Perturbative $O(α_s a)$ matching in static heavy and domain-wall light quark system
Tomomi Ishikawa
备注：
7 Pages, 1 figure, talk presented at The XXVI International Symposium on Lattice Field Theory (LATTICE 2008), July 14-19, 2008, Williamsburg, Virginia, USA
We discuss the perturbative $O(α_s a)$ matching in the static heavy and domain-wall light quark system. The gluon action is the Iwasaki action and the link smearing is performed in the static heavy action. The chiral symmetry of the light quark realized by using the domain-wall fermion formulation does not prohibit the mixing of the operators at $O(a)$. The application of $O(a)$ improvement to the actual data shows that the B meson decay constant $f_B$, the matrix elements ${\cal M}_B$ and the B parameter $B_B$ have non-negligible effects, while the effect on the SU(3) breaking ratio $ξ$ is small.
We present results from a study of the renormalisation of both quark bilinear and four-quark operators for the domain wall fermion action, using the non-perturbative renormalisation technique of the Rome-Southampton group. These results are from a quenched simulation, on a 16^3 x 32 lattice, with beta=6.0 and L_s=16.
The viability of the Non-Perturbative Renormalisation (NPR) method of the Rome/Southampton group is studied, for the first time, in the context of domain wall fermions. The procedure is used to extract the renormalisation coefficients of the various quark bilinears, as well as the four-fermion operators relevant to the $ΔS=2$ effective Hamiltonian. The renormalisation of the $ΔS=1$ Hamiltonian is also discussed.
We formulate and numerically test the Schrödinger functional scheme for domain-wall QCD. We then apply it to a non-perturbative calculation of the renormalization factors for vector and axial-vector currents in quenched domain-wall QCD with plaquette and renormalization group improved gauge actions at $a^{-1}\simeq 2$ GeV.
Renormalization factors for $ΔS =1$ four-quark operators in the effective weak Hamiltonian are perturbatively evaluated in domain-wall QCD. The one-loop corrections of $ΔS=1$ four-quark operators consist of two types of diagrams: one is gluon exchange between quark lines and the other is penguin diagrams containing quark loops. Combining both contributions, our results allow a lattice calculation of the amplitude for $K\to ππ$ decays with $O(g^2)$ corrections included.
We present non-perturbative renormalization factors for $ΔS=2$ four-quark operators in quenched domain-wall QCD using the Schroedinger functional method. Non-perturbative renormalization factor for $B_K$ is evaluated at hadronic scale. Combined with the non-perturbative RG running obtained by the Alpha collaboration, our result yields renormalization factor which converts lattice bare $B_K$ to the renormalization group invariant one. We apply the renormalization factor to bare $B_K$ previously obtained by the CP-PACS collaboration with the quenched domain-wall QCD(DWQCD). We compare our result…
We calculate one-loop renormalization factors for heavy-light bilinears as well as four-fermion operators relevant for $B^{0} - \bar{B}^{0}$ mixing calculations on the lattice. We use the static approximation for heavy quarks and the domain-wall formulation for light quarks. We present results for different choices of improved gauge action.
We compute non--perturbatively the renormalization coefficients of scalar and pseudoscalar operators, local vector and axial currents, conserved vector and axial currents, and $O_{LL}^{ΔS=2}$ over a wide range of energy scales using a scaling technique that connects the results of simulations at different values of coupling $β$. We use the domain wall fermion formulation in the quenched approximation at a series of three values of $β, 6.0, 6.45, 7.05$ corresponding to lattice spacing scaling by factors of two.
We evaluate renormalization factors of the domain-wall fermion system with various improved gauge actions at one loop level. The renormalization factors are calculated for quark wave function, quark mass, bilinear quark operators, three- and four-quark operators in modified minimal subtraction (MS-bar) scheme with the dimensional reduction(DRED) as well as the naive dimensional regularization(NDR). We also present detailed results in the mean field improved perturbation theory.
We study the non-perturbative determination of the renormalization constants of flavor non-singlet quark bilinear operators on the lattice. The renormalization condition is imposed on correlation functions of bilinear operators in the coordinate space. The results are converted to the value at 2 GeV in the $\rm\overline{MS}$ scheme by a perturbative matching. The calculation is carried out on gauge configurations generated with the Mobius domain-wall fermions at two lattice spacings $a^{-1} = 2.4$ GeV and $a^{-1} = 3.6$ GeV.
We compute non-perturbatively the renormalization coefficients of scalar and pseudoscalar operators, local vector and axial currents, conserved vector and axial currents, and $O^{ΔS=2}_{LL}$ over a wide range of energy scales using a scaling technique that connects the results of simulations at different values of coupling $β$. We use the domain wall fermion formulation in the quenched approximation at a series of three values of $β= 6.0, 6.45, 7.05$ corresponding to lattice spacing scaling by factors of two.
We investigate a U(1) chiral gauge model in 4+1 dimensions formulated on the lattice via the domain-wall method. We calculate an effective action for smooth background gauge fields at a fermion one loop level. From this calculation we discuss properties of the resulting 4 dimensional theory, such as gauge invariance of 2 point functions, gauge anomalies and an anomaly in the fermion number current.
Renormalization factors for three- and four-quark operators, which appear in the low energy effective Lagrangian of the proton decay and the weak interactions, are perturbatively calculated in domain-wall QCD. We find that the operators are multiplicatively renormalizable up to one-loop level without mixing with any other operators that have different chiral structures. As an application, we evaluate a renormalization factor for $B_K$ at the parameters where previous simulations have been performed, and find one-loop corrections to $B_K$ are 1-5% in these cases.
We present a calculation of the renormalization coefficients of the quark bilinear operators and the K-Kbar mixing parameter B_K. The coefficients relating the bare lattice operators to those in the RI/MOM scheme are computed non-perturbatively and then matched perturbatively to the MSbar scheme. The coefficients are calculated on the RBC/UKQCD 2+1 flavor dynamical lattice configurations. Specifically we use a 16^3 x 32 lattice volume, the Iwasaki gauge action at beta=2.13 and domain wall fermions with L_s=16.
We present our strategy and some preliminary results for the renormalization of four-quark operators relevant for kaon physics. We follow the non-perturbative Rome-Southampton method, with both exceptional and non-exceptional kinematics. We also implement momentum sources and twisted boundary conditions. We use an (almost) unitary setup: Domain-Wall valence on nf=2+1 Domain-Wall sea and Iwasaki gauge action, at two values of the lattice spacing corresponding to approximately 0.086 fm and 0.114 fm . The chiral properties of these fermions play a crucial role in this compuation and are studied i…
We calculate one-loop renormalization factors of generic DeltaS=2 four-quark operators for domain-wall QCD with the plaquette gauge action and the Iwasaki gauge action. The renormalization factors are presented in the modified minimal subtraction (MS-bar) scheme with the naive dimensional regularization. As an important application we show how to construct the renormalization factors for the operators contributing to K^0-K^0bar mixing in the supersymmetric models with the use of our results.
We present renormalization constants of overlap quark bilinear operators on 2+1-flavor domain wall fermion configurations. Both overlap and domain wall fermions have chiral symmetry on the lattice. The scale independent renormalization constant for the local axial vector current is computed using a Ward Identity. The renormalization constants for the scalar, pseudoscalar and vector current are calculated in the RI-MOM scheme. Results in the MS-bar scheme are obtained by using perturbative conversion ratios. The analysis uses in total six ensembles with lattice sizes 24^3x64 and 32^3x64.
We calculate one-loop renormalization factors of several quark operators including bilinear, three- and four-quark operator for domain-wall fermion action. Since Green functions are constructed for external physical quark fields, our renormalization method is simple and can be easily applied to calculation of any quark operators. Our results show that these renormalized quark operators preserve several chiral properties of continuum massless QCD, which can be understood by the property of external quark line propagator.
Using the domain-wall formulation of lattice fermions, we have computed the one-loop renormalization factors of one-link operators which measure the first nontrivial moment of the unpolarized, polarized and transversity structure functions, in the flavor nonsinglet sector. The knowledge of these factors is necessary in order to extract physical numbers from domain-wall Monte Carlo simulations of parton distributions. We have automated the perturbative calculations by developing suitable FORM codes. The results show that in many instances the total renormalization factors are almost equal to on…
We present further results for one-loop matching of heavy-light axial and vector currents between continuum QCD and a lattice theory with NRQCD heavy quarks and massless clover quarks.
We calculate one-loop renormalization factors of bilinear operators made of physical quark fields for domain-wall QCD. We find that finite parts of such renormalization factors have reasonable values at 1-loop except an overlap factor between the physical quark field and the zero mode in the theory. We point out that the 1-loop estimate of overall renormalization factors becomes unreliable at the coupling where numerical simulations are currently performed, due to the presence of this overlap factor. We show that this problem disappears if the mean-field improved perturbation theory is employe…
We report on a non-perturbative evaluation of the renormalization factors for the vector and axial-vector currents, $Z_V$ and $Z_A$, in the quenched domain-wall QCD (DWQCD) with plaquette and renormalization group improved gauge actions. We take the Dirichlet boundary condition for both gauge and domain-wall fermion fields on the finite box, and introduce the flavor-chiral Ward-Takahashi identities to calculate the renormalization factors. As a test of the method, we numerically confirm the expected relation that $Z_V \simeq Z_A$ in DWQCD. Employing two different box sizes for the numerical si…
We present results for the light meson masses, the bare strange quark mass and preliminary non-perturbative renormalisation of B_K in 2+1 flavour domain wall QCD. The ensembles used were generated with the Iwasaki gauge action and have a volume of 16^3 x 32 with a fifth dimension size of 16 and an inverse lattice spacing of 1.6 GeV. These ensembles have u and d masses as low as one quarter of the strange quark mass. All data were generated jointly by the UKQCD and RBC collaborations on QCDOC machines.
We determine the mass dependent renormalization as well as improvement coefficients for the heavy-light vector and axial-vector currents consisting of the relativistic heavy and the domain-wall light quarks through the standard matching procedure. The calculation is carried out perturbatively at the one loop level to remove the systematic error of $O(α_s (am_Q)^n a\p)$ as well as $O(α_s (am_Q)^n)$ ($n\ge$0), where $\p$ is a typical momentum scale in the heavy-light system. We point out that renormalization and improvement coefficients of the heavy-light vector current agree with those of the a…
A one-loop matching calculation between Lattice NRQCD and full QCD for the heavy-light axial current is described and the effects on f_B are discussed.
We find the renormalisation coefficients of the quark field and the flavour non-singlet fermion bilinear operators for the domain wall fermion action, in the regularisation independent (RI) renormalisation scheme. Our results are from a quenched simulation, on a 16^3x32 lattice, with beta=6.0 and an extent in the fifth dimension of 16. We also discuss the expected effects of the residual chiral symmetry breaking inherent in a domain wall fermion simulation with a finite fifth dimension, and study the evidence for both explicit and spontaneous chiral symmetry breaking effects in our numerical r…
We determine the renormalization constants for flavor non-singlet fermion bilinear operators of Möbius domain-wall fermions. The renormalization condition is imposed on the correlation functions in the coordinate space, such that the non-perturbative lattice calculation reproduces the perturbatively calculated counterpart at short distances. The perturbative expansion is precise as the coefficients are available up to $O(α_s^4)$. We employ $2+1$-flavor lattice ensembles at three lattice spacings in the range 0.044--0.080~fm.
We discuss the theoretical input into the current RBC-UKQCD calculation of $f_{B_{d, s}}$ and $B_{B_{d, s}}$ using a smeared static heavy quark propagator, light domain-wall quarks and the Iwasaki gauge action. We present the complete one-loop, mean-field improved matching of heavy-light current and four-fermion lattice operators onto the static continuum theory renormalized in $\bar{\text{MS}}$(NDR). The large degeneracies present in a static calculation are addressed, and a method for extracting $f_B$ and $B_B$ using only box sources is described; implications for future calculations are dis…
We present results of a perturbative matching calculation performed at one-loop for heavy-light currents. We use the Fermilab action for the heavy quarks, the Asqtad action for the light quarks, and an improved gluon action. We also present results for heavy-heavy currents with Fermilab heavy quarks and improved glue.
We discuss perturbative O(g^2a) matching with static heavy quarks and domain-wall light quarks for lattice operators relevant to B-meson decays and $B^0$-$\bar{B}^0$ mixing. The chiral symmetry of the light domain-wall quarks does not prohibit operator mixing at O(a) for these operators. The O(a) corrections to physical quantities are non-negligible and must be included to obtain high-precision simulation results for CKM physics. We provide results using plaquette, Symanzik, Iwasaki and DBW2 gluon actions and applying APE, HYP1 and HYP2 link-smearing for the static quark action.
We propose to determine the coefficients in the Fermilab heavy quark action by matching the finite-volume, off-shell, gauge-fixed propagator and vertex functions with those determined in the exact, relativistic theory. The matching relativistic amplitudes may be determined either from short-distance perturbation theory or from finite-volume step-scaling recursion. Tree-level matching and non-perturbative off-shell amplitudes are presented.
We investigate, in the framework of perturbation theory at finite $N_s$, the effectiveness of improved gauge actions in suppressing the chiral violations of domain-wall fermions. Our calculations show substantial reductions of the residual mass when it is compared at the same value of the gauge coupling, the largest suppression being obtained when the DBW2 action is used. Similar effects can also be observed for a power-divergent mixing coefficient which is chirally suppressed. No significant reduction instead can be seen in the case of the difference between the vector and axial-vector renorm…
We calculate the one-loop matching coefficients between continuum and lattice four-fermion operators for lattice operators constructed using staggered fermions and improved by the use of fattened links. In particular, we consider hypercubic fat links and SU(3) projected Fat-7 links, and their mean-field improved versions. We calculate only current-current diagrams, so that our results apply for operators whose flavor structure does not allow ``eye-diagrams''. We present general formulae, based on two independent approaches, and give numerical results for the cases in which the operators have t…
The use of Heavy Quark Effective Theory (HQET) on the lattice as an approach to B-physics phenomenology is based on a non-perturbative matching of HQET to QCD in finite volume. As a first step to apply the underlying strategy in the three-flavor ($N_f = 2+1$) theory, we determine the renormalization constant and improvement coefficients relating the renormalized current and subtracted quark mass of (quenched) valence quarks in $\mathcal{O}(a)$ improved $N_f=3$ lattice QCD. We present our strategy and first results for the relevant parameter region towards weak couplings along a line of constan…
We present renormalization constants of overlap quark bilinear operators on 2+1-flavor domain wall fermion configurations. This setup is being used by the chiQCD collaboration in calculations of physical quantities such as strangeness in the nucleon and the strange and charm quark masses. The scale independent renormalization constant for the axial vector current is computed using the Ward Identity. The renormalization constants for scalar, pseudoscalar and vector current are calculated in the RI-MOM scheme. Results in the MS-bar scheme are also given. The step scaling function of quark masses…
We propose a strategy to non-perturbatively match the Wilson coefficients in the three- and four-flavor theories, which uses two-point Green's functions of the corresponding four-quark operators at long distances. The idea is refined by combining with the spherical averaging technique, which enables us to convert two-point functions calculated on the lattice into continuous functions of the distance $|x-y|$ between two operators. We also show the result for an exploratory calculation of two-point functions of the $ΔS=1$ operators $Q_7$ and $Q_8$ that are in the $(8_L,8_R)$ representation of ${…
We give an introduction to the special problems encountered in a treatment of HQET beyond perturbation theory in the gauge coupling constant. In particular, we report on a recent test of HQET as an effective theory for QCD and discuss how HQET can be implemented on the lattice including the non-perturbative matching of the effective theory to QCD.
We perform lattice calculations of the spatial 't Hooft k-string tensions in the deconfined phase of SU(N) gauge theories for N=2,3,4,6. These equal (up to a factor of T) the surface tensions of the domain walls between the corresponding (Euclidean) deconfined phases. For T much larger than T_c our results match on to the known perturbative result, which exhibits Casimir Scaling, being proportional to k(N-k). At lower T the coupling becomes stronger and, not surprisingly, our calculations show large deviations from the perturbative T-dependence. Despite this we find that the behaviour proporti…
We study the chiral condensate, $<\barψψ>$, and various quark bilinear vertex functions for domain wall fermions at different lattice scales, with both the Wilson and DBW2 gauge actions, in both quenched and dynamical fermion simulations. We use the vertex functions to calculate renormalization factors within a non-perturbative scheme.
We explain how the strategy of solving renormalization problems in HQET non-perturbatively by a matching to QCD in finite volume can be implemented to include dynamical fermions. As a primary application, some elements of an HQET computation of the mass of the b-quark beyond the leading order with N_f=2 are outlined. In particular, the matching of HQET and QCD requires relativistic QCD simulations in a volume with L ~ 0.5 fm, which will serve to quantitatively determine the heavy quark mass dependence of heavy-light meson observables in the continuum limit of finite-volume two-flavour lattice …
By studying an explicit analytical solution of the Coulomb problem on the lattice we demonstrate a breakdown of perturbative matching for the description of the Coulomb artifacts in lattice NRQCD, which leads to a large systematic error in the predictions for the heavy quarkonium spectrum. The breakdown is a result of a fine interplay between the short and long distance effects specific to the lattice regularization of NRQCD. We show how the problem can be solved within the Schrodinger matching procedure.
We present preliminary results of matching factors of the four-fermion operators relevant to $B_K$, which are obtained using the non-perturbative renormalization (NPR) method in the RI-MOM scheme with HYP-smeared improved staggered fermions. We use the MILC asqtad coarse ($a \cong 0.12\,$fm) ensembles with $20^3 \times 64$ geometry and $am_{\ell}/am_s = 0.01/0.05$. We compare NPR results with those of one-loop perturbative matching.
We report on our on-going project to calculate the nucleon decay matrix elements with domain-wall fermions. Operator mixing is discussed employing a non-perturbative renormalization. Bare matrix elements of all the possible decay modes induced by the dimension-six operators are calculated with the direct method, which are compared with the indirect calculation using chiral perturbation theory.
We report on our computation of the hadronic matrix elements of the four-quark operators needed for the study of $K^0-{\bar K^0}$ mixing beyond the Standard Model (SM). We consider nf=2+1 Domain-Wall fermions on Iwasaki gauge action with lightest unitary pion of 290 MeV and a single lattice spacing a=0.086 fm. The renormalization is performed non-perturbatively through the RI-MOM scheme and our results are converted perturbatively to MSbar. We have estimated the various systematic errors. Our results confirm a previous quenched study, where large ratios of non-SM to SM matrix elements were obt…
We compute the renormalization factors of four-quark operators needed for the study of $K\toππ$ decay in the $ΔI=3/2$ channel. We evaluate the Z-factors at a low energy scale ($μ_0=1.145 \GeV$) using four different non-exceptional RI-SMOM schemes on a large, coarse lattice ($a\sim 0.14\fm$) on which the bare matrix elements are also computed. Then we compute the universal, non-perturbative, scale evolution matrix of these renormalization factors between $μ_0$ and $3\GeV$. We give the numerical results for the different steps of the computation in two different non-exceptional lattice schemes, …
The temporal and spatial components of the heavy-light vector current and the spatial components of the axial current are expressed in terms of lattice-regulated operators suitable for simulations of B and D mesons. The currents are constructed by matching the appropriate scattering amplitudes in continuum QCD and a lattice model to one-loop order in perturbation theory. In the lattice theory, the heavy quarks are treated using the nonrelativistic (NRQCD) formulation and the light quarks are described by the tadpole-improved clover action. The light quarks are treated as massless. Our currents…
We give a progress report of our new $a^{-1}\approx 3$ GeV quenched calculation of kaon matrix elements with domain-wall fermion and DBW2 gauge action. Our smaller lattice spacing allows us to address the effect of charmed quark on the lattice. We show preliminary results of $B_K$ renormalized non-perturbatively and $K\toπ$ matrix elements.
Central and spin-spin potentials for charmonium, constructed from Nambu-Bethe-Salpeter amplitudes in lattice simulations of full QCD, are matched with results from perturbative QCD at an appropriate distance scale. This matching is made possible by defining the perturbative potentials through Fourier transforms with a low-momentum cutoff. The central (spin-independent) potential is compared with potentials derived from an expansion in powers of the inverse quark mass. A well-controlled continuation of the charmonium spin-spin potential from lattice QCD to short distances is performed. The mass…
We report the status of RBCK calculations on nucleon structure with quenched and dynamical domain wall fermions. The quenched results for the moments of structure functions (< x >_q), (< x>_{Δu - Δd}), and (< 1 >_{δq}) from 1.3 GeV cutoff lattices are complete with non perturbative renormalization (NPR). The dynamical results with two degenerate dynamical quark flavors from 1.7 GeV cutoff lattices are without NPR while the axial charge result is naturally renormalized.
Using staggered fermions, we calculate the perturbative corrections to the bilinear and four-fermion operators that are used in the numerical study of weak matrix elements for $ε'/ε$. We present results for one-loop matching coefficients between continuum operators, calculated in the Naive Dimensional Regularization (NDR) scheme, and gauge invariant staggered fermion operators. These results, combined with existing results for penguin diagrams, provide the complete one-loop renormalization of the staggered four-fermion operators.
Matching of the quasi parton distribution functions between continuum and lattice is addressed using lattice perturbation theory specifically with Wilson-type fermions. The matching is done for nonlocal quark bilinear operators with a straight Wilson line in a spatial direction. We also investigate operator mixing in the renormalization and possible O(a) operators for the nonlocal operators based on a symmetry argument on lattice.
We present a first calculation of the strange quark mass using domain wall fermions. This paper contains an overview of the domain wall discretization and a pedagogical presentation of the perturbative calculation necessary for computing the mass renormalization. We combine the latter with numerical simulations to estimate the strange quark mass. Our final result in the quenched approximation is 95(26) MeV in the ${\bar{MS}}$ scheme at a scale of 2 GeV. We find that domain wall fermions have a small perturbative mass renormalization, similar to Wilson quarks, and exhibit good scaling behavior.
We calculate non-perturbative renormalization factors at hadronic scale for $ΔS=2$ four-quark operators in quenched domain-wall QCD using the Schrödinger functional method. Combining them with the non-perturbative renormalization group running by the Alpha collaboration, our result yields the fully non-perturbative renormalization factor, which converts the lattice bare $B_K$ to the renormalization group invariant (RGI) $\hat{B}_K$. Applying this to the bare $B_K$ previously obtained by the CP-PACS collaboration at $a^{-1}\simeq 2, 3, 4$ GeV, we obtain $\hat{B}_K=0.782(5)(7)$ (equivalent to $B…
Heavy Quark Effective Theory (HQET) computations of semi-leptonic decays, e.g. B -> pi l nu, require the knowledge of the parameters in the effective theory for all components of the heavy-light flavor currents. So far non-perturbative matching conditions have been employed only for the time component of the axial current. Here we perform a check of matching conditions for the time component of the vector current and the spatial component of the axial vector current up to one-loop order of perturbation theory and to lowest order of the 1/m-expansion. We find that the proposed observables have …
We calculate charmonium correlators on the lattice with 2+1-flavors of sea quarks and charm valence quark both described by the Möbius domain-wall fermion. Temporal moments of the correlators are calculated and matched to perturbative QCD formulae to extract the charm quark mass $m_c(μ)$ and strong coupling constant $α_s(μ)$. Lattice data at three lattice spacings, 0.044, 0.055, and 0.080~fm, are extrapolated to the continuum limit. The correlators in the vector channel are confirmed to be consistent with the experimental data for $e^+e^-\to c\bar{c}$, while the pseudo-scalar channel is used t…
In perturbation theory, the wave function of domain-wall quarks decreases exponentially with the fifth coordinate. We show that, regardless of the quark's own momentum, the fall-off rate of the one-loop wave function is equal to the slowest rate encountered at tree-level for any lattice four-momentum. We propose new domain-wall actions involving beyond-nearest neighbor couplings in the four physical dimensions, for which the perturbative wave function decreases much faster. It is hoped that the new actions may preserve the good chiral properties of domain-wall fermions up to larger values of t…
We present preliminary results of a study of Delta S = 2 matrix elements originating from physics beyond the Standard Model. Using 2+1 flavour Domain Wall Fermions we obtain the non-perturbative renormalisation (mixing) matrix in the RI scheme. We also discuss plans for the chiral extrapolation of the renormalised matrix elements in a partially quenched set-up.
We report on the nucleon decay matrix elements with domain-wall fermions in quenched approximation. Results from direct and indirect method are compared with a focus on the process of a proton decaying to a pion and a lepton. We discuss the renormalization necessary for the matching to the continuum theory. Preliminary results for the renormalized chiral lagrangian parameters are presented.
We present results for the QED and strong isospin breaking corrections to the hadronic vacuum polarization using $N_f=2+1$ Domain Wall fermions. QED is included in an electro-quenched setup using two different methods, a stochastic and a perturbative approach. Results and statistical errors from both methods are directly compared with each other.
Perturbation theory for lattice fermions with domain wall mass terms is developed and is applied to investigate the chiral Schwinger model formulated on the lattice by Kaplan's method. We calculate the effective action for gauge fields to one loop, and find that it contains a longitudinal component even for anomaly-free cases. From the effective action we obtain gauge anomalies and Chern-Simons current without ambiguity. We also show that the current corresponding to the fermion number has a non-zero divergence and it flows off the wall into the extra dimension. Similar results are obtained fo…
