied in Ref. [7]. Since the longitudinal e ± beam polarization is difficult to realize at the circular colliders, instead we consider thepolarization effect in the final state which has been used to highlight the signals in other7
50 GeV 500 GeV 1 TeV 3 TeV 5 TeVSM nTGC SM nTGC SM nTGC SM nTGC SM nTGC N ℓ,γ cut 5490 . . . . . . . . . . M <
15 GeV 721 . . . . . . .
60 15 . .
41 6 . | cos( θ γ ) | < . . . . . . . | cos( θ γ ) | < .
95 1 .
07 13 . | cos( θ ℓ ) | < . . . . . .
77 90 . | cos( θ ℓ ) | < . .
90 12 . | cos( θ ℓ ) | < .
95 0 .
37 6 . R ℓℓ / ∈ [0 . ,
6] 0 .
87 12 . R ℓℓ / ∈ [0 . ,
6] 0 .
31 6 . previous studies [19, 20]. One can see from Eq. (4) that the Z bosons in the final states aredominantly longitudinal polarized at large √ s . This leads to a unique angular distributionof the leptons in the rest frame (the helical frame) of the Z boson. In the rest frame of ℓ + ℓ − with z -axis pointing to the direction of p ℓ + + p ℓ − , and with the zenith angle of the lepton inthis frame denoted as θ ℓ , the normalized distributions of | cos( θ ℓ ) | are shown in Fig. 4. (c).We cut off the events with a large | cos( θ ℓ ) | .Another important issue is ∆ R ℓℓ . When the Z boson is energetic, the leptons are ap-proximately collinear to each other. As a result, the ∆ R ℓℓ is small for the signal events.However, ∆ R ℓℓ is related to the isolation of leptons, therefore a none zero ∆ R ℓℓ is requiredin the experiments. To keep the signal events, the lower bound of ∆ R ℓℓ should be as smallas possible. Therefore, in the basic cuts we use ∆ R ℓℓ > . √ s = 3 TeV and √ s = 5 TeV, with ∆ R ℓℓ > . N ℓ,γ cut, approximately 72% and 90% signal events are lost, respectively. The normal-ized distributions of ∆ R ℓℓ are shown in Fig. 4. (d). We find that, for the SM backgrounds,typically ∆ R ℓℓ lies in the region 2 < ∆ R ℓℓ <
6. We keep the events with ∆ R ℓℓ outside ofthis region. 8 s = 250 GeV √ s = 500 GeV √ s = 1 TeV √ s = 3 TeV √ s = 5 TeV0 . − .
012 pb − .
047 pb − . − . − TABLE IV. The minimal luminosities required for the constraints to enter the unitarity bounds at S stat = 5. The proposed event selection strategy and the cross-sections after cuts are summarisedin Table III. The SM backgrounds can be effectively reduced by our selection strategy. -1000 -500 0 500 1000200250300350400450500550 (a) √ s = 250 GeV -200 -100 0 100 200050100150200250300 (b) √ s = 500 GeV -30 -20 -10 0 10 20 30020406080100 (c) √ s = 1 TeV -2 -1 0 1 202468101214 (d) √ s = 3 TeV -1 -0.5 0 0.5 10123456 (e) √ s = 5 TeV FIG. 5. Cross-sections as functions of sign( c ˜ BW ) / Λ BW . By scanning the parameter spaces listed in Table II, the cross-sections of the process e + e − → ℓ + ℓ − γ at different coefficients are obtained. To study how the process is af-fected by nTGCs, the diagrams in Fig. 3. (b) and interference terms are included. Asshown in Fig. 5, the cross-sections are approximately bilinear functions of the coefficient.The constraints on the coefficient can be estimated by using signal significance defined as S stat = N nT GC / √ N nT GC + N sm where N nT GC is the number of signal events, and N sm isthe number of events of the SM backgrounds. The constraints corresponding to differentluminosities (denoted as L ) are shown in Fig. 6. As a compare, the unitarity bounds are also9 FIG. 6. The constraints corresponding to different luminosities. The vertical lines are unitaritybounds. S stat √ s = 250 GeV √ s = 500 GeV √ s = 1 TeV √ s = 3 TeV √ s = 5 TeV2 [ − . , .
4] [ − . , .
7] [ − . , .
2] [ − . , .
12] [ − . , . − . , .
6] [ − . , .
2] [ − . , .
5] [ − . , .
15] [ − . , . − . , .
8] [ − . , .
5] [ − . , .
9] [ − . , .
19] [ − . , . ˜BW ) / Λ BW at L = 2 ab − . shown in Fig. 6 as vertical lines. Considering the constraints set by experiments at S stat = 5,the minimal luminosities required for the constraints to enter the unitarity bounds are listedin Table IV. Using L = 2 ab − as a representation, the expected constraints in experimentsare shown in Table V. Compare our result with the studies of other dimension-8 operatorsat the LHC [19, 22], and the study of nTGCs at FCC-hh [6], the process e + e − → ℓ + ℓ − γ atthe e + e − colliders shows competitive sensitivity to dimension-8 operators. V. SUMMARY
The nTGCs can contribute to the process e + e − → ℓ + ℓ − γ via ZV γ vertices. The
ZV γ vertices provide a unique opportunity to study the dimension-8 physics because there is no10
V γ vertices in the SM, and there is no dimension-6 operators contributing to the
ZV γ vertices. The process e + e − → ℓ + ℓ − γ can be studied in the e + e − colliders. In this work, weinvestigate how the process e + e − → ℓ + ℓ − γ is affected by nTGCs.Whether the effective theory is the valid at a large energy scale is an important issue.We study this problem by using partial wave unitarity, the unitarity bounds are obtained.Except for that, we study the kinematic features of the signal and background events byMC simulation, and the event selection strategy for nTGCs is proposed. Then the mini-mal luminosities such that the constraints obtained by experiments can enter the unitaritybounds at different c.m. energies are obtained. The expected constraints at future e + e − colliders are estimated, the process e + e − → ℓ + ℓ − γ at the e + e − colliders shows competitivesensitivity to dimension-8 operators. ACKNOWLEDGMENT
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