Final taus and initial state polarization signatures from effective interactions of Majorana neutrinos at future e + e − colliders
FFinal taus and initial state polarization signatures from effectiveinteractions of Majorana neutrinos at future e + e − colliders. Lucía Duarte ∗ Instituto de Física, Facultad de Ingeniería, Universidad de la RepúblicaJulio Herrera y Reissig 565,(11300) Montevideo, Uruguay.
Gabriel Zapata and Oscar A. Sampayo † Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)CONICET, UNMDPDepartamento de Física,Universidad Nacional de Mar del PlataFunes 3350, (7600) Mar del Plata, Argentina.
Abstract
We study the possibility of future e + e − colliders to disentangle different new physics contributionsto the production of heavy sterile Majorana neutrinos in the lepton number violating channel e + e − → l + l + +4 jets , with l = e, µ, τ . This is done investigating the final anti-tau polarization trailsand initial beam polarization dependence of the signal on effective operators with distinct Dirac-Lorentz structure contributing to the Majorana neutrino production and decay, which parameterizenew physics from a higher energy scale. We find both analyses could well disentangle possiblevectorial and scalar operators contributions. ∗ Electronic address: lduarte@fing.edu.uy † Electronic address: [email protected] a r X i v : . [ h e p - ph ] M a r . INTRODUCTION Individual lepton flavors and total lepton number are strictly conserved quantities in thestandard model (SM). However, neutrino oscillations evidence lepton flavor violation in theneutral lepton sector, suggesting the need to consider SM extensions capable of accountingfor massive light neutrinos and lepton mixing. The incorporation of light neutrino masses isstill the most compelling experimental evidence of the need to enlarge the SM electroweaksector. The extensions considering sterile right-handed neutrinos with Majorana mass terms,lead to Majorana massive states which predict the occurrence of total lepton number viola-tion (LNV). In turn, the observation of LNV would be a clear signal of new physics, and ofthe existence of Majorana fermions.The seesaw mechanism for neutrino mass generation [1–5], introducing right handedsterile neutrinos N i which can have a Majorana mass term leading to Majorana massiveneutrino states, could account for the observation of lepton number violating processes.However, in the simplest Type I seesaw implementations, for Yukawa couplings of order Y ∼ , a Majorana mass scale of order M N ∼ GeV is needed to account for a lightneutrino mass compatible with the current neutrino data ( m ν ∼ . eV )[6]. On the otherhand, for smaller Yukawa couplings, of order Y ∼ − − − , sterile neutrinos with massesaround M N ∼ (1 − GeV could exist, but this leads to negligible neutrino mixing values U lN ∼ m ν /M N ∼ − − − [7, 8]. Thus, both alternatives lead to the decoupling of theMajorana neutrinos [9].Recent approaches consider a toy-like model in which the SM is extended by incorporatinga massive Majorana sterile fermion, assumed to have non-negligible mixings with the activestates, without making any hypothesis on the neutrino mass generation mechanism [10, 11].Such a minimal SM extension leads to contributions to LNV observables which are alreadyclose, or even in conflict, with current data from meson and tau decays, for Majorana masses M N below GeV (see [10, 12] and references therein). So, also from the experimental pointof view, the simple SM extensions which attribute LNV only to the mixing between heavyMajorana states and the active neutrinos are facing increasingly stringent constraints.In this scenario, the observation of lepton number violating (LNV) processes allowed bythe existence of a Majorana neutrino mass term would be a sign of physics beyond theminimal seesaw mechanism [13] and beyond the mere existence of sterile-active neutrino2ixings.From the theoretical point of view, one can think of an alternative approach, and considerthe Majorana neutrino interactions as originating in new physics from a higher energy scale,parameterized by a model independent effective Lagrangian [13]. In this approach, weconsider that the sterile N interacts with the SM particles by higher dimension effectiveoperators, and take these interactions to be dominant in comparison with the mixing withlight neutrinos through the Yukawa couplings, which we neglect. In this sense we departfrom the usual viewpoint in which the sterile neutrinos mixing with the standard neutrinosis assumed to govern the N production and decay mechanisms [8, 14].The effective interactions we consider here for the heavy Majorana neutrinos were earlystudied in [13], where the possible phenomenology of dimension 6 effective operators wasintroduced. The dimension 5 operators extending the low-scale Type-I seesaw were inves-tigated in [15], and their phenomenology was addressed recently in [16, 17]. Dimension 7effective N operators are studied in [18, 19]. The collider phenomenology of the dimension 6effective Lagrangian used in this paper has been studied by our group and others in [13, 20–27]. Recently, the predictions of the effective interactions in leptonic decays of pseudoscalarmesons have been investigated in [28].The different operators in the effective Lagrangian, with distinct Dirac-Lorentz structure,parameterize a wide variety of UV-complete new physics models, like extended scalar andgauge sectors as the left-right symmetric model, vector and scalar leptoquarks, etc. Thus,discerning the possible contributions given by them to specific processes gives us a hinton what kind of new physics at a higher energy regime is responsible for the observedinteractions.In [27] we studied the potential of final lepton angular asymmetries and initial electronpolarization observables to disentangle the possible contributions of effective operators withdifferent Dirac-Lorentz structure to the LNV e − p → l + + 3 jets process. Now we aim to takeadvantage of the clean environment in electron-positron colliders and exploit initial statepolarization observables to distinguish the contributions from scalar and vectorial effectiveinteractions. Also, a same-sign final anti-taus state in the e + e − → l + i l + j + 4 jets channelallows to measure the final tau polarization and build observables to this end.Lepton number violating processes have been studied thoroughly in the context of seesawmodels in colliders (for comprehensive reviews on the topic see [7, 29] and references therein).3epton colliders are very well suited for the study of Majorana neutrino interactions, as theyprovide clean signals, without QCD jet backgrounds. The literature using lepton colliders-in past, existing and proposed experiments like the linear ILC [30] or circular colliders likethe FCC-ee [31] and the CEPC [32]- to study the production of heavy sterile neutrinos isvery extensive: recent studies of the two-unit LNV channel e + e − → l ± i l ± j + 4 jets , with l ± i = e, µ, τ , in electron-positron colliders can be found in e.g. [33–35], and other (notnecessarily LNV) heavy sterile neutrino mediated processes as e + e − → lν + 2 jets [35–41].The initial leptons polarization in linear e + e − colliders has been used recently in [34] toshow that the comparison of polarized and unpolarized cross-sections in the e + e − → N N channel for the left-right symmetric model can reveal the nature of the heavy neutrinointeraction with the SM sector and probe the heavy-light neutrino mixing parameters. Also,the capability to measure final tau leptons polarization has been explored in the contextof neutrino mass physics. It has been widely used to distinguish different heavy scalarmediated neutrino mass generation mechanisms as Type II seesaw and the Zee-Babu model,in which the doubly charged Higgs can couple to either left-handed or right-handed leptons(see [42, 43] and references therein).The paper is organized as follows. In Sect.II we introduce the effective Lagrangian formal-ism, present the analytical calculation of the cross section for the e + e − → l + i l + j + 4j channeland review the existing constraints on the effective couplings. In Sect.III we calculate thevectorial and scalar operators contribution to the signal cross section for different Majorananeutrino masses m N in the range m W (cid:46) m N , implementing basic trigger cuts for a bench-mark ILC operating scenario with √ s = 500 GeV , and comment on possible backgrounds.The initial beam polarization dependence of the signal is studied in Sect.IV, while the finalanti-tau polarization signatures are discussed in Sect.V. we present our final comments andconclusions in Sect.VI.
II. MAJORANA NEUTRINO INTERACTION MODELA. Effective operators and Lagrangian
The effects of the new physics involving one heavy sterile neutrino N and the SM fieldsare parameterized by a set of effective operators O J satisfying the SU (2) L ⊗ U (1) Y gauge4ymmetry [44]. The contribution of these operators to observable quantities is suppressedby inverse powers of the new physics scale Λ . The total Lagrangian is organized as follows: L = L SM + ∞ (cid:88) n =5 n − (cid:88) J α J O ( n ) J (1)where n is the mass dimension of the operator O ( n ) J .Note that we do not include the Type-I seesaw Lagrangian -the Majorana and Yukawaterms- giving rise to the mixing between the sterile and the standard left-handed neutrinos,which we are neglecting. In this work it is considered that the dominating new physicseffects leading to the lepton number violation come from the lower dimension operators thatcan be generated at tree level in the unknown underlying renormalizable theory.The dimension 5 operators in (1) were studied in detail in [15]. These include the wellknown Weinberg operator O W ∼ ( ¯ L ˜ φ )( φ † L c ) [45] contributing to the light neutrino masses,and operators: O Nφ ∼ ( ¯ N N c )( φ † φ ) contributing to the N Majorana masses and givingcouplings of the heavy neutrinos to the Higgs (its phenomenology for the LHC has beenstudied very recently in [17]), and an operator O (5) NB ∼ ( ¯ N σ µν N c ) B µν inducing magneticmoments for the heavy neutrinos, which is identically zero if we include just one sterileneutrino N in the theory. In the following, as the dimension 5 operators do not contributeto the studied processes -discarding the heavy-light neutrino mixings- we will only considerthe contributions of the dimension 6 operators, following the treatment presented in [13].We organize the effective operators in different subsets. The first one includes operatorswith scalar and vector bosons (SVB), O ( i ) LNφ = ( φ † φ )( ¯ L i N ˜ φ ) , O NNφ = i ( φ † D µ φ )( ¯ N γ µ N ) , O ( i ) Neφ = i ( φ T (cid:15)D µ φ )( ¯ N γ µ e i ) (2)and a second subset includes the baryon-number conserving 4-fermion ( − f ) contact terms: O ( i,j ) duNe = ( ¯ d i γ µ u i )( ¯ N γ µ e j ) , O ( i,j ) LNLe = ( ¯ L i N ) (cid:15) ( ¯ L j e j ) , O ( i,j ) LNQd = ( ¯ L i N ) (cid:15) ( ¯ Q j d j ) , O ( i,j ) QuNL = ( ¯ Q i u i )( ¯ N L j ) , O ( i,j ) QNLd = ( ¯ Q i N ) (cid:15) ( ¯ L j d j ) , O ( i ) fNN = ( ¯ f i γ µ f i )( ¯ N γ µ N ) , O ( i ) LN = | ¯ N L i | (3)where e i , u i , d i and L i , Q i denote, for the family labeled i (or j ), the right handed SU (2) singlets and the left-handed SU (2) doublets, respectively. The symbol f in the O ( i ) fNN γ µ are the Dirac matrices, and (cid:15) = iσ is theantisymmetric symbol. In this work we allow for family mixing, letting the family indicesto be different in the operators that can involve more than one SM fermion family.We also consider the one-loop ( − loop ) generated operators, which are naturally sup-pressed by a factor / π [13, 46]. These operators give interaction terms that are involvedin the full calculation of the Majorana neutrino total width Γ N , and the branching ratios ofits different decay channels. Their expressions can be found in [24].In order to obtain the interactions in the process e + e − → l + i l + j + 4j , we consider theeffective Lagrangian terms involved in the calculations, taking the scalar doublet after spon-taneous symmetry breaking as φ = (cid:16) v + h √ (cid:17) , with h being the Higgs field and v its v.e.v.We only write here the Lagrangian terms involved in the production and decay processesconsidered in the current calculation. For the complete dimension 6 Lagrangian, we referthe reader to Appendix A in [24].The operators in (2) contribute to a first Lagrangian piece L treeSV B = 1Λ (cid:110) α Z ( ¯ N R γ µ N R ) (cid:16) m Z v Z µ − v P ( h ) µ h + ... (cid:17) − α ( i ) W ( ¯ N R γ µ e R,i ) (cid:18) m W v √ W + µ + ... (cid:19) + h.c. (cid:27) . (4)and the 4-fermion interactions involving quarks and leptons from (3) give L tree − f = 1Λ (cid:110) α ( i,j ) V ¯ d R,j γ µ u R,j ¯ N R γ µ e R,i + α ( i ) V ¯ e R,i γ µ e R,i ¯ N R γ µ N R + α ( i ) V ¯ L i γ µ L i ¯ N R γ µ N R + α ( i,j ) S (¯ ν L,j N R ¯ e L,i e R,i − ¯ e L,j N R ¯ ν L,i e R,i ) + α ( i,j ) S (¯ u L,j u R,j ¯ N R ν L,i + ¯ d L,j u R,j ¯ N R e L,i )+ α ( i,j ) S (¯ ν L,i N R ¯ d L,j d R,j − ¯ e L,i N R ¯ u L,j d R,j ) + α ( i,j ) S (¯ u L,j N R ¯ e L,i d R,i − ¯ d L,j N R ¯ ν L,i d R,i )+ α ( i ) S ( ¯ N R ν L,i ¯ ν L,i N R + ¯ N R e L,i ¯ e L,i N R ) + · · · + h.c. (cid:111) . (5)In Eqs. (4) and (5) a sum over the family index i, j = 1 , , is understood, and the couplings α ( i,j ) O are associated to specific operators: α Z = α NNφ , α ( i ) W = α ( i ) Neφ , α ( i,j ) V = α ( i,j ) duNe , α ( i ) V = α ( i ) eNN , α ( i ) V = α ( i ) LNN α ( i,j ) S = α ( i,j ) LNLe , α ( i,j ) S = α ( i,j ) QuNL , α ( i,j ) S = α ( i,j ) LNQd , α ( i,j ) S = α ( i,j ) QNLd , α ( i ) S = α ( i ) LN . (6)The effective operators above can be classified by their Dirac-Lorentz structure into scalar , vectorial and tensorial . The scalar and vectorial operators contributing to the studied pro-cesses are those appearing in (4) and (5) with couplings named α S and α W, Z, V , respectively.6or the Majorana neutrinos production vertices, depicted in Figs.1 and 3, and the decayprocess N → l + jj in Fig.4, we have scalar and vectorial contributions from the effectiveLagrangian related to the spontaneous symmetry breaking process coming from (2) and the4-fermion interactions involving quarks and leptons from (3). The dimension 6 tensorialoperators are generated at one-loop level, and they are suppressed by the loop factor / π with respect to the considered operators. They do take part in the calculation of the to-tal width Γ N . The relative sizes between the different effective couplings are given by thecontribution of the corresponding operators to the experimental observables. B. Signal
In this work we study the possibility for future e + e − colliders to produce clear signaturesof Majorana neutrinos in the context of interactions coming from an effective Lagrangianapproach in the e + e − → l + i l + j + 4j process.In particular, here we show the calculation for the reaction with final anti-taus e + e − → τ +1 τ +2 + 4j , which is divided into two subprocesses depicted in Figs. 1 and 3. In the first casewe consider the production of two Majorana neutrinos N which will decay into one anti-tauand jets N → τ + jj as in Fig.4. In the second case, we consider the production of a singleMajorana neutrino, with the same decay as before, and a W decaying into two jets W → jj .The differential cross section for the process in Fig. 1 can be decomposed as a product: dσ NN = 18 s m N Γ N | M I | (cid:20) (2 π ) δ ( p + p − k − k ) δ ( k − m N ) δ ( k − m N ) d k (2 π ) d k (2 π ) (cid:21) | M II | (cid:34) (2 π ) δ (cid:32) k − (cid:88) i = l ,d ,u (cid:96) i (cid:33) (cid:89) i = l ,d ,u δ ( (cid:96) i − m i ) d (cid:96) i (2 π ) (cid:35) | M III | (cid:34) (2 π ) δ (cid:32) k − (cid:88) j = l ,d ,u (cid:96) j (cid:33) (cid:89) j = l ,d ,u δ ( (cid:96) j − m j ) d (cid:96) j (2 π ) (cid:35) . (7)The N N production squared amplitude | M I | involves the effective and standard Z in-teractions in Fig. 2. It can be written as | M I | = 14 1Λ (cid:104) α (1) S + 2 α ) ( p .k )( p .k ) + 16 α ( p .k )( p .k ) (cid:105) (8)7 ℓ l l ℓ d ℓ u I I I ℓ l l ℓ d ℓ u I Ie − p e + p NN k I FIG. 1: Diagrams contributing to double N production. e − e − e + e + NN NNZ ( a ) ( b ) FIG. 2: Diagrams contributing to double N production.with α (1) S the 4-fermion LN scalar coupling in (5) and the vector combinations α = α Z Π Z g R + α (1) V α = α Z Π Z g L + α (1) V . (9)Here the Z boson propagator is Π Z = (cid:16) m Z (( p + p ) − m Z ) + m Z Γ Z (cid:17) , g R = sin ( θ W ) and g L = − / ( θ W ) are the SM couplings of the Z boson in the initial vertex in Fig.2 (b). Weneglect the contribution of a Higgs mediated diagram similar to Fig.2 (b), as it scales like ( m e v ) .The differential cross section for the single N process in Fig. 3 can be decomposed as aproduct: 8 ℓ l ℓ d ℓ u W k e − p e + p ν q N k I V ℓ l l ℓ d ℓ u V FIG. 3: Diagrams contributing to single N production. dσ NW = 18 s m N m W Γ N Γ W | M IV | (cid:20) (2 π ) δ ( p + p − k − k − (cid:96) l ) δ ( k − m N ) δ ( k − m W ) δ ( (cid:96) l − m l ) d k (2 π ) d k (2 π ) d (cid:96) l (2 π ) (cid:21) | M V | (cid:34) (2 π ) δ (cid:32) k − (cid:88) j = l ,d ,u (cid:96) j (cid:33) (cid:89) j = l ,d ,u δ ( (cid:96) j − m j ) d (cid:96) j (2 π ) (cid:35) | M V I | (cid:34) (2 π ) δ (cid:32) k − (cid:88) i = d ,u (cid:96) i (cid:33) (cid:89) i = d ,u δ ( (cid:96) i − m i ) d (cid:96) i (2 π ) (cid:35) . (10)The e − e + → N ν production amplitude in Fig. 3 is governed by the scalar 4-fermioninteraction
LN Le : | M IV | = g q ) α (1 , S ( k .p ) m W (cid:2) k .(cid:96) l )( k .q )( p .q ) + 2( (cid:96) l .q )( p .q ) m W − k .(cid:96) l )( k .p ) q − m W ( (cid:96) l .p ) q (cid:3) (11)where q = ( k + (cid:96) l ) = m W + 2( k .(cid:96) l ) is the squared momentum of the neutrino, and g is the SM SU (2) L coupling . In the case of final positrons e − e + → e + e + + 4j , there is another diagram with a contribution proportionalto α (1) W to the amplitude M IV in (11). It is included in our numerical calculations. j W jj( a ) ( b ) N ℓ + i N ℓ + i FIG. 4: Diagrams contributing to the decay process N → (cid:96) + i jj .The amplitudes | M x | , with x = II, III, V in (7) and (10) represent the N decay processinto an anti-lepton and jets N → l + jj depicted in Fig.4. They can be written as: | M x | = (cid:0) | Λ Lx | + | Λ Rx | (cid:1) | Λ Lx | = 16Λ (cid:104) Π W α (3) W ( k N .(cid:96) u x )( (cid:96) l x .(cid:96) d x ) + α (3 ,j ) V ( k N .(cid:96) d x )( (cid:96) l x .(cid:96) u x ) (cid:105) | Λ Rx | = 4Λ (cid:104) ( α (3 ,j ) S + α (3 ,j ) S − α (3 ,j ) S α (3 ,j ) S )( (cid:96) d x .(cid:96) u x )( (cid:96) l x .k N )+( α (3 ,j ) S − α (3 ,j ) S α (3 ,j ) S )( (cid:96) l x .(cid:96) d x )( k N .(cid:96) u x ) + α (3 ,j ) S α (3 ,j ) S ( (cid:96) l x .(cid:96) u x )( k N .(cid:96) d x ) (cid:105) . (12)Here k N corresponds in each case to the momentum of the N : k , k , k for x = II, III, V ,as indicated in Figs. 1 and 3, and the index j = 1 , corresponds to the final quarks family.The W boson propagator is Π W = (cid:16) m W (( k N − (cid:96) lx ) − m W ) + m W Γ W (cid:17) .In fact, as we are summing over the light-quarks in the final state ( u, d, c, s ), the contri-butions from these decays can be written as | Λ Lx | = 16Λ (cid:2) Π W C ( k N .(cid:96) u x )( (cid:96) l x .(cid:96) d x ) + C ( k N .(cid:96) d x )( (cid:96) l x .(cid:96) u x ) (cid:3) | Λ Rx | = 4Λ [ C ( (cid:96) d x .(cid:96) u x )( (cid:96) l x .k N ) + C ( (cid:96) l x .(cid:96) d x )( k N .(cid:96) u x ) + C ( (cid:96) l x .(cid:96) u x )( k N .(cid:96) d x )] (13)where C = 2 α (3) W , C = (cid:88) j =1 , α (3 ,j ) V , C = (cid:88) j =1 , ( α (3 ,j ) S + α (3 ,j ) S − α (3 ,j ) S α (3 ,j ) S ) C = (cid:88) j =1 , ( α (3 ,j ) S − α (3 ,j ) S α (3 ,j ) S ) , C = (cid:88) j =1 , ( α (3 ,j ) S α (3 ,j ) S ) . (14)Each term | Λ R,Lx | in (12) gives the contribution of a ± polarized final anti-tau. We canclearly see here that the vectorial operators in C and C will give a contribution to Left-polarized ( − ) final anti-taus, and the scalar operators in C , C and C will contribute toRight-polarized (+) anti-taus. 10he amplitudes in (12) are proportional to the Majorana neutrino mass, which is the onlysource of LNV. This can be seen by taking into account that these are Lorentz invariantexpressions. When one considers the Majorana N in its rest frame, the dot products of k N with the final momenta (cid:96) ( (cid:96) u x , (cid:96) d x , (cid:96) l x ) are proportional to m N .The amplitude | M V I | in (10) represents the standard decay of the W boson into twolight-quark ( u, d, c, s ) jets: | M V I | = 2(2 g ( (cid:96) d .(cid:96) u )) . (15)As we already mentioned, the total decay width of the Majorana neutrino Γ N appearingin the denominators in Eqs. (7) and (10) is calculated considering all the possible decaychannels, as in [24]. C. Bounds on the effective couplings
The dimensionless effective couplings α J associated to the distinct operators in the La-grangian weight the contribution of the interactions parameterized by each operator. Wecan divide them into two groups: those which correspond to operators involving only oneheavy Majorana neutrino N ( α N ≡ α ( i ) W , α ( i,j ) V , α ( i,j ) S , α ( i,j ) S , α ( i,j ) S , α ( i,j ) S ) and those involvingtwo N s ( α NN ≡ α Z , α ( i ) V , α ( i ) V , α ( i ) S ) in (6). The first group of couplings α N , for each leptonfamily i, j = 1 , , appear in the N decays in Fig. 4 and/or in the total decay width Γ N [24], while the second group α NN contribute in the double N production process in Fig. 1.The numerical value of the couplings α N can be constrained exploiting the current ex-perimental bounds on the light-heavy neutrino mixing parameters in seesaw models. In theliterature [29, 40, 47–50] the existing experimental bounds are summarized in general phe-nomenological approaches considering low scale minimal seesaw models, parameterized bya single heavy neutrino mass scale M N and a light-heavy mixing U lN , with l indicating thelepton flavor. In the Majorana neutrino mass region we are considering, the most stringentconstraints are placed on the N − ν e mixing U eN by neutrinoless double beta decay ( νββ )searches. The N − ν µ and N − ν τ mixings U µN and U τN take their most stringent boundsfrom lepton flavor violating radiative decays as µ → eγ and τ → e ( µ ) γ .We interpret the current bounds on the U lN seesaw mixings comparing the effectivecouplings α N with the general structure usually taken for the interaction between the heavy11ajorana neutrinos and the W : L W = − g √ lγ µ U lN P L N W µ + h.c. (16)The term with coupling α ( i ) W in (4) can be compared to the weak charged current in (16),giving us a relation between α ( i ) W and U l i N for each fermion family i = 1 , , : U l i N (cid:39) α ( i ) W v [13]. In order to put reliable bounds on the effective couplings α N but keeping the analysisas simple as possible, we consider the bounds on the seesaw mixings to constrain all theeffective couplings α ( i ) N for each family i . In previous work [23, 24] we have presented ourprocedure, and refer the reader to those papers for a detailed discussion.For the couplings involving the first fermion family -taking indices i = 1 and j = 1 in theLagrangian terms in (4) and (5)- the most stringent are the νββ -decay bounds obtained bythe KamLAND-Zen collaboration [51]. Following the treatment made in [24, 50, 52], theygive us an upper limit α b ( e )0 νββ ≤ . × − (cid:0) m N GeV (cid:1) / , where the new physics scale is takento be Λ = 1
T eV (here and in the following) . For the second and third fermion families-taking indices i = 2 , or j = 2 , in (4) and (5)- and sterile neutrino masses in the range m W (cid:46) m N the upper limits come from radiative lepton flavor violating (LFV) decays as µ → eγ and τ → e ( µ ) γ . For the second family the constraint Br ( µ → eγ ) < . × − translates into a bound α b ( µ ) LF V ≤ . and for the third, the bound Br ( τ → µγ ) < . × − gives us α b ( τ ) LF V ≤ . [40, 47, 50].The effective couplings of the operators in the second group, involving two heavy Majo-rana neutrinos α NN can be bounded exploiting the LEP results on single Z → νN and pair Z → N N sterile neutrino production searches [53]. However, for the m N range studied inthis work ( m W (cid:46) m N ), they do not give us any restriction on the couplings.In the numerical analysis throughout this work we will take a very conservative ap-proach and consider the most possible restricting bounds: the couplings involved in neu-trinoless double beta decay ( α W , α (1 , V , α (1 , S , , ) are taken as equal to the bound α b νββ =3 . × − (cid:0) m N GeV (cid:1) / for Λ = 1
T eV , and all the others (scalar, and vectorial, involvingone or two Majorana neutrinos) will be taken as equal to the LFV bound α b ( µ ) LF V ≤ . .All the couplings of the operators generated at one loop level (which contribute to the The new physics scale
Λ = 1
T eV is taken as an illustration. One can obtain the values at any other scale Λ (cid:48) considering α (cid:48)J = ( Λ (cid:48) Λ ) α J . Γ N ) are fixed as the corresponding tree-level coupling divided by the loop factor: α − loop = α ( i ) N / π . III. NUMERICAL ANALYSIS
In our numerical analysis we aim to study the possibility of distinguishing the contri-butions from vectorial and scalar effective interactions in the process e + e − → l + l + + 4j ,mediated by Majorana N neutrinos. This signal can be studied in future lepton colliderslike the linear ILC [30] or circular colliders like the FCC-ee [31] and the CEPC [32].For concreteness, throughout the paper we will consider an e + e − collider with center ofmass energy √ s = 500 GeV and integrated luminosity L = 500 f b − for estimating thenumbers of events. These values correspond to one of the proposed ILC operation modes[54]. We will also exploit the possibility the ILC (and other) facilities offer to use initiallypolarized beams and measure final state tau polarization.For the effective interaction model, we will consider a new physics energy scale Λ = 1
T eV ,keeping αs < Λ in order to ensure the validity of the effective Lagrangian approach .In order to consider the contributions given by the scalar operators, we set the effectivecouplings corresponding to the vectorial operators α ( i ) W , α Z and α ( i,j ) V α ( i ) V , and the tensorialoperators (involved in the numerical calculation of the decay width Γ N ) equal to zero, andset the value of the scalar couplings α ( i,j ) S , , and α ( i,j ) S in (6) to the maximum allowed valuesin Sect.II C corresponding to each fermion family i, j . In the plots, the curves labeled scalar correspond to the numerical evaluation in which all the vectorial (and tensorial) couplings areset to zero, and all the scalar couplings are set to the value of the bound (at the same time).Conversely, the curves labeled vectorial study the contribution from the vectorial (plus thetensorial) operators, and we set the scalar couplings to zero, taking all the vectorial couplingsequal to the bound in Sect.II C, and the tensorial ones to this value multiplied by the loopfactor / π . The vectorial and tensorial operators are considered together, because theyinvolve the interactions of the Majorana neutrinos with the standard vector bosons ( W ± , Z , photons) and the Higgs. As we already mentioned, the tensorial operators (generated For instance, with the bounds on the effective couplings discussed in Sect.II C, the EFT expansion pa-rameter (for the second and third fermion families) is αs Λ = 0 . for the scalar and vectorial terms, and αs Λ = 0 . for the tensorial terms, with Λ = 1
T eV and √ s = 0 . T eV .
13t one-loop level in a possible UV-complete theory and therefore suppressed by the loopfactor) give their major contribution to the decay N → νγ , which is the dominant channelonly for Majorana masses m N (cid:46) GeV [24], well below the Majorana neutrino mass rangeconsidered here.
A. Acceptance cuts and SM background
For the numerical study, we calculate the cross section for the process e + e − → l + l + + 4j according to the production and decay channels presented in Sect. II B. The phase spaceintegration of the squared amplitudes is made generating the final momenta with the MonteCarlo routine RAMBO [55].In Fig.5 we show the results for the signal cross section, as a function of the Majorananeutrino mass m N , considering all same-sign anti-lepton final states with l = e, µ, τ . Wehave implemented basic trigger cuts following the generic ILC detector design [30], taking p lT > GeV and | η l | < . for the final leptons, p j T > GeV and | η j | < for the jets, anda separation ∆ R j j , ∆ R l j > . between the final leptons and jets. σ [ pb ] −7 −6 −5 −4 −3 −2 m N [GeV]
50 100 150 200 250 300
Scalar Vectorial Total
FIG. 5: Cross section for the process e + e − → l + l + + 4j It can be appreciated that as m N approaches the c.m. energy limit, the cross sectiondrops sharply. The vectorial operators give a greater contribution to the unpolarized cross14ection by nearly one order of magnitude. This behavior was previously found for othereffective N interaction signals studied in the past [25, 27].The studied signal, being a LNV process, is strictly forbidden in the SM, and it is aclean signal with practically no SM background, which appears to be mainly due to chargemisidentification of one of the final leptons. In the case of final electrons, the signal e + e + + 4j can be faked by genuine opposite sign electron SM events. This charge-flip events are final e + e − + 4j events in which an e − undergoes bremsstrahlung in the tracker volume and theassociated photon decays into an e + e − pair, and this e + is mistaken for the primary e − ifit carries a large fraction of the original energy. This effect is negligible for final muons andtaus. When considering electrons in the final state ( e + e − + 4j ) and applying the same cutsas above, the authors in [34] find a value of σ = 2 . × − pb . When multiplied by a factor expected for electron charge misidentification at the ILC, they find this backgroundis negligible.Other possible backgrounds are SM events resulting in two genuine same-sign leptons,which could fake the same-sign dilepton signal, as backgrounds coming from the productionof four on-shell W bosons, with two like-sign ones decaying leptonically (with final neutrinosescaping undetected) and the other two decaying hadronically. For √ s = 500 GeV , thisbackground can be estimated to be of order − pb adding the three possible final leptonflavors [20]. However, as these channels involve missing energy from the final neutrinos, theycan be effectively suppressed by imposing appropriate cuts on the missing energy for thefinal states with muons and electrons ( l = e, µ )[33]. As an advantage over hadron colliders,the c.m. energy in lepton colliders is precisely measurable, and this helps the reconstructionof missing energy from the total energy-momentum unbalance in each event.The Majorana neutrino mass m N could be obtained in a reconstruction of the invariantmass of its decay products M ( l jj) , if the two final leptons (and the accompanying jets) canbe isolated. This kind of reconstruction involves finding a resonant behavior of the invariantmass for these reconstructed objects [35]. The information on m N , together with possiblemeasurements of final state tau polarization can be used to give a hint on the kind of effectiveinteractions taking part in the N production and decay, as will be discussed in Sect.V.15 V. INITIAL STATE POLARIZATION
The initial electron and positron polarizations can be used to distinguish the vectorialand scalar operators contribution to the studied process. The ILC is expected to operate indifferent polarization modes depending on the physics goals for each center of mass energyvalues. In particular, for √ s = 500 GeV a running mode with opposite initial beam polar-izations (H mode in [54], table 1.1) is planned for increasing the luminosity in annihilationprocesses. In this section we consider three distinct initial polarization benchmark modesand test the ability to disentangle the vectorial and scalar operators contributions to thedominant double N production process in Fig.1.Under these conditions the relevant amplitude | M I | in (8) can be written in terms of theinitial electron ( P e − ) and positron ( P e + ) polarizations as | M e − e + P e − P e + | = 14 (1 − P e − )(1 + P e + ) | M e − e + LR | + 14 (1 + P e − )(1 − P e + ) | M e − e + RL | (17)where the LR and RL amplitudes (left-polarized electron and right-polarized positron, andvice-versa) are | M e − e + LR | = 2Λ ( α (1) S + 2 α ) ( p .k )( p .k ) | M e − e + RL | = 8Λ α ( p .k )( p .k ) (18)and | M e − e + LL | = | M e − e + RR | = 0 .We find that while the amplitude for left-polarized electrons and right-polarized positronsgets contributions from both scalar and vectorial operators, the amplitude with right-polarized electrons and left-polarized positrons only receives vectorial contributions.In Fig. 6 we show the contributions to the signal cross section for the process e + e − → N N → l + l + + 4j , with l = e, µ, τ for Majorana neutrinos with mass m N = 150 GeV given byvectorial and scalar operators, depending on the initial electron polarization P e − , for threedifferent benchmark scenarios. In Fig. 6a the initial positron is taken to be unpolarized, inFig. 6b we take both initial polarizations to be the equal, and in Fig. 6c we take them tobe opposite, as in the mentioned ILC H operation mode.We find that for the unpolarized positron option (Fig. 6a) the vectorial operators give across section value in the f b − range, mostly independent of the initial electron polarizationvalue, while the scalars contribution decreases with positive P e − . For the equal polarization16 σ [ pb ] −5 −4 −3 −2 P e- −1 −0,5 0 0,5 1 Vectorial Scalar m N =150 GeV P e+ =0 (a) σ [ pb ] −6 −4 −2 P e- −1 −0,5 0 0,5 1 Vectorial Scalar m N =150 GeV P e+ =P e- (b) σ [ pb ] −5 −4 −3 −2 P e- −1 −0,5 0 0,5 1 Vectorial Scalar m N =150 GeV P e+ =-P e- (c) FIG. 6: Signal cross section as a function of the initial electron polarization.mode (Fig. 6b) the two contributions have the same qualitative behavior, despite the dif-ference in magnitude. The opposite polarization mode (Fig. 6c) is the most promising todistinguish the kind of new physics contribution, as in this case the vectorial operators showa minimum contribution to the cross section when the initial beams are unpolarized, and17he scalar operators contribution still decreases with positive P e − . Thus we find that com-paring the cross section for different beam polarization configurations can help distinguishthe possible vectorial or scalar effective interaction contributions.In the three plots we find that for P e − = P e + = 0 (unpolarized beams), consideringan integrated luminosity L = 500 f b − it could be possible to separate the scalar andvectorial contributions up to a value of near standard deviations. In Fig. 6c we find thatfor P e − = − P e + = 0 . (opposite polarization beams) this number grows to sigma (seeEq.(19)). S p o l m N [GeV]
50 100 150 200 250 P e- = 0.8, P e+ = -0.3P e- = -0.8, P e+ = 0.3 FIG. 7: The polarization asymmetry S pol ( m N ) for different initial beam polarizations.The cross section dependence on P e ± in Eq. (17) also allows us to compare the numberof signal events produced by the vectorial and scalar effective interactions ( N vec and N sca respectively) for different values of P e − and P e + when considering the production of Majorananeutrinos with different mass m N . In order to explore the possibility of using polarizedinitial leptons to disentangle the contributions of the scalar and vectorial operators to theproduction cross section, we define the function S pol as the number of standard deviationsbetween the numbers of events produced by the vectorial and scalar operators contributions[27]: S pol = N vec − N sca √ N vec + √ N sca (19)In Fig.7 we plot the values of the initial polarization asymmetry S pol ( m N ) for two possiblefixed initial polarization settings ( P e − = 0 . , P e + = − . ) and ( P e − = − . , P e + = 0 . ) [34].We find that the two contributions could be very well separated in both beam operationmodes, with the major difference arising in the right-polarized electron beam case. As an18xample, we find that for m N = 150 GeV , taking a positive P e − (solid line) the contributionsfrom scalar and vectorial operators could be distinguished with a statistical significance ofalmost sigma, while for negative P e − this value drops to sigma. V. TAU POLARIZATION SIGNATURES
Measurements of final state leptonic polarization have been crucial for the tests of theSM electroweak sector in lepton colliders. In particular, final tau and anti-tau polarizationmeasurements at LEP and SLD experiments [56, 57] have provided a direct measurementof the chiral asymmetries of the SM neutral current. Final taus are the only fermionswhose polarization is accessible by means of the energy and angular distribution of its decayproducts. These measurements rely on the dependence of kinematic distributions of theobserved tau decay products on the helicity of the parent tau lepton. Recent studies at theLHC claim to have a statistical uncertainty comparable to similar measurements performedat LEP [58], and we expect improvements for the sensitivity in future detectors like the ILDat the ILC [59, 60].The polarization of the final anti-taus can be used to distinguish the vectorial and scalaroperators contributions. We define the leptonic final state polarization as P τ = N ++ + N + − − N − + − N −− N ++ + N + − + N − + + N −− (20)where the subscripts + and − in the number of events correspond respectively to Right andLeft polarization states of the each final anti-tau l and l in Figs. 1 and 3. Since the finalanti-taus are identical particles, and the production and decay processes considered in thesignal are the same for each of them, it is not possible to distinguish the final + − and − + polarization cases. So we expect that both numbers of events are equal: N + − = N − + , suchthat the polarization P τ in (20) is finally the ratio between the difference of the number ofevents for which both anti-taus are right-handed and left-handed, and the total number ofevents.In order to estimate the error in the final state polarization P τ , we propagate it consideringeach number of events as Poisson distributed. Under these conditions we have ∆ P τ = (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) i,j =+ , − (cid:18) ∂P τ ∂N ij (cid:19) ( δN ij ) (21)19here δN ++ = (cid:112) N ++ , δN + − = (cid:112) N + − , δN − + = (cid:112) N − + , δN −− = (cid:112) N −− . (22)Thus we estimate the final state polarization error as ∆ P τ = 2 ( N − + + N −− ) ( N ++ + N + − ) ( N ++ + N + − − N − + − N −− ) . (23)To appreciate the ability of the final anti-taus polarization to determine the kind ofeffective operators involved in the studied interaction, we define a parameter λ ∈ [0 , tomeasure the proportion of vectorial and scalar operators contributing to the process. Thuswe multiply the vector operators by λ and the scalars by (1 − λ ) , and study the dependenceof the final polarization P τ on this parameter for different Majorana neutrino masses m N .As we found in the calculation of the Majorana neutrino decay N → τ + jj in Eqs. (12) and(14), the vectorial operators contribute to states with final Left anti-taus, and we expectto find a negative final polarization P τ = − for a pure vectorial contribution ( λ = 1 ).Conversely, we expect a final P τ = 1 for a pure scalar contribution ( λ = 0 ).In Fig. 8 we plot the final state anti-tau polarization as a function of the variable λ (Fig. 8a) and m N (Fig. 8b), respectively. In both figures we include the polarization errors,calculated as in Eq.(23), in order to appreciate the possibility of disentangling the kind ofoperators involved.In the case the studied LNV signal is detected and the Majorana neutrino mass m N is reconstructed, as we discussed in Sect. III, a measurement of the final state leptonicpolarization P τ could be able to determine the value of the parameter λ . For instance, byinspection of Fig. 8 one can see a positive final polarization P τ (cid:38) for m N ≈ GeV would indicate the effective interaction to be mostly mediated by scalar operators.
VI. SUMMARY AND CONCLUSIONS
While models like the minimal seesaw mechanism lead to the decoupling of the heavyMajorana neutrinos, predicting mostly unobservable LNV, the effective Lagrangian frame-work considered in this work could serve as a means to discern between the different possiblekinds of effective interactions contributing to LNV. The heavy neutrino effective field the-ory parameterizes high-scale weakly coupled physics beyond the minimal seesaw mechanismin a model independent framework, allowing for sizable LNV effects in colliders. In this20 P τ −1−0.500.51 λ m N =50 GeVm N =100 GeV (a) P τ −1−0.500.51 m N [GeV]
50 100 150 200 λ=0.1λ=0.3λ=0.5 (b)
FIG. 8: Final anti-tau polarization P τ as defined in Eq.(20) (a) as a function of λ fordifferent m N values and (b) as a function of m N for different λ values.work we investigate the e + e − → l + l + + 4j signal, mediated by Majorana neutrino effectiveinteractions, which could be searched for in future lepton colliders [30–32].We have calculated the vectorial and scalar operators contribution to the signal crosssection for different Majorana neutrino masses m N , implementing basic trigger cuts for abenchmark ILC operating scenario with √ s = 500 GeV . In Sect.IV we calculate thesecontributions to the initially polarized cross section, for three different possible operationmodes. We find that comparing the cross section dependence for different beam polariza-tion configurations can help to the identification of the possible vectorial or scalar effectiveinteractions contributions (Fig. 6). We also define an initial polarization asymmetry S pol ,which gives the number of standard deviations between the number of events produced bythe vectorial-only or scalar-only interactions. Studying the dependence of this observablewith the Majorana mass for two benchmark initial beam polarization configurations, we findthe scalar and vectorial contributions could be well separated in both operation modes, witha greater difference in the case of a right polarized initial electron beam (Fig.7).In Sect. V we exploit the possibility to measure the final anti-taus polarization to studythe chances to distinguish the vectorial and scalar contributions to the e + e − → N N → τ + τ + +4j signal. Weighting the vectorial and scalar operators by a factor λ ∈ [0 , : with λ =1 (purely vectorial) and λ = 0 (purely scalar) contributions (Fig. 8) we find a measurement21f the final polarization P τ might be able to determine the value of the λ parameter, providedthat the mass m N can be reconstructed, possibly with the resonant invariant mass M ( τ + jj) of its decay products.Our findings show that lepton colliders -where the clean environment allows for a detailedstudy of polarization observables- can provide relevant information on the kind of new physicsresponsible for lepton number violation in the e + e − → l + l + + 4j channel, complementingprevious studies of LNV signals mediated by Majorana neutrinos with effective interactionsat the LHC [25] and in electron-proton colliders [27, 61]. The initial beam and final taupolarization measurements could well disentangle possible vectorial and scalar operatorscontributions, which parameterize different high-scale physics beyond the minimal seesawmechanism, giving us a hint on the possible physics contributing to (eventual) LNV, afundamental puzzle in particle physics, as the nature of neutrino interactions. Acknowledgements
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