The infrared divergences of QCD scattering amplitudes can be derived from an anomalous dimension Γ, which is a matrix in color space and depends on the momenta and masses of the external partons. It has recently been shown that in cases where there are at least two massive partons involved in the scattering process, starting at two-loop order Γreceives contributions involving color and momentum correlations between three (and more) partons. The three-parton correlations can be described by two universal functions F_1 and f_2. In this paper these functions are calculated at two-loop order in closed analytic form and their properties are studied in detail. Both functions are found to be suppressed like O(m^4/s^2) in the limit of small parton masses, in accordance with mass factorization theorems proposed in the literature. On the other hand, both functions are O(1) and even diverge logarithmically near the threshold for pair production of two heavy particles. As an application, we calculate the infrared poles in the q qbar --> t tbar and g g --> t tbar scattering amplitudes at two-loop order.
The possible anomaly of the tensor current divergence equation in U(1) gauge theories is calculated by means of perturbative method. It is found that the tensor current divergence equation is free of anomalies.
We summarize our recent work [1-3] concerning the formulation of two-particle-irreducible (2PI) functional techniques for abelian gauge field theories.
In theories with extra dimensions the Standard Model Higgs field can be identified with the internal components of higher-dimensional gauge fields (Higgs-gauge unification). The higher-dimensional gauge symmetry prevents the Higgs mass from quadratic divergences, but at the fixed points of the orbifold this symmetry is broken and divergences can arise if U(1) subgroups are conserved. We show that another symmetry, remnant of the internal rotation group after orbifold projection, can avoid the generation of such divergences.
Real-time classical SU($N$) gauge theories at non-zero temperature contain linear divergences. We introduce counterterms for these divergences in the equations of motion in the continuum and on the lattice. These counterterms can be given in terms of auxiliary fields that satisfy local equations of motion. We present a lattice model with 6+1D auxiliary fields that for IR-sensitive quantities yields cut-off independent results to leading order in the coupling. Also an approximation with 5+1D auxiliary fields is discussed.
Higher order effects of the two-dimensional non-Abelian gauge theories, in which the vector-meson mass is generated by chiral anomalies, will be studied. The $β$ function and the topological nature of the non-linear $σ$ model in the action and unitarity of the theory will be discussed.
The one-loop Fermi spectra (one-particle and collective ones) are found for all momenta in the $T^2$-approximation and their gauge invariance in hot and dense Abelian and non-Abelian theories is studied. It is shown that the one-particle spectrum, if the calculation accuracy is kept strictly, is gauge invariant for all momenta and has two branches as the bare one. The collective spectrum always has four branches which are gauge dependent including also their $|\q|=0$ limit. The exception is the case $m,μ=0$ for which this spectrum is gauge invariant for all momenta as well.
A powerful tool for calculations in non-Abelian gauge theories is obtained by combining the background field gauge, the helicity basis and the color decomposition methods. It has reproduced the one-loop calculation of the five-gluon amplitudes in QCD, is applicable to electroweak processes and extendable to two-loop calculations.
We present the elements of three applications of resummation methods in non-Abelian gauge theories: (1), QED X QCD exponentiation and shower/ME matching for LHC physics; (2), IR improvement of DGLAP theory; (3), resummed quantum gravity and the final state of Hawking radiation. In all cases, the extension of the YFS approach, originally introduced for Abelian gauge theory, to non-Abelian gauge theories, QCD and quantum general relativity, leads to new results and solutions which we briefly summarize.
The question of the definition of effective charges for non-abelian gauge theories is discussed, focusing in particular on both the pinch technique and background field method approaches. It is argued that there does exist a unique generalization of the QED concept of an effective charge to non-abelian theories, and that this generalization is given by the pinch technique. The discussion is set in the wider (and controversial) context of the definition of gauge-independent Green function-like quantities in general in such theories.
We consider the infrared factorisation of non-abelian multi-particle scattering amplitudes, and we study the form of the universal colour operator responsible for infrared divergences, when expressed in terms of coordinates on the `celestial sphere' intersecting the future light-cone at asymptotic distances. We find that colour-dipole contributions to the infrared operator, to all orders in perturbation theory, have a remarkably simple expression in these coordinates, with scale and coupling dependence factorised from kinematics and colour. Generalising earlier suggestions in the abelian theor…
Estimate of average energy of the non-abelian dipole in the instanton medium is calculated. This energy for the dipole in colour singlet state escalates linearly with the separation increasing for the point-like sources unscreened. And its 'tension' coefficient develops the magnitude pretty similar to one as inferred from the lattice QCD and other model approaches.
We present some classical properties for non-abelian Yang-Mills theories that we extract directly from the Maxwell's equations of the theory. We write the equations of motion for the SU(3) Yang-Mills theory using the language of Maxwell's equations in both differential and integral forms. We show that vectorial gauge fields in this theory are non-fermionic sources for non-abelian electric and magnetic fields. These vectorial gauge fields are also responsible for the existence of magnetic monopoles. We build the continuity equation and the energy-momentum tensor for the non-abelian case.
In the perturbation theory, trasnport phenomena in hot non-Abelian gauge theories like QCD are often plagued with infrared singularities or nonperturbative effects. We show, in the context of the Kadanoff & Baym formalism, that there are certain nonequilibrium processes which are free from such difficulties. For these processes, due to an interplay between the macroscopic and microscopic physics, characteristic time scale (the mesoscale) naturally enters as an infrared cutoff and purely perturbative description by the Boltzmann equation is valid.
When computed to one-loop order in resummed perturbation theory, the non-abelian Debye mass appears to be logarithmically sensitive to the magnetic scale $g^2T$. More generally, we show that in higher orders power-like infrared divergences forbid the use of perturbation theory to calculate the corrections to Debye screening. A similar infrared problem occurs in the determination of the mass-shell for the scalar propagator in 2+1-dimensional scalar electrodynamics. In this context, we provide a non-perturbative approach which solves the infrared problems and allows for an accurate calculation o…
In this work we study the critical behavior and momentum dependence of chirality-flipping fermion 4-point functions in non-abelian gauge theories. Our formalism is based on the Schwinger-Dyson system of equations. Considering the large-N limit of an SU(N) theory simplifies the problem considerably, showing us clearly which Green functions can be non-zero and which not. It also allows us to draw conclusions on chirality-flipping n-point functions for n larger than four.
As recently pointed out, regularization schemes defined in four-dimensions may also face inconsistencies in the presence of chiral fermions. In this work, we extend this analysis to two-loop order. Adopting the implicit regularization as working arena, we discuss in detail how a consistent version of the method can be envisaged, and present many of the subtleties that appear at two-loop order using an abelian chiral model as working example.
Three dimensional abelian gauge theories classically in a Coulomb phase are affected by IR divergences even when the matter fields are all massive. Using generalizations of Ward-Takahashi identities, we show that correlation functions of gauge-invariant operators are IR finite to all orders in perturbation theory. Gauge invariance is sufficient but not necessary for IR finiteness. In particular we show that specific gauge-variant correlators, including the two-point function of matter fields, are also IR finite to all orders in perturbation theory. Possible applications of these results are br…
Nonrenormalizable couplings in supergravity-coupled supersymmetric theories can give rise to power-law divergences that destabilize the weak-scale hierarchy. For the case of the standard-model gauge group, the problem can arise in theories with \321 gauge-singlet chiral superfields. The minimal supersymmetric standard model is free from such destabilizing divergences.
We calculate Sudakov-type form factors for isolated spin-1/2 particles (fermions) entering non-Abelian gauge-field systems. We consider both the on- and the off-mass-shell case using a methodology which rests on a worldline casting of field theories. The simplicity and utility of our approach derives from the fact that we are in a position to make, a priori, a more transparent separation (factorization), with respect to a given scale, between short- and long-distance physics than diagramatic methods.
I re-derive the lowest order effective Lagrangian for electro-weak symmetry breaking without the use of Goldstone's theorem for spontaneously broken global symmetries and without the assumption of a custodial symmetry. I consider the breaking of a local symmetry with gauge group G down to an Abelian sub-group K and construct a gauge-invariant functional with one free parameter v, such that v=0 corresponds to a gauge theory with gauge group G, while v --> infinity corresponds to a gauge theory with gauge group K.
I discuss different theories of leptonic flavor and their capability of describing the features of the lepton sector, namely charged lepton masses, neutrino masses, lepton mixing angles and leptonic (low and high energy) CP phases. In particular, I show examples of theories with an abelian flavor symmetry G_f, with a non-abelian G_f as well as theories with non-abelian G_f and CP.
The high-energy scattering amplitude of two colour-singlet quark-antiquark pairs is governed by the correlation function of two Wilson loops, which follow the classical straight lines for quark (antiquark) trajectories. This quantity is expected to be free of IR divergences, differently from what happens for the parton-parton scattering amplitude, described, in the high-energy limit, by the expectation value of two Wilson lines. We shall explicitly test this IR finitness by a direct non-perturbative computation of the loop-loop scattering amplitudes in the (pedagogic, but surely physically int…
In the thesis, first, the cancellation of infrared divergences in QED is reviewed. Next, two examples in QCD, the quark-quark scattering and the quark-gluon scattering are examined at one loop, from which the importance of Ward-Takahashi identities becomes manifest for the cancellation to occur. The factorization of infrared divergences in QCD is proved at all orders, by full usage of the Ward-Takahashi identities. In these proofs, the axial gauge condition is used. The cancellation of infrared divergences in QCD occurs among the gauge invariant set of graphs. In this way, the low energy theor…
We study the stability of the gauge hierarchy in hidden-sector supergravity theories. We show that a destabilizing tadpole can appear if a theory has a gauge- and global-symmetry singlet with renormalizable couplings to the visible fields. We also find a quadratically divergent two-loop contribution to the supersymmetric effective potential. This term illustrates the difficulty of using the "LHC mechanism" to control quadratic divergences in theories with Planck-scale vevs.
Asymptotically nonlocal field theories represent a sequence of higher-derivative theories whose limit point is a ghost-free, infinite-derivative theory. Here, we extend previous work on pure scalar and Abelian gauge theories to asymptotically nonlocal non-Abelian theories. In particular, we confirm that there is a limit in which the Lee-Wick spectrum can be decoupled, but where the hierarchy problem is resolved via an emergent nonlocal scale that regulates loop diagrams and that is hierarchically smaller than the lightest Lee-Wick resonance.
We give a simplified proof for the perturbative renormalizability of theories with massive vector particles. For renormalizability it is sufficient that the vector particle is treated as an gauge field, corresponding to an Abelian gauge group. Contrary to the non-Abelian case one does not need the Higgs mechanism to create the appropriate mass terms. The proof uses ``Stueckelberg's trick'' and the Ward-Takahashi identities from local Abelian gauge invariance. The simplification is due to the fact that, again contrary to the non-Abelian case, no BRST analysis is needed.
The classical approximation provides a non-perturbative approach to time-dependent problems in finite temperature field theory. We study the divergences in hot classical field theory perturbatively. At one-loop, we show that the linear divergences are completely determined by the classical equivalent of the hard thermal loops in hot quantum field theories, and that logarithmic divergences are absent. To deal with higher-loop diagrams, we present a general argument that the superficial degree of divergence of classical vertex functions decreases by one with each additional loop: one-loop contri…
We study the solution to the Slavnov-Taylor (ST) identities in spontaneously broken effective gauge theories for a non-Abelian gauge group. The procedure to extract the $β$-functions of the theory in the presence of (generalized) non-polynomial field redefinitions is elucidated.
We present a functional derivation of recursion rules for scattering amplitudes in a non-Abelian gauge theory in a form valid to arbitrary loop order. The tree-level and one-loop recursion rules are explicitly displayed.
We demonstrate electric-magnetic duality in N=1 supersymmetric non-Abelian gauge theories in four dimensions by presenting two different gauge theories (different gauge groups and quark representations) leading to the same non-trivial long distance physics. The quarks and gluons of one theory can be interpreted as solitons (non-Abelian magnetic monopoles) of the elementary fields of the other theory. The weak coupling region of one theory is mapped to a strong coupling region of the other. When one of the theories is Higgsed by an expectation value of a squark, the other theory is confined. Ma…
We study some topological aspects of non-abelian gauge theories intimately connected to the Lie algebras of the gauge groups and the homotopy theory in the generalized gauge orbit space. The physics connection to the non-perturbative solution to strong CP problem as originally proposed by the author is also discussed. Some relevant topological formulas are also given and discussed. A result from the physics application is that the usual gauge orbit space on the compactified space can contain at most a $Z_2$ monopole structure in the SP(2N) gauge theories. Some relevance to the open universe is…
In this Letter, we construct a set of order parameters for non-Abelian gauge theories which probe directly the unbroken group and are free of the deficiencies caused by quantum fluctuations and gauge fixing which have plagued all previous attempts. These operators can be used to map out the phase diagram of a non-Abelian gauge theory.
We present a procedure of differential renormalization at the one loop level which avoids introducing unnecessary renormalization constants and automatically preserves abelian gauge invariance. The amplitudes are expressed in terms of a basis of singular functions. The local terms appearing in the renormalization of these functions are determined by requiring consistency with the propagator equation. Previous results in abelian theories, with and without supersymmetry, are discussed in this context.
Gauge theory is the foundation of the particle physics Standard Model (SM). Considering the multiple gauge sectors for one gauge transformation, we study the generalized Abelian and non-Abelian (Yang-Mills theory) gauge theories. We first point out that the U(1) gauge theory has a few unique properties, which provide the motivations for the generalized Yang-Mills theory. Also, we consider the generalized Abelian gauge theory, and study the Higgs mechanism with new interesting properties. In addition, we propose the simple and generic generalizations of Yang-Mills theory. In the simple generali…
We examine the stability of the mass hierarchy in hidden-sector supergravity theories. We show that a quadratically divergent tadpole can appear at two loops, even in minimal supergravity theories, provided the theory has a gauge- and global-symmetry singlet with renormalizable couplings to the visible fields. This tadpole can destabilize the hierarchy. We also find a quadratically divergent two-loop contribution to the field-dependent vacuum energy. This result casts doubt on the efficacy of the "LHC mechanism" for controlling quadratic divergences. We carry out the two-loop calculation in a …
Owing to subtle issues concerning quantum fluctuations and gauge fixing, a formulation of a general procedure to specify the realization of non-Abelian gauge symmetry has evaded all earlier attempts. In this Letter, we discuss these subtleties and present two sets of order parameters for non-Abelian gauge theories. Such operators directly probe the manifest low energy symmetry group and are crucial for the study of the phase diagram of a non-Abelian gauge theory.
We study the Schwinger process in a uniform non-Abelian electric field using a dynamical approach in which we evolve an initial quantum state for gluonic excitations. We evaluate the spectral energy density and number density in the excitations as functions of time. The total energy density has an ultraviolet divergence which we argue gets tamed due to asymptotic freedom, leading to $g^4E^4t^4$ growth, where $g$ is the coupling and $E$ the electric field strength. We also find an infrared divergence in the number density of excitations whose resolution requires an effect such as confinement.
Using elementary considerations of Lorentz invariance, Bose symmetry and BRST invariance, we argue why the decay of a massive color-octet vector state into a pair of on-shell massless gluons is possible in a non-Abelian SU(N) Yang-Mills theory, we constrain the form of the amplitude of the process and offer a simple understanding of these results in terms of effective-action operators.
We test at the electroweak scale the recently proposed elaborate theoretical scenario for real-time dynamics of non-abelian gauge theories at high temperature. We see no sign of the predicted behavior. This indicates that perturbative concepts like color conductivity and Landau damping might be irrelevant at temperatures corresponding to the electroweak scale.
We discuss and propose a proper extension of the Abelian projection based on the Maximal Abelian Gauge to SU(N) gauge theories. Based, on that, we investigate the properties of thermal Abelian monopoles in the deconfined phase of the SU(3) pure gauge theory. Such properties are very similar to those already found for SU(2), confirming the relevance of the magnetic component close to Tc and the possible condensation of thermal monopoles as the deconfinement temperature is crossed from above. Moreover, we study the correlation functions among monopoles related to different U(1) subgroups, which …
We present a pedagogical review of our current understanding of the ultraviolet structure of N = (1,1) 6D supersymmetric Yang-Mills theory and of N = 8 4D supergravity. These theories are not renormalizable, they involve power ultraviolet divergences and, in all probability, an infinite set of higher-dimensional counterterms that contribute to on-mass-shell scattering amplitudes. A specific feature of supersymmetric theories (especially, of extended supersymmetric theories) is that these counterterms may not be invariant off shell under the full set of supersymmetry transformations. The lowest…
The study of Yang-Mills theories in three dimensions is an insightful playground to grasp important features for the four-dimensional case. Additionally, in three dimensions, the Chern-Simons term can be introduced with a mass parameter of topological nature. Quantizing such a theory in the continuum demands a gauge fixing which, in general, is plagued by Gribov copies. In this work, Yang-Mills-Chern-Simons theories are quantized in the maximal Abelian gauge and the existence of infinitesimal Gribov copies is taken into account. The elimination of copies modifies the (Abelian) gluon propagator…
We show that a criterion for confinement, based on the BRST invariance, holds in four dimensions, by solving a non-Abelian gauge theory with a set of exact solutions. The confinement condition we consider was obtained by Kugo and Ojima some decades ago. The current understanding of gauge theories permits us to apply the techniques straightforwardly for checking the validity of this criterion. In this way, we are able to show that the non-Abelian gauge theory is confining and that confinement is rooted in the BRST invariance and asymptotic freedom.
