Fixed points in QCD can appear when the number of quark flavors ($N_f$) is increased above a certain critical value as proposed by Banks and Zaks (BZ). There is also the possibility that QCD possess an effective charge indicating an infrared frozen coupling constant. In particular, an infrared frozen coupling associated to dynamical gluon mass generation (DGM) does lead to a fixed point even for a small number of quarks. We compare the BZ and DGM mechanisms, their $β$ functions and fixed points, and within the approximations of this work, which rely basically on extrapolations of the dynamical gluon masses at large $N_f$, we verify that near the so called QCD conformal window both cases exhibit fixed points at similar coupling constant values ($g^*$). We argue that the states of minimum vacuum energy, as a function of the coupling constant up to $g^*$ and for several $N_f$ values, are related to the dynamical gluon mass generation mechanism.
We analyse the critical exponents relating to the quark mass anomalous dimension and beta-function at the Banks-Zaks fixed point in Quantum Chromodynamics (QCD) in a variety of representations for the quark in the momentum subtraction (MOM) schemes of Celmaster and Gonsalves. For a specific range of values of the number of quark flavours, estimates of the exponents appear to be scheme independent. Using the recent five loop modified minimal subtraction (MSbar) scheme quark mass anomalous dimension and estimates of the fixed point location we estimate the associated exponent as 0.263-0.268 for …
The recent QCD calculations of the five-loop beta function and of R(e+e-) to O(alpha_s^4) provide one more term in the Banks-Zaks expansion in (16.5-nf). There is no longer any hope that the expansion could extend, even crudely, to low nf. Above nf=9, however, the results appear to be reasonably consistent from order to order.
The recent calculation of the four-loop beta function in QCD provides further evidence that the Banks-Zaks expansion in 16half-n_f is sufficiently well behaved to be useful even for n_f=2 light flavours. This expansion inherently predicts "freezing" of the QCD couplant at low energies as a perturbative effect. We consider the e+e- and Bjorken-sum-rule cases as examples.
We investigate the a theorem for nonsupersymmetric gauge-Yukawa theories beyond the leading order in perturbation theory. The exploration is first performed in a model-independent manner and then applied to a specific relevant example. Here, a rich fixed point structure appears including the presence of a merging phenomenon between non-trivial fixed points for which the a theorem has not been tested so far.
We discuss cosmological consequences of the existence of physics beyond the standard model that exhibits Banks-Zaks and unparticle behavior in the UV and IR respectively. We first derive the equation of state for unparticles and use it to obtain the temperature dependence of the corresponding energy and entropy densities. We then formulate the Boltzmann and Kubo equations for both the unparticles and the Banks-Zaks particles, and use these results to determine the equilibrium conditions between the standard model and the new physics. We conclude by obtaining the constraints on the effective nu…
In both QCD and supersymmetric QCD (SQCD) with N_f flavors there are conformal windows where the theory is asymptotically free in the ultraviolet while the infrared physics is governed by a non-trivial fixed-point. In SQCD, the lower N_f boundary of the conformal window, below which the theory is confining is well understood thanks to duality. In QCD there is just a sufficient condition for confinement based on superconvergence. Studying the Banks-Zaks expansion and analyzing the conditions for the perturbative coupling to have a causal analyticity structure, it is shown that the infrared fixe…
The normalization of the gluon condensate and of renormalon-related power corrections in QCD is computed under the assumption that their ``perturbative'' part dominates over any eventual extra contribution from the non-trivial vacuum. The calculation is performed in the infrared finite coupling framework, assuming an infrared fixed point is present in the perturbative coupling down to low values of N_f. The freezing perturbative coupling is reconstructed using a Banks-Zaks expansion approach. Parameter-free predictions of the low energy moments of the coupling, which determine the process-inde…
A physical characterization of Landau singularities is emphasized, which should trace the lower boundary N_f^* of the conformal window in QCD and supersymmetric QCD. A natural way to disentangle ``perturbative'' from ``non-perturbative'' contributions below N_f^* is suggested. Assuming an infrared fixed point is present in the perturbative part of the QCD coupling even in some range below N_f^* leads to the condition gamma(N_f^*)=1, where gamma is the critical exponent. This result is incompatible with the existence of an analogue of Seiberg free dual magnetic phase in QCD. Using the Banks-Zak…
We propose an ansatz for the nonperturbative beta function of a generic non-supersymmetric Yang-Mills theory with or without fermions in an arbitrary representation of the gauge group. While our construction is similar to the recently proposed Ryttov-Sannino all order beta function, the essential difference is that it allows for the existence of an unstable ultraviolet fixed point in addition to the predicted Bank-Zaks -like infrared stable fixed point. Our beta function preserves all of the tested features with respect to the non-supersymmetric Yang-Mills theories. We predict the conformal wi…
In QCD with 16\half-epsilon massless quark flavours there is an infrared fixed point with alpha_s/pi = (8/321)epsilon in the limit epsilon -> 0+. I develop the idea of Banks and Zaks to expand about N_f=16\half. This expansion is certainly useful for N_f=16, 15, 14, ..., and arguably it can reach the phenomenologically interesting case N_f=2, where it suggests that alpha_s/pi "freezes" to a value of order of magnitude 0.4 in the infrared.
The large-order behaviour of QCD is dominated by renormalons. On the other hand renormalons do not occur in conformal theories, such as the one describing the infrared fixed-point of QCD at small beta_0 (the Banks--Zaks limit). Since the fixed-point has a perturbative realization, all-order perturbative relations exist between the conformal coefficients, which are renormalon-free, and the standard perturbative coefficients, which contain renormalons. Therefore, an explicit cancellation of renormalons should occur in these relations. The absence of renormalons in the conformal limit can thus be…
We introduce a more general set of kinematic renormalization schemes than the original momentum (MOM) subtraction schemes of Celmaster and Gonsalves. These new schemes will depend on a parameter $ω$ which tags the external momentum of one of the legs of the $3$-point vertex functions in Quantum Chromodynamics (QCD). In each of the three new schemes we renormalize QCD in the Landau and maximal abelian gauges and establish the three loop renormalization group functions in each gauge. As an application we evaluate two critical exponents at the Banks-Zaks fixed point and demonstrate that their val…
We consider zero-temperature transitions from conformal to non-conformal phases in quantum theories. We argue that there are three generic mechanisms for the loss of conformality in any number of dimensions: (i) fixed point goes to zero coupling, (ii) fixed point runs off to infinite coupling, or (iii) an IR fixed point annihilates with a UV fixed point and they both disappear into the complex plane. We give both relativistic and non-relativistic examples of the last case in various dimensions and show that the critical behavior of the mass gap behaves similarly to the correlation length in th…
Perturbative QCD with nf flavours of massless quarks becomes simple in the hypothetical limit nf -> 16.5, where the leading beta-function coefficient vanishes. The Banks-Zaks (BZ) expansion in a0=(8/321)(16.5-nf) is straightforward to obtain from perturbative results in MSbar or any renormalization scheme (RS) whose nf dependence is `regular.' However, `irregular' RS's are perfectly permissible and should ultimately lead to the same BZ results. We show here that the `optimal' RS determined by the Principle of Minimal Sensitivity does yield the same BZ-expansion results when all orders of pertu…
I show that Sudakov resummation takes a transparent form if one deals with the second logarithmic derivative of the short distance coefficient functions for deep inelastic scattering and the Drell-Yan process. A uniquely defined Sudakov exponent emerges, and the constant terms not included in the exponent are conjectured to be given by the second logarithmic derivative of the massless quark form factor. The precise framework for the implementation of the dispersive approach to power corrections is set-up, yielding results in agreement with infrared renormalon expectations, but which are not ti…
A physical characterization of Landau singularities is emphasized, which should trace the lower boundary N_f^* of the conformal window in QCD and supersymmetric QCD. A natural way to disentangle ``perturbative'' from ``non-perturbative'' contributions to amplitudes below N_f^* is suggested. Assuming an infrared fixed point persists in the perturbative part of the QCD coupling even below N_f^* leads to the condition γ(N_f^*)=1, where γis the critical exponent. Using the Banks-Zaks expansion, one gets 4<N_f^*<6. This result is incompatible with the existence of an analogue of Seiberg duality in …
Solving the QCD renormalization group equation at the 2-loop and 3-loop orders we obtain explicit expressions for the coupling as a function of the scale in terms of the Lambert W function. We study the nature of the ``Landau singularities'' in the complex Q^2 plane and show that perturbative freezing can lead, in certain cases, to an analyticity structure that is consistent with causality. We analyze the Analytic Perturbation Theory (APT) approach which is intended to remove the ``Landau singularities'', and show that at 2-loops it is uniquely defined in terms of the Lambert W function, and t…
I show that Sudakov resummation takes a particularly transparent form if one deals with the second logarithmic derivative of the short distance coefficient functions for deep inelastic scattering and the Drell-Yan process. A uniquely defined Sudakov exponent emerges, and I conjecture that the leftover constant terms not included in the exponent are given by the second logarithmic derivative of the massless quark form factor. The meaning of a previously obtained large $N_f$ evidence for an infrared finite perturbative Sudakov coupling is reconsidered. This coupling is reinterpreted as a Minkows…
We investigate the relation between a postulated skeleton expansion and the conformal limit of QCD. We begin by developing some consequences of an Abelian-like skeleton expansion, which allows one to disentangle running-coupling effects from the remaining skeleton coefficients. The latter are by construction renormalon-free, and hence hopefully better behaved. We consider a simple ansatz for the expansion, where an observable is written as a sum of integrals over the running-coupling. We show that in this framework one can set a unique Brodsky-Lepage-Mackenzie (BLM) scale-setting procedure as …
We investigate the possibility that alpha_s freezes as function of N_f within perturbation theory. We use two approaches -- direct search for a zero in the effective-charge (ECH) beta function, and the Banks-Zaks (BZ) expansion. We emphasize the fundamental difference between quantities with space-like vs. those with time-like momentum. We show that within the ECH approach several space-like quantities exhibit similar behavior. In general the 3-loop ECH beta functions can lead to freezing for N_f\gsim 5, but higher-order calculations are essential for a conclusive answer. The BZ expansion beha…
Techni-dilaton (TD) was proposed long ago in the technicolor (TC) near criticality/conformality. To reveal the critical behavior of TD, we explicitly compute the nonperturbative contributions to the scale anomaly $<θ^μ_μ>$ and to the techni-gluon condensate $<G_{μν}^2>$, which are generated by the dynamical mass m of the techni-fermions. Our computation is based on the (improved) ladder Schwinger-Dyson equation, with the gauge coupling $α$ replaced by the two-loop running one $α(μ)$ having the Caswell-Banks-Zaks IR fixed point $α_*$: $α(μ) \simeq α= α_*$ for the IR region $m < μ< Λ_{TC}$, wher…
An important aspect of Weyl anomalies is that they encode information on the irreversibility of the renormalisation group flow. We consider, $Δ\bar b = \bar{b}^{\textrm{UV}} - \bar{b}^{\textrm{IR}}$, the difference of the ultraviolet and infrared value of the $\Box R$-term of the Weyl anomaly. The quantity is related to the fourth moment of the trace of the energy momentum tensor correlator for theories which are conformal at both ends. Subtleties arise for non-conformal fixed points as might be the case for infrared fixed points with broken chiral symmetry. Provided that the moment converges,…
The walking technicolor based on the ladder Schwinger-Dyson gap equation is studied, with the scale-invariant coupling being an idealization of the Caswell-Banks-Zaks infrared fixed point in the "anti-Veneziano limit", such that $N_C \rightarrow \infty$ with $N_C \cdot α(μ^2)=$ fixed and $N_F/N_C=$ fixed ($\gg 1$), of the $SU(N_C)$ gauge theory with massless $N_F$ flavors near criticality. We show that the 125 GeV Higgs can be naturally identified with the technidilaton (TD) predicted in the walking technicolor, a pseudo Nambu-Goldstone (NG) boson of the spontaneous symmetry breaking of the ap…
