Multi-particle production in proton–nucleus collisions in the color glass condensate

We compute multi-gluon production in the Color Glass Condensate approach in dilute-dense collisions, $$\hbox {p}A$$ p A , extending previous calculations up to four gluons. We include the contributions that are leading in the overlap area of the collision but keep all orders in the expansion in the number of colors. We develop a diagrammatic technique to write the numerous color contractions and exploit the symmetries to group the diagrams and simplify the expressions. To proceed further, we use the McLerran–Venugopalan and Golec–Biernat–Wüsthoff models for the projectile and target averages, respectively. We use a form of the Lipatov vertices that leads to the Wigner function approach for the projectile previously employed, that we generalise to take into account quantum correlations in the projectile wave function. We provide analytic expressions for integrated and differential two gluon cumulants and show a smooth dependence on the parameters defining the projectile and target Wigner function and dipole, respectively. For four gluon correlations we find that the second order four particle cumulant is negative, so a sensible second Fourier azimuthal coefficient can be defined. The effect of correlations in the projectile on this result results qualitatively and quantitatively large.

Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$\begin{aligned} \partial ^2_tu(t,x)- a(x,\lambda )^2\partial _x^2u(t,x)= b(x,\lambda ,u(t,x),u(t-\tau ,x),\partial _tu(t,x),\partial _xu(t,x)), \; x \in (0,1) \end{aligned}$$ ∂ t 2 u ( t , x ) - a ( x , λ ) 2 ∂ x 2 u ( t , x ) = b ( x , λ , u ( t , x ) , u ( t - τ , x ) , ∂ t u ( t , x ) , ∂ x u ( t , x ) ) , x ∈ ( 0 , 1 ) with smooth coefficient functions a and b such that $$a(x,\lambda )>0$$ a ( x , λ ) > 0 and $$b(x,\lambda ,0,0,0,0) = 0$$ b ( x , λ , 0 , 0 , 0 , 0 ) = 0 for all x and $$\lambda $$ λ . We state conditions ensuring Hopf bifurcation, i.e., existence, local uniqueness (up to time shifts), regularity (with respect to t and x) and smooth dependence (on $$\tau $$ τ and $$\lambda $$ λ ) of small non-stationary time-periodic solutions, which bifurcate from the stationary solution $$u=0$$ u = 0 , and we derive a formula which determines the bifurcation direction with respect to the bifurcation parameter $$\tau $$ τ . To this end, we transform the wave equation into a system of partial integral equations by means of integration along characteristics and then apply a Lyapunov-Schmidt procedure and a generalized implicit function theorem. The main technical difficulties, which have to be managed, are typical for hyperbolic PDEs (with or without delay): small divisors and the “loss of derivatives” property. We do not use any properties of the corresponding initial-boundary value problem. In particular, our results are true also for negative delays $$\tau $$ τ .

The renormalised $$\mathrm{O}(a)$$ improved vector current in three-flavour lattice QCD with Wilson quarks

We present the results of a non-perturbative determination of the improvement coefficient $$c_\mathrm{V}$$ c V and the renormalisation factor $$Z_\mathrm{V}$$ Z V , which define the renormalised vector current in three-flavour $$\mathrm{O}(a)$$ O ( a ) improved lattice QCD with Wilson quarks and tree-level Symanzik-improved gauge action. In case of the improvement coefficient, we consider both lattice descriptions of the vector current, the local as well as the conserved (i.e., point-split) one. Our improvement and normalisation conditions are based on massive chiral Ward identities and numerically evaluated in the Schrödinger functional setup, which allows to eliminate finite quark mass effects in a controlled way. In order to ensure a smooth dependence of the renormalisation constant and improvement coefficients on the bare gauge coupling, our computation proceeds along a line of constant physics, covering the typical range of lattice spacings $$0.04\,\mathrm{fm}\lesssim a\lesssim 0.1\,\mathrm{fm}$$ 0.04 fm ≲ a ≲ 0.1 fm that is useful for phenomenological applications. Especially for the improvement coefficient of the local vector current, we report significant differences between the one-loop perturbative estimates and our non-perturbative results.

Elastic Contacts of Randomly Rough Indenters with Thin Sheets, Membranes Under Tension, Half Spaces, and Beyond

We consider the adhesion-less contact between a two-dimensional, randomly rough, rigid indenter, and various linearly elastic counterfaces, which can be said to differ in their spatial dimension D. They include thin sheets, which are either free or under equi-biaxial tension, and semi-infinite elastomers, which are either isotropic or graded. Our Green’s function molecular dynamics simulation identifies an approximately linear relation between the relative contact area $$a_{\text {r}}$$ a r and pressure p at small p only above a critical dimension. The pressure dependence of the mean gap $$u_{\text {g}}$$ u g obeys identical trends in each studied case: quasi-logarithmic at small p and exponentially decaying at large p. Using a correction factor with a smooth dependence on D, all obtained $$u_{\text {g}}(p)$$ u g ( p ) relations can be reproduced accurately over several decades in pressure with Persson’s theory, even when it fails to properly predict the interfacial stress distribution function. Graphical Abstract

Splitting Lemma for Biholomorphic Mappings with Smooth Dependence on Parameters

We prove that the mappings obtained in Forstnerič splitting lemma vary in a $$\mathcal {C}^{\lfloor {\frac{l-1}{2}}\rfloor }$$ C ⌊ l - 1 2 ⌋ -continuous way if only the input family of biholomorphic mappings close to Id (and their domains) is $$\mathcal {C}^l$$ C l -continuous (see Theorem 1.3 for a precise formulation).