# Chiral Symmetry

New actions for minimally doubled fermions and their counterterms
Stefano Capitani
Tue, 16:20, Seminar Room D -- Parallels 4D (Slides)

Minimally doubled fermions provide a cheap and convenient way of simulating quarks which preserve chiral symmetry. It has been established that two actions of this kind (known as Borici-Creutz and Karsten-Wilczek) require the tuning of three counterterms in order to be properly renormalized. Here we construct some more general minimally doubled actions and investigate the properties of their counterterms.

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Numerical studies of Minimally Doubled Fermions
Johannes Weber, Stefano Capitani, Hartmut Wittig
Tue, 16:40, Seminar Room D -- Parallels 4D (Slides)

Minimally doubled fermions of the Karsten-Wilczek class have been studied for the first time in the quenched approximation. Non-perturbative renormalisation criteria are discovered by means of a detailed study of the parameter dependence of mesonic observables. Anisotropies are mapped out by a study of observables acquired through measurement of different euclidean components of the transfer matrix.

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Lattice simulation of SU(2) gauge theory with chirally symmetric fermions
Hideo Matsufuru, Yoshio Kikukawa, Kei-ichi Nagai, Norikazu Yamada
Tue, 17:00, Seminar Room D -- Parallels 4D (Slides)

SU(2) lattice gauge theory with chirally symmetric fermions is numerically studied. We generate dynamical configurations with two-flavors of domain-wall fermions. Using the overlap fermion operators in addition to the domain-wall, we investigate the chiral dynamics through the eigenvalue spectrum and meson correlators.

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Lattice QCD with overlap fermions
Bjoern Walk, Hartmut Wittig
Tue, 17:20, Seminar Room D -- Parallels 4D (Slides)

With the advent of Ginsparg-Wilson fermions problems involving chiral symmetry can be treated on the lattice with controllable error. One prominent solution to the Ginsparg-Wilson relation are overlap fermions. Unfortunately, simulations with chiral fermions are still extremely hard to simulate both in terms of computational demand and algorithmic reliability. In my talk, I want to present the experiences with our GPU-based implementation of overlap fermions. I both give details of our implementation and performance gain by the accelerator. I will also show results for physical applications, i.e. random matrix theory and our main goal, non-leptonic kaon decays and the $$\Delta I = 1/2$$-rule.

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Large-scale simulations with chiral symmetry
Takashi Kaneko, Sinya Aoki, Guido Cossu, Hidenori Fukaya, Shoji Hashimoto, Junichi Noaki
Tue, 17:40, Seminar Room D -- Parallels 4D (Slides)

We report on our large-scale simulations of three-flavor lattice QCD respecting chiral symmetry. We present our comparative study of HMC performance among various domain-wall-type formulations and report on our on-going large-scale simulations with our choice of lattice action.

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The kaon mass in 2+1+1 flavor twisted mass Wilson ChPT
Oliver Bär, Ben Hörz
Wed, 08:30, Seminar Room D -- Parallels 5D (Slides)

We construct the chiral low-energy effective theory for 2+1+1 flavor lattice QCD with twisted mass Wilson fermions. In contrast to existing results we assume a heavy charm quark mass such that the D mesons are too heavy to appear as degrees of freedom in the effective theory. As an application we compute the kaon mass to 1-loop order in the Aoki regime. Somewhat surprisingly, the result contains a chiral logarithm involving the neutral pion mass which has no analogue in continuum ChPT. Since the neutral pion mass is very light in actual lattice simulations our results predict significant corrections to the kaon mass, much larger than the statistical errors.

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Applications of SU(3) ChPT including lattice data close to the SU(3) symmetric point
Andreas Schäfer, Peter Bruns, Ludwig Greil, Philipp Wein
Wed, 08:50, Seminar Room D -- Parallels 5D (Slides)

Chiral perturbation theory (ChPT) is a valuable tool to controle systematic errors in LQCD calculations. However, SU(3)-ChPT has notoriously bad convergence properties. We demonstrate for a few examples, that inclusion of data close to the SU(3) symmetric point substantially improves the situation for primarily three reasons: First, and most importantly, no pseudo-Goldstone-boson masses larger than the physical eta mass are needed in the extrapolation from the symmetric point (where $$m_u=m_d=m_s$$, while $$m_u+m_d+m_s$$ is kept fixed at its physical value) to the physical point. Second, some combinations of low-energy constants which parametrize symmetry-breaking effects can be directly obtained quite accurately from the behavior of the observables close to the symmetric point. And third, the values of observables in the chiral limit (like baryon masses, or coupling parameters), which can not be directly measured on the lattice, can for most purposes be eliminated in favor of the values at the symmetric point in the extrapolation formulae. The latter values can be measured in (relatively cheap) lattice simulations, so that some uncertainty inherent in the extrapolation to the chiral limit is also eliminated.

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Partially twisted boundary conditions for scalar mesons
Akaki Rusetsky, Dimitri Agadjanov, Ulf Meissner
Wed, 09:10, Seminar Room D -- Parallels 5D (Slides)

The properties of scalar mesons with the mass around 1 GeV can be studied on the lattice by using coupled-channel Luescher equation. Analyzing synthetic data, it was shown that the use of twisted boundary conditions leads to a significant improvement of the accuracy of the method. Unfortunately, this statement refers to the full twisting, which is very expensive. In this work we discuss the possibility of partial twisting in the scalar meson sector by studying the problem within the framework of the effective field theories in a finite volume. Partially-twisted Luescher equation is derived. It is demonstrated that, despite the presence of annihilation diagrams, in the sector with the total isospin I=1 one may impose partially twisted boundary conditions so that the resulting Luescher equation coincides with the fully twisted one up to the exponentially suppressed corrections.

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Sunset integrals at finite volume
Johan Bijnens
Wed, 09:30, Seminar Room D -- Parallels 5D (Slides)

Chiral Perturbation Theory is a useful tool to aid in performing the various extrapolations needed in lattice QCD calculations of physical quantities. These include extrapolations in quark mass, finite lattice spacing and finite size of the lattice. Especially the latter will become more important when the quark masses on the lattice become smaller. The pion mass and decay constant in two-flavour QCD is known at finite volume to two loops. Here we develop the needed two-loop integrals at finite volume to do the calculations for masses and decay constants for all general mass cases and including moving frames and possible other boundary conditions. I will present results based on an expansion in Bessel functions as well as on a version using theta functions and compare their efficiency. Work is in progress to combine these results with two-loop ChPT calculations.

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Chiral Symmetry Restoration from a Boundary
Brian Tiburzi
Wed, 09:50, Seminar Room D -- Parallels 5D (Slides)

The imposition of Dirichlet boundary conditions in lattice computations obstructs the formation of a chiral condensate. We use chiral perturbation theory and meson models to address the effect of a Dirichlet boundary on chiral symmetry breaking. While pions are the longest-range modes in QCD, the restoration of chiral symmetry due to a boundary is shown not to depend upon the pion Compton wavelength but rather on that of the sigma meson. Power-law finite size corrections are exposed, and require prohibitively large lattices to overcome. We speculate on the frustration of the chiral condensate in some other cases.

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Finite volume scaling of the electro-magnetic pion form factor in the epsilon regime
Takashi Suzuki, Hidenori Fukaya
Wed, 10:10, Seminar Room D -- Parallels 5D (Slides)

We consider finite volume effects on the pion form factor near the chiral limit, in the so-called epsilon-regime. The pseudoscalar-vector-pseudoscalar three-point function is calculated in the epsilon expansion of chiral perturbation theory to the next-to-leading order. In the epsilon regime, the finite volume effects are non-perturbatively large in general. However, we find a way to remove its dominant part, inserting momenta to the correlators, and taking an appropriate ratio of them. The subleading contribution is, then, expected to be perturbatively small, and one can extract the form factor as in a similar way to that in the p regime.

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Effects of Low vs High Fermionic Modes on Hadron Mass Generation
Mikhail Denissenya, Leonid Glozman, Christian Lang, Mario Schröck
Thu, 14:00, Seminar Room D -- Parallels 7D (Slides)

We study the effects of the low-lying fermionic modes of the Dirac operator responsible for the chiral symmetry breaking. While these modes are crucial for the pion, they account only for 2/3 of the $$\rho$$ and $$N$$ masses. Unlike the high-lying modes the low-lying modes do not produce the $$b_1$$ bound state. The latter fact is interpreted as an indication that the low-lying modes do not encode the effect of confinement.

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More effects of Dirac low-mode removal
Mario Schröck, Mikhail Denissenya, Leonid Glozman, Christian Lang
Thu, 14:20, Seminar Room D -- Parallels 7D (Slides)

In previous studies we have shown that isovector hadrons, except for a pion, survive if we remove the lowest lying Dirac eigenmodes from the spectrum. The low-modes are tied to the dynamical breaking of chiral symmetry and thus we found chiral symmetry to be restored by means of matching masses of chiral partners like the vector and axialvector currents. Here we investigate the influence of removing the lowest part of the Dirac spectrum on the locality of the Dirac operator and, moreover, analyze the influence of low-mode truncation on the quark momenta and thereupon on the hadron spectrum.

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The Effect of the Low Energy Constants on the Spectral Properties of the Wilson Dirac Operator
Savvas Zafeiropoulos, Mario Kieburg, Jacobus Verbaarschot
Thu, 14:40, Seminar Room D -- Parallels 7D (Slides)

The successful application of Random Matrix Theory to the epsilon regime of QCD has led to new insights into the spectral properties of the Dirac operator. Lately random matrix ensembles were proposed to describe lattice artifacts. Especially Wilson fermions were studied with Random Matrix Theory. Quite recently we analyzed the effect of all three order $$a^2$$ terms of Wilson chiral Perturbation Theory on the infrared spectrum of the Wilson Dirac operator. In particular we calculated analytical results for the densities of the complex eigenvalues, of the real eigenvalues and of the distribution of the chiralities over the real eigenvalues. We analyzed the asymptotics of small and large lattice spacing and identified new, easily accessible observables to extract the low energy constants of Wilson chiral Perturbation Theory.

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Investigating the Sharpe-Singleton scenario on the lattice by direct eigenvalue computation
Joni Suorsa, Kari Rummukainen, Kim Splittorff, David Weir, Teemu Rantalaiho
Thu, 15:00, Seminar Room D -- Parallels 7D (Slides)

We investigate the presence of the Sharpe-Singleton scenario on the lattice. We calculate the fifty lowest lying eigenvalues of the Wilson-Dirac operator using a parallelized GPU-based arnoldi algorithm and witness the collective effect the quark mass has on the eigenvalue distribution as predicted by the Wilson chiral perturbation theory.

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Chiral condensate from the Banks-Casher relation
Georg Engel, Leonardo Giusti, Stefano Lottini, Rainer Sommer
Thu, 15:20, Seminar Room D -- Parallels 7D (Slides)

Based on the Banks-Casher relation, we determine the chiral condensate in two-flavor QCD using CLS-configurations with several quark masses and two different lattice spacings. We compute the mode number of the O(a)-improved Wilson-Dirac operator for several values of $$\Lambda$$, and we discuss different fitting strategies to extract the chiral condensate from its mass and $$\Lambda$$ dependence.

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Discretization Effects in the $$\epsilon$$ Domain of QCD
Jacobus Verbaarschot, Mario Kieburg, Kim Splittorff, Savvas Zafeiropoulos
Thu, 15:40, Seminar Room D -- Parallels 7D (Slides)

At nonzero lattice spacing the QCD partition function with the Wilson Dirac operator, $$D_W$$, undergoes either a second other phase transition to the Aoki phase for decreasing quark mass, or shows a first order jump when the quark mass changes sign. What happens depends on the value of the low-energy constants. We discuss these phase transitions in terms of Wilson Dirac spectra and show that the first order scenario can only occur in the presence of dynamical quarks while in the quenched case we can only have a transition to the Aoki phase. The exact microscopic spectral density of the nonhermitian Wilson Dirac operator with dynamical quarks is discussed as well. We conclude with some remarks on discretization effects for the overlap Dirac operator.

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Chiral behavior of pion properties from lattice QCD
Alfonso Sastre
Fri, 14:00, Seminar Room D -- Parallels 9D (Slides)

I will present preliminary results of a study of the chiral behavior of the pion mass and decay constant, based on 2+1 flavor lattice QCD simulations. Performed at four values of the lattice spacing and all the way down to the physical value of the pion mass and even below, these calculations allow a detailed comparison with the predictions of SU(2) chiral perturbation theory and a determination of some of its low energy constants.

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Overlap/Domain-wall reweighting
Hidenori Fukaya, Sinya Aoki, Guido Cossu, Shoji Hashimoto, Takashi Kaneko, Junichi Noaki
Fri, 14:20, Seminar Room D -- Parallels 9D (Slides)

We investigate the eigenvalues of nearly chiral lattice Dirac operators constructed with five-dimensional implementaions. Allowing small violation of the Ginsparg-Wilson relation, the HMC simulation is made much faster while the eigenvalues are not significantly affected. We discuss the possibility of reweighting the gauge configurations generated with domain-wall fermions by those of exactly chiral lattice fermions.

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Computation of the chiral condensate using $$N_f=2$$ and $$N_f=2+1+1$$ dynamical flavors of maximally twisted mass fermions
Elena Garcia Ramos, Karl Jansen, Krzysztof Cichy, Andrea Shindler
Fri, 14:40, Seminar Room D -- Parallels 9D (Slides)

We apply the spectral projector method, recently introduced by Lüscher and Giusti, to compute the chiral condensate using $$N_f=2$$ and $$N_f=2+1+1$$ dynamical flavours of maximally twisted mass fermions. We present our results for several quark masses at three different lattice spacings which allows us to perform the chiral and continuum extrapolations. In addition we report our analysis on finite volume effects. We also study the effect of the dynamical strange and charm quarks by comparing our results for $$N_f=2$$ and $$N_f=2+1+1$$ dynamical flavours.

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Topological susceptibility from twisted mass fermions using spectral projectors
Krzysztof Cichy, Elena Garcia Ramos, Karl Jansen
Fri, 15:00, Seminar Room D -- Parallels 9D (Slides)

We present the results of our computation of the topological susceptibility with $$N_f=0$$, $$N_f=2$$ and $$N_f=2+1+1$$ flavours of maximally twisted mass fermions. We perform a detailed study of the quark mass dependence and discretization effects. We confront our data with chiral perturbation theory and attempt an extraction of the chiral condensate from the quark mass dependence of the topological susceptibility. We compare it with the results of our direct computation from the slope of the mode number. We also compute the continuum limit of the quenched topological susceptibility and test the Witten-Veneziano formula that relates it to the masses of the kaon and the eta and eta prime mesons.

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Study of Anomalous Mass Generation in $$N_f=1$$ QCD
Luchang Jin
Fri, 15:20, Seminar Room D -- Parallels 9D (Slides)

The $$U(1)$$ axial symmetry in QCD is anomalously broken, and in the case of one flavor, a fermion mass is generated by instanton-like gauge field configurations. Conventional continuum analysis shows that this anomalously generated mass term is soft'' and goes away at large momentum due to the low density of small instantons, distinguishing it from a normal mass term. However, it may possible that there are enough lattice-scale instantons / dislocations to generate a hard'' fermion mass, at least for a class of lattice gauge actions, leading to the mass ambiguity suggested by Creutz. We report on a study of this anomalous mass generation idea using Landau-gauge-fixed fermion propagators and examining the momentum dependence of the RI/MOM-defined fermion mass.

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On the decoupling of mirror fermions
Joel Giedt, Chen Chen, Erich Poppitz
Fri, 15:40, Seminar Room D -- Parallels 9D (Slides)

An approach to the formulation of chiral gauge theories on the lattice is to start with a vector-like theory, but decouple one chirality (the "mirror" fermions) using strong Yukawa interactions with a chirally coupled "Higgs" field. While this is an attractive idea, its viability needs to be tested with nonperturbative studies. The model that we study here, the so-called "3-4-5" model, is anomaly free and the presence of massless states in the mirror sector is not required by anomaly matching arguments, in contrast to the "1-0" model that was studied previously. We have computed the polarization tensor in this theory and find a directional discontinuity that appears to be nonzero in the limit of an infinite lattice, which is equivalent to the continuum limit at fixed physical volume. We show that a similar behavior occurs for the free massless Ginsparg-Wilson fermion, where the polarization tensor is known to have a directional discontinuity in the continuum limit. We thus find support for the conclusion that in the continuum limit of the 3-4-5 model, there are massless charged modes in the mirror sector so that it does not decouple from the light sector. The value of the discontinuity we obtain allows for two interpretations: either a chiral gauge theory does not emerge and mirror-sector fermions in a chiral anomaly free representation remain massless, or a massless vectorlike mirror fermion appears. We end by discussing some questions for future study.

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Residual mass in five-dimensional fermion formulations
Shoji Hashimoto, Sinya Aoki, Guido Cossu, Hidenori Fukaya, Takashi Kaneko, Junichi Noaki, Peter Boyle
Poster Session

Using a general five-dimensional representation of the Ginsparg-Wilson fermion, we study the residual chiral symmetry violation and numerical costs of various fermion formulations including the domain-wall and overlap fermions as special cases.

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Spectral Properties of a 2d IR Conformal Theory
Wolfgang Bietenholz, B David Landa-Marban, Ivan Hip
Poster Session

The Schwinger model with two or more flavors is a simple example for an IR conformal gauge theory. We consider numerical data for two light flavors, based on simulations with dynamical overlap fermions. We test properties and predictions that were put forward for IR conformal models in the recent literature. In particular we probe the de-correlation of low lying Dirac eigenvalues, and we discuss the mass anomalous dimension and its IR extrapolation. Here we encounter subtleties, which may urge caution with analogous efforts in other models, such as multi-flavor QCD.

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A determination of the Wilson Chiral effective theory constant $$c_2$$ using $$n_f = 2$$ CLS lattices
John Bulava
Poster Session

When performing chiral extrapolations it is helpful to estimate the contributions to chiral curvature due to lattice artifacts. With Wilson-type fermions these effects are parametrized by the parameter $$c_2$$, the coefficient in the chiral lagrangian of the symmetry breaking term due to finite lattice spacing. We present results for a determination of this term using $$I=2$$ $$\pi-\pi$$ correlation functions on ensembles of $$n_f=2$$ clover-improved fermions generated through the CLS effort. For this discretization we demonstrate that $$c_2$$ is small and comment on its sign.

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