We analyze the hard scattering amplitude of the gravitational form factors (GFFs) of hadrons at one-loop, in relation to their conformal field theory (CFT) description, within the framework of QCD factorization for hard exclusive processes at large momentum transfers. These form factors play an essential role in studying the quark and gluon angular momentum of the hadrons due to their relation to the Mellin moments of the Deeply virtual Compton scattering (DVCS) invariant amplitudes. Our analysis is performed using a diffeomorphism invariant approach, applying the formalism of the gravitational effective action and conformal symmetry in momentum space for the discussion of the quark and gluon contributions. The interpolating correlator in the hard scattering of any GFF is the non-Abelian $\textit{TJJ}$ (stress-energy/gluon/gluon) 3-point function at $$O(\alpha _s^2)$$ , revealing an effective dilaton interaction in the $\textit{t}$-channel due to the trace anomaly, in the form of a massless anomaly pole in the QCD hard scattering. We investigate the role of quarks, gauge-fixing and ghost contributions in the reconstruction of the hard scattering amplitude mediated by this interaction, performed in terms of its transverse traceless, longitudinal, and trace decomposition, as identified from CFT in momentum space ( $$\hbox {CFT}_p$$ ). We present a convenient parameterization of the hard scattering amplitude relevant for future experimental investigations of the DVCS/GFF amplitudes at the electron-ion collider at BNL.

Italy

Claudio Corianò

Stefano Lionetti

Dario Melle

Riccardo Tommasi

The gravitational form factors of hadrons from CFT in momentum space and the dilaton in perturbative QCD

We discuss extension of the electroweak gauge group to <math altimg='si1.svg'><mi>S</mi><mi>U</mi><msub><mrow><mo stretchy='false'>(</mo><mn>3</mn><mo stretchy='false'>)</mo></mrow><mrow><mi>L</mi></mrow></msub><mo>&#215;</mo><mi>S</mi><mi>U</mi><msub><mrow><mo stretchy='false'>(</mo><mn>3</mn><mo stretchy='false'>)</mo></mrow><mrow><mi>R</mi></mrow></msub><mo>&#215;</mo><mi>U</mi><msub><mrow><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><mrow><mi>L</mi></mrow></msub><mo>&#215;</mo><mi>U</mi><msub><mrow><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><mrow><mi>R</mi></mrow></msub></math>, based on the principle that anomaly cancellation may inherently involve three fermion generations. The explanation for three light quark-lepton families remains, but new particles are introduced which have no TeV scale upper limit on their masses. This approach, originally realized in the framework of the 331 Model, has been consistently defined up to the TeV scale. Our approach maintains the 331 Model's natural explanation for three light quark-lepton families, while introducing new physics with no upper bound on its energy scale.

Paul H. Frampton

The 33311 left-right bilepton model

We illustrate how the conformal Ward identities in momentum space completely determine the structure of a parity-odd three-point correlator involving currents, energy-momentum tensors, and at least one scalar operator in <math><mi>d</mi><mo>=</mo><mn>4</mn></math>. Conformal invariance fixes almost all such possible correlators to vanish. The only exceptions are given by the <math><mo>⟨</mo><mi>J</mi><mi>J</mi><mi>O</mi><msub><mo>⟩</mo><mrow><mi>odd</mi></mrow></msub></math> and the <math><mo>⟨</mo><mi>T</mi><mi>T</mi><mi>O</mi><msub><mo>⟩</mo><mrow><mi>odd</mi></mrow></msub></math> which in momentum space are protected by chiral and conformal anomalies. Specifically, one can obtain a nonvanishing solution by considering scalar operators such as <math><mi>O</mi><mo>=</mo><mo>∇</mo><mo>·</mo><msub><mi>J</mi><mi>A</mi></msub></math>, <math><mi>O</mi><mo>=</mo><msub><mi>g</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><msup><mi>T</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup></math>, or their shadow transforms. We comment on the implications of these results that constrain the coupling of axions and dilatons in a conformal phase of the early universe.

CFT constraints on parity-odd interactions with axions and dilatons

We investigate the general structure of the chiral anomaly <math><mi>A</mi><mi>V</mi><mi>V</mi><mo>/</mo><mi>A</mi><mi>A</mi><mi>A</mi></math> and <math><mo>(</mo><mi>L</mi><mi>L</mi><mi>L</mi><mo>,</mo><mi>R</mi><mi>R</mi><mi>R</mi><mo>)</mo></math> vertices, in the presence of chemical potentials in perturbation theory. The study finds application in anomalous transport, whenever chirally unbalanced matter is present, with propagating external currents that are classically conserved. Examples are topological materials and the chiral magnetic effect in the plasma state of matter of the early universe. We classify the minimal number of form factors of the <math><mi>A</mi><mi>V</mi><mi>V</mi></math> parametrization, by a complete analysis of the Schouten identities in the presence of a heat bath. We show that the longitudinal (anomaly) sector in the axial-vector channel, for on-shell and off-shell photons, is protected against corrections coming from the insertion of a chemical potential in the fermion loop. When the photons are on-shell, we prove that the transverse sector, in the same channel, is also <math><mi>μ</mi></math>-independent and vanishes. The related effective action is shown to be always described by the exchange of a massless anomaly pole, as in the case of vanishing chemical potentials. The pole is interpreted as an interpolating axionlike quasiparticle generated by the anomaly. In each axial-vector channel, it is predicted to be a correlated fermion/antifermion pseudoscalar (axionlike) quasiparticle appearing in the response function, once the material is subjected to an external chiral perturbation. The cancellation of the <math><mi>μ</mi></math> dependence extends to any chiral current within the Standard Model, including examples such as <math><mi>B</mi></math> (baryon), <math><mi>L</mi></math> (lepton), and <math><mi>B</mi><mo>−</mo><mi>L</mi></math>. This holds true irrespective of whether these currents exhibit anomalies.

Mario Cretì

Axionlike quasiparticles and topological states of matter: Finite density corrections of the chiral anomaly vertex

We investigate the gravitational anomaly vertex <math><mo>⟨</mo><mi>T</mi><mi>T</mi><msub><mi>J</mi><mn>5</mn></msub><mo>⟩</mo></math> (graviton—graviton—axial current) under conditions of finite density and temperature. Through a direct analysis of perturbative contributions, we demonstrate that neither finite temperature nor finite fermion density affects the gravitational chiral anomaly. These results find application in several contexts, from topological materials to the early Universe plasma. They affect the decay of any axion or axionlike particle into gravitational waves, in very dense and hot environments.

Gravitational chiral anomaly at finite temperature and density

We illustrate how the conformal Ward identities (CWI) and the gravitational chiral anomaly completely determine the structure of the <math><mo>⟨</mo><mi>T</mi><mi>T</mi><msub><mi>J</mi><mn>5</mn></msub><mo>⟩</mo></math> (graviton-graviton-chiral gauge current) correlator in momentum space. This analysis extends our previous results on the anomaly vertices <math><mo>⟨</mo><mi>A</mi><mi>V</mi><mi>V</mi><mo>⟩</mo></math> and <math><mo>⟨</mo><mi>A</mi><mi>A</mi><mi>A</mi><mo>⟩</mo></math>, as well as the <math><mo>⟨</mo><mi>T</mi><mi>J</mi><mi>J</mi><mo>⟩</mo></math> parity-odd conformal anomaly vertex in general CFTs. The <math><mo>⟨</mo><mi>T</mi><mi>T</mi><msub><mi>J</mi><mn>5</mn></msub><mo>⟩</mo></math> plays a fundamental role in the analysis of the conformal backreaction in early Universe cosmology, affecting the particle content and the evolution of the primordial plasma. Our approach is nonperturbative and not Lagrangian based, requiring the inclusion of a single anomaly pole in the solution of the anomaly constraint. The pole and its residue, along with the CWIs, determine the entire correlator in all of its sectors (longitudinal/transverse), all of which are proportional to the same anomaly coefficient. The method does not rely on a specific expression of the <math><mi>C</mi><mi>P</mi></math>-odd anomalous current, which in free field theory can be represented either by a bilinear fermion current or by a gauge-dependent Chern-Simons current; it relies solely on the symmetry constraints. We compute the correlator perturbatively at one loop in free field theory and verify its exact agreement with the nonperturbative result. A comparison with the perturbative analysis confirms the presence of a sum rule satisfied by the correlator, similar to the parity-even <math><mo>⟨</mo><mi>T</mi><mi>J</mi><mi>J</mi><mo>⟩</mo></math> and the chiral <math><mo>⟨</mo><mi>A</mi><mi>V</mi><mi>V</mi><mo>⟩</mo></math>.

Germany

Matteo Maria Maglio

Parity-violating CFT and the gravitational chiral anomaly

We derive the conformal constraints satisfied by classical vertices of a (Einstein) Gauss-Bonnet theory around flat space, in general dimensions and at <math altimg='si1.svg'><mi>d</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>4</mn></math> (4d EGB). In 4 <italic>d</italic> EGB they are obtained by a singular limit of the integral of the Euler-Poincar&#232; density. Our analysis exploits the relation between this theory and the conformal anomaly action, which allows to uncover some interesting features of the GB vertex at cubic and quartic level. If we introduce a conformal decomposition of the metric, the resulting theory can be formulated in two different versions, which are regularization dependent, a local one which is quartic in the dilaton field, and a nonlocal one, with a quadratic dilaton. The nonlocal version is derived by a finite redefinition of the GB density with the inclusion of a <math altimg='si2.svg'><mo stretchy='false'>(</mo><mi>d</mi><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>4</mn><mo stretchy='false'>)</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math> correction, before performing the singular <math altimg='si3.svg'><mi>d</mi><mo stretchy='false'>&#8594;</mo><mn>4</mn></math> limit. In the local version of the theory, we show how the independent dynamics of the metric and of the dilaton are interwinded by a classical trace identity. Three-gravitational wave interactions can be organised in a nontrivial way by using directly the nonlocal 4d EGB version of the theory. This is possible thanks to the consistency of such formulation - only up to 3-point functions - directly inherited from the conformal anomaly (Riegert) action. The constraints satisfied by the vertices are classical, hierarchical Ward identities. At quartic level, similar relations are derived, borrowing from the analysis of the counterterms of the 4T correlators of the conformal anomaly action, as defined by a perturbative expansion. For <math altimg='si4.svg'><mi>d</mi><mo>&#8800;</mo><mn>4</mn></math> these constraints hold also for Lovelock actions. They can be extended to higher order topological invariants in such class of theories.

Mario Cretì

Three-wave and four-wave interactions in the 4 <italic>d</italic> Einstein Gauss-Bonnet (EGB) and Lovelock theories

We elaborate on anomaly induced actions of the Wess-Zumino (WZ) form and their relation to the renormalized effective action, which is defined by an ordinary path integral over a conformal sector, in an external gravitational background. In anomaly - induced actions, the issue of scale breaking is usually not addressed, since these actions are obtained only by solving the trace anomaly constraint and are determined by scale invariant functionals. We investigate the changes induced in the structure of such actions once identified in dimensional renormalization (DR) when the <math><mi>ϵ</mi><mo>=</mo><mi>d</mi><mo>−</mo><mn>4</mn><mo>→</mo><mn>0</mn></math> limit is accompanied by the dimensional reduction (DRed) of the field dependencies. We show that operatorial nonlocal modifications <math><mo>(</mo><mo>∼</mo><msup><mrow><mo>□</mo></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>)</mo></math> of the counterterms are unnecessary to justify a scale anomaly. In this case, only the ordinary finite subtractions play a critical role in the determination of the scale breaking. This is illustrated for the WZ form of the effective action and its WZ consistency condition, as seen from a renormalization procedure. Logarithmic corrections from finite subtractions are also illustrated in a pure <math><mi>d</mi><mo>=</mo><mn>4</mn></math> (cutoff) scheme. The interplay between two renormalization schemes, one based on dimensional regularization (DR) and the second on a cutoff in <math><mi>d</mi><mo>=</mo><mn>4</mn></math>, illustrates the ambiguities of DR in handling the quantum corrections in a curved background. Therefore, using DR in a curved background, the scale and trace anomalies can both be obtained by counterterms that are Weyl invariant only at <math><mi>d</mi><mo>=</mo><mn>4</mn></math>.

Broken scale invariance and the regularization of a conformal sector in gravity with Wess-Zumino actions

We analyze the parity-odd correlators  $$\langle JJO\rangle _{odd},$$  <math> <mrow> <msub> <mrow> <mo>⟨</mo> <mi>J</mi> <mi>J</mi> <mi>O</mi> <mo>⟩</mo> </mrow> <mrow> <mi>odd</mi> </mrow> </msub> <mo>,</mo> </mrow> </math>    $$\langle JJT\rangle _{odd},$$  <math> <mrow> <msub> <mrow> <mo>⟨</mo> <mi>J</mi> <mi>J</mi> <mi>T</mi> <mo>⟩</mo> </mrow> <mrow> <mi>odd</mi> </mrow> </msub> <mo>,</mo> </mrow> </math>    $$\langle TTO\rangle _{odd}$$  <math> <msub> <mrow> <mo>⟨</mo> <mi>T</mi> <mi>T</mi> <mi>O</mi> <mo>⟩</mo> </mrow> <mrow> <mi>odd</mi> </mrow> </msub> </math>   and  $$\langle TTT\rangle _{odd}$$  <math> <msub> <mrow> <mo>⟨</mo> <mi>T</mi> <mi>T</mi> <mi>T</mi> <mo>⟩</mo> </mrow> <mrow> <mi>odd</mi> </mrow> </msub> </math>   in momentum space, constrained by conformal Ward identities, extending our former investigation of the parity-odd chiral anomaly vertex. We investigate how the presence of parity-odd trace anomalies affect such correlators. Motivations for this study come from holography, early universe cosmology and from a recent debate on the chiral trace anomaly of a Weyl fermion. In the current CFT analysis, O can be either a scalar or a pseudoscalar operator and it can be identified with the trace of the stress–energy tensor. We find that the  $$\langle JJO\rangle _{odd}$$  <math> <msub> <mrow> <mo>⟨</mo> <mi>J</mi> <mi>J</mi> <mi>O</mi> <mo>⟩</mo> </mrow> <mrow> <mi>odd</mi> </mrow> </msub> </math>   and  $$\langle TTO\rangle _{odd}$$  <math> <msub> <mrow> <mo>⟨</mo> <mi>T</mi> <mi>T</mi> <mi>O</mi> <mo>⟩</mo> </mrow> <mrow> <mi>odd</mi> </mrow> </msub> </math>   can be different from zero in a CFT. This occurs when the conformal dimension of the scalar operator is  $$\Delta _3=4,$$  <math> <mrow> <msub> <mi>Δ</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>4</mn> <mo>,</mo> </mrow> </math>   as in the case of  $$O=T^{\mu }_{\mu }.$$  <math> <mrow> <mi>O</mi> <mo>=</mo> <msubsup> <mi>T</mi> <mi>μ</mi> <mi>μ</mi> </msubsup> <mo>.</mo> </mrow> </math>   Moreover, if we assume the existence of parity-odd trace anomalies, the conformal  $$\langle JJT\rangle _{odd}$$  <math> <msub> <mrow> <mo>⟨</mo> <mi>J</mi> <mi>J</mi> <mi>T</mi> <mo>⟩</mo> </mrow> <mrow> <mi>odd</mi> </mrow> </msub> </math>   and  $$\langle TTT\rangle _{odd}$$  <math> <msub> <mrow> <mo>⟨</mo> <mi>T</mi> <mi>T</mi> <mi>T</mi> <mo>⟩</mo> </mrow> <mrow> <mi>odd</mi> </mrow> </msub> </math>   are nonzero. In particular, in the case of  $$\langle JJT\rangle _{odd}$$  <math> <msub> <mrow> <mo>⟨</mo> <mi>J</mi> <mi>J</mi> <mi>T</mi> <mo>⟩</mo> </mrow> <mrow> <mi>odd</mi> </mrow> </msub> </math>   the transverse–traceless component is constrained to vanish, and the correlator is determined only by the trace part with the anomaly pole.

Matteo Maglio

CFT correlators and CP-violating trace anomalies

We illustrate how the Conformal Ward Identities (CWI) in momentum space for parity-odd correlators determine the structure of a chiral anomaly interaction, taking the example of the VVA (vector/vector/axial-vector) and AAA correlators in momentum space. Only the conservation and the anomalous WIs, together with the Bose symmetry, are imposed from the outset for the determination of the correlators. We use a longitudinal/transverse decomposition of tensor structures and form factors. The longitudinal (L) component is fixed by the anomaly content and the anomaly pole, while in the transverse (T) sector we define a new parameterization. We relate the latter both to the Rosenberg original representation of the VVA and to the longitudinal/transverse (L/T) one, first introduced in the analysis of  $$g-2$$  <math> <mrow> <mi>g</mi> <mo>-</mo> <mn>2</mn> </mrow> </math>   of the muon in the investigation of the diagram in the chiral limit of QCD. The correlators are completely identified by the conformal constraints whose solutions are fixed only by the anomaly coefficient, the residue of the anomaly pole. In both cases, our CFT result matches the one-loop perturbative expression, as expected. The CWIs for correlators of mixed chirality  $$J_L J J_R$$  <math> <mrow> <msub> <mi>J</mi> <mi>L</mi> </msub> <mi>J</mi> <msub> <mi>J</mi> <mi>R</mi> </msub> </mrow> </math>   generate solutions in agreement with the all-orders nonrenormalization theorems of perturbative QCD and in the chiral limit.