Non-degenerate Quarks Research Articles

QCD \u03b8-vacuum energy and axion properties

At low energies, the strong interaction is governed by the Goldstone bosons associated with the spontaneous chiral symmetry breaking, which can be systematically described by chiral perturbation theory. In this paper, we apply this theory to study the θ-vacuum energy density and hence the QCD axion potential up to next-to-leading order with N non-degenerate quark masses. By setting N = 3, we then derive the axion mass, self-coupling, topological susceptibility and the normalized fourth cumulant both analytically and numerically, taking the strong isospin breaking effects into account. In addition, the model-independent part of the axion-photon coupling, which is important for axion search experiments, is also extracted from the chiral Lagrangian supplemented with the anomalous terms up to mathcal{O}left({p}^6right) .

Non-perturbative determination of improvement coefficients b_mathrm{m} and b_mathrm{A}-b_mathrm{P} and normalisation factor Z_mathrm{m}Z_mathrm{P}/Z_mathrm{A} with N_mathrm{f}= 3 Wilson fermions

We determine non-perturbatively the normalisation parameter Z_mathrm{m}Z_mathrm{P}/Z_mathrm{A} as well as the Symanzik coefficients b_mathrm{m} and b_mathrm{A}-b_mathrm{P}, required in mathrm{O}(a) improved quark mass renormalisation with Wilson fermions. The strategy underlying their computation involves simulations in N_mathrm{f}=3 QCD with mathrm{O}(a) improved massless sea and non-degenerate valence quarks in the finite-volume Schrödinger functional scheme. Our results, which cover the typical gauge coupling range of large-volume N_mathrm{f}=2+1 QCD simulations with Wilson fermions at lattice spacings below 0.1,mathrm{fm}, are of particular use for the non-perturbative calculation of mathrm{O}(a) improved renormalised quark masses.

Multigrid approach in shifted linear systems for the non-degenerated twisted mass operator

Application of multigrid solvers in shifted linear systems is studied. We focus on accelerating the rational approximation needed for simulating single flavor operators. This is particularly useful, in the case of twisted mass fermions for mass non-degenerate quarks and can be employed to accelerate the Nf=1+1 sector of Nf=2+1+1 twisted mass fermion simulations. The multigrid solver is accelerated by employing suitable initial guesses. We propose a novel strategy for proposing initial guesses for shifted linear systems based on the Lagrangian interpolation of the previous solutions.

Domain wall fermion QCD with the exact one flavor algorithm

Lattice QCD calculations including the effects of one or more non-degenerate sea quark flavors are conventionally performed using the Rational Hybrid Monte Carlo (RHMC) algorithm, which computes the square root of the determinant of $\mathscr{D}^{\dagger} \mathscr{D}$, where $\mathscr{D}$ is the Dirac operator. The special case of two degenerate quark flavors with the same mass is described directly by the determinant of $\mathscr{D}^{\dagger} \mathscr{D}$ --- in particular, no square root is necessary --- enabling a variety of algorithmic developments, which have driven down the cost of simulating the light (up and down) quarks in the isospin-symmetric limit of equal masses. As a result, the relative cost of single quark flavors --- such as the strange or charm --- computed with RHMC has become more expensive. This problem is even more severe in the context of our measurements of the $\Delta I = 1/2$ $K \rightarrow \pi \pi$ matrix elements on lattice ensembles with $G$-parity boundary conditions, since $G$-parity is associated with a doubling of the number of quark flavors described by $\mathscr{D}$, and thus RHMC is needed for the isospin-symmetric light quarks as well. In this paper we report on our implementation of the exact one flavor algorithm (EOFA) introduced by the TWQCD collaboration for simulations including single flavors of domain wall quarks. We have developed a new preconditioner for the EOFA Dirac equation, which both reduces the cost of solving the Dirac equation and allows us to re-use the bulk of our existing high-performance code. Coupling these improvements with careful tuning of our integrator, the time per accepted trajectory in the production of our 2+1 flavor $G$-parity ensembles with physical pion and kaon masses has been decreased by a factor of 4.2.

Quark mass dependence of two-flavor QCD

I explore the rich phase diagram of two-flavor QCD as a function of the quark masses. The theory involves three parameters, including one that is $CP$ violating. As the masses vary, regions of both first- and second-order transitions are expected. For nondegenerate quarks, nonperturbative effects cease to be universal, leaving individual quark mass ratios with a renormalization scheme dependence. This raises complications in matching lattice results with perturbative schemes and demonstrates the tautology of attacking the strong $CP$ problem via a vanishing up-quark mass.

Staggered chiral perturbation theory and the fourth-root trick

Staggered chiral perturbation theory ($\mathrm{S}\ensuremath{\chi}\mathrm{PT}$) takes into account the ``fourth-root trick'' for reducing unwanted (taste) degrees of freedom with staggered quarks by multiplying the contribution of each sea quark loop by a factor of $1/4$. In the special case of four staggered fields (four flavors, ${n}_{F}=4$), I show here that certain assumptions about analyticity and phase structure imply the validity of this procedure for representing the rooting trick in the chiral sector. I start from the observation that, when the four flavors are degenerate, the fourth root simply reduces ${n}_{F}=4$ to ${n}_{F}=1$. One can then treat nondegenerate quark masses by expanding around the degenerate limit. With additional assumptions on decoupling, the result can be extended to the more interesting cases of ${n}_{F}=3$, 2, or 1. An apparent paradox associated with the one-flavor case is resolved. Coupled with some expected features of unrooted staggered quarks in the continuum limit, in particular, the restoration of taste symmetry, $\mathrm{S}\ensuremath{\chi}\mathrm{PT}$ then implies that the fourth-root trick induces no problems (for example, a violation of unitarity that persists in the continuum limit) in the lowest energy sector of staggered lattice QCD. It also says that the theory with staggered valence quarks and rooted staggered sea quarks behaves like a simple, partially-quenched theory, not like a mixed theory in which sea and valence quarks have different lattice actions. In most cases, the assumptions made in this paper are not only sufficient but also necessary for the validity of $\mathrm{S}\ensuremath{\chi}\mathrm{PT}$, so that a variety of possible new routes for testing this validity are opened.

Improved bilinears in lattice QCD with nondegenerate quarks

We describe the extension of the improvement program for bilinear operators composed of Wilson fermions to non-degenerate dynamical quarks. We consider two, three and four flavors, and both flavor non-singlet and singlet operators. We find that there are many more improvement coefficients than with degenerate quarks, but that, for three or four flavors, nearly all can be determined by enforcing vector and axial Ward identities. The situation is worse for two flavors, where many more coefficients remain undetermined.

Lattice QCD at maximal twist

In this review we discuss the general features of maximally twisted lattice QCD. In particular, we illustrate how automatic O(a) improvement can be achieved and how it is possible to set up a lattice regularization scheme where the problem of wrong chirality mixing (be it finite or infinite) affecting the computation of the matrix elements of the CP-conserving effective weak Hamiltonian is neatly avoided, while having at the same time a positive determinant even for non-degenerate quark pairs. The question of reducing the large cutoff effects that appear when the quark mass tends to zero as a consequence of parity and iso-spin breaking in the action is also addressed. It is shown that such dangerous lattice artifacts are strongly suppressed if the clover term is added to the action or, alternatively, the critical mass is chosen so as to enforce the restoration of parity.

Lattice QCD with two dynamical flavors of domain wall fermions

We present results from the first large-scale study of two-flavor QCD using domain wall fermions (DWF), a chirally symmetric fermion formulation which has been proven to be very effective in the quenched approximation. We work on lattices of size ${16}^{3}\ifmmode\times\else\texttimes\fi{}32$, with a lattice cutoff of ${a}^{\ensuremath{-}1}\ensuremath{\approx}1.7\text{ }\text{ }\mathrm{GeV}$ and dynamical (or sea) quark masses in the range ${m}_{\mathrm{strange}}/2\ensuremath{\lesssim}{m}_{\mathrm{sea}}\ensuremath{\lesssim}{m}_{\mathrm{strange}}$. After discussing the algorithmic and implementation issues involved in simulating dynamical DWF, we report on the low-lying hadron spectrum, decay constants, static quark potential, and the important kaon weak matrix element describing indirect $CP$ violation in the standard model, ${B}_{K}$. In the latter case we include the effect of nondegenerate quark masses (${m}_{s}\ensuremath{\ne}{m}_{u}={m}_{d}$), finding ${B}_{K}^{\overline{MS}}(2\text{ }\text{ }\mathrm{GeV})=0.495(18)$.

Quarks with twisted boundary conditions in the epsilon regime

We study the effects of twisted boundary conditions on the quark fields in the epsilon regime of chiral perturbation theory. We consider the $SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{R}$ chiral theory with nondegenerate quarks and the $SU(3{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(3{)}_{R}$ chiral theory with massless up and down quarks and massive strange quarks. The partition function and condensate are derived for each theory. Because flavor-neutral Goldstone bosons are unaffected by twisted boundary conditions chiral symmetry is still restored in finite volumes. The dependence of the condensate on the twisting parameters can be used to extract the pion decay constant from simulations in the epsilon regime. The relative contribution to the partition function from sectors of different topological charge is numerically insensitive to twisted boundary conditions.

Lattice extraction ofK→ππamplitudes to next-to-leading order in partially quenched and in full chiral perturbation theory

We show that it is possible to construct ${ϵ}^{\ensuremath{'}}/ϵ$ to next-to-leading order (NLO) using partially quenched chiral perturbation theory (PQChPT) from amplitudes that are computable on the lattice. We demonstrate that none of the needed amplitudes require 3-momentum on the lattice for either the full theory or the partially quenched theory; nondegenerate quark masses suffice. Furthermore, we find that the electro-weak penguin ($\ensuremath{\Delta}I=3/2$ and $1/2$) contributions to ${ϵ}^{\ensuremath{'}}/ϵ$ in PQChPT can be determined to NLO using only degenerate (${m}_{K}={m}_{\ensuremath{\pi}}$) $K\ensuremath{\rightarrow}\ensuremath{\pi}$ computations without momentum insertion. Issues pertaining to power divergent contributions, originating from mixing with lower dimensional operators, are addressed. Direct calculations of $K\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ at unphysical kinematics are plagued with enhanced finite volume effects in the (partially) quenched theory, but in simulations when the sea quark mass is equal to the up and down quark mass the enhanced finite volume effects vanish to NLO in PQChPT. In embedding the QCD penguin left-right operator onto PQChPT an ambiguity arises, as first emphasized by Golterman and Pallante. With one version [the ``PQS'' (patially quenched singlet)] of the QCD penguin, the inputs needed from the lattice for constructing $K\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ at NLO in PQChPT coincide with those needed for the full theory. Explicit expressions for the finite logarithms emerging from our NLO analysis to the above amplitudes also are given.

Chirally improving Wilson fermions II. Four-quark operators

In this paper we discuss how the peculiar properties of twisted lattice QCD at maximal twist can be employed to set up a consistent computational scheme in which, despite the explicit breaking of chiral symmetry induced by the presence of the Wilson and mass terms in the action, it is possible to completely bypass the problem of wrong chirality and parity mixings in the computation of the CP-conserving matrix elements of the $\Delta S=1,2$ effective weak Hamiltonian and at the same time have a positive determinant for non-degenerate quarks as well as full O($a$) improvement in on-shell quantities with no need of improving the lattice action and the operators.

Hadronic electromagnetic properties at finite lattice spacing

Electromagnetic properties of the octet mesons as well as the octet and decuplet baryons are augmented in quenched and partially quenched chiral perturbation theory to include $O(a)$ corrections due to lattice discretization. We present the results for the $\mathrm{SU}(3)$ flavor group in the isospin limit as well as the results for $\mathrm{SU}(2)$ flavor with nondegenerate quarks. These corrections will be useful for extrapolation of lattice calculations using Wilson valence and sea quarks, as well as calculations using Wilson sea quarks and Ginsparg-Wilson valence quarks.

Orderaimproved renormalization constants

We present nonperturbative results for the constants needed for on-shell O(a) improvement of bilinear operators composed of Wilson fermions. We work at {beta}=6.0 and 6.2 in the quenched approximation. The calculation is done by imposing axial and vector Ward identities on correlators similar to those used in standard hadron mass calculations. A crucial feature of the calculation is the use of nondegenerate quarks. We also obtain results for the constants needed for off-shell O(a) improvement of bilinears, and for the scale- and scheme-independent renormalization constants, Z{sub A}, Z{sub V}, and Z{sub S}/Z{sub P}. Several of the constants are determined using a variety of different Ward identities, and we compare their relative efficacies. In this way, we find a method for calculating c{sub V} that gives smaller errors than that used previously. Wherever possible, we compare our results with those of the ALPHA Collaboration (who use the Schro''dinger functional) and with one-loop tadpole-improved perturbation theory.

Erratum: Enhanced chiral logarithms in partially quenched QCD [Phys. Rev. D 56, 7052 (1997)

I discuss the properties of pions in ``partially quenched'' theories, i.e. those in which the valence and sea quark masses, $m_V$ and $m_S$, are different. I point out that for lattice fermions which retain some chiral symmetry on the lattice, e.g. staggered fermions, the leading order prediction of the chiral expansion is that the mass of the pion depends only on $m_V$, and is independent of $m_S$. This surprising result is shown to receive corrections from loop effects which are of relative size $m_S \ln m_V$, and which thus diverge when the valence quark mass vanishes. Using partially quenched chiral perturbation theory, I calculate the full one-loop correction to the mass and decay constant of pions composed of two non-degenerate quarks, and suggest various combinations for which the prediction is independent of the unknown coefficients of the analytic terms in the chiral Lagrangian. These results can also be tested with Wilson fermions if one uses a non-perturbative definition of the quark mass.

Enhanced chiral logarithms in partially quenched QCD

I discuss the properties of pions in {open_quotes}partially quenched{close_quotes} theories, i.e., those in which the valence and sea quark masses m{sub V} and m{sub S} are different. I point out that for lattice fermions which retain some chiral symmetry on the lattice, e.g., staggered fermions, the leading order prediction of the chiral expansion is that the mass of the pion depends only on m{sub V}, and is independent of m{sub S}. This surprising result is shown to receive corrections from loop effects which are of relative size m{sub S}lnm{sub V}, and which thus diverge when the valence quark mass vanishes. Using partially quenched chiral perturbation theory, I calculate the full one-loop correction to the mass and decay constant of pions composed of two nondegenerate quarks, and suggest various combinations for which the prediction is independent of the unknown coefficients of the analytic terms in the chiral Lagrangian. These results can also be tested with Wilson fermions if one uses a nonperturbative definition of the quark mass. {copyright} {ital 1997} {ital The American Physical Society}

Hadron spectrum with Wilson fermions.

We present results of a high statistics study of the quenched spectrum using Wilson fermions at $\ensuremath{\beta}=6.0$ on ${32}^{3}$\ifmmode\times\else\texttimes\fi{}64 lattices. We calculate the masses of mesons and baryons composed of both degenerate and nondegenerate quarks. Using nondegenerate quark combinations allows us to study baryon mass splittings in detail. We find significant deviations from the lowest order chiral expansion, deviations that are consistent with the expectations of quenched chiral perturbation theory. We find that there is a \ensuremath{\sim}20% systematic error in the extracted value of ${m}_{s}$, depending on the meson mass ratio used to set its value. Using the largest estimate of ${m}_{s}$ we find that the extrapolated octet mass splittings are in agreement with the experimental values, as is ${M}_{\ensuremath{\Delta}}\ensuremath{-}{M}_{N}$, while the decuplet splittings are 30% smaller than experiment. Combining our results with data from the GF11 collaboration we find considerable ambiguity in the extrapolation to the continuum limit. Our preferred values are $\frac{{M}_{N}}{{M}_{\ensuremath{\rho}}}=1.38(7)$ and $\frac{{M}_{\ensuremath{\Delta}}}{{m}_{\ensuremath{\rho}}}=1.73(10)$, suggesting that the quenched approximation is good to only \ensuremath{\sim}10-15%. We also analyze the $O(\mathrm{ma})$ discretization errors in heavy quark masses.

Testing the chiral behavior of the hadron spectrum

We analyze the chiral behavior of the hadron spectrum obtained with quenched Wilson fermions on 170 $32^3 \times 64$ lattices at $\beta = 6.0$. We calculate masses of hadrons composed of both degenerate and non-degenerate quarks. We reduce the statistical errors in mass splittings by directly fitting to the ratio of correlation functions. We find significant deviations from a linear dependence on the quark mass, deviations that are consistent with the higher order terms predicted by quenched chiral perturbation theory. Including these corrections yields splittings in the baryon octet that agree with those observed experimentally. Smaller higher order terms are also present in $m_\rho$ and $m_N$. By contrast, the decuplet baryons are well described by a linear mass term. We find the decuplet splittings to be 30% smaller than experiment. We extrapolate our data to $a \to 0$ by combining with the GF11 results, and the best fit suggests that the quenched approximation is only good to $10-15%$.

Light hadron spectrum and decay constants in quenched lattice QCD

We present results for light hadrons composed of both degenerate and non-degenerate quarks in quenched lattice QCD. We calculate masses and decay constants using 60 gauge configurations with an $O(a)$--improved fermion action at $\beta = 6.2$. Using the $\rho$ mass to set the scale, we find hadron masses within two to three standard deviations of the experimental values (given in parentheses): $m_{K^*}=868\er{9}{8}$~MeV (892~MeV), $m_{\phi}=970\err{20}{10}$~MeV (1020~MeV), $m_N=820\err{90}{60}$~MeV (938~MeV), $m_\Delta=1300\errr{100}{100}$~MeV (1232~MeV) and $m_\Omega=1650\err{70}{50}$~MeV (1672~MeV). Direct comparison with experiment for decay constants is obscured by uncertainty in current renormalisations. However, for ratios of decay constants we obtain $f_K/f_\pi=1.20\er{3}{2}$ (1.22) and $f_\phi/f_\rho=1.13\er{2}{3}$ (1.22).

H±→W±γandH±→W±Zin two-Higgs-doublet models: Large-fermion-mass limit

In two-Higgs-doublet models, the ${H}^{\ifmmode\pm\else\textpm\fi{}}{W}^{\ensuremath{\mp}}V$ vertex ($V=Z or \ensuremath{\gamma}$) does not occur at the tree level, but is generated at one-loop order. We derive explicit formulas for the contribution of a weak doublet of fermions ($t, b$) to the ${H}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{W}^{\ifmmode\pm\else\textpm\fi{}}Z$ amplitude in the limit of large fermion masses that appear in the internal loop. In this limit, the amplitude grows as the square of the fermion masses for ${m}_{t}\ensuremath{\ne}{m}_{b}$, and is constant if ${m}_{t}={m}_{b}$. In contrast, the ${H}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{W}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\gamma}$ amplitude approaches a constant (which depends on $\frac{{m}_{t}}{{m}_{b}}$) in the large-fermion-mass limit. Asymptotic results for the contribution of a heavy supersymmetric family are also given. The observability of ${H}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{W}^{\ifmmode\pm\else\textpm\fi{}}Z$ at a future supercollider requires either ${m}_{{H}^{\ifmmode\pm\else\textpm\fi{}}}<{m}_{t}$ and a large $t$-quark mass, or the existence of a fourth-generation nondegenerate quark doublet.