The present calculations in perturbative QCD reach the order $α_s^4$ for several correlators calculated to five loops, and the huge computational difficulties make unlikely the full six-loop calculation in the near future. This situation has practical consequences, in particular the treatment of the higher orders of the perturbation series for the current-current correlator of light quarks is one of the main sources of errors in the extraction of the strong coupling from hadronic $τ$ decays. Several approximate estimates of the next coefficients of the corresponding Adler function have been proposed, using various arguments. In the present paper we exploit the analytic structure of the Adler function in the Borel plane, which allows the definition of an improved perturbative expansion in powers of a conformal variable which maps the cut Borel plane onto the unit disk. The new expansions converge in a larger domain of the Borel plane and, when reexpanded in powers of the strong coupling, yield definite values for the higher perturbative coefficients. We apply the method to the Adler function in the $\bar{\rm MS}$ scheme and to a suitable weighted integral of this function in the complex $s$ plane, chosen such as to avoid model-dependent assumptions on analyticity. Our results $c_{5,1}=287 \pm 40$, $c_{6,1}=2948 \pm 208$ and $c_{7,1}=(1.89 \pm 0.75)\times 10^4$, for the six, seven and eigth-loop coefficients, respectively, agree with a recent determination from Padé approximants applied to the perturbative expansion of the hadronic $τ$ decay rate.
In the perturbative QCD with $N_c\to\infty$ the amplitude for the collision of two heavy nuclei is expressed via dipole densities in the nuclei. Coupled equations for these densities are derived in the configuration space. The equations are conformal invariant in absence of external sources. Passing to conformal basis and its possible truncation are discussed.
The technique of conformal mappings is applied to enlarge the convergence of the Borel series and to accelerate the convergence of Borel-summed Green functions in perturbative QCD. We use the optimal mapping, which takes into account the location of all the singularities of the Borel transform as well as the present knowledge about its behaviour near the first branch points. The determination of α_{s}(m_τ) from the hadronic decay rate of the τ-lepton is discussed as an illustration of the method.
The dynamics at the IR fixed point realized in the $SU(N_c)$ gauge theories with massless Dirac fermions is studied by means of the non-perturbative renormalization group. The analysis includes the IR fixed points with non-trivial Yukawa couplings. The renormalization properties of the scalar field are also discussed and it is shown that hierarchical mass scale may be allowed without intense fine-tuning due to a large anomalous dimension.
The optimal conformal mapping of the Borel plane was recently used to accelerate the convergence of the perturbation expansions in QCD. In this work we discuss the relevance of the method for the calculation of the Laplace-Borel integral expressing formally the QCD Green functions. We define an optimal expansion of the Laplace-Borel integral in the principal value prescription and establish conditions under which the expansion is convergent.
We discuss the method of conformal mappings applied to perturbative QCD. The approach is based on the Borel-Laplace integral regulated with the principal value prescription and the expansion of the Borel transform in powers of the variable which performs the conformal mapping of the cut Borel plane onto the unit disk. We write down the expression of the conformal mapping for the most general location of the singularities of the Borel transform and review the properties of the corresponding expansions of the correlators. Unlike the standard perturbative expansions, which are divergent, the modi…
The possibility that non-supersymmetric quiver theories may have a renormalization-group fixed point at which there is conformal invariance requires non-perturbative information.
We discuss how the renormalisation scheme ambiguities in QCD can be fixed, when two observables are related, by requiring the coefficients in the perturbative expansion relating the two observables to have their conformal limit values, i.e. to be independent of the $β$-function of the renormalised coupling. We show how the next-to-leading order BLM automatic scale fixing method can be extended to next-to-next-to-leading order to fix both the renormalisation scale and $β_2$ in a unique way. As an example we apply the method to the relation between Bjorken's sum rule and $R_{e+e-}$ and compare w…
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…
The lowest order pomeron loop is calculated for the leading conformal weight with full dependence of the triple pomeron vertex on intermediate conformal weights. The loop is found to be convergent. Its contribution to the pomeron Green function begins to dominate already at rapidities 10$÷$15. The pomeron pole renormalization is found to be quite small due to a rapid fall of the triple pomeron vertex with rising conformal weights.
Using the asymptotic conformal invariance of perturbative QCD we derive the expression of the coupling of external states to all conformal spin-p components of the forward elastic amplitude. Using the wave-function formalism for structure functions at small x, we derive the perturbative coupling of the virtual photon for p= 1, which is maximal for linear transverse polarization. The non-perturbative coupling to the proton is discussed in terms of ``azimuthal matching'' between the proton color dipoles and the quark- antiquark configurations of the photon. As an application, the recent conjectu…
In this work we consider the possibility of representing the perturbative series for renormalization group invariant quantities in QCD in the form of their decomposition in powers of the conformal anomaly $β(α_s)/α_s$ in the ${\rm{\overline{MS}}}$-scheme. We remind that such expansion is possible for the Adler function of the process of $e^+e^-$ annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule for the deep-inelastic electron-nucleon scattering, which are both related by the Crewther-Broadhurst-Kataev relation. In addition, we study the discussed decompos…
Perturbation expansions appear to be divergent series in many physically interesting situations, including in quantum field theories like quantum electrodynamics (QED) and quantum chromodynamics (QCD), where the perturbative coefficients exhibit a factorial growth at large orders. While this feature has little impact on physical predictions in QED, it can have nontrivial consequences in applications of perturbative QCD at moderate energies. In particular, it affects the theoretical error in the extraction of the strong coupling $α_s$ from hadronic $τ$ decays, despite progress of perturbative c…
In the paper, we study the properties of the top-quark $\overline{\rm MS}$ running mass computed from its on-shell mass by using both the four-loop $\overline{\rm MS}$-on-shell relation and the principle of maximum conformality (PMC) scale-setting approach. The PMC adopts the renormalization group equation to set the correct magnitude of the strong running coupling of the perturbative series, its prediction avoids the conventional renormalization scale ambiguity, and thus a more precise pQCD prediction can be achieved. After applying the PMC to the four-loop $\overline{\rm MS}$-on-shell relati…
We analyze the heavy quark bound state spectrum using an order-dependent conformal mapping to re-sum the perturbative expansion for current correlators. The procedure consists of two main steps. Firstly, the Borel plane structure of the truncated perturbative expression is modified to ensure consistency with the operator product expansion. This is analogous to a resummation of infrared renormalon chains. Secondly, this perturbative expansion is conformally mapped to a new series with improved convergence properties. This approach may be shown to induce power corrections consistent with existin…
The power corrections in the Operator Product Expansion (OPE) of QCD correlators can be viewed mathematically as an illustration of the transseries concept, which allows to recover a function from its asymptotic divergent expansion. Alternatively, starting from the divergent behavior of the perturbative QCD encoded in the singularities in the Borel plane, a modified expansion can be defined by means of the conformal mapping of this plane. A comparison of the two approaches concerning their ability to recover nonperturbative properties of the true correlator was not explored up to now. In the p…
Integral kernels describing the pair interaction of reggeized gluons and quarks are reconstructed in terms of conformal symmetric 4-point functions in the transverse plane of impact parameters.
A complete calculation of the ${\cal O}(α_s^4)$ perturbative QCD corrections to the hadronic decay width of the $Z$-boson has recently been performed by Baikov et al.[1]. In their analysis, Baikov et al. relied on the conventional practice of simply guessing the renormalization scale and taking an arbitrary range to estimate the pQCD uncertainties. This procedure inevitably introduces an arbitrary, scheme-dependent theoretical systematic error in the predictions. In this paper, we show that the renormalization scale uncertainties for hadronic $Z$ decays can be greatly reduced by applying the p…
Conformal symmetry-based relations between concrete perturbative QED and QCD approximations for the Bjorken, the Ellis-Jaffe sum rules of polarized lepton- nucleon deep-inelastic scattering (DIS), the Gross-Llewellyn Smith sum rules of neutrino-nucleon DIS, and for the Adler functions of axial-vector and vector channels are derived. They result from the application of the operator product expansion to three triangle Green functions, constructed from the non-singlet axial-vector, and two vector currents, the singlet axial-vector and two non-singlet vector currents and the non-singlet axial-vect…
We apply the so-called conformal subtraction scheme to predict perturbatively exclusive processes beyond leading order. Taking into account evolution effects, we study the scheme dependence for the photon-to-pion transition form factor and the electromagnetic pion form factor at next-to-leading order for different pion distribution amplitudes. Relying on the conformally covariant operator product expansion and using the known higher order results for polarized deep inelastic scattering, we are able to predict perturbative corrections to the hard-scattering amplitude of the photon-to-pion trans…
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 present a general study of 3-point functions of conformal field theory in momentum space, following a reconstruction method for tensor correlators, based on the solution of the conformal Ward identities (CWI' s), introduced in recent works by Bzowski, McFadden and Skenderis (BMS). We investigate and detail the structure of the CWI's, their non-perturbative solutions and the transition to momentum space, comparing them to perturbation theory by taking QED as an example. We then proceed with an analysis of the $TJJ$ correlator, presenting independent and detailed re-derivations of the conform…
We identify a property of renormalizable SU(N)/U(1) gauge theories, the intrinsic Conformality ($iCF$), which underlies the scale invariance of physical observables and leads to a remarkably efficient method to solve the conventional renormalization scale ambiguity at every order in pQCD: the PMC$_\infty$. This new method reflects the underlying conformal properties displayed by pQCD at NNLO, eliminates the scheme dependence of pQCD predictions and is consistent with the general properties of the PMC (Principle of Maximum Conformality). We introduce a new method to identify conformal and $β$-t…
We study perturbative QCD corrections to deeply virtual Compton scattering on an unpolarized nucleon target in the flavor non-singlet sector to next-to-next-to-leading order accuracy, restricting ourselves to the kinematically dominant amplitude. The difference between the standard minimal subtraction and the conformal scheme, in which conformal symmetry is manifest, is studied to next-to-leading order. Beyond this order we employ conformal symmetry for the evaluation of perturbative corrections. Within a certain class of generalized parton distributions we find moderate radiative corrections.
The intrinsic conformality is a general property of the renormalizable gauge theory, which ensures the scale-invariance of a fixed-order series at each perturbative order. Following the idea of intrinsic conformality, we suggest a novel single-scale setting approach under the principle of maximum conformality (PMC) with the purpose of removing the conventional renormalization scheme-and-scale ambiguities. We call this newly suggested single-scale procedure as the PMC$_{\infty}$-s approach, in which an overall effective $α_s$, and hence an overall effective scale is achieved by identifying the …
This note is based on the summary of our book entitled "Non-perturbative field theoryfrom two dimensional conformal field theory to QCD in four dimensions", published recently by Cambridge University Press. It includes 436 pages. The book provides a detailed description of the tool box of non-perturbative techniques, presents applications of them to simplified systems, mainly of gauge dynamics in two dimensions, and examines the lessons one can learn from those systems about four dimensional QCD and hadron physics. In particular the book deals with conformal invariance, integrability, bosoniza…
We make a detailed study on the properties of the total decay width of Higgs decay channel $H\to gg$ up to $α_s^6$-order QCD corrections by using the newly suggested infinite-order scale-setting approach, which is based on the ideas of both the principle of maximum conformality and the intrinsic conformality. This approach is called as the PMC$_\infty$ approach. By using the PMC$_\infty$ approach, we observe that the conventional renormalization scale ambiguity in perturbative QCD calculation is eliminated and the residual scale dependence due to unknown higher-order terms can also be highly s…
The amplitude for the collision of two hadrons with the lowest order pomeron loop is calculated. Numerical calculations show that the loop contribution to the amplitude begins to dominate the single pomeron exchange at rapidities 8-10. Full dependence of the triple-pomeron vertex on intermediate conformal weights is taken into account.
We study perturbative unitarity corrections in the generalized leading logarithmic approximation in high energy QCD. It is shown that the corresponding amplitudes with up to six gluons in the t-channel are conformally invariant in impact parameter space. In particular we give a new representation for the two-to-six reggeized gluon vertex in terms of conformally invariant functions. With the help of this representation an interesting regularity in the structure of the two-to-four and the two-to-six transition vertices is found.
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 …
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…
We present a general study of 3-point functions of conformal field theory (CFT) in momentum space, following a reconstruction method for tensor correlators, based on the solution of the conformal Ward identities (CWIs), introduced in recent works. We investigate and detail the structure of the CWIs and their non-perturbative solutions, and compare them to perturbation theory, taking QED and QCD as examples. Exact solutions of CFT's in the flat background limit in momentum space are matched by the perturbative realizations in free field theories, showing that the origin the conformal anomaly is…
The general solution of the conformal Ward identities (CWI's) in momentum space, and their matching to perturbation theory, allows to uncover some specific characteristics of the breaking of conformal symmetry, induced by the anomaly. It allows to compare perturbative features of the 1-particle irreducible (1PI, nonlocal) anomaly action with the prediction of a similar (but exact) nonlocal action identified by the CWI's. The two predictions can be exactly matched at the level of 3-point functions. The analysis of the $TJJ$ and $TTT$ shows that both approaches - based either on 1PI or on the ex…
Using the renormalisation group framework we classify different extensions of the standard model according to their degree of naturality. A new relevant class of perturbative models involving elementary scalars is the one in which the theory simultaneously satisfies the Veltman conditions and is conformal at the classical level. We term these extensions perturbative natural conformal (PNC) theories. We show that PNC models are very constrained and thus highly predictive. Among the several PNC examples that we exhibit, we discover a remarkably simple PNC extension of the standard model in which…
As was discovered recently, 5-loop perturbative quenched QED approximation to the QED $β$-function consist from the rational term and the term proportional to $ζ(3)$-function. It is stressed, that this feature is also manifesting itself in the conformal invariant pqQED series for the 4-loop approximation to the anomalous mass dimension. The 4-loop pqQED expression for the singlet contribution into the Ellis-Jaffe polarized sum rule is obtained. It coincides with the similar approximation for the non-singlet coefficient function of Ellis-Jaffe sum rule and of Bjorken polarized sum rule. It is s…
A key problem in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale $μ$ of the running coupling $α_s(μ^2).$ The purpose of the running coupling in any gauge theory is to sum all terms involving the $β$ function; in fact, when the renormalization scale is set properly, all non-conformal $β\ne 0$ terms in a perturbative expansion arising from renormalization are summed into the running coupling. The remaining terms in the perturbative series are then identical to that of a conformal theory; i.e., the corresponding theory with $β=0$. The result…
Using conformal field theory (CFT) arguments we derive an infinite number of constraints on the large spin expansion of the anomalous dimensions and structure constants of higher spin operators. These arguments rely only on analiticity, unitarity, crossing-symmetry and the structure of the conformal partial wave expansion. We obtain results for both, perturbative CFT to all order in the perturbation parameter, as well as non-perturbatively. For the case of conformal gauge theories this provides a proof of the reciprocity principle to all orders in perturbation theory and provides a new "recipr…
We study non-perturbative aspects of QCD Kondo effect, which has been recently proposed for the finite density and strong magnetic field systems, using conformal field theory describing the low energy physics near the IR fixed point. We clarify the symmetry class of QCD Kondo effect both for the finite density and magnetic field systems, and show how the IR fixed point is non-perturbatively characterized by the boundary condition, which incorporates the impurity effect in Kondo problem. We also obtain the low temperature behavior of several quantities of QCD Kondo effect in the vicinity of the…
Near conformal dynamics is employed in different extensions of the standard model of particle interactions as well as in cosmology. Many of its interesting properties are either conjectured or determined using model computations. We introduce a relevant four dimensional gauge theory template allowing us to investigate such dynamics perturbatively. The gauge theory we consider is quantum chromodynamics with the addition of a meson-like scalar degree of freedom as well as an adjoint Weyl fermion. At the two-loop level, and in the Veneziano limit, we firmly establish the existence of several fixe…
We developed a general non-perturbative framework for the BFKL spectrum of planar N=4 SYM, based on the Quantum Spectral Curve (QSC). It allows one to study the spectrum in the whole generality, extending previously known methods to arbitrary values of conformal spin $n$. We show how to apply our approach to reproduce all known perturbative results for the Balitsky-Fadin-Kuraev-Lipatov (BFKL) Pomeron eigenvalue and get new predictions. In particular, we re-derived the Faddeev-Korchemsky Baxter equation for the Lipatov spin chain with non-zero conformal spin reproducing the corresponding BFKL k…
We trace the origin of the cosmological constant problem to the assumption that Newton's constant $G$ sets the scale for cosmology. And then we show that once this assumption is relaxed, the very same cosmic acceleration which has served to make the cosmological constant problem so very severe instead then serves to provide us with its potential resolution. We present an alternate cosmology, one based on conformal gravity, and show that once given only that the sign of the vacuum energy density $Λ$ is explicitly the negative one associated with spontaneous breakdown of the scale invariance of …
