<math><mrow><mi>S</mi></mrow></math>-matrix bootstrap and positivity bounds are usually viewed as constraints on low-energy theories imposed by the requirement of a standard UV completion. By considering graviton-photon scattering in the standard model, we argue that the low-energy theory can be used to put constraints on the UV behavior of the gravitational scattering amplitudes.

United Kingdom

Lasma Alberte

United States

Claudia de Rham

Sumer Jaitly

Andrew J. Tolley

Reverse Bootstrapping: IR Lessons for UV Physics

We apply positivity bounds directly to a <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> gauge theory with charged scalars and charged fermions, i.e., QED, minimally coupled to gravity. Assuming that the massless <math><mi>t</mi></math>-channel pole may be discarded, we show that the improved positivity bounds are violated unless new physics is introduced at the parametrically low scale <math><msub><mi>Λ</mi><mi>new</mi></msub><mo>∼</mo><mo>(</mo><mi>e</mi><mi>m</mi><msub><mi>M</mi><mi>Pl</mi></msub><msup><mo>)</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math>, consistent with similar results for scalar field theories, far lower than the scale implied by the weak gravity conjecture. This is sharply contrasted with previous treatments which focus on the application of positivity bounds to the low energy gravitational Euler-Heisenberg effective theory only. We emphasize that the low cutoff is a consequence of applying the positivity bounds under the assumption that the pole may be discarded. We conjecture an alternative resolution that a small amount of negativity, consistent with decoupling limits, is allowed and is not in conflict with standard UV completions, including weakly coupled ones.

QED positivity bounds

The presence of a massless spin-2 field in an effective field theory results in a <math><mi>t</mi></math>-channel pole in the scattering amplitudes that precludes the application of standard positivity bounds. Despite this, recent arguments based on compactification to three dimensions have suggested that positivity bounds may be applied to the <math><mi>t</mi></math>-channel pole subtracted amplitude. If correct, this would have deep implications for UV physics and the weak gravity conjecture. Within the context of a simple renormalizable field theory coupled to gravity we find that applying these arguments would constrain the low-energy coupling constants in a way which is incompatible with their actual values. This contradiction persists on deforming the theory. Further enforcing the <math><mi>t</mi></math>-channel pole subtracted positivity bounds on such generic renormalizable effective theories coupled to gravity would imply new physics at a scale parametrically smaller than expected, with far-reaching implications. This suggests that generically the standard positivity bounds are inapplicable with gravity, and we highlight a number of issues that impinge on the formulation of a three-dimensional amplitude which simultaneously satisfies the required properties of analyticity, positivity, and crossing symmetry. We conjecture instead a modified bound that ought to be satisfied independently of the precise details of the high energy completion.

Positivity bounds and the massless spin-2 pole

We explore the effective field theory for single and multiple interacting pseudo-linear spin-2 fields. By applying forward limit positivity bounds, we show that among the parameters contributing to elastic tree level scattering amplitude, there is no region of compatibility of the leading interactions with a standard local UV completion. Our result generalizes to any number of interacting pseudo-linear spin-2 fields. These results have significant implications for the organization of the effective field theory expansion for pseudo-linear fields.

Claudia Rham

Arshia Momeni

Justinas Rumbutis

Andrew Tolley

Positivity constraints on interacting pseudo-linear spin-2 fields

The spontaneous breaking of boost invariance is ubiquitous in nature, yet the associated Goldstone bosons are nowhere to be seen. We discuss why some subtleties are to be expected in the Goldstone phenomenon for spontaneously broken boosts, and derive the corresponding quantum mechanical, non-perturbative Goldstone theorem. Despite similarities with more standard Goldstone theorems, we show by examples that ours can be obeyed by quite unusual spectra of low-energy excitations. In particular, for non- relativistic Fermi liquids, we prove that it is obeyed by the particle-hole continuum. To the best of our knowledge, this is the first example of a Goldstone theorem obeyed by a continuum rather than by (approximately stable) single-particle Goldstone boson states in an interacting theory at zero temperature.

Alberto Nicolis

Spontaneously broken boosts and the Goldstone continuum

The consistency of the EFT of two interacting spin-2 fields is checked by applying forward limit positivity bounds on the scattering amplitudes to exclude the region of parameter space devoid of a standard UV completion. We focus on two classes of theories that have the highest possible EFT cutoff, namely those theories modelled on ghost-free interacting theories of a single massive spin-2 field. We find that the very existence of interactions between the spin-2 fields implies more stringent bounds on all the parameters of the EFT, even on the spin-2 self-interactions. This arises for two reasons. First, with every new field included in the low-energy EFT, comes the ‘knowledge’ of an extra pole to be subtracted, hence strengthening the positivity bounds. Second, while adding new fields increases the number of free parameters from the new interactions, this is rapidly overcome by the increased number of positivity bounds for different possible scattering processes. We also discuss how positivity bounds appear to favour relations between operators that effectively raise the cutoff of the EFT.

Positivity constraints on interacting spin-2 fields

We consider the effective field theory of multiple interacting massive spin-2 fields. We focus on the case where the interactions are chosen so that the cutoff is the highest possible, and highlight two distinct classes of theories. In the first class, the mass eigenstates only interact through potential operators that carry no derivatives in unitary gauge at leading order. In the second class, a specific kinetic mixing between the mass eigenstates is included non-linearly. Performing a decoupling and ADM analysis, we point out the existence of a ghost present at a low scale for the first class of interactions. For the second class of interactions where kinetic mixing is included, we derive the full Λ -decoupling limit and confirm the absence of any ghosts. Nevertheless both formulations can be used to consistently describe an EFT of interacting massive spin-2 fields which, for a suitable technically natural tuning of the EFT, have the same strong coupling scale Λ . We identify the generic form of EFT corrections in each case. By using Galileon Duality transformations for the specific case of two massive spin-2 fields with suitable couplings, the decoupling limit theory is shown to be a bi-Galileon.

EFT of interacting spin-2 fields

Phonons in solid materials can be understood as the Goldstone bosons of the spontaneously broken spacetime symmetries. As such, their low energy dynamics are greatly constrained and can be captured by standard effective field theory methods. In particular, knowledge of the nonlinear stress-strain curves completely fixes the full effective Lagrangian at leading order in derivatives. We attempt to illustrate the potential of effective methods focusing on the so-called hyperelastic materials, which allow large elastic deformations. We find that the self-consistency of the effective field theory imposes a number of bounds on physical quantities, mainly on the maximum strain and maximum stress that can be supported by the medium. In particular, for stress-strain relations that at large deformations are characterized by a power-law behavior <math><mrow><mi>σ</mi><mo>(</mo><mi>ϵ</mi><mo>)</mo><mo>∼</mo><msup><mrow><mi>ϵ</mi></mrow><mrow><mi>ν</mi></mrow></msup></mrow></math>, the maximum strain exhibits a sharp correlation with the exponent <math><mi>ν</mi></math>.

Italy

Greece

Matteo Baggioli

Spain

Víctor Cáncer Castillo

Oriol Pujolàs

Elasticity bounds from effective field theory

We present a class of holographic massive gravity models that realize a spontaneous breaking of translational symmetry—they exhibit transverse phonon modes whose speed relates to the elastic shear modulus according to elasticity theory. Massive gravity theories thus emerge as versatile and convenient theories to model generic types of translational symmetry breaking: explicit, spontaneous, and a mixture of both. The nature of the breaking is encoded in the radial dependence of the graviton mass. As an application of the model, we compute the temperature dependence of the shear modulus and find that it features a glasslike melting transition.

Germany

Martin Ammon

Amadeo Jiménez-Alba

Holographic Phonons

We study the elastic response of planar black hole (BH) solutions in a simple class of holographic models with broken translational invariance. We compute the transverse quasi-normal mode spectrum and the propagation speed of the lowest energy mode. We find that the speed of the lowest mode relates to the BH rigidity modulus as dictated by elasticity theory. This allows to identify these modes as transverse phonons — the pseudo Goldstone bosons of spontaneously broken translational invariance. In addition, we show that these modes have a mass gap controlled by an explicit source of the translational symmetry breaking. These results provide a new confirmation that the BHs in these models do exhibit solid properties that become more manifest at low temperatures. Also, by the AdS/CFT correspondence, this allows to extend the standard results from the effective field theory for solids to quantum-critical materials.