We obtain an improved determination of the normalization of the leading infrared renormalon of the Bjorken sum rule: <math><msub><mi>Z</mi><mi>B</mi></msub><mo>(</mo><msub><mi>n</mi><mi>f</mi></msub><mo>=</mo><mn>3</mn><mo>)</mo><mo>=</mo><mo>−</mo><mn>0.407</mn><mo>±</mo><mn>0.119</mn></math>. Estimates of higher order terms of the perturbative series are given. We compute the Bjorken sum rule with hyperasymptotic precision by including the leading terminant, associated with the first infrared renormalon. We fit the experimental data to the operator product expansion theoretical prediction with <math><msubsup><mover><mi>f</mi><mo>^</mo></mover><mn>3</mn><mrow><mi>PV</mi></mrow></msubsup></math> as the free parameter. We obtain a good agreement with the experiment with <math><mrow><msubsup><mrow><mover><mrow><mi>f</mi></mrow><mrow><mo>^</mo></mrow></mover></mrow><mrow><mn>3</mn></mrow><mrow><mi>PV</mi></mrow></msubsup><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mn>3</mn><msubsup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mn>196</mn></mrow><mrow><mo>+</mo><mn>187</mn></mrow></msubsup><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> for <math><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≥</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>.

Chile

Cesar Ayala

Spain

Antonio Pineda

Bjorken sum rule with hyperasymptotic precision

We give the hyperasymptotic expansion of the plaquette with a precision that includes the terminant associated to the leading renormalon. Subleading effects are also considered. The perturbative series is regulated using the principal value prescription for its Borel integral. We use this analysis to give a determination of the gluon condensate in SU(3) pure gluodynamics that is independent of the scale and renormalization scheme used for the coupling constant:   <math> <msub> <mfenced> <msup> <mi>G</mi> <mn>2</mn> </msup> </mfenced> <mi>PV</mi> </msub> <mfenced> <mrow> <msub> <mi>n</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mfenced> <mo>=</mo> <mn>3.15</mn> <mfenced> <mn>18</mn> </mfenced> <msubsup> <mi>r</mi> <mn>0</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msubsup> </math>  $$ {\left\langle {G}^2\right\rangle}_{\mathrm{PV}}\left({n}_f=0\right)=3.15(18){r}_0^{-4} $$ .

Xabier Lobregat

Hyperasymptotic approximation to the plaquette and determination of the gluon condensate

We give the hyperasymptotic expansion of the energy of a static quark-antiquark pair with a precision that includes the effects of the subleading renormalon. The terminants associated to the first and second renormalon are incorporated in the analysis when necessary. In particular, we determine the normalization of the leading renormalon of the force and, consequently, of the subleading renormalon of the static potential. We obtain   <math> <msubsup> <mi>Z</mi> <mn>3</mn> <mi>F</mi> </msubsup> </math>  $$ {Z}_3^F $$  (n  f  = 3) =   <math> <mn>2</mn> <msubsup> <mi>Z</mi> <mn>3</mn> <mi>V</mi> </msubsup> </math>  $$ 2{Z}_3^V $$  (n  f  = 3) = 0.37(17). The precision we reach in strict perturbation theory is next-to-next-to-next-to-leading logarithmic resummed order both for the static potential and for the force. We find that the resummation of large logarithms and the inclusion of the leading terminants associated to the renormalons are compulsory to get accurate determinations of   <math> <msub> <mi>Λ</mi> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </msub> </math>  $$ {\Lambda}_{\overline{\mathrm{MS}}} $$  when fitting to short-distance lattice data of the static energy. We obtain   <math> <msubsup> <mi>Λ</mi> <mover> <mi>MS</mi> <mo>¯</mo> </mover> <mfenced> <mrow> <msub> <mi>n</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </mfenced> </msubsup> </math>  $$ {\Lambda}_{\overline{\mathrm{MS}}}^{\left({n}_f=3\right)} $$  = 338(12) MeV and α(M  z ) = 0.1181(9). We have also MS found strong consistency checks that the ultrasoft correction to the static energy can be computed at weak coupling in the energy range we have studied.

Determination of α(M  z ) from an hyperasymptotic approximation to the energy of a static quark-antiquark pair

We construct hyperasymptotic expansions for the heavy quark pole mass regulated using the principal value (PV) prescription. We apply such hyperasymptotic expansions to the <math><mi>B</mi><mo>/</mo><mi>D</mi></math> meson masses, and <math><mover><mi>Λ</mi><mo>¯</mo></mover></math> computed in the lattice. The issue of the uncertainty of the (top) pole mass is critically reexamined. The present theoretical uncertainty in the relation between <math><msub><mover><mi>m</mi><mo>¯</mo></mover><mi>t</mi></msub></math>, the <math><mrow><mover><mrow><mi>MS</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> top mass, and <math><msub><mi>m</mi><mrow><mi>t</mi><mo>,</mo><mi>PV</mi></mrow></msub></math>, the top pole mass regulated using the PV prescription, is numerically assessed to be <math><mi>δ</mi><msub><mi>m</mi><mrow><mi>t</mi><mo>,</mo><mi>PV</mi></mrow></msub><mo>=</mo><mn>28</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> for <math><msub><mover><mi>m</mi><mo>¯</mo></mover><mi>t</mi></msub><mo>=</mo><mn>163</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>.

Hyperasymptotic approximation to the top, bottom, and charm pole mass

We compute hadronic transitions between heavy quarkonium states with two, or one, pion/eta particles in the final state. We use the multipole expansion but not the twist expansion. The latter cannot be justified for the energy release of hadronic transitions between heavy quarkonium states with different principal quantum numbers. Instead, we use a counting based on the dimension of the interpolating field of the hybrid. This alternative counting allows us to still use chiral low-energy theorems to compute the pion production by local gluonic operators. We explore the phenomenological impact of this counting. Remarkably enough, for the two-pion transitions, we obtain the same predictions for the normalized differential decay rate as those obtained assuming the twist expansion. We implement this computational scheme using the hadronic representation of the effective theory potential NRQCD. We assume that the inverse Bohr radius of the heavy quarkonium is much larger than <math><msub><mi>Λ</mi><mi>QCD</mi></msub></math> but do not impose any constraint on the relative size of <math><msub><mi>Λ</mi><mi>QCD</mi></msub></math> and the typical kinetic energy of the bound state.

Jaume Tarrús Castellà

Novel implementation of the multipole expansion to quarkonium hadronic transitions

Given an observable and its operator product expansion, we present expressions that carefully disentangle truncated sums of the perturbative series in powers of <math><mi>α</mi></math> from the nonperturbative (NP) corrections. This splitting is done with NP power accuracy. Analytic control of the splitting is achieved and the organization of the different terms is done along an super/hyperasymptotic expansion. As a test we apply the methods to the static potential in the large <math><msub><mi>β</mi><mn>0</mn></msub></math> approximation. We see the superasymptotic and hyperasymptotic structure of the observable in full glory.

Superasymptotic and hyperasymptotic approximation to the operator product expansion

We compute the heavy quarkonium mass of <math><mi>l</mi><mo>≠</mo><mn>0</mn></math> (angular momentum) states, with otherwise arbitrary quantum numbers, with next-next-to-next-to-leading logarithmic (<math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>LL</mi></mrow></math>) accuracy. This constitutes the first observable in heavy quarkonium for which two orders of the weak-coupling expansion sensitive to the ultrasoft scale are known and the resummation of ultrasoft logarithms is made. We also obtain, for the first time, resummed <math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>LL</mi></mrow></math> expressions for the different fine and hyperfine energy splittings of these states, which are not sensitive to the ultrasoft scale but still require resummation of (hard) logarithms. We do this analysis for the equal and non-equal mass cases. We also study an alternative computational scheme that treats the static potential exactly. We then perform a comprehensive phenomenological analysis: we apply these results to the <math><mi>n</mi><mo>=</mo><mn>2</mn></math>, <math><mi>l</mi><mo>=</mo><mn>1</mn></math> bottomonium, <math><msub><mi>B</mi><mi>c</mi></msub></math> and charmonium systems and study their convergence.

Germany

Clara Peset

Jorge Segovia

<math><mrow><mi>P</mi></mrow></math>-wave heavy quarkonium spectrum with next-to-next-to-next-to-leading logarithmic accuracy

We determine the charm and bottom quark masses using the N3LO perturbative expression of the ground state (pseudoscalar) energy of the bottomonium, charmonium and B  c  systems: the η  b , η  c  and B  c  masses. We work in the renormalon subtracted scheme, which allows us to control the divergence of the perturbation series due to the pole mass renormalon. Our result for the  <math><mover><mrow><mi>M</mi><mi>S</mi></mrow><mo>¯</mo></mover></math> $$ \overline{\mathrm{MS}} $$ masses reads  <math><msub><mover><mi>m</mi><mo>¯</mo></mover><mi>c</mi></msub><mfenced><msub><mover><mi>m</mi><mo>¯</mo></mover><mi>c</mi></msub></mfenced><mo>=</mo><mn>1223</mn><mfenced><mn>33</mn></mfenced></math> $$ {\overline{m}}_c\left({\overline{m}}_c\right)=1223(33) $$ MeV and  <math><msub><mover><mi>m</mi><mo>¯</mo></mover><mi>b</mi></msub><mfenced><msub><mover><mi>m</mi><mo>¯</mo></mover><mi>b</mi></msub></mfenced><mo>=</mo><mn>4186</mn><mfenced><mn>37</mn></mfenced></math> $$ {\overline{m}}_b\left({\overline{m}}_b\right)=4186(37) $$ MeV. We also extract a value of α  s  from a renormalon-free combination of the η  b , η  c  and B  c  masses: α  s (M  z ) = 0.1195(53). We explore the applicability of the weak coupling approximation to bottomonium n = 2 states. Finally, we consider an alternative computational scheme that treats the static potential exactly and study its convergence properties.

The charm/bottom quark mass from heavy quarkonium at N3LO

We obtain the renormalization group improved expressions of the Wilson coefficients of the heavy quark effective theory Lagrangian with leading logarithmic approximation to <math><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><msup><mi>m</mi><mn>3</mn></msup><mo>)</mo></math> for the spin-independent sector, which includes the heavy quark chromopolarizabilities. Our analysis includes the effects induced by spectator quarks. We observe that the numerical impact of these logarithms is very large in most cases.

Daniel Moreno

Chromopolarizabilities of a heavy quark at weak coupling

We consider the $${\mathcal {O}}(1/m)$$ <math><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>m</mi><mo>)</mo></mrow></math>  and the spin-independent momentum-dependent $${\mathcal {O}}(1/m^2)$$ <math><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><msup><mi>m</mi><mn>2</mn></msup><mo>)</mo></mrow></math>  quasi-static energies of heavy quarkonium (with unequal masses). They are defined nonperturbatively in terms of Wilson loops. We determine their short-distance behavior through $$\mathcal{O}(\alpha ^3)$$ <math><mrow><mi>O</mi><mo>(</mo><msup><mi>α</mi><mn>3</mn></msup><mo>)</mo></mrow></math>  and $$\mathcal{O}(\alpha ^2)$$ <math><mrow><mi>O</mi><mo>(</mo><msup><mi>α</mi><mn>2</mn></msup><mo>)</mo></mrow></math> , respectively. In particular, we calculate the ultrasoft contributions to the quasi-static energies, which requires the resummation of potential interactions. Our results can be directly compared to lattice simulations. In addition, we also compare the available lattice data with the expectations from effective string models for the long-distance behavior of the quasi-static energies.