Mesons upon low-lying Dirac mode exclusion
MMesons upon low-lying Dirac mode exclusion
M. Denissenya ∗ Institut für Physik, FB Theoretische Physik, Universität GrazE-mail: [email protected]
L. Ya. Glozman
Institut für Physik, FB Theoretische Physik, Universität GrazE-mail: [email protected]
C. B. Lang
Institut für Physik, FB Theoretische Physik, Universität GrazE-mail: [email protected]
We study the isoscalar and isovector J = , J = SU ( ) L × SU ( ) R and U ( ) A sym-metries. The ground states of the π , σ , a , η mesons do not survive this truncation. All possible J = SU ( ) symmetry of a dynamical QCD-like string. The 32nd International Symposium on Lattice Field Theory,23-28 June, 2014Columbia University New York, NY ∗ 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 esons upon low-lying Dirac mode exclusion M. Denissenya
1. Introduction in our previous study [1] we discovered a degeneracy of all isovector mesons of spin J = ρ , ρ (cid:48) , a , b , upon truncation of the quasi-zero modes from the valence quark propagators withthe manifestly chirally-invariant overlap Dirac operator ( for a previous study with the chirallyimproved Dirac operator see Refs. [2, 3]). The density of the quasi-zero modes is directly related tothe quark condensate of the vacuum [4]. Via such a truncation we artificially restore ( "unbreak")the chiral symmetry. All J = ρ and a has been observed, but actually a degeneracy of all four states. This degeneracy signalsnot only a simultaneous restoration of both SU ( ) L × SU ( ) R and U ( ) A symmetries, but of somehigher symmetry that includes SU ( ) L × SU ( ) R × U ( ) A as a subgroup. This symmetry wouldrequire a degeneracy of all possible chiral multiplets with J = J = π , σ , a , η mesons upon unbreaking.
2. Lattice setup
Our ensemble includes 100 gauge configurations generated by JLQCD with n f = Q top = ×
32, the lattice spacing a ∼ .
12 and the pion mass in this ensemble is M π = ( ) MeV.
The quark propagators, obtained from JLQCD, consist of two parts. The contribution of thefirst 100 Dirac eigenmodes was computed exactly, and the effect of all higher-lying modes wastaken into account via a stochastic estimate [7]. Our unbreaking procedure means consequently aremoval of the first k modes from the quark propagators: S k ( x , y ) = ∑ n = k + λ n u n ( x ) u † n ( y ) + S Stoch ( x , y ) . (2.1)The resulting full ( k =
0) and reduced ( k >
0) quark propagators are then used in the constructionof the meson correlation functions.
We use the standard variational approach [8–10]. Our basis of operators is enlarged by intro-ducing an exponential type of smearing at source/sink. We construct the cross-correlation matrices C i j ( t ) = (cid:104) | O i ( t ) O † j ( ) | (cid:105) (2.2)with the size up to 10 ×
10 with a subsequent solution of the generalized eigenvalue problem C ( t ) (cid:126) υ n ( t ) = ˜ λ ( n ) ( t ) C ( t ) (cid:126) υ n ( t ) . (2.3)The masses of the eigenstates are obtained by identifying the exponential decay of the eigenvalues˜ λ ( n ) ( t ) . Such states are extracted for each low-lying mode truncation level k in a given quantumchannel. 2 esons upon low-lying Dirac mode exclusion M. Denissenya π ση a S U ( ) A U ( ) A S U ( ) A U ( ) A Figure 1:
Symmetry relations between J = -0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 4 8 12 16 20 24 28tk = 0 k = 2 k = 10 C π -4.0e-03-3.0e-03-2.0e-03-1.0e-030.0e+001.0e-032.0e-033.0e-034.0e-03 4 8 12 16 20 24 28tk = 0 k = 2 k = 10 D η Figure 2:
Connected (left) and disconnected (right) parts of the η correlator upon the low-mode exclusion.
3. Results J = mesons The J = C ) contributions. Both the con-nected and disconnected ( D ) contributions are important for the isoscalar correlators. In our casewe have two degenerate quark flavours, i.e., there is no distinction between the u and d quark prop-agators. Hence, the connected part of the σ correlator is identical to the a correlator, the sameargument applies to η and π : F η ( σ ) = C π ( a ) + D η ( σ ) , (3.1)where F represents the full correlator with given quantum numbers.In Fig. 2 we show the connected and disconnected contributions in the η channel. In theuntruncated case k = k ∼
10 lowest eigen-modes the disconnected contribution essentially vanishes.Figure 3 shows π , σ , a , η point-to-point correlators at different truncation levels k . The SU ( ) L × SU ( ) R and U ( ) A symmetries are strongly broken at k =
0, which makes all four point-to-point correlators very different.Upon elimination of a small amount of lowest-lying Dirac eigenmodes all correlators becomeidentical. We conclude that both SU ( ) L × SU ( ) R and U ( ) A symmetries get simultaneously3 esons upon low-lying Dirac mode exclusion M. Denissenya -4 -3 -2
4 8 12 16 20 24 28 t σ a η π k=0 -4 -3 -2
4 8 12 16 20 24 28 t σ a η π k=30 Figure 3:
Two-point correlators of π , η , σ , a mesons. π (k=0) π (k=10) π (k=60) Figure 4: π effective masses at k = , , restored. The same quasi-zero modes are responsible for both symmetry breakings which is con-sistent with the instanton induced mechanism. I , J PC O R π ( , − + ) ¯ q γ τ q ( / , / ) a η ( , − + ) ¯ q γ q ( / , / ) b a ( , ++ ) ¯ q τ q ( / , / ) b σ ( , ++ ) ¯ qq ( / , / ) a ρ ( , −− ) ¯ q γ i τ q ( , ) ⊕ ( , ) ¯ q γ i γ t τ q ( / , / ) b ω ( , −− ) ¯ q γ i q ( , ) ¯ q γ i γ t q ( / , / ) a a ( , ++ ) ¯ q γ i γ τ q ( , ) ⊕ ( , ) f ( , ++ ) ¯ q γ i γ q ( , ) b ( , + − ) ¯ q γ i γ j τ q ( / , / ) a h ( , + − ) ¯ q γ i γ j q ( / , / ) b Table 1: J = J = O : R denotes an index of the chiralmultiplet within each J [11]. The next question is whether the J = π statescan be easily identified in the untruncated case. Theeffective mass for π is shown on Fig. 4. For theuntruncated case k = SU ( ) L × SU ( ) R and U ( ) A symmetries get simul-taneously restored upon unbreaking. Hence theground states of σ , a , η mesons should disappear4 esons upon low-lying Dirac mode exclusion M. Denissenya ρ (k=0) ρ (k=10) a (k=0) a (k=10) h (k=0) h (k=10) Figure 5: ρ , a and h effective masses at k=0,10. from the spectrum simultaneously with the pion. The opposite would contradict the restoration ofboth symmetries. J = mesons The interpolating fields and the respective chiral representations are given in Table 1. In con-trast to the J = J = J = J = J = ∼
10 modes. This indicates some larger symmetry that includes SU ( ) L × SU ( ) R × U ( ) A as a subgroup. This symmetry can be reconstructed and turns out to be a SU ( ) [13], mixingcomponents of the fundamental four-component vector ( u L , u R , d L , d R ) . This symmetry is not asymmetry of the QCD Lagrangian but should be considered as an emergent symmetry that appearsfrom the QCD dynamics upon elimination of the quasi-zero Dirac eigenmodes. It operates only for J ≥ esons upon low-lying Dirac mode exclusion M. Denissenya σ , MeVk ρω h b a ρ′ f ω′ b ′ ρ′′ ω′′ a ′ ω′′′ρ′′′ Figure 6:
Mass evolution of J = σ denotes theenergy gap. magnetic field in the system suggesting that we observe quantum levels of a dynamical QCD-likestring.
4. Acknowledgements
We thank S. Aoki, S. Hashimoto and T. Kaneko for supplying us with the JLQCD gauge con-figurations and quark propagators and their help and hospitality. The calculations were performedon computing clusters of the University of Graz (NAWI Graz). Support from the Austrian ScienceFund (FWF) through the grants DK W1203-N16 and P26627-N16 is acknowledged.
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