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PHYSICAL REVIEW D 97, 126005 (2018)

Gauge assisted quadratic : A framework for UV complete

John F. Donoghue* Department of Physics, University of Massachusetts Amherst, Massachusetts 01003, USA † Gabriel Menezes Department of Physics, University of Massachusetts Amherst, Massachusetts 01003, USA and Departamento de Física, Universidade Federal Rural do Rio de Janeiro, 23897-000, Serop´edica, RJ, Brazil

(Received 23 April 2018; published 12 June 2018)

We discuss a variation of quadratic gravity in which the gravitational interaction remains weakly coupled at all energies, but is assisted by a Yang-Mills gauge theory which becomes strong at the Planck scale. The Yang-Mills interaction is used to induce the usual Einstein-Hilbert term, which was taken to be small or absent in the original action. We study the spin-two propagator in detail, with a focus on the high mass resonance which is shifted off the real axis by the coupling to real decay channels. We calculate scattering in the J ¼ 2 partial wave and show explicitly that unitarity is satisfied. The theory will in general have a large and we study possible solutions to this, including a unimodular version of the theory. Overall, the theory satisfies our present tests for being a ultraviolet completion of quantum gravity.

DOI: 10.1103/PhysRevD.97.126005

I. INTRODUCTION matter fields coupled to the metric yield divergences propor- tional to the second power of the . Therefore, the There are many exotic approaches to quantum gravity, fundamental action must have terms in it which are quadratic but comparatively little present work exploring the option in the curvatures in order to renormalize the theory. This also of describing gravity by a renormalizable quantum field explains why such QFT treatments are often overlooked. theory (QFT). Nature has shown that the other funda- Curvatures involve second derivatives of the metric, so that mental interactions, i.e. those of the standard model, are quadratic gravity involves metric propagators which are described by renormalizable QFTs at present energies, so quartic in the momentum. Quartic propagators are generally this should be the most conservative approach. There is a considered problematic, for reasons which will be reviewed modest body of recent work [1–13] attempting to revive 1 below. However, quadratic gravity does have the positive this possibility. As a subfield, this literature is somewhat feature that it is renormalizable [14], and can be asymptoti- diffuse, with different groups pursuing different, although cally free [15–17]. Moreover, to recover in related, variants. However, the key question is to deter- the low energy limit one must arrange to have the usual mine if any renormalizable QFT can serve as a UV quadratic propagators at low energy. The challenge for such completion for gravity. In this paper, we explore such a QFT treatments then is to deal with fundamental quartic variation which we feel is particularly well suited for propagators at high energy while recovering usual gravity at being a controlled approach using present techniques, low energy. with encouraging results. The variation which we explore will involve a Yang-Mill The distinctive feature of a renormalizable QFT treat- gauge theory plus weakly coupled quadratic gravity. ment of gravity is simple to diagnose. Loops involving For the purposes of this introduction one can consider this to be defined by the action (in units of ℏ ¼ c ¼ kB ¼ 1, *[email protected] † which will be consistently employed throughout the paper) [email protected] Z   1For the much larger body of older research please see the ffiffiffiffiffiffi 1 1 4 p μα νβ a a μναβ references within [1–13]. S ¼ d x −g − g g FμνF − CμναβC 4g2 αβ 2ξ2 Published by the American Physical Society under the terms of ð1Þ the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, although we also discuss other variants below. The Weyl and DOI. Funded by SCOAP3. tensor is given by

2470-0010=2018=97(12)=126005(18) 126005-1 Published by the American Physical Society JOHN F. DONOGHUE and GABRIEL MENEZES PHYS. REV. D 97, 126005 (2018)

1 ð2Þ − − − iDμναβ ¼ iPμναβD2ðqÞ Cμναβ ¼ Rμναβ 2 ðRμαgνβ Rναgμβ Rμβgμα þ RνβgμαÞ   q2 iϵ q4 q4N −q2 − iϵ RðgÞ −1 þ − − eff − D2 ðqÞ¼ 2 2 2 ln 2 þ 6 ðgμαgνβ gναgμβÞ: ð2Þ κ˜ ðqÞ 2ξ ðμÞ 640π μ 4   q N ðq2Þ2 a − q : The quantity Fμν is the usual field strength tensor for the 1280π2 ln μ4 ð5Þ Yang-Mills theory. The index a is summed over the generators of the gauge group G. One has that ð2Þ Here, Pμναβ is the spin-two projector to be described below. The function 1=κ˜2ðqÞ is induced by the gauge theory and a ∂ a − ∂ a abc b c Fμν ¼ μAν νAμ þ gf AμAν ð3Þ can be described by techniques related to QCD sum rules. It will have the limits where fabc are the structure constants of G and g is the Yang-Mills coupling constant. 1 1 ξ → q ≪ MP Both couplings g and are asymptotically free, so this is κ˜2ðqÞ κ2 a renormalizable asymptotically free theory at high energy. → 0 ≫ We consider the limit where the gauge coupling g is larger q MP ð6Þ than the gravitational coupling ξ. This means that the gauge κ2 32π theory becomes strongly interacting at a higher energy. where ¼ G is the coupling in the Einstein-Hilbert The energy scale of the gauge theory will be taken to be the action. The coefficient of the first logarithm, Neff,isa Planck scale. Indeed, this is the role of the helper gauge number that depends on the number of light degrees of theory in this construction—it defines the Planck scale, and freedom (d.o.f.) with the usual couplings to gravity and Nq also induces the Einstein-Hilbert action at lower energies is a number due to coupled through the quadratic- [18–22]. We have elsewhere [4] calculated the induced action. These numbers will be defined in detail gravitational constant due to QCD, below but for now we can note that at very high energies Neff ¼ N∞ ¼ D þ NSM, where D is the number of gen- 1 erators in the gauge theory and N is due to the particles 0 0095 0 0030 2 SM 16π ¼ . . GeV ð4Þ 3 199 3 Gind of the standard model and beyond while Nq ¼ = . This propagator is described by three regions. At low 1019 2 ≪ ξ2 2 4 and so a QCD-like theory would need an energy scale energy q MP, where Mp is the Planck mass, the times greater to generate the observed Planck scale. quadratic propagator dominates, and one gets the usual 2 2 Left on its own, the Weyl-squared term would also coupling of general relativity. At high energy q >MP, one become strong at some energy. However, because we take it has a purely quartic propagator and the running gravita- to be still weakly coupled at the Planck scale, it would tional coupling is become strong only at a very low energy scale. (For ξ 0 1 ξ2 μ 320π2 example, for a pure Weyl theory if ¼ . at the Planck ξ2 ð Þ Λ 10−1006 ðqÞ¼ ξ2 μ ¼ 2 2 scale, it would become strong at ξ ¼ eV.) In ð ÞðN∞þNqÞ 2 2 ðN∞ þ N Þlnðq =Λ Þ 1 þ 2 lnðq =μ Þ q ξ practice, this running of ξ is interrupted by the induced 320π gravitational effects due to the gauge field, and hence is ð7Þ never relevant at low energy2. Indeed we will see that the gravitational interaction can remain weakly coupled at all where as usual we have defined the scale factor Λξ via 2 2 2 2 energies. This helps give us control over the predictions of 1=ξ ðμÞ¼ðN∞ þ NqÞðln μ =ΛξÞ=320π . In the inter- ξ2 2 ≤ 2 2 the theory. Our goal is to explore the structure of this theory mediate regime MP q

126005-2 GAUGE ASSISTED QUADRATIC GRAVITY: A FRAMEWORK … PHYS. REV. D 97, 126005 (2018)

10 2.0

8 1.5

6

1.0 4

0.5 2

0 0.0 -6 -5 -4 10 10 10 0.001 0.010 0.100 1 10-6 10-5 10-4 0.001 0.010 0.100 1

FIG. 1. The absolute value of the spin-two propagator for FIG. 3. The Euclidean spin-two propagator corresponding to ξ ¼ 0.1, normalized to the standard propagator of general the same conditions as shown in Fig. 1, again normalized to the relativity. The x-axis is the momentum jqj in the time-like region, standard propagator of general relativity. in units of the Planck mass. The imaginary parts have been calculated with loops of standard model particles and gravitons. unitary at all energies. This occurs despite the sign of the propagator near the pole being the opposite from normal Δ 2 ∼ ξ2 2 320π coupling the width is roughly q Neff mr = .Itis expectation. Careful readers should pay attention to this fair to think of this as an unstable spin-two resonance, and point below. we will explore this interpretation more fully. There are some approximations which have gone into We have used this propagator in the calculation of a the form of the propagator. One is that we have included the physical process which goes through the resonance region, logarithmic behavior from loops but not any residual which is the scattering in the spin-two s-channel partial nonlogarithmic constants. These in practice can be wave. The cross-section is shown in Fig. 2. The usual absorbed in a redefinition of 1=ξ2, but we have not growth of the amplitude, which normally would become explicitly calculated these terms. More importantly, we strong at the Planck scale, is tamed by the quartic part of the 2 have treated each region using only the dominant action propagator beyond s ¼ mr, and remains weakly coupled. appropriate for that region. In particular this means that We will explicitly confirm that the scattering amplitude is when we are displaying results from the highest energies, we use the quadratic curvature action only. This feature is related to the lack of an imaginary part in the ln q4 terms. We also note that the propagator in the spacelike region, 10 -5 and equally the Euclidean propagator, is featureless and well behaved. In particular the Euclidean propagator involves 10 -8    2 4 4 2 −1 qE qE qEðNeff þ NqÞ qE D2 ðq Þ¼ þ þ ln E E κ˜2ðq Þ 2ξ2ðμÞ 640π2 μ2  E   -11 2 4 2 −1 10 qE qEðNeff þ NqÞ qE ¼ κ˜2 þ 640π2 ln Λ2 ð8Þ ðqEÞ ξ

-14 2 0 10 where qE > . It is shown in Fig. 3, again normalized to the usual quadratic propagator. Note that without the helper gauge interaction inducing the usual gravitational coupling at low energy, the quartic terms would have blown up 10 -6 10 -5 10 -4 0.001 0.010 0.100 1 2 Λ2 at qE ¼ ξ but this point is innocuous in the present FIG. 2. The absolute value of the unitary J ¼ 2 partial wave framework. amplitude, calculated with ξ ¼ 0.1. The x-axis is the center of With this as introduction, we turn to the more detailed mass energy in units of the Planck mass. investigation of the theory. In Sec. II we describe some of

126005-3 JOHN F. DONOGHUE and GABRIEL MENEZES PHYS. REV. D 97, 126005 (2018) the issues of quadratic gravity in general, including some and perhaps tachyons if μ2 is negative. When considering aspects of describing the d.o.f. In Sec. III we turn to the quadratic curvature gravity, the decomposition of the d.o.f. spin-two propagator and explore some of the results varies depending on the gauge condition imposed and also mentioned above. Section IV describes a test of unitarity on the choice of field parametrization, and there are in scattering in the spin-two channel. Section V gives a generally gauge-variant unphysical states [23]. However, discussion of a broader class of theories, including the it is generally agreed that the free field propagating modes cosmological constant and also the spin-zero sector. In are a massive scalar and its massless scalar ghost, and a Sec. VI, we make some comments on the unimodular massless spin-two and its massive spin-two ghost. version of this theory, which also may be useful to explore The scalar massive mode arises uniquely from the R2 term 2 more fully. Finally Sec. VII gives a summary and dis- and its mass is proportional to f0. The massless scalar ghost cussion. In the Appendix, we calculate the magnitude of the is interesting because it is the ghost found in most 2 R term in the action which is also induced by the Yang- quantization schemes of pure Einstein gravity [24]. With Mills interaction. some work, it can be shown to be harmless. The massive spin-two ghost arises uniquely from the C2 term, and its II. QUADRATIC GRAVITY mass is proportional to ξ2. In order to see what ultimately is a problem with ghost In addition to the Einstein-Hilbert action Z states, one can draw on the work on Lee-Wick models 2 ffiffiffiffiffiffi [25–30]. In these theories, states with negative norms are 4 p− SEH ¼ κ2 d x gR ð9Þ introduced much like the Pauli-Villars regulators, which combine with regular fields to produce a q−4 fall-off of there in general can be three combinations at quadratic the combined propagators. While one might worry about order in the curvature which are generally covariant unitarity and negative energies with the ghost states, it is Z   not these features which are problematic. Due to the pffiffiffiffiffiffi 1 1 4 − 2 − μναβ − η interactions of the theory, the ghost states are unstable Squad ¼ d x g 6 2 R 2ξ2 CμναβC G ð10Þ f0 and do not appear in the asymptotic spectrum5. The theories where can be shown to be unitary. However, what does occur is microscopic violations of causality due to the ghost states. μναβ μν 2 G ¼ RμναβR − 4RμνR þ R ð11Þ While there are not gross large scale violations of causality, because the ghost states only propagate for a short time, the is the Gauss-Bonnet invariant. This latter term is a total causal properties are uncertain on small scales by amounts derivative in four dimensions, and so it cannot influence the of the order of the ghost width. This has been explicitly classical equations of motion nor graviton propagation. demonstrated in Refs. [29,30]. For example, by forming □ One can also introduce a surface term R in the above initial state localized wave packets, the final arrival times action. The counterterm associated with it in the calculation will have portions of the wave packet arriving earlier than of the one-loop effective action is gauge-dependent. We usual expectations by amounts of order of the width. will drop the surface term as well as the Gauss-Bonnet In our calculation, a similar effect in the propagator contribution in the rest of this paper. In any case, we remark holds. Interactions shift the massive spin-two effect in the that topological and surface terms should be included in propagator away from the real axis, and it appears as a order to provide renormalizability. We also note that resonance rather than an asymptotic state. The sign of the    propagator near the pole is ghostlike. However, the result- 1 μναβ 1 μν 1 2 1 − CμναβC − RμνR − R G 2ξ2 ¼ ξ2 3 þ 2 ing amplitude near the pole will be explicitly shown to be unitary. We should also expect microscopic violations of ð12Þ causality as in the Lee-Wick models. To have this happen near the Planck scale in gravity does not seem outrageous. Our conventions are: the Minkowski metric is given as η 1 −1 −1 −1 We would expect that the causal properties would become μν ¼ diagð ; ; ; Þ and the Riemann curvature ten- fuzzy in any quantum theory of gravity because the λ ∂ Γλ Γη Γλ − ν ↔ κ sor is given by R μνκ ¼ κ μν þ μν κη ð Þ. quantum fluctuations of whatever defines would In theories with fundamental curvature-squared terms, lead to uncertainties in the causal structure. In our case the graviton propagator will be quartic in the momentum. these effects would be proportional to the , This is generally considered to be problematic. With a although somewhat larger by an amount 1=ξ4 as the quartic propagator in free field theory one expects negative norm ghost states, using for example   − − 1 5For other recent discussions regarding the emergence of real i ∼ i i − i 4 2 2 2 ¼ 2 2 2 2 ð13Þ and complex ghosts in higher-derivative quantum gravity, see q q ðq − μ Þ μ q q − μ Refs. [31,32].

126005-4 GAUGE ASSISTED QUADRATIC GRAVITY: A FRAMEWORK … PHYS. REV. D 97, 126005 (2018)

μν resonance width is narrow. We are willing to accept this second order in hμν, and G is the differential operator feature as a property of our theory. associated with the gravitational gauge-fixing term. The other complaint about theories with extra derivatives The one-loop divergences for the effective action evalu- in the fundamental Lagrangian is that they lead to a ated from this theory have been calculated elsewhere, see classical Hamiltonian which is not bounded from below, for instance Refs. [15–17,23,41–43].Itisgivenby via the Ostrogradsky construction [33]. It is not clear how Z μd−4 α α ffiffiffiffiffiffi relevant this classical result is. The path integral treatment ð 1 þ 2Þ d p ¯ ¯ μναβ Γ ¼ − d x −g¯CμναβC : ð15Þ based on the Lagrangian does not find the effect of such an div d − 4 instability. The Lee-Wick theories also seem not to be ¯ bothered by it. Our calculations do not show such an effect The Cμναβ is built from the background metric either. One notes that the classical Dirac Hamiltonian is g¯μν. We will conveniently specialize this result to a flat also not bounded from below. The Dirac case is rendered background in due course. In addition, in the above stable by a modification of the quantization rules. There are equation, α1 is the contribution coming from graviton at least three variations on modified quantization rules loops, whereas α2 is the contribution coming from the which lead to a well-behaved quantum Hamiltonian in the non-Abelian gauge field. Specifically: case of quartic derivative theories [34–36]. We do not take a position on these alternative rules. However, gauge theories 199 Nq α1 ¼ ¼ and gravity are most simply quantized by the path integral 480π2 160π2 formalism using the action, and we adopt these techniques. α D They seem to lead to a well-behaved quantum theory. 2 ¼ 160π2 : ð16Þ Finally, theories with quartic propagators at high energy violate the Källen-Lehmann bound on the asymptotic In order to include standard model particles, as well as behavior of propagators [37,38]. However, this bound does interactions beyond the standard model, one would have to not apply for gauge fields—indeed it is violated in QCD consider additional shifts in the α2 coefficient due to one- [39,40]. Moreover, the microscopic violations of causality loop divergencies associated with such couplings. One also violated the assumptions for the theorem. For both finds reasons, this bound is not relevant for the present theory. Let us briefly discuss the quantization of the theory given 0 D þ NSM N∞ α2 → α ¼ ¼ ; ð17Þ by the action (1) within the path integral treatment. Using 2 160π2 160π2 the background-field method, one considers the paramet- h λ at very high energies. rization gμν ¼ g¯μλðe Þ ν, where g¯μν is a smooth background metric. One should add the necessary gauge-fixing terms concerning the Weyl part, SGF½h, and for the Yang-Mills III. THE SPIN-TWO PROPAGATOR a term, S ½Aμ;h. Moreover, one should also take into YM;GF Here we focus on the study of the spin-2 propagator. As account the associated Faddeev-Popov ghost contributions discussed above, the free field propagating modes emerg- η η ¯ a SFP½ ; ;h (for the Weyl part) and Sgh½c; c; Aμ;h (for the ing in the model depend on the choice of gauge parameters η Yang-Mills part). In such contributions, is the ghost field as well as on the choice of field parametrization. However, related to the Weyl term and c is the ghost field associated the spin-2 part of the propagator of the hμν field maintains with the Yang-Mills term. For the Weyl contribution, one the same form irrespective of gauge parameters and field can consider the construction given in Refs. [23,41]. On the parametrizations [23]. The most interesting feature of this other hand, the Yang-Mills part can be obtained from the exploration is the finite-width resonance in the propagator. usual Minkowski counterpart by the usage of general This emerges from a competition of the q2 and q4 terms in covariance and then one expands the metric as above. In the propagator. It appears with an opposite sign from usual this way, in the background-field method the partition expectation, and the imaginary part also changes sign from function for our theory is given by expectation, in a way that is consistent with the optical Z theorem. It is the remnant of a would-be ghost state, Z½g¯¼ðdet GμνÞ1=2 DhDηDηDADc¯Dc although because it is unstable it does not appear as a physical state in the asymptotic spectrum. Also of interest is η η × expfifSW½hþSGF½hþSFP½ ; ;h the existence of three energy regions, in each of which the a a a propagator has a distinct structure and/or different active þ S ½Aμ;hþS ½Aμ;hþS ½c; c;¯ Aμ;hgg; YM YM;GF gh d.o.f. In the highest energy region, the quadratic curvature ð14Þ terms are dominant, and this is reflected in the structure of the propagator. a where SW½h (SYM½Aμ;h) is obtained from (1) by solely In this section we set the cosmological constant to zero, considering the Weyl (Yang-Mills) term expanded up to and return to its influence in Sec. V. This allows us to

126005-5 JOHN F. DONOGHUE and GABRIEL MENEZES PHYS. REV. D 97, 126005 (2018)

2 expand around flat space. In this paper we will use the Πμν;αβðq Þ   exponential parameterization of the fluctuation, gμν ¼ ð0Þ 2 λ 1 λ − 1 1 η h η Neff q 4 μλðe Þ ν ¼ μν þ hμν þ 2 hμλh ν þ. One obtains the − q q q q q Iμναβ ¼ 320π2 ln μ2 3 μ ν α β þ 2 following free propagator (in momentum space) 1 2 1 4 2 ξ2 þ q ðqμqνηαβ þ qαqβημνÞ − q ημνηαβ 0 i ð2Þ 6 6 iDμναβðqÞ¼− Pμναβ: ð18Þ  ðq2Þ2 1 − 2 η η η η 4 q ðqμqβ να þ qμqα νβ þ qνqα μβ þ qνqβ μαÞ ; We will give an explicit expression for the spin-2 projector ð2Þ Pμναβ below. ð20Þ First let us work at low energies and include loop corrections. We treat all fields as massless—the small where Iμναβ is given by masses of standard model fields make no difference in 1 the energy region of interest to us. In this case, the one-loop Iμναβ ¼ ðημαηνβ þ ημβηναÞ: ð21Þ vacuum polarization can be written schematically as 2 (temporarily suppressing the Lorentz indices), for d → 4 ð0Þ   Here Neff is defined by the contributions of various matter μ 4−d Π 2 4 c1 fields to the one-loop divergence (within the standard ðqEÞ¼qE − 4 ; qE d model and beyond), normalized to that for gauge bosons. For the fields with spin J ≤ 1, this is where we considered dimensional regularization for regu- larizing the integrals and this result was obtained after a ð0Þ 1 1 N ¼ N þ N1 2 þ N ð22Þ Wick rotation to the Euclidean space (qE is the Euclidean eff V 4 = 6 S momentum). The scale factor μ, with dimensions of mass, is inserted for dimensional reasons, and c1 is a constant where NV, N1=2, NS are the number of gauge bosons, that does not contain poles as d → 4. By rearranging the fermions and scalars respectively. The contribution coming expression in terms of an exponential of a logarithm, one from graviton loops will be discussed in due course. The easily sees that, as d → 4, the one-loop vacuum polariza- one-loop spin-2 propagator is then given by tion will consist of a divergent part plus a finite part. In 2 0 2 0 2 addition, the knowledge of the form of the divergence term iDμναβðq Þ¼iDμναβðq ÞþiDμνρτðq Þ gives us straightforwardly the logarithmic finite part, since Πρτ;γδ 2 0 2 they share the same coefficient.6 × ½i ðq ÞiDγδαβðq Þ: ð23Þ All such discussions imply that the results presented at 2 the end of the previous section allows us to determine Now we rewrite Πμν;αβðq Þ by introducing the set of the explicitly the finite part of the one-loop vacuum polariza- following projectors for symmetric second-rank tensors in tion. By employing the following expansions of the differ- momentum space [23] ent invariants up to second order in h (in Minkowski 1 1 background) Pð2Þ θ θ θ θ − θ θ μνρσ ¼ 2 ð μρ νσ þ μσ νρÞ 3 μν ρσ 2 μν αβ μν R ∂μ∂νh ∂α∂βh − 2□h∂μ∂νh □h□h 1 ¼ þ Pð1Þ θ ω ω θ θ ω ω θ μνρσ ¼ 2 ð μρ νσ þ μρ νσ þ μσ νρ þ μσ νρÞ μν 1 1 μν 1 μν αβ RμνR ¼ □h□h þ □hμν□h þ ∂μ∂νh ∂α∂βh 4 4 2 ð0Þ 1 Pμνρσ ¼ θμνθρσ 1 1 3 − □ ∂ ∂ μν − ∂ ∂ μα∂β∂ν h μ νh μ νh hβα ¯ ð0Þ 2 2 Pμνρσ ¼ ωμνωρσ ð24Þ μναβ μν μα β ν RμναβR ¼ □hμν□h − 2∂μ∂νh ∂ ∂ hβα μν αβ where þ ∂μ∂νh ∂α∂βh ; ð19Þ qμqν θμν ¼ ημν − where some integrations by parts were carried out, one q2 obtains the following one-loop contribution to the vacuum qμqν ωμν ¼ : ð25Þ polarization q2

6We keep only the logarithmic terms. Additional finite terms Such projectors do not form a complete basis in the can be absorbed into the finite parts of the coupling constants and corresponding space, and hence we must also add the do not change our analysis. following transfer operators

126005-6 GAUGE ASSISTED QUADRATIC GRAVITY: A FRAMEWORK … PHYS. REV. D 97, 126005 (2018)   1 2 ϵ 4 − 2 − ϵ ð0Þ −1 2 q þ i q Neff 4 q i T μνρσ ¼ pffiffiffi θμνωρσ D ðq Þ¼ − − q ln 3 κ˜2 2ξ2ðμÞ 640π2 μ2 1 ¯ ð0Þ T μνρσ ¼ pffiffiffi ωμνθρσ ð26Þ ð32Þ 3 and κ˜2 ¼ κ2 at the low energies which we are working in order to obtain a complete basis. In terms of such presently. The pole at q2 ¼ 0 carries the usual iϵ pre- projectors the one-loop vacuum polarization can be scription because it was induced via the Yang-Mills written as interaction. Here Neff needs to include the graviton contribution, calculated within effective field theory   ð0Þ 2 N −q − iϵ 2 [46,47], such that Π 2 − eff 4 Pð Þ μν;αβðq Þ¼ 2 q ln 2 μναβ: ð27Þ 640π μ 1 1 21 Neff ¼ NV þ 4 N1=2 þ 6 NS þ 6 The imaginary part of the vacuum polarization is found via 21 the usual prescription ¼ 6 þ NSM þ NBSM ð33Þ − 2 − ϵ 2 − πθ 2 lnð q i Þ¼lnðjq jÞ i ðq Þ: ð28Þ with NSM being the contribution from standard model particles and NBSM that of new physics beyond the Using the usual orthogonality relations satisfied by the standard model but below the scale of gravity. We neglect above projectors, one can iterate the vacuum polarization the latter in what follows. In the standard model with one and find an intermediate form for the spin-2 propagator Higgs doublet and three generations of fermions, one finds 2 45 12 that NS ¼ , N1=2 ¼ ,andNV ¼ ,sothatNSM ¼ 283 12 Pð2Þ = . 2 − μναβ The effect of graviton loops is somewhat different at high Dμναβðq Þ¼ h 0  i : ð29Þ ð Þ − 2− ϵ 2 2 1 Neff q i energies. Here the curvature-squared terms in the action are ðq Þ 2 þ 2 ln 2 2ξ ðμÞ 640π μ dominant, and the graviton propagator is quartic in the momentum. We treat this region by considering only the The factor in square brackets defines the running coupling curvature-squared effect—the induced Einstein term is 2 constant ξðq Þ at these energy scales (that is, below the subdominant and is neglected. Here again the logarithmic resonance). That is, we define effects are tied to one-loop divergences. However, the difference comes in that there is no imaginary part induced ξ2 μ 320π2 by these loops. This arises because the matrix element ξ2 ð Þ   ðqÞ¼ 0   ¼ : ð30Þ ξ2 μ ð Þ 2 ð0Þ jq2j for the production of on-shell gravitons vanishes. This is ð ÞNeff jq j 1 þ 2 ln 2 N ln Λ2 320π μ eff ξ known quite generally, in that the Weyl action produces no on-shell scattering amplitudes in flat space [48–52]. In addition, there are the effects of gravitational loops. At In our situation, this can be seen directly. The vertex involved is a triple graviton coupling between low energies these are treated using the effective field 2 theory for gravity. As briefly discussed in the Introduction an off-shell graviton with invariant mass q and two on-shell section, the Einstein-Hilbert gravitational action will be gravitons. The latter will be massless, transverse and trace- induced at low energies by the helper gauge interaction less since the spectrum is defined by the induced Einstein [18–22,44,45]. In other words, an induced Einstein-Hilbert action. Let us show that this triple graviton vertex vanishes. 2 term is present in the effective action due to the scale- Consider the triple gravitonvertex arising from the R term in 2 1 μν invariance breaking generated by loop effects. We have R − 3 RμνR . The curvature can be expanded around flat calculated this effect for QCD-like theories in a previous space in powers of the number of gravitational fields involved paper, finding a positive value for κ2 [4]. In addition to leading to a q2 term in the propagator, the effects of loops of R ¼ Rð1Þ þ Rð2Þ þ Rð3Þ þ ð34Þ gravitons also provide logarithmic factors in the propaga- tor. The loops carry the usual iϵ convention. Taking into such that we can pull out the triple graviton term as account such modifications, one obtains that ffiffiffiffiffiffi 1 p 2 λ ð1Þ ð1Þ ð1Þ ð2Þ −gR ¼ … þ hλR R þ 2R R þ ð35Þ ˜ 2 ð2Þ 2 2 Dμναβðq Þ¼PμναβDðq Þ; ð31Þ The on-shell condition corresponding to transverse, traceless 2 where fields at q ¼ 0 is that Rμν ¼ 0. This holds order by order

126005-7 JOHN F. DONOGHUE and GABRIEL MENEZES PHYS. REV. D 97, 126005 (2018) in the expansion in the number of graviton fields. The off- D−1ðqÞ shell graviton will be taken from one of the terms in Eq. (35).    m2 1 N m2 However, no matter how this field is chosen, there always ¼ r − m4 þ eff ln r κ2 r 2ξ2ðμÞ 640π2 μ2 remains another curvature which satisfies RðiÞ ¼ 0 leading       to the vanishing of the vertex. This can be verified by direct 1 1 N m2 1 þ δq2 − 2m2 þ eff ln r þ computation. This argument also easily generalizes the κ2 r 2ξ2ðμÞ 640π2 μ2 2 μν RμνR . π 4 Neff The vanishing of this vertex implies that the logarithms þ i mr 640π2 do not pick up an imaginary part. This can be accom- 1 2 2 4 2 2 ð39Þ plished by using 2 ln½ðq Þ =μ instead of lnð−q − iϵÞ=μ . This form can also be achieved by regularizing the propagator via Using   1 2 1 1 Neff mr D ∼ : ð36Þ 2ξ2 μ þ 640π2 ln μ2 ¼ 2ξ2 ð40Þ q4 þ ϵ2 ð Þ ðmrÞ the real part of the pole location is then determined by the This then implies that the final propagator involves condition two logarithmic factors and takes the form quoted previously 2 2 2ξ ðmrÞ mr ¼ κ2 ð41Þ ð2Þ iDμναβ ¼ iPμναβD2ðqÞ   2 ϵ 4 4 − 2 − ϵ and we have −1 q þ i − q − q Neff q i D2 ðqÞ¼ 2 2 2 ln 2   κ˜ ðqÞ 2ξ ðμÞ 640π μ δ 2 ξ2 2ξ2 2   −1 q Neff ðmrÞ mr Neff 4 2 2 D ðqÞ¼− 1 þ þ i : ð42Þ q N ðq Þ κ2 320π2 κ2 640π − q : 1280π2 ln μ4 ð37Þ If we define the positive number γ by In the low energy region below the resonance, where the 0 2 2 Neff Einstein action dominates, we have Nq ¼ ,andNeff is γ 2ξ   ¼ mr ξ2 ð43Þ 640π 1 Neff ðmrÞ givenbyEq.(33). Above the resonance but below the þ 320π2 Planck scale, the Weyl action is dominant and we have 199 3 Nq ¼ = and Neff ¼ NSM. Finally above the Planck and pull out an overall normalization (which can be scale the gauge bosons of the helper gauge theory absorbed in the normalization of the field), we have the 199 3 become active and we have Nq ¼ = and Neff ¼ propagator near the pole involving D þ NSM ¼ N∞,whereD is the number of gauge bosons.   Now let us study the poles of the propagator given by κ2 1 DðqÞ¼ 2 2 Eq. (31). We will show that the ghost state contained in the Neff ξ ðmrÞ −δ γ 1 þ 2 q þ i free propagator becomes unstable due to loop corrections.  320π  Clearly the spin-2 propagator (31) has the standard pole at κ2 −1 2 ¼ ξ2 2 : ð44Þ q ¼ 0, which means that the graviton remains massless at Neff ðmrÞ δ − γ 1 þ 2 q i 2 − 2 320π one-loop order. Note also that, for Euclidean qE ¼ q (or −1 2 spacelike q), D ðqEÞ has no other real zeros apart from This behavior is different from the usual structure for a 2 0 2 Λ2 2 0 −1 qE ¼ for qE > ξ. In turn, for timelike q, q > and D propagator of an unstable particle. In the standard case we picks up an imaginary part. Because we are in the weakly use coupled region ξ2 ≪ 1, the pole will occur below the Planck scale. 1 DðqÞ¼ 2 Γ 2 ð45Þ Consider the propagator near the pole. If we expand q − ðM − i 2Þ using and we would have near the pole, m2 ¼ M2 − Γ2=4, the 2 2 2 q ¼ mr þ δq ð38Þ behavior 1 and note that in this energy region κ˜ ¼ κ, we get the DðqÞ¼ 2 : ð46Þ expansion of the inverse propagator δq þ iMΓ

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We note that, aside from the overall normalization, the Below this energy scale the theory is satisfactorily residue right on the pole is exactly the same as usual, with described by the effective field theory approach. the correspondence In this analysis we have made an approximation of using only the dominant gravitational action in each region of 1 1 interest. Specifically, at high energy where the Weyl action ¼ : ð47Þ iγ iMΓ dominates, we have used only the quartic propagators and dropped the effect of the subdominant q2 terms. In addition, However when the form of the propagators is defined by we have considered only the one-loop corrections. It seems the sign of δq2, this is composed of two unusual signs. clear that a more complete treatment would use the full Comparing the second version in Eq. (44) with that of propagator self-consistently within the graviton contribu- Eq. (46) we see that the overall normalization is the tion to the vacuum polarization. This would indeed be opposite of the expectation (i.e., ghostlike), and also the interesting to explore, and we hope to return to this imaginary part has the opposite sign from expectation. calculation in the future. These two signs are connected in an important way. We can summarize these by a factor Z, such that IV. UNITARITY OF THE SCATTERING AMPLITUDE 2 ∼ iZ iDðq Þ 2 2 ð48Þ q − mr þ iZγ In this section we show, with explicit calculations, that the J ¼ 2 scattering amplitude in this theory is unitary at all the where Z ¼ −1. As discussed in Refs. [29,30], the overall energy regions. This proceeds through the graviton propa- minus sign in Z is significant; such a sign is compensated gator in the s-channel and is the one which might be by the unusual sign of the ghost propagator in such a way expected to be problematic due to the lowest order expect- that the imaginary part of the forward scattering amplitude ation for the ghost state. However, this feature turns out not is positive, as it should be in order to obey the optical to violate unitarity, and indeed it is the special properties theorem. These signs also play a very important role in the of the resonance and the imaginary parts from loops which unitarity of the scattering amplitude which we will discuss enforce unitarity. We proceed in three steps. First we in the next section. describe the scattering of a single . This demon- Note that we find a single pole, in contrast with previous strates how unitarity occurs in this channel and highlights the work in Ref. [9] which claims a pair of complex-conjugated role of the various signs in the propagator and the imaginary poles. The imaginary part in the region of the pole comes parts. Then we generalize to multiple fields. Finally, there is from the loops of light particles with usual quadratic a discussion of the graviton interactions at the highest propagators. These carry the usual iϵ prescription, and energies due to the quadratic curvature terms in the action. the imaginary part comes from the property of the loga- This calculation is based on previous work by Han and rithm, Eq. (28). The fact that we are at weak coupling Willenbrock [53] and Aydemir et al. [54]. As a first step, we facilitates the analysis of the pole location, as described work with a single real massless scalar field minimally above, but the determining feature that yields a single pole coupled to gravity and consider the reaction ϕ þ ϕ → is the iϵ prescription of Eq. (28). Also note that in Ref. [9] ϕ þ ϕ through s-channel graviton exchange. We recall that the issue of the gauge dependence of the pole location has it is through the s-channel that resonances and new unstable been raised. The location is gauge independent at one loop particles are usually probed. In addition, in the s-channel, which is the order that we are working, as can be seen in the the intermediate state satisfies s>0, where s is the work of Ref. [23]. We hope to explore this issue in future Mandelstam variable describing the square of the total work when we look at loop processes using the full spin- energy of the particles in the center-of-mass frame (invari- two propagator. ant rest mass). Our calculations show the emergence of three energy The Feynman rules associated with the interacting vertex scales, defined by Λξ, Λg, which is the gauge field scale for scattering amplitudes is to be extracted from the 1 2 μν mass, and mr, the value for which the propagator presents expression ð = ÞhμνT . The s-channel amplitude for the ϕ ϕ → ϕ ϕ a resonance, as discussed above. We take Λg ≫ Λξ.For process þ þ is given by energies below Λ , the gauge field becomes strongly     g 1 1 coupled and confined. In addition, we recall that we μναβ 2 iM ¼ VμνðqÞ ½iD ðq Þ Vαβð−qÞ ð49Þ identify the Planck scale with the Yang-Mills scale, 2 2 Λg ≡ Mp. For energies ∼mr, the ghost particle goes as a resonance; since it is unstable, it will not appear in where the energy-momentum of matter has the following asymptotic states. Below such a scale one should take into on-shell matrix element (at the lowest order) account the contribution to the vacuum polarization coming 0 0 0 − 0η from the gravitons in the effective-field-theory calculations. VμνðqÞ¼hp jTμνjpi¼pμpν þ pμpν p · p μν; ð50Þ

126005-9 JOHN F. DONOGHUE and GABRIEL MENEZES PHYS. REV. D 97, 126005 (2018) with p · p0 ¼ q2=2, p þ p0 ¼ q. Since we are only con- unitarity is obtained in this case because we are here simply sidering the effect of the scalar field, the logarithmic terms expanding a unitary S matrix in perturbation theory. in the propagator only reflect the loop of that particle and If we now allow not only a single scalar but also other are described by Nq ¼ 0; on the other hand, effective field light fields plus gravitons in the theory, we are in principle 1 6 theory calculations yield the value Neff ¼ = for a single faced with a multichannel problem. An initial state of scalar field [54]. Employing the usual Mandelstam varia- scalars can scatter into a final state of gravitons, and visa bles, one finds versa. The solution is given by Han and Willenbrock [53],   and consists of diagonalizing the scattering matrix. Having 1 s2 performed this diagonalization, the problem is back into M ¼ 2tu − D¯ ðsÞ 8s 3 that of a single channel elastic scattering. Let us first     consider this in the effective field theory limit, below the 1 κ˜2 κ˜2 κ˜2 ¯ −1 1 − s − sNeff s i sNeff resonance where graviton scattering is determined by the D ðsÞ¼κ˜2 2ξ2 μ 640π2 ln μ2 þ 640π ð Þ Einstein action only (i.e. Nq ¼ 0). The diagonalization of ð51Þ Han and Willenbrock can be extended to include the graviton channel, and the result is identical to that of Eq. (53) except with a generalized value of N , given by where we used that s þ t þ u ¼ 0. Now we perform a eff Eq. (33). So we see that unitarity is again satisfied. partial wave expansion with respect to the angular momen- Finally, we turn to the high energy region where the tum J graviton propagator arises dominantly from the terms X∞ quadratic in the curvature. Consistent with the approxima- M ¼ 16π ð2J þ 1ÞTJðsÞPJðcos θÞð52Þ tion given above, we here drop consideration of the J¼0 Einstein term in the gravitational action and only consider the quadratic terms. As we described, these give no where PJðcos θÞ are the Legendre polynomials that satisfy coupling for on-shell gravitons. Likewise in the propagator, PJð1Þ¼1. Furthermore, we employ the parametrization the imaginary parts only arise from the matter particles and t ¼ −sð1 − cos θÞ=2 and u ¼ −sð1 þ cos θÞ=2. Hence not the gravitons. The scattering amplitude then takes the 2 using that P2ðxÞ¼ð3x − 1Þ=2, one finds the following form expression for the partial wave amplitude T2:      − Neffs − s − sNeff s − π T2ðsÞ¼ 2 2 ln 2 i Neffs ¯ 640π 2ξ ðμÞ 640π μ T2ðsÞ¼− DðsÞ: ð53Þ   640π sN 2 −1 − q s 1280π2 ln μ4 : ð56Þ In order to satisfy elastic unitarity, the scattering in the 2 elastic channel must have ImT2 ¼jT2j . This is satisfied In this region, we have when the amplitude has the form 199 AðsÞ AðsÞ½fðsÞþiAðsÞ Neff ¼ N∞ ¼ D þ NSM Nq ¼ : ð57Þ T2ðsÞ¼ ¼ ð54Þ 3 fðsÞ − iAðsÞ f2ðsÞþA2ðsÞ Again, unitarity is satisfied. The consistent decoupling of for any real functions fðsÞ, AðsÞ. Since the imaginary part the on-shell graviton states in both the numerator and in the denominator comes from the logarithmic factor, the denominator was important in this regard. unitarity condition is a relation between the tree-level We repeat the comment from the previous section that an scattering amplitude which determines the AðsÞ in the improved treatment would include the full propagator self- numerator and the logarithm in the vacuum polarization consistently in loops, as we hope to do in the near future. which determines the imaginary part in the denominator. This will be especially instructive in the unitarity calcu- For the elastic scattering of a single scalar, this relation is lation as it has the potential to couple in on-shell gravitons satisfied with even at the highest energies. However it appears from the general construction of the elastic channel that we would − Neffs expect unitarity to be satisfied in such a treatment also. AðsÞ¼ 640π : ð55Þ V. MORE GENERAL RESULTS We note that there is an important correlation between the unusual sign of the imaginary part in the propagator and the The discussion above has focussed on what we feel is the sign of the scattering amplitude which allows unitarity to be most important and novel aspects of the present theory. satisfied. It is however perhaps not that surprising that There are two main topics connected to other terms in the

126005-10 GAUGE ASSISTED QUADRATIC GRAVITY: A FRAMEWORK … PHYS. REV. D 97, 126005 (2018)   2 2 1 5ξ4 5 action, (1) the cosmological constant and (2) the R term in df0 2 2 4 μ ¼ þ 5ξ f0 þ f0 : ð60Þ the quadratic action which is associated with the scalar dμ 16π2 3 6 d.o.f. in quadratic gravity. Let us start with the latter effect. Even if excluded from Observe that the coupling f0 is not asymptotically free, 2 0 the initial action, the R2 interaction will be induced by the unless f0 < , which would lead to a tachyonic instabil- 2 0 helper gauge interaction, much like the Einstein action is ity M0 < . induced. This effect was first described by Brown and Zee It also would not be a problem for this theory if we were [22]. We discuss this in our Appendix below and calculate to give up the initial scale/conformal invariance, as long as −1 it within QCD. The result, in the notation of Eq. (10),is the energy scale of the intrinsic G and cosmological positive and has the value constant were small compared to the Planck scale deter- mined by the helper gauge interaction. For example for 1 0 00079 0 00030 weakly coupled Weyl gravity, we have seen that the 2 ¼ . . : ð58Þ Λ 6f0 intrinsic scale in the running coupling is ξ, which can be 10−1006 eV or even much lower. If the intrinsic Newton’s Because this is dimensionless, it would hold in a scaled-up constant and cosmological constant were govern by that version of QCD. However, the result has an unusual feature scale, the phenomenology of this model would be essen- that this value is only to be applied when working below the tially unchanged. Planck energy. It can be seen from the derivation of the Finally, there is the problem of the induced cosmological Adler-Brown-Zee formalism that the calculation is accom- constant. In a QCD-like theory, this is given by plished by Taylor expanding the gravitational field on 1 1 D β E longer wavelengths than the Planck scale arising from the μ ðgÞ a aμν Λ ¼ h0jT μj0i¼ 0j FμνF j0 : ð61Þ helper gauge interaction. The induced effect vanishes at ind 4 4 2g energies above the Planck scale, much like the induced value of the Newton constant also vanishes there. For the standard value of the gluonic condensate, this yields “ ” Λ −0 0034 4 It is also possible to have a bare value of f0 in the ind ¼ . GeV . Scaling a QCD-like theory up to original action. In this case the original symmetry is scale the Planck mass would yield a very large negative value of invariance rather than local conformal invariance7 In this the cosmological constant, of order of the Planck scale. case there is a scalar sector to analyze. The interesting In the presence of this cosmological constant, we need to feature here is that the massive scalar that appears is not a expand about anti-de Sitter space rather than about flat ghost state. It has the usual positive residue, and is not space. We can show that the AdS solution is unchanged by problemmatic. Indeed, the scalar propagator has been the presence of quadratic terms. Using the exponential calculated elsewhere and it is given by [2,24] parametrization around a background field   4 2 −1 h λ 0 q 2q 0 gμν ¼ g¯μλðe Þ ν ð62Þ Dð Þ ðq2Þ¼ − Pð Þ μναβ f2 κ˜2 μναβ 0  κ˜2 1 1 andffiffiffiffiffiffi discarding total derivatives, we find the expansion of ð0Þ p− L L ¼ − P ð59Þ gð EH þ qÞ where 2 q2 − M2 q2 μναβ 0 ffiffiffiffiffiffi p− L 2 2 2 g EH where M0 ¼ 2f0=κ˜ . One can easily identify the massive    ffiffiffiffiffiffi 2 1 Λκ2 scalar mode, as well as a (massless) ghost state. However, p ¯ μν ¯ ¯ μν μν ¼ −g¯ −Λ þ R þ g¯ R − 2R − g¯ hμν with some effort one can show that the latter is the standard κ2 κ2 2 ghost that emerges also in general relativity, so it is a Λ ¯ − λ 2 R λ 2 − ¯ μν λ − β harmless nonpropagating state which can be removed by a 8 ðhλÞ þ 4κ2 ðhλÞ R ðhμνhλ hμβhνÞ   gauge transformation [24]. In turn, the first term describes 1 1 1 the massive scalar excitation, which is in fact a propagating λ ν;β − β ν;α β α;ν − λ β;ν þ 2 hλ;νhβ hα;νhβ þ hα;νhβ hλ;νhβ : mode, as alluded to above. In this case one can also define a κ 2 2 2 running coupling constant f0ðqÞ through the renormaliza- ð63Þ tion group equation [2]: ¯ ð1Þ For the quadratic terms, Lq, we use R ¼ R þ R þ, 7 ð1Þ Indeed calculations exist that show that divergences in f0 is where R is linear in the quantum fluctuation generated perturbatively at higher loop order even if one starts from a conformally invariant initial action. See the discussion of ð1Þ λ; ν − ;ν;μ − ¯ μ ν Salvio and Strumia [2]. It would be interesting to understand if R ¼ hλ;ν hμν Rνhμ ð64Þ there were regularization schemes which could prevent such divergences. resulting in

126005-11 JOHN F. DONOGHUE and GABRIEL MENEZES PHYS. REV. D 97, 126005 (2018)    ffiffiffiffiffiffi pffiffiffiffiffiffi 1 One of the conditions under which the vacuum energy p− 2 −¯ ¯ 2 2 ¯ ð1Þ ¯ 2 λ gR ¼ g R þ RR þ 2 R hλ would certainly vanish in a Yang-Mills theory is that the    1 beta function could vanish at zero energy. This can happen ð1Þ 2 ¯ ð1Þ λ λ 2 ¯ 2 if there is an infrared fixed point in the beta function at a þ ðR Þ þ RR hλ þ 8 ðhλÞ R þ : finite coupling—the Caswell-Banks-Zaks fixed point ð65Þ [56,57]. This can emerge from a competition of terms of different order in the beta function, i.e. βðgÞ¼−bg3 þ cg5 ¯ However, for constant curvature with Rμν ¼ with b, c>0. However, the phenomenology of the IR ¯ βgμν, R ¼ 4β, with β being a constant, the linear term phase is less-well understood. There is still a dimensional ¯ ð1Þ 1 ¯ 2 λ parameter in the running coupling at higher energies, and ð2RR þ R hλÞ vanishes aside from a total derivative. 2 the perturbative contribution to the induced Einstein action Moreover, for such a spacetime the background Weyl ¯ pffiffiffiffiffiffi 2 would still exist. However it is not clear if a positive tensor vanishes Cμναβ ¼ 0. While in general −gC would Newton’s constant would result. This option deserves have a similar expansion to Eq. (65), in this spacetime there further study. will be no linear term in the expansion because of the One of the present authors has proposed that the spin vanishing of the background Weyl tensor. Therefore the connection, which appears naturally as a gauge field in lowest order Einstein equation remains unchanged and we gravitational theories, could form an asymptotically free 1 κ2Λ¯ recover the AdS curvature Rμν ¼ 4 gμν and the back- gauge theory if treated as an independent field [3]. ground dependent terms of Eq. (63) reduce to The Euclidean version of such a theory would involve 4 Λ the symmetry Oð Þ and would be confined. This involves − λ 2 − 2 β α ½a;b − 8 ½ðhλÞ hαhβ: ð66Þ six field strengths, Fμν where ½a; b¼a, b b, a is the antisymmetric combination of Lorentz indices a, b ¼ 0,1, The expansion of L is then 2, 3. By symmetry, each of the fields contributes equally to q 2   the Euclidean vacuum value of F . When continued back to ffiffiffiffiffiffi ffiffiffiffiffiffi 1 1 Lorentzian space-time, the symmetry becomes SOð3; 1Þ. p p ¯ 2 ð1Þ 2 ¯ ð1Þ λ −gL −g¯ R R RR h ½01 ½02 q ¼ 2 þ 2 ð Þ þ λ In the vacuum energy, the products F F 01 , F F 02 , 6f0 6f0 ½ ½   ½03 1 1 F F½03 change sign due to the Minkowski metric, while λ 2 ¯ 2 ð1Þ ð1Þμναβ þ ðhλÞ R − CμναβC : ð67Þ ½12 ½13 ½23 8 2ξ2 the spacelike ones F F½12, F F½13, F F½23 carry the same sign. The Lorentzian symmetry then forces the

The bilinear terms in the background field expansion of Lq vacuum expectation value of the trace anomaly to vanish. can be recovered by use of the identity of Eq. (12) and If this non-compact group makes sense as the helper gauge interaction, then it would not suffer from the large 1 cosmological constant problem. ð1Þ λ λ λ ;λ Rμν h − h − h hμν 68 ¼ 2 ½ λ;μ;ν μ;ν;λ ν;μ;λ þ ;λðÞ We have been exploring this theory in the limit where the Weyl coupling constant ξ stays small, or more precisely that as well as Eq. (64). the scale involved in the running coupling ξðqÞ is very While probably all quantum theories of gravity suffer much smaller than the Planck scale. This is chosen mainly from having a Planck-scale cosmological constant, it is for our convenience in analysing the theory. However, if the particularly embarrassing for a theory which starts from a scale in the gravitational running constant is of the same scale invariant initial action. In such a case, one cannot add order as the Planck scale, it will also become strongly a “bare” cosmological constant in order to cancel the coupled. In this case, it would induce extra contributions to induced one. We need to remove or severely suppress the induced Newton constant and to the cosmological the induced cosmological constant. We present a set of constant. Almost nothing is known about this situation, possible solutions without having a perfectly compelling although Adler [21] has developed some of the relevant choice. formalism. In the absence of knowledge, one can engage in Holdom [55] has raised the question of whether the wishful thinking that perhaps a mechanism can be found vacuum energy in massless QCD could in fact vanish. This such that the gravitational contribution to the cosmological is in part because the present lattice estimates are uncertain constant cancels the gauge contribution. enough to include a vanishing value. The calculations are Another approach could be to have the helper gauge made more difficult by the presence of a hard dimensional theory be supersymmetric. This would lead to a vanishing cutoff—the lattice spacing—which adds a dimensionful of the cosmological constant if the supersymmetry is still ingredient to the theory beyond just the running coupling. exact at the Planck scale. One would need supersymmetry It would be an important step for theories if breaking at a lower scale, and potentially a much smaller this question could be definitively answered. value of Λ.

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Finally, one can have a version of the theory in which the symmetry, and it has one less propagating d.o.f. compared energy density of the vacuum does not gravitate. This is to general quadratic gravity. In the exponential parameter- λ accomplished by fixing the determinant of the metric to a ization, this is the scalar trace h ¼ h λ. This “conformal constant, such that the vacuum energy term in the action mode” is missing from the action to all orders in pure 4D, does not involve the gravitational field. We discuss this not just in the expansion about flat space. unimodular option in the next section. This set-up differs from a unimodular version in the path integral measure. In a general theory, the conformal VI. UNIMODULAR GAQ GRAVITY mode is included in the path integration, whereas in a unimodular one it is not included. When one regulates the Because of the difficulties associated with the cosmo- general theory, this mode can in principle get fed back into logical constant, it is worth considering a unimodular the action. If one regulates dimensionally, the conformal version of the theory. In unimodular general relativity mode actually appears in the action when the dimension D [58–65], the determinant of the metric is set equal to a is not equal to 4. If one regulates with a cutoff, then the constant, which can be chosen to be unity, mode is presumably sensitive to the cutoff when done in a pffiffiffiffiffiffi gauge invariant manner. So the path integral measure is −g ¼ 1 ð69Þ one way that the general action differs from a unimod- ular one. and the symmetry of the theory is reduced to volume- Let us then remove the conformal mode from the path preserving diffeomorphisms which preserve this condition. integration. The background field expansion for the final The initial equations of motion differ from those of general pffiffiffiffiffiffi theory changes. From the analysis shown above using the relativity, because the −g factor is not varied, but there is λ exponential parametrization, we must set h ¼ h λ ¼ 0. The an additional constraint on the solutions associated with 2 energy conservation. When this constraint is imposed, the R action could still be induced, or even appear initially. equations are exactly those of general relativity, including The remaining background field expansion has a slightly the possibility of a cosmological constant term which modified structure in the unimodular case. Generally, when emerges as an integration constant of the constraint we expand to second order in the fluctuation, we take the equation. Percacci [64] has a recent treatment of unim- first order term to vanish by the equations of motion. But odular theories which makes it clear that the equivalence of for unimodular gravity the first order term is not equivalent the unimodular theory to the general one is not just a to the final equations of motion. In particular it does not property of the Einstein action, but would also be true for a include the constraint that brings in the parameter that plays theory quadratic in the curvature, such as we are consid- the role of the cosmological constant. In particular, the first ering here. order term in the expansion is   However, the striking advantage of a unimodular theory 1 is the different status of the cosmological constant. In usual ˆ μν − ¯μν hμν R 4 g R ð70Þ metric theories, the cosmological constant is tied to the energy density of the vacuum, and appears as a constant where the fluctuating field with the conformal mode term in the Lagrangian. The many large contributions to removed is this energy density form part of the “cosmological constant problem.” In unimodular theories, this energy density in the 1 ˆ − ¯μν λ Lagrangian (which we have called Λ above) does not hμν ¼ hμν 4 g h λ: ð71Þ couple to gravity and is irrelevant. The cosmological constant which appears in the equations of motion arises This is the equation of motion before the constraint is from the initial condition of the constraint equation. We can imposed. However, both this original equations of motion and the constraint are separately satisfied. This implies that call this constraint parameter Λu. Of course, we do not yet have a good theory of this initial condition. For our the first order variation vanishes as usual. When we reach second order in the fluctuation there are also background purposes Λu can have any magnitude and either sign. Unimodular theories solve this part of the cosmological curvatures here. Here we should use the background constant problem by decoupling the constant that appears curvatures which satisfy the final equations of motion. This leads to two differences in the final result. One is that in Einstein’s equation Λu from the vacuum energy Λ. In the theory considered in this paper, the unimodular constraint the fluctuating field is traceless in the exponential repre- would remove the worry about the large energy density sentation. This removes the vacuum energy Λ from the found when one scales up QCD-like theories as the helper quadratic action. The other is that the equations of motion gauge theory. will involve the constraint-generated cosmological constant Our starting point can be close to the unimodular case. rather than the one connected with the vacuum energy. That 1 2 Consider first the theory with a gauge interaction and the is, one uses Rμν ¼ 4 κ Λugμν. For example, these changes Weyl-squared action, as in Eq. (1). This has local conformal yield the leading low energy action for AdS or dS spaces as

126005-13 JOHN F. DONOGHUE and GABRIEL MENEZES PHYS. REV. D 97, 126005 (2018) Z   1 1 Λ 4 ˆ β ˆ ν;α ˆ β ˆ α;ν u ˆ μ ˆ ν the induced coupling is only present below the S d x −hα νh hα νh hνhμ EH ¼ κ2 ; β þ 2 ; þ 4 ð72Þ Planck scale, and disappears above it. The role of the helper gauge interaction in the theory is primarily to dynamically generate the Planck scale, while instead of Eqs. (63), (66). still keeping the gravitational interaction weakly coupled. This is the feature which allows the theory to be under VII. SUMMARY reasonable theoretical control. We understand the dynamics of confined gauge theories from decades of study of QCD There are several features of this theory which differ and related theories. By keeping the gravitational inter- from usual expectation. We list these here: action weakly coupled, we can use perturbation theory to (i) Although the initial theory was scale invariant, the describe it and to bypass a strongly coupled theory of Planck scale and the Einstein term in the action were gravity. One can imagine other limits of this same theory in induced by the helper gauge theory interaction via which the gravitational interaction is also strongly interact- dimensional transmutation. This allows the low ing. Or one could consider other methods for inducing the energy spectrum of gravitons to be the usual one. Planck scale, perhaps with scalar fields. However, the (ii) The spin-two graviton propagator has a resonance at variation considered here, with weakly coupled gravity ξ a scale MP. This is the remnant of the ghost state, and a strongly interacting gauge theory, seems to be a very but is unstable and does not exist as an asymptotic good laboratory for UV complete quantum field theories of state in the spectrum. Related work on Lee-Wick gravity. models indicates that we should expect microscopic The theory has passed the tests that we have explored thus violations of causality on timescales given by the far. The naive fears concerning ghosts and unitarity violation inverse of the width of the resonance, which is have not been realized. The mechanisms for avoiding these 1 ξ2 = MP. have been nontrivial, and required the signs and amplitudes (iii) The propagator sign at the resonance is opposite to to work out in a coordinated manner. Further work is needed the usual expectation, and likewise the imaginary in order to study amplitudes with gravitational loops in this part has the opposite sign from usual expectation. theory, and we hope to turn to this in the future. In addition, These two features combine to give the residue at the the cosmological constant problem remains, although we resonance the proper sign, consistent with the optical have suggested possible solutions, including a unimodular theorem. version of the theory. Overall, the possibility of using this (iv) There are three kinematic regions with different theory as a QFT-based ultraviolet completion of quantum physical content. At low energies, below the reso- gravity remains promising. nance, the usual effective field theory description is valid. In the intermediate energies between the resonance and the Planck scale, the quadratic cur- ACKNOWLEDGMENTS vature terms dominate for the gravitons, but the We thank Alberto Salvio, Alessandro Strumia, Roberto helper gauge interaction is not yet active. At high energies, the theory can be asymptotically free with Percacci, Guy Moore, David Kastor, Larry Ford, J. J. both the gauge and graviton interactions active. Carrasco and David Kosower for useful discussions. (v) When treated using only the interactions from the This work has been supported in part by the National quadratic curvatures at ultrahigh energies, the propa- Science Foundation under Grant No. NSF PHY15-20292. gator does not pick up imaginary parts from the graviton loops, because the coupling to on-shell APPENDIX: THE CALCULATION OF THE states vanishes. INDUCED R2 TERM IN THE ACTION (vi) The spin-two partial wave amplitude—which is the dangerous one due to the original ghost pole—has Here we derive the induced R2 term within the Adler- been shown to be unitary in all energy regions. This Zee-Brown approach. The gravitational effective action is nontrivial and is tied to the special properties of reads Z the propagator. ϕ (vii) The gravitational interaction remains weakly coupled eiSeff ½gμν ¼ dϕeiS½ ;gμν; ðA1Þ at all energies. The usual growth of amplitudes with increasing energy is tamed by the transition to quartic where ϕ represents generically the matter fields and propagators at high energy. S½ϕ;gμν describes matter fields on a curved background. (viii) The R2 term in the action is also induced by the As discussed above, we are considering the Yang-Mills helper gauge interaction even if it was not present in field as the matter field responsible for inducing the the initial theory. We calculate its coefficient in effective gravitational action. Since Seff½gμν is a scalar QCD-like theories. However, this has the feature that under general-coordinate transformations, it may be

126005-14 GAUGE ASSISTED QUADRATIC GRAVITY: A FRAMEWORK … PHYS. REV. D 97, 126005 (2018) ffiffiffiffiffiffi p− L represented as the integral over the manifold of a i g eff½gμν scalar density, which for slowly varying metrics can be     i 1 i 1 formallydevelopedinaseriesexpansioninpowersof ≈− 2 τμ α 2hðxÞ 4hTðxÞi þ 16ðhðxÞÞ 4h μ αðxÞi ∂λgμν Z   Z 1 2 4 1 1 2 4 2 Z − h x d z K z ∂μh d zz K z ffiffiffiffiffiffi 16ð ð ÞÞ 8 ð Þ þ 210 ð Þ ½ ð Þ 4 p− L Seff½gμν¼ d x g eff½gμν Z 1 2 2 4 2 2 − ð∂μhÞ d zðz Þ ½KðzÞ ðA6Þ L Lð0Þ Lð2Þ Lð4Þ O ∂ 6 24576 eff½gμν¼ eff ½gμνþ eff ½gμνþ eff ½gμνþ ½ð λgμνÞ Lð0Þ Λ eff ½gμν¼ ind; where integrations by parts in the two last terms were μνρσ ¯ μν ¯ ρσ μν − ημν ð2Þ R performed, K ðx yÞ¼hTfT ðxÞT ðyÞgi, T¼ T L ½gμν¼ ; μνρσ eff 16π and K ¼ ημνηρσK .Inturn Gind ð2Þ 1 2      L ½gμν¼ R : ðA2Þ pffiffiffiffiffiffi 1 1 1 eff 6f2 − ≈ ∂ ∂α λκ − ηλκ −2∂μ − η ∂ ind gR 4 αhλκ h 2 h hλμ 2 λμh κ   2 1 3 Let us derive a representation of the induced f in terms λκ λκ 2 × h − η h ¼ − ð∂μhÞ ; of the vacuum expectation value of products of the stress- 2 32 ffiffiffiffiffiffi energy tensor Tμν of the matter fields. For that, let us study p 2 μν αβ μν −gR ≈∂μ∂νh ∂α∂βh −2□h∂μ∂νh þ□h□h the response of the action functional to an external 9 2 2 classical gravitational field gμν ¼ ημν þ hμν treating hμν ∂ ¼ 16ð μhÞ ðA7Þ as a small perturbation. The response of quantum fields to an arbitrary external field is described by the generating pffiffiffiffiffiffi 2 and also −g ¼ 1 þð1=2Þh þð1=16Þh . Equipped with functional of connected Green’s functions: such results one is able to derive representations for the Z induced cosmological constant and induced Newton’s i − 4 μν constant as described in Refs. [18–22,44,45] and recently iW½h¼ 2 d xhμνðxÞhT ðxÞi Z investigated in Ref. [4] for the gluonic QCD case. In the i 4 μναβ present context, we need an expression for the induced þ d xhμνðxÞhαβðxÞhτ ðxÞi 4 coupling constant appearing before the R2 term. One finds   Z Z 2 2 i 1 4 4 Z þ d x d yhμνðxÞhρσðyÞ 1 i 2! 2 ¼ d4zðz2Þ2hTfT¯ ðzÞT¯ ð0Þgi: ðA8Þ 6f2 13824 × hTfT¯ μνðxÞT¯ ρσðyÞgi þ ðA3Þ ind Working in Euclidean space, one obtains … 0 … 0 where h i¼h j j i denotes vacuum expectation value, Z Tμν is the energy-momentum tensor of the Yang-Mills 1 1 d4x x2 2 T T¯ x T¯ 0 : field, given by 6 2 ¼ 13824 ð Þ h f ð Þ ð Þgi ðA9Þ find

1 2 a aλ a a αβ Tμν ¼ −FλμF ν þ ημνFαβF : ðA4Þ After performing a change of variables x ¼ t, we split the 4 integration into an ultraviolet part and an infrared part as follows: and T¯ μνðxÞ¼TμνðxÞ − hTμνðxÞi. The second term on the right-hand side of Eq. (A3) is needed for consistency; 1 π2 it plays a role in the cosmological constant sum rule [4]. ¼ ðI þ I Þ 6f2 13824 UV IR We consider, for arithmetical simplicity, that hμν ¼ ind Z t0 ð1=4Þημνh. We choose h to be a slowly varying over 3Ψ IUV ¼ dtt ðtÞ the scale of the gauge interaction so that one can expand Z0 ∞ 3Ψ 1 IIR ¼ dtt ðtÞðA10Þ μ∂ μ ν∂ ∂ … t0 hðy þ zÞ¼hðyÞþz μhðyÞþ2 z z μ νhðyÞþ ðA5Þ where R ffiffiffiffiffiffi ¯ ¯ 4 p− L ΨðxÞ¼hTfTðxÞTð0Þgi: ðA11Þ In this way, with W ¼ d x g eff, one finds

126005-15 JOHN F. DONOGHUE and GABRIEL MENEZES PHYS. REV. D 97, 126005 (2018) Z u0 Θ Now let us use the same approach as in Ref. [4] to evaluate L du ðuÞ 2 IUV ¼ CΨ 2 find which would arise in QCD. We will employ the same 0 u ðln uÞ ingredients, namely perturbation theory and the operator X∞ Xn −1 ½lnðln u Þm product expansion (OPE) at short distances and modern ΘðuÞ¼1 þ a mn u−1 n lattice glueball studies at long distances. n¼1 m¼0 ðln Þ Using the same expression for the infrared contribution 96 CΨ to ΨðxÞ as presented in Ref. [4], one gets the following ¼ π4 ðA16Þ result for the IIR part:   where the QCD scale parameter is given by (at one-loop 16λ2 2 1 3;0 MGt0 I ¼ G ðA12Þ order) IR π2 6 1;3 4 0 3 4 MG ; ; Λ μ μ μ −1=½bg2ðμ2Þ QCDðgð Þ; Þ¼ e where b0   … … b ¼ 2 : ðA17Þ m;n a1; ;an; ;ap 8π Gp;q z b1; …;bm; …;bq The coefficients amn are loop corrections of higher order. Λ2 1 is the Meijer G-function [66], Mg is the glueball mass and We employ the restriction that u0 ¼ QCDt0 < . Let us rewrite IL as ¯ UV λ ¼h0jTð0ÞjSi; ðA13Þ Z ∞ Θ −v L ðe Þ I ¼ CΨ dv 2 : ðA18Þ is the glueball coupling, with jSi being the normalized UV −1 ln u v scalar glueball state. 0 As for IUV portion, one finds perturbative contributions At leading order the integral can be easily evaluated and the coming from short-distance scales as well as contributions result is coming from intermediate energies which are going to be evaluated through the OPE technique. Regarding these L CΨ IUV ¼ : ðA19Þ latter terms, the results presented in Ref. [4] (and references x0 cited therein) allows one to obtain −1  where x0 ¼ ln u0 . Hence collecting our results, one has 2 2 2 b0α 2b0t0 that IOPE ¼ s hα F2i UV 256π4 π s          1 π2 16λ2 2 2 1 3 3 6 3;0 MGX0 2t0 29αs t0μ 3 ¼ G1 3 þ 4 þ ln þ 6γ − 1 hgF i ; 6f2 13824 π2M6 ; 4 0;3;4 3 3 64 ind G  2α2 2 4 CΨ b0 s b0X0 α 2 ðA14Þ þ −2 þ 4 h sF i ln½ðΛ X0Þ 256π π QCD      α 2 4π 6 6 6 where s ¼ g = and 2X 29α X μ þ 0 4 þ s ln 0 þ 6γ − 1 hgF3i : 11 2 3 3 64 b0 ¼ N − N ; 3 c 3 f ðA20Þ with N ¼ 0 and N ¼ 3 for gluonic QCD. In addition, μ is f c The above expression gives the induced f2 as a function of an arbitrary subtraction point, γ ¼ 0.5772 is the Euler- pffiffiffiffi X0 ¼ t0. Using the lattice data given by Ref. [67], as well Mascheroni constant and hð Þi are gluon condensate as the values given by Refs. [67–69] for the OPE coef- terms ficients, following Ref. [4] we quote our result for a −1 2 a aμν matching scale of X0 ¼ 2 GeV: hαsF i¼hαsFμνF i 3 abc a bν cρμ 1 hgF i¼hgf FμνFρ F i 0 00079 0 00030 2 ¼ . . : ðA21Þ 2 4 abc a bρ 2 abc a b 2 6f hαs F i¼14hðαsf FμρFν Þ i − hðαsf FμνFρλÞ i ind ðA15Þ The error bar is determined by examining changes in the input parameters. As one can easily see, QCD predicts a As for the perturbative contribution, with the change of positive shift for this coupling constant (in the energy range Λ2 2 variables u ¼ QCDt and again using the results quoted in of interest). This suggests that find should be a positive Ref. [4], one gets quantity.

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