Finite Volume Corrections Research Articles

Finite-volume corrections to charge radii

The finite-volume nature of lattice QCD entails a variety of effects that must be handled in the process of performing chiral extrapolations. Since the pion cloud that surrounds hadrons becomes distorted in a finite volume, hadronic observables must be corrected before one can compare with the experimental values. The electric charge radius of the nucleon is of particular interest when considering the implementation of finite-volume corrections. It is common practice in the literature to transform electric form factors from the lattice into charge radii prior to analysis. However, there is a fundamental difficulty with using these charge radii in a finite-volume extrapolation. The subtleties are a consequence of the absence of a continuous derivative on the lattice. A procedure is outlined for handling such finite-volume corrections, which must be applied directly to the electric form factors themselves rather than to the charge radii.

Exploring the Aoki regime

We compute next-to-leading order (NLO) corrections in the \epsilon-regime of Wilson (WChPT) and Staggered Chiral Perturbation Theory (SChPT). A difference between the two is that in WChPT already at NLO, that is at O(\epsilon^2), new low energy constants (LECs) contribute, whereas in SChPT they only enter at O(\epsilon^4). We first determine the NLO corrections in WChPT for SU(2), and for U(N_f) at fixed index. This implies finite-volume corrections to the phase boundary between the Aoki phase and the Sharpe-Singleton scenario via corrections to the mean field potential. We also compute NLO corrections to the two-point function in the scalar and pseudo-scalar sector in WChPT. Turning to SChPT we determine the NLO corrections to the LECs and their effect on the taste splitting. Here the NLO partition function can be written as the leading order one with renormalized couplings, thus preserving the equivalence to staggered chiral random matrix theory at NLO for any number of flavors N_f. In WChPT this relation only appears to hold for SU(2).

Virtual decuplet effects on octet baryon masses in covariant baryon chiral perturbation theory

We extend a previous analysis of the lowest-lying octet baryon masses in covariant baryon chiral perturbation theory (ChPT) by explicitly taking into account the contribution of the virtual decuplet baryons. Up to next-to-next-to-next-to-leading order (N$^3$LO), the effects of these heavier degrees of freedom are systematically studied. Their effects on the light-quark mass dependence of the octet baryon masses are shown to be relatively small and can be absorbed by the available low-energy constants up to N$^3$LO. Nevertheless, a better description of the finite-volume corrections of the lattice QCD data can be achieved, particularly those with small $M_\phi L$ ($<4$), which is demonstrated by a careful study of the NPLQCD and QCDSF-UKQCD small-volume data. Finally, we show that the predicted pion- and strangeness-baryon sigma terms are only slightly changed by the inclusion of the virtual decuplet baryons.

Method to extract charged hadron properties from lattice QCD in magnetic fields

By analyzing the external field dependence of correlation functions, the magnetic properties of hadrons can be determined using lattice QCD in magnetic fields. To compute the magnetic moments and polarizabilities of charged hadrons, for example, one requires sufficiently weak magnetic fields. Such field strengths, however, lead to closely spaced Landau levels that are not straightforwardly resolved using standard lattice spectroscopy. Focusing on charged spinless hadrons, we introduce a simple projection technique that can be used to isolate the lowest Landau level. As the technique requires the explicit coordinate-space wave-function, we investigate the extent to which the continuum, infinite volume wave-function can be employed. We find that, in practice, the effects of discretization can be handled using a perturbative expansion about the continuum. Finite volume corrections are taken into account by using the discrete magnetic translational invariance of the torus. We show that quantized magnetic fields can lead to pernicious volume effects which depend on the magnetic flux quantum, rather than on the lattice volume.

On the energy momentum dispersion in the lattice regularization

For a free scalar boson field and for U(1) gauge theory finite volume (infrared) and other corrections to the energy-momentum dispersion in the lattice regularization are investigated calculating energy eigenstates from the fall off behavior of two-point correlation functions. For small lattices the squared dispersion energy defined by $ E_{{\mathrm{dis}\hbox{,}\overrightarrow{k}}}^2=E_{\overrightarrow{k}}^2-E_0^2-4\sum {_{i=1}^{d-1 }} \sin {{\left( {{k_i}/2} \right)}^2} $ is in both cases negative (d is the Euclidean space-time dimension and $ {E_{\overrightarrow{k}}} $ the energy of momentum $ \overrightarrow{k} $ eigenstates). Observation of $ E_{{\mathrm{dis}\hbox{,}\overrightarrow{k}}}^2=0 $ has been an accepted method to demonstrate the existence of a massless photon ( E 0 = 0) in 4D lattice gauge theory, which we supplement here by a study of its finite size corrections. A surprise from the lattice regularization of the free field is that infrared corrections do not eliminate a difference between the groundstate energy E 0 and the mass parameter M of the free scalar lattice action. Instead, the relation $ {E_0}={\cosh^{-1 }}\left( {1+{M^2}/2} \right) $ holds independently of the spatial lattice size.

Baryon electric dipole moments from strong CP violation

The electric dipole form factors and moments of the ground state baryons are calculated in chiral perturbation theory at next-to-leading order. We show that the baryon electric dipole form factors at this order depend only on two combinations of low-energy constants. We also derive various relations that are free of unknown low-energy constants. We use recent lattice QCD data to calculate all baryon EDMs. In particular, we find d_n = -2.9\pm 0.9 and d_p = 1.1\pm 1.1 in units of 10^{-16} e \theta_0 cm. Finite volume corrections to the moments are also worked out. We show that for a precision extraction from lattice QCD data, the next-to-leading order terms have to be accounted for.

Octet baryon masses in next-to-next-to-next-to-leading order covariant baryon chiral perturbation theory

We study the ground-state octet baryon masses and sigma terms using the covariant baryon chiral perturbation theory (ChPT) with the extended-on-mass-shell (EOMS) renormalization scheme up to next-to-next-to-next-to-leading order (N3LO). By adjusting the available 19 low-energy constants (LECs), a reasonable fit of the n f = 2+1 lattice quantum chromodynamics (LQCD) results from the PACS-CS, LHPC, HSC, QCDSF-UKQCD and NPLQCD collaborations is achieved. Finite-volume corrections to the lattice data are calculated self-consistently. Our study shows that the N3LO BChPT describes better the light quark mass evolution of the lattice data than the NNLO BChPT does and the various lattice simulations seem to be consistent with each other. We also predict the pion and strangeness sigma terms of the octet baryons using the LECs determined in the fit of their masses. The predicted pion- and strangeness-nucleon sigma terms are σ πN = 43(1)(6) MeV and σ sN = 126(24)(54) MeV, respectively.

The Yang-Mills gradient flow in finite volume

The Yang-Mills gradient flow is considered on the four dimensional torus T 4 for SU(N) gauge theory coupled to N f flavors of massless fermions in arbitrary representations. The small volume dynamics is dominated by the constant gauge fields. The expectation value of the field strength tensor squared TrF μν F μν (t) is calculated for positive flow time t by treating the non-zero gauge modes perturbatively and the zero modes exactly. The finite volume correction to the infinite volume result is found to contain both algebraic and exponential terms. The leading order result is then used to define a one parameter family of running coupling schemes in which the coupling runs with the linear size of the box. The new scheme is tested numerically in SU(3) gauge theory coupled to N f = 4 flavors of massless fundamental fermions. The calculations are performed at several lattice spacings with a controlled continuum extrapolation. The continuum result agrees with the perturbative prediction for small renormalized coupling as expected.

Benchmark calculations for elastic fermion-dimer scattering

We present continuum and lattice calculations for elastic scattering between a fermion and a bound dimer in the shallow binding limit. For the lattice calculation we use the finite-volume method of L\"uscher. We take into account topological finite-volume corrections to the dimer binding energy which depend on the momentum of the dimer. After subtracting these effects, we find from the lattice calculation $\ensuremath{\kappa}{a}_{fd}=1.174(9)$ and $\ensuremath{\kappa}{r}_{fd}=\ensuremath{-}0.029(13)$. These results agree well with the continuum values $\ensuremath{\kappa}{a}_{fd}=1.17907(1)$ and $\ensuremath{\kappa}{r}_{fd}=\ensuremath{-}0.0383(3)$ obtained from the STM equation. We discuss applications to cold atomic Fermi gases, deuteron-neutron scattering in the spin-quartet channel, and lattice calculations of scattering for nuclei and hadronic molecules at finite volume.

Background field approach to electromagnetic properties of baryons

We investigate the self-energies of particles in an external magnetic field $B$. The dependence is generally of the type $\sqrt{P(B)}$ with $P$ a polynomial in $B$ and the participating masses. The non-analytic point depends on the mass and charge constellations, is unproblematic for stable particles but constrains the linear energy shift approximation for resonances. We calculate the $B$ dependent self-energies of the nucleon and $\Delta(1232)$-isobar in the SU(2) covariant chiral perturbation theory and outline a way to obtain finite volume corrections to the nucleon anomalous magnetic moment without using the three point function method. We show that finite volume corrections might explain present discrepancies of lattice QCD and chiral perturbation theory results in the small pion mass region.

Finite volume corrections to the electromagnetic current of the nucleon

We compute corrections to both the isovector anomalous magnetic moment and the isovector electromagnetic current of the nucleon to $O(p^3)$ in the framework of covariant two-flavor Baryon Chiral Perturbation Theory. We then apply these corrections to lattice data for the anomalous magnetic moment from the LHPC, RBC & UKQCD and QCDSF collaborations.

Study of the conformal hyperscaling relation through the Schwinger-Dyson equation

We study corrections to the conformal hyperscaling relation in the conformal window of the large ${N}_{f}$ QCD by using the ladder Schwinger-Dyson (SD) equation as a concrete dynamical model. From the analytical expression of the solution of the ladder SD equation, we identify the form of the leading mass correction to the hyperscaling relation. We find that the anomalous dimension, when identified through the hyperscaling relation neglecting these corrections, yields a value substantially lower than the one at the fixed point ${\ensuremath{\gamma}}_{m}^{*}$ for large mass region. We further study finite-volume effects on the hyperscaling relation, based on the ladder SD equation in a finite space-time with the periodic boundary condition. We find that the finite-volume corrections on the hyperscaling relation are negligible compared with the mass correction. The anomalous dimension, when identified through the finite-size hyperscaling relation neglecting the mass corrections as is often done in the lattice analyses, yields almost the same value as that in the case of the infinite space-time neglecting the mass correction, i.e., a substantially lower value than ${\ensuremath{\gamma}}_{m}^{*}$ for large mass. We also apply the finite-volume SD equation to the chiral-symmetry-breaking phase and find that when the theory is close to the critical point such that the dynamically generated mass is much smaller than the explicit breaking mass, the finite-size hyperscaling relation is still operative. We also suggest a concrete form of the modification of the finite-size hyperscaling relation by including the mass correction, which may be useful to analyze the lattice data.

Thermodynamic potentials from shifted boundary conditions: the scalar-field theory case

In a thermal field theory, the cumulants of the momentum distribution can be extracted from the dependence of the Euclidean path integral on a shift in the fields built into the temporal boundary condition. When combined with the Ward identities associated with the invariance of the theory under the Poincaré group, thermodynamic potentials such as the entropy or the pressure can be directly inferred from the response of the system to the shift. Crucially the argument holds, up to harmless finite-size and discretization effects, even if translational and rotational invariance are broken to a discrete subgroup of finite shifts and rotations such as in a lattice box. The formulas are thus applicable at finite lattice spacing and volume provided the derivatives are replaced by their discrete counterpart, and no additive or multiplicative ultraviolet-divergent renormalizations are needed to take the continuum limit. In this paper we present a complete derivation of the relevant formulas in the scalar field theory, where several technical complications are avoided with respect to gauge theories. As a by-product we obtain a recursion relation among the cumulants of the momentum distribution, and formulæ for finite-volume corrections to several well-known thermodynamic identities.

Finite-volume effects on octet-baryon masses in covariant baryon chiral perturbation theory

We study finite-volume effects on the masses of the ground-state octet baryons using covariant baryon chiral perturbation theory (ChPT) up to next-to-leading order by analyzing the latest $n_f=2+1$ lattice Quantum ChromoDynamics (LQCD) results from the NPLQCD collaboration. Contributions of virtual decuplet baryons are taken into account using the "consistent" coupling scheme. We compare our results with those obtained from heavy baryon ChPT and show that, although both approaches can describe well the lattice data, the underlying physics is different: In HBChPT, virtual decuplet baryons play a more important role than they do in covariant ChPT. This is because the virtual octet baryon contributions to finite-volume corrections are larger in covariant ChPT than in HBChPT, while the contributions of intermediate decuplet baryons are smaller, because of relativistic effects. We observe that for the octet baryon masses, at fixed $m_\pi L$ ($\gg1$) finite-volume corrections decrease as $m_\pi$ approaches its physical value, provided that the strange quark mass is at or close to its physical value, as in most LQCD setups.

Pion mass dependence of the K l3 semileptonic scalar form factor within finite volume

We calculate the scalar semileptonic kaon decay in finite volume at the momentum transfer $t_{m} = (m_{K} - m_{\pi})^2$, using chiral perturbation theory. At first we obtain the hadronic matrix element to be calculated in finite volume. We then evaluate the finite size effects for two volumes with $L = 1.83 fm$ and $L= 2.73 fm$ and find that the difference between the finite volume corrections of the two volumes are larger than the difference as quoted in \cite{Boyle2007a}. It appears then that the pion masses used for the scalar form factor in ChPT are large which result in large finite volume corrections. If appropriate values for pion mass are used, we believe that the finite size effects estimated in this paper can be useful for Lattice data to extrapolate at large lattice size.

Stochastic theory of large-scale enzyme-reaction networks: Finite copy number corrections to rate equation models

Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtoliters. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small subcellular compartment. This is achieved by applying a mesoscopic version of the quasisteady-state assumption to the exact Fokker-Planck equation associated with the Poisson representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing subcellular volume, decreasing Michaelis-Menten constants, and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.

Challenges in uncovering the origin of the proton's spin

One of the most fascinating challenges facing modern strong interaction physics is to understand the origin of the spin of the nucleon in terms of the spin and orbital angular momentum of the quarks and gluons. We review recent progress on this problem as well as some of the uncertainties associated with state of the art lattice QCD simulations. In particular, we explain the importance of the corrections associated with chiral extrapolation and finite volume corrections, especially for the term B(0) extracted from the appropriate low moment of the deeply virtual Compton scattering amplitude.

Finite volume effects for nucleon and heavy meson masses

We apply the resummed version of the L\"uscher formula to analyze finite volume corrections to the mass of the nucleon and of heavy mesons. We show that by applying the subthreshold expansion of the scattering amplitudes one can express the finite volume corrections in terms of only a few physical observables and the size of the box. In the case of the nucleon, the available information about the quark mass dependence of these physical quantities is discussed and used to assess the finite volume corrections to the nucleon mass as a function of the quark mass including a detailed analysis of the remaining uncertainties. For heavy mesons, the L\"uscher formula is derived both fully relativistically and in a nonrelativistic approximation and a first attempt at a numerical analysis is made.

Twisted mass finite volume effects

We calculate finite-volume effects on the pion masses and decay constant in twisted mass lattice QCD at finite lattice spacing. We show that the lighter neutral pion in twisted mass lattice QCD gives rise to finite-volume effects that are exponentially enhanced when compared to those arising from the heavier charged pions. We demonstrate that the recent two flavor twisted mass lattice data can be better fitted when twisted mass effects in finite-volume corrections are taken into account.

The epsilon expansion at next-to-next-to-leading order with small imaginary chemical potential

We discuss chiral perturbation theory for two and three quark flavors in the epsilon expansion at next-to-next-to-leading order (NNLO) including a small imaginary chemical potential. We calculate finite-volume corrections to the low-energy constants Σ and F and determine the non-universal modifications of the theory, i.e., modifications that cannot be mapped to random matrix theory (RMT). In the special case of two quark flavors in an asymmetric box we discuss how to minimize the finite-volume corrections and non-universal modifications by an optimal choice of the lattice geometry. Furthermore we provide a detailed calculation of a special version of the massless sunset diagram at finite volume.