Renormalized Gauge Coupling Research Articles

Dislocations under gradient flow and their effect on the renormalized coupling

Non-zero topological charge is prohibited in the chiral limit of gauge-fermion systems because any instanton would create a zero mode of the Dirac operator. On the lattice, however, the geometric $Q_\text{geom}=\langle F{\tilde F}\rangle /32\pi^2$ definition of the topological charge does not necessarily vanish even when the gauge fields are smoothed for example with gradient flow. Small vacuum fluctuations (dislocations) not seen by the fermions may be promoted to instanton-like objects by the gradient flow. We demonstrate that these artifacts of the flow cause the gradient flow renormalized gauge coupling to increase and run faster. In step-scaling studies such artifacts contribute a term which increases with volume. The usual $a/L\to 0$ continuum limit extrapolations can hence lead to incorrect results. In this paper we investigate these topological lattice artifacts in the SU(3) 10-flavor system with domain wall fermions and the 8-flavor system with staggered fermions. Both systems exhibit nonzero topological charge at the strong coupling, especially when using Symanzik gradient flow. We demonstrate how this artifact impacts the determination of the renormalized gauge coupling and the step-scaling $\beta$ function.

Extended investigation of the twelve-flavor β-function

We report new results from high precision analysis of an important BSM gauge theory with twelve massless fermion flavors in the fundamental representation of the SU(3) color gauge group. The range of the renormalized gauge coupling is extended from our earlier work [1] to probe the existence of an infrared fixed point (IRFP) in the β-function reported at two different locations, originally in [2] and at a new location in [3]. We find no evidence for the IRFP of the β-function in the extended range of the renormalized gauge coupling, in disagreement with [2,3]. New arguments to guard the existence of the IRFP remain unconvincing [4], including recent claims of an IRFP with ten massless fermion flavors [5,6] which we also rule out. Predictions of the recently completed 5-loop QCD β-function for general flavor number are discussed in this context.

The twelve-flavor β-function and dilaton tests of the sextet scalar

We discuss near-conformal gauge theories beyond the standard model (BSM) where interesting results on the twelve-flavor β-function of massless fermions in the fundamental representation of the SU(3) color gauge group and dilaton tests of the light scalar with two massless fermions in the two-index symmetric tensor (sextet) representation can be viewed as parts of the same BSM paradigm under investigation. The clear trend in the decreasing size of β-functions at fixed renormalized gauge coupling is interpreted as a first indicator how the conformal window (CW) is approached in correlation with emergent near-conformal light scalars. BSM model building close to the CW will be influenced by differing expectations on the properties of the emergent light 0++ scalar either as a σ-particle of chiral symmetry breaking (ΧS B), or as a dilaton of scale symmetry breaking. The twelve-flavor β-function emerges as closest to the CW, perhaps near-conformal, or perhaps with an infrared fixed point (IRFP) at some unexplored strong coupling inside the CW. It is premature to speculate on dilaton properties of the twelveflavor model since the near-conformal realization remains an open question. However, it is interesting and important to investigate dilaton tests of the light sextet scalar whose β-function is closest to the CW in the symmetry breaking phase and emerges as the leading candidate for dilaton tests of the light scalar. We report results from high precision analysis of the twelve-flavor β-function [1] refuting its published IRFP [2, 3]. We present our objections to recent claims [4, 5] for non-universal behavior of staggered fermions used in our analysis. We also report our first analysis of dilaton tests of the light 0++ scalar in the sextet model and comment on related post-conference developments. The dilaton test is the main thrust of this conference contribution including presentation #405 on the nf = 12 β-function and presentation #260 on dilaton tests of the sextet model. They are both selected from the near-conformal BSM paradigm.

A quenched study of the Schrödinger functional with chirally rotated boundary conditions: Non-perturbative tuning

The use of chirally rotated boundary conditions provides a formulation of the Schrödinger functional that is compatible with automatic O(a) improvement of Wilson fermions up to O(a) boundary contributions. The elimination of bulk O(a) effects requires the non-perturbative tuning of the critical mass and one additional boundary counterterm. We present the results of such a tuning in a quenched setup for several values of the renormalized gauge coupling, from perturbative to non-perturbative regimes, and for a range of lattice spacings. We also check that the correct boundary conditions and symmetries are restored in the continuum limit.

Nearly conformal gauge theories in finite volume

We report new results on nearly conformal gauge theories with fermions in the fundamental representation of the SU(3) color gauge group as the number of fermion flavors is varied in the Nf=4–16 range. To unambiguously identify the chirally broken phase below the conformal window we apply a comprehensive lattice tool set in finite volumes which includes the test of Goldstone pion dynamics, the spectrum of the fermion Dirac operator, and eigenvalue distributions of random matrix theory. We also discuss the theory inside the conformal window and present our first results on the running of the renormalized gauge coupling and the renormalization group beta function. The importance of understanding finite volume zero momentum gauge field dynamics inside the conformal window is illustrated. Staggered lattice fermions are used throughout the calculations.

Lattice gauge action suppressing near-zero modes ofHW

We propose a lattice action including unphysical Wilson fermions with a negative mass ${m}_{0}$ of the order of the inverse lattice spacing. With this action, the exact zero mode of the Hermitian Wilson-Dirac operator ${H}_{W}({m}_{0})$ cannot appear and near-zero modes are strongly suppressed. By measuring the spectral density $\ensuremath{\rho}({\ensuremath{\lambda}}_{W})$, we find a gap near ${\ensuremath{\lambda}}_{W}=0$ on the configurations generated with the standard and improved gauge actions. This gap provides a necessary condition for the proof of the exponential locality of the overlap-Dirac operator by Hernandez, Jansen, and L\"uscher. Since the number of near-zero modes is small, the numerical cost to calculate the matrix sign function of ${H}_{W}({m}_{0})$ is significantly reduced, and the simulation including dynamical overlap fermions becomes feasible. We also introduce a pair of twisted mass pseudofermions to cancel the unwanted higher mode effects of the Wilson fermions. The gauge coupling renormalization due to the additional fields is then minimized. The topological charge measured through the index of the overlap-Dirac operator is conserved during continuous evolutions of gauge field variables.

D-term strings

We study the embedding of cosmic strings, related to the Abrikosov-Nielsen-Olesen vortex solution, into d=4, N=1 supergravity. We find that the local cosmic string solution which saturates the BPS bound of supergravity with $D$-term potential for the Higgs field and with constant Fayet--Iliopoulos term, has 1/2 of supersymmetry unbroken. We observe an interesting relation between the gravitino supersymmetry transformation, positive energy condition and the deficit angle of the cosmic string. We argue that the string solutions with magnetic flux with F-term potential cannot be supersymmetric, which leads us to a conjecture that D1-strings (wrapped D(1+q)-branes) of string theory in the effective 4d supergravity are described by the D-term strings that we study in this paper. We give various consistency checks of this conjecture, and show that it highlights some generic properties of non-BPS string theory backgrounds, such as brane-anti-brane systems. Supersymmetry breaking by such systems can be viewed as FI D-term breaking, which implies, under certain conditions, the presence of gauged R-symmetry on such backgrounds. The D-term nature of the brane-anti-brane energy can also provide information on the superpotential for the tachyon, which Higgses the R-symmetry. In this picture, the inter-brane force can be viewed as a result of the world-volume gauge coupling renormalization by the open string loops.

Gauge coupling renormalization in orbifold field theories

We investigate the gauge coupling renormalization in orbifold field theories preserving 4-dimensional N=1 supersymmetry in the framework of 4-dimensional effective supergravity. As a concrete example, we consider the 5-dimensional Super-Yang-Mills theory on a slice of AdS_5. In our approach, one-loop gauge couplings can be determined by the loop-induced axion couplings and the tree level properties of 4-dimensional effective supergravity which are much easier to be computed.

Gauge coupling renormalization in RS1

We compute the 4D low energy effective gauge coupling at one-loop order in the compact Randall–Sundrum scenario with bulk gauge fields and charged matter, within controlled approximations. While such computations are subtle, they can be important for studying phenomenological issues such as grand unification. Ultraviolet divergences are cut-off using Pauli–Villars regularization so as to respect 5D gauge and general coordinate invariance. The structure of these divergences on branes and in the bulk is elucidated by a 5D position-space analysis. The remaining finite contributions are obtained by a careful analysis of the Kaluza–Klein spectrum. We comment on the agreement between our results and expectations based on the AdS/CFT correspondence, in particular logarithmic sensitivity to the 4D Planck scale.

Higher loop effects on unification via Kaluza–Klein thresholds

We discuss higher loop corrections to gauge coupling renormalization in the context of gauge coupling unification via Kaluza–Klein thresholds. We show that in the case of N=1 supersymmetric compactifications the one-loop threshold contributions are dominant, while the higher loop correction are subleading. This is due to the fact that at heavy Kaluza–Klein levels the spectrum as well as the interactions are N=2 supersymmetric. In particular, we give two different arguments leading to this result – one is field theoretic, while the second one utilizes the power of string perturbation techniques. To illustrate our discussions we perform explicit two-loop computations of various corrections to gauge couplings within this framework. We also remark on phenomenological applications of our discussions in the context of TeV-scale brane world.

The renormalized gauge coupling and non-perturbative tests of dimensional reduction

In 4d lattice simulations of Standard Model like theories, the renormalized gauge coupling in the broken phase can be determined from the prefactor of the Yukawa term in the static potential. We compute the same quantity in terms of the conventional MSbar scheme gauge coupling. The result allows for a further non-perturbative test of finite temperature dimensional reduction, by a comparison of the critical temperatures for the electroweak phase transition as obtained with 4d lattice simulations and with 3d effective theory simulations.

Supersymmetry-breaking loops from analytic continuation into superspace

We extend to all orders in perturbation theory a method to calculate supersymmetry-breaking effects by analytic continuation of the renormalization group into superspace. A central observation is that the renormalized gauge coupling can be extended to a real vector superfield, thereby including soft breaking effects in the gauge sector. We explain the relation between this vector superfield coupling and the "holomorphic" gauge coupling, which is a chiral superfield running only at 1 loop. We consider these issues for a number of regulators, including dimensional reduction. With this method, the renormalization group equations for soft supersymmetry breaking terms are directly related to supersymmetric beta functions and anomalous dimensions to all orders in perturbation theory. However, the real power of the formalism lies in computing finite soft breaking effects corresponding to high-loop component calculations. We prove that the gaugino mass in gauge-mediated supersymmetry breaking is ``screened'' from strong interactions in the messenger sector. We present the complete next-to-leading calculation of gaugino masses (2 loops) and sfermion masses (3 loops) in minimal gauge mediation, and several other calculations of phenomenological relevance.

Noncompact lattice QED with two charges: Phase diagram and renormalization group flow.

The phase diagram of non-compact lattice QED in four dimensions with staggered fermions of charges 1 and $-1/2$ is investigated. The renormalized charges are determined and found to be in agreement with perturbation theory. This is an indication that there is no continuum limit with non-vanishing renormalized gauge coupling, and that the theory has a validity bound for every finite value of the renormalized coupling. The renormalization group flow of the charges is investigated and an estimate for the validity bound as a function of the cut-off is obtained. Generalizing this estimate to all fermions in the Standard Model,it is found that a cut-off at the Planck scale implies that $\alpha_R$ has to be less than $1/80$. Due to spontaneous chiral symmetry breaking, strongly bound fermion-antifermion composite states are generated. Their spectrum is discussed.

Dissipative gauge theory of fermions and its marginal-Fermi-liquid-like behavior

A non-relativistic fermion system interacting with a gauge field is studied by wilsonian renormalization-group analysis. This model is motivated by the work of Halperin, Lee and Read, which discussed a two-dimensional electron system in a magnetic field at the Landau filling factor v = 1 2 . A dissipative term of the gauge field (the Landau damping factor) is generated under the renormalization-group transformations, and it plays a very important role in the low-energy behaviors of the system. Renormalization constants are calculated in the one-loop level. In some parameter region, the renormalized gauge coupling constant has a non-trivial infrared fixed point, and the low-energy renormalized fermion propagator has a branch cut rather than a pole, just like the Luttinger liquid in one dimension. At a special value of the parameter, the weight of quasi-particles decreases at low energy as 1/ln ω, where ω is the energy of the fermon. This behavior is nothing but that of the marginal Fermi liquid, which is proposed to describe the anomalous metallic phase of high- T c cuprates.

Wormhole effects on Yang-Mills theory

In this paper wormhole effects on SO(3) YM theory are examined. The wormhole wave functions for the scalar, the vector and the tensor expansion modes are computed assuming a small gauge coupling which leads to an effective decoupling of gravity and YM theory. These results are used to determine the wormhole vertices and the corresponding effective operators for the lowest expansion mode of each type. For the lowest scalar mode we find a renormalization of the gauge coupling from the two-point function and the operators tr(F 3), tr(F 2 F ̃ ) from the three-point function. The two-point function for the lowest vector mode contributes to the gauge coupling renormalization only, whereas the lowest tensor mode can also generate higher derivative terms.

Monte—Carlo methods and results for the electro—weak phase transition

The lattice SU(2) Higgs model is simulated at T = O and T ⋍ T c . Simple and efficient heatbath and reflection updates for the Higgs field are presented. It is shown how to determine a renormalized gauge coupling from Polyakov line correlations. For physical parameters g R 2 ⋍ 0.55 and m H / m W ⋍ 0.6, the surface tension at the finite temperature transition is obtained.

Proposed tests for minimal SU(5) supergravity at Fermilab, Gran Sasso, SuperKamiokande and LEP

A series of predictions are worked out in order to put the minimal SU(5) supergravity model under experimental test. Using the two-loop gauge coupling renormalization group equations, with the inclusion of supersymmetric threshold corrections, we calculate a new value for the proton decay rate in this model and find that SuperKamiokande and Gran Sasso should see the proton decay mode p→ νK + for most of the allowed parameter space. A set of physically sensible assumptions and the cosmological requirement of a not too young Universe give us a rather restrictive set of allowed points in the parameter space, which characterizes this model. This set implies the existence of interesting correlations among various masses: either the lightest chargino and the next-to-lightest neutralino are below ≈ 100 GeV (and therefore observable at the Tevatron) or the lightest Higgs boson is below ≈50 GeV (and therefore observable at LEP I-II). These tests are crucial steps towards selecting the correct low-energy effective supergravity model. We also comment on the compatibility of the model with sin 2 θ W( M Z ) measurements as a function of α 3( M Z ).

The Schrödinger functional — a renormalizable probe for non-abelian gauge theories

Following Symanzik we argue that the Schrödinger functional in lattice gauge theories without matter fields has a well-defined continuum limit. Due to gauge invariance no extra counterterms are required. The Schrödinger functional is, moreover, accessible to numerical simulations. It may hence be used to study the scaling properties of the theory and in particular the evolution of the renormalized gauge coupling from low to high energies. A concrete proposition along this line is made and the necessary perturbative analysis of the Schrödinger functional is carried through to one-loop order.

Towards a better quantitative understanding of the SU(2) Higgs model

We describe the results of a high-statistics simulation of the SU(2) Higgs model at λ=∞, β=2.4, 2.7 and 3.0, on lattices of sizes ranging between 8 3×16 and 14 3×28. The measured quantities are the Higgs and W masses, the potential between two static sources from which the renormalized gauge coupling is extracted, and the vacuum overlap order parameter which defines the Higgs expectation value in a gauge invariant way. Strong finite-size effects are observed. They seem to come under control for the larger lattices, at least at β=2.7. Renormalized tree level relations are found to hold remarkably well at β=2.7 and 3.0.

A numerical study of the SU(2) Higgs model

Preliminary results of a simulation of the SU(2) theory with a Higgs field in the fundamental representation are presented. For an 8 3 × 16 and a 12 3 × 24 lattice we investigated the parameter range 2.4 ≤ β ≤ 3.0, λ = ∞. In the confinement region huge autocorrelations prevent us from obtaining reliable results. In the Higgs region the W mass m w turns out to have larger finite size effects than previously expected. Besides the W and Higgs masses, we estimate for β = 3.0 on the 12 3 × 24 lattice the renormalized gauge coupling g т and (unfortunately only up to an unknown waave function renormalization) the Higgs expectation value ν. The latter is obtained from the vacuum overlap order parameter. Technical problems encountered in its computation are discussed in some detail. Finally we discuss our results; in particular we show that the tree level relation m w 2 = g τ 2 ν 2/2 is fulfilled remarkably well.