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 effects of the discretisation.
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.
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 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.
We present results for the non-perturbative renormalisation of four-fermion operators with two flavours of dynamical quarks. We consider both fully relativistic left current-left current operators, and a full basis for $ΔB=2$ operators with static heavy quarks. The renormalisation group running of the operators to high energy scales is computed in the continuum limit for a family of Schroedinger Functional renormalisation schemes, via standard finite size scaling techniques. The total renormalisation factors relating renormalisation group invariant to bare operators are computed for a choice o…
We outline a general strategy for the non-perturbative renormalisation of composite operators in discretisations based on Neuberger fermions, via a matching to results obtained with Wilson-type fermions. As an application, we consider the renormalisation of the four-quark operators entering the Delta S=1 and Delta S=2 effective Hamiltonians. Our results are an essential ingredient for the determination of the low-energy constants governing non-leptonic kaon decays.
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 compute the renormalised four-fermion operator $O^{ΔS=2}$ using a non-perturbative method recently introduced for determining the renormalisation constants of generic lattice composite operators. Because of the presence of the Wilson term, $O^{ΔS=2}$ mixes with operators of different chiralities. A projection method to determine the mixing coefficients is implemented. The numerical results for the renormalisation constants have been obtained from a simulation performed using the SW-Clover quark action, on a $16^3 \times 32$ lattice, at $β=6.0$. We show that the use of the constants determin…
By considering the local vector current between nucleon states and imposing charge conservation, we determine its renormalisation constant and quark mass improvement coefficient for Symanzik $O(a)$ improved Wilson fermions. The computation is first performed for quenched fermions (and for completeness also with unimproved fermions) and compared against known results. The two-flavour unquenched case is then considered.
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.
Some basic concepts are discussed to derive renormalisation factors of local lattice operators relevant to deep inelastic structure functions and to other measurable quantities. These $Z$ factors can be used to relate matrix elements measured by lattice techniques to their continuum counterparts. We discuss the $O(a)$ improvement of point and one-link lattice quark operators. Suitable bases of improved operators are derived. Tadpole improvement is applied to get more reliable perturbative results.
Using Schroedinger Functional methods, we compute the non-perturbative renormalisation and renormalisation group running of several four-fermion operators, in the framework of lattice simulations with two dynamical Wilson quarks. Two classes of operators have been targeted: (i) those with left-left current structure and four propagating quark fields; (ii) all operators containing two static quarks. In both cases, only the parity-odd contributions have been considered, being the ones that renormalise multiplicatively. Our results, once combined with future simulations of the corresponding latti…
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 calculate lattice renormalisation constants of local and one-link quark operators for overlap fermions and improved gauge actions in one-loop perturbation theory. For the local operators we stout smear the SU(3) links in the fermionic action. Using the popular tadpole improved Lüscher-Weisz actions at $β=8.45$ and $β=8.0$ we present numerical values for the Z factors in the $\bar{MS}$ scheme (partly as function of the stout smearing strength). We compare various levels of mean field (tadpole) improvement which have been applied to our results.
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…
We study the chiral Gross-Neveu model with Wilson fermions. In the framework of the Schroedinger functional we show that in general not only the bare mass has to be tuned to achieve chiral symmetry in the continuum, but also coupling constants multiplying chirally non-invariant four fermion terms. The strategy for fixing the parameters of the theory is explained in perturbation theory and the results of a first order calculation are presented. The results are shown to agree with the infinite flavour limit.
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 perform a non-perturbative study of the scale-dependent renormalisation factors of a complete set of dimension-six four-fermion operators. The renormalisation-group (RG) running is determined in the continuum limit for a specific Schrdinger Functional (SF) renormalisation scheme in the framework of lattice QCD with two dynamical flavours ( $N_f = 2$ ). The theory is regularised on a lattice with a plaquette Wilson action and $\mathcal{O}(a)$-improved Wilson fermions. For one of these operators, the computation had been performed in ref. [1]; the present work completes the study for the rest…
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 non-perturbatively the renormalisation constants of composite operators on a $16^3 \times 32 $ lattice with lattice spacing $a$ = 0.093 fm for the overlap fermion action by using the regularisation independent (RI) scheme. The quenched gauge configurations are generated by tadpole improved plaquette plus rectangle action. We test the perturbative continuum relation $Z_A = Z_V$ and $ Z_S=Z_P$ and find that they agree well above $μ$ = 1.6 GeV. We also perform a Renormalisation Group analysis at the next-to-next-to-leading order and convert the renormalisation constants to the $\bar{MS…
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).
We discuss the renormalisation properties of the complete set of $ΔB = 2$ four-quark operators with the heavy quark treated in the static approximation. We elucidate the role of heavy quark symmetry and other symmetry transformations in constraining their mixing under renormalisation. By employing the Schroedinger functional, a set of non-perturbative renormalisation conditions can be defined in terms of suitable correlation functions. As a first step in a fully non-perturbative determination of the scale-dependent renormalisation factors, we evaluate these conditions in lattice perturbation t…
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 determine from first principles the quark mass anomalous dimension in Nf=3 QCD between the electroweak and hadronic scales. This allows for a fully non-perturbative connection of the perturbative and non-perturbative regimes of the Standard Model in the hadronic sector. The computation is carried out to high accuracy, employing massless O(a)-improved Wilson quarks and finite-size scaling techniques. We also provide the matching factors required in the renormalisation of light quark masses from lattice computations with O(a)-improved Wilson fermions and a tree-level Symanzik improved gauge a…
The renormalisation constants for local bilinear quark operators are calculated using the Sheikholeslami-Wohlert improved action. In addition we compute the renormalisation constant of the leading gluon operator for different group representations and discuss the mixing of the operators E^2 and B^2.
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…
When several four-quark operators are allowed to mix through renormalisation, this can considerably amplify the problems coming from perturbative truncation and discretisation effects. In this work we investigate whether our previous $B_K$ strategy can conveniently be generalised to a wider set of operators, corresponding to the so-called "SUSY $B_K$" basis of four-quark operators. We show that the discretisation effects, when plotted as a function of $ap$, are surprisingly independent of the lattice spacing. They appear reasonably under control up to very large energy scales. This allows us t…
We report on recent results for the pion and nucleon matrix element of the twist-2 operator corresponding to the average momentum of non-singlet quark densities. We discuss finite size effects for the nucleon matrix element and present first preliminary results for the non-perturbative renormalisation from full dynamical simulations.
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 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 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…
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…
Tensor currents are the only quark bilinear operators lacking a non-perturbative determination of their renormalisation group (RG) running between hadronic and electroweak scales. We develop the setup to carry out the computation in lattice QCD via standard recursive finite-size scaling techniques, and provide results for the RG running of tensor currents in $N_f = 0$ and $N_f = 2$ QCD in the continuum for various Schrödinger Functional schemes. The matching factors between bare and renormalisation group invariant currents are also determined for a range of values of the lattice spacing relev…
Viable candidate theories for electroweak dynamical symmetry breaking are expected to show (near) conformal behaviour in order to accommodate current phenomenological constraints. In principle, renormalisation group studies using finite volume renormalisation schemes are well-suited to verify this property in a given model. The most practical schemes are based on the Schrodinger functional (SF), but suffer from potentially large O(a) effects. Some care has to be taken to remove these effects and to set up a scheme where cutoff effects are small and under control. We here take a step in this di…
We discuss the renormalisation properties of parity-odd Delta B=2 operators with the heavy quark treated in the static approximation. Via twisted mass QCD (tmQCD), these operators provide the matrix elements relevant for the B0-B0bar mixing amplitude. The layout of a non-perturbative renormalisation programme for the operator basis, using Schroedinger Functional techniques, is described. Finally, we report our results for a one-loop perturbative study of various renormalisation schemes with Wilson-type lattice regularisations, which allows, in particular, to compute the NLO anomalous dimension…
We present the results of a perturbative calculation to match the axial and vector currents for semileptonic $B\to D^{(\ast)}$ decays in lattice NRQCD to the continuum \MSb scheme. The matching is performed to $O(α_s\frac{\vec{p}}{M})$ in Feynman gauge and in the on-shell renormalisation scheme. The spatial and temporal components of the currents renormalise differently; to this order the matching involves a straightforward renormalisation for the $V_0$ and $A_k$ currents, and a rank two and four mixing matrix for the $A_0$ and $V_k$ currents respectively. The resultant one loop corrections ar…
We calculate the 1-loop renormalization of a set of extended fermionic bilinears which form a basis corresponding to moments of the parton distribution functions. We use the overlap action for fermions and Luescher-Weisz (LW) action for gluons. Our results are presented as a function of the overlap parameter rho and the parameters entering the LW action.
A gauge invariant, non-local observable is constructed in pure gauge theory, which is identical to the gluon propagator in a particular gauge, permitting to define a non-perturbative parton mass for the gluon. This mass can be shown to be related to the 1P-1S mass splitting of heavy quarkonia. Preliminary numerical results for 3d SU(2) yield m_A=0.37(6)g^2, while from the \bar{b}b spectrum one infers m_A\approx 420 MeV for QCD.
We compute non-perturbatively the evolution of the twist-2 operators corresponding to the average momentum of non-singlet quark densities. The calculation is based on a finite-size technique, using the Schrödinger Functional, in quenched QCD. We find that a careful choice of the boundary conditions, is essential, for such operators, to render possible the computation. As a by-product we apply the non-perturbatively computed renormalization constants to available data of bare matrix elements between nucleon states.
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 calculate the 1-loop renormalization of the fermion self-energy, all local fermion bilinears, as well as a set of extended bilinears which form a basis corresponding to moments of the parton distribution functions. We use the overlap action for fermions and Symanzik improved action for gluons. Our results are presented as a function of the overlap parameter rho and the parameters entering the Symanzik action.
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 determine non-perturbatively the anomalous dimensions of the second moment of non-singlet parton densities from a continuum extrapolation of results computed in quenched lattice simulations at different lattice spacings. We use a Schrödinger functional scheme for the definition of the renormalization constant of the relevant twist-2 operator. In the region of renormalized couplings explored, we obtain a good description of our data in terms of a three-loop expression for the anomalous dimensions. The calculation can be used for exploring values of the coupling where a perturbative expansion…
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…
The renormalisation group running of the quark mass is determined non-perturbatively for a large range of scales, by computing the step scaling function in the Schroedinger Functional formalism of quenched lattice QCD both with and without O(a) improvement. A one-loop perturbative calculation of the discretisation effects has been carried out for both the Wilson and the Clover-improved actions and for a large number of lattice resolutions. The non-perturbative computation yields continuum results which are regularisation independent, thus providing convincing evidence for the uniqueness of the…
Matching Non-Perturbative Renormalisation on the lattice and perturbative renormalisation would benefit from higher matching scales, which are needed for observables entering their permile era such as BK. In this work we lay down a strategy, within the Rome-Southampton framework, to push this scale higher across the charm threshold, and apply it to an exploration of the BK running from 3 GeV to 9 GeV. This is done on Nf = 2+1+1 ensembles generated by the RBC-UKQCD collaboration, and features a close study of the discretisation effects.
We apply the Schroedinger Functional (SF) formalism to determine the renormalisation group running of four-fermion operators which appear in the effective weak Hamiltonian of the Standard Model. Our calculations are done using Wilson fermions and the parity-odd components of the operators. Preliminary results are presented for the operator $O_{VA}=(\bar s γ_μd)(\bar s γ_μγ_5 d)$.
The HPQCD collaboration has a program for determining the fundamental constants of the Standard Model Lagrangian from lattice QCD. The most accurate method of doing this uses the n_f=2+1 improved staggered MILC ensembles with chiral fitting and multi-loop perturbative renormalisation to connect to the continuum \msbar scheme. This program has already been very successful with the recent strong coupling constant determination at three-loops from 28 observables at three lattice spacings, and the one-loop light quark mass calculation last year. Here a preliminary result is presented for the first…
We present details of a lattice QCD calculation of the $B_s\to D_s^*$ axial form factor at zero recoil using the Highly Improved Staggered Quark (HISQ) formalism on the second generation MILC gluon ensembles that include up, down, strange and charm quarks in the sea. Using the HISQ action for all valence quarks means that the lattice axial vector current that couples to the $W$ can be renormalized fully non-perturbatively, giving a result free of the perturbative matching errors that previous lattice QCD calculations have had. We calculate correlation functions at three values of the lattice s…
We discuss how lattice calculations can be a useful tool for the study of structure functions. Particular emphasis is given to the perturbative renormalization of the operators.
The $B_K$ parameter is computed in quenched lattice QCD with Wilson twisted mass fermions. Two variants of tmQCD are used; in both of them the relevant $ΔS = 2$ four-fermion operator is renormalised multiplicatively. The renormalisation adopted is non-perturbative, with a Schroedinger functional renormalisation condition. Renormalisation group running is also non-perturbative, up to very high energy scales. In one of the two tmQCD frameworks the computations have been performed at the physical $K$-meson mass, thus eliminating the need of mass extrapolations. Simulations have been performed at …
We present lattice results for the isovector unpolarized parton distribution with nonperturbative RI/MOM-scheme renormalization on the lattice. In the framework of large-momentum effective field theory (LaMET), the full Bjorken-$x$ dependence of a momentum-dependent quasi-distribution is calculated on the lattice and matched to the ordinary lightcone parton distribution at one-loop order, with power corrections included. The important step of RI/MOM renormalization that connects the lattice and continuum matrix elements is detailed in this paper. A few consequences of the results are also addr…
The hadronic decay rate of the tau lepton serves as one of the most precise determinations of the QCD coupling alpha_s. The dominant theoretical source of uncertainty at present resides in the seeming disparity of two approaches to improving the perturbative expansion with the help of the renormalisation group, namely fixed-order and contour-improved perturbation theory. In this work it is demonstrated that in fact both approaches yield compatible results. However, the fixed-order series is found to oscillate around the contour-improved result with an oscillation frequency of approximately six…
In recent years, the quasi parton distribution has been introduced for extracting the parton distribution functions from lattice QCD simulations. The quasi and standard distribution share the same perturbative collinear singularity and the renormalized quasi distribution can be factorized into the standard distribution with a perturbative matching factor. The quasi parton distribution is known to have power-law UV divergences, which do not exist in the standard distribution. We discuss in this talk the nonperturbative renormalization scheme for the power divergence. We also demonstrate the per…
