arXiv subjects
Publications and source records attributed to Hiromasa Takaura.
To improve accuracy in calculating QCD effects, we propose a method for renormalon subtraction in the context of the operator-product expansion. The method enables subtracting renormalons of various powers in $Λ_{\rm QCD}$ efficiently and simultaneously from single-scale observables. We apply it to different observables and examine consistency with theoretical expectations.
We perform a numerical computation of the anomalous magnetic moment ($g-2$) of the electron in QED by using the stochastic perturbation theory. Formulating QED on the lattice, we develop a method to calculate the coefficients of the perturbative series of $g-2$ without the use of the Feynman diagrams. We demonstrate the feasibility of the method by performing a computation up to the $α^3$ order and compare with the known results. This program provides us with a totally independent check of the results obtained by the Feynman diagrams and will be useful for the estimations of not-yet-calculated higher order values. This work provides an example of the application of the numerical stochastic perturbation theory to physical quantities, for which the external states have to be taken on-shell.
We present a formulation to give renormalon-free predictions consistently with fixed order perturbative results. The formulation has a similarity to Lee's method in that the renormalon-free part consists of two parts: one is given by a series expansion which does not contain renormalons and the other is the real part of the Borel integral of a singular Borel transform. The main novel aspect is to evaluate the latter object using a dispersion relation technique, which was possible only in the large-$β_0$ approximation. Here, we introduce an "ambiguity function," which is defined by inverse Mellin transform of the singular Borel transform. With the ambiguity function, we can rewrite the Borel integral in an alternative formula, which allows us to obtain the real part using analytic techniques similarly to the case of the large-$β_0$ approximation. We also present detailed studies of renormalization group properties of the formulation. As an example, we apply our formulation to the fixed-order result of the static QCD potential, whose closest renormalon is already visible.
We present additional observations to previous studies on the infrared (IR) renormalon in $SU(N)$ QCD(adj.), the $SU(N)$ gauge theory with $n_W$-flavor adjoint Weyl fermions on~$\mathbb{R}^3\times S^1$ with the $\mathbb{Z}_N$ twisted boundary condition. First, we show that, for arbitrary finite~$N$, a logarithmic factor in the vacuum polarization of the "photon" (the gauge boson associated with the Cartan generators of~$SU(N)$) disappears under the $S^1$~compactification. Since the IR renormalon is attributed to the presence of this logarithmic factor, it is concluded that there is no IR renormalon in this system with finite~$N$. This result generalizes the observation made by Anber and~Sulejmanpasic [J. High Energy Phys.\ \textbf{1501}, 139 (2015)] for $N=2$ and~$3$ to arbitrary finite~$N$. Next, we point out that, although renormalon ambiguities do not appear through the Borel procedure in this system, an ambiguity appears in an alternative resummation procedure in which a resummed quantity is given by a momentum integration where the inverse of the vacuum polarization is included as the integrand. Such an ambiguity is caused by a simple zero at non-zero momentum of the vacuum polarization. Under the decompactification~$R\to\infty$, where $R$ is the radius of the $S^1$, this ambiguity in the momentum integration smoothly reduces to the IR renormalon ambiguity in~$\mathbb{R}^4$. We term this ambiguity in the momentum integration "renormalon precursor". The emergence of the IR renormalon ambiguity in~$\mathbb{R}^4$ under the decompactification can be naturally understood with this notion.
It was conjectured that bions, semi-classical objects found in a compactified spacetime, are responsible for the cancellation of the so-called renormalon ambiguities. Contrary to the conjecture, we argue that the ambiguity due to the bion corresponds to the proliferation of Feynman diagrams. We point out that the amplitudes of almost all Feynman diagrams are enhanced due to modifications of the infrared structure of perturbation theory upon an $S^1$ compactification and twisted boundary conditions. Our findings clarify the role of the semi-classical object in resurgence structure, which has been a controversial issue in recent years.
We investigate the $u=1/2$ [$\mathcal{O}(Λ_{\rm QCD})$] and $u=3/2$ [$\mathcal{O}(Λ_{\rm QCD}^3)$] 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 $u=3/2$ renormalons. We clarify how the $u=3/2$ renormalon is suppressed in the momentum-space potential in relation with the Wilson coefficient $V_A(r)$. 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 $α_s(M_Z)$ determination from the QCD potential.
In the leading order of the large-$N$ approximation, we study the renormalon ambiguity in the gluon (or, more appropriately, photon) condensate in the 2D supersymmetric $\mathbb{C}P^{N-1}$ model on~$\mathbb{R}\times S^1$ with the $\mathbb{Z}_N$ twisted boundary conditions. In our large~$N$ limit, the combination $ΛR$, where $Λ$ is the dynamical scale and $R$~is the $S^1$ radius, is kept fixed (we set $ΛR\ll1$ so that the perturbative expansion with respect to the coupling constant at the mass scale~$1/R$ is meaningful). We extract the perturbative part from the large-$N$ expression of the gluon condensate and obtain the corresponding Borel transform~$B(u)$. For~$\mathbb{R}\times S^1$, we find that the Borel singularity at~$u=2$, which exists in the system on the uncompactified~$\mathbb{R}^2$ and corresponds to twice the minimal bion action, disappears. Instead, an unfamiliar renormalon singularity \emph{emerges\/} at~$u=3/2$ for the compactified space~$\mathbb{R}\times S^1$. The semi-classical interpretation of this peculiar singularity is not clear because $u=3/2$ is not dividable by the minimal bion action. It appears that our observation for the system on~$\mathbb{R}\times S^1$ prompts reconsideration on the semi-classical bion picture of the infrared renormalon.
We study the infrared renormalon in the gluon condensate in the $SU(N)$ gauge theory with $n_W$-flavor adjoint Weyl fermions (QCD(adj.)) on~$\mathbb{R}^3\times S^1$ with the $\mathbb{Z}_N$ twisted boundary conditions. We rely on the so-called large-$β_0$ approximation as a conventional tool to analyze the renormalon, in which only Feynman diagrams that dominate in the large-$n_W$ limit are considered while the coefficient of the vacuum polarization is set by hand to the one-loop beta function~$β_0=11/3-2n_W/3$. In the large~$N$ limit within the large-$β_0$ approximation, the W-boson, which acquires the twisted Kaluza--Klein momentum, produces the renormalon ambiguity corresponding to the Borel singularity at~$u=2$. This provides an example that the system in the compactified space~$\mathbb{R}^3\times S^1$ possesses the renormalon ambiguity identical to that in the uncompactified space~$\mathbb{R}^4$. We also discuss the subtle issue that the location of the Borel singularity can change depending on the order of two necessary operations.
We point out that the location of renormalon singularities in theory on a circle-compactified spacetime $\mathbb{R}^{d-1} \times S^1$ (with a small radius $R Λ\ll 1$) can differ from that on the non-compactified spacetime $\mathbb{R}^d$. We argue this under the following assumptions, which are often realized in large $N$ theories with twisted boundary conditions: (i) a loop integrand of a renormalon diagram is volume independent, i.e. it is not modified by the compactification, and (ii) the loop momentum variable along the $S^1$ direction is not associated with the twisted boundary conditions and takes the values $n/R$ with integer $n$. We find that the Borel singularity is generally shifted by $-1/2$ in the Borel $u$-plane, where the renormalon ambiguity of $\mathcal{O}(Λ^k)$ is changed to $\mathcal{O}(Λ^{k-1}/R)$ due to the circle compactification $\mathbb{R}^d \to \mathbb{R}^{d-1} \times S^1$. The result is general for any dimension $d$ and is independent of details of the quantities under consideration. As an example, we study the $\mathbb{C} P^{N-1}$ model on $\mathbb{R} \times S^1$ with $\mathbb{Z}_N$ twisted boundary conditions in the large $N$ limit.
We propose a clear definition of the gluon condensate within the large-$β_0$ approximation as an attempt toward a systematic argument on the gluon condensate. We define the gluon condensate such that it is free from a renormalon uncertainty, consistent with the renormalization scale independence of each term of the operator product expansion (OPE), and an identical object irrespective of observables. The renormalon uncertainty of $\mathcal{O}(Λ^4)$, which renders the gluon condensate ambiguous, is separated from a perturbative calculation by using a recently suggested analytic formulation. The renormalon uncertainty is absorbed into the gluon condensate in the OPE, which makes the gluon condensate free from the renormalon uncertainty. As a result, we can define the OPE in a renormalon-free way. Based on this renormalon-free OPE formula, we discuss numerical extraction of the gluon condensate using the lattice data of the energy density operator defined by the Yang--Mills gradient flow.
We determine the strong coupling constant $α_s$ from the static QCD potential by matching a theoretical calculation with a lattice QCD computation. We employ a new theoretical formulation based on the operator product expansion, in which renormalons are subtracted from the leading Wilson coefficient. We remove not only the leading renormalon uncertainty of $\mathcal{O}(Λ_{\rm QCD})$ but also the first $r$-dependent uncertainty of $\mathcal{O}(Λ_{\rm QCD}^3 r^2)$. The theoretical prediction for the potential turns out to be valid at the static color charge distance $Λ_{\rm \overline{MS}} r \lesssim 0.8$ ($r \lesssim 0.4$ fm), which is significantly larger than ordinary perturbation theory. With lattice data down to $Λ_{\rm \overline{MS}} r \sim 0.09$ ($r \sim 0.05$ fm), we perform the matching in a wide region of $r$, which has been difficult in previous determinations of $α_s$ from the potential. Our final result is $α_s(M_Z^2) = 0.1179^{+0.0015}_{-0.0014}$ with 1.3 % accuracy. The dominant uncertainty comes from higher order corrections to the perturbative prediction and can be straightforwardly reduced by simulating finer lattices.
Recently, Harlander et al.\ [Eur.\ Phys.\ J.\ C {\bf 78}, 944 (2018)] have computed the two-loop order (i.e., NNLO) coefficients in the gradient-flow representation of the energy--momentum tensor (EMT) in vector-like gauge theories. In this paper, we study the effect of the two-loop order corrections (and the three-loop order correction for the trace part of the EMT, which is available through the trace anomaly) on the lattice computation of thermodynamic quantities in quenched QCD. The use of the two-loop order coefficients generally reduces the $t$~dependence of the expectation values of the EMT in the gradient-flow representation, where $t$~is the flow time. With the use of the two-loop order coefficients, therefore, the $t\to0$ extrapolation becomes less sensitive to the fit function, the fit range, and the choice of the renormalization scale; the systematic error associated with these factors is considerably reduced.
We determine the strong coupling constant $α_s(M_Z)$ from the static QCD potential by matching a lattice result and a theoretical calculation. We use a new theoretical framework based on operator product expansion (OPE), where renormalons are subtracted from the leading Wilson coefficient. We find that our OPE prediction can explain the lattice data at $Λ_{\rm QCD} r \lesssim 0.8$. This allows us to use a larger window in matching, which leads to a more reliable determination. We obtain $α_s(M_Z)=0.1179^{+0.0015}_{-0.0014}$.
Perturbative calculations of the static QCD potential have the $u=3/2$ renormalon uncertainty. In the multipole expansion performed within pNRQCD, this uncertainty at LO is known to get canceled against the ultrasoft correction at NLO. To investigate the net contribution remaining after this renormalon cancellation, we propose a formulation to separate the ultrasoft correction into renormalon uncertainties and a renormalon independent part. We focus on very short distances $Λ_{\rm QCD} r \lesssim 0.1$ and investigate the ultrasoft correction based on its perturbative evaluation in the large-$β_0$ approximation. We also propose a method to examine the local gluon condensate, which appears as the first nonperturbative effect to the static QCD potential, without suffering from the $u=2$ renormalon.
In the context of OPE and using the large-$β_0$ approximation, we propose a method to define Wilson coefficients free from uncertainties due to IR renormalons. We first introduce a general observable $X(Q^2)$ with an explicit IR cutoff, and then we extract a genuine UV contribution $X_\mathrm{UV}$ as a cutoff-independent part. $X_\mathrm{UV}$ includes power corrections $\sim (Λ_\mathrm{QCD}^2/Q^2)^n$ which are independent of renormalons. Using the integration-by-regions method, we observe that $X_\mathrm{UV}$ coincides with the leading Wilson coefficient in OPE and also clarify that the power corrections originate from UV region. We examine scheme dependence of $X_\mathrm{UV}$ and single out a specific scheme favorable in terms of analytical properties. Our method would be optimal with respect to systematicity, analyticity and stability. We test our formulation with the examples of the Adler function, QCD force between $Q \bar{Q}$, and $R$-ratio in $e^{+} e^{-}$ collision.
We formulate a way to separate UV and IR contributions to the Adler function and discuss how $Λ_\mathrm{QCD}^2/Q^2$ dependence is encoded in the UV contribution within perturbative QCD.