Long-distance Part Research Articles

Long-distance contribution to εK from lattice QCD

A lattice QCD approach to the calculation of the long-distance contributions to εK is presented. This parameter describes indirect CP violation in K→ππ decay. While the short-distance contribution to εK can be accurately calculated in terms of standard model parameters and a single hadronic matrix element, BK, there is a long-distance part which is estimated to be approximately 5% of the total and is more difficult to determine. A method for determining this small but phenomenologically important contribution to εK using lattice QCD is proposed and a complete exploratory calculation of the contribution is presented. This exploratory calculation uses an unphysical light quark mass corresponding to a 339 MeV pion mass and an unphysical charm quark mass of 968 MeV, expressed in the MS¯ scheme at 2 GeV. This calculation demonstrates that future work should be able to determine this long-distance contribution from first principles with a controlled error of 10% or less. Published by the American Physical Society 2024

Forward light-by-light scattering and electromagnetic correction to hadronic vacuum polarization

Lattice QCD calculations of the hadronic vacuum polarization (HVP) have reached a precision where the electromagnetic (e.m.) correction can no longer be neglected. This correction is both computationally challenging and hard to validate, as it leads to ultraviolet (UV) divergences and to sizeable infrared (IR) effects associated with the massless photon. While we precisely determine the UV divergence using the operator-product expansion, we propose to introduce a separation scale Λ ~ 400 MeV into the internal photon propagator, whereby the calculation splits into a short-distance part, regulated in the UV by the lattice and in the IR by the scale Λ, and a UV-finite long-distance part to be treated with coordinate-space methods, thereby avoiding power-law finite-size effects altogether. In order to predict the long-distance part, we express the UV-regulated e.m. correction to the HVP via the forward hadronic light-by-light (HLbL) scattering amplitude and relate the latter via a dispersive sum rule to γ∗γ∗ fusion cross-sections. Having tested the relation by reproducing the two-loop QED vacuum polarization (VP) from the tree-level γ∗γ∗→ e+e− cross-section, we predict the expected lattice-QCD integrand resulting from the γ∗γ∗→ π0 process.

New determination of |Vcb| using the three-loop QCD corrections for the B → D⁎ semi-leptonic decays

We present a new determination of the Cabibbo-Kobayashi-Maskawa matrix element |Vcb| by using the three-loop perturbative QCD corrections for the B→D⁎ semi-leptonic decay. The decay width of B→D⁎ semi-leptonic decay can be factorized as perturbatively calculable short-distance part and the non-perturbative but universal long-distance part. We adopt the principle of maximum conformality (PMC) single-scale setting approach to deal with the perturbative series so as to achieve a precise fixed-order prediction for the short-distance parameter ηA. By applying the PMC, an overall effective αs value is achieved by recursively using the renormalization group equation, which inversely results in a precise scale-invariant pQCD series. Such scale-invariant series also provides a reliable basis for predicting the contributions from uncalculated perturbative terms. We then obtain ηA=0.9225−0.0168+0.0117, where the error is the squared average of those from Δαs(MZ)=±0.0010 and the uncertainties caused by the uncalculated higher-order perturbative terms. By using the data of B→D⁎ℓν¯ℓ, we finally obtain |Vcb|PMC=(40.60−0.57+0.53)×10−3, which is consistent with the PDG value within errors.

Anisotropic quantum Hall states in the presence of interactions with fourfold rotational symmetry

We study the effects of anisotropic interactions in the quantum Hall effect in the presence of a fourfold discrete rotational ($C_4$) symmetry. Employing the density matrix renormalization group technique on an infinite cylinder geometry (iDMRG), we calculate the anisotropy response of the Laughlin state at $\nu=1/3$ and the composite-Fermi liquid (CFL) state at $\nu=1/2$. We find that the anisotropy transferred from the interaction potential to the $\nu=1/3$ state is stronger when compared to the complementary case of an anisotropic band. Further, the strength of anisotropy reduces as the interaction is made shorter ranged. Quite surprisingly, at $\nu=1/2$, we find that the deformation in the CF Fermi-surface changes sign as the interaction range is reduced. Our results imply that the short-distance and long-distance parts of the interaction potential have opposite effects on the CFL state in the presence of $C_4$-symmetric anisotropy.

Rho resonance, timelike pion form factor, and implications for lattice studies of the hadronic vacuum polarization

We study isospin-1 P-wave $\pi\pi$ scattering in lattice QCD with two flavours of O($a$) improved Wilson fermions. For pion masses ranging from $m_\pi=265$ MeV to $m_\pi=437$ MeV, we determine the energy spectrum in the centre-of-mass frame and in three moving frames. We obtain the scattering phase shifts using L\"uscher's finite-volume quantisation condition. Fitting the dependence of the phase shifts on the scattering momentum to a Breit-Wigner form allows us to determine the resonance parameters $m_\rho$ and $g_{\rho\pi\pi}$. By combining the scattering phase shifts with the decay matrix element of the vector current, we calculate the timelike pion form factor, $F_\pi$, and compare the results to the Gounaris-Sakurai representation of the form factor in terms of the resonance parameters. In addition, we fit our data for the form factor to the functional form suggested by the Omn\`es representation, which allows for the extraction of the charge radius of the pion. As a further application, we discuss the long-distance behaviour of the vector correlator, which is dominated by the two-pion channel. We reconstruct the long-distance part in two ways: one based on the finite-volume energies and matrix elements and the other based on $F_\pi$. It is shown that this part can be accurately constrained using the reconstructions, which has important consequences for lattice calculations of the hadronic vacuum polarisation contribution to the muon anomalous magnetic moment.

Exploratory Lattice QCD Study of the Rare Kaon Decay K^{+}→π^{+}νν[over ¯

We report a first, complete lattice QCD calculation of the long-distance contribution to the K^{+}→π^{+}νν[over ¯] decay within the standard model. This is a second-order weak process involving two four-Fermi operators that is highly sensitive to new physics and being studied by the NA62 experiment at CERN. While much of this decay comes from perturbative, short-distance physics, there is a long-distance part, perhaps as large as the planned experimental error, which involves nonperturbative phenomena. The calculation presented here, with unphysical quark masses, demonstrates that this contribution can be computed using lattice methods by overcoming three technical difficulties: (i)a short-distance divergence that results when the two weak operators approach each other, (ii)exponentially growing, unphysical terms that appear in Euclidean, second-order perturbation theory, and (iii)potentially large finite-volume effects. A follow-on calculation with physical quark masses and controlled systematic errors will be possible with the next generation of computers.

Emerging lattice approach to the K-unitarity triangle

It has been clear for past many years that in low energy observables new physics can only appear as a perturbation. Therefore precise theoretical predictions and precise experimental measurements have become mandatory. Here we draw attention to the significant advances that have been made on the lattice in recent years in $K\to \pi \pi$, $\Delta M_K$, the long-distance part of $\varepsilon$ and rare K-decays. Thus, in conjunction with experiments, the construction of a unitarity triangle purely from Kaon physics should soon become feasible. Along with the B-unitarity triangle, this should allow for more stringent tests of the Standard Model and tighter constraints on new physics.

Further understanding of the non-DD¯decays ofψ(3770)

We provide details of the study of $\ensuremath{\psi}(3770)$ non-$D\overline{D}$ decays into $VP$, where $V$ and $P$ denote light vector meson and pseudoscalar meson, respectively. We find that the electromagnetic interaction plays little role in these processes, while the strong interaction dominates. The strong interaction can be separated into two parts, i.e., the short-distance part probing the wave function at origin and the long-distance part reflecting the soft gluon exchanged dynamics. The long-distance part is thus described by the intermediate charmed meson loops. We show that the transition of $\ensuremath{\psi}(3770)\ensuremath{\rightarrow}VP$ can be related to $\ensuremath{\psi}(3686)\ensuremath{\rightarrow}VP$ such that the parameters in our model can be constrained by comparing the different parts in $\ensuremath{\psi}(3770)\ensuremath{\rightarrow}VP$ to those in $\ensuremath{\psi}(3686)\ensuremath{\rightarrow}VP$. Our quantitative results confirm the findings of Zhang et al. [Phys. Rev. Lett. 102, 172001 (2009)] that the OZI-rule-evading long-distance strong interaction via the intermediate meson loops plays an important role in $\ensuremath{\psi}(3770)$ decays and could be a key towards a full understanding of the mysterious $\ensuremath{\psi}(3770)$ non-$D\overline{D}$ decay mechanism.

Practical Calculational Scheme Implementing the Wilsonian RG Results for Nuclear Effective Field Theory Including Pions

On the basis of the Wilsonian renormalization group (WRG) analysis of nuclear effective field theory (NEFT) including pions, we propose a practical calculational scheme in which the short-distance part of one-pion exchange (S-OPE) is removed and represented as contact terms. The long-distance part of one-pion exchange (L-OPE) is treated as perturbation. The use of dimensional regularization (DR) for diagrams consisting only of contact interactions considerably simplifies the calculation of scattering amplitude and the renormalization group equations. NLO results for nucleon-nucleon elastic scattering in the S-waves are obtained and compared with experiments. A brief comment on NNLO calculations is given.

Pions in Nuclear Effective Field Theory: How They Behave Differently at Different Scales and How They Decouple at Very Low Energies

We explain how the Wilsonian renormalization group (RG) can determine the power counting of the nuclear effective field theory (NEFT) including pions. We emphasize that the separation of pion exchange into the short-distance part and the long-distance part is essential since they behave differently in the RG analysis; we found that the latter is perturbative whereas the a part of the former is nonperturbative. As for the contact interactions power counting turns out to be the same as that for pionless NEFT: pion exchange does not affect the scaling property of contact operators. Our RG equations for NEFT including pions connect smoothly with those for the pionless NEFT: pions decouple at very low energies as we expect.

Pions are neither perturbative nor nonperturbative: Wilsonian renormalization-group analysis of nuclear effective field theory including pions

Pionful nuclear effective field theory (NEFT) in the two-nucleon sector is examined from the Wilsonian renormalization group point of view. The pion exchange is cut off at the floating cutoff scale, $\Lambda$, with the short-distance part being represented as contact interactions in accordance with the general principle of renormalization. We derive the nonperturbative renormalization group equations in the leading order of the nonrelativistic approximation in the operator space including up to $\mathcal{O}(p^2)$ and find the nontrivial fixed points in the $^1S_0$ and $^3S_1$--$^3D_1$ channels which are identified with those in the pionless NEFT. The scaling dimensions, which determine the power counting, of the contact interactions at the nontrivial fixed points are also identified with those in the pionless NEFT. We emphasize the importance of the separation of the pion exchange into the short-distance and the long-distance parts, since a part of the former is nonperturbative while the latter is perturbative.

Low-Energy Scattering of Heavy-Light Mesons from Nucleons

We study the low-energy scattering of charmed (D) and strange (K) mesons by nucleons. The short-distance part of the interaction is due to quark-gluon interchanges derived from a model that realizes dynamical chiral symmetry breaking and confines color. The quark-gluon interaction incorporates a confining Coulomb-like potential extracted from lattice QCD simulations in Coulomb gauge and a transverse hyperfine interaction consistent with a finite gluon propagator in the infrared. The long-distance part of the interaction is due to single vector (ρ, ω) and scalar (σ) meson exchanges. We show results for scattering cross-sections for isospin I = 0 and I = 1.

Chiral potentials, perturbation theory and the 1S0 channel of NN scattering

We use nucleon–nucleon phase shifts obtained from experimental data, together with the chiral expansion for the long-distance part of the NN interaction, to obtain information about the short-distance piece of the NN potential that is at work in the 1S0 channel. We find that if the scale R that defines the separation between ‘long-’ and ‘short-’ distance is chosen to be ≲1.8 fm then the energy dependence produced by short-distance dynamics is well approximated by a two-term polynomial for Tlab ⩽ 200 MeV. We also find that a quantitative description of NN dynamics is possible, at least in this channel, if one treats the long-distance parts of the chiral NN potential in perturbation theory. However, in order to achieve this we have to choose a separation scale R that is larger than 1.0 fm.

Improvement of the Hot QCD Pressure by the Minimal Sensitivity Criterion

The principles of minimal sensitivity (PMS) criterion is applied to the perturbative free energy density, or pressure, of hot QCD, which include the $\sim g_s^6 \ln g_s$ and part of the $\sim g_s^6$ terms. Applications are made separately to the short- and long-distance parts of the pressure. Comparison with the lattice results, at low temperatures, shows that the resultant `` optimal'' approximants are substantially improved when compared to the $\bar{MS}$ results. In particular, for the realistic case of three quark flavors, the `` optimal'' approximants are comparable with the lattice results.

On the existence of heavy pentaquarks: The largeNcand heavy quark limits and beyond

We present a very general argument that the analogue of a heavy pentaquark (a state with the quantum numbers of a baryon combined with an additional light quark and a heavy antiquark $\overline{Q}$) must exist as a particle stable under strong interactions in the combined heavy quark and large ${N}_{c}$ limits of QCD. Moreover, in the combined limit these heavy pentaquark states fill multiplets of $\mathrm{SU}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(8)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$. We explore the question of whether corrections in the combined $1/{N}_{c}$ and $1/{m}_{Q}$ expansions are sufficiently small to maintain this qualitative result. Since no model-independent way is known to answer this question, we use a class of realistic hadronic models in which a pentaquark can be formed via nucleon-heavy-meson binding through a pion-exchange potential. These models have the virtue that they necessarily yield the correct behavior in the combined limit, and the long-distance parts of the interactions are model independent. If the long-distance attraction in these models were to predict bound states in a robust way (i.e., largely insensitive to the details of the short-range interaction), then one could safely conclude that heavy pentaquarks do exist. However, in practice the binding does depend very strongly on the details of the short-distance physics, suggesting that the real world is not sufficiently near the combined large ${N}_{c}$, ${m}_{Q}$ limit to use it as a reliable guide. Whether stable heavy pentaquarks exist remains an open question.

DecayΣ+→pℓ+ℓ−within the standard model

The HyperCP collaboration has recently reported the observation of three events for the decay ${\ensuremath{\Sigma}}^{+}\ensuremath{\rightarrow}p{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$. They have suggested that new physics may be required to understand the implied decay rate and the observed ${M}_{\ensuremath{\mu}\ensuremath{\mu}}$ distribution. Motivated by this result, we reexamine this mode within the standard model, considering both the short-distance and long-distance contributions. The long-distance part depends on four complex form factors. We determine their imaginary parts from unitarity, fix two of the real parts from the ${\ensuremath{\Sigma}}^{+}\ensuremath{\rightarrow}p\ensuremath{\gamma}$ measurements, and estimate the other two with vector-meson-dominance models. Taking into account constraints from ${\ensuremath{\Sigma}}^{+}\ensuremath{\rightarrow}p{e}^{+}{e}^{\ensuremath{-}}$, we find that ${\ensuremath{\Sigma}}^{+}\ensuremath{\rightarrow}p{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ is long-distance dominated and its rate falls within the range suggested by the HyperCP measurement.

Quenched Approximation to Δ S = 1 K Decay

The importance of explicit quark loops in the amplitudes contributing to ΔS = 1, K meson decays raises potential ambiguities when these amplitudes are evaluated in the quenched approximation. Using the factorization of these amplitudes into short- and long-distance parts provided by the standard low-energy effective weak Hamiltonian, we argue that the quenched approximation can be conventionally justified if it is applied to the long-distance portion of each amplitude. The result is a reasonably well-motived definition of the quenched approximation that is close to that employed in the RBC and CP-PACS calculations of these quantities.

Padé-related resummations of the pressure of quark-gluon plasma by approximate inclusion ofgs6terms

We perform various resummations of the hot QCD pressure based on the actual knowledge of the perturbation series which includes the g**6 ln(1/g) and part of the g**6 terms. Resummations are performed separately for the short- and long-distance parts. The g**6 term of the short-distance pressure is estimated on the basis on the known UV cutoff dependence of the long-distance part. The resummations are of the Pade and Borel-Pade type, using in addition the (Pade-)resummed expression for the squared screening mass mE**2 and for the EQCD coupling parameter gE**2. The resummed results depend weakly on the yet unknown g**6 terms and on the the short-range renormalization scale, at all temperatures. The dependence on the long-range renormalization scale is appreciable at low temperatures T < 1 GeV. The resulting dependence of pressure on temperature T is compatible with the results of the lattice calculations at low T.

Resummations of free energy at high temperature

We discuss resummation strategies for free energy in quantum field theories at nonzero temperatures T. We point out that resummations should be performed for the short- and long-distance parts separately in order to avoid spurious interference effects and double-counting. We then discuss and perform Pade resummations of these two parts for QCD at high T. The resummed results are almost invariant under variation of the renormalization and factorization scales. We perform the analysis also in the case of the massless scalar $\phi^4$ theory.

Critical behavior of the two-dimensional Ising susceptibility.

We report computations of the short- and long-distance (scaling) contributions to the square-lattice Ising susceptibility. Both computations rely on summation of correlation functions, obtained using nonlinear partial difference equations. In terms of a temperature variable tau, linear in T/Tc-1, the short-distance terms have the form tau(p)(ln/tau/)q with p> or =q2. A high- and low-temperature series of N = 323 terms, generated using an algorithm of complexity O(N6), are analyzed to obtain the scaling part, which when divided by the leading /tau/(-7/4) singularity contains only integer powers of tau. Contributions of distinct irrelevant variables are identified and quantified at leading orders /tau/(9/4) and /tau/(17/4).