Quark Bilinear Operators Research Articles

Controlling excited-state contamination in nucleon matrix elements

We present a detailed analysis of methods to reduce statistical errors and excited-state contamination in the calculation of matrix elements of quark bilinear operators in nucleon states. All the calculations were done on a 2+1 flavor ensemble with lattices of size $32^3 \times 64$ generated using the rational hybrid Monte Carlo algorithm at $a=0.081$~fm and with $M_\pi=312$ MeV. The statistical precision of the data is improved using the all-mode-averaging method. We compare two methods for reducing excited-state contamination: a variational analysis and a two-state fit to data at multiple values of the source-sink separation $t_{\rm sep}$. We show that both methods can be tuned to significantly reduce excited-state contamination and discuss their relative advantages and cost-effectiveness. A detailed analysis of the size of source smearing used in the calculation of quark propagators and the range of values of $t_{\rm sep}$ needed to demonstrate convergence of the isovector charges of the nucleon to the $t_{\rm sep} \to \infty $ estimates is presented.

η′meson mass from topological charge density correlator in QCD

The flavor-singlet component of the eta prime meson is related to the topological structure of the SU(3) gauge field through the chiral anomaly. We perform a 2+1-flavor lattice QCD calculation and demonstrate that the two-point function of a gluonically defined topological charge density after a short Yang-Mills gradient flow contains the propagation of the eta prime meson, by showing that its mass in the chiral and continuum limit is consistent with the experimental value. The gluonic correlator does not suffer from the contamination of the pion contribution, and the clean signal is obtained at significantly lower numerical cost compared to the conventional method with the quark bilinear operators.

Renormalization of local quark-bilinear operators forNf=3flavors of stout link nonperturbative clover fermions

The renormalization factors of local quark-bilinear operators are computed non-perturbatively for $N_f=3$ flavors of SLiNC fermions, with emphasis on the various procedures for the chiral and continuum extrapolations. The simulations are performed at a lattice spacing $a=0.074$ fm, and for five values of the pion mass in the range of 290-465 MeV, allowing a safe and stable chiral extrapolation. Emphasis is given in the subtraction of the well-known pion pole which affects the renormalization factor of the pseudoscalar current. We also compute the inverse propagator and the Green's functions of the local bilinears to one loop in perturbation theory. We investigate lattice artifacts by computing them perturbatively to second order as well as to all orders in the lattice spacing. The renormalization conditions are defined in the RI$'$-MOM scheme, for both the perturbative and non-perturbative results. The renormalization factors, obtained at different values of the renormalization scale, are translated to the ${\bar{\rm MS}}$ scheme and are evolved perturbatively to 2 GeV. Any residual dependence on the initial renormalization scale is eliminated by an extrapolation to the continuum limit. We also study the various sources of systematic errors. Particular care is taken in correcting the non-perturbative estimates by subtracting lattice artifacts computed to one loop perturbation theory using the same action. We test two different methods, by subtracting either the ${\cal O}(g^2\,a^2)$ contributions, or the complete (all orders in $a$) one-loop lattice artifacts.

Nonperturbative renormalization of overlap quark bilinears on2+1-flavor domain wall fermion configurations

We present renormalization constants of overlap quark bilinear operators on $2+1$-flavor domain wall fermion configurations. This setup is being used by the $\ensuremath{\chi}\mathrm{QCD}$ 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 currents are calculated in the RI-MOM scheme. Results in the $\overline{\mathrm{MS}}$ scheme are also given. The step scaling function of quark masses in the RI-MOM scheme is computed as well. The analysis uses, in total, six different ensembles of three sea quarks, each on two lattices with sizes ${24}^{3}\ifmmode\times\else\texttimes\fi{}64$ and ${32}^{3}\ifmmode\times\else\texttimes\fi{}64$ at spacings $a=(1.73\text{ }\text{ }\mathrm{GeV}{)}^{\ensuremath{-}1}$ and $(2.28\text{ }\text{ }\mathrm{GeV}{)}^{\ensuremath{-}1}$, respectively.

Nucleon charges and electromagnetic form factors from2+1+1-flavor lattice QCD

We present lattice-QCD results on the nucleon isovector axial, scalar and tensor charges, the isovector electromagnetic Dirac and Pauli form factors, and the connected parts of the isoscalar charges. The calculations have been done using two ensembles of highly improved staggered quarks lattices generated by the MILC collaboration with $2+1+1$ dynamical flavors at a lattice spacing of 0.12 fm and with light-quark masses corresponding to pions with masses 310 and 220 MeV. We perform a systematic study including excited-state degrees of freedom and examine the dependence of the extracted nucleon matrix elements on source-sink separation. This study demonstrates with high-statistics data that including excited-state contributions and generating data at multiple separations is necessary to remove contamination that would otherwise lead to systematic error. We also determine the renormalization constants of the associated quark bilinear operators in the RI-sMOM scheme and make comparisons of our renormalized results with previous dynamical-lattice calculations.

Non-perturbative renormalization in coordinate space for [formula omitted] maximally twisted mass fermions with tree-level Symanzik improved gauge action

We present results of a Lattice QCD application of a coordinate space renormalization scheme for the extraction of renormalization constants for flavour non-singlet bilinear quark operators. The method consists in the analysis of the small-distance behaviour of correlation functions in Euclidean space and has several theoretical and practical advantages, in particular: it is gauge invariant, easy to implement and has relatively low computational cost. The values of renormalization constants in the X-space scheme can be converted to the MS¯ scheme via 4-loop continuum perturbative formulae. Our results for Nf=2 maximally twisted mass fermions with tree-level Symanzik improved gauge action are compared to the ones from the RI-MOM scheme and show full agreement with this method.

Towards holographic walking from $ \mathcal{N} = 4 $ super Yang-Mills

We propose that a holographic description of ‘walking’ behavior, namely quasi-conformal dynamics relevant for technicolor models, can be obtained from relevant deformations of \( \mathcal{N} = 4 \) super Yang-Mills. We consider deformations which drive the theory close to the \( \mathcal{N} = 1 \) Leigh-Strassler fixed point, eventually deviating from it in the deep IR. We use the Pilch-Warner dual supergravity description of the flow between the \( \mathcal{N} = 4 \) and the \( \mathcal{N} = 1 \) fixed points to focus on observables that only require knowledge of the walking region. These include large anomalous dimensions of quark bilinear operators, which we study via probe D7-branes. We also make a first attempt at describing the theory beyond the walking region by introducing an infrared cut-off, in the spirit of hard-wall models. In this case we find a light, dilaton-like scalar state, but whether this mode persists in the exact theory remains an open question.

Completing the framework of AdS/QCD: h 1/b 1 mesons and excited ω/ρ’s

We extend the "hard wall" gravity dual of QCD by including tensor fields b_{MN} that correspond to the QCD quark bilinear operators qbar sigma^{mu nu} q. These fields give rise to a spectrum of states which include the h_1 and b_1 mesons, as well as a tower of excited omega/rho meson states. We also identify the lowest-dimension term which leads to mixing between the new rho states and the usual tower of rho mesons when chiral symmetry is broken.

Renormalization factors of quark bilinears using the D CI operator with dynamical quarks

Non-perturbative renormalization factors of bilinear quark operators are computed for the Chirally Improved lattice action with two dynamic quarks. The analysis is based on five different parameter sets with lattice size 12^3 x 24 and four parameter sets with lattice size 16^3 x 32. For the pseudoscalar renormalization factor the pion pole contribution is subtracted and chiral extrapolations are performed. Results are given in RIprime- and MSbar-scheme as well as in RGI-form.

Two-loop matching factors for light quark masses and three-loop mass anomalous dimensions in the regularization invariant symmetric momentum-subtraction schemes

Light quark masses can be determined through lattice simulations in regularization invariant momentum-subtraction(RI/MOM) schemes. Subsequently, matching factors, computed in continuum perturbation theory, are used in order to convert these quark masses from a RI/MOM scheme to the MS-bar scheme. We calculate the two-loop corrections in quantum chromodynamics(QCD) to these matching factors as well as the three-loop mass anomalous dimensions for the RI/SMOM and RI/SMOM_gamma_mu schemes. These two schemes are characterized by a symmetric subtraction point. Providing the conversion factors in the two different schemes allows for a better understanding of the systematic uncertainties. The two-loop expansion coefficients of the matching factors for both schemes turn out to be small compared to the traditional RI/MOM schemes. For nf=3 quark flavors they are about 0.6-0.7% and 2%, respectively, of the leading order result at scales of about 2 GeV. Therefore, they will allow for a significant reduction of the systematic uncertainty of light quark mass determinations obtained through this approach. The determination of these matching factors requires the computation of amputated Green's functions with the insertions of quark bilinear operators. As a by-product of our calculation we also provide the corresponding results for the tensor operator.

Non-perturbative renormalization of quark bilinear operators with N f = 2 (tmQCD) Wilson fermions and the tree-level improved gauge action

We present results for the renormalization constants of bilinear quark operators obtained by using the tree-level Symanzik improved gauge action and the Nf=2 twisted mass fermion action at maximal twist, which guarantees automatic O(a)-improvement. Our results are also relevant for the corresponding standard (un-twisted) Wilson fermionic action since the two actions only differ, in the massless limit, by a chiral rotation of the quark fields. The scale-independent renormalization constants ZV, ZA and the ratio ZP/ZS have been computed using the RI-MOM approach, as well as other alternative methods. For ZA and ZP/ZS, the latter are based on both standard twisted mass and Osterwalder-Seiler fermions, while for ZV a Ward Identity has been used. The quark field renormalization constant Zq and the scale dependent renormalization constants ZS, ZP and ZT are determined in the RI-MOM scheme. Leading discretization effects of O(g^2 a^2), evaluated in one-loop perturbation theory, are explicitly subtracted from the RI-MOM estimates.

Nonperturbative renormalization of bilinear operators with dynamical overlap fermions

Using the nonperturbative renormalization technique, we calculate the renormalization factors for quark bilinear operators made of overlap fermions on the lattice. The background gauge field is generated by the JLQCD and TWQCD Collaborations, including dynamical effects of two or $2+1$ flavors of light quarks on a ${16}^{3}\ifmmode\times\else\texttimes\fi{}32$ or ${16}^{3}\ifmmode\times\else\texttimes\fi{}48$ lattice at lattice spacing around 0.1 fm. By reducing the quark mass close to the chiral limit, where the finite volume system enters the so-called $ϵ$ regime, the unwanted effect of spontaneous chiral symmetry breaking on the renormalization factors is suppressed. On the lattices in the conventional $p$ regime, this effect is precisely subtracted by separately calculating the contributions from the chiral condensate.

𝒪(a2) corrections to the one-loop propagator and bilinears of clover fermions with Symanzik improved gluons

We calculate corrections to the fermion propagator and to the Green's functions of all fermion bilinear operators of the form , to one-loop in perturbation theory. We employ the Wilson/clover action for fermions and the Symanzik improved action for gluons. The novel aspect of our calculations is that they are carried out to second order in the lattice spacing, (a2). Consequently, they have addressed a number of new issues, most notably the appearance of loop integrands with strong IR divergences (convergent only beyond 6 dimensions). Such integrands are not present in (a1) improvement calculations; there, IR divergent terms are seen to have the same structure as in the (a0) case, by virtue of parity under integration, and they can thus be handled by well-known techniques. We explain how to correctly extract the full (a2) dependence; in fact, our method is generalizable to any order in a. The (a2) corrections to the quark propagator and Green's functions computed in this paper are useful to improve the nonperturbative RI-MOM determination of renormalization constants for quark bilinear operators. Our results depend on a large number of parameters: coupling constant, number of colors, lattice spacing, external momentum, clover parameter, Symanzik coefficients, gauge parameter. To make these results most easily accessible to the reader, we have included them in the distribution package of this paper, as an ASCII file named: Oa2results.m; the file is best perused as Mathematica input.

Renormalization of quark bilinear operators in a momentum-subtraction scheme with a nonexceptional subtraction point

We extend the Rome-Southampton regularization independent momentum-subtraction renormalization scheme(RI/MOM) for bilinear operators to one with a nonexceptional, symmetric subtraction point. Two-point Green's functions with the insertion of quark bilinear operators are computed with scalar, pseudoscalar, vector, axial-vector and tensor operators at one-loop order in perturbative QCD. We call this new scheme RI/SMOM, where the S stands for "symmetric". Conversion factors are derived, which connect the RI/SMOM scheme and the MSbar scheme and can be used to convert results obtained in lattice calculations into the MSbar scheme. Such a symmetric subtraction point involves nonexceptional momenta implying a lattice calculation with substantially suppressed contamination from infrared effects. Further, we find that the size of the one-loop corrections for these infrared improved kinematics is substantially decreased in the case of the pseudoscalar and scalar operator, suggesting a much better behaved perturbative series. Therefore it should allow us to reduce the error in the determination of the quark mass appreciably.

Two-loop renormalization of vector, axial-vector, and tensor fermion bilinears on the lattice

We compute the two-loop renormalization functions, in the $R{I}^{\ensuremath{'}}$ scheme, of local bilinear quark operators $\overline{\ensuremath{\psi}}\ensuremath{\Gamma}\ensuremath{\psi}$, where $\ensuremath{\Gamma}$ corresponds to the vector, axial-vector, and tensor Dirac operators, in the lattice formulation of QCD. We consider both the flavor nonsinglet and singlet operators. We use the clover action for fermions and the Wilson action for gluons. Our results are given as a polynomial in ${c}_{\mathrm{SW}}$, in terms of both the renormalized and bare coupling constant, in the renormalized Feynman gauge. Finally, we present our results in the $\overline{MS}$ scheme, for easier comparison with calculations in the continuum. The corresponding results, for fermions in an arbitrary representation, together with some special features of superficially divergent integrals, are included in the appendices.

Erratum: Two-loop renormalization of scalar and pseudoscalar fermion bilinears on the lattice [Phys. Rev. D76, 094514 (2007)

We compute the two-loop renormalization functions, in the RI $^\prime$ scheme, of local bilinear quark operators $\bar{\psi}\Gamma\psi$, where $\Gamma$ denotes the Scalar and Pseudoscalar Dirac matrices, in the lattice formulation of QCD. We consider both the flavor non-singlet and singlet operators; the latter, in the scalar case, leads directly to the two-loop fermion mass renormalization, $Z_m$. As a prerequisite for the above, we also compute the quark field renormalization, $Z_{\psi}$, up to two loops. We use the clover action for fermions and the Wilson action for gluons. Our results are given as a polynomial in $c_{SW}$, in terms of both the renormalized and bare coupling constant, in the renormalized Feynman gauge. We also confirm the 1-loop renormalization functions, for generic gauge. Finally, we present our results in the $\bar{MS}$ scheme, for easier comparison with calculations in the continuum. The corresponding results, for fermions in an arbitrary representation, are included in an Appendix.

Nonperturbative renormalization of quark bilinear operators andBKusing domain wall fermions

We present a calculation of the renormalization coefficients of the quark bilinear operators and the $K\ensuremath{-}\overline{K}$ mixing parameter ${B}_{K}$. The coefficients relating the bare lattice operators to those in the RI/MOM scheme are computed nonperturbatively and then matched perturbatively to the $\overline{\mathrm{MS}}$ scheme. The coefficients are calculated on the RBC/UKQCD $2+1$ flavor dynamical lattice configurations. Specifically we use a ${16}^{3}\ifmmode\times\else\texttimes\fi{}32$ lattice volume, the Iwasaki gauge action at $\ensuremath{\beta}=2.13$ and domain wall fermions with ${L}_{s}=16$.

Two-loop renormalization of scalar and pseudoscalar fermion bilinears on the lattice

We compute the two-loop renormalization functions, in the $R{I}^{\ensuremath{'}}$ scheme, of local bilinear quark operators $\overline{\ensuremath{\psi}}\ensuremath{\Gamma}\ensuremath{\psi}$, where $\ensuremath{\Gamma}$ denotes the scalar and pseudoscalar Dirac matrices, in the lattice formulation of QCD. We consider both the flavor nonsinglet and singlet operators; the latter, in the scalar case, leads directly to the two-loop fermion mass renormalization, ${Z}_{m}$. As a prerequisite for the above, we also compute the quark field renormalization, ${Z}_{\ensuremath{\psi}}$, up to two loops. We use the clover action for fermions and the Wilson action for gluons. Our results are given as a polynomial in ${c}_{\mathrm{SW}}$, in terms of both the renormalized and bare coupling constants, in the renormalized Feynman gauge. We also confirm the one-loop renormalization functions, for generic gauge. Finally, we present our results in the $\overline{MS}$ scheme, for easier comparison with calculations in the continuum. The corresponding results, for fermions in an arbitrary representation, are included in the Appendix.

Nonperturbative renormalization of composite operators with overlap fermions

We compute nonperturbatively the renormalization constants of composite operators on a quenched ${16}^{3}\ifmmode\times\else\texttimes\fi{}28$ lattice with lattice spacing $a=0.20\text{ }\text{ }\mathrm{fm}$ for the overlap fermion by using the regularization-independent (RI) scheme. The quenched gauge configurations were generated with the Iwasaki action. We test the relations ${Z}_{A}={Z}_{V}$ and ${Z}_{S}={Z}_{P}$ and find that they agree well (less than 1%) above $\ensuremath{\mu}=1.6\text{ }\text{ }\mathrm{GeV}$. We also perform a Renormalization Group (RG) analysis at the next-to-next-to-leading order and match the renormalization constants to the $\overline{\mathrm{MS}}$ scheme. The wave function renormalization ${Z}_{\ensuremath{\psi}}$ is determined from the vertex function of the axial current and ${Z}_{A}$ from the chiral Ward identity. Finally, we examine the finite quark mass behavior for the renormalization factors of the quark bilinear operators. We find that the $(pa{)}^{2}$ errors of the vertex functions are small and the quark mass dependence of the renormalization factors to be quite weak.

Renormalization of bilinear quark operators for overlap fermions

We present nonperturbative renormalization constants of fermionic bilinears on the lattice in the quenched approximation at $\ensuremath{\beta}=6.1$ using an overlap [H. Neuberger, Phys. Lett. B 417, 141 (1998)] fermion action with hypercubic (HYP)-blocked links. We consider the effects of the exact zero modes of the Dirac operator and find they are important in calculating the renormalization constants of the scalar and pseudoscalar density. The results are given in the RI' and $\overline{MS}$ schemes and compared to perturbative calculations.