A Novel Feature of Valence Quark Distributions in Hadrons

Examining the evolution of the maximum of valence quark distribution, qV, weighted by Bjorken x, h(x,t)≡xqV(x,t), it is observed that h(x,t) at the peak becomes a one-parameter function; h(xp,t)=Φ(xp(t)), where xp is the position of the peak, t=logQ2, and Q2 is the resolution scale. This observation is used to derive a new model-independent relation which connects the partial derivative of the valence parton distribution functions (PDFs) in xp to the quantum chromodynamics (QCD) evolution equation through the xp derivative of the logarithm of the function Φ(xp(t)). A numerical analysis of this relation using empirical PDFs results in an observation of the exponential form of the Φ(xp(t))=h(xp,t)=CeDxp(t) for leading to next-to-next leading order approximations of PDFs for the range of Q2, covering four orders in magnitude. The exponent, D, of the observed “height-position” correlation function converges with the increase in the order of approximation. This result holds for all the PDF sets considered. A similar relation is observed also for the pion valence quark distribution, indicating that the obtained relation may be universal for any non-singlet partonic distribution. The observed “height-position” correlation is used also to indicate that no finite number of exchanges can describe the analytic behavior of the valence quark distribution at the position of the peak at fixed Q2.

On a Tauberian theorem of Ingham and Euler–Maclaurin summation

We discuss two theorems in analytic number theory and combinatory analysis that have seen increased use in recent years. A corollary to a Tauberian theorem of Ingham allows one to quickly prove asymptotic formulas for arithmetic sequences, so long as the corresponding generating function exhibits exponential growth of a certain form near its radius of convergence. Two common methods for proving the required analytic behavior are modular transformations and Euler–Maclaurin summation. However, these results are sometimes stated without certain technical conditions that are necessary for the complex analytic techniques that appear in Ingham’s proof. We carefully examine the precise statements and proofs of these results, and find that in practice, the technical conditions are satisfied for those cases appearing in recent applications. We also generalize the classical approach of Euler–Maclaurin summation in order to prove asymptotic expansions for series with complex values, simple poles, or multi-dimensional summation indices.

Non-analyticity of the Correlation Length in Systems with Exponentially Decaying Interactions

We consider a variety of lattice spin systems (including Ising, Potts and XY models) on $$\mathbb {Z}^d$$ Z d with long-range interactions of the form $$J_x = \psi (x) e^{-|x|}$$ J x = ψ ( x ) e - | x | , where $$\psi (x) = e^{{\mathsf o}(|x|)}$$ ψ ( x ) = e o ( | x | ) and $$|\cdot |$$ | · | is an arbitrary norm. We characterize explicitly the prefactors $$\psi $$ ψ that give rise to a correlation length that is not analytic in the relevant external parameter(s) (inverse temperature $$\beta $$ β , magnetic field $$h$$ h , etc). Our results apply in any dimension. As an interesting particular case, we prove that, in one-dimensional systems, the correlation length is non-analytic whenever $$\psi $$ ψ is summable, in sharp contrast to the well-known analytic behavior of all standard thermodynamic quantities. We also point out that this non-analyticity, when present, also manifests itself in a qualitative change of behavior of the 2-point function. In particular, we relate the lack of analyticity of the correlation length to the failure of the mass gap condition in the Ornstein–Zernike theory of correlations.

Essential regularity of the model space for the Weil–Petersson metric

This is the second in a series of papers ([7] and [6] are the others) that studies the behavior of harmonic maps into the Weil–Petersson completion {\overline{\mathcal{T}}} of Teichmüller space. The boundary of {\overline{\mathcal{T}}} is stratified by lower-dimensional Teichmüller spaces and the normal space to each stratum is a product of copies of a singular space {\overline{\bf H}} called the model space. The significance of {\overline{\bf H}} is that it captures the singular behavior of the Weil–Petersson geometry of {\overline{\mathcal{T}}} . The main result of the paper is that certain subsets of {\overline{\bf H}} are essentially regular in the sense that harmonic maps to those spaces admit uniform approximation by affine functions. This is a modified version of the notion of essential regularity introduced by Gromov–Schoen in [12] for maps into Euclidean buildings and is one of the key ingredients in proving superrigidity. In the process, we introduce new coordinates on {\overline{\bf H}} and estimate the metric and its derivatives with respect to the new coordinates. These results form the technical core for studying the analytic behavior of harmonic maps into the completion of Teichmüller space and are utilized in our subsequent paper [6], where we prove the holomorphic rigidity of the Teichmüller space and several rigidity results for the mapping class group.

Vacuum Condensate Picture of Quantum Gravity

In quantum gravity perturbation theory in Newton’s constant G is known to be badly divergent, and as a result not very useful. Nevertheless, some of the most interesting phenomena in physics are often associated with non-analytic behavior in the coupling constant and the existence of nontrivial quantum condensates. It is therefore possible that pathologies encountered in the case of gravity are more likely the result of inadequate analytical treatment, and not necessarily a reflection of some intrinsic insurmountable problem. The nonperturbative treatment of quantum gravity via the Regge–Wheeler lattice path integral formulation reveals the existence of a new phase involving a nontrivial gravitational vacuum condensate, and a new set of scaling exponents characterizing both the running of G and the long-distance behavior of invariant correlation functions. The appearance of such a gravitational condensate is viewed as analogous to the (equally nonperturbative) gluon and chiral condensates known to describe the physical vacuum of QCD. The resulting quantum theory of gravity is highly constrained, and its physical predictions are found to depend only on one adjustable parameter, a genuinely nonperturbative scale ξ in many ways analogous to the scaling violation parameter Λ M ¯ S of QCD. Recent results point to significant deviations from classical gravity on distance scales approaching the effective infrared cutoff set by the observed cosmological constant. Such subtle quantum effects are expected to be initially small on current cosmological scales, but could become detectable in future high precision satellite experiments.

Vacuum Condensate Picture of Quantum Gravity

In quantum gravity perturbation theory in Newton's constant $G$ is known to be badly divergent, and as a result not very useful. Nevertheless, some of the most interesting phenomena in physics are often associated with non-analytic behavior in the coupling constant and the existence of nontrivial quantum condensates. It is therefore possible that pathologies encountered in the case of gravity are more likely the result of inadequate analytical treatment, and not necessarily a reflection of some intrinsic insurmountable problem. The nonperturbative treatment of quantum gravity via the Regge-Wheeler lattice path integral formulation reveals the existence of a new phase involving a nontrivial gravitational vacuum condensate, and a new set of scaling exponents characterizing both the running of $G$ and the long-distance behavior of invariant correlation functions. The appearance of such a gravitational condensate is viewed as analogous to the (equally nonperturbative) gluon and chiral condensates known to describe the physical vacuum of QCD. The resulting quantum theory of gravity is highly constrained, and its physical predictions are found to depend only on one adjustable parameter, a genuinely nonperturbative scale $\xi$ in many ways analogous to the scaling violation parameter $\Lambda_{\bar MS} $ of QCD. Recent results point to significant deviations from classical gravity on distance scales approaching the effective infrared cutoff set by the observed cosmological constant. Such subtle quantum effects are expected to be initially small on current cosmological scales, but could become detectable in future high precision satellite experiments.