It is recalled why, as already stated in a previous paper, there seems to be an inconsistency in identifying the non perturbative ZA,MOM as the renormalisation of the axial current, or equivalently, in setting as normalisation condition that the renormalised vertex=1 at p^2 = mu^2 at some renormalisation scale mu, where p is the momentum in the legs. Indeed, unlike the vector case, the Ward-Takahashi (WT) identity for the axial current is shown to imply both the renormalisation scale independence of ZA and a mu2 dependence of ZA,MOM. This mu^2 dependence is simply related to certain invariants in the pseudoscalar vertex and can persist in the chiral limit due to the spontaneous breaking of chiral symmetry (pion pole). It is seen clearly in the mu^2 dependence of some lattice calculations of ZA,MOM/ZV,MOM near the chiral limit.
It is shown that the renormalisation constants of two quark operators can be accurately determined (to a precision of a few per-cent using 18 gluon configurations) using Chiral Ward identities. A method for computing renormalisation constants of generic composite operators without the use of lattice perturbation theory is proposed.
We study the influence of Gribov copies, in the Landau gauge, on the lattice renormalisation constant of the axial current, obtained from a Ward identity on quark state correlation functions, with the Clover action, in quenched SU(3) gauge theory. A comparison between the gauge invariant determination of the renorma= lization constant and the gauge dependent one is discussed. We find that the residual gauge freedom associated to Gribov copies induces observable effects, which, at the level of numerical accuracy of our simulation, are included in the statistical uncertainty inherent in a Monte …
Using chiral Ward identities, we determine the renormalisation constants of bilinear quark operators for the Sheikholeslami-Wohlert action lattice at beta=6.2. The results are obtained with a high degree of accuracy. For the vector current renormalisation constant we obtain Z_V=0.817(2)(8), where the first error is statistical and the second is due to mass dependence of Z_V. This is close to the perturbative value of 0.83. For the axial current renormalisation constant we obtain Z_A = 1.045(+10 -14), significantly higher than the value obtained in perturbation theory. This is shown to reduce t…
As the only lattice vector current that does not require renormalisation is the point-split conserved current it is convenient to have a robust, precise and computationally cheap methodology for the calculation of vector current renormalisation factors, $Z_V$. Momentum subtraction schemes, such as RI-SMOM, implemented nonperturbatively on the lattice provide such a method if it can be shown that the systematic errors, e.g. from condensates, are well controlled. We present $Z_V$ calculations for the conserved current in both the RI-SMOM and RI$'$-MOM momentum subtraction schemes as well as loca…
For Wilson and clover fermions traditional formulations of the axial vector current do not respect the continuum Ward identity which relates the divergence of that current to the pseudoscalar density. Here we propose to use a point-split or one-link axial vector current whose divergence exactly satisfies a lattice Ward identity, involving the pseudoscalar density and a number of irrelevant operators. We check in one-loop lattice perturbation theory with SLiNC fermion and gauge plaquette action that this is indeed the case including order $O(a)$ effects. Including these operators the axial Ward…
We calculate the non-forward quark matrix elements for operators with two covariant derivatives in one-loop lattice perturbation theory using Wilson fermions. These matrix elements are needed in the renormalisation of the second moment of generalised parton distributions measured in lattice QCD. For some commonly used representations of the hypercubic group we determine the sets of all mixing operators and find the matrices of mixing and renormalisation factors.
We calculate the non-forward quark matrix elements of operators with two covariant derivatives needed for the renormalisation of the second moment of generalised parton distributions in one-loop lattice perturbation theory using Wilson fermions. For some representations of the hypercubic group commonly used in simulations we determine the sets of all possible mixing operators. For those representations the one-loop mixing matrices of renormalisation factors are found. Due to non-vanishing contributions of operators with external ordinary derivatives the number of contributing operators increas…
We present the non-forward quark matrix elements of operators with one and two covariant derivatives needed for the renormalisation of the first and second moments of generalised parton distributions in one-loop lattice perturbation theory using clover fermions. For some representations of the hypercubic group commonly used in simulations we define the sets of possible mixing operators and compute the one-loop mixing matrices of renormalisation factors. Tadpole improvement is applied to the results and some numerical examples are presented.
We derive chiral Ward identities for lattice QCD with Wilson quarks and $N_\mathrm{f} \geq 3$ flavours, on small lattices with Schrödinger functional boundary conditions and vanishingly small quark masses. These identities relate the axial variation of the non-singlet pseudoscalar density to the scalar one, thus enabling the non-perturbative determination of the scale-independent ratio $Z_\mathrm{S}/Z_\mathrm{P}$ of the renormalisation parameters of these operators. We obtain results for $N_\mathrm{f}=3$ QCD with tree-level Symanzik-improved gluons and Wilson-Clover quarks, for bare gauge coup…
In the holographic approach to cosmology, cosmological observables are described in terms of correlators of a three-dimensional boundary quantum field theory. As a concrete model, we study the 3$d$ massless $SU(N)$ scalar matrix field theory. In this work, we focus on the renormalisation of the energy-momentum tensor 2-point function, which can be related to the CMB power spectra. Here we present a non-perturbative procedure to remove divergences resulting from the loss of translational invariance on the lattice, by imposing Ward identities. This will allow us to make predictions for the CMB p…
We present a determination of the strange quark mass for two flavours (nf=2) of light dynamical quarks using the axial Ward identity. The calculations are performed on the lattice using O(a) improved Wilson fermions and include a fully non-perturbative determination of the renormalisation constant. In the continuum limit we find in the MSbar scheme at 2GeV, ms = 111(6)(4)(6)MeV using the force scale r0 = 0.467fm, where the first error is statistical, the second and third are systematic due to the fit and scale uncertainties respectively. Results are also presented for the light quark mass and …
A completely non-perturbative estimate is given for gA using both quenched and unquenched O(a) improved Wilson fermions. Particular attention is paid to the determination of the axial renormalisation constant, ZA, using the Ward identity for the propagator. For the quenched case, we have results at three lattice spacings allowing a continuum extrapolation.
We investigate the viability of the quasi-temporal gauge on the lattice. This is a complete gauge fixing condition that can be implemented on the lattice at a very low computational cost. As a test case, using the Clover action, we have evaluated the (gauge invariant) renormalisation constant of the non-singlet axial current, using Ward identities extracted from quark states. Our result is in reasonable but not complete agreement with previous values obtained from Ward identities both on hadronic states and on quark states in the Landau gauge. We observe large fluctuations due to lattice Gribo…
We consider the renormalisation of composite quark-antiquark operators with one and two lattice covariant derivatives related to the lowest moments of generalised parton distributions (GPDs) and meson distribution amplitudes (DAs). Their matrix elements are calculated in one-loop lattice perturbation theory for non-zero momentum transfer. Using clover and overlap fermions we present the resulting matrices of mixing and renormalisation factors. For overlap fermions we explicitly check the absence of mixing with lower-dimensional operators of different chirality in particular representations of …
We present preliminary results of the non-perturbative computation of the RI-MOM renormalisation constants in a mass-independent scheme for the action with Iwasaki glue and four dynamical Wilson quarks employed by ETMC. Our project requires dedicated gauge ensembles with four degenerate sea quark flavours at three lattice spacings and at several values of the standard and twisted quark mass parameters. The RI-MOM renormalisation constants are obtained from appropriate O(a) improved estimators extrapolated to the chiral limit.
We present results for the first two moments of the light-cone distribution amplitudes of the pion and kaon pseudo-scalar mesons and of the rho, K* and phi vector mesons. The calculations are performed on the RBC/UKQCD collaborations' ensembles generated with the Iwasaki gauge action and with 2+1 flavours of domain wall fermions. In addition we also provide some results on the necessary non-perturbative renormalisation which we perform using the Rome-Southampton method. We discuss the benefits of the momentum source approach such as much smaller statistical errors and the possibility to see ef…
We present results for the renormalisation constants of bilinear quark operators, using the Nf=2 twisted mass Wilson action at maximal twist (which guarantees automatic O(a) improvement) and the tree-level Symanzik improved gauge action. The scale-independent renormalisation constants are computed with a new method, which makes use of both standard twisted mass and Osterwalder-Seiler fermions. Moreover, the results from an RI-MOM calculation are presented for both scale independent and scale dependent renormalisation constants.
We make use of twisted boundary conditions for off-shell Rome-Southampton renormalisation. This allows to define the vertex function precisely, at a fixed physical momentum that need not be one of the Fourier modes of a simulation. This definition includes choosing the orientation with respect to lattice axes, and so lattice artefacts including ${\cal O}(4)$ breaking then have a valid Symanzik expansion. Excellent statistical precision is afforded by volume plane-wave sources, enabling both a theoretically and statistically clean continuum limit to be taken. Thereafter all $p^2$ dependence can…
We compute lattice renormalisation constants of one-link quark operators ({\it i.e.} operators with one covariant derivative) for overlap fermions and Lüscher-Weisz gauge action in one-loop perturbation theory. Among others, such operators enter the calculation of moments of polarised and unpolarised hadron structure functions. Results are given for β=8.45, β=8.0 and mass parameter ρ=1.4, which are commonly used in numerical simulations. We apply mean field (tadpole) improvement to our results.
The perturbative and nonperturbative renormalisation of quark-antiquark operators in lattice QCD with two flavours of clover fermions is investigated within the research programme of the QCDSF collaboration. Operators with up to three derivatives are considered. The nonperturbative results based on the RI-MOM scheme are compared with estimates from one- and two-loop lattice perturbation theory.
We examine the renormalisation of flavour-diagonal vector currents in lattice QCD with the aim of understanding and quantifying the systematic errors from nonperturbative artefacts associated with the use of intermediate momentum-subtraction schemes. Our study uses the Highly Improved Staggered Quark (HISQ) action on gluon field configurations that include $n_f=2+1+1$ flavours of sea quarks, but our results have applicability to other quark actions. Renormalisation schemes that make use of the exact lattice vector Ward-Takahashi identity for the conserved current also have renormalisation fact…
We consider the renormalisation of lattice QCD operators with one and two covariant derivatives related to the first and second moments of generalised parton distributions and meson distribution amplitudes. Employing the clover fermion action we calculate their non-forward quark matrix elements in one-loop lattice perturbation theory. For some representations of the hypercubic group commonly used in simulations we determine the sets of all possible mixing operators and compute the matrices of renormalisation factors in one-loop approximation. We describe how tadpole improvement is applied to t…
We perform a comparative study of the ratio of lattice (Wilson fermion) renormalization constants Z_P/Z_S, obtained non-perturbatively from the RI/MOM renormalization conditions and from Ward Identities of on- and off-shell Green's functions. The off-shell Ward Identity used in this work relies on correlation functions with non-degenerate quark masses. We find that, due to discretization effects, there is a 10-15% discrepancy between the two Ward Identity determinations at current bare couplings (beta values). The RI/MOM result is in the same range and has a similar systematic error of 10-15%.…
We present the results of a non-perturbative determination of the improvement coefficient $c_\mathrm{V}$ and the renormalisation factor $Z_\mathrm{V}$, which define the renormalised vector current in three-flavour $\mathrm{O}(a)$ improved lattice QCD with Wilson quarks and tree-level Symanzik-improved gauge action. In case of the improvement coefficient, we consider both lattice descriptions of the vector current, the local as well as the conserved (i.e., point-split) one. Our improvement and normalisation conditions are based on massive chiral Ward identities and numerically evaluated in the …
Using non-perturbative techniques we have found the renormalisation factor, Z, in the RI-MOM scheme for quark bilinear operators in quenched QCD. We worked with overlap fermions using the Luescher-Weisz gauge action. Our calculation was performed at beta=8.45 at a lattice spacing of 1/a=2.1 GeV using a value of rho=1.4. Our results show good agreement between the vector and the axial vector in the zero mass limit. This shows that overlap fermions have good chiral properties. To attempt to improve the discretisation errors in our results we subtracted the O(a^2) terms in one-loop lattice pertur…
We compute the renormalisation factors (Z-matrices) of the $ΔF=2$ four-quark operators needed for Beyond the Standard Model (BSM) kaon mixing. We work with nf=2+1 flavours of Domain-Wall fermions whose chiral-flavour properties are essential to maintain a continuum-like mixing pattern. We introduce new RI-SMOM renormalisation schemes, which we argue are better behaved compared to the commonly-used corresponding RI-MOM one. We find that, once converted to MS, the Z-factors computed through these RI-SMOM schemes are in good agreement but differ significantly from the ones computed through the RI…
We present new results for the renormalisation and subtraction constants for the four fermion Delta F=2 operators, computed non-perturbatively in the RI-MOM scheme (in the Landau gauge). From our preliminary analysis of the lattice data at beta=6.45, for the B-bar B mixing bag-parameter we obtain B_B^{RGI} = 1.46(7)(1).
Lattice QCD calculations of form factors for rare Standard Model processes such as $B \to K \ell^+ \ell^-$ use tensor currents that require renormalisation. These renormalisation factors, $Z_T$, have typically been calculated within perturbation theory and the estimated uncertainties from missing higher order terms are significant. Here we study tensor current renormalisation using lattice implementations of momentum-subtraction schemes. Such schemes are potentially more accurate but have systematic errors from nonperturbative artefacts. To determine and remove these condensate contributions w…
A lattice QCD calculation of the kaon $B$ parameter $B_K$ is carried out with the Wilson quark action in the quenched approximation at $β=6/g^2=5.9-6.5$. The mixing problem of the $Δs=2$ four-quark operators is solved non-perturbatively with full use of chiral Ward identities employing four external quarks with an equal off-shell momentum in the Landau gauge. This method, without invoking any effective theory, enables us to construct the weak four-quark operators exhibiting good chiral behavior. Our results for $B_K$ with the non-perturbative mixing coefficients show small scaling violation be…
We discuss the renormalisation properties of the full set of $ΔF=2$ operators involved in BSM processes, including the definition of RGI versions of operators that exhibit mixing under RG transformations. As a first step for a fully non-perturbative determination of the scale-dependent renormalization factors and their runnings, we introduce a family of appropriate Schrödinger Functional schemes, and study them in perturbation theory. This allows, in particular, to determine the NLO anomalous dimensions of all $ΔF=1,2$ operators in these schemes. Finally, we discuss the systematic uncertaintie…
Single fermionic degrees of freedom together with standard chiral symmetry at finite lattice spacing, correct continuum limit and local interactions only are precluded by the Nielsen-Ninomiya no-go theorem. The class of minimally doubled fermion actions exhibits exactly two chiral modes. Recent interest in these actions has been sparked by the investigation of fermionic actions defined on "hyperdiamond" lattices. Due to the necessity of breaking hypercubic symmetry explicitly, radiative corrections generate operator mixings with relevant and marginal operators that should vanish in continuum Q…
The chiral susceptibility is given by the scalar vacuum polarisation at zero total momentum. This follows directly from the expression for the vacuum quark condensate so long as a nonperturbative symmetry preserving truncation scheme is employed. For QCD in-vacuum the susceptibility can rigorously be defined via a Pauli-Villars regularisation procedure. Owing to the scalar Ward identity, irrespective of the form or Ansatz for the kernel of the gap equation, the consistent scalar vertex at zero total momentum can automatically be obtained and hence the consistent susceptibility. This enables ca…
We calculate the intrinsic quark spin contribution to the total proton spin using overlap valence quarks on three ensembles of $2+1$-flavor RBC/UKQCD domain-wall configurations with different lattice spacings. The lowest pion mass of the ensembles is around 171 MeV which is close to the physical point. With overlap fermions and topological charge derived from the overlap operator, we verify the anomalous Ward identity between nucleon states with momentum transfer. Both the connected and disconnected insertions of the axial-vector current are calculated. For the disconnected-insertion part, the…
We discuss a general strategy to compute the coefficients of the QCD chiral Lagrangian using lattice QCD with Wilson fermions. This procedure requires the introduction of a lattice chiral Lagrangian as an intermediate step in the calculation. The QCD chiral Lagrangian is then obtained by expanding the lattice effective theory in increasing powers of the lattice spacing and the external momenta. In order to investigate the consequences of the chiral symmetry breaking induced by the Wilson term, we study the lattice chiral Lagrangian at the leading order of the strong coupling and large $N$ expa…
We establish that QED3 can possess a critical number of flavours, N_f^c, associated with dynamical chiral symmetry breaking if, and only if, the fermion wave function renormalisation and photon vacuum polarisation are homogeneous functions at infrared momenta when the fermion mass function vanishes. The Ward identity entails that the fermion-photon vertex possesses the same property and ensures a simple relationship between the homogeneity degrees of each of these functions. Simple models for the photon vacuum polarisation and fermion-photon vertex are used to illustrate these observations. Th…
