QCD Β-function Research Articles

Exact β-Function in Abelian and non-Abelian $${\cal N} = 1$$ Supersymmetric Gauge Models and Its Analogy with the QCD β-Function in the C-scheme

For $${\cal N} = 1$$ supersymmetric Yang—Mills theory without matter it is demonstrated that there is a class of renormalization schemes, in which the exact Novikov, Shifman, Vainshtein, and Zakharov (NSVZ) formula for the renormalization group β-function, defined in terms of the renormalized coupling constant, is valid. These schemes are related with each other by finite renormalizations forming a one-parameter commutative subgroup of general renormalization group transformations. The analogy between the exact β-function in $${\cal N} = 1$$ supersymmetric Yang-Mills theory without matter and the β-function of quantum chromodynamics in the C-scheme is discussed.

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.

Four-loop QCD β-function with different fermion representations of the gauge group

We present analytical results at four-loop level for the $\beta$-function of the coupling of a generic gauge group and any number of different quark representations. From this we can directly derive the gluino contribution to the strong coupling $\beta$-function of supersymmetric extensions of the Standard Model.

Top-Yukawa effects on the β-function of the strong coupling in the SM at four-loop level

Abstract We present analytical results for the QCD β-function extended to the gaugeless limit of the unbroken phase of the Standard Model at four-loop level. Apart from the strong coupling itself we include the top-Yukawa contribution and the Higgs self-coupling. We observe a numerically small non-naive γ 5 contribution at order y t 4 g s 4 , a feature not encountered in lower loop orders. We discuss the treatment of γ5 which is more involved than in previous calculations at three-loop level.

The generalized BLM approach to fix scale- dependence in QCD: the current status of investigations

I present a brief review of the generalized Brodsky-Lepage-McKenzie (BLM) approaches to fix the scale-dependence of the renormalization group (RG) invariant quantities in QCD. At first, these approaches are based on the expansions of the coefficients of the perturbative series for the RG-invariant quantities in the products of the coefficients βi of the QCD β-function, which are evaluated in the MS-like schemes. As a next step all βi-dependent terms are absorbed into the BLM-type scale(s) of the powers of the QCD couplings. The difference between two existing formulations of the above mentioned generalizations based on the seBLM approach and the Principle of Maximal Conformality (PMC) are clarified in the case of the Bjorken polarized deep-inelastic scattering sum rule. Using the conformal symmetry-based relations for the non-singlet coefficient functions of the Adler D-function and of Bjorken polarized deep-inelastic scattering sum rules CBjpNS (as) the βi-dependent structure of the NNLO approximation for CBjpNS (as) is predicted in QCD with ngl-multiplet of gluino degrees of freedom, which appear in SUSY extension of QCD. The importance of performing the analytical calculation of the N3LO additional contributions of ngl gluino multiplet to CBjpNS (as) for checking the presented in the report NNLO prediction and for the studies of the possibility to determine the discussed {β}-expansion pattern of this sum rule at the O(a4s)-level is emphasised.

The [formula omitted] correlator at three loops in perturbative QCD

It is known that the correlator of one axial and two vector currents, that receives leading contributions through one-loop fermion triangle diagrams, is not modified by QCD radiative corrections at two loops. It was suggested that this non-renormalization of the 〈VVA〉 correlator persists in higher orders in perturbative QCD as well. To check this assertion, we compute the three-loop QCD corrections to the 〈VVA〉 correlator using the technique of asymptotic expansions. We find that these corrections do not vanish and that they are proportional to the QCD β-function.

The absence of QCD β-function factorization property of the generalized Crewther relation in the ʼt Hooft [formula omitted]-based scheme

We apply the ʼt Hooft MS¯-based scheme to study the scheme-dependence of the QCD generalization of Crewther relation for the product of the normalized non-singlet perturbative contributions to the e+e−-annihilation Adler function and to the Bjorken sum rule of the polarized lepton–nucleon deep-inelastic scattering process. We prove that after the transformations from the pure MS¯-scheme to the ʼt Hooft scheme the characteristic MS¯-scheme theoretical property of this relation, namely the factorization of the β-function in its conformal symmetry breaking part, disappears. Another “non-comfortable” theoretical consequence of the application of this prescription in N=1 SUSY QED model is mentioned. It is shown, that within the ʼt Hooft scheme the expansions of Green functions in terms of the Lambert function is simplified in high orders of perturbation theory. This may be considered as the attractive feature of the ʼt Hooft scheme, which manifest itself in high-order perturbative phenomenological applications.

Three loop QCD MOM β-functions

We present the full expressions for the QCD β-function in the MOMggg, MOMq and MOMh renormalization schemes at three loops for an arbitrary colour group in the Landau gauge. The results for all three schemes are in very good agreement with the SU(3) numerical estimates provided by Chetyrkin and Seidensticker.

AdS/QCD, LIGHT-FRONT HOLOGRAPHY, AND THE NONPERTURBATIVE RUNNING COUPLING

The combination of Anti-de Sitter space (AdS) methods with light-front (LF) holography provides a remarkably accurate first approximation for the spectra and wavefunctions of meson and baryon light-quark bound states. The resulting bound-state Hamiltonian equation of motion in QCD leads to relativistic light-front wave equations in terms of an invariant impact variable ζ which measures the separation of the quark and gluonic constituents within the hadron at equal light-front time. These equations of motion in physical space-time are equivalent to the equations of motion which describe the propagation of spin-J modes in anti–de Sitter (AdS) space. The eigenvalues give the hadronic spectrum, and the eigenmodes represent the probability distributions of the hadronic constituents at a given scale. A positive-sign confining dilaton background modifying AdS space gives a very good account of meson and baryon spectroscopy and form factors. The light-front holographic mapping of this model also leads to a non-perturbative effective coupling [Formula: see text] which agrees with the effective charge defined by the Bjorken sum rule and lattice simulations. It displays a transition from perturbative to nonperturbative conformal regimes at a momentum scale ~ 1 GeV. The resulting β-function appears to capture the essential characteristics of the full β-function of QCD, thus giving further support to the application of the gauge/gravity duality to the confining dynamics of strongly coupled QCD.

The QCD β-function from global solutions to Dyson–Schwinger equations

We study quantum chromodynamics from the viewpoint of untruncated Dyson–Schwinger equations turned to an ordinary differential equation for the gluon anomalous dimension. This non-linear equation is parameterized by a function P( x) which is unknown beyond perturbation theory. Still, very mild assumptions on P( x) lead to stringent restrictions for possible solutions to Dyson–Schwinger equations. We establish that the theory must have asymptotic freedom beyond perturbation theory and also investigate the low energy regime and the possibility for a mass gap in the asymptotically free theory.

Three loop [formula omitted] anomalous dimension for renormalizable gauge invariant non-local gluon mass operator in QCD

The three loop anomalous dimension for the gauge invariant, renormalizable, non-local mass operator for a gluon is computed in the MS¯ scheme. In addition the anomalous dimensions of the associated localizing ghost fields are also deduced at the same order and it is shown that the three loop QCD β-function correctly emerges from the gluon localizing ghost vertex renormalization.

Exploring improved holographic theories for QCD: part I

Various holographic approaches to QCD in five dimensions are explored using input both from the putative five-dimensional non-critical string theory as well as QCD. It is argued that a gravity theory in five dimensions coupled to a dilaton and an axion may capture the important qualitative features of pure YM theory. A part of the effects of higher α'-corrections is resummed into a dilaton potential. The potential is shown to be in one-to-one correspondence with the exact β-function of QCD, and its knowledge determines the full structure of the vacuum solution. The geometry near the UV boundary is that of AdS5 with logarithmic corrections reflecting the asymptotic freedom of QCD. We find that all relevant confining backgrounds have an IR singularity of the ``good kind that allows unambiguous spectrum computations. Near the singularity the 't Hooft coupling is driven to infinity. Asymptotically linear glueball masses can also be achieved. The classification of all confining asymptotics, the associated glueball spectra and meson dynamics are addressed in a companion paper arXiv:0707.1349

O(a) improved QCD: The 3-loop beta-function, and the critical hopping parameter

We calculate the 3-loop bare β-function of QCD, formulated on the lattice with the clover fermionic action. The dependence of our result on the number of colors N, the number of fermionic flavors N f , and the clover parameter c SW , is shown explicitly. A direct outcome of our calculation is the two-loop relation between the bare coupling constant g 0 and the one renormalized in the MS scheme. Further, we can immediately derive the three-loop correction to the relation between the lattice Λ-parameter and g 0, which turns out to be very pronounced. We also calculate the critical value of the hopping parameter, κ c , in the clover action, to two loops in perturbation theory. This quantity is an additive renormalization; as such, it exhibits a linear divergence in the lattice spacing. We compare our results to non perturbative evaluations of κ c coming from MC simulations.

The three-loop β-function of QCD with the clover action

We calculate, to 3 loops in perturbation theory, the bare β-function of QCD, formulated on the lattice with the clover fermionic action. The dependence of our result on the number of colors N, the number of fermionic flavors N f , as well as the clover parameter c SW, is shown explicitly. A direct outcome of our calculation is the two-loop relation between the bare coupling constant g 0 and the one renormalized in the MS scheme. Further, we can immediately derive the three-loop correction to the relation between the lattice Λ-parameter and g 0, which is important in checks of asymptotic scaling. For typical values of c SW, this correction is found to be very pronounced.

Probing the Infrared Structure of Gauge Theories: A Padé-Approximant Approach

Padé-approximant treatments of the known terms of the QCD β-function are seen to develop possible infrared fixed point structure only if the number of fermion flavours is sufficiently large. This flavour threshold is seen to be between six and nine flavours, depending upon both the specific choice of approximant as well as on the presently-unknown five-loop β-function contribution. Below this flavour threshold, Padé approximants based upon the QCD β-function manifest the same infrared attractor structure as that which characterizes the exact NSVZ β-function of supersymmetric gluodynamics. Such infrared attractor structure is also seen to characterize Padé-approximant treatments of vector SU(N) gauge theory in the large N limit, suggesting common infrared dynamics for the strong and weak phases of this theory.

AN ANALYTIC APPROACH TO PERTURBATIVE QCD

The two-loop invariant (running) coupling of QCD is written in terms of the Lambert W function. The analyticity structure of the coupling in the complex Q2-plane is established. The corresponding analytic coupling is reconstructed via a dispersion relation. We also consider some other approximations to the QCD β-function, when the corresponding couplings are solved in terms of the Lambert function. The Landau gauge gluon propagator has been considered in the renormalization group invariant analytic approach (IAA). It is shown that there is a nonperturbative ambiguity in determination of the anomalous dimension function of the gluon field. Several analytic solutions for the propagator at the one-loop order are constructed. Properties of the obtained analytical solutions are discussed.

A VARIATIONAL APPROACH TO THE QCD WAVEFUNCTIONAL: CALCULATION OF THE QCD β-FUNCTION

The β-function is calculated for an SU (N) Yang–Mills theory from an ansatz for the vacuum wavefunctional. Direct comparison is made with the results of calculations of the β-function of QCD. In both cases the theories are asymptotically free, the only difference being in the numerical coefficient of the β-function, which is found to be -4 from the ansatz and [Formula: see text] from other QCD calculations. This is because, due to the constraint of Gauss' law applied to the wavefunctional, some interactions between gluons (which contribute the [Formula: see text]) are omitted. The renormalisation procedure is understood in terms of "tadpole" and "horseshoe" Feynman diagrams which must be interpreted with a nonlocal propagator.

Higher orders and infrared renormalon phenomenology in deeply virtual Compton scattering

We present results for higher-order perturbative corrections to Compton scattering in the generalized Bjorken kinematics. The approach we have used is based on the combination of two techniques: conformal operator product expansion on the one side, and resummation of the fermion vacuum insertions with consequent restoration of the full QCD β-function via the naive non-abelianization assumption, on the other. These are terms which are lost in the former approach. Due to the presence of the infrared renormalon poles in the Borel transform of the resummed amplitude the latter suffers from ambiguities which reflect the asymptotic character of perturbation series. The residues of these IR renormalon poles give an estimate for the size of power corrections in deeply virtual Compton scattering.

Resummation of (− b0αs) n corrections to the photon-meson transition form factor [formula omitted

We have resummed all the (− b 0 α s ) n contributions to the photon-meson transition form factor F πγ . To do this, we have used the assumption of ‘naive non-abelianization’ (NNA). Within NNA, a series in (N ƒα s) n is interpreted as a series in (− b 0 α s ) n by means of the restoration of the full first QCD β-function coefficient − b 0 by hand. We have taken into account corrections to the leading order coefficient function and to the evolution of the distribution function. Due to conformal symmetry constraints, it is possible to find the eigenfunctions of the evolution kernel. It turns out that the non-diagonal corrections are small, and neglecting them we obtained a representation for the distribution function with multiplicatively renormalized moments. For a simple shape of the distribution function, which is close to the asymptotic shape, we find that the radiative correction decrease the LO by 30%, and the uncertainty in the resummation lies between 10% and 2% for Q 2 between 2 and 10 GeV 2.

αs(mZ) from τ decays with matching conditions at three loops

Abstract Using the recent four-loop calculations of the QCD β-function and the three-loop matching coefficients we study the induced errors in αs(mZ) obtained from αs(mτ) due to the evolution procedure. We show that, when consistent matching and running is used at this order, these errors are pushed below 0.0005 in αs(mZ).