Spain

Nikolai Husung

Germany

Peter Marquard

Rainer Sommer

Corrigendum to “The asymptotic approach to the continuum of lattice QCD spectral observables” [Phys. Lett. B 829 (2022) 137069]

We consider spectral quantities in lattice QCD and determine the asymptotic behaviour of their discretization errors. Wilson fermion with <math><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></math>-improvement, (Möbius) Domain wall fermion (DWF), and overlap Dirac operators are considered in combination with the commonly used gauge actions. Wilson fermions and DWF with domain wall height <math><msub><mrow><mi>M</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>=</mo><mn>1</mn><mo>+</mo><mi>O</mi><mo>(</mo><msubsup><mrow><mi>g</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></math> have the same, approximate, form of the asymptotic cutoff effects: <math><mi>K</mi><mspace width="0.2em"></mspace><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>[</mo><msup><mrow><mover><mrow><mi>g</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo><mo>]</mo></mrow><mrow><mn>0.760</mn></mrow></msup></math>. A domain wall height <math><msub><mrow><mi>M</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>=</mo><mn>1.8</mn></math>, as often used, introduces large mass-dependent <math><msup><mrow><mi>K</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>m</mi><mo>)</mo><mspace width="0.2em"></mspace><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>[</mo><msup><mrow><mover><mrow><mi>g</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo><mo>]</mo></mrow><mrow><mn>0.518</mn></mrow></msup></math> effects. Massless twisted mass fermions have the same form as Wilson fermions when the Sheikholeslami-Wohlert term [1] is included. For their mass-dependent cutoff effects we have information on the exponents <math><msub><mrow><mover><mrow><mi>Γ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub></math> of <math><msup><mrow><mover><mrow><mi>g</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math> but not for the pre-factors. For staggered fermions there is only partial information on the exponents.We propose that tree-level <math><mi>O</mi><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> improvement, which is easy to do [2], should be used in the future – both for the fermion and the gauge action. It improves the asymptotic behaviour in all cases.

The asymptotic approach to the continuum of lattice QCD spectral observables

The QCD/HQET matching coefficient for the heavy-quark field is calculated up to four loops. It must be finite; this requirement produces analytical results for some terms in the four-loop on-shell heavy-quark field renormalization constant which were previously only known numerically. The effect of a nonzero lighter-flavor mass is calculated up to three loops. A class of on-shell integrals with two masses is analyzed in detail. By specifying our result to QED, we obtain the relation between the electron field and the Bloch–Nordsieck field with four-loop accuracy.

Russia

Andrey G. Grozin

Alexander V. Smirnov

Vladimir A. Smirnov

Matthias Steinhauser

Matching the heavy-quark fields in QCD and HQET at four loops

Discretization effects of lattice QCD are described by Symanzik’s effective theory when the lattice spacing, a, is small. Asymptotic freedom predicts that the leading asymptotic behavior is $$\sim a^{n_{\mathrm{min}}}[{\bar{g}}^2(a^{-1})]^{\hat{\gamma }_1} \sim a^{n_{\mathrm{min}}}\left[ \frac{1}{-\log (a\Lambda )}\right] ^{\hat{\gamma }_1}$$ <math><mrow><mo>∼</mo><msup><mi>a</mi><msub><mi>n</mi><mi>min</mi></msub></msup><msup><mrow><mo>[</mo><msup><mrow><mover><mrow><mi>g</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mn>2</mn></msup><mrow><mo>(</mo><msup><mi>a</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mo>]</mo></mrow><msub><mover><mi>γ</mi><mo>^</mo></mover><mn>1</mn></msub></msup><mo>∼</mo><msup><mi>a</mi><msub><mi>n</mi><mi>min</mi></msub></msup><msup><mfenced><mfrac><mn>1</mn><mrow><mo>-</mo><mo>log</mo><mo>(</mo><mi>a</mi><mi>Λ</mi><mo>)</mo></mrow></mfrac></mfenced><msub><mover><mi>γ</mi><mo>^</mo></mover><mn>1</mn></msub></msup></mrow></math> . For spectral quantities, $${n_{\mathrm{min}}}=d$$ <math><mrow><msub><mi>n</mi><mi>min</mi></msub><mo>=</mo><mi>d</mi></mrow></math>  is given in terms of the (lowest) canonical dimension, $$d+4$$ <math><mrow><mi>d</mi><mo>+</mo><mn>4</mn></mrow></math> , of the operators in the local effective Lagrangian and $$\hat{\gamma }_1$$ <math><msub><mover><mi>γ</mi><mo>^</mo></mover><mn>1</mn></msub></math>  is proportional to the leading eigenvalue of their one-loop anomalous dimension matrix $$\gamma ^{(0)}$$ <math><msup><mi>γ</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></msup></math> . We determine $$\gamma ^{(0)}$$ <math><msup><mi>γ</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></msup></math>  for Yang–Mills theory ($${n_{\mathrm{min}}}=2$$ <math><mrow><msub><mi>n</mi><mi>min</mi></msub><mo>=</mo><mn>2</mn></mrow></math> ) and discuss consequences in general and for perturbatively improved short distance observables. With the help of results from the literature, we also discuss the $${n_{\mathrm{min}}}=1$$ <math><mrow><msub><mi>n</mi><mi>min</mi></msub><mo>=</mo><mn>1</mn></mrow></math>  case of Wilson fermions with perturbative $$\mathrm{O}(a)$$ <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math>  improvement and the discretization effects specific to the flavor currents. In all cases known so far, the discretization effects are found to vanish faster than the naive $$\sim a^{n_{\mathrm{min}}}$$ <math><mrow><mo>∼</mo><msup><mi>a</mi><msub><mi>n</mi><mi>min</mi></msub></msup></mrow></math>  behavior with rather small logarithmic corrections – in contrast to the two-dimensional O(3) sigma model.

Asymptotic behavior of cutoff effects in Yang–Mills theory and in Wilson’s lattice QCD

We compute the on-shell wave function renormalization constant to four-loop order in QCD and present numerical results for all coefficients of the <math><mi>SU</mi><mo>(</mo><msub><mi>N</mi><mi>c</mi></msub><mo>)</mo></math> color factors. We extract the four-loop Heavy Quark Effective Theory anomalous dimension of the heavy-quark field and also discuss the application of our result to QED.

Four-loop wave function renormalization in QCD and QED

The heavy-quark contribution to the polarization function <math><mrow><mi>Π</mi></mrow></math> at higher perturbative orders is presently only known approximately. We scrutinize the accuracy of state-of-the-art approximations at three- and four-loop order. At three loops, we present for the first time a result with arbitrary numerical precision for general kinematics and compare to the best Padé estimate. At four loops, we calculate the fourth (inverse) moment of the nonsinglet heavy-quark vacuum polarization in order to test the prediction for this moment based on the Padé approximation.

Andreas Maier

Validity of Padé approximations in vacuum polarization at three- and four-loop order

We focus on a non-abelian gauge field coupled to a single (but general) representation of a family of N f fermions. By using the same machinery that had allowed us to evaluate the sub-leading large- N f term of the five-loop Beta function earlier, we here report on a confirmation of the all- N f result that has in the meantime been published by another group. Furthermore, in order to push forward the 5-loop renormalization program regarding gauge parameter dependence, we present the linear terms of the complete set of anomalous dimensions, in an expansion in the covariant gauge parameter around the Feynman gauge.

Thomas Luthe

United Kingdom

Chile

York Schröder

The five-loop Beta function for a general gauge group and anomalous dimensions beyond Feynman gauge

We present new analytical five-loop Feynman-gauge results for the anomalous dimensions of ghost field and -vertex, generalizing the known values for SU(3) to a general gauge group. Together with previously published results on the quark mass and -field anomalous dimensions and the Beta function, this completes the 5-loop renormalization program of gauge theories in that gauge.

Complete renormalization of QCD at five loops

We present analytical five-loop results for the quark mass and quark field anomalous dimensions, for a general gauge group and in the M S ¯ $$ \overline{\mathrm{MS}} $$ scheme. We confirm the values known for the gauge group SU(3) from an independent calculation, and find full agreement with results available from large- N f studies.

Five-loop quark mass and field anomalous dimensions for a general gauge group

We present analytical results for the N f 4 and N f 3 terms of the five-loop Beta function, for a general gauge group. While the former term agrees with results available from large- N f studies, the latter is new and extends the value known for SU(3) from an independent calculation.