<p>We perform a precise calculation of physical quantities related to the axial structure of the nucleon using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor lattice QCD gauge configuration (PACS10 configuration) generated at the physical point with lattice volume larger than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mn>10</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn></mrow></msup></mrow></math> by the PACS Collaboration. The nucleon matrix element of the axial-vector current has two types of the nucleon form factors, the axial-vector (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>A</mi></msub></math>) form factor and the induced pseudoscalar (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>P</mi></msub></math>) form factor. Recently, lattice QCD simulations have succeeded in reproducing the experimental value of the axial-vector coupling, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>g</mi><mi>A</mi></msub></math>, determined from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> at zero momentum transfer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math>, at a percent level of statistical accuracy. However, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>P</mi></msub></math> form factor so far has not reproduced the experimental values well due to strong <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mi>N</mi></math> excited-state contamination. Therefore, we propose a simple subtraction method for removing the so-called leading <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mi>N</mi></math>-state contribution. This method succeeds in reproducing the values obtained by two experiments of muon capture on the proton and pion electroproduction for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>P</mi></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>. The novel approach can also be applied to determine the pseudoscalar (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>G</mi><mi>P</mi></msub></math>) form factor from the nucleon pseudoscalar matrix element with the help of the axial Ward-Takahashi identity. The resulting form factors, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>P</mi></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>G</mi><mi>P</mi></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>, are in good agreement with the prediction of the pion-pole dominance model. In the new analysis, the induced pseudoscalar coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>g</mi><mi>P</mi><mo>*</mo></msubsup></math> and the pion-nucleon coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>g</mi><mrow><mi>π</mi><mi>N</mi><mi>N</mi></mrow></msub></math> can be evaluated with a few percent accuracy including systematic uncertainties using existing data calculated at two lattice spacings.</p>

Japan

Yasumichi Aoki

Ken-Ichi Ishikawa

Yoshinobu Kuramashi

Shoichi Sasaki

Kohei Sato

Eigo Shintani

Ryutaro Tsuji

Hiromasa Watanabe

Takeshi Yamazaki

Method for high-precision determination of the nucleon axial structure using lattice QCD: Removing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>π</mi><mi>N</mi></mrow></math>-state contamination

We investigate the phase diagram of the three-dimensional SU(3) spin model with finite chemical potential, which is an effective Polyakov loop model for finite density QCD, using the tensor renormalization group method. We successfully determine the location of the critical endpoints being free from the complex action problem in the standard Monte Carlo approach. The critical values of the parameters show the consistency with previous ones obtained by other analytic and numerical methods.

Xiao Luo

Critical endpoints of three-dimensional finite density SU(3) spin model with tensor renormalization group

We study the quantum phase transition of the (1+1)-dimensional O(3) nonlinear sigma model at finite density using the tensor renormalization group method. This model suffers from the sign problem, which has prevented us from investigating the properties of the phase transition. We investigate the properties of the phase transition by changing the chemical potential $\textit{μ}$ at a fixed coupling of $\textit{β}$. We determine the transition point $\textit{μ}$ and the critical exponent $\textit{ν}$ from the $\textit{μ}$ dependence of the number density in the thermodynamic limit. The dynamical critical exponent $\textit{z}$ is also extracted from the scaling behavior of the temporal correlation length as a function of $\textit{μ}$.

Quantum phase transition of (1+1)-dimensional O(3) nonlinear sigma model at finite density with tensor renormalization group

We investigate the phase structure of the (1+1)-dimensional U(1) gauge-Higgs model with a θ term, where the U(1) gauge action is constructed with Lüscher’s admissibility condition. Using the tensor renormalization group, both the complex action problem and topological freezing problem in the standard Monte Carlo simulation are avoided. We find the first-order phase transition with sufficiently large Higgs mass at θ = π, where the ℤ2 charge conjugation symmetry is spontaneously broken. On the other hand, the symmetry is restored with a sufficiently small mass. We determine the critical endpoint as a function of the Higgs mass parameter and show the critical behavior is in the two-dimensional Ising universality class.

Shinichiro Akiyama

Tensor renormalization group study of (1 + 1)-dimensional U(1) gauge-Higgs model at θ = π with Lüscher’s admissibility condition

We present results for the nucleon form factors: electric (<math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>E</mi></mrow></msub></mrow></math>), magnetic (<math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></math>), axial (<math><msub><mi>F</mi><mi>A</mi></msub></math>), induced pseudoscalar (<math><msub><mi>F</mi><mi>P</mi></msub></math>), and pseudoscalar (<math><msub><mi>G</mi><mi>P</mi></msub></math>) form factors, using the second PACS10 ensemble that is one of three sets of <math><mn>2</mn><mo>+</mo><mn>1</mn></math> flavor lattice QCD configurations at physical quark masses in large spatial volumes [exceeding <math><mrow><mo>(</mo><mn>10</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup></mrow></math>]. The second PACS10 gauge configurations are generated by the PACS Collaboration with the six stout-smeared <math><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></math> improved Wilson quark action and Iwasaki gauge action at the second gauge coupling <math><mi>β</mi><mo>=</mo><mn>2.00</mn></math> corresponding to the lattice spacing of <math><mi>a</mi><mo>=</mo><mn>0.063</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. We determine the isovector electric, magnetic and axial radii, and magnetic moment from the corresponding form factors, as well as the axial-vector coupling <math><msub><mi>g</mi><mi>A</mi></msub></math>. Combining our previous results for the coarser lattice spacing [E. Shintani et al., Phys. Rev. D 99, 014510 (2019); Phys. Rev. D 102, 019902(E) (2020)], the finite lattice spacing effects on the isovector radii, magnetic moment, and axial-vector coupling are investigated using the difference between the two results. It was found that the effect on <math><msub><mi>g</mi><mi>A</mi></msub></math> is kept smaller than the statistical error of 2% while the effect on the isovector radii was observed as a possible discretization error of about 10%, regardless of the channel. We also report the partially conserved axial-vector current relation using a set of nucleon three-point correlation functions in order to verify the effect by <math><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></math> improvement of the axial-vector current.

Nucleon form factors in <math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> lattice QCD at the physical point: Finite lattice spacing effect on the root-mean-square radii

We investigate the entanglement and Rényi entropies for the (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method. The central charge is determined from the asymptotic scaling properties of both entropies. We also examine the consistency between the entanglement entropy and the nth-order Rényi entropy with n → 1.

Entanglement and Rényi entropies of (1+1)-dimensional O(3) nonlinear sigma model with tensor renormalization group

The critical endpoint of the (3+1)-dimensional ℤ3 gauge-Higgs model at finite density is determined by the tensor renormalization group method. This work is an extension of the previous one on the ℤ2 model. The vital difference between them is that the ℤ3 model suffers from the sign problem, while the ℤ2 model does not. We show that the tensor renormalization group method allows us to locate the critical endpoint for the ℤ3 gauge-Higgs model at finite density, regardless of the sign problem.

Critical endpoint of (3+1)-dimensional finite density ℤ3 gauge-Higgs model with tensor renormalization group

We apply the tensor renormalization group method to the (<math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional SU(2) principal chiral model at finite chemical potential with the use of the Gauss-Legendre quadrature to discretize the SU(2) Lie group. The internal energy at vanishing chemical potential <math><mi>μ</mi><mo>=</mo><mn>0</mn></math> shows good consistency with the prediction of the strong and weak coupling expansions. This indicates an effectiveness of the Gauss-Legendre quadrature for the partitioning of the SU(2) Lie group. In the finite density region with <math><mi>μ</mi><mo>≠</mo><mn>0</mn></math> at the strong coupling we observe the Silver-Blaze phenomenon for the number density.

Tensor renormalization group approach to (<math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional SU(2) principal chiral model at finite density

We present results for the scalar and tensor isovector-couplings (<math><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></math> and <math><msub><mi>g</mi><mi>T</mi></msub></math>) of the nucleon measured at the physical point (<math><msub><mi>M</mi><mi>π</mi></msub><mo>=</mo><mn>135</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>) with a single lattice spacing of 0.085 fm in <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor QCD. Our calculations are carried out with two ensembles of gauge configurations generated by the PACS Collaboration with nonperturbatively <math><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></math> improved Wilson quark action and Iwasaki gauge action on <math><mo>(</mo><mn>10.9</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi><msup><mo>)</mo><mn>4</mn></msup></math> and <math><mo>(</mo><mn>5.5</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi><msup><mo>)</mo><mn>4</mn></msup></math> lattices, where the finite-size effect on the nucleon mass was not shown at the level of statistical precision less than 0.5%. We compute the nucleon three-point correlation functions in the vector, axial, scalar, and tensor channels. We confirm that our previous result of the nucleon axial coupling on the large spatial volume of <math><mo>(</mo><mn>10.9</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi><msup><mo>)</mo><mn>4</mn></msup></math> has no finite-size effect at the level of the statistical precision of 1.9%. For the renormalization, we first renormalize <math><msub><mi>g</mi><mi>S</mi></msub></math> and <math><msub><mi>g</mi><mi>T</mi></msub></math> nonperturbatively using the <math><mrow><mi>RI</mi><mo>/</mo><msub><mrow><mi>SMOM</mi></mrow><mrow><mo>(</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>)</mo></mrow></msub></mrow></math> scheme, a variant of Rome-Southampton RI/MOM scheme with reduced systematic errors, as the intermediate scheme. We evaluate our final results at the renormalization scale of 2 GeV in the <math><mover><mi>MS</mi><mo>¯</mo></mover></math> scheme through the matching procedure between the <math><mi>RI</mi><mo>/</mo><msub><mrow><mi>SMOM</mi></mrow><mrow><mo>(</mo><msub><mi>γ</mi><mi>μ</mi></msub><mo>)</mo></mrow></msub></math> and <math><mover><mi>MS</mi><mo>¯</mo></mover></math> schemes with the help of perturbation theory, and then obtain <math><msub><mi>g</mi><mi>S</mi></msub><mo>=</mo><mn>0.927</mn><mo>(</mo><mn>83</mn><msub><mo>)</mo><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>22</mn><msub><mo>)</mo><mrow><mi>syst</mi></mrow></msub></math> and <math><msub><mi>g</mi><mi>T</mi></msub><mo>=</mo><mn>1.036</mn><mo>(</mo><mn>6</mn><msub><mo>)</mo><mrow><mi>stat</mi></mrow></msub><mo>(</mo><mn>20</mn><msub><mo>)</mo><mrow><mi>syst</mi></mrow></msub></math>.

Natsuki Tsukamoto

Nucleon isovector couplings in <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> lattice QCD at the physical point

We present updated results for the form factors of the kaon semileptonic <math><mo>(</mo><msub><mi>K</mi><mrow><mo>ℓ</mo><mn>3</mn></mrow></msub><mo>)</mo></math> decay process calculated with <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> nonperturbatively <math><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></math>-improved Wilson quark action and Iwasaki gauge action at the physical point on large volumes of more than <math><mrow><mo>(</mo><mn>10</mn><mtext> </mtext><mi>fm</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup></mrow></math>. In addition to our previous calculation at the lattice spacing <math><mi>a</mi><mo>=</mo><mn>0.085</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, we perform a calculation at the second lattice spacing of 0.063 fm. Using the results for the form factors extracted from 3-point functions with the local and also conserved vector currents at the two lattice spacings, continuum extrapolation and interpolation of the momentum transfer are carried out simultaneously to obtain the value of the form factor <math><msub><mi>f</mi><mo>+</mo></msub><mo>(</mo><mn>0</mn><mo>)</mo></math> at the zero momentum transfer in the continuum limit. After investigation of stability of <math><msub><mi>f</mi><mo>+</mo></msub><mo>(</mo><mn>0</mn><mo>)</mo></math> against several fit forms and different data, we obtain <math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0.9615</mn><mo>(</mo><mn>10</mn><mo>)</mo><mo>(</mo><msubsup><mrow><mtext> </mtext></mrow><mrow><mo>−</mo><mn>3</mn></mrow><mrow><mo>+</mo><mn>47</mn></mrow></msubsup><mo>)</mo><mo>(</mo><mn>5</mn><mo>)</mo></mrow></math>, where the first, second, and third errors are statistical, systematic errors from choice of the fit forms and isospin breaking effect, respectively. Furthermore, we obtain the slope and curvature of the form factors, and the phase space integral from the momentum transfer dependence of the form factors. Combining our value of <math><msub><mi>f</mi><mo>+</mo></msub><mo>(</mo><mn>0</mn><mo>)</mo></math> and experimental input of the <math><msub><mi>K</mi><mrow><mo>ℓ</mo><mn>3</mn></mrow></msub></math> decay, one of the Cabibbo-Kobayashi-Maskawa matrix elements <math><mo>|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>s</mi></mrow></msub><mo>|</mo></math> is determined as <math><mrow><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>u</mi><mi>s</mi></mrow></msub><mo>|</mo><mo>=</mo><mn>0.2252</mn><mo>(</mo><msubsup><mrow><mtext> </mtext></mrow><mrow><mo>−</mo><mn>12</mn></mrow><mrow><mo>+</mo><mn>5</mn></mrow></msubsup><mo>)</mo></mrow></math>, whose error contains the experimental one as well as that in the lattice calculation. This value is reasonably consistent with the ones determined from recent lattice QCD results of <math><msub><mi>f</mi><mo>+</mo></msub><mo>(</mo><mn>0</mn><mo>)</mo></math> and also the one determined through the kaon leptonic decay process. We observe some tension between our value and <math><mo>|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>s</mi></mrow></msub><mo>|</mo></math> evaluated from the unitarity of the CKM matrix with <math><mo>|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>d</mi></mrow></msub><mo>|</mo></math>, while it depends on the size of the error of <math><mo>|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>d</mi></mrow></msub><mo>|</mo></math>. It is also found that <math><mo>|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>s</mi></mrow></msub><mo>|</mo></math> determined with our phase space integrals through six <math><msub><mi>K</mi><mrow><mo>ℓ</mo><mn>3</mn></mrow></msub></math> decay processes is consistent with the above one using <math><msub><mi>f</mi><mo>+</mo></msub><mo>(</mo><mn>0</mn><mo>)</mo></math>.

Ken-ichi Ishikawa

Naruhito Ishizuka

Yusuke Namekawa

Yusuke Taniguchi

Naoya Ukita

Tomoteru Yoshié

<math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mo>ℓ</mo><mn>3</mn></mrow></msub></mrow></math> form factors at the physical point: Toward the continuum limit