An inverse seesaw model with U(1 ) R gauge symmetry
aa r X i v : . [ h e p - ph ] A ug An inverse seesaw model with U (1) R gauge symmetry Takaaki Nomura ∗ School of Physics, KIAS, Seoul 02455, Korea
Hiroshi Okada † Asia Pacific Center for Theoretical Physics, Pohang, Geyoengbuk 790-784, Republic of Korea (Dated: August 21, 2018)We propose a natural realization of inverse seesaw model with right-handed and flavor dependent U (1) gauge symmetries, in which we formulate the neutrino mass matrix to reproduce currentneutrino oscillation data in a general way. Also we study a possibility to provide predictions to theneutrino sector by imposing an additional flavor dependent U (1) L µ − L τ gauge symmetry that alsosatisfies the gauge anomaly cancellation conditions associated with U (1) R . Then we analyze colliderphysics on an extra gauge boson and show a possibility of detection. I. INTRODUCTION U (1) B − L [1] and U (1) R [2–6] symmetries require threefamilies of neutral right-handed (or left-handed) fermionsin order to cancel the gauged anomalies. A feature ofthese two symmetries is very similar each other aboutYukawa sector. In fact, once a lepton Yukawa model isconstructed in a gauge symmetry, the other symmetrycan also reproduce the same one. And these symmetriesare known as a natural extension of the standard model(SM) to realize various seesaw mechanisms such as canon-ical seesaw model [7–10], inverse seesaw model [11, 12],linear seesaw model [12–14], etc.On the other hand, nature of these two gauge sectorsare so different each other, and one might be able to testtheir differences via current or future experiments so asto make use of the polarized electron/positron beam ate.g., ILC [15]. Indeed, U (1) B − L is chirality-universal in akinetic term, while U (1) R has right-handed chirality only.In this sense, it would be worthwhile for us to constructmodels with gauged U (1) B − L and/or U (1) R symmetryas many as possible, so that we can distinguish these twoextra symmetries in a various phenomenological pointsof view.In this paper, we construct an inverse seesaw modelwith U (1) R symmetry, in which we formulate the neu-trino mass matrix to reproduce current neutrino oscil-lation data [16] in a general way. Inverse seesaw re-quires a left-handed neutral fermions S L in addition tothe right-handed ones N R , and provides us more compli-cated neutrino mass matrix. Therefore, each of mass hi-erarchies are softer than the other models such as canon-ical seesaw and it could provide abundant phenomenolo-gies such as unitarity constraints. Note that we expect S L has nonzero U (1) B − L charge, because it is a kind ofpartner of N R . In that case, however, U (1) B − L can notbe gauged since anomaly cancellation condition can not ∗ Electronic address: [email protected] † Electronic address: [email protected] Q aL u aR d aR L aL e aR N aR S aL H ϕ ϕ SU (3) C SU (2) L U (1) Y
16 23 − − − U (1) R − − SU (3) C × SU (2) L × U (1) Y × U (1) R , where the upper index a is the number of family that runs over 1-3. Singlet scalar ϕ is required when we add U (1) L µ − L τ gauge symmetry. be satisfied. Therefore introduction of left-handed singletfermion is more natural in gauged U (1) R symmetry casecompared with gauged U (1) B − L symmetry case since theleft-handed singlet fermion cannot have lepton number inthe latter case. Also we study a possibility to provide pre-dictions to the neutrino sector by imposing an additionalflavor dependent U (1) L µ − L τ gauge symmetry that alsosatisfies the gauge anomaly cancellations among U (1) R .[39] Then we analyze collider physics on an extra gaugeboson and show a possibility of detection.This letter is organized as follows. In Sec. II, we re-view our model and formulate the lepton sector. Thenwe discuss phenomenologies of neutrinos and an extraneutral gauge boson at colliders. Finally we devote thesummary of our results and the conclusion. II. MODEL SETUP AND CONSTRAINTS
In this section we formulate our model. At first, weadd three families of right(left)-handed fermions N R ( S L )with 1(0) charge under the U (1) R gauge symmetry, andan isospin singlet boson ϕ with 1 charge under the samesymmetry. Here we denote each of vacuum expectationvalue to be h H i ≡ v H / √
2, and h ϕ i ≡ v ϕ / √
2. Further-more, the SM Higgs boson H also has 1 charge to inducethe masses of SM fermions from the Yukawa Lagrangianafter the spontaneously symmetry breaking. [40] All thefield contents and their assignments are summarized inTable I. The relevant Yukawa Lagrangian under thesesymmetries is given by [41] − L ℓ = y ℓ aa ¯ L aL He aR + y D ab ¯ L aL ˜ HN bR + y SN aa ¯ S aL N aR ϕ ∗ + µ ab ¯ S aL ( S cL ) b + h . c ., (1)where ˜ H ≡ iσ H , and upper indices ( a, b ) = 1-3 arethe number of families, and y ℓ and y SN can be diag-onal matrix without loss of generality due to the re-definitions of the fermions. Each of the mass matrixis defined by m ℓ = y ℓ v H / √ m D = y D v H / √
2, and M SN = y SN v ϕ / √
2. Notice that S L is singlet un-der all gauge symmetry and it does not interact withany gauge interactions without mixing among neutralfermions. Here we denote S L as left-handed in a senseit dose not have U (1) R charge. As we discuss belowheavy extra neutral fermion mass is approximately givenby M NS which is taken to be TeV scale. In addition,Majorana mass term of S L breaks lepton number as weassign lepton number to S L . A. Neutrino sector without U (1) L µ − L τ After the spontaneously symmetry breaking, neutralfermion mass matrix with 9 × M N = m D m TD M TNS M NS µ . (2)Then the active neutrino mass matrix can approximatelybe found as m ν ≈ m D M − NS µ ( M TNS ) − m TD , (3)where µ << m D . M NS is expected [42]. The neutrinomass matrix is diagonalized by unitary matrix U MNS ; D ν = U TMNS m ν U MNS , where D ν ≡ diag( m , m , m ).One of the elegant ways to reproduce the current neu-trino oscillation data [16] is to apply the Casas-Ibarraparametrization [18] without loss of generality, and findthe following relation m D = U ∗ MNS p D ν O mix p I N ( L TN ) − . (4)Here O mix is an arbitrary 3 by 3 orthogonal matrix withcomplex values, I N is a diagonal matrix, and L N is alower unit triangular [32], which can uniquely be decom-posed to be M − NS µ ( M TNS ) − = L TN I N L N , since it is sym-metric matrix. Note here that all the components of m D should not exceed 246 GeV, once perturbative limit of y D is taken to be 1.Mass scale of heavy neutral fermions is approximatelygiven by M NS which is taken to be O (1) TeV in ourscenario. Then neutrino mass scale is m ν ∼ − (cid:16) m D GeV (cid:17) (cid:18) M NS TeV (cid:19) µ GeV GeV . (5)Thus we can realize neutrino mass scale ∼ . µ ∼ . m D ∼ . v ′ is as large as &
18 TeV. L L , N R , S L , e R , ϕ L L , N R , S L , e R U (1) L µ − L τ − U (1) L µ − L τ , where the other fields do not have L µ − L τ charge. B. Neutrino sector with U (1) L µ − L τ Here we introduce local U (1) L µ − L τ symmetry to re-strict neutrino mass structure in inverse seesaw sce-nario [19, 20] where we add SM singlet scalar ϕ with L µ − L τ charge 1 to break the symmetry spontaneously.Then Yukawa interactions and Majorana masses are con-strained, and we have new Yukawa interactions; − L new = y ij ϕ ¯ S iL ( S cL ) j + h.c., (6)where index i ( j ) is determined to satisfy gauge invari-ance. Thus once we impose U (1) L µ − L τ gauge symmetryas shown in table II [43], the mass matrices m D , M SN , µ are specified to be M SN = m SN m SN
00 0 m SN ,m D = m d m d
00 0 m d , µ = µ µ µ µ µ µ µ , (7)where µ , is induced only after the U (1) L µ − L τ spon-taneously symmetry breaking. Therefore, the neutrinomass matrix directly reflects the form of µ as m ν = µ m d m SN µ m d m d m SN m SN µ m d m d m SN m SN µ m d m d m SN m SN µ m d m d m SN m SN µ m d m d m SN m SN µ m d m d m SN m SN . (8)Thus we can predict inverted neutrino ordering and spe-cific value of Dirac phase by analyzing the two-zero tex-ture [19, 21]. Here the number of parameters in theneutrino mass matrix is nine real parameters (that areequivalent of four complexes and one real). Then onemore phase is there in addition to the Dirac phase andtwo Majorana phases. Note here that this two-zero tex-ture originates from µ in the inverse seesaw model thatcannot be reproduced by a canonical seesaw model. C. Non-unitarity
Here, let us briefly discuss non-unitarity matrix U ′ MNS .This is typically parametrized by the form U ′ MNS ≡ (cid:18) − F F † (cid:19) U MNS , (9)where F ≡ ( m TNS ) − m D is a hermitian matrix, and U ′ MNS represents the deviation from the unitarity. Theglobal constraints are found via several experimental re-sults such as the SM W boson mass M W , the effectiveWeinberg angle θ W , several ratios of Z boson fermionicdecays, invisible decay of Z , electroweak universality,measured Cabbibo-Kobayashi-Maskawa, and lepton fla-vor violations [37]. The result is then given by [38] | F F † | ≤ . × − . × − . × − . × − . × − . × − . × − . × − . × − . (10)Once we conservatively take F ≈ − , we find µ ≈ M NS ∼ y D ∼ − which is slightly larger than caseof Type-I seesaw [23]. D. Collider physics
Here we discuss collider physics of our model mainlyfocusing on Z ′ R boson from U (1) R which obtain its massvia the vacuum expectation value of ϕ . The gauge in-teraction associated with Z ′ R is given by L ⊃ g R (cid:0) ¯ u R γ µ u R − ¯ d R γ µ d R − ¯ ℓ R γ µ ℓ R + ¯ N R γ µ N R (cid:1) Z ′ µR , (11)where g R is gauge coupling constant for U (1) R , and flavorindex is omitted. Z ′ R physics at the LHC : In our model Z ′ R can be pro-duced via q ¯ q → Z ′ R process, and it will decay into SMfermions and N R if kinematically allowed. Then stringentconstraint is given by di-lepton resonance search at theLHC. We estimate the cross section with CalcHEP [24]by use of the CTEQ6 parton distribution functions(PDFs) [25], implementing relevant interactions. In ad-dition, we find branching ratio (BR) for the decay mode Z ′ R → e + e − /µ + µ − is ∼ .
8% for both electron and muonwhen we assume ¯ N R N R mode is not kinematically al-lowed; even if we include ¯ N R N R mode the BR does notchange much as BR ( Z ′ R → e + e − /µ + µ − ) & . σ ( pp → Z ′ R ) BR ( Z ′ R → ℓ + ℓ − ) as a func-tion of m Z ′ R for several values of g R where the BR is sumof electron and muon mode and the red curve indicate theLHC limit obtained from ref. [26]. We find that Z ′ R massshould be heavier than ∼ . g R = 0 . σ ( pp → Z ′ R ) . ν H atthe LHC via Z ′ R boson. If masses of ν H i are sufficientlylighter than m Z ′ R / BR ( Z ′ R → ν H ν H ) is around 4% foreach mass eigenstate. Then ν H decays such that ν H → W ± ℓ ∓ and ν H → Zν L via mixing in neutrino sector.As we discussed above, Z ′ R production cross section isless than ∼ m Z ′ R being several TeV scale, and ν H production cross section will be σ · BR . .
04 fb for g R = LHC limit - - m Z ' R @ TeV D Σ × BR H Z ' R ™ l + l - L @ pb D FIG. 1: The product of Z ′ R production cross section and BR ( Z ′ R → ℓ + ℓ − ) where region above red curve is excludedby the latest data [26]. each mass eigenstate. Thus large integrated luminosity isrequired to obtain sufficient number of events to analyzethe signal. It is also important to confirm the ratio ofthe BR of each decay mode of Z ′ R to distinguish it fromother Z ′ boson like that from U (1) B − L gauge symmetrywhere the approximated BR s are given in Table. III.Here we also comment on Z ′ µ − τ boson from U (1) L µ − L τ gauge symmetry. Gauge interactions among Z ′ µ − τ andfermions are written by L ⊃ g µ − τ (¯ µγ α µ − ¯ τ γ α τ + ¯ ν µ γ α P L ν µ − ¯ ν τ γ α P L ν τ + ¯ N γ α P R N − ¯ N γ α P R N + ¯ S γ α P L S − ¯ S γ α P L S ) Z ′ αµ − τ , (12)where g µ − τ is the gauge coupling constant of U (1) L µ − L τ and fermions are flavor eigenstates. It is difficult to detect Z ′ µ − τ when we consider it to be light as O (10)- O (100)MeV so that muon g − Z ′ µ − τ interaction induces flavor violating decay of heavy neu-trino such as ν H i → Z ′ µ − τ ν H j where m ν Hi > m ν Hj . Thusphenomenology of heavy neutrino at the LHC would beaffected by the gauge boson. However detailed analy-sis is beyond the scope of the paper. We note that the Z ′ µ − τ gauge interaction with charged lepton is flavor di-agonal and do not induce any flavor violations(LFVs)even at loop levels, once we do not seriously take mixingsamong neutral gauge bosons into consideration. Contri-bution via W ± and neutral fermions at one-loop levelalso gives no LFVs, considering negligible mixing amongneutral fermions. Even when we consider the mixingamong gauge bosons or neutral fermions, their mixingsare so small because they are respectively proportionalto ( m Z ′ µ − τ /m Z ( Z ′ R ) ) . − (10 − ) and ( m D /M SN ) . − [28]. Mode ℓ − ℓ + q ¯ q ν H ¯ ν H BR BR s for Z ′ R decay under an approxima-tion assuming m Z ′ R >> m f where m f is mass of final statefermion. Z ′ R physics at lepton collider : Although it would bedifficult to produce Z ′ R directly at lepton colliders wecan explore the effective interaction induced from Z ′ R ex-change; L eff = 11 + δ eℓ g R m Z ′ R (¯ eγ µ P R e )( ¯ f γ µ P R f ) , (13)where f indicates all the fermions in the model, andonly the right-handed chirality appears due to the na-ture of U (1) R symmetry. For example, the analysis ofdata by LEP experiment in ref. [27] provides the con-straint m Z ′ R g R & . e + e − → f ¯ f at the International Linear Collider (ILC) using polarizedinitial state. The partially-polarized differential crosssection can be defined as [29] dσ ( P e − , P e + ) d cos θ = X σ e − ,σ e + = ± σ e − P e − σ e + P e − dσ σ e − σ e + d cos θ , (14)where P e − ( e + ) is the degree of polarization for the elec-tron(positron) beam and σ σ e − σ e + indicates the crosssection when the helicity of initial electron(positron) is σ e − ( e + ) and the helicity of final states is summed up;more detailed form is found in ref [29]. The polarizedcross sections σ L,R is given by following two cases as re-alistic values at the ILC [30]: dσ R d cos θ = dσ (0 . , − . d cos θ , dσ L d cos θ = dσ ( − . , . d cos θ . (15)Then we apply σ R to study the sensitivity to Z ′ R since itis sensitive to right-handed current interactions [5]. Toinvestigate the effect of the new interaction we considerthe measurement of a forward-backward asymmetry atthe ILC which is given by A F B = N F − N B N F + N B ,N F ( B ) = ǫL Z c max (0)0( − c max ) d cos θ dσd cos θ , (16) where a kinematical cut c max = 0 . .
95) is chosen tomaximize the sensitivity for electron(muon) [31], L is anintegrated luminosity and ǫ is an efficiency depending onthe final states which is assumed to be ǫ = 1 for electronand muon final states. The sensitivity to Z ′ R contributionis estimated by∆ A F B = | A SM + Z ′ R F B − A SMF B | , (17)where A SM + Z ′ R F B and A SMF B are forward-backward asym-metry for ”SM + Z ′ R ” and SM cases respectively. Wecompare ∆ A F B with a statistical error of the asymme-try in only SM case δ SMA FB = s − ( A SMF B ) N SMF + N SMB , (18)and we focus on muon final state which is the most sensi-tive one. We find that it is difficult to get ∆ A F B > δ
SMA FB for √ s = 250 or 500 GeV in the region which satisfythe LHC constraint even if the integrated luminosityis O (10) ab − . On the other hand, for √ s = 1 TeV,∆ A F B ∼ δ SMA FB can be obtained with the integrated lu-minosity of 5 ab − with m Z ′ R /g R = 40 TeV. Therefore toinvestigate the chirality structure, we need √ s = 1 TeVwith large integrated luminosity which would be achievedif the ILC is upgraded [15]. III. SUMMARY AND CONCLUSIONS
We have constructed an inverse seesaw model with U (1) R symmetry, in which we have formulated the neu-trino mass matrix to reproduce current neutrino oscilla-tion data in a general way. Also we have found a pre-dictive two-zero neutrino mass matrix, by imposing anadditional flavor dependent U (1) L µ − L τ gauge symmetrythat also satisfies the gauge anomaly cancellations among U (1) R . Then we have analyzed collider physics on an ex-tra gauge boson and show a possibility of detection, Al-though the result of collider physics is almost the sameas the one of our canonical seesaw model [5]. the neu-trino predictions originate from the inverse seesaw modelthat could be difficult to reproduce any canonical seesawmodels. Acknowledgments
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