Yang-Mills Coupling Research Articles

Large-N integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM: when resurgence meets modularity

Exact expressions for certain integrated correlators of four half-BPS operators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills theory with gauge group SU(N) have been recently obtained thanks to a beautiful interplay between supersymmetric localisation and modular invariance. The large-N expansion at fixed Yang-Mills coupling of such integrated correlators produces an asymptotic series of perturbative terms, holographically related to higher derivative interactions in the low energy expansion of the type IIB effective action, as well as exponentially suppressed corrections at large N, interpreted as contributions from coincident (p, q)-string world-sheet instantons. In this work we define a manifestly modular invariant Borel resummation of the perturbative large-N expansion of these integrated correlators, from which we extract the exact non-perturbative large-N sectors via resurgence analysis. Furthermore, we show that in the ’t Hooft limit such modular invariant non-perturbative completions reduce to known resurgent genus expansions. Finally, we clarify how the same non-perturbative data is encoded in the decomposition of the integrated correlators based on SL(2, ℤ) spectral theory.

Exact results for giant graviton four-point correlators

We study the four-point correlator O2O2DD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left\\langle {\\mathcal{O}}_2{\\mathcal{O}}_2\\mathcal{DD}\\right\\rangle $$\\end{document} in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory (SYM) with SU(N) gauge group, where O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{O}}_2 $$\\end{document} represents the superconformal primary operator with dimension two, while D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{D} $$\\end{document} denotes a determinant operator of dimension N, which is holographically dual to a giant graviton D3-brane extending along S5. We analyse the integrated correlator associated with this observable, obtained after integrating out the spacetime dependence over a supersymmetric invariant measure. Similarly to other classes of integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM, this integrated correlator can be computed through supersymmetric localisation on the four-sphere. Employing matrix-model recursive techniques, we demonstrate that the integrated correlator can be reformulated as an infinite sum of protected three-point functions with known coefficients. This insight allows us to circumvent the complexity associated with the dimension-N determinant operator, significantly streamlining the large-N expansion of the integrated correlator. In the planar limit and beyond, we derive exact results for the integrated correlator valid for all values of the ’t Hooft coupling, and investigate the resurgent properties of their strong coupling expansion. Additionally, in the large-N expansion with fixed (complexified) Yang-Mills coupling, we deduce the SL(2, ℤ) completion of these results in terms of the non-holomorphic Eisenstein series. The proposed modular functions are confirmed by explicit instanton calculations from the matrix model, and agree with expectations from the holographic dual picture of known results in type IIB string theory.

Four-derivative Yang-Mills couplings in heterotic theory through T-duality

This study delves into the dimensional reduction of the classical effective action of heterotic string theory on a circle, along with its T-duality symmetry, with the aim of identifying the bosonic couplings. To achieve this, we propose a truncation scheme for the generalized Buscher rules and the reduced action, specifically targeting the truncation of the nonlinear appearance of the scalar component of the Yang-Mills field in the base space. By imposing this truncated T-duality on the reduced action, we successfully determine the four- derivative bosonic couplings in the minimal basis, where field redefinition is imposed. Notably, these couplings, which are associated with the Lorentz Chern-Simons coupling HΩ, exhibit an exact correspondence with the NS-NS couplings found in the Metsaev-Tseytlin action.Furthermore, we investigate the bosonic couplings in the maximal basis, where field redefinition is not imposed. In this scenario, the truncated T-duality fixes the effective action up to 17 arbitrary parameters. By assigning specific values to these parameters, we establish a framework in which the NS-NS couplings align with those in the Meissner action. Remarkably, within this scheme, the Yang-Mills couplings precisely coincide with those obtained through the S-matrix method.

Large N master field optimization: the quantum mechanics of two Yang-Mills coupled matrices

We study the large N dynamics of two massless Yang-Mills coupled matrix quantum mechanics, by minimization of a loop truncated Jevicki-Sakita effective collective field Hamiltonian. The loop space constraints are handled by the use of master variables. The method is successfully applied directly in the massless limit for a range of values of the Yang-Mills coupling constant, and the scaling behaviour of different physical quantities derived from their dimensions are obtained with a high level of precision. We consider both planar properties of the theory, such as the large N ground state energy and multi-matrix correlator expectation values, and also the spectrum of the theory. For the spectrum, we establish that the U(N) traced fundamental constituents remain massless and decoupled from other states, and that bound states develop well defined mass gaps, with the mass of the two degenerate lowest lying bound states being determined with a particularly high degree of accuracy. In order to confirm, numerically, the physical interpretation of the spectrum properties of the U(N) traced constituents, we add masses to the system and show that, indeed, the U(N) traced fundamental constituents retain their “bare masses”. For this system, we draw comparisons with planar results available in the literature.

BRST view of spontaneous symmetry breaking

After the phase transition of a Grand Unified Theory (GUT), its single coupling splits into the couplings of the basic interactions. On the other side, the measurement of the Weinberg mixing angle in the eletroweak theory determines that the couplings associated to the SU(2) and U(1) symmetries were already distinct in its symmetric phase, before the electroweak breaking. This picture is consistent as we understand that such couplings evolve independently with the energy scale, meeting a common point together with the strong interaction coupling at the GUT scale. From a BRST symmetry point of view, this implies that each independent coupling should be associated to independent cocycles defined on a cohomological basis. The problem is that, before the GUT symmetry breaking, the BRST operator associated to this symmetric phase predicts a single Yang-Mills coupling. This sets the question of what should be the BRST operator of the asymmetric phase, that would allow for the splitting of this coupling but at the same time remain compatible with the BRST operator of the symmetric phase. In this work we proceed to answer this question.

Perturbative linearization of super-Yang-Mills theories in general gauges

Supersymmetric Yang-Mills theories can be characterized by a non-local and non-linear transformation of the bosonic fields (Nicolai map) mapping the interacting functional measure to that of a free theory, such that the Jacobi determinant of the transformation equals the product of the fermionic determinants obtained by integrating out the gauginos and ghosts at least on the gauge hypersurface. While this transformation has been known so far only for the Landau gauge and to third order in the Yang-Mills coupling, we here extend the construction to a large class of (possibly non-linear and non-local) gauges, and exhibit the conditions for all statements to remain valid off the gauge hypersurface. Finally, we present explicit results to second order in the axial gauge and to fourth order in the Landau gauge.

Spontaneous chiral symmetry breaking in the massive Landau gauge: Realistic running coupling

We investigate the spontaneous breaking of chiral symmetry in QCD by means of a recently proposed approximation scheme in the Landau-gauge Curci-Ferrari model, which combines an expansion in the Yang-Mills coupling and in the inverse number of colors, without expanding in the quark-gluon coupling. The expansion allows for a consistent treatment of ultraviolet tails via renormalization group techniques. At leading order, it leads to the resummation of rainbow diagrams for the quark propagator, with, however, a trivial running of both the gluon mass and the quark-gluon coupling. In a previous work, by using a simple model for a more realistic running of these parameters, we could reproduce the known phenomenology of chiral symmetry breaking, including a satisfactory description of the lattice data for the quark mass function. Here, we get rid of this model-dependence by taking our approximation scheme to next-to-leading order. This allows us to consistently include the realistic running of the parameters and to access the unquenched gluon and ghost propagators to first nontrivial order, which we can compare to available lattice data for an even more stringent test of our approach. In particular, our results for the various two-point functions compare well with lattice data while the parameters of the model are strongly constrained.

Exact properties of an integrated correlator in mathcal{N} = 4 SU(N) SYM

We present a novel expression for an integrated correlation function of four superconformal primaries in SU(N) mathcal{N} = 4 supersymmetric Yang-Mills ( mathcal{N} = 4 SYM) theory. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. In this paper the correlator is re-expressed as a sum over a two dimensional lattice that is valid for all N and all values of the complex Yang-Mills coupling tau =theta /2pi +4pi i/{g}_{mathrm{YM}}^2 . In this form it is manifestly invariant under SL(2, ℤ) Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the SU(N) correlator to the SU(N + 1) and SU(N − 1) correlators. For any fixed value of N the correlator can be expressed as an infinite series of non-holomorphic Eisenstein series, Eleft(s;tau, overline{tau}right) with s ∈ ℤ, and rational coefficients that depend on the values of N and s. The perturbative expansion of the integrated correlator is an asymptotic but Borel summable series, in which the n-loop coefficient of order (gYM/π)2n is a rational multiple of ζ(2n + 1). The n = 1 and n = 2 terms agree precisely with results determined directly by integrating the expressions in one-loop and two-loop perturbative mathcal{N} = 4 SYM field theory. Likewise, the charge-k instanton contributions (|k| = 1, 2, . . .) have an asymptotic, but Borel summable, series of perturbative corrections. The large-N expansion of the correlator with fixed τ is a series in powers of {N}^{frac{1}{2}-mathrm{ell}} (ℓ ∈ ℤ) with coefficients that are rational sums of Eleft(s;tau, overline{tau}right) with s ∈ ℤ + 1/2. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider the ’t Hooft topological expansion of large-N Yang-Mills theory in which lambda ={g}_{mathrm{YM}}^2N is fixed. The coefficient of each order in the 1/N expansion can be expanded as a series of powers of λ that converges for |λ| < π2. For large λ this becomes an asymptotic series when expanded in powers of 1/sqrt{lambda } with coefficients that are again rational multiples of odd zeta values, in agreement with earlier results and providing new ones. We demonstrate that the large-λ series is not Borel summable, and determine its resurgent non-perturbative completion, which is Oleft(exp left(-2sqrt{lambda}right)right) .

New modular invariants in mathcal{N} = 4 Super-Yang-Mills theory

We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the mathcal{N} = 4 SU(N) super-Yang-Mills theory, in the limit where N is taken to be large while the complexified Yang-Mills coupling τ is held fixed. The specific four-point functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the mathcal{N} = 2∗ theory with respect to the squashing parameter b and mass parameter m, evaluated at the values b = 1 and m = 0 that correspond to the mathcal{N} = 4 theory on a round sphere. At each order in the 1/N expansion, these fourth derivatives are modular invariant functions of (τ, overline{tau} ). We present evidence that at half-integer orders in 1/N , these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in 1/N, they are certain “generalized Eisenstein series” which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the low-energy expansion of the four-graviton amplitude in type IIB superstring theory in ten-dimensional flat space and have interesting implications for the structure of the analogous expansion in AdS5× S5.

Five-dimensional gauge theories on spheres with negative couplings

We consider supersymmetric gauge theories on S5 with a negative Yang-Mills coupling in their large N limits. Using localization we compute the partition functions and show that the pure SU(N) gauge theory descends to an SU(N/2)+N/2× SU(N/2)−N/2× SU(2) Chern-Simons gauge theory as the inverse ’t Hooft coupling is taken to negative infinity for N even. The Yang-Mills coupling of the SU(N/2)±N/2 is positive and infinite, while that on the SU(2) goes to zero. We also show that the odd N case has somewhat different behavior. We then study the SU(N/2)N/2 pure Chern-Simons theory. While the eigenvalue density is only found numerically, we show that its width equals 1 in units of the inverse sphere radius, which allows us to find the leading correction to the free energy when turning on the Yang-Mills term. We then consider USp(2N) theories with an antisymmetric hypermultiplet and Nf< 8 fundamental hypermultiplets and carry out a similar analysis. Along the way we show that the one-instanton contribution to the partition function remains exponentially suppressed at negative coupling for the SU(N) theories in the large N limit.

Modular invariance in superstring theory from mathcal{N} = 4 super-Yang-Mills

We study the four-point function of the lowest-lying half-BPS operators in the mathcal{N} = 4 SU(N) super-Yang-Mills theory and its relation to the flat-space four-graviton amplitude in type IIB superstring theory. We work in a large-N expansion in which the complexified Yang-Mills coupling τ is fixed. In this expansion, non-perturbative instanton contributions are present, and the SL(2, ℤ) duality invariance of correlation functions is manifest. Our results are based on a detailed analysis of the sphere partition function of the mass-deformed SYM theory, which was previously computed using supersymmetric localization. This partition function determines a certain integrated correlator in the undeformed mathcal{N} = 4 SYM theory, which in turn constrains the four-point correlator at separated points. In a normalization where the two-point functions are proportional to N2− 1 and are independent of τ and overline{tau} , we find that the terms of order sqrt{N} and 1/sqrt{N} in the large N expansion of the four-point correlator are proportional to the non-holomorphic Eisenstein series Eleft(frac{3}{2},tau, overline{tau}right) and Eleft(frac{5}{2},tau, overline{tau}right) , respectively. In the flat space limit, these terms match the corresponding terms in the type IIB S-matrix arising from R4 and D4R4 contact inter-actions, which, for the R4 case, represents a check of AdS/CFT at finite string coupling. Furthermore, we present striking evidence that these results generalize so that, at order {N}^{frac{1}{2}-m} with integer m ≥ 0, the expansion of the integrated correlator we study is a linear sum of non-holomorphic Eisenstein series with half-integer index, which are manifestly SL(2, ℤ) invariant.

Minimal geometric deformation of Yang-Mills-Dirac stellar configurations

The method of minimal geometric deformation (MGD) is used to derive static, strongly gravitating, spherically symmetric, compact stellar distributions that are solutions of the Yang-Mills-Einstein-Dirac coupled field equations, on fluid membranes with finite tension. Their solutions characterize MGD Yang-Mills-Dirac stars, whose mass has order of the Chandrasekhar mass, once the range of both the fermionic self-interaction and the Yang-Mills coupling constants is suitably chosen. Physical features of MGD Yang-Mills-Dirac stars are then discussed, and their ADM mass are derived, as a function of the fermion coupling constant, the finite brane tension, and the Yang-Mills running parameter as well.

Light hadron masses from a matrix model for QCD

The $SU(3)$ Yang-Mills matrix model coupled to fundamental fermions is an approximation of quantum chromodynamics (QCD) on a 3-sphere of radius $R$. The spectrum of this matrix model Hamiltonian is estimated using standard variational methods, and is analyzed in the strong coupling limit. By employing a matching prescription to determine the dependence of the Yang-Mills coupling constant $g$ on $R$, we relate the asymptotic values of the energy eigenvalues in the $R \rightarrow \infty$ (flat space) limit to the masses of light hadrons. We find that the matrix model estimates the light hadron spectrum fairly accurately, with the light baryon masses falling within $10\%$, and most light meson masses falling within about $30\%$ of their observed values.

On scalar radiation

We discuss radiation in theories with scalar fields. Our key observation is that even in flat spacetime, the radiative fields depend qualitatively on the coupling of the scalar field to the Ricci scalar: for non-minimally coupled scalars, the radiative energy density is not positive definite, the radiated power is not Lorentz invariant and it depends on the derivative of the acceleration. We explore implications of this observation for radiation in conformal field theories. First, we find a relation between two coefficients that characterize radiation, that holds in all the conformal field theories we consider. Furthermore, we find evidence that for a 1/2-BPS probe coupled to mathcal{N} = 4 super Yang-Mills, and following an arbitrary trajectory, the spacetime dependence of the one-point function of the energy momentum tensor is independent of the Yang-Mills coupling.

Localized rainbows in the QCD phase diagram

We study the phase diagram of strongly interacting matter with light quarks using a recently proposed, small parameter approach to infrared QCD in the Landau gauge. This is based on an expansion with respect to both the inverse number of colors and the pure Yang-Mills coupling in the presence of a Curci-Ferrari mass term. At leading order, this leads to the well-known rainbow equation for the quark propagator with a massive gluon propagator. We solve the latter at nonzero temperature and chemical potential using a simple semi-analytic approximation known to capture the essence of chiral symmetry breaking in the vacuum. In the chiral limit, we find a tricritical point which becomes a critical endpoint in the presence of a nonzero bare quark mass, in agreement with the results of nonperturbative functional methods and model calculations. This supports the view that the present approach allows for a systematic study of the QCD phase diagram in a controlled expansion scheme.

U(N) Yang-Mills in non-commutative space time

We present an approach to U⋆(N) Yang-Mills theory in non-commutative space based upon a novel phase-space analysis of the dynamical fields with additional auxiliary variables that generate Lorentz structure and colour degrees of freedom. To illustrate this formalism we compute the quadratic terms in the effective action focusing on the planar divergences so as to extract the β-function for the Yang-Mills coupling constant. Nonetheless the method presented is general and can be applied to calculate the effective action at arbitrary order of expansion in the coupling constant and is well suited to the computation of low energy one-loop scattering amplitudes.

The one-loop spectral problem of strongly twisted mathcal{N} = 4 Super Yang-Mills theory

We investigate the one-loop spectral problem of γ-twisted, planar mathcal{N} = 4 Super Yang-Mills theory in the double-scaling limit of infinite, imaginary twist angle and vanishing Yang-Mills coupling constant. This non-unitary model has recently been argued to be a simpler version of full-fledged planar mathcal{N} = 4 SYM, while preserving the latter model’s conformality and integrability. We are able to derive for a number of sectors one-loop Bethe equations that allow finding anomalous dimensions for various subsets of diagonalizable operators. However, the non-unitarity of these deformed models results in a large number of non-diagonalizable operators, whose mixing is described by a very complicated structure of non-diagonalizable Jordan blocks of arbitrarily large size and with a priori unknown generalized eigenvalues. The description of these blocks by methods of integrability remains unknown.

Glueball spectra from a matrix model of pure Yang–Mills theory

We present variational estimates for the low-lying energies of a simple matrix model that approximates SU(3) Yang–Mills theory on a three-sphere of radius R. By fixing the ground state energy, we obtain the (integrated) renormalization group (RG) equation for the Yang–Mills coupling g as a function of R. This RG equation allows to estimate the mass of other glueball states, which we find to be in excellent agreement with lattice simulations.

Deformation of the cubic open string field theory

We study a consistent deformation of the cubic open bosonic string theory in such a way that the non-planar world sheet diagrams of the perturbative string theory are mapped onto their equivalent planar diagrams of the light-cone string field theory with some length parameters fixed. An explicit evaluation of the cubic string vertex in the zero-slope limit yields the correct relationship between the string coupling constant and the Yang–Mills coupling constant. The deformed cubic open string field theory is shown to produce the non-Abelian Yang–Mills action in the zero-slope limit if it is defined on multiple D-branes. Applying the consistent deformation systematically to multi-string world sheet diagrams, we may be able to calculate scattering amplitudes with an arbitrary number of external open strings.

Phases of planar 5-dimensional supersymmetric Chern-Simons theory

In this paper we investigate the large-$N$ behavior of 5-dimensional $\mathcal{N}=1$ super Yang-Mills with a level $k$ Chern-Simons term and an adjoint hypermultiplet. As in three-dimensional Chern-Simons theories, one must choose an integration contour to completely define the theory. Using localization, we reduce the path integral to a matrix model with a cubic action and compute its free energy in various scenarios. In the limit of infinite Yang-Mills coupling and for particular choices of the contours, we find that the free-energy scales as $N^{5/2}$ for $U(N)$ gauge groups with large values of the Chern-Simons 't\,Hooft coupling, $\tilde\lambda\equiv N/k$. If we also set the hypermultiplet mass to zero, then this limit is a superconformal fixed point and the $N^{5/2}$ behavior parallels other fixed points which have known supergravity duals. We also demonstrate that $SU(N)$ gauge groups cannot have this $N^{5/2}$ scaling for their free-energy. At finite Yang-Mills coupling we establish the existence of a third order phase transition where the theory crosses over from the Yang-Mills phase to the Chern-Simons phase. The phase transition exists for any value of $\tilde\lambda$, although the details differ between small and large values of $\tilde\lambda$. For pure Chern-Simons theories we present evidence for a chain of phase transitions as $\tilde\lambda$ is increased. We also find the expectation values for supersymmetric circular Wilson loops in these various scenarios and show that the Chern-Simons term leads to different physical properties for fundamental and anti-fundamental Wilson loops. Different choices of the integration contours also lead to different properties for the loops.