Finite Volume Effects Research Articles

Consistency checks for two-body finite-volume matrix elements. II. Perturbative systems

Using the general formalism presented in Refs. [1,2], we study the finite-volume effects for the $\mathbf{2}+\mathcal{J}\to\mathbf{2}$ matrix element of an external current coupled to a two-particle state of identical scalars with perturbative interactions. Working in a finite cubic volume with periodicity $L$, we derive a $1/L$ expansion of the matrix element through $\mathcal O(1/L^5)$ and find that it is governed by two universal current-dependent parameters, the scalar charge and the threshold two-particle form factor. We confirm the result through a numerical study of the general formalism and additionally through an independent perturbative calculation. We further demonstrate a consistency with the Feynman-Hellmann theorem, which can be used to relate the $1/L$ expansions of the ground-state energy and matrix element. The latter gives a simple insight into why the leading volume corrections to the matrix element have the same scaling as those in the energy, $1/L^3$, in contradiction to earlier work, which found a $1/L^2$ contribution to the matrix element. We show here that such a term arises at intermediate stages in the perturbative calculation, but cancels in the final result.

Estimating the integrated bispectrum from weak lensing maps

We use a recently introduced statistic called Integrated Bispectrum (IB) to probe the gravity-induced non-Gaussianity at the level of the bispectrum from weak lensing convergence or κ maps. We generalize the concept of the IB to spherical coordinates, {This result is next connected to the response function approach.} Finally, we use the Euclid Flagship simulations to compute the IB as a function of redshift and wave number. We also outline how the IB can be computed using a variety of analytical approaches including the ones based on Effective Field Theory (EFT), Halo models and models based on the Separate Universe approach in projection or two-dimension (2D). Comparing these results against simulations we find that the existing theoretical models tend to over-predict the numerical value of the IB. We emphasize the role of the finite volume effect in numerical estimation of the IB. We introduced the concept of squeezed and collapsed tripsectrum for 2D κ maps. We derive the IB for many paramterized theories of modified gravity including the Horndeskii and beyond-Horndeskii theories specifically for the non-degenerate scenarios that are also known as the Gleyzes-Langlois-Piazza-Venizzi or GPLV theories. In addition the cosmological models with clustering quintessence and models involving massive neutrinos are also derived.

The surface tension of surfactant-containing, finite volume droplets

Surface tension influences the fraction of atmospheric particles that become cloud droplets. Although surfactants are an important component of aerosol mass, the surface tension of activating aerosol particles is still unresolved, with most climate models assuming activating particles have a surface tension equal to that of water. By studying picoliter droplet coalescence, we demonstrate that surfactants can significantly reduce the surface tension of finite-sized droplets below the value for water, consistent with recent field measurements. Significantly, this surface tension reduction is droplet size-dependent and does not correspond exactly to the macroscopic solution value. A fully independent monolayer partitioning model confirms the observed finite-size-dependent surface tension arises from the high surface-to-volume ratio in finite-sized droplets and enables predictions of aerosol hygroscopic growth. This model, constrained by the laboratory measurements, is consistent with a reduction in critical supersaturation for activation, potentially substantially increasing cloud droplet number concentration and modifying radiative cooling relative to current estimates assuming a water surface tension. The results highlight the need for improved constraints on the identities, properties, and concentrations of atmospheric aerosol surfactants in multiple environments and are broadly applicable to any discipline where finite volume effects are operative, such as studies of the competition between reaction rates within the bulk and at the surface of confined volumes and explorations of the influence of surfactants on dried particle morphology from spray driers.

Scale setting for N_mathrm{f}=3+1 QCD

We present the scale setting for a new set of gauge configurations generated with N_mathrm{f}=3+1 Wilson quarks with a non-perturbatively determined clover coefficient in a massive O(a) improvement scheme. The three light quarks are degenerate, with the sum of their masses being equal to its value in nature and the charm quark has its physical mass. We use open boundary conditions in time direction to avoid the problem of topological freezing at small lattice spacings and twisted-mass reweighting for improved stability of the simulations. The decoupling of charm at low energy allows us to set the scale by measuring the value of the low-energy quantity t_0^star /a^2, which is the flow scale t_0 at our mass point, and comparing it to an N_mathrm{f}=2+1 result in physical units. We present the details of the algorithmic setup and tuning procedure and give the bare parameters of ensembles with two lattice spacings a=0.054fm and a=0.043fm. We discuss finite volume effects and lattice artifacts and present physical results for the charmonium spectrum. In particular the hyperfine splitting between the eta _c and J/psi mesons agrees very well with its physical value.

Moments of nucleon generalized parton distributions from lattice QCD simulations at physical pion mass

We present results for the moments of nucleon isovector vector and axial generalised parton distribution functions computed within lattice QCD. Three ensembles of maximally twisted mass clover-improved fermions simulated with a physical value of the pion mass are analyzed. Two of these ensembles are generated using two degenerate light quarks. A third ensemble is used having, in addition to the light quarks, strange and charm quarks in the sea. A careful analysis of the convergence to the ground state is carried out that is shown to be essential for extracting the correct nucleon matrix elements. This allows a controlled determination of the unpolarised, helicity and tensor second Mellin moments. The vector and axial-vector generalised form factors are also computed as a function of the momentum transfer square up to about 1 GeV$^2$. The three ensembles allow us to check for unquenching effects and to assess lattice finite volume effects.

Universal renormalons in principal chiral models

Perturbative expansions in many physical systems yield 'only' asymptotic series which are not even Borel resummable. Interestingly, the corresponding ambiguities point to nonperturbative physics. We numerically verify this renormalon mechanism for the first time in two-dimensional sigma models, that, like four-dimensional gauge theories, are asymptotically free and generate a strong scale through dimensional transmutation. We perturbatively expand the energy through a numerical version of stochastic quantization. In contrast to the first energy coefficients, the high order coefficients are independent on the rank of the model. Technically, they require a sophisticated analysis of finite volume effects and the continuum limit of the discretized model. Although the individual coefficients do not grow factorially (yet), but rather decrease strongly, the ratio of consecutive coefficients clearly obey the renormalon asymptotics.

Finite volume effects on chiral phase transition and pseudoscalar mesons properties from the Polyakov-Nambu-Jona-Lasinio model

Within the framework of Polyakov-Nambu-Jona-Lasinio model and by means of Multiple Reflection Expansion, we study the finite volume effects on chiral phase transition, especially its influence on the location of the possible critical end point (CEP) and masses of mesons. Our result shows that as the radius of the sphere decreases, the location of CEP shifts toward smaller temperature while remains almost a constant in chemical potential. As for the finite volume effects on the masses of mesons, the masses of π and K increase with decreasing volume, while for σ, η and η′ the situation is the opposite. Especially, the masses of chiral parters π and σ get closer as the volume decreases, indicating that the dynamical chiral symmetry breaking effect reduces with decreasing volume.

Absorbed dose measurements from a 90Y radionuclide liquid solution using LiF:Mg,Cu,P thermoluminescent dosimeters

In the last few years there has been an increasing interest in the measurement of the absorbed dose from radionuclides, with special attention devoted to molecular radiotherapy treatments. In particular, the determination of the absorbed dose from beta emitting radionuclides in liquid solution poses a number of issues when dose measurements are performed using thermoluminescent dosimeters (TLD). Finite volume effect, i.e. the exclusion of radioactivity from the volume occupied by the TLD is one of these. Furthermore, TLDs need to be encapsulated into some kind of waterproof envelope that unavoidably contributes to beta particle attenuation during the measurement.The purpose of this study is twofold: I) to measure the absorbed dose to water, Dw, using LiF:Mg,Cu,P chips inside a PMMA cylindrical phantom filled with a homogenous 90YCl3 aqueous solution II) to assess the uncertainty budget related to Dw measurements. To this purpose, six cylindrical PMMA phantoms were manufactured at ENEA. Each phantom can host a waterproof PMMA stick containing 3 TLD chips encapsulated by a polystyrene envelope. The cylindrical phantoms were manufactured so that the radioactive liquid environment surrounds the whole stick. Finally, Dw measurements were compared with Monte Carlo (MC) calculations.The measurement of absorbed dose to water from 90YCl3 radionuclide solution using LiF:Mg,Cu,P TLDs turned out to be a viable technique, provided that all necessary correction factors are applied. Using this method, a relative combined standard uncertainty in the range 3.1–3.7% was obtained on each Dw measurement. The major source of uncertainty was shown to be TLDs calibration, with associated uncertainties in the range 0.7–2.2%. Comparison of measured and MC-calculated absorbed dose per emitted beta particle provided good results, with the two quantities being in the ratio 1.08.

Lattice QCD study of the rare kaon decay K+→π+νν¯ at a near-physical pion mass

The rare kaon decay $K^+\to\pi^+\nu\bar{\nu}$ is an ideal process in which to search for signs of new physics and is the primary goal of the NA62 experiment at CERN. In this paper we report on a lattice QCD calculation of the long-distance contribution to the $K^+\to\pi^+\nu\bar{\nu}$ decay amplitude at the near-physical pion mass $m_\pi=170$ MeV. The calculations are however, performed on a coarse lattice and hence with a lighter charm quark mass ($m_c^{\bar{\mathrm{MS}}}(\mbox{3 GeV})=750$ MeV) than the physical one. The main aims of this study are two-fold. Firstly we study the momentum dependence of the amplitude and conclude that it is very mild so that a computation at physical masses even at a single kinematic point would provide a good estimate of the long-distance contribution to the decay rate. Secondly we compute the contribution to the branching ratio from the two-pion intermediate state whose energy is below the kaon mass and find that it is less than 1% after its exponentially growing unphysical contribution has been removed and that the corresponding non-exponential finite-volume effects are negligibly small.

Sp (4) gauge theories on the lattice: Nf = 2 dynamical fundamental fermions

We perform lattice studies of the gauge theory with Sp(4) gauge group and two flavours of (Dirac) fundamental matter. The global SU(4) symmetry is spontaneously broken by the fermion condensate. The dynamical Wilson fermions in the lattice action introduce a mass that breaks the global symmetry also explicitly. The resulting pseudo-Nambu-Goldstone bosons describe the SU(4)/Sp(4) coset, and are relevant, in the context of physics beyond the Standard Model, for composite Higgs models. We discuss scale setting, continuum extrapolation and finite volume effects in the lattice theory. We study mesonic composite states, which span representations of the unbroken Sp(4) global symmetry, and we measure masses and decay constants of the (flavoured) spin-0 and spin-1 states accessible to the numerical treatment, as a function of the fermion mass. With help from the effective field theory treatment of such mesons, we perform a first extrapolation towards the massless limit. We assess our results by critically comparing to the literature on other models and to the quenched results, and we conclude by outlining future avenues for further exploration. The results of our spectroscopic analysis provide new input data for future phenomenological studies in the contexts of composite Higgs models, and of dark matter models with a strongly coupled dynamical origin.

Long-distance contributions to neutrinoless double beta decay π−→π+ee

Neutrinoless double beta decay, if detected, would prove that neutrinos are Majorana fermions and provide the direct evidence for lepton number violation. If such decay would exist in nature, then $\pi^-\pi^-\to ee$ and $\pi^-\to\pi^+ ee$ (or equivalently $\pi^-e^+\to\pi^+ e^-$) are the two simplest processes accessible via first-principle lattice QCD calculations. In this work, we calculate the long-distance contributions to the $\pi^-\to\pi^+ee$ transition amplitude using four ensembles at the physical pion mass with various volumes and lattice spacings. We adopt the infinite-volume reconstruction method to control the finite-volume effects arising from the (almost) massless neutrino. Providing the lattice QCD inputs for chiral perturbation theory, we obtain the low energy constant $g_\nu^{\pi\pi}(m_\rho)=-10.89(28)_\text{stat}(74)_\text{sys}$, which is close to $g_\nu^{\pi\pi}(m_\rho)=-11.96(31)_\text{stat}$ determined from the crossed-channel $\pi^-\pi^-\to ee$ decay.

QED self-energies from lattice QCD without power-law finite-volume errors

Using the infinite-volume photon propagator, we developed a method which allows us to calculate electromagnetic corrections to stable hadron masses with only exponentially suppressed finite-volume effects. The key idea is that the infinite volume hadronic current-current correlation function with large time separation between the two currents can be reconstructed by its value at modest time separation, which can be evaluated in finite volume with only exponentially suppressed errors. This approach can be extended to other possible applications such as QED corrections to (semi-)leptonic decays and some rare decays.

⟨A2⟩ asymmetry in lattice SU(2) gluodynamics at T>Tc

We study numerically the chromoelectric-chromomagnetic asymmetry of the dimension two ${A}^{2}$ gluon condensate at $T>{T}_{c}$ in the Landau-gauge $SU(2)$ lattice gauge theory with a particular emphasis on finite-volume effects. We show that previously found so called symmetric point at which asymmetry changes sign is an artifact of the finite volume effects. We find that with increasing temperature the asymmetry decreases approaching zero value from above in agreement with the perturbative result.

Saving the universe with finite volume effects

Setting aside anthropic arguments, there is no reason why the universe should initially favour a net expanding phase rather than one experiencing a net contraction. However, a collapsing universe containing "normal" matter will end at a singularity in a finite time. We point out that there is a mechanism, derived from non-perturbative effects in Quantum Field Theory in a finite volume, which may provide a bias towards expansion when the spacetime volume shrinks, by dynamically violating the null energy condition, without the need for modified gravity or exotic matter. We describe a scalar field component subjected to this non-perturbative effect in a cosmological background and consider its impact on a contracting phase. We discuss how this could dynamically generate the necessary initial conditions for inflation to get started, or form part of the mechanism for a non-singular cosmological bounce.

Finite-Volume Effects in (g-2)_{μ}^{HVP,LO}.

An analytic expression is derived for the leading-order finite-volume effects arising in lattice QCD calculations of the hadronic-vacuum-polarization contribution to the muon's magnetic moment a_{μ}^{HVP,LO}≡(g-2)_{μ}^{HVP,LO}/2. For calculations in a finite spatial volume with periodicity L, a_{μ}^{HVP,LO}(L) admits a transseries expansion with exponentially suppressed L scaling. Using a Hamiltonian approach, we show that the leading finite-volume correction scales as exp[-M_{π}L] with a prefactor given by the (infinite-volume) Compton amplitude of the pion, integrated with the muon-mass-dependent kernel. To give a complete quantitative expression, we decompose the Compton amplitude into the spacelike pion form factor F_{π}(Q^{2}) and a multiparticle piece. We determine the latter through next-to leading order in chiral perturbation theory and find that it contributes negligibly and through a universal term that depends only on the pion decay constant, with all additional low-energy constants dropping out of the integral.

Lattice QCD and Three-Particle Decays of Resonances

Most strong-interaction resonances have decay channels involving three or more particles, including many of the recently discovered X, Y, and Z resonances. In order to study such resonances from first principles using lattice QCD, one must understand finite-volume effects for three particles in the cubic box used in calculations. We review efforts to develop a three-particle quantization condition that relates finite-volume energies to infinite-volume scattering amplitudes. We describe in detail the three approaches that have been followed, and present new results on the relationship between the corresponding results. We show examples of the numerical implementation of all three approaches and point out the important issues that remain to be resolved.

Finite-volume effects on the chiral phase transition of thermal QED3

The influence of finite-volume effects on dynamical chiral symmetry breaking of thermal ${\mathrm{QED}}_{3}$ is investigated. We show that the chiral symmetry gets restored at all temperatures when the size of system is less than some certain value ${L}_{c}^{\ensuremath{'}}=189$; the critical temperature of chiral phase transition generally increases with the size of the system and the critical temperature when the system volume tends to infinity is slightly less than that in the case of infinite volume.

Hadronic vacuum polarization contribution to the muon g−2 with ( 2+1 )-flavor lattice QCD on a larger than (10 fm)4 lattice at the physical point

We study systematic uncertainties in the lattice QCD computation of hadronic vacuum polarization (HVP) contribution to the muon $g-2$. We investigate three systematic effects; finite volume (FV) effect, cutoff effect, and integration scheme dependence. We evaluate the FV effect at the physical pion mass on two different volumes of (5.4 fm$)^4$ and (10.8 fm$)^4$ using the PACS10 configurations at the same cutoff scale. For the cutoff effect, we compare two types of lattice vector operators, which are local and conserved (point-splitting) currents, by varying the cutoff scale on a larger than (10 fm$)^4$ lattice at the physical point. For the integration scheme dependence, we compare the results between the coordinate- and momentum-space integration schemes at the physical point on a (10.8 fm$)^4$ lattice. Our result for the HVP contribution to the muon $g-2$ is given by $a_\mu^{\rm hvp} = 737(9)(^{+13}_{-18})\times 10^{-10}$ in the continuum limit, where the first error is statistical and the second one is systematic.

Effects of long-wavelength fluctuations in large galaxy surveys

ABSTRACT In order to capture as much information as possible large galaxies surveys have been increasing their volume and redshift depth. To face this challenge theory has responded by making cosmological simulations of huge computational volumes with equally increasing numbers of dark matter particles and supercomputing resources. Thus, it is taken for granted that the ideal situation is when a single computational box encompasses the whole volume of the observational survey, e.g. $\sim 50\, h^{-3}\,{\rm Gpc}^3$ for the DESI and Euclid surveys. Here we study the effects of missing long waves in a finite volume using several relevant statistics: the abundance of dark matter haloes, the probability distribution function (PDF), the correlation function and power spectrum, and covariance matrices. Finite volume effects can substantially modify the results if the computational volumes are less than $\sim (500\mbox{$\, h^{-1}$Mpc})^3$. However, the effects become extremely small and practically can be ignored when the box size exceeds ∼1 Gpc3. We find that the average power spectra of dark matter fluctuations show remarkable lack of dependence on the computational box size with less than 0.1 per cent differences between $1$ and $4\mbox{$\, h^{-1}\,$Gpc}$ boxes. No measurable differences are expected for the halo mass functions for these volumes. The covariance matrices are scaled trivially with volume, and small corrections due to supersample modes can be added. We conclude that there is no need to make those extremely large simulations when a box size of $1-1.5\mbox{$\, h^{-1}$Gpc}$ is sufficient to fulfil most of the survey science requirements.

Form factors of two-hadron states from a covariant finite-volume formalism

In this work we develop a Lorentz-covariant version of the previously derived formalism for relating finite-volume matrix elements to $\textbf 2 + \mathcal J \to \textbf 2$ transition amplitudes. We also give various details relevant for the implementation of this formalism in a realistic numerical lattice QCD calculation. Particular focus is given to the role of single-particle form factors in disentangling finite-volume effects from the triangle diagram that arise when $\mathcal J$ couples to one of the two hadrons. This also leads to a new finite-volume function, denoted $G$, the numerical evaluation of which is described in detail. As an example we discuss the determination of the $\pi \pi + \mathcal J \to \pi \pi$ amplitude in the $\rho$ channel, for which the single-pion form factor, $F_\pi(Q^2)$, as well as the scattering phase, $\delta_{\pi\pi}$, are required to remove all power-law finite-volume effects. The formalism presented here holds for local currents with arbitrary Lorentz structure, and we give specific examples of insertions with up to two Lorentz indices.