Fixing the conformal window in QCD

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 \gamma(N_f^*)=1, where \gamma is 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 QCD. The presence of a negative ultraviolet fixed point is required both in QCD and in supersymmetric QCD to preserve causality within the conformal window. Evidence for the existence of such a fixed point in QCD is provided.Comment: 10 pages, 1 figure, extended version of a talk given at the QCDNET2000 meeting, Paris, September 11-14 2000; main new material added is evidence for negative ultraviolet fixed point in QC

Next-to-next-to-leading soft-gluon corrections for the top quark cross section and transverse momentum distribution

I present results for top quark production in hadronic collisions at LHC and Tevatron energies. The soft-gluon corrections to the differential cross section are resummed at next-to-next-to-leading-logarithm (NNLL) accuracy via the two-loop soft anomalous dimension matrices. Approximate next-to-next-to-leading-order (NNLO) differential and total cross sections are calculated. Detailed theoretical predictions are shown for the t tbar cross section and the top quark p_T distribution at the Tevatron and the LHC.Comment: 23 pages, 14 figures; additional results and figure

On the renormalization of multiparton webs

We consider the recently developed diagrammatic approach to soft-gluon exponentiation in multiparton scattering amplitudes, where the exponent is written as a sum of webs - closed sets of diagrams whose colour and kinematic parts are entangled via mixing matrices. A complementary approach to exponentiation is based on the multiplicative renormalizability of intersecting Wilson lines, and their subsequent finite anomalous dimension. Relating this framework to that of webs, we derive renormalization constraints expressing all multiple poles of any given web in terms of lower-order webs. We examine these constraints explicitly up to four loops, and find that they are realised through the action of the web mixing matrices in conjunction with the fact that multiple pole terms in each diagram reduce to sums of products of lower-loop integrals. Relevant singularities of multi-eikonal amplitudes up to three loops are calculated in dimensional regularization using an exponential infrared regulator. Finally, we formulate a new conjecture for web mixing matrices, involving a weighted sum over column entries. Our results form an important step in understanding non-Abelian exponentiation in multiparton amplitudes, and pave the way for higher-loop computations of the soft anomalous dimension.Comment: 60 pages, 15 figure

Probing QCD dynamics in two-photon interactions at high energies

In this paper the two-photon interactions at high energies are investigated considering different approaches for the QCD dynamics. In particular, we calculate the $\gamma^* \gamma^*$ total cross section in different theoretical approches and present a comparison among the predictions of the BFKL dynamics at leading and next-to-leading order with those from saturation physics. We analyze the possibility that the future linear colliders could discriminate between these different approaches.Comment: 14 pages, 2 figures. Version to be published in Journal of Physics G: Nuclear and Particle Physic

Pade Approximants, Optimal Renormalization Scales, and Momentum Flow in Feynman Diagrams

We show that the Pade Approximant (PA) approach for resummation of perturbative series in QCD provides a systematic method for approximating the flow of momentum in Feynman diagrams. In the large-$\beta_0$ limit, diagonal PA's generalize the Brodsky-Lepage-Mackenzie (BLM) scale-setting method to higher orders in a renormalization scale- and scheme-invariant manner, using multiple scales that represent Neubert's concept of the distribution of momentum flow through a virtual gluon. If the distribution is non-negative, the PA's have only real roots, and approximate the distribution function by a sum of delta-functions, whose locations and weights are identical to the optimal choice provided by the Gaussian quadrature method for numerical integration. We show how the first few coefficients in a perturbative series can set rigorous bounds on the all-order momentum distribution function, if it is positive. We illustrate the method with the vacuum polarization function and the Bjorken sum rule computed in the large-$\beta_0$ limit.Comment: 28 pages, LaTeX, including 6 figures requires epsfig.st

An infrared approach to Reggeization

We present a new approach to Reggeization of gauge amplitudes based on the universal properties of their infrared singularities. Using the "dipole formula", a compact ansatz for all infrared singularities of massless amplitudes, we study Reggeization of singular contributions to high-energy amplitudes for arbitrary color representations, and any logarithmic accuracy. We derive leading-logarithmic Reggeization for general cross-channel color exchanges, and we show that Reggeization breaks down for the imaginary part of the amplitude at next-to-leading logarithms and for the real part at next-to-next-to-leading logarithms. Our formalism applies to multiparticle amplitudes in multi-Regge kinematics, and constrains possible corrections to the dipole formula starting at three loops.Comment: 4 page

Why Pad\'e Approximants reduce the Renormalization-Scale dependence in QFT?

We prove that in the limit where the beta function is dominated by the 1-loop contribution (``large beta_0 limit'') diagonal Pad\'e Approximants (PA's) of perturbative series become exactly renormalization scale (RS) independent. This symmetry suggest that diagonal PA's are resumming correctly contributions from higher order diagrams that are responsible for the renormalization of the coupling-constant. Non-diagonal PA's are not exactly invariant, but generally reduce the RS dependence as compared to partial-sums. In physical cases, higher-order corrections in the beta function break the symmetry softly, introducing a small scale and scheme dependence. We also compare the Pad\'e resummation with the BLM method. We find that in the large-N_f limit using the BLM scale is identical to resumming the series by a $x[0/n]$ non-diagonal PA.Comment: 25 pages, LateX. Replaced so that the figures would fit into the page siz

On the Structure of Infrared Singularities of Gauge-Theory Amplitudes

A closed formula is obtained for the infrared singularities of dimensionally regularized, massless gauge-theory scattering amplitudes with an arbitrary number of legs and loops. It follows from an all-order conjecture for the anomalous-dimension matrix of n-jet operators in soft-collinear effective theory. We show that the form of this anomalous dimension is severely constrained by soft-collinear factorization, non-abelian exponentiation, and the behavior of amplitudes in collinear limits. Using a diagrammatic analysis, we demonstrate that these constraints imply that to three-loop order the anomalous dimension involves only two-parton correlations, with the possible exception of a single color structure multiplying a function of conformal cross ratios depending on the momenta of four external partons, which would have to vanish in all two-particle collinear limits. We argue that such a function does not appear at three-loop order, and that the same is true in higher orders. Our formula predicts Casimir scaling of the cusp anomalous dimension to all orders in perturbation theory, and we explicitly check that the constraints exclude the appearance of higher Casimir invariants at four loops. Using known results for the quark and gluon form factors, we derive the three-loop coefficients of the 1/epsilon^n pole terms (with n=1,...,6) for an arbitrary n-parton scattering amplitude in massless QCD. This generalizes Catani's two-loop formula proposed in 1998.Comment: 46 pages, 9 figures; v2: improved treatment of collinear limits, references added; v3: improved discussion of non-abelian exponentiation, references updated; v4: typo in eq. (17) fixed, references updated; v5: additional term in (17

From Webs to Polylogarithms

We compute a class of diagrams contributing to the multi-leg soft anomalous dimension through three loops, by renormalizing a product of semi-infinite non-lightlike Wilson lines in dimensional regularization. Using non-Abelian exponentiation we directly compute contributions to the exponent in terms of webs. We develop a general strategy to compute webs with multiple gluon exchanges between Wilson lines in configuration space, and explore their analytic structure in terms of $\alpha_{ij}$, the exponential of the Minkowski cusp angle formed between the lines $i$ and $j$. We show that beyond the obvious inversion symmetry $\alpha_{ij}\to 1/\alpha_{ij}$, at the level of the symbol the result also admits a crossing symmetry $\alpha_{ij}\to -\alpha_{ij}$, relating spacelike and timelike kinematics, and hence argue that in this class of webs the symbol alphabet is restricted to $\alpha_{ij}$ and $1-\alpha_{ij}^2$. We carry out the calculation up to three gluons connecting four Wilson lines, finding that the contributions to the soft anomalous dimension are remarkably simple: they involve pure functions of uniform weight, which are written as a sum of products of polylogarithms, each depending on a single cusp angle. We conjecture that this type of factorization extends to all multiple-gluon-exchange contributions to the anomalous dimension.Comment: 64 pages, 8 figure

A novel series solution to the renormalization group equation in QCD

Recently, the QCD renormalization group (RG) equation at higher orders in MS-like renormalization schemes has been solved for the running coupling as a series expansion in powers of the exact 2-loop order coupling. In this work, we prove that the power series converges to all orders in perturbation theory. Solving the RG equation at higher orders, we determine the running coupling as an implicit function of the 2-loop order running coupling. Then we analyze the singularity structure of the higher order coupling in the complex 2-loop coupling plane. This enables us to calculate the radii of convergence of the series solutions at the 3- and 4-loop orders as a function of the number of quark flavours $n_{\rm f}$. In parallel, we discuss in some detail the singularity structure of the ${\bar{\rm MS}}$ coupling at the 3- and 4-loops in the complex momentum squared plane for $0\leq n_{\rm f} \leq 16$. The correspondence between the singularity structure of the running coupling in the complex momentum squared plane and the convergence radius of the series solution is established. For sufficiently large $n_{\rm f}$ values, we find that the series converges for all values of the momentum squared variable $Q^2=-q^2>0$. For lower values of $n_{\rm f}$, in the ${\bar{\rm MS}}$ scheme, we determine the minimal value of the momentum squared $Q_{\rm min}^2$ above which the series converges. We study properties of the non-power series corresponding to the presented power series solution in the QCD Analytic Perturbation Theory approach of Shirkov and Solovtsov. The Euclidean and Minkowskian versions of the non-power series are found to be uniformly convergent over whole ranges of the corresponding momentum squared variables.Comment: 29 pages,LateX file, uses IOP LateX class file, 2 figures, 13 Tables. Formulas (4)-(7) and Table 1 were relegated to Appendix 1, some notations changed, 2 footnotes added. Clarifying discussion added at the end of Sect. 3, more references and acknowledgments added. Accepted for publication in Few-Body System