aa r X i v : . [ g r- q c ] A p r Conformal Standard Model
Luca Fabbri
DIPTEM Sez. Metodi e Modelli Matematici dell’Università di Genova &INFN Sez. di Bologna and Dipartimento di Fisica dell’Università di Bologna
Abstract
In recent papers we have constructed the conformal theory of metric-torsional gravitation, and in this paper we shall include the gauge fields tostudy the conformal U (1) × SU (2) Standard Model; we will show that themetric-torsional degrees of freedom give rise to a potential of conformal-gauge dynamical symmetry breaking: consequences are discussed.
Introduction
The reasons for which conformal Weyl gravity is remarkable are two: Weyl grav-ity possesses solutions that, for the solar system, are able to approximate theEinstein gravity solutions, while, as the scales increase to galaxies, they are alsoable to fit the rotation curves [1, 2], and for the universe in its entirety, theyaddress the cosmological constant problem [3], so to be able of replacing darkforms of matter and energy [4]; the equations of Weyl gravity are fourth-orderdifferential equations with dimensionless constants whose renormalizability isuseful to address the gravitational quantization [5]. Thus far, the entire dis-cussion has been carried out in terms of conformal Weyl gravity in the purelymetric case, about which a general discussion can be found in [6], but on theother hand one has to keep in mind that in order to build a complete quantumtheory not only the metric but also the torsional degrees of freedom have to beconsidered, because according to the Wigner classification of quantum particlesin terms of their mass and spin, matter fields possess not only energy but alsospin density, the former being related to the curvature while the latter beinglinked to the torsion of the spacetime: thus for a conformal metric-torsionalgravitation to be constructed one needs to define, beside the conformal trans-formations for the metric, also the corresponding conformal transformations forthe torsion tensor [7]. Once the conformal properties of the metric as well astorsion are settled, one needs to define a conformally covariant metric-torsionalcurvature in (1 + 3) -dimensional spacetimes upon which to build the conformalgravitational field theory as it was done in [8]. This theory has further beenapplied to the special case of the conformal massless Dirac field theory as in [9].Then it is important to recall that not only the metric-torsional degrees offreedom but also the gauge degrees of freedom have to be considered becausematter fields possess not only energy-spin but also current density: for theconformal metric-torsional gravitation to include gauge fields one should defineconformal transformations for the gauge fields, which nonetheless are trivial asit is widely known. Because of this fact we have that the conformally covariant1auge strength in (1 + 3) -dimensions is unchanged with respect to the standardcase in which the conformal invariance of gauge field theories was already acharacter of the model [10]. The inclusion of scalar fields then follows [11].However when in the purely metric curvature the torsion is accounted thenthe metric-torsional curvature changes its properties, and consequently the ef-fects of its coupling to other fields, so that while the background is decoupledfrom the gauge strength, and therefore such a modification has no influence onthe gauge sector, the background has peculiar coupling to the scalar fields, andhenceforth this modification greatly affects the scalar fields as we shall show inthe present paper. This allows us to merge the results of [12] and [13] obtaininga metric-torsion conformal U (1) × SU (2) -gauge symmetric interaction for theDirac and scalar fields to get the conformal Standard Model. In this paper we shall follow all notation and conventions of [8, 9] about con-formal gravity. In particular and as usual, the metric is given by g µν whilethe connection is given by Γ αρσ whose antisymmetric part in the lower indicesis torsion Γ αρσ − Γ ασρ = Q αρσ also known as Cartan tensor; in the following ofthe paper we will assume the metric-compatibility condition for the connection,that is we shall insist on the fact that the metric is the constant for the co-variant derivatives D α g µν = 0 spelling out that the metric properties and thedifferential features of the space have characters that are compatible, althoughbecause of the presence of torsion, they remain independent, as it is clear fromthe fact the metric-compatibility condition is equivalent to the decomposition Γ σρα = g σθ [ Q ραθ + Q αρθ + Q θρα + ( ∂ ρ g αθ + ∂ α g ρθ − ∂ θ g ρα )] (1)in terms of both metric and Cartan torsion tensor. An equivalent formalismcan be introduced. In it we consider the constant Minkowskian metric η ij anda basis of vierbein e iα such that we have the relationship e pα e iν η pi = g αν togetherwith the spin-connection ω ipα for which no torsion tensor can possibly be definedbecause of the different type of Latin and Greek indices; vierbein-compatibilityconditions are imposed accordingly, leading to the conditions D α e jµ = 0 andtherefore D α η ij = 0 respectively yielding the formula ω ipα = e iσ (Γ σρα e ρp + ∂ α e σp ) (2)with the property ω ipα = − ω piα , showing that it is therefore possible to employthe connection and the vierbein to give rise to the spin-connection, which isantisymmetric in the two Latin indices, and again vierbein and spin-connectionare taken independent. The former formalism indicated with Greek letters andthe latter formalism indicated with Latin letters are respectively denoted asspacetime formalism and world formalism, and they are equivalent, althoughin the last formalism there is the advantage for which the introduction of thespinorial structure is possible. To this extent, we introduce the set of γ a matricesverifying the Clifford algebra { γ a , γ b } = 2 I η ab from which it is further possibleto define the σ ab matrices σ ab = [ γ a , γ b ] such that { γ a , σ bc } = iε abcd γγ d whichare the generators of the spinorial transformation defining spinor fields ψ , whosedynamics defined in terms of the spinor-connection Ω ρ = ω ijρ σ ij is encoded2hrough the spinor-covariant derivatives D ρ ψ = ∂ ρ ψ +Ω ρ ψ with respect to whichthe constancy of the γ a matrices is automatic. Thus the geometrical backgroundis given, and conformal properties have next to be assigned according to theusual metric conformal transformation as in the following g αθ → σ g αθ (3)and by defining Q σσα = Q α as the torsion trace vector we postulate the torsionalconformal transformations to be given by Q σρα → Q σρα + qσ − ( δ σρ ∂ α σ − δ σα ∂ ρ σ ) (4)for a given parameter q , and as it is clear by taking the contraction it is on thetorsion trace alone that the conformal transformation is loaded, thus implyingthat Q βρµ ε βρµα = − V α known as torsion dual axial vector and the remainingirreducible part of torsion are conformally covariant; the curvature tensor G ρξµν = ∂ µ Γ ρξν − ∂ ν Γ ρξµ + Γ ρσµ Γ σξν − Γ ρσν Γ σξµ (5)or equivalently in the form G ρξµν e iρ e ξj = G ijµν given by G ijµν = ∂ µ ω ijν − ∂ ν ω ijµ + ω ikµ ω kjν − ω ikν ω kjµ (6)contains torsion implicitly through the connection, and from it it is possible todefine a modified metric-torsional curvature tensor M αθµν = G αθµν + ( − q q )( Q θ Q αµν − Q α Q θµν ) (7)containing torsion also explicitly, and which has the same symmetries, and it issuch that its irreducible decomposition T αθµν = M αθµν − ( M αµ g νθ − M θµ g να − M αν g µθ + M θν g µα ) ++ M ( g αµ g νθ − g θµ g να ) (8)does not only have the same symmetries and is traceless but it is also conformallycovariant; and finally, it is in terms of the set of three free parameters A , B , C that it is possible to define the parametric curvature tensor P αθµν = AT αθµν + BT µναθ ++ C ( T αµθν − T θµαν + T θναµ − T ανθµ ) (9)having the same symmetries and being traceless and also conformally covariantin (1+3) -dimensional spacetimes, whose usefulness will turn out in the following.Next we will follow the notation [12, 13] for the standard model. In particu-lar the fields B µ and ~A µ are vectors having gauge transformations given by theabelian U (1) and the simplest non-abelian SU (2) group and with trivial con-formal transformation, from which it is possible to define the gauge-covariantderivatives D α and the Maxwell and Yang-Mills gauge curvatures B µν = ∂ µ B ν − ∂ ν B µ (10) ~A µν = ∂ µ ~A ν − ∂ ν ~A µ + g ~A µ × ~A ν (11)3ntisymmetric for indices transposition and traceless for indices contraction andconformally covariant in (1 + 3) -dimensions, as expected for gauge fields.The fermion fields are introduced as a single right-handed spinor ψ R and adoublet of left-handed spinors ψ L defined in terms of their transformation lawunder the same U (1) × SU (2) group and with scaling σ − while the scalar fieldis a doublet of complex scalar fields φ set by its transformation law under thesame U (1) × SU (2) group and with scaling σ − as usual. For the Standard Model as we know it [12], we have that the action is a scalar andgauge symmetric under the U (1) × SU (2) group and not conformally invariant asthe Lagrangian has terms that do not scale by the σ − factor, but instead theyscale by the σ − factor; these terms are given by the Ricci curvature G necessaryfor the gravitational dynamics and the scalar quadratic potential φ essentialto bring the trivial vacuum in a non-stable configuration that is supposed toeventually move toward a stable configuration with non-trivial vacuum.For a conformal version of the Standard Model instead [13], the action mustbe a scalar and gauge symmetric under the U (1) × SU (2) group and also confor-mally invariant with a Lagrangian that scales by the σ − factor; consequentlywe need to have, on the one hand, terms like the square of the T αθµν tensor usedto determine the gravitational dynamics, while, on the other hand, the productbetween the curvature M and the quadratic potential φ † φ = φ may be used asa potential for the conformal-gauge dynamical symmetry breaking.Actually the approach enjoys a particular elegance that can be appreciatedby noticing the following fact: under a global conformal transformation both thescalar dynamical term D ρ φ † D ρ φ and the scalar potential term φ M scale by thecorrect σ − factor, although under a local more general conformal transforma-tion these two terms will be accompanied by extra pieces that would spoil theinvariance, unless a proper fine-tuning is chosen so to have them all cancellingexactly, yielding a conformally invariant scalar action; on the other hand how-ever, general conformal transformation in presence of metric and torsion widenthe range of possibilities because beyond the usual term φ M there are addi-tional terms like φ Q α Q α as well as D ν φ Q ν that may be taken into accountbeside the dynamical term D ρ φ † D ρ φ and, since expressions with derivatives ofthe scalar and torsion such as for instance D ρ φ + q Q ρ φ are conformally covari-ant, then it is possible to restore the metric-torsional conformal invariance of thescalar action as a whole. Under this point of view, all potentials of conformal-gauge symmetry breaking are not just added for generality, but because suchscalar potentials beside the scalar dynamical term are necessary to maintain theconformal-gauge symmetry before its breakdown: after the most general scalaraction is found, the total action is given in the following form S SM = R [ T αθµν P αθµν − B µν B µν − ~A µν · ~A µν ++ i (cid:0) ψ R γ µ D µ ψ R − D µ ψ R γ µ ψ R (cid:1) + i (cid:0) ψ L γ µ D µ ψ L − D µ ψ L γ µ ψ L (cid:1) ++ D ρ φ † D ρ φ + (cid:16) − k (1 − q )3 q (cid:17) D ν φ Q ν + (cid:16) − k (1 − q )(1+2 q )9 q (cid:17) φ Q α Q α ++ kφ M − Y (cid:0) ψ R φ † ψ L + ψ L φψ R (cid:1) − λ φ ] p | g | dV (12)4n terms of the k , λ and Y parameters and such that under the most generalcoordinate U (1) × SU (2) conformal transformation it is invariant; we vary thisaction with respect to the spin-connection and the vierbein taking into accountthat these variations are transferred through (2) and (6) onto the variationsof the torsion and curvature according to the identities given by the followingformulas δQ iρα = − (cid:0) D ρ δe iα − D α δe iρ − δe iσ Q σρα (cid:1) + (cid:0) δω ipα e pρ − δω ipρ e pα (cid:1) togetherwith δG ijµν = (cid:0) D µ δω ijν − D ν δω ijµ − δω ijρ Q ρµν (cid:1) showing in particular that thevariation of the curvature does not depend on the variation of the vierbeins,then for the variation of the action with respect to the gauge fields we useformulas given in (10-11), and the variation of the spinor and the scalar fieldsis straightforward, so that variation with respect to the spin-connection andvierbeins gives field equations for the spin and energy densities according to − q q )( Q σρβ g µ [ α P θ ] σρβ − Q ρ P ρ [ αθ ] µ ) ++ D ρ P αθµρ + Q ρ P αθµρ − Q µρβ P αθρβ ] = S µαθ (13) − q q )( Q ν (2 P µραν Q ρ − g µα P νθρσ Q θρσ − P µνρσ Q αρσ ) + D ν (2 P µραν Q ρ − g µα P νθρσ Q θρσ + g µν P αθρσ Q θρσ )) ++ P θσρα T µθσρ − g αµ P θσρβ T θσρβ + P µσαρ M σρ ] ++ [( B αρ B µρ − B g αµ ) + ( ~A αρ · ~A µρ − A g αµ )] = T αµ (14)whereas the variation with respect to the pair of gauge potentials gives thecouple of field equations for the two currents according to D ρ B ρµ + Q ρ B ρµ + Q µβρ B βρ = J µ (15) D ρ ~A ρµ + Q ρ ~A ρµ + Q µβρ ~A βρ = ~J µ (16)where the spin and energy densities are given by S µαθ = i ψ R { γ µ , σ αθ } ψ R + i ψ L { γ µ , σ αθ } ψ L ++ (cid:16) k − q (cid:17) (cid:0) D θ φ g αµ − D α φ g θµ (cid:1) ++ (cid:16) − k − kq q (cid:17) φ (cid:0) Q α g θµ − Q θ g αµ (cid:1) − kφ Q µαθ (17) T αµ = i (cid:0) ψ R γ α D µ ψ R − D µ ψ R γ α ψ R (cid:1) + i (cid:0) ψ L γ α D µ ψ L − D µ ψ L γ α ψ L (cid:1) ++ (cid:0) D α φ † D µ φ + D µ φ † D α φ − g αµ D ρ φ † D ρ φ (cid:1) −− (cid:16) − k (1 − q )3 q (cid:17) (cid:0) D µ D α φ − D φ g αµ − Q α D µ φ (cid:1) −− (cid:16) − k (1 − q )9 q (cid:17) (cid:0) D µ φ Q α − g αµ D ν φ Q ν + φ D µ Q α − g αµ φ D ρ Q ρ (cid:1) ++ (cid:16) − k (1 − q )9 q (cid:17) g αµ φ Q ν Q ν − k (cid:16) − q q (cid:17) φ (cid:0) Q α Q µ + Q αµθ Q θ (cid:1) ++2 kφ (cid:0) M αµ − g αµ M (cid:1) + λ g αµ φ (18)and the currents are given by J µ = − g ′ ψ R γ µ ψ R − g ′ ψ L γ µ ψ L − ig ′ (cid:0) D µ φ † φ − φ † D µ φ (cid:1) (19) ~J µ = − g ψ L γ µ ~σψ L + ig (cid:0) D µ φ † ~σφ − φ † ~σD µ φ (cid:1) (20)5hile varying with respect to the spinor we get the spinorial field equations iγ µ D µ ψ R + i Q µ γ µ ψ R − Y φ † ψ L = 0 (21) iγ µ D µ ψ L + i Q µ γ µ ψ L − Y φψ R = 0 (22)and varying with respect to the scalar we have the scalar field equations D φ + Q ρ D ρ φ + (cid:16) − k (1 − q )3 q (cid:17) D ν Q ν φ ++ (cid:16) − q +6 k − qk +6 kq q (cid:17) Q ν Q ν φ − kM φ + λ φ φ + Y ψ R ψ L = 0 (23)as the system of field equations of the conformal Standard Model. Finally itis possible to see that in this set of field equations when the Dirac and scalarfield equations are considered for the energy and spin and the two currents thenthe conserved quantities satisfy the following conservation laws arising from theinvariance under gauge phase transformations D µ J µ + Q µ J µ = 0 (24) D µ ~J µ + Q µ ~J µ = 0 (25)the spacetime frame transformations D µ T µρ + Q µ T µρ − T µσ Q σµρ + S θµσ G σµθρ + J µ B µρ + ~J µ · ~A µρ = 0 (26) D ρ S ρµν + Q ρ S ρµν + T [ µν ] = 0 (27)and spacetime conformal scaling (1 − q )( D µ S νµν + Q µ S νµν ) + T µµ = 0 (28)and for which the Jacobi-Bianchi identities are verified identically.We have to notice two important issues: the first is that in the energydensity of the spinor field there is no explicit torsional contribution, and thisis due to the fact that the variation with respect to the vierbein of the spinor-covariant derivative of the spinor field vanishes identically therefore developingno term that need to be integrated by parts; the second is that the scalar fieldcontributes to the spin density, and this is due to the coupling between torsionand scalar fields, forced by conformal invariance. Notice also that the gauge fieldequations for the currents are the same in the non-conformal and the conformalversion of the standard model. A further step consists in decomposing thefull connection into the torsionless connection plus torsional contributions, sothat the torsionless connection known as Levi-Civita connection gives covariantderivatives ∇ µ and curvature tensors R αβµν whose irreducible part C αβµν is theWeyl conformal curvature while torsion itself in its three irreducible parts is Q αµν = ( g αµ Q ν − g αν Q µ ) + ε αµνσ V σ + T αµν (29)where T αµν is the non-completely antisymmetric irreducible part: then in thespinorial field equations (21-22) because of the extra term, torsion trace contri-butions cancel exactly leaving only the torsional dual axial contributions iγ µ ∇ µ ψ R − V µ γ µ ψ R − Y φ † ψ L = 0 (30) iγ µ ∇ µ ψ L + V µ γ µ ψ L − Y φψ R = 0 (31)6s usual and where the different sign highlights the chirality of this type ofself-interactions, while in the scalar field equations (23) we have that ∇ φ + (cid:16) − k q (cid:17) ∇ ν Q ν φ − (cid:16) − k q (cid:17) Q ν Q ν φ − k V ν V ν φ −− k T νπα T νπα φ − kRφ + λ φ φ + Y ψ R ψ L = 0 (32)and where we see that the configuration φ ≡ is not a solution in presence ofspinorial fields. We have that instead if spinorial fields are present more generalsolutions with a non-vanishing vacuum expectation value must be sought. The Conformal Standard Model built so far has an advantage with respectto the ordinary Standard Model [12], which consists in the fact that becauseof the presence of conformal gravitational degrees of freedom there is a morecomplicated form for the Higgs potential inducing now a dynamical symmetrybreaking phenomenon: in fact after a direct calculation we have that once thevacuum expectation value for the Higgs is given by v = φ the condition forthe stable stationary point for which the symmetry is broken is given by λ v = (cid:16) k − q (cid:17) ∇ ν Q ν − (cid:16) k − q (cid:17) Q ν Q ν + k V ν V ν + k T νπα T νπα + kR (33)which we are now going to discuss in some detail: notice first of all that if theconstant k has the value it would have in the torsionless case k = then (33)would reduce to the simpler expression given by λv = V ν V ν + T νπα T νπα + R (34)in which we see that even in absence of torsion non-trivial solutions are possiblewhenever R = 3 λv is satisfied, which it is in a de Sitter spacetime of negativespatial curvature, as reported for instance in [13]; however, this condition reliesupon the curvature of the spacetime and its purely spatial projection, whichis certainly considerable in cosmology but it might turn out to be small inparticle physics, and therefore we shall retain the presence of torsion in order toensure that (34) has non-trivial solutions even in a context in which the metricis approximately flat. Actually, even if the metric were not flat and curvaturelarge, the study of torsional effects for the symmetry breaking would still be ofsome interest: if however the metric is flat or curvature small, we would have λv = V ν V ν + T νπα T νπα (35)and the presence of torsion would allow new ways to provide the dynamicalsymmetry breaking mechanism in the conformal Standard Model. As it is widely known, after the vacuum expectation value is gotten by the Higgsfield a symmetry breaking occurs because the new ground state of the Higgs fieldis no longer invariant, and the new Higgs field is seen as a fluctuation over thespecial ground state, therefore producing two mechanisms: on the one hand,7here is generation of the masses of the particles that couple to the Higgs,taking place in two ways: both as a transfer of degrees of freedom from theHiggs to the massless bosons which then become massive bosons, and as theresult of the presence of the potential of interaction between Higgs and fermionsand of self-interaction of the Higgs with itself; on the other hand, there is theappearance of the cosmological constant, again as the result of the presence ofthe same potential of self-interaction of the Higgs with itself. In the following,we shall not take into account the mechanism of the generation of the masses ofthe bosons, thoroughly discussed in the literature; we will instead focus on thegeneration of the masses of the fermion and Higgs field and of the cosmologicalconstant, whose origin is due to the presence of the Yukawa and Higgs potentials.After a straightforward calculation, it is easy to see that the values of themass of the fermion and the Higgs and also the cosmological constant m fermion = Y v m = λv Λ = λv (36)are given by the Yukawa coupling Y and the Higgs parameter λ in terms of thevacuum expectation value v of the Higgs field itself. Notice that as the Yukawacoupling is unknown the knowledge of the fermion mass does not give any clueabout v whose value of about
350 GeV is determined when the low-energy limitof the fermion scattering is compared to the effective Fermi scattering, and thisvalue is used to evaluate from the Higgs mass the cosmological constant.In fact by combining the two definitions above we have that
Λ = (cid:16) m Higgs v √ (cid:17) (37)which with the vacuum expectation value of about
350 GeV and the Higgs massat least of the order of magnitude of GeV gives a cosmological constant atleast of order of magnitude of GeV which is far from the upper limit of theorder of magnitude of − GeV as astrophysical experiments tell.This situation, in which the ground state of the Higgs field gives reasonablemasses only at the price of having a largely wrong cosmological constant, givesrise to what is usually known as the cosmological constant problem: this prob-lem may be solved by a mechanism for which the vacuum expectation value ofthe Higgs field diminishes as we move from particle physics to cosmology, andthis is precisely what happens in this approach. Nevertheless, the cosmologicalproblem might be circumvented if we think that conformal Weyl gravity andstandard Einstein gravity are quite different in their structure, and correspond-ingly there are two different treatments for the cosmological constant problem,so that what appears to be a wrong prediction in standard cosmologies mighthave no conflict with observations in conformal cosmologies [14]. Conclusion
In the present paper, we have considered the fully endowed metric-torsional con-formal Standard Model writing the most generally invariant action and derivingthe field equations: we have isolated the Higgs sector determining the groundstate for the stable stationary potential that determines dynamical symmetrybreaking generating masses and cosmological constant; we have discussed the8acuum expectation value in some cases, calculating the generated masses andthe cosmological constant; we have discussed how the cosmological constantproblem arises. The results we have found indicate that the cosmological prob-lem may be solved by a model in which the vacuum expectation value tends tovanish as we move the energy scale from particle physics toward cosmologicalsystems: we have discussed how this is impossible where the Higgs vacuum ex-pectation value is constant as in the Standard Model, possible but might needfine-tunings where the Higgs vacuum expectation value is a parameter as inthe metric conformal Standard Model, possible in a dynamical way where theHiggs vacuum expectation value is a field as in the metric-torsional conformalStandard Model; we also reported that in conformal gravity the possibility toexploit scale transformation may be use to control the value of the cosmologicalconstant, and the cosmological constant problem might have consequences lesssevere than those it has in the non-conformal Standard Model, and the cos-mological constant problem might not even arise [14]. It is then easy to arguethat, if beside the metric also torsion is considered and if gravity receives a con-formal treatment, when discussing the Standard Model, then both advantagespresented here and in [14] can be used simultaneously, in order either to solve oravoid the cosmological constant problem. The discussion presented is thereforeessential to address one of the most important problems in physics.
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