Gluonic-Excitation Energies and Abelian Dominance in SU(3) QCD
aa r X i v : . [ h e p - l a t ] J u l Gluonic-excitation energies and Abelian dominance in SU(3) QCD
Hiroki Ohata
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Hideo Suganuma
Department of Physics, Kyoto University, Kitashirakawaoiwake, Sakyo, Kyoto 606-8502, Japan (Dated: July 27, 2020)We present the first study of the Abelian-projected gluonic-excitation energies for the staticquark-antiquark (Q ¯Q) system in SU(3) lattice QCD at the quenched level, using a 32 lattice at β = 6 .
0. We investigate ground-state and three excited-state Q ¯Q potentials, using smeared linkvariables on the lattice. We find universal Abelian dominance for the quark confinement force ofthe excited-state Q ¯Q potentials as well as the ground-state potential. Remarkably, in spite of theexcitation phenomenon in QCD, we find Abelian dominance for the first gluonic-excitation energyof about 1 GeV at long distances in the maximally Abelian gauge. On the other hand, no Abeliandominance is observed for higher gluonic-excitation energies even at long distances. This suggeststhat there is some threshold between 1 and 2 GeV for the applicable excitation-energy region ofAbelian dominance. Also, we find that Abelian projection significantly reduces the short-distance1 /r -like behavior in gluonic-excitation energies. I. INTRODUCTION
Since quantum chromodynamics (QCD) has been es-tablished as the fundamental theory of the strong interac-tion, analytical derivation of quark confinement directlyfrom QCD has been an open problem. The difficulty orig-inates from non-Abelian dynamics and nonperturbativefeatures of QCD, which are largely different from QED.In 1970’s, Nambu, ’t Hooft, and Mandelstam pro-posed an interesting idea that quark confinement mightbe physically interpreted with the dual version of the su-perconductivity [1]. In the dual-superconductor picturefor the QCD vacuum, there takes place one-dimensionalsqueezing of the color-electric flux among (anti)quarksby the dual Meissner effect, as a result of condensationof color-magnetic monopoles.As for the possible connection between QCD and thedual-superconductor theory, ’t Hooft proposed an inter-esting concept of Abelian gauge fixing, a partial gaugefixing which diagonalizes some gauge-dependent quantity[2]. In particular, in the maximally Abelian (MA) gauge[3–9], which is a special Abelian gauge, the off-diagonalgluon has a large effective mass of about 1 GeV [6], andAbelian dominance of quark confinement is observed inlattice QCD [4, 5, 8, 9]. Then, infrared QCD in the MAgauge becomes an Abelian gauge theory including thecolor-magnetic monopoles, of which condensation leadsto the dual superconductor [10].For other nonperturbative QCD quantities such asspontaneous chiral-symmetry breaking, Abelian domi-nance is observed in lattice QCD [11]. However, it is non-trivial whether Abelian dominance holds for excitationphenomena in QCD or not, because this Abelianizationscheme is conjectured to be valid only for low energiesand long distances. For instance, Abelian dominance isshown to be decreasing with larger momentum or smallerdistance from the gluon propagator in the MA gauge inboth SU(2) and SU(3) lattice QCD [6, 7], although the propagator itself is not physical observable.Then, in this paper, we study Abelian dominancefor excited-state inter-quark potentials and gluonic-excitation energies in the MA gauge in SU(3) color QCDat the quenched level. Here, the excited-state potentialsare important for the description of excitation phenom-ena of QCD [12–14], and the gluonic-excitation energiesare interesting physical observables appearing in hybridhadrons [15]. They have been investigated in lattice QCD[12–14], and the lattice results have been compared asstringy modes in the string picture of hadrons for thestatic quark-antiquark system. In fact, apart from thelinear confinement part, the excited-state potential has1 /r part with a positive coefficient in long distances of r ≥ /r behavior can be a signal of thestringy mode, although the stringy behavior is signifi-cantly suppressed in shorter distances than 2 fm [12, 14].The gluonic-excitation energies are defined by the dif-ferences between the ground-state and excited-state po-tentials, and the lowest gluonic-excitation energy takesa larger value than about 1 GeV both for static quark-antiquark (Q ¯Q) and 3Q systems in lattice QCD [12, 13].This large gluonic-excitation energy explains success ofthe quark model [13].The organization of this paper is as follows. In Sec. II,we briefly review the Abelian projection in lattice QCDin the MA gauge. In Sec. III, we present our calculationprocedure for the ground- and excited-state potentialsin the static Q ¯Q system. In Sec. IV, we show the lat-tice QCD result for the excited-state potentials and thegluonic-excitation energies. Section V is devoted to sum-mary and conclusion. II. MAXIMALLY ABELIAN GAUGE FIXINGAND ABELIAN PROJECTION
We perform SU(3) lattice QCD simulations at thequenched level with the standard plaquette action [16].In lattice QCD, the gauge variable is described as thelink variable U µ ( s ) ≡ e iagA µ ( s ) ∈ SU(3), with the gluonfield A µ ( s ) ∈ su(3), QCD gauge coupling g and the lat-tice spacing a . In this paper, we use the lattice size of L × L t = 32 at β ≡ /g = 6 . a , we take a ≃ a is determined to reproduce thestring tension σ ≃ R MA [ U µ ( s )] ≡ X s X µ =1 tr (cid:16) U † µ ( s ) ~HU µ ( s ) ~H (cid:17) = X s X µ =1 − X i = j | U ij | (1)under the SU(3) gauge transformations for each gaugeconfiguration. Here, ~H ≡ ( T , T ) is the Cartan sub-algebra of SU(3), i.e., T = diag (1 , − ,
0) and T = √ diag (1 , , − R MA / (4 L L t ) = 1.We numerically perform MA gauge fixing using theover-relaxation method for rapid achievement in the max-imization algorithm. As for the stopping criterion, westop the maximization algorithm when the deviation∆ R MA / (4 L L t ) becomes smaller than 10 − . The con-verged value h R MA / (4 L L t ) i is 0 . u µ ( s ) = exp (cid:8) iθ µ ( s ) T + iθ µ ( s ) T (cid:9) ∈ U(1) × U(1) (2)from the link variable in the MA gauge, U MA µ ( s ) ∈ SU(3),by maximizing the norm R Abel ≡ Re tr (cid:8) U MA µ ( s ) u † µ ( s ) (cid:9) ∈ (cid:20) − , (cid:21) , (3)so that the distance between u µ ( s ) and U MA µ ( s ) becomesthe smallest in the SU(3) manifold. We thus find “mi-croscopic Abelian dominance”, i.e., h R Abel i = 0 . U µ ( s ) by the Abelian part u µ ( s )for each gauge configuration, i.e., O [ U µ ( s )] → O [ u µ ( s )]for QCD operators. III. LATTICE QCD CALCULATION OFEXCITED-STATE INTER-QUARK POTENTIALS
In this section, we briefly mention the lattice QCDformalism to obtain the excited-state Q ¯Q potentials andour numerical procedure.
A. Formalism
We explain the variational and diagonalization methodto calculate the ground- and excited-state potentials,originally reported in Ref. [18], in the same manner withRef. [13]. We denote the QCD Hamiltonian ˆ H and thephysical eigenstates | n i ( n = 0 , , , . . . ) for the static Q ¯Qsystem. As the eigenvalues ˆ H | n i = V n | n i , we define the n th excited-state potential V n with V ≤ V ≤ V ≤ · · · in the static Q ¯Q system, where the eigenvalues physicallymean ground and excited potentials between the quarkand the antiquark. For the simple notation, the groundstate is often regarded as the “0th excited state”.As sample states for the static Q ¯Q system, we pre-pare arbitrary given independent Q ¯Q states, | Φ k i ( k =0 , , , . . . ). In general, each state can be expressed witha linear combination of the eigenstates | n i as | Φ k i = c k | i + c k | i + c k | i + · · · . (4)Let us consider time evolution of the sample states | Φ k i with the spatial locations of the quark and the antiquarkbeing fixed. The Euclidean time evolution of the Q ¯Qstate | Φ k ( t ) i is expressed with the operator e − ˆ Ht , andwe introduce the generalized Wilson loop W jkT ≡ h Φ j ( T ) | Φ k (0) i (5)= h Φ j | e − ˆ HT | Φ k i = ∞ X m =0 ∞ X n =0 ¯ c jm c kn h m | e − ˆ HT | n i = ∞ X n =0 ¯ c jn c kn e − V n T . (6)Here, we define the matrix C and the diagonal matrix Λ T by C nk = c kn , Λ mnT = e − V n T δ mn , and rewrite the aboverelation as W T = C † Λ T C . Note that C is not a unitarymatrix, and hence this relation does not mean the simplediagonalization by the unitary transformation.In the lattice QCD simulations, we numerically cal-culate the generalized Wilson loop W jkT , and extract thepotentials V , V , V , . . . from W T and W T +1 by using theformula W − T W T +1 = (cid:0) C † Λ T C (cid:1) − C † Λ T +1 C = C − diag (cid:0) e − V , e − V , . . . (cid:1) C. (7)In fact, e − V , e − V , e − V , . . . can be obtained as the eigen-values of the matrix W − T W T +1 , i.e., W − T W T +1 ψ n = e − V n ψ n , and they are also the solutions of the secularequationdet (cid:0) W − T W T +1 − t (cid:1) = Y n (cid:0) e − V n − t (cid:1) = 0 . (8)In the lattice QCD calculation, we use the above-mentioned method and extract low-lying excited-statepotentials numerically for SU(3) QCD and Abelian-projected QCD, respectively. B. Numerical procedure
In the practical calculation, we prepare four samplestates | Φ k i for each gauge configuration by using theAPE smearing method [19]. Originally, the smearingmethod was developed as a useful technique to reducethe higher excitation components in a gauge-invariantmanner. Here, the gauge-invariant Q ¯Q system consistsof a quark, an antiquark, and gauge-covariant product oflink variables connecting them, and, in the APE smearingmethod, gauge-covariant smeared link variables are usedinstead of original link variables [19, 20]. We note thatthe smeared link variables depends on the iteration num-ber N smr of the smearing, and the N smr -times smearedstates are generally linear independent each other when N smr is different [20]. Furthermore, the smeared stateshave only small components for highly excited states, andtherefore they are appropriate as the sample states | Φ k i for the study of low-lying excitations [13].Next, let us consider the quantum number of the Q ¯Qstates obtained in this procedure with the APE smear-ing method, in terms of their parity (P), charge conju-gation (C) and angular momentum (Λ), using the stan-dard notation from the physics of diatomic molecules.Here, the parity transformation means spatial inversionabout the midpoint between the static quark and anti-quark. The excitation modes obtained in this procedureare even ( g ) under charge-parity (CP) conjugation op-eration [12], since the generalized Wilson loop W T withthe reflection-symmetrically smeared states is invariantunder CP transformation and all the CP-odd ( u ) com-ponents are cancelled in calculating the W T . For theCP-odd potentials, one needs to prepare CP-odd sample states, as is done in Ref. [12]. As for the total angularmomentum J g of gluons, the projection J g · ˆ R onto themolecular axis R gives a good quantum number and themagnitude of the eigenvalue of J g · ˆ R is denoted by Λ [12].In this procedure with the APE smearing method, onlyΣ states with Λ = 0 are expected to be obtained, sincethe smeared states are constructed in an axial-symmetricmanner on the lattice. Also, there is a sign quantumnumber ± under a reflection in a plane containing themolecular axis [12], and our axial-symmetric proceduremakes only even (+) states. In fact, in our procedure,we generate only CP-even ( g ), reflection-even (+) andΛ = 0 states, which are denoted by the Σ + g states [12] inthe notation of the physics of diatomic molecules.In the actual calculation, we prepare 8 , , ,
32 timessmeared states with the smearing parameter α = 2 . | Φ k i seem to be practically linear independent with dif-ferent patterns of the coefficients c kn , since the secularequation (8) can be numerically solved. (If two of thesample states are not linearly independent, Eq. (8) can-not be solved.) In this analysis, we make an assumptionthat higher excitation components | n i with n ≥ | Φ k i are small enough and can be droppedoff, and solve the eigenvalue problem of the 4 × W − T W T +1 .As for the Abelian projection, we repeat just the sameprocedure by using Abelian link variables u µ ( s ) insteadof SU(3) link variables U µ ( s ). Hereafter, we add the label“Abel” for the Abelian-projected physical quantities.In this way, we obtain the effective masses V eff ( r, t ) , V Abeleff ( r, t ) for the 0th, 1st, 2nd, 3rd excitedstate, respectively. The measurement is done for the on-axis and off-axis inter-quark directions as (1,0,0), (1,1,0),(2,1,0), (1,1,1), (2,1,1), and (2,2,1). As the statisticalerror estimate, we adopt the jack-knife error estimate.In calculating the potentials, higher excited states suf-fer larger systematic errors because the assumption ofabsence of higher modes becomes relatively more sub-tle. Hence, we do not make quantitative analysis ofthe third-excited-state potentials, although preparing thefour sample states definitely contributes to the significanterror reduction for all the results. To reduce the system-atic errors further, we pick effective masses V eff ( r, t ) atlarger t as long as the error is small. IV. LATTICE QCD RESULT
In this section, we show the excited-state potentialsand their Abelian projection in the static Q ¯Q system.Figure 1 shows the effective mass plots V eff ( r, t ) for theSU(3) potentials V n ( r ) with n = 0 , ,
2. Owing to thevariational and diagonization method, for the low-lyingstates, t dependence is small and an approximate plateauis observed even in small t region, although higher excited t e ff ec t i v e m a ss V e ff ( r , t ) [ G e V ] (a) ground state r = 15 r = 12 r = 9 r = 6 r = 3 t (b) first excited state t (c) second excited state FIG. 1: Effective mass plots V eff ( r, t ) of the SU(3) potential V ( r ) for (a) the ground state, (b) the first-excited state, and (c)the second-excited state in the static Q ¯Q system. Here, we display on-axis data of r = 3 , , , ,
15 in the lattice unit. Larger r data are a bit shifted horizontally for visibility. t e ff ec t i v e m a ss V A b e l e ff ( r , t ) [ G e V ] (a) ground state r = 15 r = 12 r = 9 r = 6 r = 3 t (b) first excited state t (c) second excited state FIG. 2: Effective mass plots V Abeleff ( r, t ) of the Abelian potential V Abel ( r ) for (a) the ground state, (b) the first-excited state,and (c) the second-excited state in the static Q ¯Q system. Here, we display on-axis data of r = 3 , , , ,
15 in the lattice unit.Larger r data are a bit shifted horizontally for visibility. state suffers larger statistical errors. In this paper, we donot show the meaningless data with too large errors infigures. Here, we pick effective masses at t = 4 , , V Abeleff ( r, t ) forthe Abelian potentials V Abel n ( r ) with n = 0 , ,
2. Com-pared with the SU(3) case, V Abeleff ( r, t ) is slightly increas-ing as a function of t , and this might cause a systematicerror of about 0.1 GeV on the choice of t . On the otherhand, the statistical errors are smaller, because Abelianprojection enhances the expectation value of the Wilsonloop. We pick effective masses at t = 4 , , V n ( n = 0 , ,
2) in the CP-evenQ ¯Q system, and first and second gluonic-excitation ener-gies ∆ E n ≡ V n − V ( n = 1 ,
2) for both SU(3) and Abeliancases in Fig. 3.In the SU(3) case, the lattice results of V n ( n = 0 , , + g states inRefs.[12, 14], in terms of the overall behavior of V n ( r ),the infrared slope σ n ∼ V n ( r ), and theinterval V n +1 − V n ≃ r =1 fm, except thatthe short-distance behavior of V ( r ) is somehow differ-ent from Ref.[14] because of the 0 ++ glueball mixture.As shown in Fig. 3 (a), all the SU(3) and Abelian po-tentials have approximately the same linear slope (thestring tension) in long distances, which is also observedfor the third-excited potential. This indicates universalAbelian dominance for the quark confinement force of the . . . . . . . . . r [fm] V , V A b e l [ G e V ] (a) ground and excited state potentials V , V , V V Abel0 , V
Abel1 , V
Abel2 − A/r + σr + Cσr + C . . . . . . . . . r [fm] . . . . . . . ∆ E , ∆ E A b e l [ G e V ] (b) gluonic excitation energies ∆ E , ∆ E ∆ E Abel1 , ∆ E Abel2 a/r + E th FIG. 3: (a) The ground-state potential and two excited-state potentials in the static Q ¯Q system. The circles and the squaresdenote the SU(3) potentials V , V , V and the Abelian potentials V Abel0 , V
Abel1 , V
Abel2 , respectively. The solid curves are thebest fits with the Cornell potential − A/r + σr + C for V and V Abel0 for r > a . The dashed lines are the best fits with σr + C forexcited-state potentials V , V , V Abel1 , V Abel2 at long distances. For the clear display, an irrelevant overall constant (+0.2 GeV)is added to V Abel n . (b) Gluonic-excitation energies ∆ E n ( r ) ≡ V n ( r ) − V ( r ). The circles and squares denote gluonic-excitationenergies ∆ E , ∆ E and the Abelian parts ∆ E Abel1 , ∆ E Abel2 , respectively. The curves are the best fits with the Ansatz a/r + E th for SU(3) gluonic-excitations. excited-state Q ¯Q potentials as well as the ground-statepotential.For more quantitative argument, we evaluate the stringtension σ n and the Abelian string tension σ Abel n from V n and V Abel n , respectively, for n = 0 , ,
2. For theground-state potentials, V and V Abel0 , we consider thebest fits with the Cornell potential − A/r + σr + C (curves in Fig. 3). From the fit for 2 a < r ≤ a , weevaluate the string tension σ ≃ A ≃ σ Abel0 ≃ A Abel ≃ σ Abel ≃ σ from the long-distance data witha large number of gauge configurations.) For the excited-state potentials V n , we evaluate the string tensions σ n from the fit with σr + C for large r > σ ≃ σ ≃ V Abel n , we evaluate theAbelian string tensions σ Abel n from the fit with σr + C forlarge r > σ Abel1 ≃ σ Abel2 ≃ σ Abel n & . σ n for n = 0 , , E n ( r ) ≡ V n ( r ) − V ( r ). Therefore theirabsolute values are physically meaningful, while all po-tentials have ambiguity of an overall constant shift. Forthe gluonic-excitation energies, we expect cancellation ofsystematic errors on V n , especially for the Abelian part.From Fig. 3(b), the SU(3) gluonic-excitation ener-gies ∆ E n ( r ) seem to be roughly approximated with theAnsatz a n /r + E th n (the curves), and the best fit param-eters are a = 0.54(2), E th1 = 0.98(2) GeV for ∆ E ( r ),and a = 0.451(9), E th2 = 1.818(7) GeV for ∆ E ( r ). On the other hand, the Abelian-projected gluonic-excitation energies ∆ E Abel n ( r ) seem to be approximatelyconstant: ∆ E Abel1 ( r ) ≃ E Abel2 ( r ) ≃ E Abel n ( r ) is forced to be fit with the Ansatz a Abel n /r + E th , Abel n , the best fit parameters are a Abel1 = 0.14(2), E th , Abel1 = 1.00(1) GeV for ∆ E Abel1 ( r ), and a Abel2 =0.10(1), E th , Abel2 = 1.323(6) GeV for ∆ E Abel2 ( r ).Thus, we find three significant features for the gluonic-excitation energies ∆ E n ( r ) and ∆ E Abel n ( r ) as follows:1. Abelian dominance is observed in the first gluonic-excitation energy in longer distances than about0.7 fm: ∆ E Abel1 ≃ ∆ E ≃ E Abel2 < ∆ E . (This feature is also found for thethird gluonic-excitation energy as ∆ E Abel3 < ∆ E .)3. The short-distance 1 /r -like behavior is significantlyreduced in the Abelian-projected gluonic-excitationenergies ∆ E Abel n ( r ).From the first two features, we conjecture that thereis some threshold between 1 and 2 GeV for the ap-plicable excitation-energy region of Abelian dominance,and Abelian dominance holds below the threshold. Thisseems to be qualitatively consistent with the behavior ofthe MA-gauge gluon propagator, which shows decreasingof Abelian dominance with larger momentum or smallerdistance[6, 7].Here, Abelian dominance holds for nonperturbativeproperties such as confinement and spontaneous chiral-symmetry breaking, but does not hold for perturbativeQCD. Then, as an interesting conjecture, we expect thatthe first gluonic-excitation energy of about 1 GeV in longdistances is nonperturbative, since it exhibits Abeliandominance.On the other hand, the higher gluonic-excitation ener-gies, which do not show Abelian dominance, might haveperturbative ingredients, which would obey ∆ E AbelpQCD ≃ ∆ E pQCD , according to the gluon-number reductionthrough Abelianization.Finally, let us consider the significant reduction of theshort-distance 1 /r -like behavior in the Abelian-projectedgluonic-excitation energies. If one considers the above-mentioned best fit with the Ansatz a/r + E th for ∆ E n and∆ E Abel n to be serious, one finds a Abel n ≃ a n , which agreeswith the gluon-number reduction through Abelianiza-tion, as is also seen in the perturbative one-gluon ex-change. Then, as an interesting possibility, the short-distance 1 /r -like behavior might originate from pertur-bative QCD, instead of nonperturbative QCD. In anycase, this finding would be a key to understanding theshort-distance 1 /r behavior in the excited SU(3) poten-tials for the static Q ¯Q system. V. SUMMARY AND CONCLUSION
In this paper, we have presented the first study ofthe Abelian-projected gluonic-excitation energies in thestatic Q ¯Q system in SU(3) lattice QCD at the quenchedlevel. Using smeared link variables on the lattice, we haveexamined four low-lying CP-even Q ¯Q potentials. Wehave found universal Abelian dominance for the quarkconfinement force also in the excited-state Q ¯Q potentials.As a remarkable fact, we have found Abelian dom-inance in the first gluonic-excitation energy of about1 GeV in long distances in the maximally Abelian gauge,although it is an excitation phenomenon in QCD. In con- trast, no Abelian dominance has been observed in the sec-ond and higher gluonic-excitation energies. From thesetwo findings, we have conjectured that there is somethreshold for the applicable excitation-energy region ofAbelian dominance between 1 and 2 GeV.In addition, we have found that the short-distance 1 /r behavior in gluonic-excitation energies is significantly re-duced by the Abelian projection. This finding would bea key to understand the short-distance 1 /r behavior inthe excited SU(3) potentials for the static Q ¯Q system.As a future work, it is interesting to perform the similarstudy for the baryonic 3Q system. It is also meaningful toexamine Abelian projection for CP-odd excited-state Q ¯Qpotentials, using asymmetric sample states as in Ref. [12].It is also interesting to investigate the long-distancebehavior of the gluonic-excitation energies in Abelian-projected QCD and to compare with the stringy modeof the Q ¯Q flux tube. In SU(3) lattice QCD, the stringymodes grow up and appear in longer distances than 2 fm[12]. Since Abelian-projected QCD also exhibits quarkconfinement and the flux-tube formation [8], the stringymodes are expected also in the Abelian part in longerdistances.Also, it is meaningful to analyze our result in termsof low and high momentum gluon modes, since the gluonpropagator shows that Abelian dominance decreases withlarger momentum or smaller distance in the MA gaugein both SU(2) and SU(3) lattice QCD [6, 7]. Acknowledgments
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