aa r X i v : . [ h e p - t h ] J a n A unitary renormalizable model of composite gravitons
Peter Orland ab ∗ a. Baruch College, The City University of New York, 17 Lexington Avenue, New York, NY 10010, U.S.A. andb. The Graduate School and University Center, The City University of New York, 365 Fifth Avenue, New York, NY 10016, U.S.A. A four-dimensional SU ( ) confining Yang-Mills field, coupled to a fundamental fermion field and a bi-fundamental scalar field, has excitations with spin-2, but no other quantum numbers. These spin-2 excitationscan be light or can condense, depending upon the scalar coupling. If condensation occurs, there is a masslessspin-2 Goldstone boson with (possibly weakly) broken Lorentz invariance in the effective theory. The low-lyingspectrum contains additional spin-0 and spin-1 particles. We discuss how to couple these new fields to othermatter fields. To our knowledge, this is the only explicit proposal for a unitary and perturbatively-renormalizablelocal field theory of gravity. The nonrenormalizibility of Einstein’s theory of gravity means that it must be modified at high energies. It is conceivablethat in nature the modification implies that the graviton is a bound state of more fundamental particles. We introduce a unitaryand renormalizable model, whose spectrum contains two spin-2 excitations. The parameters of this model may be tuned, so thatthe bare mass-squared of at least one of the excitations is small or negative. The consequence is the appearance of either lightmassive gravitons, and/or massless gravitons as Goldstone bosons of spontaneously-broken Lorentz symmetry. The theoremof Weinberg and Witten forbids massless composite spin-2 excitations in a Lorentz-invariant theory with a conserved stress-energy-momentum tensor [1]. Thus, in the spontaneously-broken phase of our model, the spectrum cannot be exactly Lorentzinvariant.Historically, gravity theories with arbitrarily small graviton masses were thought ruled out by experiment [4]. Later it wasshown this is not the case [5]; massive gravitons may be experimentally viable, provided the mass is sufficiently small. Amassive gravity theory should have an action similar to that of de Rham, Gabadadze and Tolley [2], a construction withoutBoulware-Deser ghosts [3]). The viability of a spontaneously-broken phase depends upon the Lorentz symmetry breaking in theeffective theory being sufficiently weak.The study of composite gauge bosons began with the remarkable paper of Bjorken [6], who proposed that photons wereGoldstone bosons of spontaneously-broken Lorentz invariance. Later this idea was extended to massless spin-2 Goldstonebosons [7]. The coupling of any massless spin-2 particles to matter is through the energy-momentum tensor [8]. This leadsnaturally to Einstein’s gravity as an effective theory. Banks and Zaks noticed that the vacuum of a model utilizing Bjorken’smechanism cannot be exactly Lorentz invariant [9], as Bjorken himself later observed [10]. A very general analysis was doneby Kraus and Tomboulis, showing that approximate Lorentz invariance can be present, including deSitter solutions [11]. Theyalso pointed out that the effective Lagrangian for the gravitational Goldstone boson could not have a tadpole term, meaningthat a large cosmological constant is absent. Kostelecky and Potting studied effective gravity theories based on spontaneousbreaking of Lorentz symmetry more generally [12]. Unfortunately, the explicit models presented in References [6, 7, 10, 11]have nonrenormalizable four- or higher-fermion interactions. The spin-1 or spin-2 vacuum condensates are bilinears in thefermion field. One may hope that the long-range behavior of these models does not strongly depend on the cut-off at the criticalcoupling, but this seems unlikely (Kraus and Tomboulis suggested that models with 4-fermi interactions for photons or gravitonscould be effective descriptions of renomalizable gauge theories. To the author’s knowledge, this has not been established, butsee Reference [13] for a large- N , strong-coupling analysis of a lattice gauge model).In the light of the difficulty above, the approach using four-fermion interactions does not improve upon naively quantizedEinstein gravity, which Weinberg speculated to have an “aymptotically-safe” critical point [14]. The lack of perturbative renor-malizability, however, hobbles the analytic predictive power of such a theory, although numerical studies [16] may eventuallybe successful. A serious concern is the appearance of higher-derivative, unitarity-violating, ultraviolet-divergent counterterms(the inducing of nonrenormalizable and higher-derivative terms is not pathological, provided the coefficients of these terms arefinite).Our unitary and renormalizable model is an SU ( ) gauge theory, with fundamental fermions, henceforth called aces , with fourtypes of charges, henceforth called suits (which may be labeled ♣ , ♦ , ♥ , ♠ , and the four corresponding antisuits ♣ , ♦ , ♥ , ♠ ).We denote suit indices by lower case Roman letters, a , b , etc. The degrees of freedom are a traceless hermitian gauge field A ¯ a , b µ , µ = , , ,
3, a fundamental four-component complex ace (Dirac) field ψ a , ψ ¯ a and a bi-fundamental complex scalar field Φ ab , Φ ∗ ¯ a ¯ b . The ace fields have no other quantum numbers. As we discuss below, the beta function of the gauge coupling isnegative (although those of the other couplings are not), and we expect confinement of aces into suit singlets, with a stringtension proportional to the dimensional-transmutation scale.In the confined phase, the lowest lying particle states are meson-like ace-antiace bound states and assemblies of four aces,analogous to baryons, which we call quadyons . We expect that the lightest quadyon is of orbital angular momentum zero. As asuit singlet it must be suit-antisymmetric and therefore spin-symmetric [15]. Thus the lightest quadyon and the correspondingantiquadyon have spin-2. Note that the Weinberg-Witten theorem implies that the spin-2 quadyon must be massive [1], even ifthe constituent aces are massless. By additional (possibly charge-conjugation breaking) interactions with bi-fundamental scalars,the quadyon and antiquadyon mix to produce a spin-2 condensate, with no quantum numbers except angular momentum. Four-fermion condensates in relativistic models were studied in References [17], although the context considered here is different.Gravitons are not quadyons. There are distinct spin-2 quadyons and antiquadyons. Furthermore, if the dimensional-transmutation scale Λ of the SU ( ) gauge theory is large, the quadyons are too heavy to be gravitons; but if Λ is small, thereare many other light excitations which are not observed. We propose a mechanism by which quadyons and antiquadyons mix toproduce light gravitons, with Λ significantly larger than the scales of the Standard Model.Other suit-singlet bound states can consist of one scalar with two aces, two scalars, and a scalar with an antiscalar. These, andmeson-like bound states, ultimately couple to each other and gravitation, but not to other matter.We briefly summarize our conventions. The diagonal Minkowski metric is η µν , with η = , η = η = η = −
1. Theelements of the basis of the Lie algebra su (4) will be denoted t α , α = , . . . ,
15, with normalization Tr t α t β = δ αβ and structurecoefficients f αβρ , defined by [ t α , t β ] = i f αβρ t ρ . The gauge field may be written as A ¯ ab = ∑ α A α t α ¯ ab . We work with the standardDirac matrices γ µ , satisfying [ γ µ , γ ν ] + = η µν , where [ , ] + is the anticommutator. The generators of proper orthochronousLorentz transformations are σ µν = i [ γ µ , γ ν ] / F αµν = ∂ µ A αν − ∂ ν A αµ + G f αβγ A βµ A γν , where, G is the gauge coupling and the summation convention for repeated indices is used. We will drop the bars over antisuit indicesin the remainder of our discussion, for convenience.We propose the Lagrangian for the gravitational sector: L = − F αµν F α µν + ψ a ( i / D ab − m δ ab ) ψ b + [( D µ ) ab ; cd Φ cd ] ∗ ( D µ ) ab ; jk Φ jk + M Φ ∗ ab Φ ab + λ abcd jklm Φ ab Φ cd Φ jk Φ lm + λ abcd jklm ∗ Φ ∗ ab Φ ∗ cd Φ ∗ jk Φ ∗ lm + Y Φ ba ψ a ψ Cb − Y ∗ Φ ∗ ab ψ Ca ψ b , (1)where the covariant derivatives are / D ab = / ∂δ ab − i G γ µ A µ ab = γ µ ∂ µ δ ab − i G γ µ A µ ab , (2)and ( D µ ) ab ; jk Φ jk = ( ∂ µ δ a j δ bk − i G A µ a j δ bk − i G δ a j A µ bk ) Φ jk , (3)the superscript C denotes charge conjugation, defined on the ace field up to a phase e i ϕ C : ψ Ca = C ψ a C − = e i ϕ C C ψ Ta , ψ Ca = C ψ a C − = γ ( ψ Ca ) † , (4)where C γ µ C − = − γ µ T and complex constants Y and Y ∗ are Yukawa couplings and λ abcd jklm and λ abcd jklm ∗ project tensors ofeight suits and eight antisuits, respectively, into singlets: λ abcd jklm = ∑ P ∈ S λ P ε P ( a ) P ( b ) P ( c ) P ( d ) ε P ( j ) P ( j ) P ( l ) P ( m ) , λ abcd jklm ∗ = ∑ P ∈ S λ ∗ P ε P ( a ) P ( b ) P ( c ) P ( d ) ε P ( j ) P ( j ) P ( l ) P ( m ) , (5)where P denotes a permutation in the symmetric group of eight objects S , and λ P is a coupling constant depending on thepermutation P . The number of independent complex values of λ P is 8! / ( ) . The Yukawa terms are gauge and Lorentzinvariant, but violate charge conjugation and parity symmetries, unless Y has vanishing real part, Y = − Y ∗ .In our discussion above of the Lagrangian (1), we have assumed a flat background spacetime with g µν = η µν , but findinggeneralizations with other background metrics is straightforward. In the this paper we consider only the flat-spacetime case indetail.Under a gauge transformation U ∈ SU ( ) , A µ → U − A µ U + i G U − ∂ µ U , ψ → U ψ , ψ → ψ U − , Φ ab → U ac U bd Φ cd , and Φ ∗ ab → Φ ∗ cd U − ca U − db . The reader should be able to see that the quartic couplings in scalars, being suit-singlet operators, areSU ( ) gauge invariant.Even if the ace mass m vanishes, chiral tranformations ψ a → e i αγ ψ a , ψ a → ψ a e i αγ , are not a symmetry of the Lagrangian(1). Not only is this U (1) symmetry broken by Yang-Mills configurations of nontrivial topological index [18], but also by theYukawa terms.The gauge-field part of the Lagrangian (1) is asymptotically free. The one-loop contribution to the beta function is β ( G ) = ∂ G ∂ ln Λ = − π N s G + π N s G + π G + · · · = − π G + · · · , (6)where the successive terms are contributions from gauge bosons, bi-fundamental scalars and aces, respectively, and the numberof suits is N s =
4. As we stated above, we assume that suits are confined with physical masses proportional to Λ . The acecontent of the quadyon field, which is the complex symmetric tensor Q µν ( x ) , with engineering dimension one, is illustrated bythe matrix element h | Q µν ( x ) | p , s , a ; p , s , a , ; p , s , a ; p , s , p i in = Λ − e i x · ∑ j p j ε a a a a × [( γ µ κ ) s s ( γ ν κ ) s s + ( γ ν κ ) s s ( γ µ κ ) s s ] K (cid:16) p j · p k Λ (cid:17) , (7)where p , . . . , p , s , . . . , s and a , . . . , a are the momenta, spinor and suit indices, respectively, of the four aces, κ is a 4 × γ µ κ and σ µν κ are symmetric [19], and K is a dimensionless Lorentz-invariant function of momenta. We em-phasize that (7) only makes sense at momenta greater than Λ ; otherwise the in-state ket must include the effects of confinement, e.g. , virtual partons, nonabelian Wilson lines, etc. For simplicity, we have assumed that the ace masses are small, which is whywe have included the factor of Λ − in (7), although our conclusions do not depend on this assumption.The effective action for quadyons is induced by integrating out aces. This can be seen in the three-loop process shown in Figure1.a and the six-loop process shown in Figure 1.b In the latter, quadyon-antiquadyon mixing occurs. The nonlocal quadyon-acecoupling is given by (7). We do not seriously suggest that these diagrams yield good approximations for these processes, butthey illustrate how the effective quadyon action emerges. A numerical calculation may be possible by lattice methods. If themass of the scalar particles, M is much larger than Λ , we can ignore mixing between quadyons and bound states containingscalars.Processes like those in Figure 1.a and Figure 1.b lead to the terms of the effective Lagrangian for Q µν (suitably normalized), L eff 1 . a = − A Q ∗ µν (cid:3) µναβ Q αβ + B ( Q ∗ µν Q µν − Q αα ∗ Q ββ ) , (8)and L eff 1 . b = − C e − i δ Q µν ∗ (cid:3) µναβ Q αβ ∗ − C e i δ Q µν (cid:3) µναβ Q αβ + D e − i χ ( Q ∗ µν Q µν ∗ − Q αα ∗ Q ββ ∗ )+ D e i χ ( Q µν Q µν − Q αα Q ββ ) , (9)respectively, where A , B , C , D are real and positive, 0 ≤ δ , χ < π and (cid:3) µναβ = ( δ µα δ νβ − η µν η αβ ) ∂ λ ∂ λ − δ { µ { α ∂ ν } ∂ β } + η αβ ∂ µ ∂ ν + η µν ∂ α ∂ β , (10)where the brackets {} denote symmetrization of Lorentz indices. The mass terms, which in (8) are of order B and which in (9)of are of order D , have the only form consistent with the absence of ghosts [3, 4, 20]. Terms of second order in the quadyonfield with more than two derivatives are induced, and the additional derivatives appear as ∂ µ / Λ ; we ignore these for now, butwill return to them later.If Q µν is normalized so that the coefficients A and C are dimensionless, then A and B do not strongly depend on couplingsin involving scalar fields, whereas C and D are proportional to λ | Y | (we are not distinguishing the different scalar couplings λ P , but only considering rough dependence of the parameters in the effective Lagrangian). The effective Lagrangian is L eff = L eff 1 . a + L eff 1 . b + L eff 2 ( { F } ) + V ( Q µν ∗ , Q µν , { F } ) , (11)where L eff 2 ( { F } ) is the Lagrangian for other bound states { F } , of the fundamental fields and the last term contains cubic- andhigher-order parts of the potential as well as interactions with { F } . No ghosts are present in (11), as our original system (1) isrenormalizable and unitary.Diagonalizing the quadratic form of the kinetic terms of (11), rescaling the fields appropriately, then diagonalizing thequadratic form of the mass matrix yields L eff = ∑ ± (cid:20) − H ± µν (cid:3) µναβ H ± αβ + m ± ( H ± µν H ± µν − H ± αα H ± ββ ) (cid:21) + L eff 2 ( { F } ) + V ( H + µν , H µν − , { F } ) , (12)where H ± µν are real symmetric tensor fields with masses m ± = A B + C D cos ( δ − χ ) A − C ± s(cid:20) A B + C D cos ( δ − χ ) A − C (cid:21) − B − D A . (13)The excitations of the fields H µν ± have spin equal to two, but no other quantum numbers.In principle, the phases of the original model (1) are: i) m + > , m − >
0, ii) m + > , m − <
0, iii) m + < , m − > m + < , m − <
0. The Weinberg-Witten theorem forbids m + or m − being exactly equal to zero [1]. Thus any phase boundaryis a hypersurface of first-order transitions, with no critical points. For sufficiently large λ | Y | , we will have B < D , and eitherphase ii) or iii) occurs. For small λ | Y | , phase i) must exist. Only a nonperturbative calculation of A , B , C , D and δ − χ , candetermine whether Phase iv) exists in our model. It is clear, however, that Phase i) and either Phase ii) or Phase iii) (or both) arepossible. In the remainder of this letter, we consider only these possibilities.Suppose that one of the two symmetric tensor fields condenses. Without loss of generality, we assume h H + µν i = h H − µν i 6 = i.e. , Phase ii). The proper orthochronous subgroup SO ( , ) of the Lorentz group is thereby spontaneously brokento the trivial group. A coordinate transformation is making h H − µν i diagonal and constant over spacetime exists.The Goldstone field H − µν has six components, only two of which are the helicity states of the graviton [11, 12]. The potential V does not depend on these components. The effective Lagrangian in H − µν may have terms with more than two derivatives,but no non-derivative terms can be present. By general arguments [21], the long-range form of the effective action of H − µν isthe Einstein-Hilbert action, coupled to dark suit-singlet effective fields ( H + µν , bound states with scalars or suit-antsuit pairs areamong these).We next discuss coupling to other fields. Coupling suitless two-component fermions Ψ A requires the existence of suitedtwo-component fermions Π A ab , coupling through new Yukawa terms in the Lagrangian, L = y Ψ A Φ ∗ ab Π A ab + y ∗ Π A ab Φ ab Ψ A . (14)If the fermi fields included in (14) are coupled to a gauge field B µ AB , there are further terms: L = Ψ A i / D AB Ψ B + Π A ab ( i / D A ab ; B cd − µ A ab ; B cd ) Π B cd (15)where µ A ab ; B cd is the mass matrix of Π A ab , and the differential operators in (15) are / D AB = / ∂δ AB − i g / B AB , and / D A ab ; B cd = / ∂δ AB δ ac δ bd − i g / B AB δ ac δ bd − i G / A ac δ bd − i G δ ac / A bd , (16)where g is the gauge coupling associated with B µ AB .A two-component Higgs field φ A can also be coupled to Φ ab through a quartic term, L = Ωφ ∗ A φ A Φ ∗ ab Φ ab . (17)Thus, elementary quarks, leptons and the Higgs can be coupled to our model of gravity through L = L + L + L + L . Unfortunately, we have no direct renormalizable coupling to the Standard Model gauge fields, in particular the gluon field, whichpossesses most of the mass of visible matter. Furthermore, for the case of spontaneous Lorentz symmetry breaking, we cannotexpect that such a coupling results from the arguments of Reference [8], because Lorentz invariance is not exact in the effectivetheory. To couple gravitons to all the fields of the Standard model may require grand unification, such as embedding the gaugesymmetry SU ( ) × SU ( ) × SU ( ) × U ( ) into a larger gauge group, such as SU(9). Work on this idea is in progress.At sufficiently high temperature, there is a deconfining phase transition. Our argument for spin-2 bound states of aces shouldremain valid in the high-temperature phase, although it is not clear whether light gravitons are present in this phase. This phasetransition should have implications for the early universe.Our model is formulated with a background metric (taken here to be the Minkowski metric). It is conceivable that our modelof gravity could be at least approximately background independent, if our quantum field theory is asymptotically topological inthe infrared. This would mean that after integrating out fast modes of the functional integral with Lagrangian (1), `a la Wilson,the result is a topological quantum field theory. It is unclear to the author whether this is the case. Acknowledgements
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