As a straightforward application of the recently calculated two-loop heavy quarkonium Hamiltonian, we evaluate the two-loop <math><mi>O</mi><mo>(</mo><mi>ϵ</mi><mo>)</mo></math> term of the <math><mn>1</mn><mo>/</mo><mo>(</mo><mi>m</mi><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> heavy quarkonium potential. Compared to a previous calculation we find a small difference in the coefficient of the maximally non-abelian color factor <math><msub><mrow><mi>C</mi></mrow><mrow><mi>F</mi></mrow></msub><msubsup><mrow><mi>C</mi></mrow><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math>. We further examine this coefficient carefully.

Japan

Yukinari Sumino

Two-loop ( ) Term of 1/( ) Heavy Quarkonium Potential

We calculate the two-loop heavy quarkonium Hamiltonian within potential-nonrelativistic-QCD effective field theory in the nonannihilation channel. This calculation represents the first nontrivial step toward determining the <math><msup><mi>N</mi><mn>4</mn></msup><mi>LO</mi></math> Hamiltonian in the weak coupling regime. The large amount of computation is systematically handled by employing the <math><mi>β</mi></math> expansion, differential equations for master integrals, and adopting a single-step matching procedure, in contrast to the conventional two-step approach.

Go Mishima

Hiromasa Takaura

Two-Loop Quarkonium Hamiltonian in Nonannihilation Channel

For the heavy quarkonium system we examine <math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>QCD</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>/</mo><mi>m</mi><mo>)</mo></math> renormalons, which are expected to be included in the perturbative series of the pole mass and <math><mn>1</mn><mo>/</mo><mo>(</mo><mi>m</mi><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> interquark potential. We find indications of existence and cancellation of these renormalons, from examinations of stability and convergence properties of the perturbative series and their resummations, as well as by comparison with the known properties of the <math><mi>O</mi><mo>(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>QCD</mi></mrow></msub><mo>)</mo></math> renormalons.

Takuya Agemura

On order Renormalons in Quarkonium System

We determine <math><mo>|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo>|</mo></math> from the inclusive semileptonic decay width of the <math><mi>B</mi></math> meson with the known <math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>LO</mi></mrow></math> perturbative coefficients for the first time in the <math><mover><mi>MS</mi><mo>¯</mo></mover></math> mass scheme. We make use of a recently developed method, dual-space-renormalon-subtraction method, to separate and subtract the order <math><mrow><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>QCD</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>/</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></math> infrared renormalon. This allows us to perform the analysis accurately in the <math><mover><mi>MS</mi><mo>¯</mo></mover></math> mass scheme, in which otherwise the perturbative series does not converge well up to the currently calculated perturbation order. Our result reads <math><mrow><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo>|</mo><mo>=</mo><mn>0.0415</mn><mo>(</mo><msubsup><mrow><mtext> </mtext></mrow><mrow><mo>−</mo><mn>12</mn></mrow><mrow><mo>+</mo><mn>10</mn></mrow></msubsup><mo>)</mo></mrow></math>, which is consistent with the previous results based on other mass schemes and shows an independent cross-check of the current theoretical evaluation of the inclusive decay width.

Yuuki Hayashi

Inclusive <math><mrow><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo>|</mo></mrow></math> determination in <math><mover><mrow><mi>MS</mi></mrow><mo>¯</mo></mover></math> mass scheme using dual-space-renormalon-subtraction method

We determine <math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo>|</mo></math> using the third-order perturbative series for the inclusive semileptonic B decay width and for the masses of the bottomonium 1S states. We use the masses of <math><msub><mrow><mi>η</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mn>1</mn><mi>S</mi><mo>)</mo></math> and <math><mi>ϒ</mi><mo>(</mo><mn>1</mn><mi>S</mi><mo>)</mo></math> as short-distance masses and point out that there is a sizable difference of <math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo>|</mo></math> between the two 1S mass schemes. This is the dominant error of our determination and stems from insufficiency to describe theoretically the observed mass splitting of the bottomonium 1S states. We also study the significance of the <math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>Λ</mi></mrow><mrow><mover><mrow><mrow><mi>MS</mi></mrow></mrow><mo>‾</mo></mover></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo></math> non-perturbative effects in HQET with respect to the current perturbative accuracy. Our result <math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo>|</mo><mo>=</mo><mn>0.0425</mn><mo>(</mo><mn>11</mn><mo>)</mo></math> is consistent with the PDG value determined from the inclusive decays and has a slightly larger error.

Determination of | | using N LO perturbative corrections to Γ( → ) and 1S masses

It is becoming more important to subtract renormalons efficiently from perturbative calculations, in order to achieve high precision QCD calculations. We propose a new framework “Dual Space Approach” for renormalon separation, which enables subtraction of multiple renormalons simultaneously. Using a dual transform which suppresses infrared renormalons, we derive a one-parameter integral representation of a general observable. We investigate systematically how renormalons emerge and get canceled in the entire operator product expansion (OPE) of an observable, by applying the expansion-by-regions (EBR) method to this one-parameter integral expression. In particular we investigate in detail OPEs in a solvable model, the 2-dimensional O(N) nonlinear σ model, by the dual space approach. A nontrivial mechanism of renormalon cancellation in this model can be understood from an integration identity on which the EBR method is founded. We demonstrate that the dual space approach can be useful by a simulation study imitating the QCD case. Application of this method to QCD calculations is also discussed.

 HiromasaTakaura

Renormalon subtraction in OPE by dual space approach: nonlinear sigma model and QCD

Properly separating and subtracting renormalons in the framework of the op- erator product expansion (OPE) is a way to realize high precision computation of QCD effects in high energy physics. We propose a new method (FTRS method), which enables to subtract multiple renormalons simultaneously from a general observable. It utilizes a property of Fourier transform, and the leading Wilson coefficient is written in a one-parameter integral form whose integrand has suppressed (or vanishing) renormalons. The renormalon subtraction scheme coincides with the usual principal-value prescription at large orders. We perform test analyses and subtract the   <math> <mi>O</mi> <mfenced> <msubsup> <mi>Λ</mi> <mi>QCD</mi> <mn>4</mn> </msubsup> </mfenced> </math>  $$ \mathcal{O}\left({\Lambda}_{\mathrm{QCD}}^4\right) $$  renormalon from the Adler function, the   <math> <mi>O</mi> <mfenced> <msubsup> <mi>Λ</mi> <mi>QCD</mi> <mn>2</mn> </msubsup> </mfenced> </math>  $$ \mathcal{O}\left({\Lambda}_{\mathrm{QCD}}^2\right) $$  renormalon from the B → X  u  l   <math> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\nu} $$  decay width, and the   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (ΛQCD) and   <math> <mi>O</mi> <mfenced> <msubsup> <mi>Λ</mi> <mi>QCD</mi> <mn>2</mn> </msubsup> </mfenced> </math>  $$ \mathcal{O}\left({\Lambda}_{\mathrm{QCD}}^2\right) $$  renormalons from the B, D meson masses. The analyses show good consistency with theoretical expectations, such as improved convergence and scale dependence. In particular we obtain   <math> <mover> <mi>Λ</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\Lambda} $$  FTRS = 0.495 ± 0.053 GeV and (  <math> <msubsup> <mi>μ</mi> <mi>π</mi> <mn>2</mn> </msubsup> </math>  $$ {\mu}_{\pi}^2 $$ )FTRS = −0.12 ± 0.23 GeV2 for the non-perturbative parameters of HQET. We explain the formulation and analyses in detail.

Renormalon subtraction in OPE using Fourier transform: formulation and application to various observables

We investigate the $$ u=1/2\left[\mathcal{O}\left({\Lambda}_{\mathrm{QCD}}\right)\right] $$ and $$ u=3/2\left[\mathcal{O}\left({\Lambda}_{\mathrm{QCD}}^3\right)\right] $$ renormalons in the static QCD potential in position space and momentum space using the OPE of the potential-NRQCD effective field theory. This is an old problem and we provide a formal formulation to analyze it. In particular we present detailed examinations of the = 3/2 renormalons. We clarify how the = 3/2 renormalon is suppressed in the momentum-space potential in relation with the Wilson coefficient ( ). We also point out that it is not straightforward to subtract the IR renormalon and IR divergences simultaneously in the multipole expansion. Numerical analyses are given, which clarify the current status of our knowledge on the perturbative series. The analysis gives a positive reasoning to the method for subtracting renormalons used in recent ( ) determination from the QCD potential.

On renormalons of static QCD potential at

We calculate the static quark potential for an SU(N) gauge theory in the Maximal Abelian gauge as well as its Abelian projection up to two loops in perturbation theory. We discuss its renormalization properties. The result is compared with a recent lattice result at <math><mi>r</mi><mo>≲</mo><mn>0.5</mn></math> fm.

Matthias Berwein

Perturbative static quark potential in Maximal Abelian gauge

The total energy of a static quark-antiquark pair <math><msub><mrow><mi>E</mi></mrow><mrow><mi>tot</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><mn>2</mn><msub><mrow><mi>m</mi></mrow><mrow><mi>pole</mi></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>QCD</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo></math> is known to include <math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>QCD</mi></mrow><mrow><mn>3</mn></mrow></msubsup><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> and <math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>QCD</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>/</mo><msup><mrow><mover><mrow><mi>m</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> renormalon uncertainties in the large-<math><msub><mrow><mi>β</mi></mrow><mrow><mn>0</mn></mrow></msub></math> approximation, after canceling <math><mi>O</mi><mo>(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>QCD</mi></mrow></msub><mo>)</mo></math> renormalons. We compute the (renormalon-free) genuine UV part in terms of the <math><mover><mrow><mrow><mi>MS</mi></mrow></mrow><mo>‾</mo></mover></math> mass <math><mover><mrow><mi>m</mi></mrow><mo>‾</mo></mover></math>, extending a recently-proposed method which conforms with OPE. In particular the r-independent part is determined. The result would help understanding the nature of <math><msub><mrow><mi>E</mi></mrow><mrow><mi>tot</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo></math> in the context of OPE with renormalon subtraction.