Bare Coupling Research Articles

Relation between two-point Green’s functions of $$\mathcal{N} = 1$$ SQED with N f flavors, regularized by higher derivatives, in the three-loop approximation

We verify the identity which relates the two-point Green’s functions of \(\mathcal{N} = 1\) SQED with N f flavors, regularized by higher derivatives, by explicit calculations in the three-loop approximation. This identity explains why, in the limit of the vanishing external momentum, the two-point Green’s function of the gauge superfield is given by integrals of double total derivatives in the momentum space. It also makes it possible to exactly derive the NSVZ β-function in all loops if the renormalization group functions are defined in terms of the bare coupling constant. In order to verify the considered identity, we use it to construct integrals giving the three-loop β-function starting from the two-point Green’s functions of matter superfields in the two-loop approximation. Then we demonstrate that the results for these integrals coincide with the sums of the corresponding three-loop supergraphs.

Towards the equation of state of quark-gluon matter from lattice QCD with Wilson twisted mass fermions

This contribution gives an overview of a long-term study of quark-gluon matter under extreme conditions employing Euclidean lattice discretized Quantum Chromodynamics (QCD) with Wilson twisted mass fermions and improved gauge actions. The analysis relies on methods and results of the European Twisted Mass Collaboration at zero temperature. We will start for two mass-degenerate light flavour degrees of freedom with a thorough investigation of the three-dimensional phase diagram of the bare coupling and the two mass-related parameters. Then for the same case we are going to describe the search for the pseudo-critical behaviour at various quark mass values. We shall discuss the behaviour of the (pseudo-)critical temperature Tc as a function of the (charged) pion mass mπ and the difficulty to extract the chiral limit in order to establish or reject the conjectured O(4)-universality. For the two-flavour case we shall further present very recent results for the thermodynamic Equation of State at non-zero temperature. Finally we will provide a view on an ongoing first exploration of QCD thermodynamics under the inclusion of virtual strange and charm quarks at their realistic mass values.

On the particle spectrum and the conformal window

We study the SU(3) gauge theory with twelve flavours of fermions in the fundamental representation as a prototype of non-Abelian gauge theories inside the conformal window. Guided by the pattern of underlying symmetries, chiral and conformal, we analyze the two-point functions theoretically and on the lattice, and determine the finite size scaling and the infinite volume fermion mass dependence of the would-be hadron masses. We show that the spectrum in the Coulomb phase of the system can be described in the context of a universal scaling analysis and we provide the nonperturbative determination of the fermion mass anomalous dimension γ∗ = 0.235(46) at the infrared fixed point. We comment on the agreement with the four-loop perturbative prediction for this quantity and we provide a unified description of all existing lattice results for the spectrum of this system, them being in the Coulomb phase or the asymptotically free phase. Our results corroborate the view that the fixed point we are studying is not associated to a physical singularity along the bare coupling line and estimates of physical observables can be attempted on either side of the fixed point. Finally, we observe the restoration of the U(1) axial symmetry in the two-point functions.

The NSVZ β-function in supersymmetric theories with different regularizations and renormalization prescriptions

We briefly review the calculations of quantum corrections related with the exact NSVZ $\beta$-function in ${\cal N}=1$ supersymmetric theories, paying especial attention to the scheme dependence of the results. It is explained, how the NSVZ relation is obtained for the renormalization group functions defined in terms of the bare coupling constant if a theory is regularized by higher derivatives. Also we describe, how to construct a special renormalization prescription which gives the NSVZ relation for the renormalization group functions defined in terms of the renormalized coupling constant exactly in all orders for Abelian supersymmetric theories, regularized by higher derivatives. The scheme dependence of the NSVZ $\beta$-function (for the renormalization group functions defined in terms of the renormalized coupling constant) is discussed in the non-Abelian case. It is shown that in this case the NSVZ $\beta$-function leads to a certain scheme-independent equality.

SUSY FLAVOR v2.5: A computational tool for FCNC and CP-violating processes in the MSSM

SUSY FLAVOR v2.5: A computational tool for FCNC and CP-violating processes in the MSSM

Inflationary parameters in renormalization group improved ϕ4 theory

Inflation models can be examined by the cosmological observations, WMAP, Planck, BICEP2 and so on. These observations directly constrain the spectral index, $n_s$, and the tensor-to-scalar ratio, $r$. Besides, from a theoretical point of view, it has been shown that any inflation models asymptote a universal attractor in $(n_s,r)$ plane for a larger scalar-gravity coupling. In this work we consider a simple chaotic inflation model with a scalar quartic and a scalar-curvature interactions. The quantum corrections are introduced for these interactions through the renormalization group. The inflationary parameters, $n_s$ and $r$, are numerically calculated with shifting the bare scalar-gravity coupling $\xi_0$, the quartic scalar bare coupling $\lambda_0$, the renormalization scale $\mu$ and the e-folding number $N$. The Planck data is consistent with the $\phi^4$ theory with a finite scalar-curvature coupling. It is found that the RG running induces a non-universal contribution for $n_s$ and $r$. It can increase the tensor-to-scalar ratio, $r$, which is consistent with Planck data so that it may approach the BICEP2 data for a small renormalization scale.

Why Does Graphene Behave as a Weakly Interacting System?

We address the puzzling weak-coupling perturbative behavior of graphene interaction effects as manifested experimentally, in spite of the effective fine structure constant being large, by calculating the effect of Coulomb interactions on the quasiparticle properties to next-to-leading order in the random phase approximation (RPA). The focus of our work is graphene suspended in vacuum, where electron-electron interactions are strong and the system is manifestly in a nonperturbative regime. We report results for the quasiparticle residue and the Fermi velocity renormalization at low carrier density. The smallness of the next-to-leading order corrections that we obtain demonstrates that the RPA theory converges rapidly and thus, in contrast to the usual perturbative expansion in the bare coupling constant, constitutes a quantitatively predictive theory of graphene many-body physics for any coupling strength.

The NSVZ β-function and the Schwinger-Dyson equations for N $$ \mathcal{N} $$ = 1 SQED with N f flavors, regularized by higher derivatives

The effective diagram technique based on the Schwinger-Dyson equations is constructed for N=1 SQED with N_f flavors, regularized by higher derivatives. Using these effective diagrams, it is possible to derive the exact NSVZ relation between the beta-function and the anomalous dimension of the matter superfields exactly in all loops, if the renormalization group functions are defined in terms of the bare coupling constant. In particular, we verify that all integrals which give the beta-function defined in terms of the bare coupling constant are integrals of double total derivatives and prove some identities relating Green functions.

Residue formulas for prepotentials, instanton expansions and conformal blocks

We study the extended prepotentials for the S-duality class of quiver gauge theories, considering them as quasiclassical tau-functions, depending on gauge theory condensates and bare couplings. The residue formulas for the third derivatives of extended prepotentials are proven, which lead to effective way of their computation, as expansion in the weak-coupling regime. We discuss also the differential equations, following from the residue formulas, including the WDVV equations, proven to be valid for the $SU(2)$ quiver gauge theories. As a particular example we consider the constrained conformal quiver gauge theory, corresponding to the Zamolodchikov conformal blocks by 4d/2d duality. In this case part of the found differential equations turn into nontrivial relations for the period matrices of hyperelliptic curves.

Triviality of [formula omitted] theory in a finite volume scheme adapted to the broken phase

We study the standard one-component φ4-theory in four dimensions. A renormalized coupling is defined in a finite size renormalization scheme which becomes the standard scheme of the broken phase for large volumes. Numerical simulations are reported using the worm algorithm in the limit of infinite bare coupling. The cutoff dependence of the renormalized coupling closely follows the perturbative Callan Symanzik equation and the triviality scenario is hence further supported.

Magnetic catalysis (and inverse catalysis) at finite temperature in two-color lattice QCD

Two-color lattice QCD with N_f=4 staggered fermion degrees of freedom (no rooting trick is applied) with equal electric charge q is studied in a homogeneous magnetic background field B and at non-zero temperature T. In order to circumvent renormalization as a function of the bare coupling we apply a fixed-scale approach. We study the influence of the magnetic field on the critical temperature. At rather small pseudo-scalar meson mass ($m_{\pi} \approx 175 \mathrm{MeV} \approx T_c(B=0)$) we confirm a monotonic rise of the quark condensate $<\bar{\psi} \psi>$ with increasing magnetic field strength, i.e. magnetic catalysis, as long as one is staying within the confinement or deconfinement phase. In the transition region we find indications for a non-monotonic behavior of $T_c(B)$ at low magnetic field strength ($qB<0.8 \mathrm{GeV}^2$) and a clear rise at stronger magnetic field. The conjectured existence of a minimum value $T_c(B^{*}) < T_c(B=0)$ would leave a temperature window for a decrease of $<\bar{\psi} \psi>$ with rising $B$ (inverse magnetic catalysis) also in the present model.

Higgs mass bounds from renormalization flow for a simple Yukawa model

We study the functional renormalization group flow of a Higgs-Yukawa toy model mimicking the top-Higgs sector of the standard model. This approach allows for treating arbitrary bare couplings. For the class of standard bare potentials of \phi^4-type at a given ultraviolet cut-off, we show that a finite infrared Higgs mass range emerges naturally from the renormalization group flow itself. Higgs masses outside the resulting bounds cannot be connected to any conceivable set of bare parameters in this standard-model \phi^4 class. By contrast, more general bare potentials allow to diminish the lower bound considerably. We identify a simple renormalization group mechanism for this depletion of the lower bound. If active also in the full standard model, Higgs masses smaller than the conventional infrared window do not necessarily require new physics at low scales or give rise to instability problems.

Scheme independent consequence of the NSVZ relation for [formula omitted] SQED with [formula omitted] flavors

The exact NSVZ β-function is obtained for N=1 SQED with Nf flavors in all orders of the perturbation theory, if the renormalization group functions are defined in terms of the bare coupling constant and the theory is regularized by higher derivatives. However, if the renormalization group functions are defined in terms of the renormalized coupling constant, the NSVZ relation between the β-function and the anomalous dimension of the matter superfields is valid only in a certain (NSVZ) scheme. We prove that for N=1 SQED with Nf flavors the NSVZ relation is valid for the terms proportional to (Nf)1 in an arbitrary subtraction scheme, while the terms proportional to (Nf)k with k⩾2 are scheme dependent. These results are verified by an explicit calculation of a three-loop β-function and a two-loop anomalous dimension made with the higher derivative regularization in the NSVZ and MOM subtraction schemes. In this approximation it is verified that in the MOM subtraction scheme the renormalization group functions obtained with the higher derivative regularization and with the dimensional reduction coincide.

Dynamical mean field approximation applied to quantum field theory

We apply the dynamical mean field theory (DMFT) approximation to the real, scalar ${\ensuremath{\varphi}}^{4}$ quantum field theory. By comparing to lattice Monte Carlo calculations, perturbation theory and standard mean field theory, we test the quality of the approximation in two, three, four and five dimensions. The quantities considered in these tests are the critical coupling for the transition to the ordered phase and the associated critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$. We also map out the phase diagram in the most relevant case of four dimensions. In two and three dimensions, DMFT incorrectly predicts a first-order phase transition for all bare quartic couplings, which is problematic, because the second-order nature of the phase transition of lattice ${\ensuremath{\varphi}}^{4}$ theory is crucial for taking the continuum limit. Nevertheless, by extrapolating the behavior away from the phase transition, one can obtain critical couplings and critical exponents. They differ from those of mean field theory and are much closer to the correct values. In four dimensions the transition is second order for small quartic couplings and turns weakly first order as the coupling increases beyond a tricritical value. In dimensions five and higher, DMFT gives qualitatively correct results, predicts reasonable values for the critical exponents and considerably more accurate critical couplings than standard mean field theory. The approximation works best for small values of the quartic coupling. We investigate the change from first- to second-order transition in the local limit of DMFT which is computationally much cheaper. We also discuss technical issues related to the convergence of the nonlinear self-consistency equation solver and the solution of the effective single-site model using Fourier-space Monte Carlo updates in the presence of a ${\ensuremath{\varphi}}^{4}$ interaction.

Running coupling constant and mass anomalous dimension of six-flavor SU(2) gauge theory

In the exploration of viable models of dynamical electroweak symmetry breaking, it is essential to locate the lower end of the conformal window and know the mass anomalous dimensions there for a variety of gauge theories. We calculate, with the Schr\"odinger functional scheme, the running coupling constant and the mass anomalous dimension of SU(2) gauge theory with six massless Dirac fermions in the fundamental representation. The calculations are performed on ${6}^{4}--{24}^{4}$ lattices over a wide range of lattice bare couplings to take the continuum limit. The discretization errors for both quantities are removed perturbatively. We find that the running slows down and comes to a stop at $0.06\ensuremath{\lesssim}1/{g}^{2}\ensuremath{\lesssim}0.15$ where the mass anomalous dimension is estimated to be $0.26\ensuremath{\lesssim}{\ensuremath{\gamma}}_{m}^{*}\ensuremath{\lesssim}0.74$.

NSVZ scheme with the higher derivative regularization for [formula omitted] SQED

The exact NSVZ relation between a β-function of N=1 SQED and an anomalous dimension of the matter superfields is studied within the Slavnov higher derivative regularization approach. It is shown that if the renormalization group functions are defined in terms of the bare coupling constant, this relation is always valid. In the renormalized theory the NSVZ relation is obtained in the momentum subtraction scheme supplemented by a special finite renormalization. Unlike the dimensional reduction, the higher derivative regularization allows to fix this finite renormalization. This is made by imposing the conditions Z3(α,μ=Λ)=1 and Z(α,μ=Λ)=1 on the renormalization constants of N=1 SQED, where Λ is a parameter in the higher derivative term. The results are verified by the explicit three-loop calculation. In this approximation we relate the DR¯ scheme and the NSVZ scheme defined within the higher derivative approach by the finite renormalization.

Tau-functions for quiver gauge theories

The prepotentials for the quiver supersymmetric gauge theories are defined as quasiclassical tau-functions, depending on two different sets of variables: the parameters of the UV gauge theory or the bare compexified couplings, and the vacuum condensates of the theory in IR. The bare couplings are introduced as periods on the UV base curve, and the consistency of corresponding gradient formulas for the tau-functions is proven using the Riemann bilinear relations. It is shown, that dependence of generalised prepotentials for the quiver gauge theories upon the bare couplings turns to coincide with the corresponding formulas for the derivatives of tau-functions for the isomonodromic deformations. Computations for the SU(2) quiver gauge theories with bi- and tri-fundamental matter are performed explicitly and analysed in the context of 4d/2d correspondence.

Four fermion interactions in non-Abelian gauge theory

We continue our earlier study of the phase structure of a $SU(2)$ gauge theory whose action contains additional chirally invariant four fermion interactions. Our lattice theory uses a reduced staggered fermion formalism to generate two Dirac flavors in the continuum limit. In the current study we have tried to reduce lattice spacing and taste breaking effects by using an improved fermion action incorporating stout smeared links. As in our earlier study we observe two regimes; for weak gauge coupling the chiral condensate behaves as an order parameter differentiating a phase at small four Fermi coupling where the condensate vanishes from a phase at strong four Fermi coupling in which chiral symmetry is spontaneously broken. This picture changes qualitatively when the gauge coupling is strong enough to cause confinement; in this case we observe a phase transition for some critical value of the four Fermi coupling associated with a strong enhancement of the chiral condensate. We present evidence that this transition is likely first order. Furthermore, we observe that the critical four Fermi coupling varies monotonically with bare gauge coupling---decreasing, as expected, as the gauge coupling is increased. We have checked that these results remain stable under differing levels of smearing. These results argue against the appearance of new fixed points associated with chirally invariant four fermion interactions in confining non-Abelian gauge theories.

Four-fermion interaction in relativistic heavy ion collisions

A (nearly) perfect liquid discovered in experiments with ultrarelativistic heavy ion collisions is investigated by studying the quark ensembles with four-fermion interaction considered as a fundamental theoretical approach. A comparative analysis of several quantum liquid models is performed, allowing one to conclude that the presence of gas-liquid phase transition is a characteristic feature of those models. The problem of the instability of the quark droplets with the small number of quarks, related with a possible formation of a chiral soliton, is discussed. A mixed phase of vacuum and baryon matter, as a possible scenario of its stability, is studied. Some aspects of the color superconductivity are considered. In addition, a recently proposed nontrivial thermodynamic state, called the fermion condensate, is studied. An analysis of unexpected opportunity to link the bare and effective coupling constants is performed within the framework of a simple model. It is pointed out that a simple subtraction procedure leads to the finite result without a typical logarithmic singularity for the observed coupling constant as a function of the transferred energy.

Deformed $ \mathcal{N}=2 $ theories, generalized recursion relations and S-duality

Abstract We study the non-perturbative properties of $ \mathcal{N}=2 $ super conformal field theories in four dimensions using localization techniques. In particular we consider SU(2) gauge theories, deformed by a generic ϵ-background, with four fundamental flavors or with one adjoint hypermultiplet. In both cases we explicitly compute the first few instanton corrections to the partition function and the prepotential using Nekrasov’s approach. These results allow us to reconstruct exact expressions involving quasi-modular functions of the bare gauge coupling constant and to show that the prepotential terms satisfy a modular anomaly equation that takes the form of a recursion relation with an explicitly ϵ-dependent term. We then investigate the implications of this recursion relation on the modular properties of the effective theory and find that with a suitable redefinition of the prepotential and of the effective coupling it is possible, at least up to the third order in the deformation parameters, to cast the S-duality relations in the same form as they appear in the Seiberg-Witten solution of the undeformed theory.