The second Mellin moments <math><mo>⟨</mo><mi>x</mi><mo>⟩</mo></math> of the nucleon’s unpolarized, polarized, and transversity parton distribution functions are computed. Two lattice QCD ensembles at the physical pion mass are used: these were generated using a tree-level Symanzik-improved gauge action and <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor tree-level improved Wilson Clover fermions coupling via 2-level HEX-smearing. The moments are extracted from forward matrix elements of local leading twist operators. We determine renomalization factors in RI-(S)MOM and match to <math><mover><mi>MS</mi><mo>¯</mo></mover></math> at scale 2 GeV. Our findings show that operators that exhibit vanishing kinematics at zero momentum can have significantly reduced excited-state contamination. The resulting polarized moment is used to quantify the longitudinal contribution to the quark spin-orbit correlation. All our results agree within two sigma with previous lattice results.

Germany

Marcel Rodekamp

United States

Michael Engelhardt

Jeremy R. Green

Stefan Krieg

Simonetta Liuti

Stefan Meinel

John W. Negele

Andrew Pochinsky

Sergey Syritsyn

Moments of nucleon unpolarized, polarized, and transversity parton distribution functions from lattice QCD at the physical point

In this paper, the existence of a massive dark photon, associated with a new gauge group is considered. The dark photon can be kinetically mixed with the photon. To study some applications, the thermo field dynamics formalism is used. Exploring the topological structure of this approach, the influence of dark photons on the Stefan–Boltzmann law and the Casimir effect at zero and finite temperature is calculated.

Brazil

V. Prata

A. Santos

Canada

Faqir Khanna

Stefan–Boltzmann law and Casimir effect for dark photons

We determine the <math><mi>Δ</mi><mo>(</mo><mn>1232</mn><mo>)</mo></math> resonance parameters using lattice QCD and the Lüscher method. The resonance occurs in elastic pion-nucleon scattering with <math><mrow><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi></mrow></msup><mo>=</mo><mn>3</mn><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> in the isospin <math><mi>I</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math>, <math><mi>P</mi></math>-wave channel. Our calculation is performed with <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> flavors of clover fermions on a lattice with <math><mi>L</mi><mo>≈</mo><mn>2.8</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. The pion and nucleon masses are <math><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>255.4</mn><mo>(</mo><mn>1.6</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and <math><msub><mi>m</mi><mi>N</mi></msub><mo>=</mo><mn>1073</mn><mo>(</mo><mn>5</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, respectively, and the strong decay channel <math><mi>Δ</mi><mo>→</mo><mi>π</mi><mi>N</mi></math> is found to be above the threshold. To thoroughly map out the energy dependence of the nucleon-pion scattering amplitude, we compute the spectra in all relevant irreducible representations of the lattice symmetry groups for total momenta up to <math><mrow><mover><mrow><mi>P</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mi>L</mi></mrow></mfrac><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math>, including irreps that mix <math><mi>S</mi></math> and <math><mi>P</mi></math> waves. We perform global fits of the amplitude parameters to up to 21 energy levels, using a Breit-Wigner model for the <math><mi>P</mi></math>-wave phase shift and the effective-range expansion for the <math><mi>S</mi></math>-wave phase shift. From the location of the pole in the <math><mi>P</mi></math>-wave scattering amplitude, we obtain the resonance mass <math><msub><mi>m</mi><mi>Δ</mi></msub><mo>=</mo><mn>1378</mn><mo>(</mo><mn>7</mn><mo>)</mo><mo>(</mo><mn>9</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and the coupling <math><msub><mi>g</mi><mrow><mi>Δ</mi><mo>−</mo><mi>π</mi><mi>N</mi></mrow></msub><mo>=</mo><mn>23.8</mn><mo>(</mo><mn>2.7</mn><mo>)</mo><mo>(</mo><mn>0.9</mn><mo>)</mo></math>.

Giorgio Silvi

Srijit Paul

Cyprus

Constantia Alexandrou

Luka Leskovec

John Negele

Marcus Petschlies

Gumaro Rendon

Italy

Antonino Todaro

<math><mi>P</mi></math>-wave nucleon-pion scattering amplitude in the <math><mi>Δ</mi><mo>(</mo><mn>1232</mn><mo>)</mo></math> channel from lattice QCD

The Dirac field, spin 1/2 particles, is investigated in phase space. The Dirac propagator is defined. The Thermo Field Dynamics (TFD) formalism is used to introduce finite temperature. The energy-momentum tensor is calculated at finite temperature. The Stefan-Boltzmann law is established, and the Casimir effect is calculated for the Dirac field in phase space at zero and finite temperature. A comparative analysis with these results in standard quantum mechanics space is realized.

S.C. Ulhoa

R.G.G. Amorim

J.S. da Cruz Filho

A.F. Santos

F.C. Khanna

Spin-1/2 Particles in Phase Space: Casimir Effect and Stefan-Boltzmann Law at Finite Temperature

It would be very useful to find a way of reducing excited-state effects in lattice QCD calculations of nucleon structure that has a low computational cost. We explore the use of hybrid interpolators, which contain a nontrivial gluonic excitation, in a variational basis together with the standard interpolator with tuned smearing width. Using the clover discretization of the field strength tensor, a calculation using a fixed linear combination of standard and hybrid interpolators can be done using the same number of quark propagators as a standard calculation, making this a cost-effective option. We find that such an interpolator, optimized by solving a generalized eigenvalue problem, reduces excited-state contributions in two-point correlators. However, the effect in three-point correlators, which are needed for computing nucleon matrix elements, is mixed: for some matrix elements such as the tensor charge, excited-state effects are suppressed, whereas for others such as the axial charge, they are enhanced. The results illustrate that the variational method is not guaranteed to reduce the net contribution from excited states except in its asymptotic regime, and suggest that it may be important to use a large basis of interpolators capable of isolating all of the relevant low-lying states.

Nesreen Hasan

Andrew V. Pochinsky

Sergey N. Syritsyn

Excited-state effects in nucleon structure on the lattice using hybrid interpolators

We report on lattice QCD calculations of the nucleon isovector axial, scalar, and tensor charges. Our calculations are performed on two <math><mn>2</mn><mo>+</mo><mn>1</mn></math>-flavor ensembles generated using a 2-HEX-smeared Wilson-clover action at the physical pion mass and lattice spacings <math><mi>a</mi><mo>≈</mo><mn>0.116</mn></math> and 0.093 fm. We use a wide range of source-sink separations—eight values ranging from roughly 0.4 to 1.4 fm on the coarse ensemble and three values from 0.9 to 1.5 fm on the fine ensemble—which allows us to perform an extensive study of excited-state effects using different analysis and fit strategies. To determine the renormalization factors, we use the nonperturbative Rome-Southampton approach and compare <math><mrow><msup><mrow><mi>RI</mi></mrow><mrow><mo>′</mo></mrow></msup><mtext>−</mtext><mi>MOM</mi></mrow></math> and RI-SMOM intermediate schemes to estimate the systematic uncertainties. Our final results are computed in the <math><mrow><mover><mrow><mi>MS</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> scheme at scale 2 GeV. The tensor and axial charges have uncertainties of roughly 4%, <math><msub><mi>g</mi><mi>T</mi></msub><mo>=</mo><mn>0.972</mn><mo>(</mo><mn>41</mn><mo>)</mo></math> and <math><msub><mi>g</mi><mi>A</mi></msub><mo>=</mo><mn>1.265</mn><mo>(</mo><mn>49</mn><mo>)</mo></math>. The resulting scalar charge, <math><msub><mi>g</mi><mi>S</mi></msub><mo>=</mo><mn>0.927</mn><mo>(</mo><mn>303</mn><mo>)</mo></math>, has a much larger uncertainty due to a stronger dependence on the choice of intermediate renormalization scheme and on the lattice spacing.

Jeremy Green

Nucleon axial, scalar, and tensor charges using lattice QCD at the physical pion mass

Gravitons are described by the propagator in teleparallel gravity in nearly flat space-time. Finite temperature is introduced by using Thermofield Dynamics formalism. The gravitational Casimir effect and Stefan-Boltzmann law are calculated as a function of temperature. Then an equation of state for gravitons is determined.

T.F. Furtado

On Gravitational Casimir Effect and Stefan-Boltzmann Law at Finite Temperature

We use the scalar field constructed in phase space to analyze the analogous Stefan-Boltzmann law and Casimir effect, both of them at finite temperature. The temperature is introduced by Thermo Field Dynamics (TFD) formalism and the quantities are analyzed once projected in the space of coordinates. We show that using the framework of phase space it is possible to introduce a thermal energy which is related to temperature as it vanishes when the temperature tends to zero. In fact given such a correlation the formalism of TFD is equivalent when project is in momenta space when compared to coordinates space.

Stefan-Boltzmann Law and Casimir Effect for the Scalar Field in Phase Space at Finite Temperature

We describe a procedure for extracting momentum derivatives of nucleon matrix elements on the lattice directly at <math><msup><mi>Q</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math>. This is based on the Rome method for computing momentum derivatives of quark propagators. We apply this procedure to extract the nucleon isovector magnetic moment and charge radius as well as the isovector induced pseudoscalar form factor at <math><msup><mi>Q</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math> and the axial radius. For comparison, we also determine these quantities with the traditional approach of computing the corresponding form factors, i.e. <math><mrow><msubsup><mrow><mi>G</mi></mrow><mrow><mi>E</mi></mrow><mrow><mi>v</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></math> and <math><msubsup><mi>G</mi><mi>M</mi><mi>v</mi></msubsup><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> for the case of the vector current and <math><msubsup><mi>G</mi><mi>P</mi><mi>v</mi></msubsup><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> and <math><msubsup><mi>G</mi><mi>A</mi><mi>v</mi></msubsup><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> for the axial current, at multiple <math><msup><mi>Q</mi><mn>2</mn></msup></math> values followed by <math><mi>z</mi></math>-expansion fits. We perform our calculations at the physical pion mass using a 2HEX-smeared Wilson-clover action. To control the effects of excited-state contamination, the calculations were done at three source-sink separations and the summation method was used. The derivative method produces results consistent with those from the traditional approach but with larger statistical uncertainties especially for the isovector charge and axial radii.

Computing the nucleon charge and axial radii directly at <math><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>0</mn></mrow></math> in lattice QCD

It has recently been suggested (S.B. Giddings (2016) [2] ) that the Hawking black-hole radiation spectrum originates from an effective quantum “atmosphere' which extends well outside the black-hole horizon. In particular, comparing the Hawking radiation power of a (3+1) -dimensional Schwarzschild black hole of horizon radius rH with the familiar Stefan–Boltzmann radiation power of a (3+1) -dimensional flat space perfect blackbody emitter, Giddings concluded that the source of the Hawking semi-classical black-hole radiation is a quantum region outside the Schwarzschild black-hole horizon whose effective radius rA is characterized by the relation Δr≡rA−rH∼rH . It is of considerable physical interest to test the general validity of Giddings's intriguing conclusion. To this end, we study the Hawking radiation of (D+1) -dimensional Schwarzschild black holes. We find that the dimensionless radii rA/rH which characterize the black-hole quantum atmospheres, as determined from the Hawking black-hole radiation power and the (D+1) -dimensional Stefan–Boltzmann radiation law, are a decreasing function of the number D+1 of spacetime dimensions. In particular, it is shown that radiating (D+1) -dimensional Schwarzschild black holes are characterized by the relation (rA−rH)/rH≪1 in the large D≫1 regime. Our results therefore suggest that, at least in some physical cases, the Hawking emission spectrum originates from quantum excitations very near the black-hole horizon.

Israel

Shahar Hod

Hawking radiation and the Stefan–Boltzmann law: The effective radius of the black-hole quantum atmosphere