We discuss a new channel of photon emission from the quark-gluon plasma, which opens up as photon acquires a tachyonic mass in the presence of the CP-odd topological domains generated by the chiral anomaly. It allows photon radiation through the decay $q\to qγ$ and annihilation $q\bar q\to γ$ processes closely related to the chiral Cherenkov radiation. Unlike previous proposals this mechanism does not require an external magnetic field. The differential photon emission rate per unit volume is computed and shown to be comparable to the rate of photon emission in conventional processes.
Considering that the chiral kinetic equations of motion (CEOM) can be derived from the spin kinetic equations of motion (SEOM) for massless particles with approximations, we simulate the chiral anomalies by using the latter in a box system with the periodic boundary condition under a uniform external magnetic field. We found that the chiral magnetic effect is weaker while the damping of the chiral magnetic wave is stronger from the SEOM compared with that from the CEOM. In addition, effects induced by chiral anomalies from the SEOM are less sensitive to the decay of the magnetic field than fro…
A new mechanism of photon emission in the quark-gluon plasma is proposed. Photon dispersion relation in the presence of the $CP$-odd topological regions generated by the chiral anomaly acquires an imaginary mass. It allows photon radiation through the decay $q\to qγ$ and annihilation $q\bar q\to γ$ processes closely related to the chiral Cherenkov radiation. Unlike previous proposals this mechanism does not require an external magnetic field. The differential photon emission rate per unit volume is computed and shown to be comparable to the rate of photon emission in conventional processes.
Polarization and chirality are direct manifestations of quantum mechanics in the Quark Gluon Plasma as a relativistic fluid. This is one of the reasons why they are intriguing phenomena, that have attracted so much attention lately. In this talk I will review what is, in my view, the current theoretical understanding and highlight the most interesting recent result.
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.
Time evolution of electromagnetic field created in heavy-ion collisions strongly depends on the electromagnetic response of the quark-gluon plasma, which can be described by the Ohmic and chiral conductivities. The later is intimately related to the Chiral Magnetic Effect. I argue that a solution to the classical Maxwell equations at finite chiral conductivity is unstable due to the soft modes $k<σ_χ$ that grow exponentially with time. In the kinematical region relevant for the relativistic heavy-ion collisions, I derive analytical expressions for the magnetic field of a point charge. I show t…
Effective chiral Lagrangians involving constituent quarks, Goldstone bosons and long-distance gluons are believed to describe the strong interactions in an intermediate energy region between the confinement scale and the chiral symmetry breaking scale. Baryons and mesons in such a description are bound states of constituent quarks. We discuss the combined use of the techniques of effective chiral field theory and of the field theoretic method known as Fock-Tani representation to derive effective hadron interactions. The Fock-Tani method is based on a change of representation by means of a unit…
In this work, we investigate the gluon distribution functions for the pion and kaon, in addition to the improved result of the valence-quark ones, in the gauge-invariant nonlocal chiral-quark model (NL$χ$QM), in which the momentum dependence of the quark interactions is properly taken into account. We then analyze the gluon distribution functions, generated dynamically through the splitting functions in the DGLAP QCD evolution. By comparing with the recent lattice QCD results and JAM global analyses, it is found that the present numerical results for the gluon parton distribution functions for…
The spin polarization functions $(Δu, ~Δd, ~Δs)$ for proton are calculated in the chiral quark model ($χ$QM) with SU(3) symmetry breaking as well as configuration mixing generated by one gluon exchange forces for the NMC and the most recent E866 data. Besides reproducing the spin polarization functions $Δu,Δd, Δs$ as well as $G_A/G_V$, it can accomodate nucleon magnetic moments and neutron charge radius as well, thus resolving the compatibility problem of these parameters which could not be achieved in constituent quark models.
A recent proposal by Kaplan for a chiral gauge theory on the lattice is tested with background gauge fields. The spectrum of the finite lattice Hamiltonian is calculated and the existence of a chiral fermion is demonstrated. Lattice doublers are found to decouple. The flavor anomalies, which are in agreement with the continuum anomaly relation, are obeserved on a finite lattice. Non-trivial anomaly cancellation is observed in a chiral gauge current.
Axial anomalies give rise to interesting new transport phenomena such as the "chiral magnetic effect". We discuss how the associated transport coefficients can be studied via Kubo formulas at weak and strong coupling, the latter via gauge gravity duality. We argue for a new "chiral gravito-magnetic" (or vortical) effect sensitive to the presence of mixed gauge-gravitational anomalies.
We study the mechanism responsible for the onset of instabilities in a chiral phase transition at nonzero temperature and baryon chemical potential. As a low-energy effective model, we consider an expanding relativistic plasma of quarks coupled to a chiral field, and obtain a phenomenological chiral hydrodynamics from a variational principle. Studying the dispersion relation for small fluctuations around equilibrium, we identify the role played by chiral waves and pressure waves in the generation of instabilities. We show that pressure modes become unstable earlier than chiral modes.
Parametrizing the possible underlying theory or new physics' decoupling effects in the most general way we reexamined the validity of canonical trace relation and chiral symmetry in certain one-loop two-point functions. The anomalies and the relation between chiral and trace Ward identities are investigated and interpreted in the perspective of a complete underlying theory or new physics instead of regularization, with some new phenomena found as by-products.
We study the gluon polarization contribution at the quark model renormalization scale to the proton spin, $Γ$, in the chiral bag model. It is evaluated by taking the expectation value of the forward matrix element of a local gluon operator in the axial gauge $A^+=0$. It is shown that the confining boundary condition for the color electric field plays an important role. When a solution satisfying the boundary condition for the color electric field, which is not the conventionally used but which we favor, is used, the $Γ$ has a positive value for {\it all} bag radii and its magnitude is comparab…
I consider global transformations of a Dirac fermion field, that are generated by the generators of Poincar'e transformations, but with a γ_5 appended. Such chiral translations and chiral Lorentz transformations are usually not symmetries of the Lagrangian, but naively they are symmetries of the fermionic measure. However, by using proper time regularization in Minkowski space, I find that they in general give rise to a nontrivial Jacobian. In this sense they have "anomalies". I calculate these anomalies in a theory of a massive fermion coupled to an external Abelian vector field. My motivatio…
First part of the paper is devoted to applying the confinement mechanism proposed earlier by the author to estimate the possible parameters of the confining SU(3)-gluonic field in neutral kaons. The estimates obtained are consistent with the widths of the electromagnetic decays $K^0,\bar{K}^0\to2γ$ too. The corresponding estimates of the gluon concentrations, electric and magnetic colour field strengths are also adduced for the mentioned field at the scales of the mesons under consideration. The second part of the paper takes into account the results obtained previously by the author to estima…
We show that the quark (not nucleon) based chiral linear $σ$ model which receives strong support as a mean field theory for the strong interactions can actually be made asymptotically free by coupling to gluons. It has the further bonus of providing confinement.Alternatively, this theory is QCD that is chirally extended by the addition of a $(σ,π)$ chiral multiplet. It could therefore be a candidate theory for the strong interactions which is somewhat different from QCD.
The Dual Meissner Effect description of QCD in the confining region provides $\frac{1}{q^4}$ behaviour for the gluon propagator and involves the dual gluon mass $m$ as a parameter. This is used in the Schwinger-Dyson equation for the quarks in the infrared region to exhibit chiral symmetry breaking for light quarks. Using the light quark condensate as input, the dual gluon mass is determined and its importance in showing the asymptotic free behaviour of the extrinsic curvature coupling in the rigid QCD string is discussed.
I review some aspects of the interplay between anomalies and chiral symmetry. The quantum anomaly that breaks the U(1) axial symmetry of massless QCD leaves behind a flavor-singlet discrete chiral invariance. When the mass is turned on this residual symmetry has a close connection with the strong CP violating parameter theta. One result is that a first order transition is usually expected when the strong CP violating angle passes through pi. This symmetry can be understood either in terms of effective chiral Lagrangians or in terms of the underlying quark fields.
In a simple abelian spinor field theory, the canonical trace identities for certain axial-vector and axial-scalar operators are reexamined in dimensional regularization, some disagreements with previous results are found and an interesting new phenomenon is observed and briefly discussed.
Heavy-ion collision experiments at RHIC and the LHC have found a new emergent phase of QCD, a strongly coupled quark-gluon plasma (sQGP) that is distinctively different from either the low temperature hadron phase or the very high temperature weakly coupled plasma phase. Highly nontrivial emergent phenomena occur in such sQGP and two examples will be discussed in this contribution: the magnetic component of sQGP that stems from topologically nontrivial configurations in the gluon sector; and the anomalous chiral transport that arises as macroscopic manifestation of microscopic chiral anomaly i…
Motivated by the search for possible CP violating terms in the trace of the energy-momentum tensor in theories coupled to gravity we revisit the problem of trace anomalies in chiral theories. We recalculate the latter and ascertain that in the trace of the energy-momentum tensor of theories with chiral fermions at one-loop the Pontryagin density appears with an imaginary coefficient. We argue that this may break unitarity, in which case the trace anomaly has to be used as a selective criterion for theories, analogous to the chiral anomalies in gauge theories. We analyze some remarkable consequ…
This is a brief report of work performed in arXiv:1106.3576. We consider the chiral transport terms in a relativistic charged superfluid, and their relation to triangle anomalies. The terms allowed by the Second Law of thermodynamics have been worked out. A simplified form is proposed on heuristic grounds, from an analysis of the better-understood chiral effects in normal fluids. We point out the appearance of a "chiral electric conductivity", and relate it through an educated guess to the axial anomaly coefficient.
We compute the critical coupling constant for the dynamical chiral-symmetry breaking in a model of quantum chromodynamics, solving numerically the quark self-energy using infrared finite gluon propagators found as solutions of the Schwinger-Dyson equation for the gluon, and one gluon propagator determined in numerical lattice simulations. The gluon mass scale screens the force responsible for the chiral breaking, and the transition occurs only for a larger critical coupling constant than the one obtained with the perturbative propagator. The critical coupling shows a great sensibility to the g…
Martinus Veltman was the first to point out the inconsistency of the experimental value for the decay rate of $π^0\rightarrowγγ$ and its calculation by J. Steinberger with the very successful concept of the pion as the (pseudo)Nambu-Goldstone boson of the spontaneously broken global axial symmetry of strong interactions. That inconsistency has been resolved by J. Bell and R. Jackiw in their famous paper on the chiral anomalies. We review the connection between the decay amplitudes of an axion into two gauge bosons in Abelian vector-like and chiral gauge theories. The axion is the Nambu-Goldsto…
We consider here chiral symmetry breaking in quantum chromodynamics arising from gluon condensates in vacuum. Through coherent states of gluons simulating a mean field type of approximation, we show that the off-shell gluon condensates of vacuum generate a mass-like contribution for the quarks, giving rise to chiral symmetry breaking. We next note that spontaneous breaking of global chiral symmetry links the four component quark field operator to the pion wave function. This in turn yields many hadronic properties in the light quark sector in agreement with experiments, leading to the conclusi…
We exploit the recent lattice results for the infrared gluon propagator with light dynamical quarks and solve the gap equation for the quark propagator. Chiral symmetry breaking and confinement (intimately tied with the analytic properties of QCD Schwinger functions) order parameters are then studied.
Chiral Magnetic Wave (CMW) is a gapless collective excitation of quark-gluon plasma in the presence of external magnetic field that stems from the interplay of Chiral Magnetic (CME) and Chiral Separation Effects (CSE); it is composed by the waves of the electric and chiral charge densities coupled by the axial anomaly. We consider CMW at finite baryon density and find that it induces the electric quadrupole moment of the quark-gluon plasma produced in heavy ion collisions: the "poles" of the produced fireball (pointing outside of the reaction plane) acquire additional positive electric charge,…
Considering a QCD chiral symmetry breaking model where the gap equation contains an effective confining propagator and a dressed gluon propagator with a dynamically generated mass, we verify that the chiral symmetry is restored for a large number of quarks $n_{f}\approx 7-13$. We discuss the uncertainty in the results, that is related to the determination of the string tension ($K_{F}$), appearing in the confining propagator, and the effective gluon mass ($m_{g}$) at large $n_{f}$.
We show the existence of a new gapless collective excitation in a rotating fluid system with chiral fermions, named as the Chiral Vortical Wave (CVW). The CVW has its microscopic origin at the quantum anomaly and macroscopically arises from interplay between vector and axial charge fluctuations induced by vortical effects. The wave equation is obtained both from hydrodynamic current equations and from chiral kinetic theory and its solutions show nontrivial CVW-induced charge transport from different initial conditions. Using the rotating quark-gluon plasma in heavy ion collisions as a concrete…
It is possible to formulate fluid dynamics in terms of group-valued variables. This is particularly suited to the cases where the fluid has nonabelian charges and is coupled to nonabelian gauge fields. We explore this formulation further in this paper. An action for a fluid of relativistic particles (with and without spin) is given in terms of the Lorentz and Poincare (or de Sitter) groups. Considering the case of particles with flavor symmetries, a general fluid action which also incorporates all flavor anomalies is given. The chiral magnetic and chiral vorticity effects as well as the conseq…
The Standard Model (SM) is amended by one generation of quarks and leptons which are vector-like (VL) under the SM gauge group but chiral with respect to a new $\mathrm{U}(1)_{3-4}$ gauge symmetry. We show that this model can simultaneously explain the deviation of the muon $g-2$ as well as the observed anomalies in $b\rightarrow sμ^+μ^-$ transitions without conflicting with the data on Higgs decays, lepton flavor violation, or $B_s-\bar{B}_s$ mixing. The model is string theory motivated and GUT compatible, i.e. UV complete, and fits the data predicting VL quarks, leptons and a massive $Z'$ at…
We develop a color bosonization approach to treatment of QCD gauge field (''gluons'') at low energies in order to derive an effective color action of QCD taking into account the quark chiral anomaly in the case of SU(2) color.. We have found that there exists such a region in the chiral sector of color space, where a gauge field coincides with of chirally rotated vector field, while an induced axial vector field disappears. In this region, the unit color vector of chiral field plays a defining role, and a gauge field is parametrized in terms of chiral parameters, so that no additional degrees …
In the framework of the chiral quark-soliton model of the nucleon we investigate the properties of the polarized quark distribution. In particular we analyse the so called anomalous difference between the representations of the quark distribution functions in terms of occupied and non-occupied quark states. By an explicit analytical calculation it is shown that this anomaly is absent in the polarized isoscalar distribution Δu + Δd, which is ultaviolet finite. In the case of the polarized isovector quark distribution Δu - Δd the anomaly can be cancelled by a Pauli-Villars subtraction which is a…
The effect of transverse gluons on the chiral symmetry patterns of excited mesons is studied in a Coulomb gauge QCD model. The linear rising static quark-antiquark potential and the transverse gluon propagator known from lattice studies are input into the model. The non-perturbative quark propagator, which enters the meson bound state equations, is derived from the Dyson--Schwinger equations and a complete set of mesons for general spin quantum number is presented. From analyzing the bound state equations for large spins it is demonstrated, that chiral and axial symmetry are restored. In this …
We give an elementary derivation of the chiral magnetic effect based on a strong magnetic field lowest-Landau-level projection in conjunction with the well-known axial anomalies in two- and four-dimensional space-time. The argument is general, based on a Schur decomposition of the Dirac operator. In the dimensionally reduced theory, the chiral magnetic effect is directly related to the relativistic form of the Peierls instability, leading to a spiral form of the condensate, the chiral magnetic spiral. We then discuss the competition between spin projection, due to a strong magnetic field, and …
A low energy effective field theory model for QCD with a scalar color octet field is discussed. The model relates the gluon mass, the constituent quark masses and the quark condensate. The gluon mass comes about $\sqrt{N_c}\, Λ_{QCD}$ with the quark condensate being proportional to the gluon mass squared. The model suggests that the restoration of chiral symmetry and the deconfinement transition occur at the same temperature and that, near the transition, the critical exponent for the condensate is twice the gluon mass one. The model also favors the decoupling like solution for the gluon propa…
Gaussian QCD sum-rules are used to analyze all possible two-point correlation functions of scalar gluonic and quark currents. The independent predictions of the masses and relative coupling strengths from the different correlators are remarkably consistent with a scenario of two scalar states that couple to nearly-maximal mixtures of quark and gluonic currents.
The quantum anomaly that breaks the U(1) axial symmetry of massless multi-flavored QCD leaves behind a discrete flavor-singlet chiral invariance. With massive quarks, this residual symmetry has a close connection with the strong CP-violating parameter theta. One result is that if the lightest quarks are degenerate, then a first order transition will occur when theta passes through pi. The resulting framework helps clarify when the rooting prescription for extrapolating in the number of flavors is valid.
We study UV-complete Froggatt-Nielsen-like models for the generation of mass and mixing hierarchies, assuming that the integrated heavy fields are chiral with respect to an abelian Froggatt-Nielsen symmetry. It modifies the mixed anomalies with respect to the Standard Model gauge group, which opens up the possibility to gauge the Froggatt-Nielsen symmetry without the need to introduce additional spectator fermions, while keeping mass matrices usually associated to anomalous flavour symmetries. We give specific examples where this happens, and we study the flavourful axion which arises from an …
In the chiral magnetic effect an imbalance in the number of left- and right-handed quarks gives rise to an electromagnetic current parallel to the magnetic field produced in noncentral heavy-ion collisions. The chiral imbalance may be induced by topologically nontrivial gluon configurations via the QCD axial anomaly, while the resulting electromagnetic current itself is a consequence of the QED anomaly. In the Sakai-Sugimoto model, which in a certain limit is dual to large-N_c QCD, we discuss the proper implementation of the QED axial anomaly, the (ambiguous) definition of chiral currents, and…
We analyze the chiral transport terms in relativistic superfluid hydrodynamics. In addition to the spontaneously broken symmetry current, we consider an arbitrary number of unbroken symmetries and extend the results of arXiv:1105.3733. We suggest an interpretation of some of the new transport coefficients in terms of chiral and gravitational anomalies. In particular, we show that with unbroken gauged charges in the system, one can observe a chiral electric conductivity - a current in a perpendicular direction to the applied electric field. We present a motivated proposal for the value of the a…
If the phase transition of $QCD$ at nonzero temperature is dominated by the (approximate) restoration of chiral symmetry, then the transition might be characterized using a gauged linear sigma model. Assuming that vector meson dominance holds, such sigma models predict that at the temperature of chiral restoration, the pole mass of the thermal $ρ$ meson is greater than that at zero temperature; in the chiral limit and in weak coupling this mass is $\sim 962 \, MeV$. The width of the thermal $ρ-a_1$ peak is estimated to be about $200 - 250 \, MeV$.
Using perturbation theory in the Euclidean (imaginary time) formalism as well as the non-perturbative Fujikawa method, we verify that the chiral anomaly equation remains unaffected in the presence of nonzero chemical potential, $μ$. We extend our considerations to fermions with exact chiral symmetry on the lattice and discuss the consequences for the recent Bloch-Wettig proposal for the Dirac operator at finite chemical potential. We propose a new simpler method of incorporating $μ$ and compare it with the Bloch-Wettig idea.
After many years of investigations, our understanding of the dynamics of strongly-coupled chiral gauge theories is still quite unsatisfactory today. Conventional wisdom about strongly-coupled gauge theories, successfully applied to QCD, is not always as useful in chiral gauge theories. Recently some new ideas and techniques have been developed, which involve concepts of generalized symmetries, of gauging a discrete center symmetry, and of generalizing the 't Hooft anomaly matching constraints to include certain mixed symmetries. This new development has been applied to chiral gauge theories, l…
A large approximate degeneracy of the excited hadron spectra, appears in the light meson spectra measured at LEAR, suggesting a novel principal quantum number n+j in QCD spectra. We recently showed that the large degeneracy could not be understood with state of the art confining and chiral invariant quark models, derived in a truncated Coulomb gauge. To search for a solution to this problem, here we add the gluon or string degrees of freedom. Although independently the quarks or the gluons would lead to a 2 n +j or 2 n +l spectrum, adding them may lead to the desired n+j pattern.
The gauging of axionic, Stueckelberg-type, shift symmetries with generalized Chern-Simons terms and the Green-Schwarz mechanism of anomaly cancellation has recently been studied in the context of certain string compactifications and with regard to the phenomenology of unusual variants of Z'-bosons. In this talk, following recent work with De Rydt, Rosseel, Schmidt and Van Proeyen, I summarize how such theories can be reconciled with global and local N=1 supersymmetry. Two simple consistency conditions are shown to encode strong constraints on the allowed anomalies for different types of gauge …
We calculate the hadron tensor related to the photon-induced Drell-Yan process, with incoming nucleon being transversally polarised. We predict the new single spin asymmetry which probes gluon poles and which is expressed in terms of photon (both chiral-odd and chiral-even) distribution amplitudes and chiral-odd nucleon function stemming from the nucleon transverse polarization.
The gluon propagator is one of the fundamental Green's functions of QCD. It is an essential ingredient in, for example, the modeling of the Schwinger-Dyson equation used to describe hadronic phenomenology. From the Landau gauge gluon propagator, computed with lattice QCD methods, we discuss its interpretation as a massive propagator and measure the gluon mass as a function of the momenta. Special attention is given to the mass at infrared scales. In the last part of the talk, the gluon mass and chiral symmetry breaking are related via an effective model for QCD.
We propose a variant of the Pati-Salam model, with gauge group $SU(4)_C\times SU(2)_L\times U(1)_{Y'}$, in which the chiral left-handed quarks and leptons are unified into a $\underline{4}$ of $SU(4)_C$, while the right-handed quarks and leptons have quite a distinct treatment. The $SU(4)_C$ leptoquark gauge bosons can explain the measured deviation of lepton flavour universality in the rare decays: $\bar B \to \bar K^{(*)}\bar\ell\ell, \ell=μ,e$ (taken as a hint of new physics). The model satisfies the relevant experimental constraints and makes predictions for the important $B$ and $τ$ decay…
We study two well-known $SU(N)$ chiral gauge theories with fermions in the symmetric, anti-symmetric and fundamental representations. We give a detailed description of the global symmetry, including various discrete quotients. Recent work argues that these theories exhibit a subtle mod 2 anomaly, ruling out certain phases in which the theories confine without breaking their global symmetry, leaving a gapless composite fermion in the infra-red. We point out that no such anomaly exists. We further exhibit an explicit path to the gapless fermion phase, showing that there is no kinematic obstructi…
The restrictions imposed by anomaly cancellation on the chiral fermion content of nonsupersymmetric gauge theories based on various groups are studied in spacetime dimension D=6, 8, and 10. In particular, we show that the only mathematically consistent chiral SU(5) theory in D=6 contains three nonidentical generations.
We discuss the effect of $η$--$η'$ mixing and of axial current anomalies on various decay processes. We describe the 2$γ$ decays of $η$ and $η'$, the radiative decays of low-lying vector mesons as well as the decays $η\to 3π$, $J/ψ\toηγ$ and $J/ψ\toη'γ$ in one unified framework, in which they are related to the electromagnetic and strong axial current anomalies, respectively. We also discuss briefly the enhancement of $η'$ in $D_s$ decays and suggest a possible relation to gluon-mediated processes. The importance of the decays of the glueball candidates $f_0(1500)$ and $ f_0(1590)$ into $ηη'$ …
The confinement mechanism proposed earlier by the author is employed for to compute the decay constants $f_P$ corresponding to leptonic decays $P\to l^\pm+ν_l$, $l=μ, e$, where $P$ stands for any meson from $π^\pm$, $K^\pm$. For this aim the weak axial form factor of $P$-meson is nonperturbatively calculated. The study entails estimates for parameters of the confining SU(3)-gluonic field in charged pions and kaons. The corresponding estimates of the gluon concentrations, electric and magnetic colour field strengths are also adduced for the mentioned field at the scales of the mesons under cons…
Chiral symmetry breaking in QCD is studied introducing a confining effective propagator, as proposed recently by Cornwall, and considering the effect of dynamically massive gluons. The effective confining propagator has the form $1/(k^2+m^2)^2$ and we study the bifurcation equation finding limits on the parameter $m$ below which a satisfactory fermion mass solution is generated. Since the coupling constant and gluon propagator are damped in the infrared, due to the presence of a dynamical gluon mass, the major part of the chiral breaking is only due to the confining propagator and related to t…
We explore the idea that in some class of strongly-coupled chiral $SU(N)$ gauge theories the infrared dynamics might be characterized by a bifermion condensate in the adjoint representation of the color gauge group. As an illustration, in this work we revisit an $SU(N)$ chiral gauge theory with Weyl fermions in a symmetric ($ψ$) and anti-antisymmetric ($χ$) tensor representations, together with eight fermions in the anti-fundamental representations ($η$), which we called $ψχη$ model in the previous investigations. We study the infrared dynamics of this system more carefully, by assuming dynami…
