119144040
RG flows, cycles, and c-theorem folklore
Monotonic renormalization group flows of the “c” and “a” functions are often cited as reasons why cyclic or chaotic coupling trajectories cannot occur. It is argued here, based on simple examples, that this is not necessarily true. Simultaneous monotonic and cyclic flows can be compatible if the flow-function is multi-valued in the couplings. Exact general results for renormalization group (RG) flows are important as they may provide physical insight for strongly coupled systems. The c-theorem for 2D systems [1] and the a-theorem for 4D systems [2, 3] are two such results that have been established for very broad classes of models [4]. The c-theorem shows the existence of a monotonically decreasing function of the length scale, c (L), which interpolates between 2D Virasoro central charges of theories at conformal fixed points, and thereby provides an intuitively correct count of system degrees of freedom — fewer in the infrared than in the ultraviolet. The a-theorem establishes similar monotonic flow for the induced coefficient of the Euler density, a (L), for a 4D theory in a curved spacetime background. It is a common conclusion — a “folk theorem”— based on these monotonically evolving “observables” that the underlying couplings can not have RG trajectories which are limit cycles or undergo more exotic (e.g. chaotic) oscillations (e.g. see 2nd bullet item under §6 in [5]). The point of this note is to explain and illustrate with just one coupling, as simply as possible, why this conclusion is unwarranted. (Somewhat similar criticism of the monotonic folklore has been proffered in other contexts, involving degenerate Morse function counterexamples for models with vorticity in the flow of several couplings [6].) In principle, we believe cyclic or perhaps even chaotic coupling trajectories are not ruled out by either the cor a-theorems, nor are they necessarily excluded by other monotonic “potential flow-functions.” To illustrate our reasoning, we begin with a very simple example based on a mechanical analogy. While this example does indeed exhibit both monotonic flow and a cycling trajectory, it has the peculiar feature — insofar as intuitively counting degrees of freedom is concerned — that the monotonic flow is unbounded both above and below. Nevertheless, we recall there is a field theory model that produces just such behavior [8]. We then exhibit another example where the monotonic flow is bounded below and the coupling trajectory is not only cyclic but, in fact, chaotic. The essential ideas, expressed for a single coupling x (t), where t = lnL, are given by general statements for a locally gradient RG flow, dx (t) dt = β (x (t)) = − (x (t)) dx (t) , (1) dC (x (t)) dt = dx dt dC dx = β dC dx = − (
115164400
Classical and quantal ternary algebras
We consider several ternary algebras relevant to physics. We compare and contrast the quantal versions of the algebras, as realized through associative products of operators, with their classical counterparts, as realized through classical Nambu brackets. In some cases involving infinite algebras, we show the classical limit may be obtained by a contraction of the quantal algebra, and then explicitly realized through classical brackets. We illustrate this classical-contraction method by the Virasoro-Witt example.
119142727
Approximate Solutions of Functional Equations
Approximate solutions to functional evolution equations are constructed through a combination of series and conjugation methods, and relative errors are estimated. The methods are illustrated, both analytically and numerically, by construction of approximate continuous functional iterates for , sin x, and λx(1 − x). Simple functional conjugation by these functions, and their inverses, substantially improves the numerical accuracy of formal series approximations for their continuous iterates.
115167324
Multi-operator brackets acting thrice
We generalize an identity first found by Bremner for Nambu 3-brackets. For odd N-brackets built from associative operator products, we show that for any fixed A, when totally antisymmetrized over all the Bs.
117928601
Critical end point of Nf=3 QCD at finite temperature and density
Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, JapanWe investigate the phase structure of 3-ﬂavor QCD in the presence of ﬁnite quark chemical po-tential aµ=0.1 by using the Wilson-Clover fermion action. Especially, wefocus on locating thecritical end point that characterizes the phase structure. We do this by the kurtosis intersectionmethod for the quark condensate. For Wilson-type fermions,the correspondence between bareparameters and physical parameters is indirect. Hence we present a strategy to transfer the bareparameter phase structure to the physical one.32nd International Symposium on Lattice Field Theory LATTICE 2014June 23 - 28, 2014New York, USA
119211023
Critical endpoint of the finite temperature phase transition for three flavor QCD
We investigate the critical endpoint of finite temperature phase transition of $N_f=3$ QCD at zero chemical potential. We employ the renormalization-group improved Iwasaki gauge action and non-perturbatively $O(a)$-improved Wilson-clover fermion action. The critical endpoint is determined by using the intersection point of kurtosis for the temporal size $N_t$=4, 6, 8. Spatial sizes of $N_l$=6-16 ($N_t$=4), 10-24 ($N_t$=6), and 12-24 ($N_t$=8) are employed. We find that $N_t$=4 is out of the scaling region. Using results for $N_t$=6 and 8, and making linear extrapolations in $1/N_t^2$, we obtain $\sqrt{t_0}T_{\rm E}=0.0975(14)(8)$, $\sqrt{t_0}m_{\rm PS,E}=0.2254(52)(105)$ and $m_{\rm PS,E}/T_{\rm E}=2.311(63)(13)$, where the first error is statistical error, the second error is systematic error, and $m_{\rm PS}$ is the pseudo scalar meson mass. If one uses $1/\sqrt{t_0}=1.347(30)$ GeV reported by Borsanyi et al., one finds $T_{\rm E}=131(2)(1)(3)$ MeV, $m_{\rm PS,E}=304(7)(14)(7)$ MeV and $m_{\rm PS,E}/m_{\rm PS,E}^{\rm phys, sym}=0.739(17)(34)(17)$, where the third error comes from error of $\sqrt{t_0}$ and $m_{\rm PS}^{\rm phys, sym}=\sqrt{(m_\pi^2+2m_K^2)/3}$. Our current estimation of $\sqrt{t_0}m_{\rm PS,E}$ in the continuum limit is about 25% smaller than the SU(3) symmetric point.
118617324
Zeros of QCD partition function from finite density lattices
Email: xjin@riken.jpPartition function zeros steer the critical behavior of a system. Studying four-ﬂavor lattice QCDat ﬁnite temperature and density with the Wilson-clover fermion action and the Iwasaki gauge ac-tion using a phase-quenched fermion determinant, we combine statistics from multiple chemicalpotentials to improve sampling of the conﬁguration space, and aim at unraveling the movementof zeros in ﬁnite systems. Preparing for further investigations, we discuss methods and criteriaused to sieve through complex parameter space spanned by (Rem;Imm) and (Rem;Imb), andpresent statistically robust zeros of the partition function.31st International Symposium on Lattice Field Theory LATTICE 2013July 29 – August 3, 2013Mainz, Germany
118449163
Lattice QCD with 12 Quark Flavors: A Careful Scrutiny
With a substantial amount of simulations, we have explored the system across a wide range of lattice scales. We have located a lattice artifact, first order bulk transition, have studied its properties, and found that the flavor-singlet scalar meson mass vanishes at the critical endpoint. We will discuss the lattice phase diagrams and the continuum limits for both a spontaneous chiral symmetry breaking phase and an infrared conformal phase, and compare results with other groups.