Absolute X-distribution and self-duality
AAbsolute X-distribution and self-duality
Andrei Alexandru ∗ The George Washington University, Washington, DC, USAE-mail: [email protected]
Ivan Horváth
University of Kentucky, Lexington, KY, USAE-mail: [email protected]
Various models of QCD vacuum predict that it is dominated by excitations that are predominantlyself-dual or anti-self-dual. In this work we look at the tendency for self-duality in the case ofpure-glue SU(3) gauge theory using the overlap-based definition of the field-strength tensor. Togauge this property, we use the absolute X-distribution method which is designed to quantify the dynamical tendency for polarization for arbitrary random variables that can be decomposed in apair of orthogonal subspaces.
XXIX International Symposium on Lattice Field TheoryJuly 10-16 2011Squaw Valley, Lake Tahoe, California ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] N ov bsolute X-distribution and self-duality Andrei Alexandru
1. Motivation
Various models of QCD vacuum use semi-classical arguments to describe the mechanism re-sponsible for confinement or chiral-symmetry breaking. The semi-classical arguments start byexpanding QCD partition function around extremal points of the action, i.e., (cid:10) Ω | e − H τ | Ω (cid:11) ≈ e − S cl (cid:90) D x ( τ ) exp (cid:32) − δ x δ S δ x (cid:12)(cid:12)(cid:12)(cid:12) x cl δ x + · · · (cid:33) . (1.1)The first task is then to find the extremal points of the action and then take into account gaussianfluctuations around these extrema.The action for pure-glue QCD can be expressed in terms of the self-dual and anti-self-dualcomponents of the field strength tensor S = g (cid:90) d x F a µν F a µν = g (cid:90) d x (cid:20) ± F a µν ˜ F a µν + (cid:0) F a µν ∓ ˜ F a µν (cid:1) (cid:21) . (1.2)The integral of the F a µν ˜ F a µν term is a boundary term that is related to the topological charge of theconfiguration. If we keep the boundary values fixed, the integral is minimized when the quantityin the parenthesis vanishes. This happens when the field is self-dual, F a µν = ˜ F a µν , or anti-self-dual F a µν = − ˜ F a µν . A more sophisticated analysis leads to the conclusion that all the extremal points ofthe classical action that are not saddle points satisfy this condition [1].It is then natural to expect that if QCD vacuum is correctly described by a semi-classical model,the field strength in a typical lattice QCD ensemble will exhibit a high degree of self-duality . Togauge this tendency we decompose the field strength at every point on the lattice into its self-dualcomponents and analyze their polarization properties. To do this, we use the method of absolute X-distribution designed to analyze the dynamical aspects of polarization [2]. A more detailed accountof this work is given in Ref. [3].
2. Dynamical polarization
We start by reviewing the method of absolute X-distribution. A first version of this approachwas introduced in a study of the local chirality of the low-lying eigenmodes of the Dirac opera-tor [4]. In general, for an arbitrary observable that can be split in two components Q = Q + Q , wesay that Q is polarized when it tends to be aligned with either one of the components. More pre-cisely, if we look at the magnitude of components, q i = (cid:107) Q i (cid:107) , we tend to think that the observable Q is polarized when the probability distribution P b ( q , q ) , with support in the positive quadrantof the q q -plane, is peaked in the vicinity of the q , axes.The raw distribution P b ( q , q ) is difficult to characterize. A more direct measure is offeredby the induced distribution of the polarization angle . In Fig. 1 we plot the raw distribution ofchirality components as determined in a previous study [2] and the corresponding polarizationangle distribution (the curve indicated by α = X-distribution . We see thatthe X-distribution tends to be concentrated towards the middle of the graph, suggesting an anti-polarization tendency. 2 bsolute X-distribution and self-duality
Andrei Alexandru
Figure 1:
Sample pair distribution generated by chirality components of the lowest eigenmodes of ensemble E from [2]. Right: the associated X-distribution, i.e., the induced distribution of the polarization angle. A more careful analysis reveals that the conclusions based on this method can be misleading.The X-distribution is determined by the choice of parametrization for the angles measured in the q q -plane. The definition we used to plot Fig. 1 is x = π arctan (cid:107) Q (cid:107)(cid:107) Q (cid:107) − . (2.1)We will refer to this choice as the reference polarization [4]. However, this choice is not unique.Alternative definitions were used in various studies. Using t ≡ (cid:107) Q (cid:107) / (cid:107) Q (cid:107) , one class of valid anglevariables is given by a generalization of the above definition¯ x = π arctan ( t α ) − , (2.2)where α > α = x is the referencepolarization defined above, while the definition based on ¯ x with α = x with α =
4. The qualitative behavior of the distribution changes dramatically, whilethe dynamics producing the original distribution is unchanged. It is clear then that conclusionsbased on X-distributions alone cannot be trusted.To address this problem we define the absolute X-distribution , a measure of the pair correlationinduced by the underlying dynamics [2, 6]. The basic idea is to compare the correlated distribu-tion P b ( q , q ) with a similar distribution where the components are statistically independent, toisolate the effect of the dynamics. The uncorrelated distribution is constructed from the marginaldistributions P ( q ) = (cid:90) dq P b ( q , q ) and P ( q ) = (cid:90) dq P b ( q , q ) . (2.3)For our application, symmetry guarantees that P = P . The uncorrelated distribution is P u ( q , q ) ≡ P ( q ) P ( q ) . We define an angle variable that has constant angular density for the uncorrelated3 bsolute X-distribution and self-duality Andrei Alexandru
Figure 2:
X-distribution using the reference polarization for the correlated and uncorrelated distributions(left) and the absolute X-distribution (right). distribution. This is the absolute polarization . The histogram of this angle variable for the un-correlated distribution is flat. In our figures this is indicated by a horizontal dashed line. TheX-distribution in terms of the absolute polarization is the absolute X-distribution .In the left panel of Fig. 2 we present the X-distribution for the reference polarizations for bothcorrelated distribution, P b , and the uncorrelated one, P u , for the ensemble presented in Fig. 1.Notice that these two distributions are almost identical indicating that there is little dynamicalcorrelation. In the right panel we plot the absolute polarization histogram, which is almost flat.There is a small enhancement towards the edges indicating that the dynamics induces a slightpolarization. This is consistent with the plots in the right panel, where we see that the uncorrelateddistribution is more prominent towards the center of the histogram.Based on the absolute polarization distribution, P A ( x ) , we construct a more compact measureof the polarization tendency, the correlation coefficientC A = Γ − Γ = (cid:90) − dx P A ( x ) | x | . (2.4)The coefficient Γ measures the probability that a sample drawn from distribution P b is more po-larized than one drawn from P u . When we have no dynamical correlation this probability is 0 . C A =
3. Field strength definition
In this study, we will use a definition of the field strength based on the overlap operator.Compared to the ultra-local definitions, the overlap definition is less susceptible to ultra-violetfluctuations, so no arbitrary link smearing or cooling is needed. Moreover, this definition providesa natural expansion in terms of eigenmodes of the Dirac operator which allows us to define asmoothed version of field strength tensor controlled by the value of the eigenvalue cutoff.If we denote with S F = ¯ ψ D ( x , y ) ψ the fermionic contribution to the action in the overlap for-mulation, it is easy to show that tr s σ µν D ( x , x ) has the same quantum numbers as the field strength4 bsolute X-distribution and self-duality Andrei Alexandru F µν [7]. Here tr s denotes the trace over the spinor index. It was shown by explicit calculation thaton smooth fields in the limit a → s σ µν D ( x , x ) = c T F µν ( x ) + O ( a ) . (3.1)Above, c T is a constant that depends on the kernel used to define the overlap operator. The latticeversion of the field strength operator used in this study is F ov µν ( x ) ≡ c T tr s σ µν D ( x , x ) = − c T tr s σ µν [ ρ − D ( x , x )] , (3.2)where 2 ρ is the largest eigenvalue of D , the eigenvalue associated with the zero modes’ partners.We used the fact that tr s σ µν = F Λ µν ( x ) ≡ − c T ∑ | λ | < Λ a tr s σ µν ( ρ − λ ) ψ λ ( x ) ψ λ ( x ) † . (3.3)This definition has the property that lim Λ → ∞ F Λ = F and that the contribution of the largest eigen-modes is suppressed. The self-dual and anti-self-dual parts of the field strength are defined usingthe dual of the field strength ˜ F µ , ν = ε µναβ F αβ F S = ( F + ˜ F ) F A = ( F − ˜ F ) . (3.4)
4. Numerical results
For our study we used a set of pure-glue ensembles generated using Iwasaki action [10]. Theparameters for these ensembles are presented in Table 1. To study the continuum limit we have aset of 5 ensembles with the same volume. To determine the finite volume effects we also generatedone ensemble with a larger volume.In Fig. 3 we plot the histogram for the absolute polarization for all ensembles with volume ( .
32 fm ) . We find a small tendency for polarization that decreases as we make the lattice spacingsmaller. To understand whether this tendency survives the continuum limit, we compute the cor-relation coefficient and fit it with a quadratic polynomial in a . As we can see from the right panelEnsemble Size Lattice spacing Volume Configurations E .
110 fm 400 E .
083 fm 200 E .
066 fm ( .
32 fm ) E .
055 fm 40 E .
041 fm 20 E .
055 fm ( .
76 fm ) Table 1:
The size and lattice spacing for the ensembles used in this study. bsolute X-distribution and self-duality Andrei Alexandru
Figure 3:
Left: absolute X-distribution for self-duality components. Note that the y-scale is magnified tobetter show the difference between different lattice spacings. Right: the correlation coefficient as a functionof the lattice spacing and its continuum limit extrapolation. Error bars are present in these plots but they aresmaller than the symbol size. of Fig. 3 the polynomial fits the data well. The coefficient remains positive in the continuum limit,indicating a very small tendency for polarization. The probability that the sample drawn from thecorrelated distribution is more polarized than one drawn from the uncorrelated distribution is 51%compared to 50% when the dynamics would produce no correlation.To gauge the size of the finite volume effects, we compute the absolute polarization on twoensembles with the same lattice spacings but different volumes. Referring to Table 1, these areensembles E and E . In the left panel of Fig. 4 we compare the absolute polarizations on thesetwo ensembles. We find no difference between the two histograms and we conclude that the finitevolume effects are negligible.We also computed a set of eigenmodes of the overlap Dirac operators on ensembles E , E and E and used them to compute the smoothed field strength operator F Λ . To study the continuumlimit, a consistent definition of the smoothed operator sums over all modes smaller than a physical Figure 4:
Left: absolute X-distribution for ensemble E (circles) and E (crosses) which have the samelattice spacing but different volume. Right: correlation coefficient for the smoothed strength field (diamonds)compared to the full version (circles). Error bars are included in both plots. bsolute X-distribution and self-duality Andrei Alexandru −1.0 −0.5 0.0 0.5 1.0 R H(R)
100 % 50 % 10 %
Figure 5:
Left: X-distribution for self-duality components of a smooth field strength based on the low-lyingmodes of the chirally-improved Dirac operator [5]. The curved marked with 100% is the relevant one forour comparison. Right: absolute X-distribution P A and X-distribution P r based on two different polarizationvariables (see Eq. 2.2) for ensemble E . cutoff. We set the cutoff Λ = E which is similar to the ensemble used in Ref. [5]. The discrepancyis due to the fact that Ref. [5] uses a polarization measure dominated by kinematical effects. Toshow this, in the right panel of Fig. 5 we also plot the X-distribution measured using the referencepolarization, α =
1, and the polarization angle used in Ref. [5], α =
2. To better compare ourresults, for these plots we used, as in the referenced study, a smoothed F Λ constructed using thesame number of modes. We see then that when using the same angle definition, our results areconsistent with those of Ref. [5]. However, using another valid angle parametrization producesqualitatively different results due to kinematical effects. We conclude that the strong polarizationobserved in Ref. [5] is mainly due to the specific choice of angle variable rather than the underlyingdynamics.
5. Conclusions
In this work we studied the dynamical polarization propertied of self-duality components in-duced by pure-glue QCD dynamics. We found a very mild polarization tendency that survives in thecontinuum limit. This results has negligible finite-volume corrections. The self-duality tendencyis very small making it unlikely that the vacuum fluctuations are well-described by semi-classical7 bsolute X-distribution and self-duality
Andrei Alexandru models. Our findings are at variance with the results of a previous study [5]. We conclude that thediscrepancy is the result of kinematical effects.
Acknowledgments : Andrei Alexandru is supported in part under DOE grant DE-FG02-95ER-40907. The computational resources for this project were provided in part by the George Wash-ington University IMPACT initiative. Ivan Horváth acknowledges warm hospitality of the BNLTheory Group during which part of this work has been completed.
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