Presymmetry in the Standard Model with adulterated Dirac neutrinos
aa r X i v : . [ h e p - ph ] O c t Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
Presymmetry in the Standard Model with adulterated Dirac neutrinos
Ernesto A. Matute
Departamento de F´ısica, Universidad de Santiago de Chile,Usach, Casilla 307 – Correo 2, Santiago, [email protected]
Recently we proposed a model for light Dirac neutrinos in which two right-handed (RH)neutrinos per generation are added to the particles of the Standard Model (SM), imple-mented with the symmetry of fermionic contents. The ordinary one is decoupled via thehigh scale type-I seesaw mechanism, while the extra pairs off with its left-handed (LH)partner. The symmetry of lepton and quark contents was merely used as a guideline tothe choice of parameters because it is not a proper symmetry. Here we argue that theunderlying symmetry to take for this correspondence is presymmetry, the hidden elec-troweak symmetry of the SM extended with RH neutrinos defined by transformationswhich exchange lepton and quark bare states with the same electroweak charges and noMajorana mass terms in the underlying Lagrangian. It gives a topological character tofractional charges, relates the number of families to the number of quark colors, and nowguarantees the great disparity between the couplings of the two RH neutrinos. Thus,Dirac neutrinos with extremely small masses appear as natural predictions of presym-metry, satisfying the ’t Hooft’s naturalness conditions in the extended seesaw where theextra RH neutrinos serve to adulterate the mass properties in the low scale effective the-ory, which retains without extensions the gauge and Higgs sectors of the SM. However,the high energy threshold for the seesaw implies new physics to stabilize the quantumcorrections to the Higgs boson mass in agreement with the naturalness requirement.
Keywords : Dirac neutrinos; extra right-handed neutrinos; seesaw mechanism; presym-metry.PACS Nos.: 14.60.St, 14.60.Pq, 11.30.Hv, 11.30.Ly
1. Introduction
The nonzero mass of neutrinos is one of the most compelling evidences for physicsbeyond the Standard Model (SM) based on the gauge groups SU(3) c × SU(2) L × U(1) Y and Higgs fields in a single doublet.1–3 In the SM, neutrinos do not haveDirac mass as in the case of charged leptons and quarks because only left-handed(LH) neutrinos are included. They do not possess Majorana mass either, as B − L isan (accidental) exact global symmetry of the SM, where B and L denote the baryonand lepton numbers.In order to produce generic mass terms, one must then extend the SM by addingthree right-handed (RH) neutrinos to arrange Dirac mass terms and also break the B − L symmetry through RH Majorana mass terms, which are allowed by the E. A. Matute gauge symmetry of the SM. It is a minimal expansion where the gauge and Higgssectors of the SM are maintained. Invoking the naturalness criterion of ’t Hooft,4this breakdown may be small since Majorana mass terms equal to zero recoverthe B − L symmetry and the model becomes more symmetric. Actually, this is thewell-known pseudo-Dirac scenario,5–9 where the dominant contribution to neutrinomasses comes from Dirac mass terms with small corrections from Majorana massterms. It appears as an alternative to the popular seesaw approach,10–13 whereneutrino mass terms are preponderantly of Majorana type, assuming a high scaleof new physics which leads to very different masses for light LH and heavy RHneutrinos. This scenario also makes natural in the sense of ’t Hooft the tiny massof neutrinos since the symmetry of lepton number is restored in the limit of smallmass terms equal to zero.Thus far, the problem on the Dirac or Majorana nature of massive neutrinosremains unresolved. As a matter of fact, no signals in the search for neutrinolessdouble-beta decay of nuclei have been observed,14–16 which is at present the mostfeasible process capable of establishing the Majorana nature of neutrinos. Hence,light neutrinos can be Dirac-like fermions. In this work we restrict ourselves to thisplausible possibility, which is not so much explored in the literature.Within the framework of the SM just extended with three RH neutrinos, thereis no known natural mechanism to accounting for the smallness of Dirac neutrinomasses in comparison with the charged lepton masses. The inclusion of small Majo-rana mass terms, in addition to Dirac mass terms, as in the pseudo-Dirac scenario,does not explain why neutrino masses are so light compared to those of chargedleptons. The inclusion of heavy Majorana mass terms, as done in the type-I seesawmechanism, only leads to light Majorana neutrinos, but not to Dirac neutrinos.Hence, explaining the Dirac nature of light neutrinos requires more physics, beyondadding three RH neutrinos with small or large Majorana mass terms. In other words,in the extension of the SM where a RH neutrino per generation is introduced, thesmall value of the Dirac mass of neutrinos compared to the charged leptons is notnatural, in the sense of the ’t Hooft’s principle of naturalness, so that a fine-tuningis badly needed.On the other hand, the current experimental status magnifies the disturbingpossibility that the successes of the SM may continue at the TeV scale, so that thenew runs of the Large Hadron Collider (LHC) do not produce any significant hint ofthe new physics which introduces extra gauge and Higgs fields with breaking scalesat the TeV range. If this simple view of the experimental situation is adopted, oneis left with the SM and the presumed Dirac neutrinos with extremely small masses,for which there is no explanation. This scenario can be expanded to accommodatedark matter, for instance, by introducing extra sterile neutrinos, although this pointis not addressed here.Even assuming this very conservative scenario, however, we proposed for thefirst time in Ref. 17 a simple approach to understanding the small masses of Diracneutrinos in comparison with charged leptons where masses are adulterated by resymmetry with adulterated Dirac neutrinos adding in each generation of the SM a second, almost inert, RH neutrino with smallMajorana mass. It is a minimal extension in which the gauge and Higgs fields ofthe SM are kept, consistent with the fact that the experimental evidence at theTeV scale may still support the physics of the SM with massive neutrinos of Diractype. It applies the seesaw mechanism to suppress the ordinary RH states withheavy Majorana masses and uses the surviving extra RH neutrinos to generate thetiny Dirac masses, i.e. the actual nature of light neutrinos would be of adulteratedDirac type, where the usual RH neutrino is replaced by the extra one of muchsmaller couplings. The key ingredient of this model to making natural the largedifference between the Majorana masses of the two RH neutrinos in each generationis the correspondence between lepton and quark contents when one of the RHneutrinos is introduced in each generation. But this relation of particle contentsis not a proper symmetry to invoking the ’t Hooft’s principle of naturalness, i.e.there is no symmetry transformation between leptons and quarks that maintainsthe Lagrangian in the model invariant as they have different charges and Majoranamass terms in the quark sector are absent. Therefore such a correspondence wasregarded just as a guideline to the choice of parameters, conceding that the propersymmetry behind it should be founded within the SM with RH neutrinos itself.From another standpoint, the SM extended with three RH Dirac neutrinos hasbeen considered to re-establishing the electroweak lepton–quark symmetry and in-corporate presymmetry,18 ,
19 the symmetry hidden by the nontrivial topology ofthe weak gauge fields which goes with the lepton–quark symmetry from weak toelectromagnetic interactions, where a symmetry of lepton and quark contents is de-manded. More specifically, the symmetric electroweak patterns have been explainedin terms of underlying bare states of leptons and quarks having the same chargesand no Majorana mass terms, but located in a topologically-nontrivial vacuum ofthe weak gauge fields in a manner that the charge shifts are induced, in theory, viavacuum tunneling weak instantons. Consequently, fractional charges get a topolog-ical character and the number of families becomes associated with the number ofquark colors. And presymmetry transformations exchange the bare states of leptonsand quarks keeping the underlying Lagrangian invariant.Our aim in this work is to build up a consistent model for light Dirac-type neutri-nos, establishing presymmetry as the underlying symmetry required to substantiatethe symmetry of lepton and quark contents used in the SM extended with two RHneutrinos per generation, first discussed in Ref. 17. This unifies models of neutri-nos and presymmetry, showing that Dirac-like neutrinos with masses exceptionallysmall compared to charged leptons are natural predictions of presymmetry, in thesense of ’t Hooft. As described above, all of these motivated by the successes of theSM well above the TeV scale and the actual, though not so well explored possibilitythat light neutrinos have a Dirac nature.Specifically, the motivation for presymmetry is the finding of a proper symme-try which distinguishes the two RH neutrinos. Allowing this concrete symmetryentails that the Dirac and Majorana mass terms of the extra RH neutrinos are not
E. A. Matute free parameters, independent of the original RH neutrinos. And they are not madesmall by fine-tuning. Their smallness compared to the original RH neutrinos areguaranteed by the presymmetry imposed on the theory with these ones at the high-energy seesaw scale, much heavier than the electroweak symmetry breaking scale.The ’t Hooft’s argument of naturalness for the small values of the Dirac and Majo-rana masses of the extra RH neutrinos in the Lagrangian relies on this presymmetrywith the usual RH neutrinos; as the couplings of the extra RH neutrinos tend tozero, the underlying theory only involving the original RH neutrinos becomes moresymmetric. This symmetry guarantees the quantum corrections of such parametersto be proportional to the parameters themselves and its interplay with the seesawmechanism leading to the low-energy effective theory with the original RH neutrinosdecoupled only introduces omissible tiny corrections to the mass parameters. In par-ticular, the smallness of the Dirac mass with the extra RH neutrino in comparisonwith the charged leptons appears robust.Yet, there is no natural protection of the Higgs boson mass of the SM againstthe large quantum corrections introduced by the high scale of the seesaw for neu-trino masses. The problem of naturalness arises from the disparity between theenergy scales for the seesaw threshold and its upper value allowed by the naturalcondition.20 In order to maintain the stability of the Higgs mass, the new physicsassociated with the seesaw must then suppress the new contributions. The onlybest known manner of having this cancellation in agreement with the naturalnessrequirement is through supersymmetry. What is more, it can be realized partially,as recently explored in Ref. 21, where the SM is considered non-supersymmetric inthe first stage. The implementation of our extended seesaw with the new physicsable to control the quantum corrections to the Higgs mass, however, is beyond thescope of this work.The paper is organized as follows. In Sec. 2, we recall for completeness and sub-sequent discussion the needed results about neutrino masses in the mixed scenarioof Dirac and Majorana neutrinos where two RH neutrinos per generation are added.In Sec. 3, we look into the SM expanded with the adulterated Dirac neutrinos ofnaturally small masses, where ordinary RH neutrinos are decoupled. In Sec. 4, we goover presymmetry in the extension of the SM with RH neutrinos and its relevancein the model of light Dirac neutrinos. In Sec. 5, we refer to phenomenological impli-cations of the unified model of massive neutrinos and presymmetry. The conclusionsare summarized in Sec. 6.
2. Generic Neutrino Masses with Two Right-Handed Neutrinosper Generation
The addition of RH neutrinos and the violation of lepton number conservation aremodifications of the SM in order to produce neutrino masses in a generic way. Itsexpansion with two RH neutrinos in each generation is to construct light Diracneutrinos with the extra one. In the following we review, for completeness and resymmetry with adulterated Dirac neutrinos subsequent discussion, the basic results for the SM extended with two RH neutrinosper generation, preserving its gauge and Higgs structures.17 The first RH neutrinois the ordinary one, which may carry a B − L charge and form a doublet with itsRH charged lepton partner, as in models of left–right symmetry.22–24 The other isa secondary singlet with generally small couplings and no local charges. The crucialelement of the model is the symmetry of lepton and quark contents when just oneof the RH neutrinos is added. In this manner, invoking the ’t Hooft’s criterion,4the smallness of couplings of the second RH neutrino appears natural since thesymmetry of quark and lepton contents with the first RH neutrino is re-establishedif these couplings are set to be zero.The symmetry of fermionic contents, however, is merely used as a guideline tothe choice of parameters because it is not a proper symmetry in the Lagrangian.This means that one cannot define a symmetry transformation between leptons andquarks to keep the Lagrangian invariant, as these particles have different chargesand Majorana mass terms are not present in the quark sector. In Sec. 4, we discusson the proper symmetry that must be attached to this symmetry of lepton andquark contents when one of the RH neutrinos is introduced for each generation.The mass terms after spontaneous electroweak symmetry breaking are17 − L ν = 12 (cid:0) ¯ ν L ¯ ν cL ¯ ν ′ cL (cid:1) M D M ′ D M TD M R M ′ T M ′ TD M ′ M ′ R ν cR ν R ν ′ R + h.c., (1)where ν R ( ν ′ R ) is a three-component vector denoting the ordinary (extra) RH neu-trinos, and M D , M ′ D are 3 × M R , M ′ R to the RH Majorana mass terms, and M ′ to the mixing terms; the phaseconvention is ν cR = C ¯ ν TL .The values of masses and couplings of RH neutrinos should be understood.Because they are not part of the SM, whose origin in turn is not known yet, theform of their findings may not be well defined. In the SM every LH charged leptonand quark has its RH charged lepton or quark partner, while the RH partner ofthe neutrino is absent. This content of chiral fermions is clearly not symmetric.The simplest manner of having such a symmetry between leptons and quarks isthrough the introduction of ordinary RH neutrinos, ν R .25 ,
26 Within the formalismof Eq. (1), it corresponds to M ′ D = 0, M ′ = 0 and M ′ R = 0, but keeping M D and M R as nontrivial. Here the proposal embraces the rationale of the type-I seesawmechanism based on the assumptions that M D has the same mass scale as chargedleptons, and M R is sufficiently large to suppress M D according to M D M − R M TD .Thus, the lepton–quark symmetry of particle content together with a large M R and M ′ D = M ′ = M ′ R = 0 mimic a high scale type-I seesaw, decoupling the ordinaryRH neutrinos.The lepton–quark correspondence, however, is broken when the extra RH neutri-nos, ν ′ R , are added. This is regarded as a reason for having small couplings M ′ D , M ′ , M ′ R for ν ′ R , as the ’t Hooft’s naturalness criterion applied to this symmetry of lepton E. A. Matute and quark contents in the Lagrangian gives a ready explanation. Indeed these extraRH neutrinos establish an alternative, exceptionally weak, lepton–quark correspon-dence. Therefore the symmetry of fermionic contents distinguishes ν R from ν ′ R byrequiring a large difference between M D , M R and M ′ D , M ′ R , respectively, whichparametrize the two forms of the symmetry. This other lepton–quark symmetry ofparticle content together with M D = M ′ = M R = 0 and small M ′ R mimic a lowscale pseudo-Dirac scenario, but pairing off the LH with the extra RH neutrinos. Asemphasized above, the correspondence of leptons and quarks is only considered as aguideline to coupling selections since it is not a proper symmetry in the electroweakLagrangian.The mass matrix in Eq. (1) is diagonalizable by the unitary transformation U † MU ∗ = D L D R
00 0 D ′ R , (2)where D L , D R , and D ′ R are diagonal, real, and non-negative 3 × U † = V † L V † R
00 0 V ′† R ( √ I + W † LL ) V † RL ( − √ I + W ′† RL ) V † LR I V ′† RL ( √ I + W ′† LR ) V ′† LR ( √ I + W ′† RR ) , (3)where V L , V R , and V ′ R are unitary 3 × M R and M ′ D are non-singular and symmetric matrices, and that M ′ R , M ′ , M ′ D , M D ≪ M R and M ′ R , M D M − R M TD , M ′ M − R M ′ T , M ′ M − R M TD ≪ M ′ D , as arguedabove, and the constraints from unitarity and the matrix MU ∗ are used as in theordinary seesaw mechanism, the following expressions are obtained: W † LL ≃ √ M ′ R M ′− D + √ ( M D − M ′ ) M − R ( M TD + M ′ T ) M ′− D ,W ′† RR ≃ √ M ′ R M ′− D + √ ( M D + M ′ ) M − R ( M TD − M ′ T ) M ′− D ,W ′† RL ≃ W † LL ,W ′† LR ≃ − W ′† RR ,V † RL ≃ − ( √ I + W † LL ) M D M − R + ( √ I − W † LL ) M ′ M − R ,V ′† LR ≃ − ( √ I − W ′† RR ) M D M − R − ( √ I + W ′† RR ) M ′ M − R ,V † LR ≃ M − † R M † D ,V ′† RL ≃ M − † R M ′† . (4) resymmetry with adulterated Dirac neutrinos These lead to D L ≃ V † L [ − M ′ D + 12 M ′ R −
12 ( M D − M ′ ) M − R ( M TD − M ′ T )] V ∗ L ≃ − V † L M ′ D V ∗ L ,D R ≃ V † R M R V ∗ R , (5) D ′ R ≃ V ′† R [ M ′ D + 12 M ′ R −
12 ( M D + M ′ ) M − R ( M TD + M ′ T )] V ′∗ R ≃ V ′† R M ′ D V ′∗ R . In the mixed pseudo-Dirac and seesaw regimes with M ′ R = 0 and M D , M ′ keptdown, there are three light almost degenerate pairs of mass eigenstates with smallmass differences, having almost maximal mixing of LH neutrinos ν L and adulterantRH neutrinos ν ′ R , and three heavy, mostly ordinary RH neutrinos ν R with massmatrix M R . The mass of light neutrinos are of the order of M ′ D instead of M D ,which are down by the seesaw mechanism. The incidence of matrices V LR , V RL , V ′ LR , and V ′ RL are reduced by M R , while W LL , W ′ RR , W ′ LR , and W ′ RL are by M R and/or M ′ D .
3. The Standard Model with Adulterated Dirac Neutrinos
The RH neutrinos with large masses can be integrated out by means of the equationof motion d L ν dν R = 0 , (6)which gives¯ ν cL = − ¯ ν L M D M − R − ¯ ν ′ cL M ′ M − R , ν R = − M − R M TD ν cR − M − R M ′ T ν ′ R . (7)The effective Lagrangian is then written as − L ν = 12 (cid:0) ¯ ν L ¯ ν ′ cL (cid:1) (cid:18) M LL M ′ LR M ′ TLR M ′ RR (cid:19) (cid:18) ν cR ν ′ R (cid:19) + h.c., (8)where M LL ≃ − M D M − R M TD , M ′ LR ≃ M ′ D − M D M − R M ′ T ,M ′ RR ≃ − M ′ M − R M ′ T . (9)The mass matrix in Eq. (8) is diagonalized by the approximately unitary matrix U † ≃ V † L V ′† R ( √ I + W † LL ) ( − √ I + W † LL )( √ I − W ′† RR ) ( √ I + W ′† RR ) , (10)such that U † MU ∗ = (cid:18) D L D ′ R (cid:19) , (11) E. A. Matute where W † LL and W ′† RR , D L and D ′ R have the expressions given in Eqs. (4) and (5)with M ′ R = 0.Now, the mass hierarchies M D M − R M TD , M D M − R M ′ T , M ′ M − R M ′ T ≪ M ′ D ≪ M D ≪ M R , (12)lead to the mass matrix M ≃ (cid:18) M ′ D M ′ TD (cid:19) , (13)which is consistent with the SM extended with the extra RH neutrinos, ν ′ R , havinga Dirac nature. Actually, in the vanishing limit of the small values M D M − R M TD , M D M − R M ′ T , M ′ M − R M ′ T , a lepton number conservation and a lepton–quark sym-metry are set up at low energies. It is the adulterated lepton–quark symmetry of par-ticle content in terms of ν ′ R with all couplings of ν R removed ( M D = M R = M ′ = 0).Dirac neutrino masses much smaller than those of charged leptons now appear nat-ural because M ′ D = 0 (with M ′ = M ′ R = 0) restores an enhanced symmetry in theoriginal Lagrangian, specifically, the symmetry of lepton and quark contents includ-ing the ordinary neutrino partners ν R . Thus adulterated Dirac neutrinos with tinymasses can be accommodated naturally. Once more, we emphasize that the corre-spondence between lepton and quark contents just serves as a guideline to the choiceof parameters since it is not a proper symmetry in the electroweak Lagrangian.
4. Presymmetry in the Standard Model with Right-HandedNeutrinos
We now substantiate the lepton–quark symmetry of particle content through aproper symmetry of the SM extended with RH neutrinos, based on the gauge groupsSU(3) c × SU(2) L × U(1) Y and Higgs fields in a doublet. An available simple optionfor this symmetry is presymmetry, where the crucial elements to have a well-definedsymmetry transformation appear naturally. In fact, presymmetry is conceived as asymmetry of an electroweak Lagrangian under transformations on underlying barestates of leptons and quarks having the same charges and no associated Majoranamass terms. To make the whole in a consistent way, we review in the following thekey arguments for presymmetry when one RH neutrino is added in each generation.On the one hand, there is the following hypercharge symmetry between chiralleptons and quarks within each of their three families:18 , Y ( ν L ) = Y ( u L ) + ∆ Y ( u L ) = − , Y ( e L ) = Y ( d L ) + ∆ Y ( d L ) = − ,Y ( ν R ) = Y ( u R ) + ∆ Y ( u R ) = 0 , Y ( e R ) = Y ( d R ) + ∆ Y ( d R ) = − , (14)and, on the other hand, Y ( u L ) = Y ( ν L ) + ∆ Y ( ν L ) = 13 , Y ( d L ) = Y ( e L ) + ∆ Y ( e L ) = 13 ,Y ( u R ) = Y ( ν R ) + ∆ Y ( ν R ) = 43 , Y ( d R ) = Y ( e R ) + ∆ Y ( e R ) = − , (15) resymmetry with adulterated Dirac neutrinos with the ∆ Y equal to 4 / − / Y = 43 ( L − B ) , (16)where the conventional relation Q = T + Y / Y , but the charge symmetry described in Eqs. (14)and (15) is preserved.27Presymmetry is associated with the equality of lepton and quark charges whenthe global part ∆ Y is set apart. The inclusion of RH neutrinos is essential to com-pleting this correspondence between charges, which in turn allows a symmetry oflepton and quark contents when one RH neutrino per generation is introduced;the symmetric pattern in terms of the extra, adulterant RH neutrinos considers ν ′ R instead of ν R in Eqs. (14) and (15).The charge symmetry and charge dequantization underlying Eqs. (14) and (15)can be understood if prelepton and prequark states are taken into account. Theseare defined by the quantum numbers of leptons and quarks, respectively, exceptcharge values, and denoted by a hat accent over the corresponding flavor symbol.The hypercharges of preleptons and prequarks are the same as their respective quarkand lepton weak partners. From Eqs. (14) and (15) one is then led, in the first case,to18 , Y ( ν L ) = Y (ˆ ν L ) + ∆ Y (ˆ ν L ) , Y ( e L ) = Y (ˆ e L ) + ∆ Y (ˆ e L ) ,Y ( ν R ) = Y (ˆ ν R ) + ∆ Y (ˆ ν R ) , Y ( e R ) = Y (ˆ e R ) + ∆ Y (ˆ e R ) , (17)and, in the other case, to Y ( u L ) = Y (ˆ u L ) + ∆ Y (ˆ u L ) , Y ( d L ) = Y ( ˆ d L ) + ∆ Y ( ˆ d L ) ,Y ( u R ) = Y (ˆ u R ) + ∆ Y (ˆ u R ) , Y ( d R ) = Y ( ˆ d R ) + ∆ Y ( ˆ d R ) , (18)with prelepton–quark charge symmetry established as Y (ˆ ν L ) = Y ( u L ) , ∆ Y (ˆ ν L ) = ∆ Y ( u L ) ,Y (ˆ ν R ) = Y ( u R ) , ∆ Y (ˆ ν R ) = ∆ Y ( u R ) ,Y (ˆ e L ) = Y ( d L ) , ∆ Y (ˆ e L ) = ∆ Y ( d L ) ,Y (ˆ e R ) = Y ( d R ) , ∆ Y (ˆ e R ) = ∆ Y ( d L ) , (19) E. A. Matute and prequark–lepton charge symmetry given by Y (ˆ u L ) = Y ( ν L ) , ∆ Y (ˆ u L ) = ∆ Y ( ν L ) ,Y (ˆ u R ) = Y ( ν R ) , ∆ Y (ˆ u R ) = ∆ Y ( ν R ) ,Y ( ˆ d L ) = Y ( e L ) , ∆ Y ( ˆ d L ) = ∆ Y ( e L ) ,Y ( ˆ d R ) = Y ( e R ) , ∆ Y ( ˆ d R ) = ∆ Y ( e L ) , (20)where the relation of ∆ Y with the lepton and baryon numbers for preleptons andprequarks is now ∆ Y = 43 (3 L − B ) , (21)with ∆ Y equal to − / / L = − / B = − Z group whichexchange topological preleptons and topological quarks on the one hand, ˆ ν L ( R ) ↔ u L ( R ) , ˆ e L ( R ) ↔ d L ( R ) , and prequarks and bare leptons on the other hand, ˆ u L ( R ) ↔ ν L ( R ) , ˆ d L ( R ) ↔ e L ( R ) .The charge shifts are originated by the nonstandard hypercharges of the newfermionic states, which produce gauge anomalies in the couplings by fermion triangleloops of three currents related to the chiral U(1) Y and SU(2) L gauge symmetries.In fact, in the scenario of topological preleptons and quarks, for example, the U(1) Y resymmetry with adulterated Dirac neutrinos gauge current in all representationsˆ J µY = ˆ ℓ L γ µ Y ℓ L + ˆ ℓ R γ µ Y ℓ R + q L γ µ Y q L + q R γ µ Y q R , (22)exhibits the U(1) Y [SU(2) L ] and [U(1) Y ] anomalies due to the nonvanishing of thefollowing sums which include one RH preneutrino per generation: X L Y = 8 , X LR Y = − , (23)where the first runs over the LH and the second over the LH and RH topologicalpreleptons and quarks, with ( −
1) for the RH contributions. Their cancellationsneed a counterterm which contains topological currents or Chern–Simons classesassociated with the U(1) Y and SU(2) L gauge groups: J µT = 14 K µ X L Y + 116 L µ X LR Y = 2 K µ − L µ , (24)where K µ = g π ǫ µνλρ tr (cid:18) W ν ∂ λ W ρ − igW ν W λ W ρ (cid:19) ,L µ = g ′ π ǫ µνλρ A ν ∂ λ A ρ , (25)so that the new current J µY = ˆ J µY + J µT is anomaly free, gauge noninvariant, andalso symmetric under the exchange of topological preleptons and quarks. Moreover,its charge is not conserved due to the topological charge which gives the change inthe topological winding number of the asymptotic, pure gauge field configurations,assuming that the space–time region of nonzero energy density is bounded. Indeed,advocating the principle of equality for all preleptons of the system in the partitionof the topological charge,18 the change in each charge, using Eqs. (24) and (25) forthe pure gauge fields, is ∆ Q Y = 16 ( n + − n − ) = 16 n , (26)with the topological charge given by n = Z d x ∂ µ K µ = g π Z d x tr( W µν ˜ W µν ) . (27)These topological numbers vanish in the U(1) Y case.Vacuum states labeled by different topological numbers are then tunneled bySU(2) L instantons carrying topological charges, making possible in principle thecharge shifts and transitions from fermions with nonstandard to those with standardcharges. Each hypercharge is changed by a same amount: Y (ˆ ℓ ) → Y (ˆ ℓ ) + n . (28)The value n = − E. A. Matute invariance and renormalizability of the theory; the coefficient 3 L − B in Eq. (21) isjust a counting number.19According to the presymmetry model, topological preleptons have a vacuumgauge field configuration of winding number n − = 4, if gauge freedom is used toset n + = 0 for that of leptons. The transformation of topological preleptons intobare leptons is via a Euclidean topological weak instanton with topological charge n = −
4, conceived in Minkowski space–time as a quantum mechanical tunnelingevent between vacuum states of weak SU(2) L gauge fields with different topologicalwinding numbers. Thus, topological preleptons and bare leptons are differentiatedby the topological vacua of their weak gauge configurations, tunneled by a weakfour-instanton bearing the topological charge and inducing the global fractionalpiece of charge required for normalization.However, the transitions from topological preleptons to bare leptons by means ofthe weak SU(2) L instantons, as well as those from prequarks to topological quarks,do not occur in the actual world because topological preleptons, prequarks and topo-logical quarks are not physical dynamical entities. They are bare states of leptonsand quarks considered as mathematical entities out of which the observed parti-cle states are constructed. In a sense, such transformations are frustrated by theextreme smallness of the instanton transition probability at zero temperature, andthe charge normalization eliminates the extraordinarily large time scale for them,leading to leptons and quarks with trivial topology and standard charges. The re-placement of bare leptons and quarks with normalized charges by the standard onesin the electroweak Lagrangian is straightforward as they have the same quantumnumbers,18 ,
19 and the insertion of Majorana mass terms for RH neutrinos givesthe shape of the extended effective theory.Despite this, one still has a proper symmetry transformation at the level ofpreleptons defined by the exchange of all topological preleptons and quarks in theelectroweak sector of the Lagrangian, which requires a correspondence betweenfermionic contents at the stages of preleptons and leptons. But this is preciselywhat is needed to have a natural framework which allows light Dirac-like neutrinosin the low-energy theory. In fact, as seen in Secs. 2 and 3, the addition of a sec-ond RH neutrino per generation coupling `a la Dirac to the LH neutrino providesthe seed for that as the relation between light neutrino masses and the chargedlepton masses is broken. By means of this, the naturalness problem is solved, i.e.the question why light Dirac neutrinos are so much lighter than charged leptons isanswered.The well-defined presymmetry now validates the sequence of hierarchies assumedin Secs. 2 and 3. The first hierarchy, M D ≪ M R , mimics the standard high scaletype-I seesaw scenario in which only one RH neutrino per generation, ν R , is intro-duced. The symmetry of lepton and quark contents and the hypercharge symme-try in Eq. (14) are re-established. At the underlying level of preleptons defined inEqs. (17) and (19), one has the presymmetry transformations that exchange topo- resymmetry with adulterated Dirac neutrinos logical prelepton and quark fields in a Lagrangian where Majorana mass terms areabsent since all preleptons have electroweak charges.By introducing the second RH neutrino for each generation of leptons andquarks, which breaks presymmetry, the second hierarchy, M ′ D ≪ M D , is under-stood naturally since if this new parameter is set equal to zero, the presymmetryinvolving the first RH neutrino is recovered.The last hierarchies, ( M D M − R M TD , M ′ M − R M ′ T , M D M − R M ′ T ) ≪ M ′ D (entail-ing M D , M ′ ≪ M R ) and M ′ R ≪ M ′ D , mimic the low scale pseudo-Dirac scenario.If the small ratios of mass parameters are neglected, for all practical purposes thefirst RH neutrino is decoupled from the low energy model. As such, the symmetryof lepton and quark contents, the symmetric charge relations in Eq. (14) and inEqs. (17) and (19) at the underlying level, as well as presymmetry, now engage ν ′ R instead of ν R . In this way, the residual presymmetry connecting ν ′ R makes the lowscale model more symmetric. Thus all the hierarchies are natural in the sense oft’ Hooft.
5. Phenomenological Implications of the Model
In order to have light neutrinos of adulterated Dirac type, our extension of the SMassumes two RH neutrinos in each generation and the hierarchy of masses given inEq. (12). As discussed above, the first choice is to decouple through the extendedseesaw mechanism the original RH neutrinos from the others by making them muchheavier than the other mass parameters and ratios of mass parameters. In a secondstep, the light neutrino masses are effectively controlled by the new Dirac mass ofthe extra RH neutrinos. This requires a hierarchy between the extra Dirac massesand the other mass parameters and ratios of mass parameters. Only this secondhierarchy gives rise to the light pseudo-Dirac neutrinos.The Dirac mass M D is supposed to be of order the charged lepton mass. Inthe approximation of one generation, if a mass M D = O (1 MeV) is for the chargedlepton of first generation and the upper bounds M LL , M ′ RR ≤ O (10 − eV) (29)obtained from data analyzes on solar neutrino oscillations28 ,
29 are considered,then Eq. (9) leads to an energy threshold for the type-I seesaw equal to M R = O (10 GeV). Besides, using the experimental data on neutrino mass,30 M ′ LR = O (10 − eV) , (30)the indicative values for the parameters in the model become M R ≥ O (10 GeV) , M D = O (1 MeV) , M ′ D = O (10 − eV) ,M ′ R ≤ O (10 − eV) , M ′ ≤ O (1 MeV) , (31)which are consistent with the hierarchy of masses M LL , M ′ RR ≪ M ′ LR ≪ M D ≪ M R (32) E. A. Matute adopted in the model, so ratifying the Dirac type assumed for light neutrinos.Thus the parameter region being considered excludes the pseudo-Dirac limit,but not the Dirac character for light neutrinos. Their masses or Yukawa couplingsmay have exceptionally small values because of the adulterant character of RHpartners. Also, there is consistency between this Dirac picture and the vanishing ofthe Majorana mass M ′ R assumed above. Besides, the Dirac nature of lighter neutri-nos forbids the neutrinoless double-beta decay, in accordance with recent precisionexperiments.14–16 Even more, no significant departures from the SM predictionsare expected at the TeV region, leading to substantive tensions with models whichassume extensions of the gauge and Higgs sectors with breaking scales at the TeVrange. Thus, the model can be tested through the successes of the SM and the Diracnature of light neutrinos. Although we do not address here the phenomenon of darkmatter, we note that our model can be extended to accommodate its particles, forinstance, by adding extra sterile neutrinos.Yet, the model maintains the expectations of the high scale type-I seesawmechanism.10–13 This includes the new physics to be introduced with its energythreshold O (10 GeV), required to solving the problem of naturalness generated bythe quantum corrections to the Higgs boson mass. In fact, the type-I seesaw is natu-ral up to M R = O (10 GeV),20 but demands an unnatural fine-tuning cancellationfor M R = O (10 GeV). The only best known way of persisting is through supersym-metry. This can be implemented in stages, starting with the non-supersymmetriclow-energy theory,21 although the high seesaw threshold calls for the supersym-metrization of the Higgs, electroweak and fermionic sectors. The no observation ofsupersymmetric phenomena may imply that the quantum corrections to the Higgsmass would be suppressed by another kind of new physics.On the other hand, the model is crucially based on the hypothesis of a symmetryof quark and lepton contents, which, however, is validated by presymmetry. LightDirac neutrinos are then predictions of presymmetry in the SM extended with twoRH neutrinos per generation, implemented with a high scale seesaw mechanism.Moreover, presymmetry of the SM with adulterated Dirac neutrinos appears asa residue after removal of the heavy RH neutrinos of Majorana type. Thus, thesignatures of presymmetry are also marks of the proposed SM with adulteratedDirac neutrinos. Besides light Dirac neutrinos, they include explanations of thefractional charge of quarks and quark–lepton charge relations, understanding ofthe equality between the number of generations and the number of quark colors,accounting for the topological charge conservation in quantum flavor dynamics, andelucidation of the charge quantization and the no observation of fractionally chargedhadrons.18 ,
6. Conclusions
In the scenario of the SM extended with one RH neutrino per generation, a sim-ple paradigm for understanding the small value of neutrino masses compared to the resymmetry with adulterated Dirac neutrinos charged leptons is the type-I seesaw mechanism where RH neutrinos have Majoranamasses in addition to Dirac masses. A distinguishing feature of this mechanism isthe Majorana nature of light neutrinos, which, however, is not favored by recentexperimental data on double-beta decay of nuclei. Moreover, the current experi-mental situation shows an agreement with the SM predictions well above the TeV,apart from the presumed Dirac neutrinos with small masses for which it gives noexplanation.Assuming no serious departures from the SM expectations at the TeV rangeand the Dirac character of light neutrinos, recently we proposed an extended see-saw in which two RH neutrinos per generation are added, implemented with thehypothesis of the symmetry of lepton and quark contents in order to restrain thenumber of RH neutrinos from freedom, produce Dirac neutrinos and naturally givethem tiny masses. The first one is the usual RH neutrino which re-establishes thecorrespondence between quarks and leptons at high energies with weak couplingshaving order of magnitudes as those of its weak charged partner and a Majoranamass term whose coupling is assumed to be large, as in the canonical high-scaletype-I seesaw scenario. The second RH neutrino, which breaks the quark–leptonsymmetry founded with the first one, has small masses and couplings, as explainedby the ’t Hooft’s naturalness criterion applied to this symmetry of contents. Thefirst RH neutrino is decoupled at the high scale, while the second one survives downto the low scale to pair off in a Dirac-like fashion with the corresponding LH neu-trino, driving its pattern of the symmetry of fermionic content. These symmetries ofparticle content were only regarded as guidelines to the choice of parameters sincethey cannot be understood as symmetry transformations that exchange lepton andquark fields in the Lagrangian of the model. It was supposed, however, that theproper symmetry to take on has to be hidden in the SM with RH neutrinos itself.From another viewpoint, presymmetry was assumed as a symmetry that under-lies the SM extended with three RH neutrinos having Dirac mass terms, so restoringlepton–quark symmetry of particle content, unifying the electroweak properties ofleptons and quarks, and explaining the observed charge relations and chiral struc-ture of weak interactions. However, this scenario cannot account naturally for thesmallness of Dirac neutrino mass terms relative to those of charged leptons. Be-sides, if it is pointed out that the B − L symmetry is an accidental symmetry of theSM and that presymmetry is a hidden electroweak symmetry at a bare level, theinclusion of Majorana mass terms for neutrinos seems natural. Thus, both Diracand Majorana mass terms are included in the minimal SM extension of only threeRH neutrinos, explicitly breaking the conservation of the lepton number. Presym-metry still appears as a basic symmetry of the model with neutrinos having genericmass terms, since it is an extra enhanced symmetry established at the bare step;such presymmetry transformations perform at the underlying level of topologicalbare states which have the same electroweak charges and no Majorana mass terms.Yet, the tiny mass of Dirac-like neutrinos respect to charged leptons still remainsunnatural, so that the addition of just three RH neutrinos with generic masses to E. A. Matute the SM is not enough for understanding such smallness.The addition of a second RH neutrino per generation, however, provides theseed for the expected light Dirac neutrinos. We have shown that presymmetry de-fines properly the symmetry transformations required by the symmetry of leptonand quark contents and the assumed sequence of hierarchies. This means that theextra Dirac and Majorana masses are not free parameters, independent of the firstRH neutrinos. We emphasize that these are not made small by fine-tuning. Theirsmallness compared to the first RH neutrinos are guaranteed by the presymmetrydefined with the usual RH neutrinos, at the high-energy seesaw scale. The ’t Hooft’sargument of naturalness for the small values of the Dirac and Majorana mass termsof the extra RH neutrinos in the Lagrangian relies on this presymmetry with thefirst RH neutrinos; as the couplings of the extra RH neutrinos tend to zero, theunderlying theory only involving the first RH neutrinos becomes more symmetric.This symmetry guarantees the quantum corrections of such parameters to be pro-portional to the parameters themselves and its interplay with the seesaw mechanismleading to the low-energy effective theory with the original RH neutrinos decoupledonly introduces negligible corrections to the mass parameters. In particular, thesmallness of the Dirac mass of neutrinos compared to the charged leptons is stable.Now a low scale Dirac scenario with symmetry of particle content and smallneutrino masses appears natural, satisfying ’t Hooft’s naturalness conditions. But,they involve the additional and not the standard RH neutrinos, which are decoupled.The claim is that the SM extended with extra RH neutrinos and implemented withthe seesaw mechanism and presymmetry, or the presymmetry model implementedwith extra RH neutrinos and the seesaw mechanism, makes natural the existence oflight Dirac-like neutrinos. Neutrinos with extremely small masses are predicted tobe of adulterated Dirac nature in the sense that the ordinary RH components arereplaced by the almost inert extra ones. Besides, the parameter region consideredin this approach makes irrelevant to low energy processes the perturbation of theseesaw mechanism on a description given in terms of light Dirac neutrinos, foreseeingthat experiments will not have sensitiveness to the Majorana character of neutrinospredicted by the seesaw mechanism, as in the case of the neutrinoless double-betadecay.On the other hand, the signatures of presymmetry are also noticeable featuresof the model of Dirac neutrinos, such as the topological character of fractionalcharges and the relationship between the number of generations and the numberof quark colors. Although there are no hard predictions for the masses and mixingof light neutrinos, the model does provide a new line of physics beyond the SM forexploration.Nevertheless, the high energy seesaw threshold established by our approachraises the issue on the naturalness of the renormalization of the Higgs mass dueto the quantum corrections introduced by the new physics states associated withthe seesaw. The best known way to solve it is via supersymmetry. An eventual ten-sion between experimental data and supersymmetry expectations at the TeV range resymmetry with adulterated Dirac neutrinos to be tested by the LHC may imply that the solution to this naturalness problemwould be given by another, still unknown new physics. Acknowledgment
This work was supported by Vicerrector´ıa de Investigaci´on, Desarrollo e Innovaci´on,Universidad de Santiago de Chile, Usach, Proyecto DICYT No. 041431MC.
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