The anomalous magnetic moment of the muon, aμ, has been measured with an overall precision of 540ppb by the E821 experiment at BNL. Since the publication of this result in 2004 there has been a ...persistent tension of 3.5 standard deviations with the theoretical prediction of aμ based on the Standard Model. The uncertainty of the latter is dominated by the effects of the strong interaction, notably the hadronic vacuum polarisation (HVP) and the hadronic light-by-light (HLbL) scattering contributions, which are commonly evaluated using a data-driven approach and hadronic models, respectively. Given that the discrepancy between theory and experiment is currently one of the most intriguing hints for a possible failure of the Standard Model, it is of paramount importance to determine both the HVP and HLbL contributions from first principles. In this review we present the status of lattice QCD calculations of the leading-order HVP and the HLbL scattering contributions, aμhvp andaμhlbl. After describing the formalism to express aμhvp andaμhlbl in terms of Euclidean correlation functions that can be computed on the lattice, we focus on the systematic effects that must be controlled to achieve a first-principles determination of the dominant strong interaction contributions to aμ with the desired level of precision. We also present an overview of current lattice QCD results for aμhvp andaμhlbl, as well as related quantities such as the transition form factor for π0→γ∗γ∗. While the total error of current lattice QCD estimates of aμhvp has reached the few-percent level, it must be further reduced by a factor ∼5 to be competitive with the data-driven dispersive approach. At the same time, there has been good progress towards the determination of aμhlbl with an uncertainty at the 10−15%-level.
1We study isospin-1 P-wave ππ scattering in lattice QCD with two flavors of O(a) improved Wilson fermions. For pion masses ranging from mπ = 265 MeV to mπ = 437 MeV, we determine the energy spectrum ...in the center-of-mass frame and in three moving frames. We obtain the scattering phase shifts using Lüscher's finite-volume quantization condition. Fitting the dependence of the phase shifts on the scattering momentum to a Breit-Wigner form allows us to determine the corresponding ρ mass mρ and gρππ coupling. By combining the scattering phase shifts with the decay matrix element of the vector current, we calculate the timelike pion form factor, Fπ, and compare the results to the Gounaris-Sakurai representation of the form factor in terms of the resonance parameters. In addition, we fit our data for the form factor to the functional form suggested by the Omnès representation, which allows for the extraction of the charge radius of the pion. As a further application, we discuss the long-distance behavior of the vector correlator, which is dominated by the two-pion channel. We reconstruct the long-distance part in two ways: one based on the finite-volume energies and matrix elements, and the other based on Fπ. It is shown that this part can be accurately constrained using the reconstructions, which has important consequences for lattice calculations of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment.
In the framework of quantum chromodynamics (QCD), parton distribution functions (PDFs) quantify how the momentum and spin of a hadron are divided among its quark and gluon constituents. Two main ...approaches exist to determine PDFs. The first approach, based on QCD factorization theorems, realizes a QCD analysis of a suitable set of hard-scattering measurements, often using a variety of hadronic observables. The second approach, based on first-principle operator definitions of PDFs, uses lattice QCD to compute directly some PDF-related quantities, such as their moments. Motivated by recent progress in both approaches, in this document we present an overview of lattice-QCD and global-analysis techniques used to determine unpolarized and polarized proton PDFs and their moments. We provide benchmark numbers to validate present and future lattice-QCD calculations and we illustrate how they could be used to reduce the PDF uncertainties in current unpolarized and polarized global analyses. This document represents a first step towards establishing a common language between the two communities, to foster dialogue and to further improve our knowledge of PDFs.
A
bstract
We compute the hadronic running of the electromagnetic and weak couplings in lattice QCD with
N
f
= 2 + 1 flavors of
O
(
a
) improved Wilson fermions. Using two different discretizations of ...the vector current, we compute the quark-connected and –disconnected contributions to the hadronic vacuum polarization (HVP) functions
Π
¯
γγ
and
Π
¯
γZ
for Euclidean squared momenta
Q
2
≤ 7 GeV
2
. Gauge field ensembles at four values of the lattice spacing and several values of the pion mass, including its physical value, are used to extrapolate the results to the physical point. The ability to perform an exact flavor decomposition allows us to present the most precise determination to date of the SU(3)-flavor-suppressed HVP function
Π
¯
08
that enters the running of sin
2
θ
W
. Our results for
Π
¯
γγ
,
Π
¯
γZ
and
Π
¯
08
are presented in terms of rational functions for continuous values of
Q
2
below 7 GeV
2
. We observe a tension of up to 3
.
5 standard deviation between our lattice results for
∆
α
had
5
(−
Q
2
) and estimates based on the
R
-ratio for space-like momenta in the range 3–7 GeV
2
. The tension is, however, strongly diminished when translating our result to the
Z
pole, by employing the Euclidean split technique and perturbative QCD, which yields
∆
α
had
5
M
Z
2
= 0
.
02773(15) and agrees with results based on the
R
-ratio within the quoted uncertainties.
We study the performance of all-mode-averaging (AMA) when used in conjunction with a locally deflated SAP-preconditioned solver, determining how to optimize the local block sizes and number of ...deflation fields in order to minimize the computational cost for a given level of overall statistical accuracy. We find that AMA enables a reduction of the statistical error on nucleon charges by a factor of around two at the same cost when compared to the standard method. As a demonstration, we compute the axial, scalar and tensor charges of the nucleon in Nf=2 lattice QCD with non-perturbatively O(a)-improved Wilson quarks, using O(10,000) measurements to pursue the signal out to source-sink separations of ts∼1.5 fm. Our results suggest that the axial charge is suffering from a significant amount (5–10%) of excited-state contamination at source-sink separations of up to ts∼1.2 fm, whereas the excited-state contamination in the scalar and tensor charges seems to be small.
The hadronic contribution to the eight forward amplitudes of light-by-light scattering (γ*γ*→γ*γ*) is computed in lattice QCD. Via dispersive sum rules, the amplitudes are compared to a model of the ...γ*γ*→hadrons cross sections in which the fusion process is described by hadronic resonances. Our framework can provide an important test for the model estimates of hadronic light-by-light scattering in the anomalous magnetic moment of the muon, aμHLbL. Using simple parametrizations of the resonance M→γ*γ* transition form factors, we determine the corresponding monopole and dipole masses by performing a global fit to all eight amplitudes. Together with a previous dedicated calculation of the π0→γ*γ* transition form factor, our calculation provides valuable information for phenomenological estimates of aμHLbL. The presented calculations are performed in two-flavor QCD with pion masses extending down to 190 MeV at two different lattice spacings. In addition to the fully connected Wick contractions, on two lattice ensembles we also compute the (2+2) disconnected class of diagrams, and find that their overall size is compatible with a parameter-free, large-N inspired prediction, where N is the number of colors. Motivated by this observation, we estimate in the same way the disconnected contribution to aμHLbL.
A
bstract
We present results for the short-distance window observable of the hadronic vacuum polarization contribution to the muon
g
– 2, computed via the time-momentum representation (TMR) in ...lattice QCD. A key novelty of our calculation is the reduction of discretization effects by a suitable subtraction applied to the TMR kernel function, which cancels the leading
-behaviour at short distances. To compensate for the subtraction, one must substitute a term that can be reliably computed in perturbative QCD. We apply this strategy to our data for the vector current collected on ensembles generated with 2 + 1 flavours of O(
a
)-improved Wilson quarks at six values of the lattice spacing and pion masses in the range 130 – 420 MeV. Our estimate at the physical point contains a full error budget and reads
= 68
.
85(14)
stat
(42)
syst
·10
−
10
, which corresponds to a relative precision of 0.7%. We discuss the implications of our result for the observed tensions between lattice and data-driven evaluations of the hadronic vacuum polarization.
A
bstract
We present a comprehensive study of the electromagnetic form factor, the decay constant and the mass of the pion computed in lattice QCD with two degenerate O(
a
)-improved Wilson quarks at ...three different lattice spacings in the range 0
.
05 – 0
.
08 fm and pion masses between 280 and 630 MeV at
m
π
L
≥ 4. Using partially twisted boundary conditions and stochastic estimators, we obtain a dense set of precise data points for the form factor at very small momentum transfers, allowing for a model-independent extraction of the charge radius. Chiral Perturbation Theory (ChPT) augmented by terms which model lattice artefacts is then compared to the data. At next-to-leading order the effective theory fails to produce a consistent description of the full set of pion observables but describes the data well when only the decay constant and mass are considered. By contrast, using the next-to-next-to-leading order expressions to perform global fits result in a consistent description of all data. We obtain
fm
2
as our final result for the charge radius at the physical point. Our calculation also yields estimates for the pion decay constant in the chiral limit,
F
π
/F
= 1
.
080(16)(6), the quark condensate,
MeV and several low-energy constants of SU(2) ChPT.
We present an exploratory study of leading isospin breaking effects in mesonic masses using
O
(
a
) improved Wilson fermions. Isospin symmetry is explicitly broken by distinct masses and electric ...charges of the up and down quarks. In order to be able to make use of existing isosymmetric QCD gauge ensembles we apply reweighting techniques. The path integral describing QCD+QED is expanded perturbatively in powers of the light quark’ mass deviations and the electromagnetic coupling. We employ QED
L
as a finite volume formulation of QED.
Renormalization of minimally doubled fermions Capitani, Stefano; Creutz, Michael; Weber, Johannes ...
The journal of high energy physics,
09/2010, Letnik:
2010, Številka:
9
Journal Article
Recenzirano
Odprti dostop
We investigate the renormalization properties of minimally doubled fermions, at one loop in perturbation theory. Our study is based on the two particular realizations of Boriçi-Creutz and ...Karsten-Wilczek. A common feature of both formulations is the breaking of hyper-cubic symmetry, which requires that the lattice actions are supplemented by suitable counterterms. We show that three counterterms are required in each case and determine their coefficients to one loop in perturbation theory. For both actions we compute the vacuum polarization of the gluon. It is shown that no power divergences appear and that all contributions which arise from the breaking of Lorentz symmetry are cancelled by the counterterms. We also derive the conserved vector and axial-vector currents for Karsten-Wilczek fermions. Like in the case of the previously studied Boriçi-Creutz action, one obtains simple expressions, involving only nearest-neighbour sites. We suggest methods how to fix the coefficients of the counterterms non-perturbatively and discuss the implications of our findings for practical simulations.
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