Frozen up Dilaton and the GUT/Planck Mass Ratio
aa r X i v : . [ g r- q c ] J un Frozen up Dilaton and the GUT/Planck Mass Ratio
Aharon Davidson ∗ and Tomer Ygael † Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel (Dated: June 20, 2017)By treating modulus and phase on equal footing, as prescribed by Dirac, local scale invariancecan consistently accompany any Brans-Dicke ω -theory. We show that in the presence of a soft scalesymmetry breaking term, the classical solution, if it exists, cannot be anything else but generalrelativistic. The dilaton modulus gets frozen up by the Weyl-Proca vector field, thereby constitutinga gravitational quasi-Higgs mechanism. Assigning all grand unified scalars as dilatons, they enjoyWeyl universality, and upon symmetry breaking, the Planck (mass) becomes the sum of all theirindividual (VEV) s. The emerging GUT/Planck (mass) ratio is thus ∼ ωg GUT / π . Critical local Weyl invariance
The Brans-Dicke theory [1] is described by the action I BD = − Z d x √− g (cid:0) φ R + 4 ωg µν φ ; µ φ ; ν (cid:1) . (1)The theory, characterized by a dimensionless parameter ω , the coefficient of the dilaton kinetic term, is invariantunder the combined global scaling transformation g µν ( x ) → e − χ g µν ( x ) , φ ( x ) → e χ φ ( x ) . (2)Consistent with the latter global symmetry is the quarticscalar potential term V ( φ ) = λφ . As is well known, itis only the critical case ω = − which further enjoys thefull local scale symmetry g µν ( x ) → e − χ ( x ) g µν ( x ) , φ ( x ) → e χ ( x ) φ ( x ) . (3)In the ’unitary’ gauge, often called Einstein gauge, de-fined by fixing φ ( x ) = v , for some arbitrary constant v ,the theory resembles general relativity chracterized byan arbitrary Planck mass M P l = 16 πv , and further-more accompanied by a matching cosmological constantΛ E = λv . This by itself, however, does not make Ein-stein theory of gravity a gauge-fixed version of the criticalBran-Dicke theory. Non-critical local Weyl invariance
Recalling the profound success of the standard elec-tro/nuclear theory, a local scale symmetry is most wel-come and currently quite popular [2, 3]. The relativeminus sign between the gravitational and the kineticscalar terms, a characteristic feature of the critical Brans-Dicke theory, appears to be problematic on ghost relatedgrounds. However, as we were guided by Dirac [4], lo-cal scale invariance can be extended to accompany anyBrans-Dicke ω -theory, including in particular the ω > I D = − Z d x √− g (cid:0) φ R ∗ + 4 ωg µν φ ∗ µ φ ∗ ν (cid:1) , (4)instructing us to replace the various tensors involved bytheir (starred) co-tensor substitutes. The procedure re-quires the presence of the Weyl vector field κ µ , subject to the familiar transformation law κ µ ( x ) → κ µ ( x ) − χ ( x ) ; µ . (5)An optional universal coupling constant has been mo-mentarily absorbed within κ µ redefinition (to be justifiedlater on universality grounds).The Ricci scalar replacement R ∗ takes the explicit form R ∗ = R − g µν κ µ ; ν + 6 g µν κ µ κ ν . (6)Under scale transformations it behaves as a co-scalar ofpower -2, that is R ∗ → e χ ( x ) R ∗ . While the dilaton field φ is by construction a co-scalar of power -1, its covariantderivative φ ; µ is not a co-vector at all. It is only theco-covariant Weyl derivative [5] φ ∗ µ = φ ; µ + κ µ φ , (7)which constitutes a co-vector of power -1, thereby mak-ing the kinetic term replacement g µν φ ∗ µ φ ∗ ν a legitimateco-scalar of power -4. The Weyl co-covariant derivativeconceptually differs from the Stueckelberg [6] covariantderivative φ ; µ + mκ µ , but is in full analogy with theMaxwell covariant derivative φ ; µ + ieA µ φ of an electricallycharged (and hence necessarily complex) scalar field. Theimaginary electromagnetic coupling constant ie has beenforcefully traded for a real (currently absorbed as notedearlier) coupling constant.A mandatory ingredient is a kinetic term for the Weylvector field. Truly, it is not directly required on plainlocal scale symmetry grounds, but in its absence κ µ would have stayed non-dynamical in nature. The trans-formation law eq.(5) dictates the exact Maxwell struc-ture, with the corresponding anti-symmetric differential2-form given by X µν = κ µ ; ν − κ ν ; µ . (8)Altogether, up to the total derivative 6( φ κ µ ) ; µ , and afull re-arrangement of the various terms floating around,the non-critical (arbitrary ω ) local Weyl invariant theorycan be described in a somewhat more familiar languageby the action I = − Z d x √− g (cid:2) φ ( R − sg µν κ µ κ ν ) ++ 4 ωg µν D µ φD ν φ + λφ + g µν g λσ X µλ X νσ (cid:3) , (9)where we have used the shorthand notation s = 3 + 2 ω ω = 1 . (10)The latter action eq.(9) looks deceptively conventional,so a word of caution is necessary. Note that D µ φ = φ ; µ + sκ µ φ (11)is in fact a fake co-covariant derivative, and should not beconfused with the genuine co-covariant derivative eq.(7).In the Einstein gauge φ ( x ) = v (with v still beingan arbitrary constant at this stage), the theory resem-bles a particular Einstein-Proca [8] theory accompaniedas before by a cosmological constant Λ E = λv (notethat the corresponding Proca/Planck mass ratio is v -independent). However, this by itself does not makeEinstein-Proca theory of gravity a gauge-fixed versionof the non-critical Bran-Dicke theory. A gravitationalHiggs-like mechanism capable of singling out the ’uni-tary’ Einstein gauge on physical (local scale) symmetrybreaking grounds is in order. In this paper, however, onlya pseudo-Higgs mechanism is offered. A fake scale symmetry?
The notion of a fake symmetry has been coined byJackiw and Pi [9] to address a situation where the con-served Noether/Weyl current vanishes identically. Theirassertion was that in certain cases the correspondingWeyl symmetry does not actually have any dynamicalrole. The critical Brans-Dicke theory, as well as someof its currently proposed derivative models [3], fall intosuch a category. Following the Jackiw-Pi analysis, wenow calculate the Noether/Weyl current stemming fromthe action eq.(9). The result is non-trivial, owing to thepresence of κ µ , thereby implying in our case a genuinelocal Weyl symmetry.First, without using the Euler-Lagrange equations, weperform the variation with respect to the combined sym-metry transformations eqs.(3,5), and find δ L = L µ ; µ √− g , L µ = − φ g µν χ ; ν . (12)Utilizing a previous Jackiw-Pi calculation, the Weyl vec-tor field κ µ and its antisymmetric derivative X µν simplydo not enter at this stage. Next, however, we do invokethe equations of motion, and following the Noether pro-cedure, conventionally use them to eliminate ∂ L ∂φ , ∂ L ∂κ µ ,and ∂ L ∂g µν from the variation. Doing so, we arrive at analternate divergence formula for δ L which can be writtenin the form δ L = ( L µ + J µ ) ; µ √− g . (13)Equating eqs.(12,13), the Weyl/Noether conservation lawmakes its appearance J µ ; µ = 0 . (14) But this time, contrary to the fake symmetry case, theclassically conserved symmetry current J µ does not van-ish. To be specific, it is explicitly given by J µ = 8 ωφ κ µ χ + sX µν χ ; ν , (15)emphasizing the role played by the Weyl gauge field. As-sociated with the Lagrangian eq.(9) is thus a genuinelocal scale symmetry. Frozen up dilaton
A spontaneously scale symmetry breaking mechanismin four (generically in more than two) dimensions is stillat large. The emergence of the Planck mass scale withinthe framework of a theory which does not tolerate the in-troduction of any dimensional parameter at the level ofthe bare Lagrangian is quite challenging [10]. For a re-cent attempt, based on a Coleman-Weinberg like mecha-nism in a framework similar to ours, see Ref.([11]). Withthis in mind, we leave the kinetic part of the Lagrangianabsolutely intact, and thus fully scale symmetric, andsupplement the potential part by a soft scale symmetrybreaking piece. By ’soft’ we mean • Terms whose coefficients have a positive power of mass, • Terms whose transformation law do not involve deriva-tives of the gauge function χ ( x ).In particular, while a scalar field mass term is welcome,a vector field mass term will not do. Following ’tHooft[12], adding a non-conformal part such as a scalar fieldmass term does not have any effect on the dangerouslydivergent term in the effective action. The more so inthe context of this paper, adding a scalar field mass termshould be regarded merely a technical tool primarily de-signed to single out the Einstein gauge. We will showthat the corresponding classical solution, if it exists, can-not be anything else but general relativistic.We thus return to the action eq.(9), and would like totrade the strictly quartic potential λφ for a more generalpotential of the type V ( φ ) = λφ + pφ + q . On peda-gogical grounds, however, to appreciate the fact that ourresults are generic, and are not that sensitive to the exactstructure of the potential, we keep momentarily workingwith a general V ( φ ). Associated with the non-criticalWeyl invariant Lagrangian contaminated by a generalscalar potential V ( φ ) are the following field equations,corresponding to variations with respect to φ, κ µ , g µν , re-spectively:4 ωg µν φ ; µν = V ′ ( φ ) ++ φ R + 4 ωsφg µν ( κ µ κ ν − κ µ ; ν ) , (16a) X νµ ; ν = 4 sω (cid:0) φ µ + 2 κ µ φ (cid:1) , (16b) φ (cid:0) R µν − g µν R (cid:1) == − φ µν + g µν g αβ φ αβ + g µν V ( φ ) −− γ µν + g µν g αβ γ αβ −− X λµ X νλ + g µν X αβ X αβ , (16c)where we have used the notation γ µν = − sφ κ µ κ ν + 4 ωD µ φD ν φ . (17)We can now trace eq.(16c) and subsequently substitutethe Ricci scalar R into eq.(16a). Re-organizing the vari-ous terms, we arrive at a generalized Klein-Gordon equa-tion for φ , namely g µν (cid:0) φ µ + 2 κ µ φ (cid:1) ; ν = ∂W eff ( φ ) ∂φ . (18)The effective potential W eff ( φ ) which governs the φ -evolution, defined by means of ∂W eff ( φ ) ∂φ = 13 + 2 ω (cid:18) φV ′ ( φ ) − V ( φ ) (cid:19) , (19)is known to play a central role [13] in scalar-tensor the-ories. One may verify that, owing to its conformal na-ture, the quartic term λφ in V ( φ ) does not contributeto W eff ( φ ).By no coincidence, the same current which sourceseq.(16b), and whose divergence must therefore vanishidentically (thereby defining a superpotential), is exactlythe current whose divergence we meet again on the l.h.s.of eq.(18). In turn, on self-consistency grounds, a classi-cal solution can exist only provided12 φV ′ ( φ ) − V ( φ ) = 0 (20)We remark in passing that the chain of arguments leadingto eq.(20) breaks down in the absence of the κ µ gaugefield. In which case, the theory resembles Zee’s brokensymmetric theory of gravity [14]. The above equationcan also be written in the form V ′ E ( φ ) = 0, where V E ( φ ) = φ − V ( φ ) (21)stands for the associated Einstein frame scalar poten-tial. The crucial point now is that eq.(20) constitutesan algebraic (rather than differential ) equation for φ . Itssolutions, if exist, are constants rather than functions of x . Other classical solutions, associated with (say) φ ( x ) evolving along the scalar potential (e.g. oscillationsaround the VEV), simply cannot exist. Depending onthe number of solutions, our discussion trifurcates:(1) No solutions - The classical equations of motion areself-inconsistent. Not too many examples of this sortappear in the literature.(2) A single solution - The classical configuration comesthen with a frozen dilaton field φ cl ( x ) = v , (22)and thus, is exclusively general relativistic. The Planckmass being identified as M P l = 16 πv , and the accom-panying cosmological constant is given byΛ = 12 v V ( v ) = 18 v V ′ ( v ) . (23)(3) Multiple solutions - With eq.(20) strictly enforced,one cannot classically tell in this case a minimum froma maximum. In turn, each solution comes with its ownPlanck scale, a situation to be avoided or hopefully re-solved at the quantum mechanical level.Resembling the phrase ’eaten up’ borrowed from Higgsterminology, the dilaton degree of freedom has been to-tally ’frozen up’, thereby converting the massless Weylgauge field into a massive Proca field m κ = − sv + 4 ωs v = 4 ωsv , (24)leaving the physical spectrum scalar particle free. Alto-gether, starting from an explicit (soft) scale symmetrybreaking, we have eventually encountered a gravitationalquasi-Higgs mechanism. We note that both sphericallysymmetric [15] as well as cosmological [16] solutions ofthe Einstein-Proca theory have already been studied inthe literature.Where does the consistency requirement eq.(20) actu-ally come from? The points to notice are that (i) By con-struction, the κ µ field equation does not directly ’know’about V ( φ ). In particular, its associated conserved cur-rent stays independent of V and V ′ . Hence, the κ µ equation still captures the full local scale invariance ofthe kinetic part of the Lagrangian. (ii) Treating eq.(20)as a differential equation for V ( φ ) results in the confor-mal V ( φ ) = λφ . But once a different potential entersthe game, affecting only the φ, g µν -equations of motion,the local scale symmetry identity turns an algebraic con-straint which is solely respected by the Einstein gauge..Clearly, the local structure of V ( φ ) is irrelevant at thislevel. It is only a global feature, namely the discretespectrum of the ∂W eff ( φ ) ∂φ roots, which actually mat-ters. Insisting on soft scale symmetry breaking, we re-strict ourselves to the class of bi-quadratic polynomials.It is practical to parametrize the potential as follows V ( φ ) = λφ + (2Λ − λv )(2 φ − v ) . (25)Such a potential has the further advantage that its effec-tive potential companion (up to a non-physical additiveconstant) W eff ( φ ) = λv − ω ( φ − v ) , (26)admits a single extremum (as a function of φ ). Note thatthe positivity of W is correlated with the negativity ofthe φ mass term added. While classically, as explained,we cannot tell a minimum from a maximum, quantummechanical stability would require Λ < Λ E = λv for aghost free (positive) ω . This includes the special Λ = 0case. CP-violating Weyl-Maxwell mixing
By construction, the local scale invariant non-criticalBrans-Dicke theory can easily accommodate a complexdilaton field. The action eq.(9) is simply traded for I = − R d x √− g (cid:2) φ † φ ( R − sg µν κ µ κ ν ) ++4 ωg µν ( D µ φ ) † ( D ν φ ) + λ ( φ † φ ) + g µν g λσ X µλ X νσ (cid:3) , (27)leaving the door open for the incorporation of Abelianand non-Abelian gauge fields. Using the notation φ ( x ) = ρ ( x ) e iθ ( x ) , (28)with eq.(11) becoming now D µ φ = e iθ ( ρ ; µ + sρκ µ + iρθ ; µ ) . (29)In turn, the action eq.(27) takes now the exact form ofeq.(9), with ρ replacing φ of course, to which the general-ized kinetic term 4 ωρ g µν θ ; µ θ ; ν √− g is added. Note thatthe realization of the Einstein gauge ρ ( x ) = v is achievedwithout restricting θ ( x ) whatsoever, leaving the latter toplay the role of a free massless scalar field in the Ein-stein frame. Still, using the notion of a Goldstone boson,while quite tempting, is unjustified here since no globalsymmetry has actually been spontaneously violated.The subsequent incorporation of a U (1) gauge interac-tion is achieved naturally and flawlessly. However, froman obvious reason (soon to be clarified), it should be em-phasized from the outset that it cannot be electromag-netism we are talking about. Associated with a U (1)-charged dilaton is now the unified co-covariant deriva-tive φ ⋆µ = φ ; µ + κ µ φ + ieA µ φ . By the same token, themodified fake co-covariant derivative splits into D µ ( ρe iθ ) = e iθ [ ρ ; µ + sρκ µ + iρ ( θ ; µ + eA µ )] , (30)with the non-Abelian generalization being straight for-wards (and relevant for a variant class of grand unifiedtheories). A key element in allowing for this construc-tion is the fact that A µ constitutes a co-vector of powerzero. In turn, as was shown by Dirac [4], its co-covariant derivative A µ⋆ν differs from the covariant derivative A µ ; ν only by a symmetric term, namely A µ⋆ν = A µ ; ν + κ µ A ν + κ ν A µ − g µν κ α A α , (31)so that also F µν ≡ A µ⋆ν − A ν⋆µ stays power zero.The Maxwell kinetic term, on the other hand, can beaccompanied by a novel mixed kinetic term14 g µν g λσ X µλ X νσ + 14 g µν g λσ F µλ F νσ + ξ g µν g λσ F µλ X νσ , (32)parametrized by a dimensionless coefficient ξ , which can-not be ruled out solely on local symmetry grounds. Ongroup theoretical grounds, a non-Abelian analogue sim-ply cannot exist. Note that a U (1) ⊗ U (1) kinetic mixinghas already been studied by Holdom [17] as a mecha-nism for shifting electromagnetic charges by a calculableamount.Soft scale symmetry breaking, the advocated mecha-nism for singling out general relativity at the classicallevel, is governed now by the U (1)-invariant scalar po-tential V ( φ ) = λ ( φ † φ ) + (2Λ − λv )(2 φ † φ − v ) . (33)A closer inspection reveals that once the dilaton modulus ρ ( x ) gets frozen up and the Goldstone boson θ ( x ) eatenup, one encounters a diagonal (mass) matrix for the twovector fields involved, namely m κ = 2(3 + 2 ω ) v , m A = 4 ωe v . (34)Notably, unlike the conventional Higgs mechanism, thephysical spectrum does not contain a massive free scalarparticle, to be regarded a fingerprint of the pseudo-Higgsmechanism. The non-diagonal kinetic mixing eq.(32)gives rise to the Holdom effect. The equations of mo-tion involve two conserved currents X νµ ; ν + ξF νµ ; ν = 8 ωs (cid:20)
12 ( φ † φ ) ; µ + κ µ φ † φ (cid:21) , (35) F νµ ; ν + ξX νµ ; ν = 4 ωeg µν h − iφ † ←→∇ µ φ + 2 eA µ φ † φ i . (36)In particular, a residual non-vanishing U (1) source cur-rent, proportional to (1 − ξ ) − ξv κ µ , survives the e → A µ → − A µ , κ µ → + κ µ . (37)The masses m κ,A are proportional to a common VEV,the one which sets the Planck scale M P l = 16 πv in thepresent theory. This is the reason why A µ cannot repre-sent here electromagnetism. From the same reason, thestandard model Higgs doublet cannot serve as a dilaton(like in the Higgs inflation scenario [19]). It is more likelythat a grand unified theory (GUT) is involved. In whichcase, m GUT and M P l share a common origin, and henceacquire the one and the same mass scale. Up to somepotentially large group theoretical factor ∼ − , asso-ciated with the dimensions and multiplicity of the scalarfield representations involved, the typical (mass) ratioshould be m GUT M P l ∝ ω g GUT π (38)where g GUT stands for the coupling constant of the grandunifying group.
Weyl universality and the Planck scale
A grand unified theory generically introduces a vari-ety of scalar fields. And once local scale symmetry joinsthe game, the question is whether such a unified theorycan tolerate the coexistence of several kinds of dilatons φ i , differing from each other not only by their grand uni-fied representation r i but also by their scaling powers.The answer of course is negative, and the reason is quiteobvious. We may have several scalars at our disposal,but just one underlying metric to govern the dynamicsof the spacetime they live in. To be specific, owing tothe identical structure of their kinetic terms (exhibitinga single g µν ), all minimally coupled scalar fields mustconstitute co-scalars of order -1. By a similar token, allmassless fermions involved constitute co-spinors of order-3/2, transforming according to ψ → e χ ψ . Their kineticterms Z d x √− gi ¯ ψγ µ ( x )[ ∂ µ + Γ µ ( x )] ψ , (39)with Γ µ ( x ) denoting the Levi-Civita spin connection,have the further advantage that they are automaticallyconformally invariant. In turn, unlike the universal min-imal coupling of the scalar fields, fermions simply do notcouple to the Weyl gauge vector field κ µ .We now attempt to go one step further and suggesta variant grand unified theory where all scalar fields arein fact dilatons. Following the Dirac prescription, theirindividual local scale symmetric contributions to the ki-netic term in the Lagrangian sum up into − Z d x √− g " R ⋆ X i φ † i φ i + 4 g µν X i ω i φ † i⋆µ φ i⋆ν , (40)where φ i⋆µ = ( ∇ µ + Iκ µ + igT ki A kµ ) φ i . Note that localscale symmetry can tolerate ω i = ω j for i = j . Such anarbitrariness in the Weyl sector remind us, in a remoteway, of a similar arbitrariness which characterizes theYukawa sector. While the latter formula is just a straightforwards generalization of eq.(4), its overall message ispleasing and is by no means conventional: The Planck (mass) which governs the general relativistic couplingof matter to geometry is nothing but the sum over allindividual (VEV) s which have been invoked to give massto the variety of particles in the first place. To be morespecific, M P l = 16 π X i v † i v i (41)This formula is clearly in accord with the GUT/Planckmass ratio eq.(38), and by being sensitive to the underly-ing group theoretical structure (expressed via the sum),can hopefully be used to tell one such grand unified the-ory from the other. A particular SO (10) grand unifiedmodel incorporating the latter idea is currently in themake. Epilogue