Gauge Invariant Operators Research Articles

Perturbative renormalization of quasi-parton distribution functions

In this paper we present results for the renormalization of gauge invariant nonlocal fermion operators which contain a Wilson line, to one-loop level in lattice perturbation theory. Our calculations have been performed for Wilson/clover fermions and a wide class of Symanzik improved gluon actions. The extended nature of such long-link operators results in a nontrivial renormalization, including contributions which diverge linearly as well as logarithmically with the lattice spacing, along with additional finite factors. On the lattice there is also mixing among certain subsets of these nonlocal operators; we calculate the corresponding finite mixing coefficients, which are necessary in order to disentangle individual matrix elements for each operator from lattice simulation data. Finally, extending our perturbative setup, we present non-perturbative prescriptions to extract the linearly divergent contributions.

Gauge invariants, correlators and holography in bosonic and fermionic tensor models

Motivated by the close connection of tensor models to the SYK model, we use representation theory to construct the complete set of gauge invariant observables for bosonic and fermionic tensor models. Correlation functions of the gauge invariant operators in the free theory are computed exactly. The gauge invariant operators close a ring. The structure constants of the ring are described explicitly. Finally, we construct a collective field theory description of the bosonic tensor model.

Far-from-equilibrium initial conditions probed by a nonlocal observable

Using the gauge/gravity duality, we investigate the evolution of an out-of-equilibrium strongly-coupled plasma from the viewpoint of the two-point function of scalar gauge-invariant operators with large conformal dimension. This system is out of equilibrium due to the presence of anisotropy and/or a massive scalar field. Considering various functions for the initial anisotropy and scalar field, we conclude that the effect of the anisotropy on the evolution of the two-point function is considerably more than the effect of the scalar field. We also show that the ordering of the equilibration time of the one-point function for the non-probe scalar field and the correlation function between two points with a fixed separation can be reversed by changing the initial configuration of the plasma, when the system is out of the equilibrium due to the presence of at least two different sources like our problem. In addition, we find the equilibration time of the two-point function to be linearly increasing with respect to the separation of the two points with a fixed slope, regardless of the initial configuration that we start with. Finally we observe that, for larger separations the geodesic connecting two points on the boundary crosses the event horizon after it has reached its final equilibrium value, meaning that the two-point function can probe behind the event horizon.

Lepton number violation at colliders from kinematically inaccessible gauge bosons

We reevaluate the necessity of W_R gauge bosons being kinematically accessible to test the left–right symmetric model (LRSM) at hadron colliders. In the limit that W_R are too heavy, resonant production of sub-TeV Majorana neutrinos N can still proceed at the Large Hadron Collider (LHC) via the process pprightarrow W_R^{pm *}rightarrow N ell ^pm rightarrow ell ^pm ell ^pm +nj if mediated by a far off-shell W_R. Traditional searches strategies are insensitive to this regime as they rely on momenta of final states scaling with TeV-scale M_{W_R}. For such situations, the process is actually kinematically and topologically identical to the direct production (DP) process pprightarrow W_mathrm{SM}^{pm *} rightarrow N ell ^pm rightarrow ell ^pm ell ^pm +nj. In this context, we reinterpret sqrt{s}=8 TeV LHC constraints on DP rates for the minimal LRSM. For m_N = 200–500 GeV and right–left coupling ratio kappa _R = g_R/g_L, we find (M_{W_R} / kappa _R)> 1.1–1.8 TeV at 95% CLs. Expected sensitivities to DP at 14 (100) TeV are also recast: with mathscr {L}=1~(10) ab^{-1}, one can probe (M_{W_R} / kappa _R) < 7.9–8.9 (14–40) TeV for m_N = 100–700 (1200) GeV, well beyond the anticipated sensitivity of resonant W_R searches. Findings in terms of gauge invariant dimension-six operators with heavy N are also reported.

Feynman rules for the Standard Model Effective Field Theory in R\u03be-gauges

We assume that New Physics effects are parametrized within the Standard Model Effective Field Theory (SMEFT) written in a complete basis of gauge invariant operators up to dimension 6, commonly referred to as “Warsaw basis”. We discuss all steps necessary to obtain a consistent transition to the spontaneously broken theory and several other important aspects, including the BRST-invariance of the SMEFT action for linear Rξ-gauges. The final theory is expressed in a basis characterized by SM-like propagators for all physical and unphysical fields. The effect of the non-renormalizable operators appears explicitly in triple or higher multiplicity vertices. In this mass basis we derive the complete set of Feynman rules, without resorting to any simplifying assumptions such as baryon-, lepton-number or CP conservation. As it turns out, for most SMEFT vertices the expressions are reasonably short, with a noticeable exception of those involving 4, 5 and 6 gluons. We have also supplemented our set of Feynman rules, given in an appendix here, with a publicly available Mathematica code working with the FeynRules package and producing output which can be integrated with other symbolic algebra or numerical codes for automatic SMEFT amplitude calculations.

Right-handed charged currents in the era of the Large Hadron Collider

We discuss the phenomenology of right-handed charged currents in the frame-work of the Standard Model Effective Field Theory, in which they arise due to a single gauge-invariant dimension-six operator. We study the manifestations of the nine complex couplings of the W to right-handed quarks in collider physics, flavor physics, and low-energy precision measurements. We first obtain constraints on the couplings under the assumption that the right-handed operator is the dominant correction to the Standard Model at observable energies. We subsequently study the impact of degeneracies with other Beyond-the-Standard-Model effective interactions and identify observables, both at colliders and low-energy experiments, that would uniquely point to right-handed charged currents.

Exact holography of the mass-deformed M2-brane theory

We test the holographic relation between the vacuum expectation values of gauge invariant operators in {mathcal {N}} = 6 U_k(N)times mathrm{U}_{-k}(N) mass-deformed ABJM theory and the LLM geometries with {mathbb {Z}}_k orbifold in 11-dimensional supergravity. To do so, we apply the Kaluza–Klein reduction to construct a 4-dimensional gravity theory and implement the holographic renormalization procedure. We obtain an exact holographic relation for the vacuum expectation values of the chiral primary operator with conformal dimension Delta = 1, which is given by langle {mathcal {O}}^{(Delta =1)}rangle = N^{frac{3}{2}} , f_{(Delta =1)}, for large N and k=1. Here the factor f_{(Delta )} is independent of N. Our results involve an infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a non-trivial test of gauge/gravity duality away from the conformal fixed point. We extend our results to the case of kne 1 for LLM geometries represented by rectangular-shaped Young diagrams. We also discuss the exact mapping of the gauge/gravity at finite N for classical supersymmetric vacuum solutions in field theory side and corresponding classical solutions in gravity side.

Gauge invariant gluon spin operator for spinless nonlinear wave solutions

We consider nonlinear wave type solutions with intrinsic mass scale parameter and zero spin in a pure SU(2) quantum chromodynamics (QCD). A new stationary solution which can be treated as a system of static Wu–Yang monopole dressed in off-diagonal gluon field is proposed. A remarkable feature of such a solution is that it possesses a finite energy density everywhere. All considered nonlinear wave type solutions have common features: presence of the mass scale parameter, nonvanishing projection of the color fields along the propagation direction and zero spin. The last property requires revision of the gauge invariant definition of the spin density operator which is supposed to produce spin one states for the massless vector gluon field. We construct a gauge invariant definition of the classical gluon spin density operator which is unique and Lorentz frame independent.

Abelian Higgs model with charge conjugate boundary conditions

The abelian Higgs model is studied on the lattice with charge conjugate boundary conditions. A locally gauge invariant operator for the charged scalar field is constructed and the charged scalar particle mass is calculated in the Coulomb phase of the lattice model. Agreement is found with the mass calculated in Coulomb gauge. The gauge invariant scalar field operator is used to calculate the Higgs boson mass in the Higgs region and to show that the charged particle disappears from the spectrum in the confined regime.

On form factors and correlation functions in twistor space

In this paper, we continue our study of form factors and correlation functions of gauge-invariant local composite operators in the twistor-space formulation of mathcal{N}=4 super Yang-Mills theory. Using the vertices for these operators obtained in our recent papers [1, 2], we show how to calculate the twistor-space diagrams for general NkMHV form factors via the inverse soft limit, in analogy to the amplitude case. For general operators without α indices, we then reexpress the NMHV form factors from the position-twistor calculation in terms of momentum twistors, deriving and expanding on a relation between the two twistor formalisms previously observed in the case of amplitudes. Furthermore, we discuss the calculation of generalized form factors and correlation functions as well as the extension to loop level, in particular providing an argument promised in [3].

Three loop form factors of a massive spin-2 particle with nonuniversal coupling

We investigate the interaction of spin-2 fields with those of Standard Model in a model independent framework. We have considered interactions where the spin-2 fields couple to two sets of gauge invariant tensorial operators that are not conserved unlike the energy momentum tensor with different coupling strengths. Such interactions not only change the ultraviolet behaviour of the couplings but also expand the scope of the searches of spin-2 particles at the colliders. We present all the relevant renormalisation constants up to three loop level in QCD and also the form factors that contribute to potential observables. This sets the ground to investigate the phenomenological consequences of these interactions with spin-2 fields through more than one tensorial operator.

Electric Properties of One-Neutron Halo Nuclei in Halo EFT

We exploit the separation of scales in weakly-bound nuclei to compute E2 transitions and electric form factors in a Halo EFT framework. The relevant degrees of freedom are the core and the halo neutron. The EFT expansion is carried out in powers of $R_{core}/R_{halo}$, where $R_{core}$ and $R_{halo}$ denote the length scales of the core and halo, respectively. We include the strong $s$-wave and $d$-wave interactions by introducing dimer fields. The dimer propagators are regulated by employing the power divergence subtraction scheme and matched to the effective range expansion in the respective channel.Electromagnetic interactions are included via minimal substitution in the Lagrangian. We demonstrate that, depending on the observable and respective partial wave, additional local gauge-invariant operators contribute in LO, NLO and higher orders.

Quantum phase transition in many-flavor supersymmetricQED3

We study $\mathcal{N}=4$ supersymmetric QED in three dimensions, on a three-sphere, with 2N massive hypermultiplets and a Fayet-Iliopoulos parameter. We identify the exact partition function of the theory with a conical (Mehler) function. This implies a number of analytical formulas, including a recurrence relation and a second-order differential equation, associated with an integrable system. In the large N limit, the theory undergoes a second-order phase transition on a critical line in the parameter space. We discuss the critical behavior and compute the two-point correlation function of a gauge invariant mass operator, which is shown to diverge as one approaches criticality from the subcritical phase. Finally, we comment on the asymptotic 1/N expansion and on mirror symmetry.

Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models

Certain models with rank-$3$ tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large $N$ limit, where $g^2 N^3$ is held fixed. In this limit the perturbative expansion in the quartic coupling constant, $g$, is dominated by a special class of "melon" diagrams. We study "uncolored" models of this type, which contain a single copy of real rank-$3$ tensor. Its three indexes are distinguishable; therefore, the models possess $O(N)^3$ symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large $N$ limit dominated by the melon diagrams. The quantum mechanics of a real anti-commuting tensor therefore has a similar large $N$ limit to the model recently introduced by Witten as an implementation of the Sachdev-Ye-Kitaev (SYK) model which does not require disorder. Gauging the $O(N)^3$ symmetry in our quantum mechanical model removes the non-singlet states; therefore, one can search for its well-defined gravity dual. We point out, however, that the model possesses a vast number of gauge-invariant operators involving higher powers of the tensor field, suggesting that the complete gravity dual will be intricate. We also discuss the quantum mechanics of a complex 3-index anti-commuting tensor, which has $U(N)^2\times O(N)$ symmetry and argue that it is equivalent in the large $N$ limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension $d$. While the quartic interaction is not positive definite, we construct the large $N$ Schwinger-Dyson equation for the two-point function and show that its solution is consistent with conformal invariance. We carry out a perturbative check of this result using the $4-\epsilon$ expansion.

On the gauge-invariant operator [formula omitted] in Euclidean Yang-Mills theories

We review our recent work on the gauge-invariant non-local dimension-two operator Amin2, whose minimization is defined along the gauge orbit. Albeit non-local, the operator Amin2 can be cast in local form through the introduction of an auxiliary Stueckelberg field. The whole procedure results into a local action which turns out to be renormalizable to all orders.

A quantum check of AdS/dCFT

We build the framework for performing loop computations in the defect version of mathcal{N} = 4 super Yang-Mills theory which is dual to the probe D5-D3 brane system with background gauge-field flux. In this dCFT, a codimension-one defect separates two regions of space-time with different ranks of the gauge group and three of the scalar fields acquire non-vanishing and space-time-dependent vacuum expectation values. The latter leads to a highly non-trivial mass mixing problem between different colour and flavour components, which we solve using fuzzy-sphere coordinates. Furthermore, the resulting space-time dependence of the theory’s Minkowski space propagators is handled by reformulating these as propagators in an effective AdS4. Subsequently, we initiate the computation of quantum corrections. The one-loop correction to the one-point function of any local gauge-invariant scalar operator is shown to receive contributions from only two Feynman diagrams. We regulate these diagrams using dimensional reduction, finding that one of the two diagrams vanishes, and discuss the procedure for calculating the one-point function of a generic operator from the SU(2) subsector. Finally, we explicitly evaluate the one-loop correction to the one-point function of the BPS vacuum state, finding perfect agreement with an earlier string-theory prediction. This constitutes a highly non-trivial test of the gauge-gravity duality in a situation where both supersymmetry and conformal symmetry are partially broken.

BPS/CFT correspondence II: Instantons at crossroads, moduli and compactness theorem

Gieseker-Nakajima moduli spaces $M_{k}(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\bf R}^{4}$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\bf C\bf P}^{2}$ with ${\rm ch}_{2}({\mathcal{E}}) = k$. They also serve as local models of the moduli spaces of instantons on general four-manifolds. We study the generalization of gauge theory in which the four dimensional spacetime is a stratified space $X$ immersed into a Calabi-Yau fourfold $Z$. The local model ${\bf M}_{k}({\vec n})$ of the corresponding instanton moduli space is the moduli space of charge $k$ (noncommutative) instantons on origami spacetimes. There, $X$ is modelled on a union of (up to six) coordinate complex planes ${\bf C}^{2}$ intersecting in $Z$ modelled on ${\bf C}^{4}$. The instantons are shared by the collection of four dimensional gauge theories sewn along two dimensional defect surfaces and defect points. We also define several quiver versions ${\bf M}_{\bf k}^{\gamma}({\vec{\bf n}})$ of ${\bf M}_{k}({\vec n})$, motivated by the considerations of sewn gauge theories on orbifolds ${\bf C}^{4}/{\Gamma}$. The geometry of the spaces ${\bf M}_{\bf k}^{\gamma}({\vec{\bf n}})$, more specifically the compactness of the set of torus-fixed points, for various tori, underlies the non-perturbative Dyson-Schwinger identities recently found to be satisfied by the correlation functions of $qq$-characters viewed as local gauge invariant operators in the ${\mathcal{N}}=2$ quiver gauge theories. The cohomological and K-theoretic operations defined using ${\bf M}_{k}({\vec n})$ and their quiver versions as correspondences provide the geometric counterpart of the $qq$-characters, line and surface defects.

Flavour singlets in gauge theory as permutations

Gauge-invariant operators can be specified by equivalence classes of permutations. We develop this idea concretely for the singlets of the flavour group $SO(N_f)$ in $U(N_c)$ gauge theory by using Gelfand pairs and Schur-Weyl duality. The singlet operators, when specialised at $N_f =6$, belong to the scalar sector of ${\cal N}=4$ SYM. A simple formula is given for the two-point functions in the free field limit of $g_{YM}^2 =0$. The free two-point functions are shown to be equal to the partition function on a 2-complex with boundaries and a defect, in a topological field theory of permutations. The permutation equivalence classes are Fourier transformed to a representation basis which is orthogonal for the two-point functions at finite $N_c , N_f$. Counting formulae for the gauge-invariant operators are described. The one-loop mixing matrix is derived as a linear operator on the permutation equivalence classes.

Quantum entanglement of locally excited states in Maxwell theory

In 4 dimensional Maxwell gauge theory, we study the changes of (Renyi) entangle-ment entropy which are defined by subtracting the entropy for the ground state from the one for the locally excited states generated by acting with the gauge invariant local operators on the state. The changes for the operators which we consider in this paper reflect the electric-magnetic duality. The late-time value of changes can be interpreted in terms of electromagnetic quasi-particles. When the operator constructed of both electric and magnetic fields acts on the ground state, it shows that the operator acts on the late-time structure of quantum entanglement differently from free scalar fields.

All-order bounds for correlation functions of gauge-invariant operators in Yang-Mills theory

We give a complete, self-contained, and mathematically rigorous proof that Euclidean Yang-Mills theories are perturbatively renormalisable, in the sense that all correlation functions of arbitrary composite local operators fulfil suitable Ward identities. Our proof treats rigorously both all ultraviolet and infrared problems of the theory and provides, in the end, detailed analytical bounds on the correlation functions of an arbitrary number of composite local operators. These bounds are formulated in terms of certain weighted spanning trees extending between the insertion points of these operators. Our proofs are obtained within the framework of the Wilson-Wegner-Polchinski-Wetterich renormalisation group flow equations, combined with estimation techniques based on tree structures. Compared with previous mathematical treatments of massless theories without local gauge invariance [R. Guida and Ch. Kopper, arXiv:1103.5692; J. Holland, S. Hollands, and Ch. Kopper, Commun. Math. Phys. 342 (2016) 385] our constructions require several technical advances; in particular, we need to fully control the BRST invariance of our correlation functions.