14029325
N ov 2 00 5 DESY / 05-139 SFB / CPP-0535 November 2005 Parton Distribution Functions with Twisted Mass Fermions
We present a first Wilson twisted mass fermion calculation of the matrix element between pion states of the twist-2 operator, which is related to the the lowest moment x of the valence quark parton distribution function in a pion. Using Wilson twisted mass fermions in the quenched approximation we demonstrate that x can be computed at small pseudoscalar meson masses down to values of order 250 MeV. We investigate the scaling behaviour of this physically important quantity by applying two definitions of the critical mass and observe a scaling compatible with the expected O(a 2) behaviour in both cases. A combined continuum extrapolation allows to obtain reliable results for x at very small pseudoscalar meson masses, which previously could not be explored by lattice QCD simulations.
195317737
Non-perturbative pion matrix element of a twist-2 operator from the lattice
Abstract.We give a continuum limit value of the lowest moment of a twist-2 operator in pion states from non-perturbative lattice calculations. We find that the non-perturbatively obtained renormalization group invariant matrix element is $\langle x\rangle_{\mathrm{RGI}} = 0.179(11)$, which corresponds to $\langle x \rangle^{\overline{\mathrm{MS}}}(2\mathrm{ GeV}) = 0.246(15)$. In obtaining the renormalization group invariant matrix element, we have controlled important systematic errors that appear in typical lattice simulations, such as non-perturbative renormalization, finite size effects and effects of a non-vanishing lattice spacing. The crucial limitation of our calculation is the use of the quenched approximation. Another question that remains not fully clarified is the chiral extrapolation of the numerical data.
15114433
Finite size effects of a pion matrix element
Abstract We investigate finite size effects of the pion matrix element of the non-singlet, twist-2 operator corresponding to the average momentum of non-singlet quark densities. Using the quenched approximation, they come out to be surprisingly large when compared to the finite size effects of the pion mass. As a consequence, simulations of corresponding nucleon matrix elements could be affected by finite size effects even stronger which could lead to serious systematic uncertainties in their evaluation.
3093984
Continuous external momenta in non-perturbative lattice simulations: a computation of renormalization factors
Abstract We discuss the usage of continuous external momenta for computing renormalization factors as needed to renormalize operator matrix elements. These kind of external momenta are encoded in special boundary conditions for the fermion fields. The method allows to compute certain renormalization factors on the lattice that would have been very difficult, if not impossible, to compute with standard methods. As a result we give the renormalization group invariant step scaling function for a twist-2 operator corresponding to the average momentum of non-singlet quark densities.
118913189
Scaling test for Wilson twisted mass QCD
Abstract We present a first scaling test of twisted mass QCD with pure Wilson quarks for a twisting angle of  π /2. We have computed the vector meson mass m V and the pseudoscalar decay constant  F PS for different values of  β at fixed value of  r 0 m PS . The results obtained in the quenched approximation are compared with data for pure Wilson and non-perturbatively O( a ) improved Wilson computations. We show that our results from Wilson twisted mass QCD show clearly reduced lattice spacing errors, consistent with O( a ) improvement and without the need of any improvement terms added. These results thus provide numerical evidence of the prediction of Frezzotti and Rossi [ hep-lat/0306014 ].
18373360
Comparison between overlap and twisted mass fermions towards the chiral limit
We compare overlap fermions, which are chirally invariant, and Wilson twisted mass fermions in the approach to the chiral limit. Our quenched simulations reveal that with both formulations of lattice fermions pion masses of O(250 MeV) can be reached in practical simulations. Our comparison is done at a fixed lattice spacing a ⋍ 0.123 fm. Several quantities are measured, such as hadron masses and pseudoscalar decay constants.
15136419
Going chiral: Overlap versus twisted mass fermions
We compare the behavior of overlap fermions, which are chirally invariant, and of Wilson twisted mass fermions at full twist in the approach to the chiral limit. Our quenched simulations reveal that with both formulations of lattice fermions pion masses of (250 MeV) can be reached in practical applications. Our comparison is done at a fixed value of the lattice spacing a 0.123 fm. A number of quantities are measured such as hadron masses, pseudoscalar decay constants and quark masses obtained from Ward identities. We also determine the axial vector renormalization constants in the case of overlap fermions.
11326298
Twisted mass QCD for the pion electromagnetic form factor
The pion form factor is computed using quenched twisted mass QCD and the GMRES-DR matrix inverter. The momentum averaging procedure of Frezzotti and Rossi is used to remove leading lattice spacing artifacts, and numerical results for the form factor show the expected improvement with respect to the standard Wilson action. Although some matrix inverters are known to fail when applied to twisted mass QCD, GMRES-DR is found to be a viable and powerful option. Results obtained for the pion form factor are consistent with the published results from other O(a) improved actions and are also consistent with the available experimental data.
29631820
Spectrum of quenched twisted mass lattice QCD at maximal twist
Hadron masses are computed from quenched twisted mass lattice QCD for a degenerate doublet of up and down quarks with the twist angle set to {pi}/2, since this maximally-twisted theory is expected to be free of linear discretization errors. Two separate definitions of the twist angle are used, and the hadron masses for these two cases are compared. The flavor breaking, that can arise due to twisting, is discussed in the context of mass splittings within the {delta}(1232) multiplet.
11832510
Quenched scaling of Wilson twisted mass fermions
We investigate the scaling behaviour of quenched Wilson twisted mass fermions at maximal twist applying two definitions of the critical mass. The first definition uses the vanishing of the pseudoscalar meson mass mPS while the second employs the vanishing of the PCAC quark mass mPCAC. We confirm in both cases the expected O(a) improvement. In addition, we show that the PCAC quark mass definition leads to substantially reduced O(a2) cut-off effects even when the pseudoscalar meson mass mPS is as small as 270 MeV. At a fixed value of mPS we perform continuum limits for the vector meson mass mV and for the pseudoscalar decay constant fPS and discuss the renormalisation constant ZV of the vector current.
19002825
The hadron spectrum from twisted mass QCD with a strange quark
Various suggestions exist for incorporating the strange quark into twisted mass QCD. One option for quenched simulations is to employ two twisted doublets, (u,d) and (c,s), with separate twist angles. Working in the isospin limit, preliminary results for the quenched spectrum of strange hadrons are presented, with both twist angles tuned to pi/2. Splittings within the mass multiplets provide some insight into the symmetry breaking effects of twisted mass lattice QCD.
14665548
Maximal twist and the spectrum of quenched twisted mass lattice QCD
Results on the hadron masses for a degenerate doublet of up and down quarks from quenched twisted mass lattice QCD at maximal twist are presented. Two definitions of maximal twist are used and the hadron masses for these definitions are compared. Mass splittings within the Delta(1232) multiplet due to flavor breaking effects are discussed.
124174166
twisted mass lattice QCD
We present first results for the pseudoscalar decay constant s fK, fD and fDs from lattice QCD with Nf = 2+ 1+ 1 flavours of dynamical quarks. The lattice simulations have been performed by the European Twisted Mass collaboration (ETMC) using maximally twisted mass quarks. For the pseudoscalar decay constants we follow a mixed action approach by using so called OsterwalderSeiler fermions in the valence sector for strange and charm quarks. The data for two values of the lattice spacing and several values of the up/down quark mass is analysed using chiral perturbation theory.
6106481
Twisted mass lattice QCD: Recent developments and results
I review recent theoretical developments and numerical results of twisted mass QCD. I argue that, combined with an efficient algorithm, twisted mass QCD can be an attractive QCD lattice action, to perform large scale simulations at small pion masses, where a matching with chiral perturbation theory can be performed. Open issues like flavour breaking effects are also addressed.
14394233
Twisted mass quarks and the phase structure of lattice QCD
Abstract.The phase structure of zero temperature twisted mass lattice QCD is investigated. We find strong metastabilities in the plaquette observable in correspondence of which the untwisted quark mass assumes positive or negative values. We provide interpretations of this phenomenon in terms of chiral symmetry breaking and the effective potential model of Sharpe and Singleton.
118923994
Exploring the phase structure of lattice QCD with twisted mass quarks
The phase structure of zero temperature twisted mass lattice QCD is investigated. We find strong metastabilities in the plaquette observable when the untwisted quark mass sweeps across zero.
15307946
The phase structure of lattice QCD with Wilson quarks and renormalization group improved gluons
The effect of changing the lattice action for the gluon field on the recently observed [1] first order phase transition near zero quark mass is investigated by replacing the Wilson plaquette action by the DBW2 action. The lattice action for quarks is unchanged: it is in both cases the original Wilson action. It turns out that Wilson fermions with the DBW2 gauge action have a phase structure where the minimal pion mass and the jump of the average plaquette are decreased, when compared to Wilson fermions with Wilson plaquette action at similar values of the lattice spacing. Taking the DBW2 gauge action is advantageous also from the point of view of the computational costs of numerical simulations.
18981832
Lattice spacing dependence of the first order phase transition for dynamical twisted mass fermions
Lattice QCD with Wilson fermions generically shows the phenomenon of a first order phase transition. We study the phase structure of lattice QCD using Wilson twisted mass fermions and the Wilson plaquette gauge action in a range of β values where such a first order phase transition is observed. In particular, we investigate the dependence of the first order phase transition on the value of the lattice spacing. Using only data in one phase and neglecting possible problems arising from the phase transition we are able to perform a first scaling test for physical quantities using this action.
42141058
Spontaneous flavor and parity breaking with Wilson fermions
We discuss the phase diagram of Wilson fermions in the m{sub 0}{minus}g{sup 2} plane for two-flavor QCD. We argue that, as originally suggested by Aoki, there is a phase in which flavor and parity are spontaneously broken. Recent numerical results on the spectrum of the overlap Hamiltonian have been interpreted as evidence against Aoki{close_quote}s conjecture. We show that they are in fact consistent with the presence of a flavor-parity broken {open_quotes}Aoki phase.{close_quotes} We also show how, as the continuum limit is approached, one can study the lattice theory using the continuum chiral Lagrangian supplemented by additional terms proportional to powers of the lattice spacing. We find that there are two possible phase structures at non-zero lattice spacing: (1) there is an Aoki phase of width {Delta}m{sub 0}{approximately}a{sup 3} with two massless Goldstone pions; (2) there is no symmetry breaking, and all three pions have an equal non-vanishing mass of order a. Present numerical evidence suggests that the former option is realized for Wilson fermions. Our analysis then predicts the form of the pion masses and the flavor-parity breaking condensate within the Aoki phase. Our analysis also applies for non-perturbatively improved Wilson fermions. {copyright} {ital 1998} {ital The American Physical Society}
120007260
On the phase structure of twisted mass lattice QCD
The introduction of a chirally twisted mass term has been proposed as an attractive approach to O(a) improvement of Quantum Chromodynamics with Wilson fermions on a lattice. For numerical simulation projects it is important to know the phase structure of the theory in the region of small quark masses. We study this question in the framework of chiral perturbation theory. Generalizing the analysis of Sharpe and Singleton we find extensions of normal and Aoki phase scenarios and a possible new phase with spontaneous breakdown of chiral symmetry to a discrete $Z_2$.
12697528
Pion mass splitting and phase structure in twisted mass QCD
Abstract.In the framework of Wilson chiral perturbation theory [1], we study the effect induced by a twisted Wilson term, as it appears in twisted mass QCD [2] (with two degenerate quarks). In particular we consider the vacuum orientation and the pion masses. The computations are done to NLO both in the mass and in the lattice spacing (i.e. to O(a2)). There are no restrictions on the relative size of lattice artifacts with respect to the physical mass, thus allowing, in principle, to bridge between the physical regime and the unphysical one, where lattice artifacts tend to dominate. The inclusion of O(a2) lattice artifacts can account for the splitting of degeneracy of the three pion masses. Moreover O(a2) terms are necessary to model non-trivial behaviors of the vacuum orientation such as possible Aoki phases. It turns out that these last two phenomena are determined by the same constant.
29185737
Phase diagram of twisted mass lattice QCD
We use the effective chiral Lagrangian to analyze the phase diagram of two-flavor twisted mass lattice QCD as a function of the normal and twisted masses, generalizing previous work for the untwisted theory. We first determine the chiral Lagrangian including discretization effects up to next-to-leading order (NLO) in a combined expansion in which m{sub {pi}}{sup 2}/(4{pi}f{sub {pi}}){sup 2}{approx}a{lambda} (a being the lattice spacing, and {lambda}={lambda}{sub QCD}). We then focus on the region where m{sub {pi}}{sup 2}/(4{pi}f{sub {pi}}){sup 2}{approx}(a{lambda}){sup 2}, in which case competition between leading and NLO terms can lead to phase transitions. As for untwisted Wilson fermions, we find two possible phase diagrams, depending on the sign of a coefficient in the chiral Lagrangian. For one sign, there is an Aoki phase for pure Wilson fermions, with flavor and parity broken, but this is washed out into a crossover if the twisted mass is nonvanishing. For the other sign, there is a first order transition for pure Wilson fermions, and we find that this transition extends into the twisted mass plane, ending with two symmetrical second order points at which the mass of the neutral pion vanishes. We provide graphs of the condensate and pion masses for both scenarios,more » and note a simple mathematical relation between them. These results may be of importance to numerical simulations.« less
31911482
Twisted mass QCD, O(a) improvement, and Wilson chiral perturbation theory
We point out a caveat in the proof for automatic O(a) improvement in twisted mass lattice QCD at maximal twist angle. With the definition for the twist angle previously given by Frezzotti and Rossi, automatic O(a) improvement can fail unless the quark mass satisfies m{sub q}>>a{sup 2}{lambda}{sub QCD}{sup 3}. We propose a different definition for the twist angle which does not require a restriction on the quark mass for automatic O(a) improvement. In order to illustrate explicitly automatic O(a) improvement we compute the pion mass in the corresponding chiral effective theory. We consider different definitions for maximal twist and show explicitly the absence or presence of the leading O(a) effect, depending on the size of the quark mass.
53558034
Twisted mass chiral perturbation theory at next-to-leading order
We study the properties of pions in twisted mass lattice QCD (with two degenerate flavors) using chiral perturbation theory ($\ensuremath{\chi}\mathrm{PT}$). We work to next-to-leading order (NLO) in a power-counting scheme in which ${m}_{q}\ensuremath{\sim}a{\ensuremath{\Lambda}}_{\mathrm{QCD}}^{2}$, with ${m}_{q}$ the physical quark mass and $a$ the lattice spacing. We argue that automatic $O(a)$ improvement of physical quantities at maximal twist, which has been demonstrated in general if ${m}_{q}\ensuremath{\gg}a{\ensuremath{\Lambda}}_{\mathrm{QCD}}^{2}$, holds even if ${m}_{q}\ensuremath{\sim}a{\ensuremath{\Lambda}}_{\mathrm{QCD}}^{2}$, as long as one uses an appropriate nonperturbative definition of the twist angle, with the caveat that we have shown this only through NLO in our chiral expansion. We demonstrate this with explicit calculations, for arbitrary twist angle, of all pionic quantities that involve no more than a single pion in the initial and final states: masses, decay constants, form factors, and condensates, as well as the differences between alternate definitions of twist angle. We also calculate the axial and pseudoscalar form factors of the pion, quantities which violate flavor and parity, and which vanish in the continuum limit. These are of interest because they are not automatically $O(a)$ improved at maximal twist. They allow a determination of the unknown low-energy constants introduced by discretization errors, and provide tests of the accuracy of $\ensuremath{\chi}\mathrm{PT}$ at NLO. We extend our results into the regime where ${m}_{q}\ensuremath{\sim}{a}^{2}{\ensuremath{\Lambda}}_{\mathrm{QCD}}^{3}$, and argue in favor of a recent proposal that automatic $O(a)$ improvement at maximal twist remains valid in this regime.
11765733
Reducing cutoff effects in maximally twisted LQCD close to the chiral limit
When analyzed in terms of the Symanzik expansion, lattice correlators of multi-local (gauge-invariant) operators with non-trivial continuum limit exhibit in maximally twisted lattice QCD "infrared divergent" cutoff effects of the type a(2k)/(m(pi)(2))(h), 2k >= h >= 1 (k, h integers), which tend to become numerically large when the pion mass gets small. We prove that, if the action is O(a) improved a la Symanzik or, alternatively, the critical mass counter-term is chosen in some "optimal" way, these lattice artifacts are reduced to terms that are at worst of the order a(2)(a(2)/m(pi)(2))(k-1), k >= 1. This implies that the continuum extrapolation of lattice results is smooth at least down to values of the quark mass, m(q), satisfying the order of magnitude inequality mq > a(2)Lambda(3)(QCD).
121307460
On the relations between irreducible representations of the hyperoctahedral group and O(4) and SO(4)
In this paper we describe the relations between the irreducible representations of the hyperoctahedral group in four dimensions and irreducible, low‐dimensional representations of the orthogonal groups O(4) and SO(4).
122865932
Representations of the rotation reflection symmetry group of the four-dimensional cubic lattice
Abstract The structure and representations of the rotation reflection symmetry group of the four-dimensional cubic lattice are described. Their connections with the representations of the three-dimensional lattice rotation reflection group, and with the representations of the continuous O(3) and O(4) groups are given.
122605075
Pion structure functions from lattice QCD
Abstract We compute 〈 x 〉 and 〈 x 2 〉 of the valence quarks in a pion, on the lattice using Wilson fermions in the quenced approximat We find 〈 x 〉 = 0.49±0.08 and 〈 x 2 〉 = 0.02 at a scale μ ≅ 7 GeV, in a good agreement with experimental data.
119086052
On the Schrödinger functional in QCD
Abstract In a series of publications (Nucl. Phys. B384 (1994) 168; B389 (1994) 247), Luscher et al. have demonstrated the usefulness of the Schrodinger functional in pure SU(2) and SU(3) gauge theory. In this paper, it is shown how their formalism can be extended to include fermions. In the framework of Wilson's lattice QCD, we define the Schrodinger functional by making use of the transfer matrix formalism. Boundary conditions for the fermions arise naturally. We then take the naive continuum limit of the action and show that no lattice peculiarities are left over. The corresponding free Dirac operator has a unique self-adjoint extension with purely discrete spectrum and no zero-modes.
118956818
Non-perturbative renormalization of lattice QCD at all scales
Abstract A general strategy to solve the non-perturbative renormalization problem in lattice QCD, using finite-size techniques and numerical simulations, is described. As an illustration we discuss the computation of the axial current normalization constant, the running coupling at zero quark masses and the scale evolution of the renormalized axial density. The non-perturbative calculation of O(a) correction terms (as they appear in Symanzik's improvement programme) is another important field of application.
15914783
Precision computation of a low-energy reference scale in quenched lattice QCD
We present results for the reference scale r0 in SU(3) Lattice Gauge Theory for β = 6g02 in the range 5.7 ⩽ β ⩽ 6.57. The high relative accuracy of 0.3–0.6% in r0/a was achieved through good statistics, the application of a multi-hit procedure and a variational approach in the computation of Wilson loops. A precise definition of the force used to extract r0 has been used throughout the calculation which guarantees that r0/a is a smooth function of the bare coupling and that subsequent continuum extrapolations are possible. The results are applied to the continuum extrapolations of the energy gap Δ in the static quark potential and the scale Lmax/r0 used in the calculation of the running coupling constant.
14949768
Cutoff schemes in chiral perturbation theory and the quark mass expansion of the nucleon mass
Abstract We discuss the use of cutoff methods in chiral perturbation theory. We develop a cutoff scheme based on the operator structure of the effective field theory that allows to suppress high momentum contributions in Goldstone boson loop integrals and by construction is free of the problems traditional cutoff schemes have with gauge invariance or chiral symmetries. As an example, we discuss the chiral expansion of the nucleon mass. We show that the mass of a nucleon in heavy baryon chiral perturbation theory has a well behaved chiral expansion up to effective Goldstone boson masses of 400 MeV when one utilizes standard dimensional regularization techniques. With the help of the here developed cutoff scheme we can demonstrate a well-behaved chiral expansion for the nucleon mass up to 600 MeV of effective Goldstone boson masses. We also discuss in detail the price, in numbers of additional short distance operators involved, that has to be paid for this extended range of applicability of chiral perturbation theory with cutoff regularization. We also compare the fourth-order result for the chiral expansion of the nucleon mass with lattice results and draw some conclusions about chiral extrapolations based on such type of representation.
6302153
Matrix elements of twist-2 operators in quenched chiral perturbation theory
Abstract We compute the leading quark mass dependence of the matrix elements of isovector twist-2 operators between octet baryon states in quenched QCD using quenched chiral perturbation theory. There are contributions of the form m q log m q and m q , in analogy with QCD, but there are also contributions of the form log m q from hairpin interactions. The nucleon does not receive such hairpin contributions.
16667438
Nucleon Properties at Finite Lattice Spacing in Chiral Perturbation Theory
Properties of the proton and neutron are studied in partially quenched chiral perturbation theory at finite lattice spacing. Masses, magnetic moments, the matrix elements of isovector twist-2 operators and axial-vector currents are examined at the one-loop level in a double expansion in the light-quark masses and the lattice spacing. This work will be useful in extrapolating the results of simulations using Wilson valence and sea quarks, as well as simulations using Wilson sea quarks and Ginsparg-Wilson valence quarks, to the continuum.
2644162
Comparing iterative methods for overlap and twisted mass fermions
We present a systematic comparison of various iterative methods to obtain the fermion propagator with both overlap and twisted mass fermions at fixed pion mass in the quenched approximation. Taking the best available algorithm in each case we find that calculations with the overlap operator are by a factor of 20-40 more expensive than with the twisted mass operator at the parameter values considered here. For the overlap operator we also compare the efficiency of various methods for calculating the topological index.
19437204
Parton distributions for the pion extracted from Drell-Yan and prompt photon experiments.
We determine the parton distributions of the pion from a consistent next-to-leading-order analysis of several high-statistics π ± N experiments including both Drell-Yan and prompt photon production
18917486
Pionic parton distributions revisited
Abstract. Using constituent quark model constraints we calculate the gluon and sea–quark content of pions solely in terms of their valence density (fixed by $\pi N$ Drell–Yan data) and the known sea and gluon distributions of the nucleon, using the most recent updated valence–like input parton densities of the nucleon. The resulting small–x dynamical QCD predictions for $g^{\pi}(x,Q^2)$ and $\bar{q} ^{\pi}(x,Q^2)$ are unique and parameter free. Simple analytic parametrizations of the resulting parton distributions of the pion are presented in LO and NLO. These results and parametrizations will be important, among other things, for updated formulations of the parton distributions of real and virtual photons.
12738945
Pion parton distribution function in the valence region
The parton distribution function of the pion in the valence region is extracted in a next-to-leading order analysis from Fermilab E-615 pionic Drell-Yan data. The effects of the parametrization of the pion's valence distributions are examined. Modern nucleon parton distributions and nuclear corrections were used and possible effects from higher twist contributions were considered in the analysis. In the next-to-leading order analysis, the high-x dependence of the pion structure function differs from that of the leading order analysis, but not enough to agree with the expectations of pQCD and Dyson-Schwinger calculations.