Chiral-spin symmetry of the meson spectral function above T c
OOU-HET-1025
Chiral-spin symmetry of the meson spectral function above T c C. Rohrhofer,
1, 2
Y. Aoki, L.Ya. Glozman, and S. Hashimoto
4, 51
Department of Physics, Osaka University, Toyonaka 560-0043, Japan Institute of Physics, University of Graz, 8010 Graz, Austria RIKEN Center for Computational Science, Kobe 650-0047, Japan KEK Theory Center, High Energy Accelerator ResearchOrganization (KEK), Tsukuba 305-0801, Japan School of High Energy Accelerator Science,The Graduate University for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan (Dated: January 30, 2020)
Abstract
Recently, via calculation of spatial correlators of J = 0 , N F = 2 QCD, it has been found that QCD at temperatures T c − T c is approximately SU (2) CS and SU (4) symmetric. The latter symmetry suggests that thephysical degrees of freedom are chirally symmetric quarks bound by the chromoelectric field intocolor singlet objects without chromomagnetic effects. This regime of QCD has been referred to asa Stringy Fluid. Here we calculate correlators for propagation in time direction at a temperatureslightly above T c and find the same approximate symmetries. This means that the meson spectralfunction is chiral-spin and SU (4) symmetric in the same temperature range. a r X i v : . [ h e p - l a t ] J a n . INTRODUCTION Artificial truncation of the near-zero modes of the Dirac operator at zero temperatureresults in the emergence of a large degeneracy in the hadron spectrum, larger than impliedby the chiral symmetry of the QCD Lagrangian [1–4]. A symmetry group of this degeneracy,the chiral-spin SU (2) CS group and its flavor extension SU (2 N F ), contains chiral symmetriesas subgroups [5, 6]. These symmetries are not symmetries of the Dirac Lagrangian. Howeverthey are symmetries of the electric interaction in a given reference frame, while the magneticinteraction as well as the quark kinetic term break them. Consequently these symmetriesallow us to separate the electric and magnetic interactions in a given frame. The emergenceof the SU (2) CS and SU (2 N F ) symmetries in the hadron spectrum upon the low modetruncation means that while the confining chromoelectric interaction is distributed amongall modes of the Dirac operator, the chromomagnetic interaction contributes only (or atleast predominantly) to the near-zero modes. Some unknown microscopic dynamics shouldbe responsible for this phenomenon.At high temperatures, above the pseudocritical temperature T c , chiral symmetry is re-stored due to the near-zero modes of the Dirac operator being naturally suppressed bytemperature effects [7–10]. Then one could expect a natural emergence of the SU (2) CS and SU (2 N F ) symmetries in QCD above T c [11].In [12, 13] we have studied a complete set of J = 0 and J = 1 isovector correlationfunctions in z -direction for a system with N F = 2 dynamical quarks in simulations with thechirally symmetric domain wall Dirac operator at temperatures up to 5 . T c . Similar ensem-bles have been used previously for the study of the U (1) A restoration in t -correlators andvia the Dirac eigenvalue decomposition of correlators [9, 14]. We have observed emergenceof approximate SU (2) CS and SU (4) symmetries in the spatial correlators in the tempera-ture range T c − T c . These symmetries of spatial correlators reflect symmetries of the QCDaction since correlation functions are driven only by the action of the theory. Observation ofapproximate SU (2) CS and SU (4) symmetries at T c − T c suggests that the physical degreesof freedom in this temperature range are chirally symmetric quarks bound by the chromo-electric field into color-singlet compounds without chromomagnetic effects. Such a systemis reminiscent of a “string”, that is why the corresponding regime of QCD at T c − T c isreferred to as a Stringy Fluid. The chemical potential term in the QCD action has precisely2he same symmetries [15], so one can expect that the symmetries observed in the latticecalculations at zero chemical potential will persist at µ > T > SU (2) CS and SU (4) symmetries in t -correlators wouldimply that the spectra of the corresponding color-singlet states in Minkowski space havethe same symmetry. The symmetries of the z -correlators do suggest the same symmetriesin the spectra, albeit indirectly. A direct observation of these symmetries in t -correlatorsin practice is a priori not obvious since on the lattice one has only a few lattice sites alongthe time direction at high T and large discretization errors as well as a small evolution timecan easily spoil the real picture. Here we use N t = 12 ensembles at T = 1 . T c and observeclear SU (2) CS and SU (4) symmetries in t -correlators. This implies that the correspondingspectral functions in Minkowski space are also SU (2) CS and SU (4) symmetric. II. CHIRAL-SPIN SYMMETRY
The SU (2) CS chiral-spin transformations for quarks are defined by [5, 6] ψ ( x ) → exp (cid:18) i Σ (cid:15) (cid:19) ψ ( x ) , ¯ ψ ( x ) → ¯ ψ ( x ) γ exp (cid:18) − i Σ (cid:15) (cid:19) γ , (1)where (cid:15) ∈ R are the rotation parameters. For the generators Σ , Σ = { γ k , − i γ γ k , γ } , (2)one has four different choices Σ = Σ k with k = 1 , , ,
4. Here γ k , k = 1 , , ,
4, are hermitianEuclidean gamma-matrices, obeying the anticommutation relations γ i γ j + γ j γ i = 2 δ ij ; γ = γ γ γ γ . (3)The su (2) algebra [Σ a , Σ b ] = 2i (cid:15) abc Σ c (4)is satisfied for any k .The choice of k is fixed by the requirement that the SU (2) CS transformation does notmix operators with different spin, i.e., respects the rotational O (3) symmetry in Minkowskispace. For propagators in time direction, defined below, this implies k = 4.3 (1) A is a subgroup of SU (2) CS . The SU (2) CS transformations mix the left- and right-handed fermions and different representations of the Lorentz group.The direct product SU (2) CS × SU ( N F ) can be embedded into a SU (2 N F ) group. Thechiral symmetry group of QCD, SU ( N F ) L × SU ( N F ) R × U (1) A , is a subgroup of SU (2 N F ).The SU (2) CS and SU (2 N F ) groups are not symmetries of the Dirac equation as well ofthe QCD Lagrangian as a whole. In a given reference frame the quark-gluon interactionLagrangian in Minkowski space can be split into temporal and spatial parts: ψγ µ D µ ψ = ψγ D ψ + ψγ i D i ψ. (5)Here D µ is a covariant derivative that includes interaction of the quark field ψ with thegluon field A µ , D µ ψ = ( ∂ µ − ig t · A µ ψ. (6)The temporal term includes an interaction of the color-octet charge density¯ ψ ( x ) γ t ψ ( x ) = ψ ( x ) † t ψ ( x ) (7)with the electric part of the gluonic gauge field. It is invariant under any unitary transfor-mation acting in the Dirac and/or flavor spaces. In particular it is a singlet under SU (2) CS and SU (2 N F ) groups. The spatial part consists of a quark kinetic term and interactionwith the magnetic part of the gauge field. It breaks SU (2) CS and SU (2 N F ). We concludethat interaction of electric and magnetic components of the gauge field with fermions canbe distinguished by symmetry.In order to discuss the notions “electric” and “magnetic” one needs to fix a referenceframe. The invariant mass of the hadron is the rest frame energy. Consequently, to discussphysics of hadron mass generation it is natural to use the hadron rest frame.The spectral density ρ ( ω ) is an integral transform C Γ ( t ) = (cid:90) dω cosh( ω ( t − T ))sinh( ω T ) ρ Γ ( ω ) (8)of the rest frame t -direction Euclidean correlator C Γ ( t ) = (cid:88) x,y,z (cid:104)O Γ ( x, y, z, t ) O Γ ( , † (cid:105) , (9)where O Γ ( x, y, z, t ) is an operator that creates a quark-antiquark pair for mesons with fixedquantum numbers. 4 (0 , ++ )Ψ( F ⊗ γ γ i )Ψ ω (0 , −− )Ψ( F ⊗ γ i )Ψ b (1 , + − )Ψ( τ a ⊗ γ γ γ i )Ψ ω (0 , −− )Ψ( F ⊗ γ γ i )Ψ ρ (1 , −− )Ψ( τ a ⊗ γ γ i )Ψ h (0 , + − )Ψ( F ⊗ γ γ γ i )Ψ ρ (1 , −− )Ψ( τ a ⊗ γ i )Ψ a (1 , ++ )Ψ( τ a ⊗ γ γ i )Ψ(0 , / , / a (1 / , / b (1 , ⊕ (0 , U (1) A U (1) A SU (2) A SU (2) A SU (2) A f (0 , ++ )Ψ( F ⊗ γ γ i )Ψ ω (0 , −− )Ψ( F ⊗ γ i )Ψ b (1 , + − )Ψ( τ a ⊗ γ γ γ i )Ψ ω (0 , −− )Ψ( F ⊗ γ γ i )Ψ ρ (1 , −− )Ψ( τ a ⊗ γ γ i )Ψ h (0 , + − )Ψ( F ⊗ γ γ γ i )Ψ ρ (1 , −− )Ψ( τ a ⊗ γ i )Ψ a (1 , ++ )Ψ( τ a ⊗ γ γ i )Ψ(0 , / , / a (1 / , / b (1 , ⊕ (0 , SU (2) CS SU (2) CS SU (4) FIG. 1. Transformations between interpolating vector operators, i = 1 , ,
3. The left columnsindicate the chiral representation for each operator. Red and blue arrows connect operators thattransform into each other under SU (2) L × SU (2) R and U (1) A , respectively. Green arrows connectoperators that form triplets of SU (2) CS , k = 4. The f and a operators are the SU (2) CS , k = 4 –singlets. Purple arrows show the 15-plet of SU (4). The f operator is a SU (4)-singlet. Transformation properties of the local J = 1 quark-antiquark bilinears O Γ ( x, y, z, t ) withrespect to SU (2) L × SU (2) R and U (1) A are given on the left side of Fig. 1 and those withrespect to SU (2) CS , k = 4 and SU (4) on the right side of Fig. 1 [6]. Emergence of therespective symmetries is signalled by the degeneracy of the correlators (9) calculated withoperators that are connected by the corresponding transformations. III. METHODOLOGY
The lattice data presented in the next section is calculated on JLQCD gauge configura-tions with N F = 2 fully dynamical domain wall fermions ([9, 16]). The length of the fifthdimension for the fermions is chosen as L = 16, to ensure good chiral symmetry [14].The quark propagators are computed on point sources after three steps of stout smearing.The fermion fields obey anti-periodic boundary conditions in time direction. For the gaugepart we use the Symanzik-improved gauge action with an inverse gauge coupling β g =4 . a = 0 .
075 fm). The time extent of the lattices is N t = 12, which corresponds to atemperature of T (cid:39)
220 MeV ( ∼ . T c ). We calculate the data on three spatial volumes, N s = 24 , ,
48, with a quark mass of m ud = 0 . O (50)independent configurations. 5he main observables are correlation functions of local isovector bilinears O Γ = ¯ ψ ( τ / ⊗ Γ) ψ, where the Γ structures from Fig. 1 determine the resulting quantum numbers. To extractcorrelation functions of states with definite, i.e. zero, momentum, we perform a momentumprojection according to Eq.( 9).Finally, the data shown in the next section is rescaled to a dimensionless variable t T = ( n t a ) / ( N t a ) = n t /N t , (10)where t is the measured lattice distance in time direction, T the temperature, a the latticeconstant, and N t the overall temporal lattice extent. For spatial correlators in z -directionthe same rescaling is done with z = n z a instead of t . IV. RESULTS
On the right side of Fig. 2 we show t -correlators (9) normalized at n t = 1 calculated on48 ×
12 lattices at T = 1 . T c . The results obtained on N s = 32 ,
24 lattices are similar andagree within statistical errors, they are omitted for clarity.Specifically we calculate the correlators of J = 0 isovector scalar ¯ ψ τ / ψ ( S ) and pseu-doscalar ¯ ψγ τ / ψ ( P S ) operators, where τ are isospin Pauli matrices as well as correlatorsof isovector operators { b , (1 / , / a } , { ρ, (1 / , / b } , { ρ, (1 , ⊕ (0 , } and { a , (1 , ⊕ (0 , } from Fig. 1. A degeneracy of scalar and pseudoscalar correlators reflects restora-tion of U (1) A symmetry, since the corresponding operators are connected by U (1) A trans-formations, observed already previously in Refs. [9, 14]. Since the { b , (1 / , / a } and { ρ, (1 / , / b } operators are also connected by U (1) A transformations, the degeneracy of thecorresponding correlators also signals U (1) A symmetry. A degeneracy of { ρ, (1 , ⊕ (0 , } and { a , (1 , ⊕ (0 , } correlators evidences the restoration of chiral SU (2) L × SU (2) R symmetry.An approximate degeneracy of { b , (1 / , / a } , { ρ, (1 / , / b } and { ρ, (1 , ⊕ (0 , } correlators signals emergence of SU (2) CS symmetry, since all three operators belong tothe same irreducible representation (triplet) of SU (2) CS . Finally a degeneracy of all fourcorrelators { b , (1 / , / a } , { ρ, (1 / , / b } , { ρ, (1 , ⊕ (0 , } and { a , (1 , ⊕ (0 , } is dueto the emergence of SU (4) symmetry. 6 -2 -1 , ρ (1/2,1/2) b a , ρ (1,0)+(0,1) non interacting C ( t ) / C ( t = ) tT -2 -1 PS, Sb , ρ (1/2,1/2) b a , ρ (1,0)+(0,1) full QCD C ( t ) / C ( t = ) tT PSS ρ (0,1)+(1,0) a ρ (1/2,1/2) b b FIG. 2. Temporal correlation functions for 48 ×
12 lattices. The l.h.s. shows correlators calculatedwith free noninteracting quarks on the same lattice, and features a symmetry pattern expected fromchiral symmetry. The r.h.s. presents full QCD data at a temperature of T = 220MeV (1 . T c ),which shows multiplets of all U (1) A , SU (2) L × SU (2) R , SU (2) CS and SU (4) groups. On the left side of Fig. 2 we show the correlators calculated with free, noninteractingquarks on the same lattice with the same Dirac action (the gauge operator U is set to 1).Dynamics of free quarks are governed by the Dirac equation and only chiral symmetriesexist. Indeed, a multiplet structure in this case is very different as compared to the rightside of Fig. 2 and only degeneracies due to U (1) A and SU (2) L × SU (2) R symmetries areseen in meson correlators calculated for free quarks. The pattern seen on the left of Fig. 2reflects correlators at a very high temperature, since due to the asymptotic freedom at veryhigh T the quark-gluon interactions can be neglected.While we observe practically exact chiral symmetries, the SU (2) CS and SU (4) symme-tries are only approximate. A degree of the symmetry breaking can be evaluated via theparameter κ , κ = C (1 , ⊕ (0 , ρ − C (1 / , / ρ C (1 , ⊕ (0 , ρ − C S , (11)that measures the splitting within the SU (2) CS multiplet relative to the distance betweendifferent multiplets. With this definition, good symmetry implies | κ | (cid:28) tT . At tT ∼ . SU (2) CS symmetry and in infinite volume | κ | ∼ κ tT, zT t-directionz-direction FIG. 3. Kappa parameter for 48 ×
12 lattice at T = 220 MeV. The blue circles show κ for t -correlations, the green triangles for longer z -correlations. Both values saturate at | κ | < . It is instructive to compare the scale dependence of the symmetry breaking parameter κ extracted from t -correlators and from z -correlators [13] since t - and z -correlators probeQCD at different dimensionless “distance” tT , zT (the time extent of the lattice is smallerthan its spatial extent). In our finite temperature setup ( T > t - and z -correlators havedifferent sensitivity to electric and magnetic fields. E.g. in the limiting case T → ∞ QCDreduces to a 3-dim magnetic theory with a vanishing coupling constant. Consequently at T ∼ . T c a possible small admixture of the magnetic field should have a larger symmetrybreaking effect in the z -direction than in the t -direction correlator. This is well visible inthe symmetry breaking parameter κ at tT = zT = 0 . V. CONCLUSIONS
We have calculated meson rest-frame correlators of J = 0 and J = 1 isovector operatorsalong the time-direction with N F = 2 QCD with physical masses with the chirally symmetricdomain wall Dirac operator at T = 1 . T c . We have observed a very clear emergence of ap-proximate chiral-spin SU (2) CS and SU (4) symmetries in these correlaros. The t -correlatorsare connected via an integral transform with the measurable spectral density in Minkowskispace. Approximate SU (2) CS and SU (4) symmetries of the t -correlators imply the samesymmetries of spectral densities. This result reinforces our findings in Refs. [12, 13].8hese symmetries are incompatible with free deconfined quarks and suggest that the phys-ical degrees of freedom are chirally symmetric quarks bound into color-singlet compoundsby the chromoelectric field without chromomagnetic effects. This result relies solely on lat-tice results and symmetry classification of the QCD Lagrangian. Such relativistic objectsare reminiscent of “strings” since they are purely electric and we refer to the correspondingregime of QCD as a Stringy Fluid. ACKNOWLEDGMENTS
The authors thank T. Cohen, H. Fukaya, C. Gattringer, C. B. Lang and R. Pisarskifor discussions of results. Numerical simulations are performed on IBM System Blue GeneSolution at KEK under a support of its Large Scale Simulation Program (No. 16/17-14)and Oakforest-PACS at JCAHPC under a support of the HPCI System Research Projects(Project ID:hp170061). This work is supported in part by JSPS KAKENHI Grant Num-ber JP26247043 and by the Post-K supercomputer project through the Joint Institute forComputational Fundamental Science (JICFuS). [1] M. Denissenya, L. Y. Glozman and C. B. Lang, Phys. Rev. D (2014) no.7, 077502doi:10.1103/PhysRevD.89.077502 [arXiv:1402.1887 [hep-lat]].[2] M. Denissenya, L. Y. Glozman and C. B. Lang, Phys. Rev. D (2015) no.3, 034505doi:10.1103/PhysRevD.91.034505 [arXiv:1410.8751 [hep-lat]].[3] M. Denissenya, L. Y. Glozman and M. Pak, Phys. Rev. D (2015) no.11, 114512doi:10.1103/PhysRevD.91.114512 [arXiv:1505.03285 [hep-lat]].[4] M. Denissenya, L. Y. Glozman and M. Pak, Phys. Rev. D (2015) no.7, 074508 Erra-tum: [Phys. Rev. D (2015) no.9, 099902] doi:10.1103/PhysRevD.92.099902, 10.1103/Phys-RevD.92.074508 [arXiv:1508.01413 [hep-lat]].[5] L. Y. Glozman, Eur. Phys. J. A (2015) no.3, 27 doi:10.1140/epja/i2015-15027-x[arXiv:1407.2798 [hep-ph]].[6] L. Y. Glozman and M. Pak, Phys. Rev. D (2015) no.1, 016001doi:10.1103/PhysRevD.92.016001 [arXiv:1504.02323 [hep-lat]].
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