Ward Identities Research Articles

Quantum kinetic theory for spin transport of quarks with background chromo-electromagnetic fields

We derive the quantum kinetic equations for massive and massless quarks coupled with the background chromo-electromagnetic fields from the Wigner-function approach with the n expansion and effective power-counting scheme. For each case, one obtains coupled color-singlet and color-octet kinetic equations, which also involve the scalar and axial-vector components for the charge and spin transport. These kinetic equations delineate entangled evolution of the corresponding distribution functions decomposed in color space. At weak coupling, we derive the close form of the color-singlet kinetic equations for spin transport, which incorporates the diffusion term and the source term that triggers dynamical spin polarization led by correlation functions of color fields. Also, the non-dynamical source term is found in the axial Wigner function. The induced spin polarization and axial charge currents by these source terms are discussed under physical assumptions for color-field correlators in near-equilibrium quark gluon plasmas. In the constant-field limit, we further obtain non-vanishing axial Ward identities, from which we extract the pseudo-scalar condensate for massive quarks at finite temperature.

Sub-subleading soft graviton theorem from asymptotic Einstein’s equations

We identify in Einstein gravity an asymptotic spin-2 charge aspect whose conservation equation gives rise, after quantization, to the sub-subleading soft theorem. Our treatment reveals that this spin-2 charge generates a non-local spacetime symmetry represented at null infinity by pseudo-vector fields. Moreover, we demonstrate that the non-linear nature of Einstein’s equations is reflected in the Ward identity through collinear corrections to the sub-subleading soft theorem. Our analysis also provides a unified treatment of the universal soft theorems as conservation equations for the spin-0,-1,-2 canonical generators, while highlighting the important role played by the dual mass.

One-loop corrections to the spectral action

We analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability in a generalized sense, while staying within the spectral framework of noncommutative geometry. Our result is based on the perturbative expansion of the spectral action in terms of higher Yang-Mills and Chern-Simons forms. In the spirit of random noncommutative geometries, we consider the path integral over matrix fluctuations around a fixed noncommutative gauge background and show that the corresponding one-loop counterterms are of the same form so that they can be safely subtracted from the spectral action. A crucial role will be played by the appropriate Ward identities, allowing for a fully spectral formulation of the quantum theory at one loop.

Multiple-SLE κ connectivity weights for rectangles, hexagons, and octagons

In previous work, two of the authors determined, completely and rigorously, a solution space for a homogeneous system of 2N + 3 linear partial differential equations (PDEs) in 2N variables that arises in conformal field theory (CFT) and multiple Schramm–Löwner evolution (SLE κ ). The system comprises 2N null-state equations and three conformal ward identities that govern CFT correlation functions of 2N one-leg boundary operators or SLE κ partition functions. M Bauer et al conjectured a formula, expressed in terms of ‘pure SLE κ partition functions,’ for the probability that the growing curves of a multiple-SLE κ process join in a particular connectivity. In a previous article, we rigorously define certain elements of , which we call ‘connectivity weights,’ argue that they are in fact pure SLE κ partition functions, and show how to find explicit formulas for them in terms of Coulomb gas contour integrals. Our formal definition of the connectivity weights immediately leads to a method for finding explicit expressions for them. However, this method gives very complicated formulas where simpler versions may be available, and it is not applicable for certain values of κ ∈ (0, 8) corresponding to well-known critical lattice models in statistical mechanics. In this article, we determine expressions for all connectivity weights in for N ∈ {1, 2, 3, 4} (those with N ∈ {3, 4} are new) and for so-called ‘rainbow connectivity weights’ in for all . We verify these formulas by explicitly showing that they satisfy the formal definition of a connectivity weight. In appendix B, we investigate logarithmic singularities of some of these expressions, appearing for certain values of κ predicted by logarithmic CFT.

BRST, Ward identities, gauge dependence, and a functional renormalization group

Basic properties of gauge theories in the framework of the Faddeev-Popov (FP) method, Batalin-Vilkovisky (BV) formalism, and functional renormalization group (FRG) approach are considered. The FP and BV quantizations are characterized by the Becchi-Rouet-Stora-Tyutin (BRST) symmetry, while the BRST symmetry is broken in the FRG approach. It is shown that the FP method, the BV formalism, and the FRG approach can be provided with the Slavnov-Taylor identity, the Ward identity, and the modified Slavnov-Taylor identity, respectively. It is proven that using the background field method the background gauge invariance of the effective action within the FP and FRG quantization procedures can be achieved in nonlinear gauges. The gauge-dependence problem within the FP, BV, and FRG quantizations is studied. Arguments allowing us to state the existence of principal problems of the FRG in the case of gauge theories are given.

Topological hydrodynamic modes and holography

We study topological modes in relativistic hydrodynamics by weakly breaking the conservation of energy momentum tensor. Several systems have been found to have topologically nontrivial crossing nodes in the spectrum of hydrodynamic modes and some of them are only topologically nontrivial with the protection of reflection symmetries in two directions. The nontrivial topology for all these systems is further confirmed from a calculation of the topological invariant. Associated transport properties and second order effects have also been studied for these systems. The nonconservation terms of the energy momentum tensor could come from an external effective symmetric tensor matter field or a gravitational field which could be generated by a specific noninertial reference frame transformation from the original inertial reference frame. Finally we introduce a possible holographic realization of one of these systems. We propose a new method to calculate the holographic Ward identities for the energy momentum tensor without calculating out all components of the Green functions and match the Ward identities of both sides.

Emergent dual holographic description as a nonperturbative generalization of the Wilsonian renormalization group

In holographic duality, dynamics along the emergent extra-dimensional space describes a renormalization group (RG) flow of the corresponding quantum field theory (QFT). Following this idea, we develop an emergent holographic description of a QFT, where not only the information of the RG flow is introduced into an IR holographic dual effective field theory (HDEFT), but also the UV information of the QFT is encoded in the HDEFT through the IR boundary condition. In particular, we argue that this dual holographic construction is self-consistent within the assumption of bulk locality, showing the following two aspects: The solution of the Hamilton-Jacobi equation is given by the IR boundary effective action, and the Ward identity involving the QFT energy-momentum tensor current is satisfied naturally. We discuss the role of the RG $\beta$-function in the bulk effective dynamics of the metric tensor near a conformally invariant fixed point. We claim that this emergent dual gravity theory generalizes the perturbative Wilsonian RG framework into a non-perturbative way.

Fulfillment of sum rules and Ward identities in the multiloop functional renormalization group solution of the Anderson impurity model

We investigate several fundamental characteristics of the multiloop functional renormalization group (mfRG) flow by hands of its application to a prototypical many-electron system: the Anderson impurity model (AIM). We first analyze the convergence of the algorithm in the different parameter regions of the AIM. As no additional approximation is made, the multiloop series for the local self-energy and response functions converge perfectly to the corresponding results of the parquet approximation (PA) in the weak- to intermediate-coupling regime. Small oscillations of the mfRG solution as a function of the loop order gradually increase with the interaction, hindering a full convergence to the PA in the strong-coupling regime, where perturbative resummation schemes are no longer reliable. By exploiting the converged results, we inspect the fulfillment of (i) sum rules associated to the Pauli principle and (ii) Ward identities related to conservation laws. For the Pauli principle, we observe a systematic improvement by increasing the loop order and including the multiloop corrections to the self-energy. This is consistent with the preservation of crossing symmetries and two-particle self-consistency in the PA. For the Ward identities, we numerically confirm a visible improvement by means of the Katanin substitution. At weak coupling, violations of the Ward identity are further reduced by increasing the loop order in mfRG. In this regime, we also determine the precise scaling of the deviations of the Ward identity as a function of the electronic interaction. For larger interaction values, the overall behavior becomes more complex, and the benefits of the higher-loop terms are mostly present in the contributions at large frequencies.

Ward Identities in the $$\mathfrak {sl}_3$$ Toda Conformal Field Theory

Toda conformal field theories are natural generalizations of Liouville conformal field theory that enjoy an enhanced level of symmetry. In Toda conformal field theories this higher-spin symmetry can be made explicit, thanks to a path integral formulation of the model based on a Lie algebra structure. The purpose of the present document is to explain how this higher level of symmetry can manifest itself within the rigorous probabilistic framework introduced by R. Rhodes, V. Vargas and the first author. One of its features is the existence of holomorphic currents that are introduced via a rigorous derivation of the Miura transformation. More precisely, we prove that the spin-three Ward identities, that encode higher-spin symmetry, hold in the $\mathfrak{sl}_3$ Toda conformal field theory; as an original input we provide explicit expressions for the descendent fields which were left unidentified in the physics literature. This representation of the descendent fields provides a new systematic method to find the degenerate fields of the $\mathfrak{sl}_3$ Toda (and Liouville) conformal field theory, which in turn implies that certain four-point correlation functions are solutions of an hypergeometric differential equation of the third order.

Yangian Ward identities for fishnet four-point integrals

We derive and study Yangian Ward identities for the infinite class of four-point ladder integrals and their Basso-Dixon generalisations. These symmetry equations follow from interpreting the respective Feynman integrals as correlation functions in the biscalar fishnet theory. Alternatively, the presented identities can be understood as anomaly equations for a momentum space conformal symmetry. The Ward identities take the form of inhomogeneous extensions of the partial differential equations defining the Appell hypergeometric functions. We employ a manifestly conformal tensor reduction in order to express these inhomogeneities in compact form, which are given by linear combinations of Basso-Dixon integrals with shifted dimensions and propagator powers. The Ward identities naturally generalise to a one-parameter family of D-dimensional integrals representing correlators in the generalised fishnet theory of Kazakov and Olivucci. When specified to two spacetime dimensions, the Yangian Ward identities decouple. Using separation of variables, we explicitly bootstrap the solution for the conformal 2D box integral. The result is a linear combination of Yangian invariant products of Legendre functions, which reduce to elliptic K integrals for an isotropic choice of propagator powers. We comment on differences in the transcendentality patterns in two and four dimensions and their relations to discontinuities.

Rebooting quarter-BPS operators in mathcal{N} = 4 super Yang-Mills

We start a systematic study of quarter-BPS operators in four-dimensional mathcal{N} = 4 Super Yang-Mills with gauge group SU(N) making use of recently developed tools in conformal field theory. We adapt the technology of embedding space tensor structures in four dimensions to the problem of computing R-symmetry tensor structures, and we use the underlying chiral algebra to obtain the superconformal Ward identities. This allows us to fix the protected part of the four-point correlators, up to few ambiguities. As applications, we use the Lorentzian inversion formula to study the leading order OPE data in the large N supergravity limit and we make contact with the OPE limit of the five-point function of half-BPS operators.

Bootstrapping holographic defect correlators in mathcal{N} = 4 super Yang-Mills

We study two-point functions of single-trace half-BPS operators in the presence of a supersymmetric Wilson line in mathcal{N} = 4 SYM. We use modern analytic bootstrap in the form of an inversion formula, that reconstructs the CFT data starting from a single discontinuity of the correlator. In the planar strong coupling limit only a finite number of conformal blocks contributes to the discontinuity, which allows us to obtain elegant closed-form expressions for two-point functions of single-trace operators mathcal{O} J of weight J = 2, 3, 4. Our bootstrap approach guarantees that the correlator satisfies all the expected properties: it has the correct discontinuity, it satisfies the superconformal Ward identities, and it has a sensible expansion in both defect and bulk OPEs. The result is also consistent with available results coming from localization. The method is completely algorithmic and can be implemented to calculate correlators of arbitrary weight.

Gauge invariant operators in the SU(2) Higgs model: Ward identities and renormalization

The renormalization properties of two local gauge invariant composite operators $(O,{R}_{\ensuremath{\mu}}^{a})$ corresponding, respectively, to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are analyzed to all orders in perturbation theory by means of the algebraic renormalization in the $SU(2)$ Higgs model, with a single scalar in the fundamental representation, when quantized in the Landau gauge in Euclidean space-time. The present analysis generalizes earlier results presented in the case of the $U(1)$ Higgs model. A powerful global Ward identity, related to an exact custodial symmetry, is derived for the first time, with deep consequences at the quantum level. In particular, the gauge invariant vector operators ${R}_{\ensuremath{\mu}}^{a}$ turn out to be the conserved Noether currents of the above-mentioned custodial symmetry. As such, these composite operators do not renormalize, as expressed by the fact that the renormalization $Z$-factors of the corresponding external sources, needed to define the operators ${R}_{\ensuremath{\mu}}^{a}$ at the quantum level, do not receive any quantum corrections. Using this Ward identity, one can also prove that the longitudinal component of the two-point correlation function $⟨{R}_{\ensuremath{\mu}}^{a}(p){R}_{\ensuremath{\nu}}^{b}(\ensuremath{-}p)⟩$ exhibits only a tree level nonvanishing contribution which, moreover, is momentum independent, thus it is not associated to any physical propagating mode. Finally, we point out that the renowned nonrenormalization theorem for the ghost-antighost-vector boson vertex in Landau gauge remains true to all orders, also in the presence of the Higgs field.

Dyson gas on a curved contour

We introduce and study a model of a logarithmic gas with inverse temperature β on an arbitrary smooth closed contour in the plane. This model generalizes Dyson’s gas (the β-ensemble) on the unit circle. We compute the non-vanishing terms of the large N expansion of the free energy (N is the number of particles) by iterating the ‘loop equation’ that is the Ward identity with respect to reparametrizations and dilatation of the contour. We show that the main contribution to the free energy is expressed through the spectral determinant of the Neumann jump operator associated with the contour, or equivalently through the Fredholm determinant of the Neumann–Poincare (double layer) operator. This result connects the statistical mechanics of the Dyson gas to the spectral geometry of the interior and exterior domains of the supporting contour.

Nonperturbative running of the quark mass for Nf=3 QCD from the chirally rotated Schrödinger functional

We study the renormalization group (RG) running of the quark mass, for ${N}_{f}=3$ QCD with Wilson fermions in a mixed action setup, with standard Schr\"odinger functional (SF) boundary conditions for sea quarks and chirally rotated Schr\"odinger functional ($\ensuremath{\chi}\mathrm{SF}$) boundary conditions for valence quarks. This necessitates the tuning of the boundary factor ${z}_{\mathrm{f}}({g}_{0}^{2})$ of the $\ensuremath{\chi}\mathrm{SF}$ valence action, in order to ensure that QCD symmetries are fully recovered in the continuum. The properties of this novel setup are monitored through the ratios ${Z}_{S}/{Z}_{P}$ and ${\mathrm{\ensuremath{\Sigma}}}_{S}/{\mathrm{\ensuremath{\Sigma}}}_{P}$ of the renormalization parameters and step scaling functions of the scalar and pseudoscalar densities. Where comparison is possible, our ${Z}_{S}/{Z}_{P}$ results are found to agree with previous determinations, based on a mass ratio method [G. M. de Divitiis et al. (ALPHA Collaboration), Eur. Phys. J. C 79, 797 (2019)] and Ward identities [ J. Heitger et al. (ALPHA Collaboration), Eur. Phys. J. C 80, 765 (2020); J. Heitger et al., Eur. Phys. J. C 81, 606 (2021)], with Schr\"odinger functional boundary conditions. The behavior of ${\mathrm{\ensuremath{\Sigma}}}_{S}/{\mathrm{\ensuremath{\Sigma}}}_{P}$ confirms the theoretical expectations of $\ensuremath{\chi}\mathrm{SF}$ QCD, related to the restoration of the theory's symmetries in the continuum limit. From the step-scaling function of the pseudoscalar density we obtain the quark mass RG-running function from hadronic to perturbative energy scales. This is fully compatible with the earlier result obtained in a similar setup for Wilson quarks with Schr\"odinger functional boundary conditions [I. Campos et al., Eur. Phys. J. C 78, 387 (2018)], and provides a strong universality test for the two lattice setups.

Current noise and Keldysh vertex function of an Anderson impurity in the Fermi-liquid regime

We present a complete microscopic Fermi-liquid description for next-to-leading order transport through an Anderson impurity under a finite bias voltage $V$. It is applicable to multilevel quantum dots without particle-hole or time-reversal symmetry and is constructed based on the nonequilibrium Keldysh formalism, taking into account the current conservation between electrons in the impurity levels and the conduction bands. Specifically, we derive the formula for the current noise generated in the steady flow up to terms of order ${(eV)}^{3}$ at zero temperature $T=0$. To this end, we calculate the Keldysh vertex functions ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}};{\ensuremath{\sigma}}^{\ensuremath{'}}\ensuremath{\sigma}}^{{\ensuremath{\nu}}_{1}{\ensuremath{\nu}}_{2};{\ensuremath{\nu}}_{3}{\ensuremath{\nu}}_{4}}(\ensuremath{\omega},{\ensuremath{\omega}}^{\ensuremath{'}};{\ensuremath{\omega}}^{\ensuremath{'}},\ensuremath{\omega})$, which depend on branches ${\ensuremath{\nu}}_{1},\phantom{\rule{4pt}{0ex}}{\ensuremath{\nu}}_{2},\phantom{\rule{4pt}{0ex}}{\ensuremath{\nu}}_{3}$, and ${\ensuremath{\nu}}_{4}$ of the time-loop contour and on spin degrees of freedom $\ensuremath{\sigma}$ and ${\ensuremath{\sigma}}^{\ensuremath{'}}$, up to linear order terms with respect to $eV$, $T$, and frequencies $\ensuremath{\omega}$ and ${\ensuremath{\omega}}^{\ensuremath{'}}$. The coefficients of these linear order terms are determined by a set of the parameters, defined with respect to the equilibrium ground state: the phase shift, static susceptibilities, and nonlinear three-body susceptibilities of the impurity electrons. The low-energy expressions of the vertex components are shown to satisfy the Ward identities with the Keldysh Green's functions expanded up to terms of order ${\ensuremath{\omega}}^{2}$, ${(eV)}^{2}$, and ${T}^{2}$. We also find that the imaginary part of the Ward identities can be described in terms of the $eV$-dependent collision integrals for a single-quasiparticle excitation and that for a single quasiparticle-quasihole pair excitation. These collision integrals ensure the current conservation of the next-to-leading order Fermi-liquid transport due to the quasiparticles with a finite damping rate.

On refined Chern\u2013Simons and refined ABJ matrix models

We consider the matrix model of U(N) refined Chern–Simons theory on S^3 for the unknot. We derive a q-difference operator whose insertion in the matrix integral reproduces an infinite set of Ward identities which we interpret as q-Virasoro constraints. The constraints are rewritten as difference equations for the generating function of Wilson loop expectation values which we solve as a recursion for the correlators of the model. The solution is repackaged in the form of superintegrability formulas for Macdonald polynomials. Additionally, we derive an equivalent q-difference operator for a similar refinement of ABJ theory and show that the corresponding q-Virasoro constraints are equal to those of refined Chern–Simons for a gauge super-group U(N|M). Our equations and solutions are manifestly symmetric under Langlands duality qleftrightarrow t^{-1} which correctly reproduces 3d Seiberg duality when q is a specific root of unity.

Anomalies and supersymmetry

We revisit quantum field theory anomalies, emphasizing the interplay with diffeomorphisms and supersymmetry. The Ward identities of the latter induce Noether currents of all continuous symmetries, and we point out how these consistent currents are replaced by their covariant form through the appearance of the Bardeen-Zumino currents, which play a central role in our study. For supersymmetry Ward identities, two systematic methods for solving the Wess-Zumino consistency conditions are discussed: anomaly inflow and anomaly descent. The simplest inflows are from supersymmetric Chern-Simons actions in one dimension higher, which are used to supersymmetrize flavor anomalies in $d=4$ and, for $d=2$ $\mathcal{N}=(p,q)$, flavor anomalies with $p,q\leq 3$ and Lorentz-Weyl anomalies with $p,q\leq 6$. Finally, we extend the BRST algebra and the subsequent descent, a necessity for the diffeomorphism anomaly in retrospect. The same modification computes the supersymmetrized anomalies, and determines the above Chern-Simons actions when these exist.

Black hole superpotential as a unifying entropy function and BPS thermodynamics

We develop an effective superpotential formalism for the SU(2)×U(1) invariant sector of mathcal{N} = 2 gauged supergravity in five dimensions with a U(1)3 Fayet-Iliopoulos gauging, and determine the exact superpotential that describes all 1/4 BPS solutions in this sector. This includes the Gutowski-Reall black holes, but also a much broader class of solutions with a squashed S3, magnetic flux and vector multiplet sources, as well as complex Euclidean BPS saddles. Some of these solutions are known only numerically, but the exact superpotential allows us to analytically evaluate the on-shell action, holographic one-point functions and conserved charges of all BPS solutions and to study their thermodynamics. In particular, by examining the supersymmetry Ward identities we show that solutions with supersymmetric vector multiplet sources break supersymmetry spontaneously. We also demonstrate the first law for black holes in the SU(2)×U(1) invariant sector and show that the conserved charges of supersymmetric solutions satisfy the generalized BPS relation derived in [1], which includes the supersymmetric Casimir energy as a consequence of the anomalous supersymmetry transformation of the mathcal{N} = 1 supercurrent at the boundary. Finally, we show that the effective superpotential provides a unifying entropy extremization principle, reproducing Sen’s entropy function for near extremal black holes and the Hosseini-Hristov-Zaffaroni functional for complex Euclidean BPS saddles.

Chiral condensates and screening masses of neutral pseudoscalar mesons in thermomagnetic QCD medium

We point out that chiral condensates at nonzero temperature and magnetic fields are in strict connection to the space-time integral of corresponding two-point neutral meson correlation functions in the pseudoscalar channel via the Ward-Takahashi identity. Screening masses of neutral pseudoscalar mesons, which are defined as the exponential decay of the corresponding spatial correlation functions in the long distance, thus are intrinsically connected to (inverse) magnetic catalysis of chiral condensates. To study this we performed lattice simulations of ($2+1$)-flavor QCD on ${32}^{3}\ifmmode\times\else\texttimes\fi{}{N}_{t}$ lattices with pion mass ${M}_{\ensuremath{\pi}}\ensuremath{\simeq}220\text{ }\text{ }\mathrm{MeV}$ in a fixed scale approach having temperature $T\ensuremath{\in}[17,281]\text{ }\text{ }\mathrm{MeV}$ and magnetic field strength $eB\ensuremath{\in}[0,2.5]\text{ }\text{ }{\mathrm{GeV}}^{2}$. We find that screening lengths, i.e., inverses of screening masses of ${\ensuremath{\pi}}^{0}$, ${K}^{0}$ and ${\ensuremath{\eta}}_{s\overline{s}}^{0}$, turn out to have the similar complex $eB$ and $T$ dependences of the corresponding chiral condensates. Although the transition temperature is found to always decrease as $eB$ grows, we show that the suppression due to magnetic fields becomes less significant for hadron screening length and chiral condensates with heavier quarks involved, and ceases to occur for ${\ensuremath{\eta}}_{s\overline{s}}^{0}$ and strange quark chiral condensate. The complex $eB$ and $T$ dependences of both screening masses and chiral condensates, reflecting the crossover nature of the QCD transition, are attributed to the competition between sea and valence quark effects. These findings could be useful to guide low-energy models and effective theories of QCD.