Chiral restoration in excited nucleons versus SU(6)
aa r X i v : . [ h e p - ph ] A p r Chiral restoration in excited nucleons versus SU (6) L. Ya. Glozman
Institute for Physics, Theoretical Physics Branch,University of Graz, Universit¨atsplatz 5, A-8010 Graz, Austria
A. V. Nefediev
Institute of Theoretical and Experimental Physics,117218, B.Cheremushkinskaya 25, Moscow, Russia
We compare axial charges of excited nucleons, as predicted by the chiral symmetryrestoration picture, with the traditional, moderately successful for the ground-statebaryons SU (6) symmetry. The axial charges of excited nucleons can (and will) bemeasured in lattice QCD simulations, and comparison of the lattice results with thetwo different symmetry schemes will give an insight on the origins of the excitedhadron masses as well as on interrelations of chiral symmetry and confinement. PACS numbers: 11.30.Rd, 12.38.Aw, 14.20.Gk
I. INTRODUCTION
There are well-known phenomenological successes of the SU (6) flavour-spin symmetry indescription of static properties of the ground states of baryons [1, 2]. Indeed, all the 1 / + and 3 / + ground states in the u -, d -, and s -quark sector, that is the octet and the decupletof SU (3), fall into the -plet of SU (6). Splittings within this -plet are due to the SU (3)breaking, that is due to different masses of the u , d , and s quarks, as well as because of the SU (6) breaking via the effective spin and/or flavour-spin interactions. Magnetic moments(more exactly, their ratios) of the ground states typically agree, within 10-15%, with the SU (6) predictions. The axial charge of the nucleon, as predicted by the SU (6), G AN = 5 / G AN = 1 .
26. Possible reasons for the deviationare: relativistic effects, SU (6) breaking, pion cloud effects, etc. Taking these effects intoaccount allows one to improve the SU (6) prediction for the G AN .It is established that indeed, in the large- N C limit of QCD, ground states satisfy thecontracted SU (6) algebra [3, 4], and both the constituent quark and the Skyrme descrip-tions become algebraically equivalent. While we do not know yet all dynamical details, weunderstand that the emergency of the SU (6) symmetry is a consequence of the spontaneousbreaking of chiral symmetry in QCD: at low momenta the valence quarks acquire a ratherlarge dynamical mass due to their coupling to the quark condensate of the QCD vacuum.Whatever particular gluonic interaction in QCD is responsible for chiral symmetry breaking,the microscopical mechanism of this breaking is related to the selfenergy dressing of quarks.We also know that this selfenergy (“constituent quark mass”) is not a constant, because atlarge space-like momenta the asymptotic freedom of QCD requires this selfenergy to vanish.What is its fate at the time-like momenta depends on microscopical details of confinementand chiral symmetry breaking and is unknown.Successes of the SU (6) symmetry for excited baryons are much more moderate. For thedescription of excited states the SU (6) must be supplemented by a dynamical assumptionthat specifies the radial and orbital structure of baryons. In the simplest case of the harmonicoscillator confining potential in a nonrelativistic 3 q system, the energy is totally fixed bythe principal quantum number N , which is the number of excitation quanta in the system.Then the excited nucleon and delta states of negative parity in the 1 . ÷ . N = 1 shell and have the flavour-spin structure prescribed by the -pletof the SU (6). The amount of the empirical negative parity states in this energy region aswell as their quantum numbers nicely fit the SU (6) × O (3) classification, which is quite anontrivial prediction, indeed. Historically it was taken as a justification for the constituentquark model for excited states.This scheme leads to a pronounced gap between the N = 0 and N = 1 shells, of theorder of 500 ÷
700 MeV. One should expect then that plenty of the positive-parity stateswhich, by parity, could belong only to the N = 2 shell, should be well above the N = 1 shelland lie in the region of around 2 GeV. In reality, however, the positive-parity states haveroughly the same excitation energies as the negative-parity states. Even more, some of them(the Roper-like states) lie below the negative-parity resonances. The SU (6) × O (3) schemepredicts three different supermultiplets in the N = 2 shell: a -plet, a -plet, and a -plet, which is a vast amount of states. Only a handful of them are observed experimentally.Although it is possible to engineer a SU (6) breaking interaction that provides a lowering ofsome of the positive-parity states from the N = 2 shell [5], still the amount of the observedpositive-parity states is much smaller than prescribed by the SU (6) × O (3) symmetry.Phenomenologically the positive- and negative-parity excited states almost systematicallyform approximate parity doublets. From the SU (6) × O (3) point of view such a doubling isunnatural and accidental. If it is not accidental, then a symmetry must be behind such aparity doubling. At the same time there must be reasons for this symmetry not to be explicitin the lowest baryons. It was suggested that this parity doubling reflects in fact a restorationin excited hadrons of the spontaneously broken chiral symmetry of QCD (effective chiraland U (1) A restorations) [6, 7, 8, 9, 10, 11, 12, 13] — see Ref. [14] for a review. The chiralsymmetry restoration in excited nucleons requires that these hadrons must decouple fromthe Goldstone bosons and that their axial charges must vanish [14, 15, 16]. At the same timeit forbids N ∗ → N π decays [17]. Experimentally, the N ∗ N π coupling constants are stronglysuppressed for all observed approximate parity doublets, indeed. There is only one state,3 / − , N (1520), whose chiral partner is certainly missing in the spectrum. This state doesstrongly decay into N π . Hence one observes a 100% correlation of the decay data with thespectroscopic parity doublets, as predicted by chiral restoration [17]. Although the diagonalaxial charges of excited states cannot be measured experimentally, they can be studied onthe lattice. The first lattice results for the lowest negative-parity state, 1 / − , N (1535) (inthis case a possible chiral partner is the Roper resonance 1 / + , N (1440)) show a very smallaxial charge thus supporting chiral restoration [18].Chiral restoration in excited hadrons means that a mass generation mechanism in thesehadrons is essentially different compared to the lowest states. In the latter, the mass is drivenby chiral symmetry breaking in the vacuum, that is by the quark condensate. Consequently,the pion coupling to the valence quarks must also be important for this mass generation.Chiral symmetry is strongly broken in these states and is realised nonlinearly [16, 19]. Incontrast, according to the effective chiral restoration scheme, the quark condensate in thevacuum is almost irrelevant for the approximate parity doublets and their mass has mostlya chirally symmetric origin.This issue of mass generation and interrelation between chiral symmetry breaking andconfinement is a key for understanding the QCD phase diagram. If chiral symmetry isindeed approximately restored in the excited hadrons, where physics should be dominatedby confinement, then it is quite likely that, above the chiral symmetry restoration point atfinite chemical potential, one would have a new phase [20] that represents a confining butchirally symmetric matter [21], rather than a deconfining quark matter. If so, there will bedramatic implications for the QCD phase diagram and astrophysics.Within the SU (6) × O (3) scheme and within related constituent quark models there areno chiral partners at all. Consequently, the whole hadron mass within this picture has to bedue to chiral symmetry breaking in the vacuum. Indeed, the only scenario that allows the u, d -hadrons in the chiral limit not to have chiral partners requires their masses to originatefrom the chiral symmetry breaking in the vacuum [22, 23, 24]. Then the axial properties ofhadrons must be essentially different as compared to the chiral restoration picture. Consider,for example, the pion decay properties of the lowest 1 / − state, N (1535), in the framework ofthe SU (6) × O (3) scheme and related constituent quark models. A very small N (1535) → N π decay width can be achieved in this scheme if one assumes the N (1535) wave function tobe a superposition of two 70-plet states with different spin configurations, correspondingto the total quark spin S = 1 / S = 3 /
2, which are mixed by the SU (6)-breakingspin–spin and spin–tensor quark–quark forces [25, 26]. Such a superposition provides acancellation of two, individually large, contributions into the N (1535) → N π decay width.At the same time, such a superposition of two configurations induces an enhancement of the N (1535) → N η constant. What is important, however, is that the orthogonal combinationof these two S = 1 / S = 3 / / − state, N (1650). It has to come as a mystery then that the decay couplingconstant for the transition N (1650) → N π is as small as that for the decay N (1535) → N π [17] . Furthermore, why are both these coupling constants much smaller than the πN N one? On the contrary, why are the decay constants for the would-be LS partners (withinthe SU (6) × O (3)) 1 / − , N (1535) and 3 / − , N (1520) so dramatically different? Why is therea clear correlation of the decay patterns with the approximate parity doublets? We are notaware of any satisfactory explanation of these facts within the SU (6) × O (3) scheme.The aim of the present paper is to perform a systematic comparison of the predictions Note that typically the constituent quark models [27], as well as the large- N C expansions [28], operatewith the nonrelativistic decay amplitudes and, as a result, with improper phase–space factors. They try tofit, with some free parameters, decay widths, rather than coupling constants. Physics is contained, however,in the coupling constants. Attempts to describe strong decays within relativistic constituent quark modelswithout fitting lead to the results that are qualitatively incompatible with phenomenology [29]. of the SU (6) × O (3) and the chiral restoration scheme for the axial charges. While theycannot be measured experimentally, they are a subject of intensive lattice calculations. Acomparison of different symmetry predictions with the future lattice results will provide aclue for our understanding of the mass generation mechanism in excited hadrons as well asinterrelations of chiral symmetry and confinement.In the next section we review predictions of chiral symmetry restoration for diagonal andoff-diagonal axial charges. The third section is devoted to the same constants evaluatedwithin the SU (6) × O (3) symmetry scheme. In the discussion/conclusion part we compareboth predictions and give an outlook. II. AXIAL CHARGES FOR CHIRAL PARITY DOUBLETS
In this section we review the results for the axial charges of chiral parity doublets [14].Assume that we have a free I = 1 / B in the (0 , / ⊕ (1 / ,
0) representationand there are no chiral symmetry breaking terms. This doublet is a column [24], B = (cid:18) B + B − (cid:19) , (1)where the bispinors B + and B − have a positive and a negative parity, respectively. Theaxial transformation law for the (0 , / ⊕ (1 / ,
0) representation provides a mixing of thefields B ± : B → exp (cid:18) i θ aA τ a σ (cid:19) B. (2)Here σ is a Pauli matrix that acts in the 2 × L = i ¯ Bγ µ ∂ µ B − m ¯ BB = i ¯ B + γ µ ∂ µ B + + i ¯ B − γ µ ∂ µ B − − m ¯ B + B + − m ¯ B − B − . (3)Alternative forms of this Lagrangian can be found in Refs. [30, 31].A crucial property of this Lagrangian is that the fermions B + and B − are strictly degen-erate and have a nonzero chiral-invariant mass m . In contrast, for usual fermions, chiralsymmetry in the Wigner–Weyl mode restricts particles to be massless. Thus there are twopossibilities to satisfy chiral symmetry with the Dirac-type fermions: Note that the axial transformation given in Ref. [24] is incorrect as it breaks chiral symmetry of thekinetic term. The correct axial transformation is given in Ref. [14].
1. The standard scenario, which is to be considered for the nucleon and other ground-state baryons in the light-quark sector: (i) fermions are massless in the Wigner–Weylmode; (ii) independent chiral partners are not required; (iii) fermion mass can begenerated in the Nambu–Goldstone mode only due to spontaneous breaking of chiralsymmetry in the vacuum, that is via the coupling of the fermion with the chiral orderparameter (quark condensate).2. The chiral symmetry restoration scenario, which applies to highly excited hadrons: (i)parity-doubled fermions are massive already in the Wigner–Weyl mode; (ii) this massis manifestly chiral-invariant and is not related at all to the quark condensate; (iii) therole of the chiral symmetry breaking in the Nambu–Goldstone mode (that is of thequark condensate) is to lift the chiral degeneracy of the opposite-parity baryons. Effec-tive chiral restoration means that these opposite-parity baryons almost entirely decou-ple from the quark condensate and most of their mass is manifestly chiral-invariant;(iv) consequently there appear approximate parity doublets in the spectrum.The global chiral symmetry properties (2) of the Lagrangian (3) imply, via the N¨othertheorem, the following form of the conserved axial-vector current: A aµ = ¯ B + γ µ τ a B − + ¯ B − γ µ τ a B + . (4)This current does not contain diagonal terms, like ¯ B + γ µ γ τ a B + or ¯ B − γ µ γ τ a B − . Conse-quently, the diagonal axial charges of the parity-doubled baryons B + and B − are exactly0. In contrast, the off-diagonal axial charges, which normalise axial transitions between themembers of the parity doublet are exactly 1. Hence general chiral symmetry properties ofthe chiral parity doublets uniquely fix the axial properties of the opposite-parity baryons: G A + = G A − = 0 , G A + − = G A − + = 1 . (5)This is another crucial property that distinguishes parity doublets from the usual Diracfermions, the latter having G A = 1. Any microscopic model of chiral parity doublets mustsatisfy these general constraints (see Ref. [32] for a particular microscopic realisation of theseconditions in the framework of the Generalised Nambu–Jona-Lasinio model).Consider now constraints implied by the conservation of the axial-vector current in thechiral limit. Since the diagonal axial charges of the B + and B − strictly vanish, then theGoldberger–Treiman relation, g πB ± B ± = G A ± m ± f π , requires that the diagonal coupling constantsto the pion must vanish, g πB + B + = g πB − B − = 0. Hence small values of the diagonal axialcharges as well as pion–baryon coupling constants taken together with the large baryonmass would tell us that the origin of this mass is not due to chiral symmetry breaking inthe vacuum. It is a challenge for lattice calculations to measure these quantities for excitedstates [18].A similar relation can be obtained for the off-diagonal coupling to the pion, g πB + B − = 0.Indeed, a generic axial-vector current matrix element between two arbitrary 1 / + and 1 / − isodoublet baryons is given as: h B − ( p f ) | A aµ | B + ( p i ) i = ¯ u ( p f ) (cid:2) γ µ H ( q ) + σ µν q ν H ( q ) + q µ H ( q ) (cid:3) τ a u ( p i ) , q = p f − p i , (6)where H , H , and H are formfactors. Then the matrix element of the divergence of theaxial-vector current is h B − ( p f ) | ∂ µ A aµ | B + ( p i ) i = i (cid:2) ( m + − m − ) H ( q ) + q H ( q ) (cid:3) ¯ u ( p f ) τ a u ( p i ) . (7)Because of the axial-vector current conservation, this matrix element must vanish. Thereare two possibilities to satisfy this:1. m + = m − . Then the Goldstone boson pole is required in H ( q ) and one arrives at ageneralised Goldberger-Treiman relation: g πB + B − = G A + − ( m + − m − )2 f π , G A + − = H (0) . (8)2. m + = m − . Then H ( q ) = 0 and, automatically, g πB + B − = 0.Note that the condition g πB + B − = 0 is a general consequence of the axial-vector currentconservation for degenerate opposite-parity baryons. Whatever reason for this degeneracy is— whether it is due to chiral restoration or something else — the off-diagonal pion couplingmust vanish.In reality, of course, chiral symmetry is never completely restored in excited baryons.The effect of chiral symmetry breaking in the vacuum is to split the masses of B + and B − .Consequently the axial charges should deviate from the limiting values given in Eq. (5).However, if the coupling to the condensate is weak, then chiral symmetry breaking is only TABLE I: Chiral multiplets of excited nucleons. Comments: (i) All these states are well establishedand can be found in the Baryon Summary Table of the Review of Particle Physics. (ii) There aretwo possibilities to assign the chiral representation: (1 / , ⊕ (0 , /
2) or (1 / , ⊕ (1 , /
2) becausethere is a possible chiral pair in the ∆ spectrum with the same spin with similar mass. (iii) Themissing chiral partner is predicted.Spin Chiral multiplet Representation Comment1/2 N + (1440) − N − (1535) (1 / , ⊕ (0 , /
2) (i)1/2 N + (1710) − N − (1650) (1 / , ⊕ (0 , /
2) (i)3/2 N + (1720) − N − (1700) (1 / , ⊕ (0 , /
2) (i)5/2 N + (1680) − N − (1675) (1 / , ⊕ (0 , /
2) (i)7/2 N + (?) − N − (2190) see comment (ii) (i),(ii),(iii)9/2 N + (2220) − N − (2250) see comment (ii) (i),(ii)11/2 N + (?) − N − (2600) see comment (ii) (i),(ii),(iii)3/2 N − (1520) no chiral partner (i) a small perturbation, and one should expect that, for approximate parity doublets, therelations (5) are approximately satisfied. A possible assignment of excited nucleons to thechiral multiplets is given in Table I. III. AXIAL CHARGES FROM THE SU (6) SYMMETRY.
First, let us review briefly the quantum numbers and the SU (6) symmetric wave functionsfor excited baryons, known from the first years of the quark model. The flavour-spin SU (6)multiplet is uniquely specified by the corresponding Young pattern [ f ] F S . In particular,the -plet is given by the completely symmetric Young pattern [3] F S , the -plet is givenby the mixed symmetry pattern [21] F S , and the -plet is specified by the antisymmetricYoung diagram, [111] F S . Different baryons within a given SU (6) multiplet are characterisedby the flavour SU (3) F and spin SU (2) S symmetries. In particular, the decuplet, octet,and singlet states have the following Young patterns: [3] F , [21] F , and [111] F , respectively.The spin symmetry [ f ] S is uniquely determined by the total spin S in the 3 q system, thatis for spins S = 3 / S = 1 / S and [21] S , respectively. The orbital state in TABLE II: The SU (6) assignments of the well established nucleons below 2 GeV. Vacant statesfrom the -plet are not shown. N ( λµ ) L [ f ] X [ f ] F S [ f ] F [ f ] S LS multiplet0(00)0[3] X [3] F S [21] F [21] S
12 + , N X [3] F S [21] F [21] S
12 + , N (1440)1(10)1[21] X [21] F S [21] F [21] S − , N (1535); − , N (1520)1(10)1[21] X [21] F S [21] F [3] S − , N (1650); − , N (1700); − , N (1675)2(20)2[3] X [3] F S [21] F [21] S
32 + , N (1720);
52 + , N (1680)2(20)0[21] X [21] F S [21] F [21] S
12 + , N (1710)2(20)0[21] X [21] F S [21] F [3] S
32 + , N (?)2(20)2[21] X [21] F S [21] F [21] S
32 + , N (?);
52 + , N (?);2(20)2[21] X [21] F S [21] F [3] S
12 + , N (?);
32 + , N (?);
52 + , N (?);
72 + , N (?) the 3 q system is specified in general by the orbital angular momentum L as well as by thespatial permutational symmetry [ f ] X . The latter is uniquely fixed by the Pauli principle,[ f ] X = [ f ] F S . Within a specific spatial basis there can appear additional quantum numbersin order to specify uniquely the given spatial basis function, as it happens, for example, tothe highly symmetric harmonic basis. In the latter case, these additional quantum numbersinclude N , which specifies the energy and is the number of the excitation quanta (it is theprincipal quantum number for the U (6) spatial symmetry of the harmonic oscillator in athree-body system), and the spatial SU (3) symmetry of the orbital wave function is fixedby the symbol ( λµ ). Given all these quantum numbers one traditionally assigns excitednucleons of positive and negative parity to the SU (6) multiplets, as is shown in Table II.Our purpose now is to calculate the diagonal and off-diagonal axial charges for excitedbaryons that can be accessed on the lattice. The SU (6) symmetry is a nonrelativisticsymmetry. Then the diagonal and off-diagonal axial charges for the SU (6) baryons are givenby the matrix elements of the nonrelativistic single-quark axial-vector charge operators, G Afi = h Ψ f (1 , , | X n =1 Q An | Ψ i (1 , , i . (9)The nonrelativistic leading order axial charge operator Q An for a point-like Dirac constituentquark (index n numerates quarks in the baryon) is given by the Gamov-Teller operator σ τ ,0where ~σ and ~τ are the spin- and isospin operators, respectively. Note that there is no depen-dence on the spatial coordinate in the nonrelativistic leading order axial charge operator .Consequently only diagonal matrix elements with respect to the spatial quantum numberscan have nonzero values. Therefore within the nonrelativistic SU (6) × O (3) symmetry allthe off-diagonal axial charges for all degenerate opposite parity states vanish identically, G A + − = G A − + = 0. This is the first crucial difference between the manifest SU (6) × O (3) andchiral symmetry of excited baryons.It is a legitimate question to consider relativistic corrections to the pure static nonrel-ativistic axial transition operator. It is well known from the nuclear physics applications,long before the naive quark model, that the lowest-order correction, ∼ /M , is given by theoperator 12 M ~σ ( ~p i + ~p f ) τ a e i~q~r , (10)where ~p i and ~p f are the initial and final quark momenta. Consequently, if the pure nonrel-ativistic (static) transition is forbidden for some reason, the contribution of the relativisticcorrections may become important. For example, this kind of operator was taken into ac-count in a four-parameter fit for the baryon resonance decays in Ref. [26], and its strengthwas actually considered as a free parameter. Within the usual nonrelativistic constituentquark models, however, v/c ∼
1, or even larger. Hence there are no reasons to consider onlythe first correction and to ignore all others.Note that the same operator leads to a nonvanishing g πB + B − . This is in an obvious conflict Strictly speaking, the nonrelativistic single-quark axial transition operator does contain a spatial de-pendence, via the exponent exp [ i~q~r ]. In the meantime, the diagonal baryon axial charge is defined at thepoint q = 0, that would yield, in the nonrelativistic limit, ~q = 0. In the case of the off-diagonal axialcharge, the role played by the spatial dependence of the transition operator strongly depends on the baryonmasses in the initial and in the final states. Indeed, for the baryons with substantially different masses,the off-diagonal transition is determined by the corresponding axial form-factor H ( q ) — see Eq. (6) —taken at the point q = 0. Notice that there is no unique and unambiguous form of the nonrelativistic axialcharge operator in this case. On the contrary, in the present work, we consider only the opposite-paritystates which are approximately degenerate in mass. Then the same point ~q = 0 is to be considered for thenonrelativistic axial charge operator. g πB + B − = 0(see the text below Eq. (8)), even for G A + − = 0. It is not clear how could it be possibleto satisfy this requirement within the naive quark model, as a matter of principle, oncedifferent relativistic corrections are taken into account to the axial transition operator. Thisissue represents an additional conceptual problem for the naive quark model of excitedparity-doubled baryons.In lattice simulations it is possible to separate the lowest negative and positive parity J = 1 / N (1535), N (1650), N (1440), and N (1710) [33]. Given explicit SU (6) wavefunctions — see, for example, Ref. [5] — it is straightforward to calculate the requireddiagonal axial charges: 12 + , N (1440) : G A = 53 , (11)12 + , N (1710) : G A = 13 , (12)12 − , N (1535) : G A = − , (13)12 − , N (1650) : G A = 59 . (14)It is also possible to access on the lattice the lowest 3 / − , N (1520) state [34]. Its diagonalaxial charge matrix element reads: h / − , N (1520); M, T | X n =1 Q An | / − , N (1520); M, T i = √ C M M C T T , (15)where M and T are its total spin and isospin projection, respectively. IV. DISCUSSION AND OUTLOOK
Now we are in a position to confront systematically the predictions of the chiral restorationand SU (6) schemes for excited nucleons. As we have already mentioned above, while chiralsymmetry restoration predicts the off-diagonal axial charges within chiral multiplets to beclose to 1 ( G A + − = G A − + ∼ SU (6) × O (3) requires all these off-diagonal axial charges tovanish, ( G A + − = G A − + = 0). For the diagonal axial charges the difference is also substantial.Indeed, if the 1 / + , N (1440) − / − , N (1535) pair is the lowest approximate parity doublet,then the diagonal axial charge of the lowest positive-parity state (the Roper state) must besmall, ∼
0. On the contrary, the SU (6) symmetry suggests for this Roper state a large axial2charge, the latter coincides with the nucleon axial charge. The axial charge of the N (1535)is predicted in both cases to be small. For the next approximate J = 1 / SU (6) × O (3) predicts for the negative-parity state, 1 / − , N (1650) a rather large diagonalaxial charge and not a small diagonal charge for 1 / + , N (1710). The chiral symmetryrestoration requires both of them to have a small axial charge. The state 3 / − , N (1520) is a LS -partner of the 1 / − , N (1535) state within the SU (6) × O (3) symmetry. According to thechiral symmetry restoration scenario, there is no chiral partner for 3 / − , N (1520), and henceits mass is due to chiral symmetry breaking in the vacuum. In this case there are no strictconstraints for its axial charge coming from chiral symmetry. Naively one would expect itsdiagonal axial charge to be of the order 1, because empirically nucleon has a rather largeaxial charge, though in actuality it can take any arbitrary value.We have systematically compared diagonal and off-diagonal axial properties of the stateswhich are expected to be approached soon within the lattice QCD calculations. There arefirst unquenched QCD results for the diagonal axial charge of the N (1535) resonance [18],that is very small. Given a combined set of empirical spectroscopic and decay data, discussedin the introduction, it provides an additional support for the chiral restoration picture. Still,specifically this small axial charge is also marginally compatible with the SU (6) picture. Itwould be very interesting to measure diagonal and off-diagonal axial charges of other excitedstates. Such a program is now under way. Acknowledgments
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