AA hidden classical symmetry of QCD
L. Ya.
Glozman ,(cid:63) Institute of Physics, University of Graz, A-8010 Graz, Austria
Abstract.
The classical part of the QCD partition function (the integrand) has, ignor-ing irrelevant exact zero modes of the Dirac operator, a local
S U (2 N F ) ⊃ S U ( N F ) L × S U ( N F ) R × U (1) A symmetry which is absent at the Lagrangian level. This symmetry isbroken anomalously and spontaneously. E ff ects of spontaneous breaking of chiral sym-metry are contained in the near-zero modes of the Dirac operator. If physics of anomalyis also encoded in the same near-zero modes, then their truncation on the lattice shouldrecover a hidden classical S U (2 N F ) symmetry in correlators and spectra. This naturallyexplains observation on the lattice of a large degeneracy of hadrons, that is higher than the S U ( N F ) L × S U ( N F ) R × U (1) A chiral symmetry, upon elimination by hands of the lowest-lying modes of the Dirac operator. We also discuss an implication of this symmetry forthe high temperature QCD. The QCD Lagrangian in Minkowski space-time has in the chiral limit the chiral symmetry: U ( N F ) L × U ( N F ) R = S U ( N F ) L × S U ( N F ) R × U (1) A × U (1) V . (1)In the following we will always drop the U (1) V symmetry that is irrelevant to our subject. The U (1) A symmetry is invariance of the Lagrangian upon the axial flavor-neutral transformation Ψ ( x ) → e i αγ Ψ ( x ); ¯ Ψ ( x ) → ¯ Ψ ( x ) e i αγ . (2)The S U ( N F ) L × S U ( N F ) R symmetry implies invariance under a pure flavor rotation S U ( N F ) ( for N F = Ψ ( x ) → e i γ (cid:126)λ · (cid:126)α Ψ ( x ); ¯ Ψ ( x ) → ¯ Ψ ( x ) e i γ (cid:126)λ · (cid:126)α , (3)where (cid:126)λ are S U ( N F ) generators.The axial U (1) A symmetry is broken anomalously, which is due to a noninvariance of the in-tegration measure in the functional integral under a local U (1) A transformation [1]. The S U ( N F ) A "symmetry" (3) is broken spontaneously, because the ground state of the theory, the vacuum, is notinvariant under the transformation (3). The latter noninvariance is encoded in the nonzero quark con-densate of the vacuum, < | ¯ Ψ ( x ) Ψ ( x ) | > (cid:44)
0. The quark condensate of the vacuum can be expressed (cid:63) e-mail: [email protected] These transformations do not form a closed subgroup of the chiral group. a r X i v : . [ h e p - l a t ] A ug hrough a density of the near-zero modes λ → m → < | ¯ Ψ ( x ) Ψ ( x ) | > = − πρ (0) . (4)The idea of the N F = J = , S U (2) A transformation wouldbecome identical. If hadrons survive this truncation, then masses of chiral partners should be equal.It has turned out that a very clean exponential decay of correlators was detected in all J = J = J = S U (2) L × S U (2) R sym-metry but also the U (1) A symmetry. One then concludes that the same lowest-lying modes are respon-sible for both S U (2) L × S U (2) R and U (1) A breakings which is consistent with the instanton-inducedmechanism of both breakings [7–10].However, a larger degeneracy that includes all possible chiral multiplets of the J = S U (2) L × S U (2) R × U (1) A . This not yet known symmetry has been reconstructed inrefs. [11, 12] and turned out to be S U (2 N F ) ⊃ S U ( N F ) L × S U ( N F ) R × U (1) A . (5)Transformations of this group include both the flavor rotations of the left- and right-handed quarksas well as transformations that mix the left- and right-handed components. This symmetry has beenconfirmed in lattice simulations with the J = S U (2 N F ) local symmetry. This symmetry is not a symmetry of theEuclidean QCD Lagrangian because the irrelevant exact zero modes of the Euclidean Dirac operatorbreak it. Consequently, we refer it as a hidden classical symmetry. This hidden classical symmetry isbroken by the anomaly to S U ( N F ) L × S U ( N F ) R . Spontaneous breaking of chiral symmetry reducesit to S U ( N F ) V . E ff ects of spontaneous and anomalous breakings is encoded in the near-zero modesof the Dirac operator. Consequently, truncation on the lattice of the lowest-lying modes restores thehidden S U (2 N F ) symmetry, which naturally explains degeneracies observed in lattice experiment.Now we will present some details. A truncation of the low-lying modes of the Dirac operator means the following. The Euclidean La-grangian with N F degenerate quarks in a given gauge background is: L = Ψ † ( x )( γ µ D µ + m ) Ψ ( x ) , (6)where Chiral-Parity group for spin-1 mesons (0 , (1 / , / a (1 / , / b (1 , (0 , f (0 , ++ ) ! (0 , ) b (1 , + ) ! (0 , ) ⇢ (1 , ) ⇢ (1 , ) h (0 , + ) a (1 , ++ ) ( ⌧ a ⌦ k ) ( ⌧ a ⌦ k ) ( ⌧ a ⌦ k ) ( ⌧ a ⌦ k ) ( F ⌦ k ) ( F ⌦ k ) ( F ⌦ k ) ( F ⌦ k ) SU (2) A SU (2) A SU (2) A U (1) A U (1) A Figure 1.
S U (2) L × S U (2) R and U (1) A classification of the J = D µ = ∂ µ + i g t a A a µ . (7)The hermitian Dirac operator, i γ µ D µ , has in a volume V a discrete spectrum with real eigenvalues λ n : i γ µ D µ Ψ n ( x ) = λ n Ψ n ( x ) . (8)We subtract from the full quark propagator S Full the lowest k eigenmodes of the Dirac operator S ( x , y ) = S Full ( x , y ) − k (cid:88) n = λ n + im Ψ n ( x ) Ψ † n ( y ) . (9)At the same time we keep the gauge configurations intact. Given these truncated quark propagatorswe apply standard procedures to extract hadron spectra using the variational approach. Then we studydependence of hadron masses on the truncation number k . We perform N F = S U (2) L × S U (2) R symmetry in correlators. If in addition the hadronstates survive this surgery, then we can expect a mass degeneracy of chiral partners. For the J = U (1) A transformation connects operators that are linked by the blue arrows on Fig. 1. Con-sequently, if the whole chiral symmetry of QCD S U (2) L × S U (2) R × U (1) A is restored, then we canexpect a degeneracy of four mesons from the (1 / , / a and (1 / , / b representations on the onehand, and on the other hand a degeneracy of the ρ and a mesons from the (1 , + (0 ,
1) chiral repre-sentation. Note that in the chirally symmetric world there two independent orthogonal ρ -mesons thatbelong to two di ff erent chiral representations. In the real world with chiral symmetry breaking thesetwo chiral representations are mixed in the meson wave function and two di ff erent ρ operators coupleto one and the same ρ -meson.The S U (2) L × S U (2) R × U (1) A symmetry does not connect, however, the four mesons from the(1 / , / a and (1 / , / b representations with the other mesons in Fig. 1. Consequently, given only -4 -3 -2 -1
0 2 4 6 8 10 12 14 16t ρ (k=10) -5 -4 -3 -2 -1
0 2 4 6 8 10 12 14 16t a (k=10) -5 -4 -3 -2 -1
0 2 4 6 8 10 12 14 16t b (k=10) Figure 2.
The eigenvalues of the cross-correlation matrix and e ff ective mass plateaus for the isovector J = k = the S U (2) L × S U (2) R × U (1) A chiral symmetry we cannot expect a degeneracy larger than it is shownby arrows on Fig. 1. We do not show here our results for the real world ( k = ff ective mass plateau at k =
10 (10 lowest Dirac eigenmodes have beenremoved) for the isovector mesons are shown in Fig. 2.A very clean exponential decay of the correlators is obvious, which means that there are physicalstates. It is much cleaner than in the untruncated (real) world. The reason for this is intuitively clear:After truncation there are no pion fluctuations in the system. We can conclude that mesons (which arebound states now) survive the truncation.Note a double degeneracy of the rho-meson eigenvalues, which is absent in the untruncated world.This double degeneracy can be obtained only if both rho-operators from Fig. 1 are used in the cross-correlation matrix. This double degeneracy tells that there are two independent orthogonal degenerate ρ -mesons. If we put two lowest rho-eigenvalues, the lowest a and the lowest b eigenvalues on thesame plot, then we will see that they all are identical (the same is true with the higher eigenvalues,but the result is less precise). This means that there is a symmetry in the system that is higher than S U (2) L × S U (2) R × U (1) A .Evolution of meson masses is shown in Fig. 3. We clearly see a larger degeneracy than the chiral S U (2) L × S U (2) R × U (1) A symmetry of the QCD Lagrangian. What does it mean?! The same resultspersist for the J = σ , MeVk ρ ω h b a ρ′ f ω′ b ′ρ′′ ω′′ a ′ Figure 3. J = k . σ shows energy gap in the Diracspectrum. S U (4) symmetry of the meson spectra
Given this degeneracy first we need to understand what symmetry group does it correspond. Thisunexpected new symmetry has been reconstructed in ref. [11].Given the standard spin, parity, etc. quantum numbers we can construct explicitly basis vectorsfor all irreducible representations of the chiral group shown in Fig. 1. (i) (0,0): | (0 , ± ; J (cid:105) = √ | ¯ RR ± ¯ LL (cid:105) J . (10) (ii) (1 / , / a and (1 / , / b : | (1 / , / a ; + ; I = J (cid:105) = √ | ¯ RL + ¯ LR (cid:105) J , (11) | (1 / , / a ; − ; I = J (cid:105) = √ | ¯ R (cid:126)τ L − ¯ L (cid:126)τ R (cid:105) J , (12) | (1 / , / b ; − ; I = J (cid:105) = √ | ¯ RL − ¯ LR (cid:105) J , (13) | (1 / , / b ; + ; I = J (cid:105) = √ | ¯ R (cid:126)τ L + ¯ L (cid:126)τ R (cid:105) J . (14) (iii) (0,1) ⊕ (1,0): | (0 , + (1 , ± ; J (cid:105) = √ | ¯ R (cid:126)τ R ± ¯ L (cid:126)τ L (cid:105) J , (15)Now we need to to find a minimal group that contains S U (2) L × S U (2) R × U (1) A as a subgroup and that com-bines all these vectors into one irreducible representation. These new symmetry transformations must connect allthese basis vectors. The latter requirement can be achieved if these new symmetry transformations mix the left-and right-handed quarks. Consequently, the required symmetry group must contain as a subgroup the S U (2) CS chiralspin rotations that act on the following doublets: = (cid:32) u L u R (cid:33) , D = (cid:32) d L d R (cid:33) . (16)A three-dimensional imaginary space where these rotations are performed is called the chiralspin space. Thechiralspin rotations mix the right- and left-handed components of the fermion fields. It is similar to the wellfamiliar isospin space: Rotations in the isospin space mix particles with di ff erent electric charges.If we combine the S U (2) CS and the isospin S U (2) group into one larger group one arrives at the
S U (4) groupwith the fundamental vector
Ψ = u L u R d L d R . (17)The dim =
15 irreducible representation of this group connects all (0 , , (1 / , / a , (1 / , / b , (1 , + (0 , ,
0) basis vectors, namely | (0 , − ; J = (cid:105) > is a singlet of S U (4).We can construct an explicit realization of the
S U (2) CS and S U (4) algebra that acts on Dirac spinors [12].Then the
S U (2) CS chiralspin rotations are generated through Σ = { γ , i γ γ , − γ } , [ Σ i , Σ j ] = i (cid:15) ijk Σ k . The Dirac spinor transforms under a global or local
S U (2) CS transformation as Ψ → Ψ (cid:48) = e i ε · Σ / Ψ . (18)The S U (4) group contains at the same time
S U (2) L × S U (2) R and S U (2) CS ⊃ U (1) A and has the following setof generators: { ( τ a ⊗ D ) , (1 F ⊗ Σ i ) , ( τ a ⊗ Σ i ) } . The global and local
S U (4) transformations of the Dirac spinor are defined through Ψ → Ψ (cid:48) = e i (cid:15) · T / Ψ . (19)The S U (2) CS and S U (4) transformations of all operators from Fig. 1 are shown in Fig. 4.
The
S U (4) symmetry is obtained in lattice simulations upon subtraction of the near-zero modes of the Diracoperator. It implies that this symmetry should be encoded in the Euclidean QCD. Obviously the Lagrangian (6)does not have this symmetry. This is because the Dirac operator does not commute with the
S U (2) CS transfor-mations. Then we should recover at which level this symmetry is hidden in Euclidean QCD. More explicitly,we have to find a part of the Euclidean QCD formalism that breaks this symmetry.Consider the zero modes of the Dirac equation, γ µ D µ Ψ ( x ) = . (20)Given standard antiperiodic boundary conditions for the quark field along the time direction, the zero modesare solutions of the Dirac equation with the gauge configurations of a nonzero global topological charge. Thedi ff erence of numbers of the left-handed and right-handed zero modes is according to the Atiyah-Singer theoremfixed by the global topological charge Q of the gauge configuration: n L − n R = Q . (21)Some S U (2) CS transformations rotate the right-handed spinor into the left-handed one and vice versa. Conse-quently, the zero modes explicitly violate the S U (2) CS and S U (2 N F ) symmetries: The zero modes introduce an In Euclidean space we have to substitute the γ matrix through the γ matrix. (0 , (1 / , / a (1 / , / b (1 , (0 , f (0 , ++ ) ( F ⌦ k ) b (1 , + ) ( ⌧ a ⌦ k ) ⇢ (1 , ) ( ⌧ a ⌦ k ) ⇢ (1 , ) ( ⌧ a ⌦ k ) ! (0 , ) ( F ⌦ k ) ! (0 , ) ( F ⌦ k ) h (0 , + ) ( F ⌦ k ) a (1 , ++ ) ( ⌧ a ⌦ k ) SU (2) CS SU (2) CS SU (4) Figure 4.
The green arrows connect operators that belong to the
S U (2) CS triplets. The f and a operators arethe S U (2) CS singlets. The purple arrows show the S U (4) 15-plet. The f operator is a singlet of S U (4).asymmetry between the left- and right-handed degrees of freedom and break the
S U (2) CS invariance. The latteris possible only if there is no asymmetry between the left and the right.It is well understood, however, that the exact zero modes are completely irrelevant since their contributionsto the Green functions and observables vanish in the thermodynamical limit V → Ψ ( x ) and Ψ † ( x ) in the Lagrangian over a complete and orthonormal set Ψ n ( x ) of theeigenvalue problem (8): Ψ ( x ) = (cid:88) n c n Ψ n ( x ) , Ψ † ( x ) = (cid:88) k ¯ c k Ψ † k ( x ) , (22)where ¯ c k , c n are Grassmannian numbers. Then the fermionic part of the QCD partition function takes the follow-ing form Z = (cid:90) (cid:89) k , n d ¯ c k dc n e (cid:80) k , n (cid:82) d x ¯ c k c n ( λ n + im ) Ψ † k ( x ) Ψ n ( x ) . (23)Now we can directly read-o ff symmetry properties of the classical part of the partition function, i.e. of theintegrand. This functional contains only a superposition of terms Ψ † k ( x ) Ψ n ( x ). It is precisely S U (2) CS and S U (2 N F ) symmetric, because ( U Ψ k ( x )) † U Ψ n ( x ) = Ψ † k ( x ) Ψ n ( x ) , (24)where U is any local or global transformation from the groups S U (2) CS and S U (2 N F ) , U † = U − . The exactzero modes, for which the equation (24) does not hold, have been subtracted from the partition function. Weconclude that classically the Euclidean QCD without the irrelevant exact zero mode contributions is invariantwith respect to both global and local S U (2) CS and S U (2 N F ) transformations.The term "hidden classical S U (2 N F ) symmetry" should be correctly understood. It is not a global symmetryof the Lagrangian and consequently there are no respective conserved Noether currents. However, it is a reallocal symmetry of the classical part of the QCD partition function ignoring irrelevant exact zero modes.How is this hidden classical symmetry broken? The integration measure in (23) is not invariant under alocal U (1) A transformation [1]. Consequently, the U (1) A anomaly breaks the classical U (1) A symmetry. Sincehe U (1) A is a subgroup of S U (2) CS , the anomaly breaks either the S U (2) CS symmetry. Hence the classical S U (2 N F ) ⊃ S U ( N F ) L × S U ( N F ) R × U (1) A symmetry is broken by anomaly to S U ( N F ) L × S U ( N F ) R .The quark condensate in Minkowski space breaks all U (1) A , S U (2) CS , S U ( N F ) L × S U ( N F ) R and S U (2 N F )symmetries to the vector flavor symmetry S U ( N F ) V . Hence, the new hidden classical S U (2) CS and S U (2 N F )symmetries are broken both by the condensate and anomalously.Spontaneous chiral symmetry breaking is encoded in the near-zero modes of the Dirac operator, as it followsfrom the Banks-Casher relation. If anomaly is also encoded in the near-zero modes, as suggested e.g. by theinstanton mechanism of both breakings, then removal on lattice of the near-zero modes should restore not onlychiral S U ( N F ) L × S U ( N F ) R and U (1) A symmetries, but also a larger S U (2 N F ) symmetry, which naturally explainslattice observations reviewed in previous sections. From the results presented in Fig. 3 it is clearly seen that the degeneracy pattern is larger than
S U (4), becausethe
S U (4) singlet ( f ) and the S U (4) 15-plet mesons ( ρ, ρ (cid:48) , a , b , h , ω, ω (cid:48) ) are also degenerate. This impliesthat actually some higher symmetry is observed that includes the S U (4) as a subgroup [12]. It was found that nohigher symmetry exists, that would connect local quark bilinears from the 15-plet and singlet of
S U (4) withinthe same irreducible representation [17].This challenging problem has been solved in ref. [13]. Hadron spectra are extracted from the correlationfunctions calculated with the gauge-invariant source operators. At each time slice "t" a meson correlator containsminimum the lowest Fock ¯ qq component with a quark and an antiquark located at di ff erent space points x and y . Both q and ¯ q interact with the same gauge configuration. Then all arguments of the previous section applyindependently for q and ¯ q . Since the S U (2 N F ) invariance is local, we can perform S U (2 N F ) rotations at points x and y with di ff erent rotation parameters. It is then clear that the meson correlation function with the ¯ qq valencecontent has a bilocal S U (2 N F ) × S U (2 N F ) symmetry. A symmetry of higher Fock components is obviouslylarger, but the whole correlator has a symmetry of the lowest ¯ qq component. Obviously, averaging over gaugeconfigurations does not change this symmetry property.The same argument applies to baryons and in this case we expect a trilocal S U (2 N F ) × S U (2 N F ) × S U (2 N F )symmetry.One of the irreducible representations of the S U (4) × S U (4) is 16-dimensional and is a direct sum of the15-plet and singlet of
S U (4). Hence a direct prediction of this bilocal symmetry is a degeneracy of the
S U (4)-singlet and of the
S U (4) 15-plet, in agreement with the lattice observations. This symmetry is bilocal and cannotbe represented by the local composite operators which is consistent with conclusions of Ref. [17].
Our main findings can be summarized as follows.1. The classical part of the partition function (the integrand), excluding irrelevant exact zero mode contri-butions, has
S U (2) CS and S U (2 N F ) local symmetries. Since these symmetries are not symmetries of the QCDLagrangian we refer them as hidden classical symmetries of QCD. There are no respective conserved Noethercurrents. These symmetries are broken at the quantum level by the axial anomaly and by the quark condensate.The physics of chiral symmetry spontaneous breaking and of anomaly is contained in the near-zero modes of theDirac operator. Their truncation on the lattice should restore not only the S U ( N f ) L × S U ( N F ) R × U (1) A chiralsymmetry but actually higher hidden classical symmetries S U (2) CS and S U (2 N F ).2. We have shown that elimination of the near-zero modes leads to S U (2 N F ) × S U (2 N F ) and S U (2 N F ) × S U (2 N F ) × S U (2 N F ) symmetries in mesons and baryons.3. The bilocal S U (4) × S U (4) symmetry explains a degeneracy of the
S U (4) singlet f correlator with the S U (4) 15-plet ρ, ρ (cid:48) , ω, ω (cid:48) , h , a , b correlators. Implications
It is natural to expect many di ff erent implications of the hidden classical symmetry. Here we will mention a mostdramatic one [18].At high temperature the quark condensate of the vacuum vanishes. There are lattice indications that abovethe critical temperature the U (1) A symmetry is restored and a gap opens in the Dirac spectrum [19, 20]. Then itfollows that the
S U (2) CS and S U (2 N F ) symmetries are manifest in Euclidean correlation functions and observ-ables. Such symmetries cannot be obtained in terms of deconfined quarks and gluons in Minkowski space, wherewe live. Hence at high temperatures QCD is also in the confining regime and elementary objects are color singlet S U (4) symmetric "hadrons". "Hadrons" with such a symmetry can be directly constructed in Minkowski space[23].We acknowledge a partial support from the Austrian Science Fund (FWF) through the grant P26627-N27.
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