Walitti qabaan ispeektaraa faankishiniiwwan walqabsiisa hadroonii, kan akka faankishinii poolaarizeeshinii vaakiyuumii Π(q2), kan bifa,. Keessattuu, walitti qabaan ispeektaraa faankishiniiwwan walqabsiisa hadroonii, kan akka faankishinii poolaarizeeshinii vaakiyuumii Π(q2), kan bifa,.
QCD in the continuum theory
Strong coupling constant
In fact, we expand a number of physical quantities by the coupling constant αs in perturbative QCD. The solid line corresponds to the coupling constant calculated by the β function at the five-loop level.
Quark mass
The dotted, dotted, and dotted lines correspond to toms(μ) where c(x) of (2.23) is truncated at 1-4 loop level. The numerical calculation of the running massαs(µ2) can be performed byRunDec [52, 53], as well as the coupling constantαs(µ2).
The Euclidean action
Dψ¯DψDA e−SQCDO, (2.32) where is the partition function defined as. 2.33) This path integral brings about analogy with the statistical mechanics. In the following sections, we review calculations of lattice QCD using the Euclidean path integral.
Gluons in lattice QCD
The discretization error can be reduced by adding other loops such as rectangles and parallelograms. At small expansion, the effect of discretization is limited due to symmetry on the mesh and operators of dimension five do not appear.
Fermions in lattice QCD
One can introduce the Wilson term into the Dirac operator to make the doublers irrelevant in the continuum limit. The Dirac operator in the relation is not against commuting withγ5; therefore, it does not satisfy the assumption of the Nielsen-Ninomiya theorem.
Measurements of meson correlators
- Point-to-all propagator
- γ 5 -hermiticity
- Correlation functions in coordinate space
- Correlation functions with zero momentum
- Noise source
The fermion determinant can be considered as part of the Boltzmann weight. we define an effective action. Therefore, we can measure the spectra of the mesons in the rest frame from the sum.
Möbius Domain-Wall fermions
To obtain physical quantities composed by 4D quark s, we need to properly project 5D fermion fields ψ(x,s) onto 4D spacetime. Using the solution (2.92) with the relation (2.95), we can measure hadron correlation functions as discussed in the previous sections.
Renormalization of lattice operators
We review the X-space method, which gives the renormalization constant [19]. we will use in Chap. In other words, the renormalization constant of the vector course ¯qγµq is equivalent to that of the axial current ¯qγµγ5q:.
PCAC relation of the Domain-wall fermion
In the ss¯ channel we focus on other chapters, the remaining mass is negligible compared to the strange mass of quarks. The integral in the dispersion relation differs, however, since the spectral function does not vanish in the limit s. On the other hand, the total cross section σ(e+e−→µ−µ+;s) to the leading order in the massless limit m2µ is expressed as
Quark-hadron duality
As we have seen in the previous section, the spectral function (3.6) appears in the ratio of the cross section up to a constant. In fact, the spectral function oscillates in the low-energy regime (√ .. 2 GeV) and disagrees with perturbative QCD. In the deep Euclidean region, this term becomes an exponentially decaying function, and the violation of quark-hadron duality would be negligible.
Two-point correlation functions in the massless limit
This cannot be explained by the truncation uncertainties of the perturbative series and the drift corrections. An expression of the solution that depends explicitly on log µ2/Q2. is more useful to calculate the Borel transform4. 4The expression is also used for correlator in the coordinate space, which is derived by the Fourier transform.
Dimension-two corrections of the correlator
The solution can be constructed in the same way as the previous section, but γΓ = −γm andγΓΓ = 0. Compared to the perturbative series of Π0(µ2;Q2), we find that the dimension two correction is less convergent. We will discuss the size of the truncation error after Borel transform in the next section.
Borel transform of the vacuum polarization function
We discuss this form of spectral sum in detail later in this chapter. The first and second terms in the integral correspond to the subtraction of the contact term. The corresponding perturbative series has been calculated up to the same order of the vector correlator, namely O(α3s)[90,98].
Borel transform from lattice correlators
Calculation of such correlators as a function of time separation is straightforward in lattice QCD. To avoid this problem, the method of [35] associates the correlator with a smeared spectral function such as (3.43) instead of the spectral function ρ(ω2) itself. The smeared kernel (4.36) has an obvious divergence problem at ω = 0, which induces divergences of the coefficientssc∗j (4.31).
Lattice calculation
Convergence of Chebyshev expansion
Such a low energy regime is dominated by the ground state and we are able to explicitly correct the error by using the mass and amplitude of the ground state. We can also analytically calculate the Borel transform of the single-pole spectrum with the modification of the low-energy spectrum. The results are compared in Fig.5.6 at three grid spacing. The expansion is almost perfect and the expansions at three grid spacings match each other.
Correction for the low-energy cut of smearing function
Continuum limit
Higher-order perturbative corrections are insignificant compared to the statistical accuracy of gridded data. In any case, the correction may introduce a systematic uncertainty at large 1/M2 since the correction is based on the OPE. Although ˜Πlat(M2) has a relatively large error on the finest lattice, the error of ˜Π(M2) is under good control in the continuum limit.
Result
Comparison with OPE
When it is assumed that the condensate is fully factored in vacuum, κ0 is equal to 1. We introduce the renormalization scales µ0 and µ2 respectively for c0 and c2, we change them in the range 2M2e−γE ≤ µ20, µ22 ≤ 8M2e−γE separately and take the maximum (smallest) c0+c2/M2 value as the upper (lower) band limit. The grid data are in good agreement with the OPE, including terms 1/M4 and 1/M6 within uncertainty, as shown in Figure 5.10.
Extraction of the gluon condensate
The Borel transform ˜ΠOPE(M2) converges well in the range 1/M2 ≤ 1 GeV−2, as can be seen from the small effect of O(1/M6), albeit with the large uncertainty due to the unknown condensates. By fixing c0 and c2 in (5.14) through the perturbative calculation, we determine c4 and c6 through a fit to the lattice data. A more precise determination of the gluon condensate will require more statistics and an improvement in the perturbative calculation.
Saturation by the ground state
Another interesting application of the lattice calculation of the Borel transform is the determination of αs. On the other hand, the perturbative series of the anomalous dimensionγVV(as) has been calculated to O(a4s) [78,79]. Hashimoto, “Lattice calculation of the Dirac eigenvalue density in the perturbative regime of QCD,” Phys.
McNeile (HPQCD), "Determination of the Quark Condensate from Heavy-Light Current Correlators in Full Lattice QCD", Phys. Trottier, "Direct determination of the strange and light quark condensates from full lattice QCD", Phys.
Running of the strange quark mass m s . The dotted, dashed dash-
The link valuable and the plaquette
This property is inherited by the domain wall fermions, as they consist of the Wilson fermions. [19] takes advantage of this chiral property and determines the renormalization constant by combining ˜ZMS/latΓ (Γ2,a2;x) to cancel some of the non-perturbative effects appearing in the fit.
The integration contour in the complex s-plane
The duality violating term can be oscillated as trigonometric functions that do not appear in the OPE, such as,ρDV(s) ∼ e−αssin(βs)where α and β are constants. Using the dispersion relation, we can write them in terms of derivatives of HVP atQ2=0,. The typical length scale is given by an inverse of the quark mass−q1, which is short enough to describe perturbatively for charm and bottom quarks.
We calculate the Borel transform of the HVP using the technique outlined in the previous section. FIGURE 5.12: Comparison of ˜Π(M2) in the continuum limit with the experimental values of the φmeson contribution. FIGUREH.4: The Borel transform with the mass correction ˜Πlat+δΠ˜ divided by the ˜Πlatat three grid spacings.
The integration contour in the complex ω-plane
The uncertainty due to the modeling of the excited state and continuum contributions is another important issue in the QCD sum rule. Solving (F.2) for ZVMS/lat, we can express the solution in the following form:. F.4) The discretization effect is incorporated into this function asC−2(Ma)2. At the finest grid spacing, the deviation is ~ 20%; therefore we correct the incorrect setting of the valence quark mass in the continuum extrapolation.
Schroder, "The five-loop Beta function for a general gauge set and anomalous dimensions beyond the Feynman gauge," JHEP. Sturm, “Four-loop moments of the heavy quark vacuum polarization function in perturbative QCD”, Eur.
As a test of our renormalization procedure, we calculate the renormalization constant of the vector current.
We define the vector current renormalization constant and discuss extensions to other current operators in Appendix F. By defining the beta function, we obtain the relation between the scale µ and the coupling constants, . From the definition of the anomalous dimension γm and the function β, the general form of the solution can be written as
There is no irregular dimension for the vector current, and the renormalization constant is independent of the renormalization scale up to the clipping error. As we will discuss later, the masses of some mesons are written in terms of quark masses.
Consequently, the Borel transform in this channel is not well described by perturbative QCD, even atM = 2 GeV. In this appendix, we propose a renormalization method based on the Borel transform following SVZ. The errors are due to the truncation of perturbative expansion, the finite mass correction, and the choice of the fit range as well as the Chebyshev expansion.
To reproduce this meson mass ratio, we need to set the bare quark mass. In classical theory, the Lagrangian is invariant with respect to the U(1)Atransformationψi →eαγ5ψi; however, the U(1) asymmetry is broken by the anomaly. We need to set the bare strange mass of the quark to satisfy this relation, since we know from experiments the value 2M2K−M2π '0.47 GeV2.
Noaki (JLQCD), "Grid computation of coordinate space vector and axial vector current correlators in QCD", Phys. Kim, "Spectral functions at small energies and the electrical conductivity in hot quenched lattice QCD", Phys. Robaina, "From deep inelastic scattering to heavy-flavor semileptonic decay: Total rates in multihadron final states from lattice QCD", Phys.
Same as Fig. 5.7 but on the fine (top panel) and finest (bottom panel)
Same as Fig. 5.8 but after including the correction δ Π. ˜
Shintani, "αs from lattice hadronic vacuum polarization", (2018). HPQCD), "High-Precision Charm-Quark Mass from Current-Current Correlators in Lattice and Continuum QCD", Phys. Hashimoto, “Charmonium short-range correlator on a Möbius domain wall fermion lattice and charm quark mass determination”, Phys. Lepage, “High-Precision c and b Masses, and QCD Coupling from Current-Current Correlators in Lattice and Continuum QCD”, Phys.
