PHYSICAL REVIEW D 89, 043515 (2014) Local in physics and

Itzhak Bars Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089-0484, USA

Paul Steinhardt Physics Department, , Princeton New Jersey 08544, USA

Neil Turok Perimeter Institute for , Waterloo ON N2L 2Y5, Canada (Received 30 July 2013; published 19 February 2014) We show how to lift a generic non-scale-invariant action in Einstein frame into a locally conformally invariant (or Weyl-invariant) theory and present a new general form for Lagrangians consistent with Weyl symmetry. Advantages of such a conformally invariant formulation of and gravity include the possibility of constructing geodesically complete . We present a conformal-invariant version of the standard model coupled to gravity, and show how Weyl symmetry may be used to obtain unprecedented analytic control over its cosmological solutions. Within this new framework, generic Friedmann-Robertson-Walker cosmologies are geodesically complete through a series of big crunch– transitions. We discuss a new scenario of cosmic evolution driven by the Higgs in a “minimal” conformal standard model, in which there is no new physics beyond the standard model at low energies, and the current Higgs vacuum is metastable as indicated by the latest LHC data.

DOI: 10.1103/PhysRevD.89.043515 PACS numbers: 98.80.-k, 98.80.Cq, 04.50.-h, 11.25.Hf

I. WHY CONFORMAL SYMMETRY? our conformal standard model given in Eq. (5).Thisfollows from the new properties of the standard theory whose is a well-known symmetry [1] that couplings to gravity includes all patches of field space that has been studied in many physical contexts. A strong are required for geodesic completeness of all cosmological physical motivation for incorporating scale symmetry in solutions for all times and any set of initial conditions1. fundamental physics [2–4] comes from low-energy The models we describe contain no mass scales—no particle physics. Namely, the classical action of the gravitational constant, no mass for the Higgs field, no standard model is already consistent with scale sym- and no mass parameters for the metry if the Higgs mass term is dropped. This invites quarks, leptons or gauge bosons. All of these are prevented the idea, which many have considered, that the mass by the local conformal symmetry combined with the term may emerge from the vacuum expectation value of SUð3Þ ×SUð2Þ ×Uð1Þ gauge symmetry of the standard an additional scalar field ϕðxÞ in a fully scale-invariant model. There is only one source of mass which follows theory. Another striking hint of scale symmetry occurs from gauge fixing (in some sense spontaneously breaking) on cosmic scales: the (nearly) scale-invariant spectrum the Weyl symmetry through a scalar field which is a singlet of primordial fluctuations, as measured by WMAP and under SUð3Þ ×SUð2Þ ×Uð1Þ or a combination of both the Planck satellite [5,6]. This amazing simplicity seems SUð3Þ ×SUð2Þ ×Uð1Þ singlet and nonsinglet fields, to cry out for an explanation in terms of a fundamental depending on the choice of gauge. This source, which is symmetry in , rather than as just the outcome of a associated with the emergence of the Planck scale, drives scalar field evolving along some particular potential the spontaneous breakdown of the electroweak symmetry, given some particular initial condition. In this paper, we consider the incorporation into funda- mental physics of local conformal symmetry (or Weyl- 1In this paper, we use the term “geodesic completeness” to symmetry): that is, classical local scaling symmetry in an refer to two notions: (a) geodesic continuation through all action that includes the standard model coupled to gravity. A singularities separating patches of and (b) avoidance new result of our approach [7–12] is that using conformal of unnatural initial conditions by requiring infinite action for symmetry we are able to solve the classical FRW equations geodesics that reach arbitrarily far in the past. The two notions are – inequivalent since one is local in time and the other is a global across big crunch big bang transitions, thus obtaining the condition. Both properties are satisfied in our conformal standard full set of geodesically complete cosmological solutions of model in the form given in Eq. (5) as described in Sec. IV.

1550-7998=2014=89(4)=043515(15) 043515-1 © 2014 American Physical Society ITZHAK BARS, , AND PHYSICAL REVIEW D 89, 043515 (2014) which in turn generates the other known masses of (iv) On the other hand, physics at cosmological scales can elementary constituents, with ratios of masses related to be quite sensitive to the conformal structures discussed dimensionless constants. in this paper as becomes apparent in certain gauges. In Our approach to conformal symmetry has new features Sec. IV we outline some of the phenomena that follow with significant physical consequences that were not from our unique conformal coupling. In the minimal considered before. While we shall work in 3 þ 1 dimen- conformal standard model of Eq. (5), guided by our sions throughout, there is a close connection with field analytic solutions (supplemented, where necessary, theories and gravity or in 4 þ 2 dimensions in with numerical methods), we propose a cyclic con- the context of 2 T physics [13–18] as will be pointed out formal cosmology driven only by the Higgs field with occasionally throughout this paper. Stated directly in 3 þ 1 no recourse to other scalar fields such as an . dimensions, some of the novel features and their conse- The Higgs-driven cosmological scenario we propose is quences are as follows: strongly motivated by the metastability of the Higgs (i) In Sec. III we provide the most general form of vacuum as implied by the renormalization-group flow coupling gravity or supergravity with any number of the minimal standard model to large Higgs vev, with of scalar fields, fermions and gauge bosons, while its parameters fixed by the LHC measurements [19]. maintaining local conformal symmetry. Only a spe- The main discussion in this paper is at the classical field cialized form of our general formalism in Sec. III theory level, but it is worth commenting briefly on the coincides with previously known methods for local important question of quantum corrections. There are two conformal symmetry. We find that in a gauge sym- enormous hierarchies in the standard model coupled to metric theory, if there is only one physical scalar field gravity: the weak scale is around 10−16 and the (for example, the Higgs field) after all gauge choices scale is around 10−30 of the Planck scale. It is hard to are exhausted, then there is a unique gauge-invariant understand how such tiny numbers enter fundamental and Weyl-invariant form for the Lagrangian represent- physics, and why quantum corrections would not spoil ing the geodesically complete coupling to gravity. these fine tunings. (ii) The minimal, realistic and locally conformal stan- In both cases, conformal symmetry has been suggested as dard model coupled to gravity includes the Higgs a solution. We will not explore this here in any detail, except ð Þ ð3Þ ð2Þ doublet H x and an additional SU ×SU × to note that the stability of such hierarchies in the perturba- ð1Þ ϕð Þ U singlet x . Taking into account all of the tive standard model, which is what attracted attention in the local symmetry, there is a single physical scalar field past for the Higgs field, should not be in jeopardy from identifiable with the observed Higgs field. Therefore, quantum corrections since dimensionless constants in a according to the statement above, this is a geodesi- conformal theory are logarithmically divergent as opposed cally complete conformal standad model provided it to the quadratic divergence of a bare Higgs mass term. To utilizes our unique coupling to gravity as shown in fully grasp how this can work at the quantum level for Eq. (5) and as proven in Sec. III A. The relative ϕ fundamental scalars and in particular the Higgs, requires a minus sign between and H that appears in this better understanding of how to perform regularization and action is obligatory and is related to geodesic renormalization consistent with local conformal symmetry in completeness as will be explained later. The unique 3 þ 1 dimensions. Elaborating on earlier suggestions structure of local scale symmetry has led to full [4,20,21], there has been recent progress [22] in developing analytic control of cosmological solutions and, in a renormalization theory consistent with Weyl symmetry. certain limits, enabled us to obtain the complete set According to this new work, local conformal symmetry of homogeneous cosmological solutions for all remains as a valid symmetry at the quantum level as possible initial conditions of the relevant fields. This anticipated in [4]. The local scale invariance survives is discussed in Sec. IV. quantization even though there is a trace in the (iii) In a particular gauge of the spontaneously broken local scale symmetry (called c gauge) we recover the usual stress tensor of the matter sector of the theory; this point that renormalizable field theory of the standard model in caused confusion in the past is clarified by noting that the the flat space limit, including a mass scale for the trace of the matter stress tensor is distinct from the generator Higgs. In this gauge, low-energy physics of our model of local scale transformations that includes the additional ϕ coincides with the familiar form of the standard model fields, such as , that implement and establish the local unless more fields are included beyond those already conformal symmetry. In passing, we note that if 3 þ 1 observed, such as right-handed neutrinos and dark dimensional conformal symmetry is treated consistently matter which can be accommodated consistently with within 2 T physics in 4 þ 2 dimensions, then scale (dilation) conformal symmetry following our methods. Thus, symmetry is a part of a linearly realized SO(4,2) in the low-energy physics at LHC scales is not sensitive to flat limit which is presumably not anomalous just as its the conformal structures suggested in our approach. Lorentz subgroup SO(3,1) in 3 þ 1 dimensions cannot be

043515-2 LOCAL CONFORMAL SYMMETRY IN PHYSICS AND COSMOLOGY PHYSICAL REVIEW D 89, 043515 (2014) anomalous. This quantum aspect of 2 T physics is under minimal single-Higgs model is naturally compatible with investigation directly in 4 þ 2 dimensions and this is the cyclic picture [26]. In general, the Weyl-invariant expected to shed additional light on these issues. approach also hints at the intriguing possibility that the A plan of this paper is as follows. In Sec. II, we will minimal, electroweak Higgs may have played a central role review and expand on the motivation for global and local in cosmic evolution as explained in detail in [27]. symmetry and show how to recast (or “lift”) the standard There is a deep connection between the ideas presented in model plus gravity into a locally gauge invariant, Weyl- this paper and theories in four-space and two-time dimen- symmetric theory by adding new fields and, at the same sions. Many of the ideas discussed here for a global or local time, introducing compensating gauge symmetries, in such conformally symmetric theory emerged progressively from a way that the theory is both Weyl invariant and geodesi- developing the 4 þ 2 dimensional formalism since 1996 (for cally complete. The gauge fixed version of this theory a recent overview see [13]), and then for the standard model reverts back to the familiar minimal form of the standard as given in [14],forgravityin[16], for SUSY in [17] and model at low energies wherever gravity is negligible. supergravity in [18]. The formulation given in [18], with However, in regions of spacetime where strong gravity many scalars coupled to gravity and supergravity in 4 þ 2 effects are important, such as in cosmology, especially and 3 þ 1 dimensions was the precursor of the general close to the singularity, the geodesically complete Weyl formalism presented in Sec. III, while the simplest specific lifted version plays a crucial role, as described in Sec. IV. models were analyzed cosmologically in some detail in In Sec. III, following [18] we present a general form for a [7–12]. There are some recent similar examples [28] that 3 þ 1 Lagrangian consistent with Weyl symmetry for any number overlap with our conformal symmetry vision in of scalar fields. We argue that, if the Weyl-invariant theory dimensions; these seem to be oblivious to our previous is equivalent to having only a single physical scalar field publications that introduced at an earlier stage the crucial 3 þ 1 after gauge fixing (like the physical Higgs), then by field conformal structures with restrictions on scalars in 4 þ 2 redefinitions it is always possible to recast the theory into dimensions as predictions from 2 T physics [18].The our version given in Eq. (5), which includes all patches in theory has more predictions of hidden symmetries and 3 þ 1 field space so as to insure geodesic completeness. dualities in dimensions which are mainly understood As another example of our general formalism, we lift the in classical and quantum mechanical contexts [13] and are Bezrukov-Shaposhnikov Higgs model [23] to a also partially developed in field theory [29]; these go well fully scale-invariant model, making it logically and physi- beyond conformal symmetry in their implications for uni- cally consistent. Based on our formalism for constructing fication and the meaning of spacetime and are bound to play general Weyl-invariant actions, we show that the model is an interesting role in future progress. not unique. In fact, there are many single-field cosmologi- We emphasize that, despite the name, the physics content ð þ 2Þ cal Higgs models with the same properties, all consistent in the 2 T-physics formalism in d dimensions is same with conformal symmetry and producing the same infla- as the physics content in the standard 1 T-physics formal- ð − 1Þþ1 tionary outcome. However, this inflationary scenario is not ism in d dimensions except that 2 T physics geodesically complete, furthermore, in view of our com- provides a holographic perspective and, due to a much plete set of solutions, it is extremely unlikely. larger set of gauge symmetries, naturally makes predictions In Sec. IV, we turn more generally to cosmology. We that are not anticipated in 1 T physics. These additional explain that a Weyl-lifted theory can resolve the singularity gauge symmetries are in phase space rather than position and enable the geodesic completion of cosmological space, and can be realized only if the formalism is , while also indicating the presence of new developed with two times. Nevertheless, the gauge invari- ant sector of 2 T physics is equivalent to a causal one-time phenomena. Our previous work in [7–12] elaborated in spacetime without any ghosts. some detail on the properties of geodesically complete In this paper, we will stick to 3 þ 1 dimensions, cosmological analytic solutions in the context of solvable advocating a coherent overall picture of a conformally examples of what was supposed to be toy models. However invariant formulation of fundamental physics and cosmol- the same models now reappear in our conformal standard ogy. However, we will occasionally remind the reader that model and hence our previous solutions are now the full set these outcomes naturally follow from 4 þ 2 dimensions of cosmological solutions for all initial conditions of the with appropriate (but unusual) gauge symmetries. familiar standard model. In particular, we point out a model independent attractor mechanism discovered in [10] that may help to determine the likely initial conditions of our II. WHY LOCAL CONFORMAL SYMMETRY? just after the big bang. The Weyl-invariant In this section, we will describe the motivation for and formulation also provides a natural framework for incor- construction of simple theories with global and local scale porating cyclic cosmology [24,25] driven only by the Higgs invariance. An important application is the lift of the field as introduced. In the case that the Higgs field is standard model plus gravity into a Weyl-invariant theory metastable, the Higgs inflation model is inoperable, but the that is also geodesically complete.

043515-3 ITZHAK BARS, PAUL STEINHARDT, AND NEIL TUROK PHYSICAL REVIEW D 89, 043515 (2014) A. Global scale invariance big bang. Here, as is the case throughout the paper, the We begin by examining a scale-invariant extension of the solutions are in the limit of negligible gauge and top quark standard model of particles and forces. For example mass coupling so that the Higgs evolution is described by consider the usual standard model with all the usual fields, classical equations of motion. This time-dependent behav- ϕ including the doublet Higgs field HðxÞ coupled to gauge ior of the Higgs is driven by the evolution of the field ,as bosons and fermions, but add also an SUð2Þ ×Uð1Þ singlet anticipated in [15] for this simple model. In fact, it is ϕðxÞ (plus right-handed neutrinos and candi- realized as the generic solution for all homogeneous dates), and take the following purely quartic renormalizable cosmological solutions if the Higgs vacuum is stable. potential involving only the minimal set of scalar fields, There are other interesting theoretical structures worth noting about this scale-invariant setup. Instead of super- λ λ0 symmetry, conformal symmetry may explain the stability ð ϕÞ¼ ð † − α2ϕ2Þ2 þ ϕ4 V H; 4 H H 4 : (1) of the hierarchy between the low mass scale of 246 GeV versus the Planck scale of 1019 GeV, as suggested in [15]. This model, discussed at some length in Sec. VI of [14],is The conformal protection of the hierarchy is not as clear cut the minimal extension of the standard model that is fully as SUSY’s protection and requires better understanding of scale invariant at the classical level, globally. See also regularization and renormalization techniques consistent [30–36], which use a similar field, and the different approach with conformal symmetry as outlined in the introduction. that also adds a Weyl vector field [37] leadingtodifferent What about the (fluctuations in ϕ) that emerges in physical consequences. The field ϕðxÞ, which we call the the broken scale invariance scenario above? At least from “dilaton,” absorbs the scale transformations and is analogous the perspective of only the standard model, the dilaton is a to the dilaton in theory. In the current context, it has a massless Goldstone boson due to the spontaneous breaking number of interesting features: Due to SUð3Þ ×SUð2Þ × of the global scale invariance. As discussed in some detail Uð1Þ gauge symmetry, the singlet ϕ is prevented from in Sec. VI of [14], the original doublet field H is the only coupling to all other fields of the standard model—except for field coupled to known matter, while ϕ is decoupled. the additional right-handed singlet neutrinos or dark matter However, after the spontaneous breaking, H must be candidates. These features of ϕ, that prevent it from rewritten as a mixture of the mass eigenstates of the model, interacting substantially with standard visible matter except which include the observed massive Higgs particle and the via the Higgs in Eq. (1), suggest naturally that ϕ itself could massless dilaton (or maybe low-mass dilaton if the scale be a candidate for dark matter [14]. symmetry is broken by some source). The mixing strength The only parameters associated with ϕ that are relevant is controlled by the dimensionless parameter for our discussion, are α and λ0. For positive λ0 the † 2 2 pffiffiffi minimum of this potential occurs at H H ¼ α ϕ : α ¼ð246 GeVÞ=ð 2ϕ0Þ; (3) Accordingly, the vacuum expectation value of the Higgs may fluctuate throughout spacetime, depending on the dynamics of ϕðxÞ, without breaking the scale symmetry. which appears in Eq. (2). Therefore, through this mixing, However, if for some reason (e.g., driven by quantum the (pseudo-) Goldstone dilaton must couple to all matter, fluctuations or gravitational interactions), ϕ develops a 0.05 vacuum expectation value ϕ0 which is constant in some Plot of Higgs[ ] as a function of time region of spacetime, then the Higgs is dominated by a 0.04 constant vacuum expectation value, 0.03 2 † ¼ α2ϕ2 ≡ v 0.02 H0H0 0 2 ; (2) 0.01 with v fixed by observation to be approximately 246 GeV. The Higgs vacuum H0 provides the source of mass for all 100 200 300 400 known elementary forms of matter, quarks, leptons, and 0.01 After phase transition gauge bosons (while ϕ0 may be the source of Majorana V H 0.02 it settles to the minimum of mass for neutrinos). The observation at the LHC of the Higgs particle, which is just the small fluctuation on top of the vacuum value v, has by now solidified the view that this FIG. 1 (color online). After the big bang, the Higgs field oscillates initially around zero with a large amplitude of the order is how nature works in our region of the Universe, at least of the Planck scale. It slowly loses energy to the gravitational up to the energy scales of the LHC. field, causing its amplitude to diminish. As it approaches the time Figure 1 illustrates how the Higgs field slowly relaxes to or energies of the electroweak scale, it undergoes the phase the spontaneously broken symmetry vacuum described by transition seen in the figure, and then slowly settles to a constant Eq. (2) beginning from large oscillations shortly after the vacuum value v at a stable minimum of the potential.

043515-4 LOCAL CONFORMAL SYMMETRY IN PHYSICS AND COSMOLOGY PHYSICAL REVIEW D 89, 043515 (2014) just like the Higgs does, namely with a coupling given by model to Einstein gravity in the conventional way makes no the mixing angle ðsinðθÞ¼α=ð1 þ α2Þ1=2Þ times the mass sense because the dimensionful Newton constant explicitly of the particle divided by v. The largest coupling is to the breaks scale invariance. If scale invariance is already top quark. If ϕ0 is much larger than 246 GeV, it is possible broken in one sector of the theory, then there is no rationale for the dilaton to hide from accelerator experiments due to for requiring that it be a good symmetry in another part of the weak coupling of α in Eq. (3). the theory. At best, it would occur as an accidental Such a (pseudo-) Goldstone boson has many other symmetry of low-energy physics and only when gravity observable consequences, including a long range force is negligible. This is not the scenario we have in mind; we that competes with gravity and contributions to quantum argue for a fully scale-invariant approach to all physics, a effects as a virtual particle in Feynman loop diagrams. If natural outcome of the larger gauge symmetries in 4 þ 2 such an additional massless (or low mass scalar) is dimensions, as formulated in 2 T physics. Happily, the idea observed in experiments, it would be strong evidence in of an underlying 4 þ 2 dimensions with appropriate extra favor of the scale-invariant scenario. However, if it is not gauge symmetry fits all known physics in 3 þ 1 observed, this could be interpreted merely as setting a lower dimensions, from dynamics of particles and field theory limit on the scale ϕ0. [13–17] all the way to supergravity [18]. So, consistency Eventually it must be understood what sets the scale ϕ0. with this larger underlying structure is well motivated. The model as presented above has no mechanism at the In fact, there is a locally scale-invariant field theory in classical level to set the scale ϕ0; its equations 3 þ 1 dimensions, compatible with 2 T physics in 4 þ 2 of motion are self consistent at the minimum of the dimensions, that couples the standard model and gravity † 2 2 0 potential, H0H0 ¼ α ϕ0; only if λ ¼ 0 [14]; then ϕ0 with no dimensionful constants. We do not mean conformal remains undetermined due to a flat direction in gravity which has ghost problems, but rather the non- the remaining term in the potential (1). The value for ϕ0 minimal conformal coupling of the curvature RðgÞ to scalar would then be obtained phenomenologically from experi- fields [3] which is invariant under Weyl transformations as ment, without a theoretical explanation. Quantum correc- a gauge symmetry. In the next section we will discuss a tions, such as those discussed in [38], may alleviate this generalized Weyl-invariant coupling with many scalar problem by removing the flat direction. However, if the fields also predicted by 2 T physics, but in this section quantum corrections are small, there is the danger that ϕ0 we begin with the well-known method of conformally would be so low that it is not possible to obtain the small coupled scalars, as follows [3,4], value of α in Eq. (3) required to protect the dilaton from 1 1 current experimental limits. ϕ2 ð Þþ μν∂ ϕ∂ ϕ 12 R g 2 g μ ν : (4) B. Local scale invariance These two terms form an invariant unit (up to a Since at present there is no sign of the dilaton in low- total derivative) under local scale transformations, −2 energy physics, suppose it does not exist at all as a degree gμν → Ω gμν, ϕ → Ωϕ, with a local parameter ΩðxÞ. of freedom. Is this incompatible with the idea that con- When there are more scalar fields, the most general way formal symmetry underlies fundamental physics? Not at of achieving local conformal symmetry is discussed in all, because any possible phenomenological problems Sec. III. However, when there is only one additional scalar associated with a Goldstone boson fluctuation of ϕ can field beyond ϕ, which is in a single representation of a be overcome if the scaling symmetry is a local gauge Yang-Mills gauge group, there is a unique way to also symmetry, known as the Weyl symmetry. Then the mass- achieve geodesic completeness as explained in more detail less fluctuations of the dilaton can be eliminated by fixing a in Secs. III A and IV. Hence, for geodesic completeness we 2 unitary gauge. require that not only the field ϕ, but also the doublet Higgs The standard model decoupled from gravity has no local field be a set of conformally coupled scalars consistent with 1 † scale symmetry that could remove a Goldstone dilaton. But SUð2Þ ×Uð1Þ. Namely using the unit 6 ðH HÞRðgÞþ μν † such a gauge symmetry can in fact be successfully g DμH DνH which is also locally invariant (up to a total incorporated as part of the standard model provided it is derivative), we can lift the globally scale-invariant standard coupled to gravity in the right way. Coupling the standard model of the last section into a locally invariant one, while also being coupled to gravity in a geodesically complete 2The interesting features of the dilaton of the previous section, theory. All other terms present in the usual standard model, including massive fluctuations, could emerge from one more namely all fermion, gauge boson and Yukawa terms, when SUð2Þ ×Uð1Þ scalar field as part of the general theory that we minimally coupled to gravity are already automatically will discuss in Sec. III A. But for simplicity in this section we will invariant under the local Weyl symmetry. R first concentrate on the minimal model that contains only the ¼ 4 Lð Þ confirmed observed degrees of freedom up to now, thus allowing Hence a Weyl-invariant action S d x x that the fluctuations of a single ϕ to be eliminated by a gauge describes the coupling of gravity and the standard model symmetry in this minimal model. is given by

043515-5 ITZHAK BARS, PAUL STEINHARDT, AND NEIL TUROK PHYSICAL REVIEW D 89, 043515 (2014) 2 1 2 † 3 12 ðϕ − 2H HÞRðgÞ 6   7 6 7 6 þ μν 1 ∂ ϕ∂ ϕ − † 7 6 g 2 μ ν DμH DνH 7 pffiffiffiffiffiffi6   7 Lð Þ¼ − 6 7 (5) x g6 0 7: 6 − λ ðH†H − α2ϕ2Þ2 þ λ ϕ4 7 6 4 4 7 6   7 4 quarks; leptons; gauge bosons; 5 þL SM Yukawa couplings to H&ϕ; dark matter:

The only Yukawa couplings of ϕ allowed by SUð3Þ × ϕðxÞ → ϕ0: (7) SUð2Þ ×Uð1Þ are to the right-handed neutrinos for which it becomes a source of mass. Note the relative minus sign In the gauge fixed version of the Lagrangian where between ϕ and H terms, which is required, as explained ϕðxÞ → ϕ0, we can express the physically important below. Here LSM is the well-known standard model dimensionful parameters, namely, the Newton constant Lagrangian minimally coupled to gravity, except for the G, the cosmological constant Λ associated with dark Higgs kinetic and potential terms, which are now modified energy, and the electroweak scale v, in terms of ϕ0, and explicitly written out in the first three lines. This action 2 2 is invariant under Weyl rescaling with an arbitrary local 1 ϕ0 Λ 0 4 † 2 2 v Ωð Þ ¼ ; ¼ λ ϕ ;HH0 ¼ α ϕ ≡ : parameter x as follows, 16πG 12 16πG 0 0 0 2 −2 (8) gμν → Ω gμν; ϕ → Ωϕ;H→ ΩH; 3=2 γ;W;Z;g 0 γ;W;Z;g ϕð Þ ψ q;l → Ω ψ q;l;Aμ → Ω Aμ ; (6) Since the field x in this gauge ceases to be a degree of freedom altogether, the massless dilaton is absent and the where ψ q;l are the fermionic fields for quarks or leptons, potential problem with global scaling symmetry is avoided. γ;W;Z;g and Aμ are the gauge fields for the photon, gluons, W Nevertheless, there still is an underlying hidden conformal and Z. Note that the gauge bosons do not change under the symmetry for the full theory. Weyl rescaling. We can explain now why it is necessary to have the We note that the Lagrangian (5) is the one obtained in the opposite signs for ϕ and H in the first two lines of the action second reference in [16] from the 4 þ 2 version of the in Eq. (5). The positive sign for ϕ is necessary in order to standard model [14,15] coupled to 2 T-gravity [16],bya obtain a positive gravitational constant in (8). However, method of gauge fixing and solving some kinematical conformal symmetry then forces the kinetic term for ϕ to equations associated with constraints related to the under- have the wrong sign, so ϕ is a ghost. This can be seen to be lying gauge symmetries. In that approach we learned that a gauge artifact, though, since in a unitary gauge the ghost the Weyl symmetry is not an option in 3 þ 1 dimensions, it ϕ is fixed to a constant or expressed in terms of other is a prediction of 2 T physics: the 4 þ 2 dimensional theory degrees of freedom. This also explains why H must have is not Weyl invariant, but yet the local Weyl symmetry in the opposite sign, since otherwise H would be a real ghost. 3 þ 1 dimensions emerges as a remnant gauge symmetry This relative sign has important consequences in cosmol- associated with the general coordinate transformations as ogy as follows. they act in the extra 1 þ 1 dimensions. So the Weyl In flat space RðgÞ → 0, where experiments such as those symmetry is a required symmetry in 3 þ 1 dimensions at the LHC are conducted, the Lagrangian above becomes as predicted in the 4 þ 2 dimensional approach; this precisely the usual standard model, including the familiar symmetry carries information and imposes properties tachyonic mass term for the Higgs. Furthermore, in related to the extra 1 þ 1 space and time dimensions [13]. weak gravitational fields, at low energies, the gravitational The Weyl gauge symmetry of the action in (5) does not effect of the Higgs field coupling to the curvature, 1 2 † allow any dimensionful constants: no mass term in the 12 ðϕ0 − 2H HÞRðgÞ, is ignorable since H (order of Higgs potential, no Einstein-Hilbert term with its dimen- v ≈ 246 GeV) is tiny compared to the Planck scale 19 sionful Newton constant, or any other mass terms. This is a (ϕ0 ≈ 10 GeV). Actually, the gravitational constant mea- very appealing starting point because it leads to the sured at low energies is corrected by the electroweak scale, −1 2 2 emergence of all dimensionful parameters from a single namely ð16πGÞ ¼ðϕ0 − v Þ=12 instead of (8), but in 2 2 source. That source is the field ϕðxÞ that motivated this practice this is a negligible correction since v ≪ ϕ0. So, discussion in the previous section, and the only scale is then at low energies there is no discernible difference between generated by gauge fixing ϕ to a constant for all spacetime, our Weyl-lifted theory and the usual standard model. The

043515-6 LOCAL CONFORMAL SYMMETRY IN PHYSICS AND COSMOLOGY PHYSICAL REVIEW D 89, 043515 (2014) practically isolated standard model appears as a renorma- scalar fields ϕiðxÞ labeled by the index i. If there are lizable theory decoupled from gravity. complex fields, we can extract their real and imaginary However, in the cosmological context, the conformally parts and treat those as part of the ϕi. We introduce a Weyl coupled H can and will be large at some stages in the factor UðϕiÞ, a sigma-model-type metric in field space i i evolution of the universe. Working out the dynamics of Gijðϕ Þ and a potential Vðϕ Þ. The general Lagrangian cosmological evolution, it turns out that, for generic initial takes the form † conditions, H H typically grows to large scales over   ðϕ2 − 2 † Þ → 0 pffiffiffiffiffiffi 1 1 cosmological times, hitting 0 H H in the vicin- i i μν i j i L¼ −g Uðϕ ÞRðgÞ− G ðϕ Þg ∂μϕ ∂νϕ −Vðϕ Þ : ity of a big bang or big crunch singularity. This behavior 12 2 ij plays an essential role in cosmological evolution, as well as (10) in the resolution of the singularity via geodesic complete- ness. In our conformal theory, the standard model is not The results given in [18] are the following constraints on isolated from gravity, and we posit that the Higgs field can these functions: play a bigger role in nature than originally anticipated. (i) UðϕiÞ must be homogeneous of degree two, UðtϕiÞ¼ We may take the Higgs doublet in the unitary gauge of t2UðϕiÞ; and VðϕiÞ must be homogeneous of degree ð2Þ ð1Þ i 4 i i the SU ×U gauge symmetry, four, Vðtϕ Þ¼t Vðϕ Þ; and Gijðϕ Þ must be homo-   geneous of degree zero, G ðtϕiÞ¼G ðϕiÞ. 0 ij ij ð Þ¼ (ii) The following differential constraints must also be H x p1ffiffi ð Þ : (9) 2 s x satisfied. These may be interpreted as homothety conditions on the geometry in field space, ð Þ¼ þ δ ð Þ δ ð Þ The field s x v h x , where h x is the Higgs field ∂ U ¼ −2G ϕj; ϕi∂ U ¼ 2U; G ϕiϕj ¼ −U: fluctuation on top of the electroweak vacuum, is then i ij i ij identified with the generic field sðxÞ that appeared in our (11) cosmological papers [7–12]. The action for sðxÞ has the The second and third equations follow from the first exact same form in those papers as here. one and the homogeneity requirements. Thus, our previous analytical cosmological solutions can (iii) Physics requirements also include that G cannot have now be applied to investigate the cosmological properties ij more than one negative eigenvalue because the local of the Higgs field. When the parameter α vanishes, our scale symmetry is just enough to remove only one previous work provides the full set of geodesically com- negative norm ghost. However, if more gauge sym- plete analytic cosmological solutions for all initial con- metry that can remove more ghosts is incorporated, ditions of the relevant fields in the standard model coupled α then the number of negative eigenvalues can increase to gravity. The nonzero is then easily taken into account accordingly. The gauged R-symmetry in supergravity with numerical methods. This much analytic control over (which is automatic in the 4 þ 2 approach) is such an cosmological properties of a realistic theory is unprec- example. edented. This became a very valuable tool that led us into a In [18] these rules emerged from gauge symmetries in new cosmological scenario driven only by the Higgs field, 2 T gravity. Since Weyl symmetry is an automatic outcome as outlined in Sec. IV and completed in [27]. from 4 þ 2 dimensions, we can check that these same requirements follow directly in 3 þ 1 dimensions by III. GENERAL WEYL SYMMETRIC THEORY imposing Weyl symmetry on the general form in Eq. (10). The gauge symmetries derived from 2 T gravity and 2 T As an example, it is easy to check that all these supergravity in 4 þ 2 dimensions lead to the general Weyl- conditions are automatically satisfied by the action given ϕ invariant coupling described below for any number of in Eq. (5), with one and four real fields in the doublet H. 2 ≡ 2 † scalar fields in 3 þ 1 dimensions [18]. This generalizes the Using the symbol s,ass H H, we write it in the form possible forms of conformally coupled scalar theories ¼ ϕ2 − 2 ¼ ϕ4 ð ϕÞ ¼ η beyond those encompassed by Eq. (4) and allows for U s ;V f s= ;Gij ij; (12) richer possibilities for model building consistent with local where η is a flat Minkowski metric in the five-dimensional scale invariance. We ignore the spinors and gauge bosons ij field space with a single negative eigenvalue—this reduces whose minimal couplings to gravity are already automati- to a two-dimensional η in the unitary gauge of Eq. (9) cally Weyl symmetric. ij since indeed there is only a single physical field s. In our work [7–12] generally we let the potential fðpzÞffiffiffiffiffiffiffiffiffiffiffiffiffito be an A. Gravity arbitrary function of its argument z ¼ s=ϕ ¼ 2H†H=ϕ, We begin with gravity without . In the as allowed by the Weyl symmetry conditions above. In next subsection we will indicate the additional constraints Eq. (1) we have a purely quartic renormalizable potential, λ 2 2 2 2 λ0 4 that emerge in supergravity. We assume any number of real V ¼ 4 ðs − α ϕ Þ þ 4 ϕ . When quantum corrections are

043515-7 ITZHAK BARS, PAUL STEINHARDT, AND NEIL TUROK PHYSICAL REVIEW D 89, 043515 (2014) included, fðzÞ contains logarithmic corrections, but by Vðϕ;sIÞ¼ϕ4fðzÞ; with any fðzIÞ; (14) reexamining the underlying symmetry, the effective quan- tum potential can again be rewritten in the form ϕ4fðs=ϕÞ, I consistent with the local conformal symmetry. We use the GIJðz Þ¼any nonsingular n × n metric; (15) quantum corrected potential in our discussion of Higgs cosmology.   1 ∂ Next we give the general solution of the requirements u K G0IðzÞ¼GI0ðzÞ¼− þ 2GIKz ; (16) above in a convenient parametrization for n þ 1 fields 2 ∂zI labeled with i ¼ 0; 1; 2; …;n; namely, ϕi ¼ðϕ;sIÞ, where ϕ0 ≡ ϕ I ¼ 1 2 … is distinguished, while s with I ; ; ;nare all ∂ I u I J the other scalar fields. Then the general solution to the G00ðzÞ¼−u þ z þ z z G ; (17) ∂ I IJ conditions above for Weyl gauge symmetry takes the z following form, where the ratio zI ≡ sI=ϕ is gauge invariant. Thus, after using the chain rule, ∂=∂zI ¼ ϕ∂=∂sI, the general Weyl- Uðϕ;sIÞ¼ϕ2uðzÞ; with any uðzIÞ; (13) invariant action becomes

0 1 1 ðϕ Þ ð Þ − ðϕ Þ B 12 U ;s R g V ;s B   C pffiffiffiffiffiffiB þ 1 − I ∂U − K L ð∂μ ϕÞð∂ ϕÞ C L ¼ − B 2 U s I s s GKL ln μ ln C (18) gB ∂s A; @   − 1 ð∂μ I∂ JÞþ 2 J þ ∂U ∂μ I∂ ϕ 2 GIJ s μs GIJs ∂sI s μ ln

with the U; V; GIJ in Eqs. (13)–(15). It should be noted that phenomenology of the dilaton-like singlet discussed in I I I GIJðz Þ;uðz Þ;fðz Þ are not determined by Weyl symmetry Sec. II A can be included with slightly more freedom on its alone. Various other symmetries in a given model could coupling parameters. Keeping such possibilities in mind for restrict them. Any choice consistent with additional sym- more general phenomenological considerations, we the metries is permitted in the construction of physical models. define the minimal model to include only the standard For example, the model in Eq. (5) has the SUð3Þ ×SUð2Þ × Higgs doublet and the singlet ϕ as in the previous section. Uð1Þ symmetry. In particular, local superconformal sym- Hence, for clarity we will write out the general Weyl- metry gives more severe restrictions by relating Gij and U invariant Lagrangian for the case of only the Higgs doublet from the beginning, as discussed below. field H plus ϕ. This generalizes Eq. (5), but we suppress the As the number of scalar fields increases, the restrictions other fields in our discussion. Furthermore, we will work imposed by Weyl symmetry become less severe. For directly with the gauge fixed version of the Higgs field in example, an additional SUð3Þ ×SUð2Þ ×Uð1Þ gauge sin- Eq. (9), so the Higgs doublet is reduced to a single field glet field beyond ϕ that would be needed to reproduce the sðxÞ as in (9). In that case, from Eqs. (13)–(17) we obtain

2 3 1 ϕ2uðs=ϕÞRðgÞ − ϕ4fðs=ϕÞ 6 12   7 6 2 ! 7 pffiffiffiffiffiffi6 1 − s 0 − s ¯ ∂ ϕ∂ ϕ 7 LðxÞ¼ −g6 2 u ϕ u ϕ2 G μ ν 7; (19) 4 þgμν   5 1 ¯ 0 s ¯ − 2 G∂μs∂νs þ u þ 2 ϕ G ∂μϕ∂νs

¯ where GIJ reduces to G11ðsÞ ≡ GðsÞ for the single field. pffiffiffiffiffiffiffiffiffiffiffi Generally, G¯ ðs=ϕÞ;uðs=ϕÞ;fðs=ϕÞ in Eq. (19) are three arbitrary functions of the Higgs field, z ¼ s=ϕ ¼ 2HH=ϕ, which may be used for model building. As an example, if we take Uðϕ;sÞ¼ϕ2 þ ξs2, and G¯ ¼ 1, we obtain a relatively simple kinetic term with an arbitrary potential,

043515-8 LOCAL CONFORMAL SYMMETRY IN PHYSICS AND COSMOLOGY PHYSICAL REVIEW D 89, 043515 (2014) 2 3 1 2 2 4 12 ðϕ þ ξs ÞRðgÞ − ϕ fðs=ϕÞ space inadvertently, and such specialized restrictions on 6   ! 7 pffiffiffiffiffiffi6 2 7 fields is what leads to geodesic incompleteness. 1 1 − s ð1 þ ξÞ ∂ ϕ∂μϕ LðxÞ¼ −g6 2 ϕ2 μ 7: Hence, the a class of well motivated models that we 4 þ 5 ¯ 1 s μ used in many of our studies amounts to Gðs=ϕÞ¼1, − ∂μs∂νs þ 2 ð1 þ ξÞ∂μϕ∂ s 2 ϕ uðs=ϕÞ¼1 − s2=ϕ2, and the low-energy renorm- (20) alizable quartic potential Vðϕ;sÞ¼ϕ4 fðs=ϕÞ, with λ 2 2 2 2 λ0 fðs=ϕÞ¼4 ðs =ϕ − α Þ þ 4. This is the model that This Lagrangian is Weyl invariant for any value of the now is identical to the conformally symmetric standard constant parameter ξ, but we see now a simple generali- model we proposed. Its homogeneous cosmological equa- zation of the special simplifying role played by ξ ¼ −1 tions have been solved analytically exactly in [11] for all that corresponds to the conformally coupled scalars possible initial conditions of the fields, including radiation of Eq. (1). and curvature, and all possible values of the parameters 0 In the general single-s Lagrangian (19), field redefi- λ; λ , but with α ¼ 0. After this much analytic control, a nitions of the gauge invariant variable z → FðzÞ (or small α is easily handled with numerical methods and still s → ϕFðs=ϕÞ) may be used to map one of these three have a full understanding of all the cosmological solutions functions, G¯ ðs=ϕÞ;uðs=ϕÞ;fðs=ϕÞ, to any desired func- of the standard model. This forms the basis of our further tion of z without changing the form of the Lagrangian in work in cosmology that is discussed in the following (19). For example it may be convenient to take G¯ ¼ 1, sections. and still obtain all single-s Weyl-symmetric models by For further discussion, a useful gauge choice is ϕðxÞ¼ using all possible uðzÞ, fðzÞ. Another option is to take all ϕ0 for all spacetime as in Eq. (7). This is the gauge called possible G¯ ðzÞ;fðzÞ with a fixed uðzÞ¼1 − z2 as in the c-gauge in our previous work. Because we use also other gauges, we attach the letter c to each field in this Eq. (5), which has certain advantages for understanding μν geodesic completeness [10] of all the fields ϕ;s;gμν,in gauge, thus ϕc;sc;gc will remind us that we are in the cosmological spacetimes, as discussed in Sec. IV.Ifwe c-gauge, where ϕcðxÞ¼ϕ0.Inc-gauge we rename scðxÞ¼ choose uðzÞ¼1 − z2 and demand renormalizability of hðxÞ to recall that in this gauge we obtain the simplest the action in the limit of ϕ2 ≫ s2 (where gravity connection to the Higgs field hðxÞ at low energy, in nearly μν μν effectively decouples, as seen in the gauge ϕ ¼ ϕ0 with flat spacetime, gc ¼ η þ, as discussed in the para- ¯ ϕ0 ≫ s, we are forced to G ¼ 1 and fðzÞ being a quartic graphs before Eq. (9). The Lagrangian in Eq. (19) with ¯ polynomial. general u; G; f, takes the following greatly simplified form The last form is actually quite unique. We now give a in the c gauge, proof that the general model of Eq. (19) with a single s  pffiffiffiffiffiffiffiffi 1 (which corresponds to our minimal realistic model) can Lð Þ¼ − ϕ2 ð ϕ Þ ð Þ always be written in the geodesically complete form that we x gc 12 0u h= 0 R gc  advocated in our previous work [8,9] and implemented in 1 proposing the action in Eq. (5). We begin in the notation − ¯ ð ϕ Þ μν∂ ∂ − ϕ4 ð ϕ Þ (21) 2 G h= 0 gc μh νh 0f h= 0 : of Eq. (10) with only two fields ϕi ¼ðϕ;sÞ labeled by i ¼ 0, 1. Using well-known results of geometry in two dimensions, the metric in field space G ðϕÞ can always be If we apply a Weyl transformation to go to the Einstein ij 3 E c diagonalized by general field reparametrizations (as in frame , gμν ¼ uðh=ϕ0Þgμν, then we obtain ) and put into the conformally flat form  pffiffiffi ¼ ðϕÞη η pffiffiffiffiffiffiffiffiffi 1 1 ð∂ Þ2 þ ¯ ð ϕ Þ Gij g ij where ij is the flat metric in two dimen- 2 h=ϕ0 u G h= 0 LðxÞ¼ −gE ϕ0RðgEÞ − sions. Using further field reparametrizations with an overall 12 2 uðh=ϕ0Þ  rescaling consistent with Weyl transformations, the factor ð ϕ Þ ðϕÞ ðϕÞ¼1 μν∂ ∂ − ϕ4 f h= 0 (22) g can also be set equal to the constant g . Having × gE μh νh 0 2 : ðuðh=ϕ0ÞÞ arrived at Gij ¼ ηij as still the most general metric in field space, the only possible form of UðϕiÞ that is consistent with the local Weyl symmetry is UðϕiÞ¼ðϕ2 − s2Þ=12. Without loss of generality, a particularly simplifying choice ¯ ð ϕ Þ This completes the general proof that the action in Eq. (5) is for G h= 0 , namely the most general Weyl-invariant theory without losing pffiffiffi ¯ ð ϕ Þ¼ ð ϕ Þ − ð∂ Þ2 generality. Then the complete analysis of homogeneous G h= 0 u h= 0 h=ϕ0 u ; (23) cosmological solutions provided in our previous work [8,9] pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi shows that this form is the general geodesically complete 3 − ð Þ¼ − ð E Þ i pTheffiffiffi Weylpffiffiffiffiffiffiffiffiffi transformationpffiffiffi is gcuR gc gER gμν version of the theory. Certain other forms for Uðϕ Þ arrived þ6 ∂ ð − μν∂ Þ u μ gEgE ν u . After an integration by parts of the at by field reparametrizations, in particular purely positive last term (or by dropping a total derivative) we obtain the given forms of UðϕiÞ, always end up putting restrictions in field result.

043515-9 ITZHAK BARS, PAUL STEINHARDT, AND NEIL TUROK PHYSICAL REVIEW D 89, 043515 (2014) is the one that yields a canonically normalized Higgs field. normalized Higgs, σ, as described more generally in In this form, the theory can be interpreted directly as an footnote (4). Einstein frame formulation of the standard model with a Hence, the BS model has a fully Weyl symmetric canonically normalized Higgs field4 and an effective Higgs formulation which was not noticed before. Its presentation ð Þ potential Veff h given by in the literature has included various ambiguities and inconsistencies, with clashing ideas on scaling symmetries ð ϕ Þ ð Þ¼ϕ4 f h= 0 at the classical level and quantum corrections. For example, Veff h 0 2 : (24) ðuðh=ϕ0ÞÞ in some cases, the coupling to gravity has a dimensionful Newton constant that is inconsistent with the scaling This potential can be crafted by various choices of fðzÞ and symmetry of the rest of the theory; a massless dilaton is uðzÞ to fit cosmological observations, while still being said to exist in cases where global scaling symmetry is consistent with an underlying local conformal symmetry. explicitly broken at low energies; unimodular gravity has However, since this gauge led to an overall positive been introduced but this is inconsistent with scale sym- coefficient in front of the Ricci scalar, it could not recover metry; and there is ambiguity about which renormalization all field configurations that are demanded in the geodesi- scheme is appropriate for computing quantum corrections. cally complete theory; inevitably the generic solutions in These issues are fully resolved with the underlying Weyl this gauge are geodesically incomplete as we have learned symmetric formulation discussed here. Also, now that we in our previous work. have recast the BS model into a fully conformally invariant For example, the best fit inflaton potentials based on form, we can see in the gauge fixed Einstein frame (22)–(24) that it is not unique. Rather, it is just a special recent Planck satellite data [5] are “plateau models” [39]. case of a larger set of conformally invariant models Using the geodesically incomplete form of the theory in including a range that have similar plateau properties. (22) it is possible to construct examples with plateaus at 2 2 Furthermore, since U ¼ ϕ þ ξh is purely positive, it large h=ϕ0, by choosing f and u such that the potential ð Þ means the fields in the BS model form a basis that is Veff h is very slowly varying and approaches a constant at ð Þ geodesically incomplete, and hence its generic solutions large h and by requiring Veff h becomes approximately can describe only a subsector of the available field space. the familiar Higgs potential needed to fit low-energy We make further comments in Sec. IV including the effects ≪ ϕ physics for h 0. of geodesic incompleteness. The Bezrukov-Shaposhnikov (BS) model for Higgs inflation [23] is a special case of our more general form B. Supergravity in Eq. (21), namely their proposal  We do not know if supersymmetry (SUSY) is a property pffiffiffiffiffiffiffiffi 1 1 of nature, but theoretically it is an attractive possibility. LBSð Þ¼ − ðϕ2 þ ξ 2Þ ð Þ − μν∂ ∂ x gc 12 0 h R gc 2 gc μh νh Therefore, it is of interest to investigate whether it is  compatible with an underlying local conformal symmetry. λ λ0 2 2 2 2 4 As a superconformal local symmetry, the generalization of − ðh − α ϕ0Þ − ϕ0 ; (25) 4 4 the Weyl-invariant formalism of the previous section produces stronger constraints on scalar fields. This was → ðϕ2 þ ξ 2Þ ¯ → 1 follows from Eq. (21) by taking U 0 h ; G , derived in [18] from the gauge symmetry formalism in → ðλ ð 2 − α2ϕ2Þ2 þ λ0 ϕ4Þ and V 4 s 0 4 0 . This leads to an effective 4 þ 2 dimensions. It is possible to arrive at the results given potential Veff of the type above in Eq. (24), below directly in 3 þ 1 dimensions by requiring super-

0 gravity with a local superconformal symmetry, but pro- λ ðh2 − α2ϕ2Þ2 þ λ ϕ4 ðσÞ¼4 0 4 0 → ðσÞ vided the usual Einstein-Hilbert term is dropped (which is Veff 2 2 2 ; with h h : (26) ð1 þ ξh =ϕ0Þ unusual in the supergravity literature), and instead a Weyl symmetric formulation like the previous sections is imple- In the BS model, because G is chosen as G ¼ 1, the field h mented. In the 4 þ 2 dimensional approach of 2 T physics, is not canonically normalized, so h must be replaced there is no option: it is a prediction that the emergent 3 þ 1 through a field redefinition [23] to obtain a canonically dimensional theory is automatically invariant under a local symmetry SUð2; 2j1Þ, where SUð2; 2Þ¼SOð4; 2Þ is the 4 þ 2 ð1 1Þ ⊂ 4Alternatively, without fixing G¯ , it is possible to do a field connection to dimensions, the subgroup SO ; redefinition such that h is written in terms of a canonically SOð3; 1Þ ×SOð1; 1Þ ⊂ SOð4; 2Þ is the Weyl subgroup that normalized field σ, where the relation between σ and hðσÞ is acts on the extra 1 þ 1 dimensions, and the supersymmet- given bypffiffiffi the first order differential equation rization in 4 þ 2 dimensions promotes SU(2,2) to dh 2 2 ¯ ð Þ ½ð∂ ϕ uÞ þ Gðh=ϕ0Þ ¼ uðh=ϕ0Þ. Then the Lagrangian dσ h= 0 SUð2; 2jN Þ for N [17], with N ¼ 1 in rewritten in terms of the canonically normalized σ is again of ð ðσÞÞ the present case [18]. standard form with the same potential Veff h as Eq. (24), except for expressing h in terms of σ. This complicated procedure The scalar-field sector of the emergent 3 þ 1 dimen- is avoided by the simple choice of G in Eq. (23). sional locally superconformal theory is presented here

043515-10 LOCAL CONFORMAL SYMMETRY IN PHYSICS AND COSMOLOGY PHYSICAL REVIEW D 89, 043515 (2014) briefly in a streamlined fashion, including some simplifi- The Kähler metric and Kähler potential structure is rem- cations and extensions. First note that a scalar field in a iniscent of general supergravity [40], however the absence chiral supermultiplet must be complex. Thus SUSY of the Einstein-Hilbert term, and the corresponding scale requires the singlet ϕ to be complexified so that it is a invariance is the important difference. Here, Uðϕ; ϕ¯ Þ is member of a supermultiplet. Then to describe all the scalar homogeneous of degree two and satisfies homothety fields we use a complex basis ϕm and denote their complex constraints similar to Eq. (11), conjugates as ϕ¯ m¯ with the barred index m¯ . The notation is ¯ 2 ¯ reminiscent of standard supergravity as reviewed in [40]. Uðtϕ;tϕÞ¼t Uðϕ; ϕÞ; However now there is no Einstein-Hilbert term; instead 2 ∂Uðϕ; ϕ¯ Þ ∂ Uðϕ; ϕ¯ Þ there is a Weyl plus an additional Uð1Þ gauge symmetry ¼ ϕn ; and complex conjugate: ∂ϕ¯ m¯ ∂ϕn∂ϕm¯ that can gauge fix a complex scalar ϕ into a dimensionful real constant ϕ0 that plays the role of the Planck scale. (30) Recalling the discussion about removing the (real) ghost ðϕ ϕ¯ Þ¼ scalar field ϕ in Sec. II B, it should be noticed that the The potential energy has two parts VFþD ; ðϕ ϕ¯ Þþ ðϕ ϕ¯ Þ ðϕ ϕ¯ Þ complex ϕ amounts to two real ghosts, and therefore two VF ; VD ; . The potential VF ; must be ðϕÞ gauge symmetries are needed to remove them. This role is derived from an analytic superpotential f that depends ϕm ϕ¯ m¯ played by the Weyl symmetry SO(1,1) that acts on the extra only on and not on , and is homogeneous of degree 1 þ 1 dimensions and the local R-symmetry U(1), both of three so that VF is homogeneous of degree four, which are included in the local SUð2; 2j1Þ outlined above.   ∂ϕ¯ ⊗ ∂ϕ nm¯ ∂ ¯ ∂ The local U(1) is a crucial ingredient as a partner of the ð ϕÞ¼ 3 ðϕÞ ¼ − f f f t t f ;VF 2 ¯ ¯ m : Weyl symmetry to remove the additional ghost from a ∂ U ∂ϕn ∂ϕ complexified dilaton field ϕ that is demanded by SUSY. (31) Like before, we have Uðϕ; ϕ¯ Þ, but SUSY requires that ¯ ðϕ ϕ¯ Þ the metric G ¯ be derived from the derivatives of Uðϕ; ϕÞ The potential VD ; is derived from another mn ðϕÞ like a Kähler metric, independent analytic function zab with adjoint group indices a, b that appear in Yang-Mills terms as − 1 ð ðϕÞÞ a bμν ðϕÞ ∂2Uðϕ; ϕ¯ Þ 4 Re zab FμνF . This zab should be homo- G ¯ ¼ : (27) ð ϕÞ¼ ðϕÞ mn ∂ϕm∂ϕn¯ geneous of degree zero zab t zab . Then VD takes the form ð− Þ    The metric must be nonsingular and Gmn¯ cannot 1 ∂U ∂U contain any more than one negative eigenvalue (because V ðϕ;ϕ¯ Þ¼ Reðz−1Þ ðt ϕÞm ðt ϕ¯ Þn¯ ; (32) D 2 ab ∂ϕm a ∂ϕ¯ n¯ b no more than one complex ghost can¯ be removed). We nm¯ ¼ð∂ϕ⊗∂ϕÞnm¯ denote its inverse formally as G ∂2 . A simple U where ta is the matrix representation of the Yang-Mills quadratic example similar to Eq. (5) is ¯ group as it acts on the scalars ðϕm; ϕ¯ mÞ: The fermionic terms are added consistently with the usual rules of ðϕ ϕ¯ Þ¼ϕmϕ¯ n¯ η ¼ η U ; mn¯ ;Gmn¯ mn¯ ; supergravity [40].

where ηmn¯ ¼ diagð1; −1; …; −1Þ: (28) This is the general setup for the Weyl-invariant matter coupled to Weyl-invariant supergravity as derived from the 4 þ 2 In this example, all scalars in the theory are conformally dimensional theory. The simplest example that coupled, and U and G are automatically invariant under a corresponds to the supersymmetric generalization of global SU(1,N) symmetry. This global symmetry, which Eq. (5) would be the minimal supersymmetric standard will continue to be a hidden symmetry in some gauges, may model extended with an additional singlet supermultiplet, ϕ be broken explicitly by some terms in the potential V, whose scalar component is the complex field . Then U depending on the model considered. Nevertheless, keeping takes the quadratic form suggested in [18] track of this (broken) symmetry in physical applications ¼ ϕϕ¯ − † − † can be useful. U HuHu HdHd; (33) The scalar field sector of the supergravity theory with local Weyl symmetry is given as follows, where Hu, Hd are the two Higgs doublets needed in the supersymmetric version of the standard model. The  superpotential fðϕ;H ;H Þ and the matrix z ðϕ;H ;H Þ pffiffiffiffiffiffi 1 ∂2U d u ab d u L ¼ − ðϕ ϕ¯ Þ ð Þþ μν ϕm ϕ¯ n bose g U ; R g ¯ g Dμ Dν are chosen to fit the usual practice of supersymmetric 6 ∂ϕm∂ϕn ¼ δ  model building [40]. An example is zab ab, 03 ¯ and fðϕ;Hd;HuÞ¼gðHuHdÞϕ þ g , where ðHuHdÞ¼ − V þ ðϕ; ϕÞ : (29) F D α βε ð2Þ ð1Þ HuHd αβ is the only SU ×U invariant which is

043515-11 ITZHAK BARS, PAUL STEINHARDT, AND NEIL TUROK PHYSICAL REVIEW D 89, 043515 (2014) −2 analytic in both Hu and Hd. In an effective (rather than and gμνðxÞ → Ω ðxÞgμνðxÞ. For our cosmological discus- renormalizable) low- energy theory, these can be modified sion, geodesics xμðλÞ in a spatially homogeneous space 0 by replacing the dimensionless coupling constants g, g by are computed when both homogeneous fields that ð Þ ϕ2 2 an arbitrary function of the ratio HuHd = , and similarly appear in the particle action S, hðτÞ and gμνðτÞ¼a ðτÞ z ϕðxÞ 0 for ab. When the complex is gauge fixed to the real ðημν þ anisotropy & curvatureÞ, where τ ≡ x is the ϕ constant 0, this approach generates all the dimensionful conformal time, correspond to consistent solutions of the parameters from the same source ϕ0 which is of order of the 0 field equations of our conformal standard model in (5). Planck scale. Then we see that the modification of g, g and The full solution to the geodesic equation (neglecting ð Þ ϕ2 zab by arbitrary functions of HuHd = is negligible at curvature and anisotropy for simplicity) is low energies. Z More general scale-invariant models with more fields τ dτ0 (such as generalizations of the minimal SUSY model) are x⃗ðτÞ¼q⃗þ p⃗ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (34) ⃗2 þ 2ðτ0Þ 2ðτ0Þ easily constructed by using the rules on scalar fields given τ0 p m a in this section. With special forms of the superpotential ⃗ fðϕÞ, it is possible to construct so called “no-scale” models where p is the conserved spatial component of the μ ¼ ∂ ∂ : μ 0 ¼ [41–43] in which the cosmological constant is guaranteed particlepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi momentum vector, p S= x , while p 2 2 2 to be zero at the classical level even after spontaneous p⃗ þ m ðτÞa ðτÞ, and q⃗¼ x⃗ðτ0Þ is the initial position. breakdown of symmetries. Simple examples of no-scale These expressions are invariant under reparametrizations of models, that are lifted to be fully Weyl invariant, are given the affine parameter λ, since they depend only on the μ in [18]. physical spacetime variables x . Then, the local continu- In this paper we will not explore any further the general ation property of a geodesic x⃗ðτÞ in a homogeneous space is superconformal structures discussed above although this satisfied automatically when one insures that all the fields, could be of interest in some future applications. Some of including mðτÞ¼ghðτÞ and aðτÞ, are given in all the our examples that are very similar to the Weyl-invariant patches of homogeneous field space and that the fields are supersymmetric cases previously discovered in [18] were smoothly continued through singularities. In [10,11] and later explored in a cosmological context in [28]. However [11], we showed how local Weyl invariance, in the special their discussion, which is focussed on very specialized Weyl frame displayed in Eq. (5) that covers all the patches, initial conditions that are so nongeneric, is difficult to be are essential ingredients to insure that all homogeneous convincing especially in the face of the complete set of cosmological solutions of our conformal standard model solutions that we now understand much better. (5) have this continuation property. The global action for a geodesic (34) is given by IV. CONFORMAL COSMOLOGY Z τ 2ðτÞ 2ðτÞ j j¼ f τ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim a In this paper we have emphasized the geodesically Sfi d 2 2 2 : (35) τ ⃗ þ ðτÞ ðτÞ complete nature of our conformal standard model. In this i p m a section we first describe what we mean by geodesic 2ðτÞ 2ðτÞ¼ 2 2ðτÞ 2ðτÞ completeness and the related notion of completeness in The combination m a g h a that appears j j field space, and then discuss some applications of our in Sfi is invariant under Weyl transformations, so it conformal standard model in cosmology. can be computed in any convenient gauge [see e.g. – γ There are two required properties for geodesic com- Eqs. (36) (38)]. In particular in the gauge used in the ðτÞ¼1 pleteness: the first is smooth local geodesic continuation solutions in [10] and [11], where aγ , we have shown 2ðτÞ 2ðτÞ¼ 2ðτÞ 2ðτÞ through singularities across all space-time patches; the that in all cyclic solutions in [11], h a hγ aγ second is infinite action for geodesics that reach arbitrarily oscillates with a fixed maximum amplitude many times far into the past so that unnatural initial conditions are within a cycle, and for an infinite number of times over an avoided. Both properties are satisfied in our theory which infinite number of cycles, leading to an infinite action, since j j is insured to also be complete in field space. Geodesics Sfi receives the same positive finite contribution in each μ cycle. For these solutions the average magnitude of ja j in x ðλÞ are computedqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by extremizing the particle action E R the E-gauge (usual Einstein frame, see Eq. (36)) is the same ¼ − λ : μ : ν ð ðλÞÞ S d m x x gμν x , where m is the mass and in each cycle. Recently, we have also constructed [27] gμν is a gravitational field. In our Weyl-invariant standard solutions with increasing entropy in each future cycle such model (5) all particle masses are generated by the Higgs that the average jaEj in the E gauge increases in each future field, so m is proportional to the Higgs field h, cycle. For these solutions, jSfij diverges to the past [27] m ¼ ghðxðλÞÞ, where g is the dimensionless coupling even faster as compared to our solutions in [11]. Hence, constant for the corresponding particle as prescribed by unlike inflation scenarios [44], we may expect that the the standard model. Then this particle action is locally universe described by our conformal standard model has no invariant under Weyl transformations hðxÞ → ΩðxÞhðxÞ unnatural initial conditions.

043515-12 LOCAL CONFORMAL SYMMETRY IN PHYSICS AND COSMOLOGY PHYSICAL REVIEW D 89, 043515 (2014)

Next, we discuss various potential applications of of field space ðϕ;s;gμνÞ that are missing in the Einstein Weyl-invariant theories to cosmology. Early universe cos- frame can then be added in order to obtain a geodesically mology near the singularity is a natural place to look for uses complete space. We have argued in [10], and provided a because the interactions imposed by Weyl invariance (more proof in Sec. III A, that geodesic completeness is accom- 2 2 profoundly 4 þ 2 dimensional gauge symmetries) produce plishedP when we bring U to the form U ¼ ϕ − s , where 2 ¼ 2 ϕ their most significant changes relative to conventional s sI , is the sum of all scalars other than . physics in the limit of strong gravity, as noted in some of Geodesics remain incomplete when they hit the singu- the examples above. phenomena is another larity in all frames in which U is always positive, or the interesting arena for study, but we will not discuss them here. equivalent Einstein frame, such as the one in Eq. (25). When The first important application was to use Weyl invariance we rewrite those, by field redefinitions, in terms of new fields to construct geodesically complete cosmological solutions. in which U ¼ ϕ2 − s2, then the patches of field space ϕ2 ≥ Conventional cosmological analysis is usually confined to s2 are equivalent to the Einstein frame, or to the other frames theories coupled to Einstein gravity with a dimensionful with positive U. Geodesic completion is achieved by Newton’s constant. In this framework, all cosmological allowing all regions of field space in the parametrization solutions of interest are geodesically incomplete. However, that has U ¼ ϕ2 − s2; for this form, U is allowed to we have observed that a Weyl-invariant theory can be cast in smoothly go negative. In these coordinates the gauge Einstein frame, as illustrated by Eq. (22), and in other invariant vanishing point of U ¼ 0, given by the gauge frames. The observation made in [8–12] is that, geodesic invariant expression js=ϕj¼1, has a special significance; it incompleteness is an artifact of an unsuitable frame choice: corresponds to the singularity of the scale factor of the geodesically incomplete solutions in Einstein frame may be universe in the Einstein gauge. This is seen by equating the completed in other frames, even though the theories are gauge invariant ðdetðgÞÞ1=4Uðϕ;sÞ in the Einstein gauge, in entirely equivalent away from the singularity. which UðϕE;sEÞ¼6, and the unimodular gauge (labeled The Einstein frame can always be reached directly from a with γ), in which det ð−gγÞ¼1, as follows Weyl-invariant theory in (19) by making the Einstein-gauge 1=4 1=4 choice, ðdet ð−gÞÞ Uðϕ;sÞ¼ðdet ð−gEÞÞ × 6

1 1 ¼ 1 × Uðϕγ;sγÞ: (38) ðϕ Þ¼ð16π Þ−1 ¼ 12 U E;sE G 2 : (36) At spacetime points or regions where Uðϕγ;sγÞ¼ 2 2 We label this the E gauge and mark the fields in this gauge ϕγ − sγ ¼ 0, which is where the gauge invariant quantity ðϕ μνÞ j ϕj j ϕ j¼ j ϕ j with the subscript E (as in E;sE;gE ) to distinguish them s= hits unity in all gauges, sγ= γ sE= E μν ¼j ϕ j¼1 from the c-gauge fields ðϕc;sc;gc Þ. Note that the E gauge h= 0 , the geometry in the Einstein gauge fails can be valid only in a patch in field space since restrictions completely since det ð−gEÞ¼0, and this is how the cos- must be imposed on the fields to require that UðϕE;sEÞ is mological singularity occurs at some point in time [10]. positive. Then, as we can see in the c-gauge, the region h=ϕ0 ∼ 1 The same theory can easily be transformed to other (Higgs of Planck size) is the region of the big crunch or big gauges. For example, the relations between the E- and bang singularity, where js=ϕj¼1 in any gauge. The c-gauge fields can be easily derived by considering the behavior of the universe in this region is governed by a Weyl gauge invariants such as s=ϕ and ðdet ð−gÞÞ1=8ϕ, universal attractor mechanism that is independent of the ðdet ð−gÞÞ1=8s, etc.; for example, we deduce scalar potential, therefore independent of the details of the model [10]. sE h 1=8 1=8 Although in the classical theory, cosmological singular- ¼ ; and ðdetð−gcÞÞ ϕ0 ¼ðdetð−gEÞÞ ϕE; etc: ϕE ϕ0 ities of FRW type are typically resolved in the Weyl-lifted 2 2 (37) theory with U ¼ ϕ − s in suitable Weyl gauges, one should still worry about quantum gravity corrections. From these, we can express ϕEðhÞ and sEðhÞ in terms of the When U vanishes, the coefficient of the Ricci scalar in single field h, so that the gauge condition (36) is satisfied. the gravitational actionvanishes so there is no suppression of Inserting these expressions into the gauge invariant action metric fluctuations and quantum gravity corrections should (19) we arrive at the same E-frame action as Eq. (22). become large. However, it is notable that for certain types of As argued in our work [8–12], classically geodesically cosmic singularities, including realistic ones, the metric and complete solutions can be obtained for all single-scalar fields possess a unique continuation around the singularity theories that can be cast in the form of Eq. (19). But, for all in the complex time plane. A complex time path can be patches of field space ðϕ;s;gμνÞ to be included, as demanded chosen to remain far from the singularity so that U is always ð Þ by the geodesics derived from Veff h in the Einstein frame, large and gravity remains weak. Thus, we are able to find an ðU; G; fÞ must be brought to an appropriate form by using analytic continuation of our classical solutions connecting the field redefinitions discussed below Eq. (20). The patches big crunches to big bangs along which quantum gravity

043515-13 ITZHAK BARS, PAUL STEINHARDT, AND NEIL TUROK PHYSICAL REVIEW D 89, 043515 (2014) corrections are small. This issue is under active investigation The solution of Fig. 1 changes drastically if the vacuum and we defer further discussion to future work. is metastable after including quantum corrections, which is A particularly interesting application of Weyl invariance a possibility suggested by the most recent LHC data for the is to Higgs cosmology, where the geodesically complete Higgs and the top quark masses [19], and assuming no new solutions show cosmology can be much more interesting physics up to the Planck scale. Metastability is incompat- than conventionally assumed. In fact, there is the possibility ible with Higgs inflation and generally causes problems for that the Higgs field alone may be sufficient to explain the inflation because the Higgs will typically escape from the large-scale features of the universe, as suggested by the metastable phase right after the big bang and roll to a state Bezrukov-Shaposhnikov Higgs inflation model [23] and of negative energy density that can prevent inflation of any our recent work on the Higgs in cyclic cosmology [26,27]. sort from occurring. In considering Higgs cosmology or other models of the The exact solutions of the Weyl-invariant theory suggest early universe, we believe that the cosmological solutions an alternative cosmology in this case. The generic solution found for geodesically complete theories provide some at first behaves as in Fig. 1 after the big bang, all the way important new insights on the question of the likely initial through the electroweak phase transition. But after some conditions just after the big bang. For example, the generic time (order of the lifetime of the universe) the Higgs behavior of the Higgs after the bang when h ∼ ϕ0 is oscillations in the electroweak vacuum grow larger and dramatically different than the contrived initial conditions larger, like the mirror image of Fig. 1, taking away energy that are commonly assumed in Higgs inflation scenarios. from the gravitational field and eventually causing a This is easily seen by examining the analytic work in collapse of the universe to a big crunch, while the Higgs [10,11], where for the simple model in Eq. (5) with α ¼ 0, does a quantum tunneling to a lower state of the potential. all homogeneous solutions including curvature and radia- At that stage our exact analysis near the singularity given in tion are obtained and the effects of anisotropy are deter- [10] takes over to describe interesting new phenomena that mined [10]. Emphasizing that this is not only the contrived occur just after the crunch and before another rebirth of the solutions, but all solutions, it serves as an example of the universe with a big bang. The result is a regularly repeating richness of the phenomena that occur in a model even with sequence in which the Higgs field is trapped in its a simple potential. The results make clear a point that is metastable state after a big bang, remains there for a long obvious, but often forgotten: for a given scalar potential, period of expansion followed by contraction, escapes as the there is an enormous range of cosmological solutions. By Universe approaches the big crunch, passes through to a big comparison, it is clear that the slow-roll initial conditions bang and becomes trapped again. The evolution can be frequently assumed in the analysis of cosmic inflation are considered a Higgs-driven cyclic theory of the universe. very special and unlikely. For example, by transforming the The details are presented in a separate paper [27]. inflationary solution in the Bezrukov-Shaposhnikov Higgs inflation model [23] to a field basis in which U ¼ ϕ2 − s2, it may be possible to trace cosmic evolution right back to ACKNOWLEDGMENTS the singularity and to judge whether the inflation is likely in This research was partially supported by the U.S. the space of geodesically complete cosmological solutions. Department of Energy under Grant No. DE-FG03- Figure 1 is an illustration of our solution for the generic 84ER40168 (IB) and under Grant No. DE-FG02- cosmological behavior of the Higgs field just after the big 91ER40671 (P. J. S.). Research at Perimeter Institute is bang if the Higgs potential has a stable nontrivial minimum, supported by the Government of Canada through Industry as in Eq. (5), as usually assumed and as required for the Canada and by the Province of Ontario through the Bezrukov-Shaposhnikov Higgs inflation model [23]. This Ministry of Research and Innovation. The work of N. T. figure describes the generic cosmological evolution of the is supported in part by a grant from the John Templeton Higgs field, that must start with fluctuations of Planck size Foundation. The opinions expressed in this publication are and energy (due to the universal attractor near the singularity those of the authors and do not necessarily reflect the views [10]), and quickly reduce its amplitude by losing energy to of the John Templeton Foundation. I. B. thanks CERN and the gravitational field; then after a phase transition, settle P. J. S. thanks the Perimeter Institute for hospitality while down to an almost constant value at the electroweak scale v this research was completed. determined by the dimensionless parameter α in Eq. (3).

043515-14 LOCAL CONFORMAL SYMMETRY IN PHYSICS AND COSMOLOGY PHYSICAL REVIEW D 89, 043515 (2014) [1] P. A. M. Dirac, Ann. Math. 37, 429 (1936); G. Mack and A. [25] P. J. Steinhardt and N. Turok, Phys. Rev. D 65, 126003 Salam, Ann. Phys. (N.Y.) 53, 174 (1969). (2002). [2] H. Weyl, Spacetime Matter (Dover, New York, 1922), Sect. 35; [26] I. Bars, in Proc. Conf. on Conformal Nature of the Universe, Sitzungsber. Preuss Acad. d. Wissensch (1918), p. 465; re- Perimeter Institute (2012); N. Turok, in Proc. CERN printed in The Principles of Relativity (Dover, New York,1923). Colloquium (2012); in Proc. T. W. B. Kibble 80th Birthday [3] S. Deser, Ann. Phys. (N.Y.) 59, 248 (1970). Conference, Imperial College (2013); in Proc. Yale Physics [4] F. Englert, E. Gunzig, C. Truffin, and P. Windey, Phys. Lett. Department Colloquium (2013); P. Steinhardt, in Proc. 57B, 73 (1975); F. Englert, C. Truffin, and R. Gastmans, Davis Conference (2013). Nucl. Phys. B117, 407 (1976). [27] I. Bars, P. J. Steinhardt, and N. Turok, Phys. Lett. B 726,50 [5] P. Ade et al. (Planck Collaboration), arXiv:1303.5082. (2013). [6] J. L. Sievers, R. A. Hlozek, M. R. Nolta, V. Acquaviva, G. E. [28] S. Ferrara, R. Kallosh, A. Linde, A. Marrani, and A. Van Addison et al., J. Cosmol. Astropart. Phys. 10 (2013) 060. Proeyen, Phys. Rev. D 83, 025008 (2011); R. Kallosh and [7] I. Bars and S.-H. Chen, Phys. Rev. D 83, 043522 (2011). A. Linde, J. Cosmol. Astropart. Phys. 06 (2013) 027; 06 [8] I. Bars, S.-H. Chen, and N. Turok, Phys. Rev. D 84, 083513 (2013) 028. (2011). [29] I. Bars, S.-H. Chen, and G. Quelin, Phys. Rev. D 76, 065016 [9] I. Bars, arXiv:1109.5872; in Proceedings of the DPF- 2011 (2007); I. Bars and G. Quelin, ibid. 77, 125019 (2008). Conference, Providence, RI (2011), http://www.slac [30] M. Shaposhnikov, arXiv:0708.3550. .stanford.edu/econf/C110809/. [31] W. D. Goldberger, B. Grinstein, and W. Skiba, Phys. Rev. [10] I. Bars, S.-H. Chen, P. J. Steinhardt, and N. Turok, Phys. Lett. 100, 111802 (2008). Lett. B 715, 278 (2012). [32] R. Foot, A. Kobakhidze, and R. R. Volkas, Phys. Lett. B [11] I. Bars, S.-H. Chen, P. J. Steinhardt, and N. Turok, Phys. 655, 156 (2007). Rev. D 86, 083542 (2012). [33] R. Foot, A. Kobakhidze, K. McDonald, and R. Volkas, [12] I. Bars, arXiv:1209.1068; in Beyond the Big Bang, Com- Phys. Rev. D 76, 075014 (2007); 77,035006 (2008). peting Scenarios for an Eternal Universe, edited by [34] J. R. Espinosa and M. Quiros, Phys. Rev. D 76, 076004 (Springer, New York, 2013). (2007). [13] I. Bars, Int. J. Mod. Phys. A 25, 5235 (2010); [35] G. t’Hooft, in Proc. Conf. Conformal Nature of the [14] I. Bars, Phys. Rev. D 74, 085019 (2006). Universe, http://www.pirsa.org/C12027. [15] I. Bars, AIP Conf. Proc. 903, 550 (2007). [36] G. Belanger, U. Ellwanger, J. F. Gunion, Y. Jiang, S. Kraml, [16] I. Bars, Phys. Rev. D 77, 125027 (2008); I. Bars and S. H. and J. H. Schwarz, J. High Energy Phys. 01 (2013) 069. Chen, ibid. 79, 085021 (2009). [37] H. Cheng, Phys. Rev. Lett. 61, 2182 (1988); H. Nishino and [17] I. Bars and Y.-C. Kuo, Phys. Rev. Lett. 99, 041801 (2007); S. Rajpoot, arXiv:hep-th/0403039; Classical Quantum Phys. Rev. D 76, 105028 (2007); 79025001 (2009); Gravity 28, 145014 (2011); AIP Conf. Proc. 881,82 arXiv:1008.4761. (2007). [18] I. Bars, Phys. Rev. D 82, 125025 (2010). [38] S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 [19] G. Degrassi, S. Di Vita, J. Elias-Miró, J. R. Espinosa, G. F. (1973). Giudice, G. Isidori, and A. Strumia, J. High Energy Phys. 08 [39] A. Ijjas, A. Loeb, and P. J. Steinhardt, Phys. Lett. B 723, 261 (2012) 098. (2013). [20] W. A. Bardeen, Report No. FERMILAB-CONF-95-391- [40] See for example S. Weinberg, The Quantum Theory of TReport No. FERMILABCONF-95-377-T. Fields (Cambridge University Press, New York, 2000), [21] K. Meissner and H. Nicolai, Phys. Lett. B 648, 312 (2007); Vol. III. 660,260 (2008). [41] E. Cremmer, S. Ferrara, C. Kounnas, and D. Nanopoulos, [22] A. Codello, G. D’Odorico, C. Pagani, and R. Percacci, Phys. Lett. 133B, 61 (1983). Classical Quantum Gravity 30, 115015 (2013). [42] J. Ellis, C. Kounnas, and D. Nanopoulos, Nucl. Phys. B247, [23] F. L. Bezrukov and M. Shaposhnikov, Phys. Lett. B 659, 373 (1984); Phys. Lett. 143B, 410 (1984). 703 (2008). [43] A. Lahanas and D. Nanopoulos, Phys. Rep. 145, 1 (1987). [24] P. J. Steinhardt and N. Turok, Science 296, 1436 (2002). [44] A. Borde, A. H. Guth, and A. Vilenkin, Phys. Rev. Lett. 90, 151301 (2003).

043515-15