Gluonic excitation energies and Abelian dominance in SU(3) QCD

We present the first study of the Abelian-projected gluonic-excitation energies for the static Q$\bar{\rm Q}$ system in SU(3) lattice QCD at the quenched level, using a $32^4$ lattice at $\beta = 6.0$. We investigate four low-lying parity-even Q$\bar{\rm Q}$ potentials, using smeared link variables on the lattice. We find universal Abelian dominance for the quark confinement force of the excited-state Q$\bar{\rm Q}$ potentials as well as the ground state potential. Remarkably, in spite of the excitation phenomenon in QCD, we find Abelian dominance for the first gluonic-excitation energy of about 1GeV at long distances in the maximally Abelian gauge. On the other hand, no Abelian dominance is observed for higher gluonic-excitation energies even at long distances. This suggests that there is some threshold between 1GeV and 2GeV for the applicable excitation-energy region of Abelian dominance. Also, we find that Abelian projection significantly reduces the short-distance $1/r$-like behavior in gluonic excitation energies.


I. INTRODUCTION
Since quantum chromodynamics (QCD) has been established as the fundamental theory of the strong interaction, analytical derivation of quark confinement directly from QCD has been an open problem. The difficulty originates from non-Abelian dynamics and nonperturbative features of QCD, which are largely different from QED.
In 1970's, Nambu, 't Hooft, and Mandelstam proposed an interesting idea that quark confinement might be physically interpreted with the dual version of the superconductivity [1]. In the dual-superconductor picture for the QCD vacuum, there takes place one-dimensional squeezing of the color-electric flux among (anti-)quarks by the dual Meissner effect, as a result of condensation of color-magnetic monopoles.
As for the possible connection between QCD and the dual-superconductor theory, 't Hooft proposed an interesting concept of Abelian gauge fixing, a partial gauge fixing which only remains Abelian gauge degrees of freedom in QCD [2]. In particular, in the maximally Abelian (MA) gauge [3][4][5][6][7][8][9], which is a special Abelian gauge, the off-diagonal gluon has a large effective mass of about 1GeV [6], and Abelian dominance of quark confinement is observed in lattice QCD [4,5,8,9]. Then, infrared QCD in the MA gauge becomes an Abelian gauge theory including the color-magnetic monopoles, of which condensation leads to the dual superconductor [10].
For other nonperturbative QCD quantities such as spontaneous chiral-symmetry breaking and instantons, Abelian dominance is observed in lattice QCD [11]. However, it is non-trivial whether Abelian dominance holds for excitation phenomena in QCD or not, because this Abelianization scheme is conjectured to be valid only for low-energies and long distances. Actually, in high energies, perturbative QCD (pQCD) is applicable, and pQCD does not exhibit Abelian dominance, that is, each colored gluon plays an equal role in pQCD.
Then, in this paper, we study Abelian dominance for excited-state inter-quark potentials and gluonic excitation energies in the MA gauge in SU(3) color QCD at the quenched level. Here, the excited state potential and the gluonic excitation are interesting excitation phenomena in QCD, and they have been investigated in lattice QCD [12,13] and have been discussed in the context of stringy modes in the string/flux-tube picture of hadrons. In fact, apart from the linear confinement part, the excited state potential has 1/r part with a positive coefficient in long distances of r ≥ 2fm, and this 1/r behavior can be a signal of the stringy mode, although the stringy behavior is significantly suppressed in shorter distances than 2 fm [12]. The gluonic excitation energies are defined by the differences between the ground-state and excited-state potentials, and the lowest gluonic-excitation energy takes a larger value than about 1GeV both for static QQ and 3Q systems in lattice QCD [12,13]. This large gluonicexcitation energy can explain the absence of gluon degrees freedom in the quark model [13]. The organization of this paper is as follows. In Sec. II, we briefly review the Abelian projection in lattice QCD in the MA gauge. In Sec. III, we present our calculation procedure for the ground and excited state potentials in the static quark-antiquark (QQ) system. In Sec. IV, we show the lattice QCD result for the excited state potentials and the gluonic-excitation energies. Section V is devoted to summary and conclusion.

II. MAXIMALLY ABELIAN GAUGE FIXING AND ABELIAN PROJECTION
We perform SU(3) lattice QCD simulations at the quenched level with the standard plaquette action [14]. In lattice QCD, the gauge variable is described as the link variable U µ (s) ≡ e iagAµ(s) ∈ SU(3), with the gluon field A µ (s) ∈ su(3), QCD gauge coupling g and the lattice spacing a. In this paper, we use the lattice size of L 3 × L t = 32 4 at β ≡ 2N c /g 2 = 6.0. The lattice spac-ing a is determined to reproduce the string tension σ ≃ 0.89GeV/fm, and we then take a ≃ 0.1022fm. Using the pseudo-heat-bath algorithm, we generate 50 gauge configurations which are taken every 500 sweeps after a thermalization of 5000 sweeps. For the estimate of the gluonic excitation energies, it is found to be enough to use such number of configurations, although much more statistics would be desired for more accurate analysis.
We perform MA gauge fixing by maximizing the norm We numerically perform MA gauge fixing using the over-relaxation method for the maximization algorithm. As for the stopping criterion, we stop the maximization algorithm when the deviation ∆R MA /(4L 3 L t ) becomes smaller than 10 −5 . The converged value R MA /(4L 3 L t ) is 0.7322(3) for 50 configurations, where the value in parentheses denotes the standard deviation. Judging from the small deviation, our procedure seems to escape bad local minima, and we expect that the Gribov ambiguity does not affect our results.
Finally, we extract the Abelian part of the link variable from the link variable in the MA gauge, U MA µ (s) ∈ SU(3), by maximizing the norm so that the distance between u µ (s) and U MA µ (s) becomes the smallest in the SU(3) manifold. We thus find "microscopic Abelian dominance", i.e., R Abel = 0.9028(1) as a local indicator [15]. Of course, this does not necessarily mean "macroscopic Abelian dominance" in long distances, because, for instance, the quark confinement is judged by the exponential damping behavior in the Wilson loop [14], and any component can be dominant if its damping is small. Actually, we will see a counterexample in the second gluonic-excitation energy.
The Abelian projection is defined by the replacement of the SU(3) link variable U µ (s) by the Abelian part u µ (s) for each gauge configuration, i.e., for QCD operators.

III. LATTICE QCD CALCULATION OF EXCITED-STATE INTER-QUARK POTENTIALS
In this section, we briefly mention the lattice QCD formalism to obtain the excited-state QQ potentials and our numerical procedure.

A. Formalism
We explain the variational and diagonalization method to calculate the ground and excited state potentials, originally reported in Ref. [16], in the same manner with Ref. [13]. For the simple notation, the ground state is often regarded as the "0th excited state". For the static QQ system, we denote QCD HamiltonianĤ and the physical eigenstates |n (n = 0, 1, 2, . . . ). Since the static QQ system has no kinetic energy of quarks, one findŝ As sample states for the static QQ system, we prepare arbitrary given independent QQ states, |Φ k (k = 0, 1, 2, . . . ). In general, each state can be expressed with a linear combination of the eigenstates |n as Let us consider time evolution of the sample states |Φ k with the spatial locations of the quark and the antiquark being fixed. Since the Euclidean time evolution of the QQ state |Φ k (t) is expressed with the operator e −Ĥt , the overlap W jk Here, we define the matrix C and the diagonal matrix Λ T by C nk = c k n , Λ mn T = e −VnT δ mn , and rewrite the above relation as W T = C † Λ T C. Note that C is not a unitary matrix, and hence this relation does not mean the simple diagonalization by the unitary transformation.
In the lattice QCD simulations, the overlap W jk T can be calculated, since it is given by the Wilson loop sandwiched by the initial state |Φ k (0) and the final state |Φ j (T ) , i.e., Φ j |W QQ |Φ k .
We extract the potentials V 0 , V 1 , V 2 , . . . from the overlaps W T and W T +1 by using the formula e −V0 , e −V1 , e −V2 , . . . can be obtained as the eigenvalues of the matrix W −1 T W T +1 , i.e., the solutions of the secular In the lattice QCD calculation, we use the abovementioned method and extract low-lying excited-state potentials numerically for SU(3) QCD and Abelian projected QCD, respectively.

B. Numerical procedure
In the practical calculation, we prepare four sample states |Φ k for each gauge configuration by using the APE smearing method [17]. Originally, the smearing method was developed as a useful technique to reduce the higher excitation components in a gauge-invariant manner. We note that the gauge-invariant QQ system composed of the smeared link variables depends on the iteration number N smr of the smearing, and the N smr -times smeared states are generally independent each other when N smr is different [18]. Furthermore, the smeared states have only small components for highly excited states, and therefore they are appropriate as the sample states |Φ k for the study of low-lying excitations [13]. Here, it should be noted that the excitation modes obtained in this procedure are all parity-even, since the APE smearing does not mix parity-odd states in the static QQ system. For the parity-odd potentials, one needs to prepare parity-odd sample states, for example, as is done in Ref. [12].
In the actual calculation, we prepare 8, 16, 24, 32 times smeared states with the smearing parameter α = 2.3, which is the standard value for the measurement of the SU(3) inter-quark potential [18,19]. Due to the 8 times interval of the smearing, each sample state |Φ k can be regarded to be independent practically. In this analysis, we make an assumption that higher excitation components |n with n ≥ 4 in the sample states |Φ k are small enough and can be dropped off, and solve the eigenvalue problem of the 4×4 matrix W −1 T W T +1 . In this way, we obtain the effective masses V eff (r, t), V Abel eff (r, t) for the 0th, 1st, 2nd, 3rd excited state, respectively. The measurement is done for the onaxis 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 statistical error estimate, we adopt the jack-knife error estimate.
In calculating the potentials, higher excited states suffer larger systematic errors because the assumption becomes relatively more subtle. Hence, we do not make quantitative analysis of the third excited-state potentials, although the preparing four sample states definitely contributes to the significant error reduction for all the results. To reduce the systematic errors further, we pick effective masses V eff (r, t) at larger t as long as the error is small.

IV. LATTICE QCD RESULT
Now, we show the excited state potentials and their Abelian projection in the static QQ system. Figure 1 shows the effective mass plots V eff (r, t) for the SU(3) potentials V n (r) with n = 0, 1, 2. Owing to the variational and diagonization method, for the lowlying states, the t dependence is small and an approximate plateau is observed even in small t region, although higher excited state suffers larger statistical errors. In this paper, we do not show the meaningless data with too large errors in figures. Here, we pick effective masses at t = 2, 2, 1 as ground, first-excited, second-excited state SU(3) potentials, respectively. Figure 2 show the effective mass plots V Abel eff (r, t) for the Abelian potentials V Abel n (r) with n = 0, 1, 2. Compared with SU(3) case, there is slightly larger t dependence, and this might cause systematic errors of about 0.1GeV. On the other hand, the statistical errors are smaller, because Abelian projection enhances the expectation value of the Wilson loop. We pick effective masses at t = 4, 3, 2 as ground, first-excited, second-excited state Abelian potentials, respectively. Now, we show the ground, first-excited and secondexcited state potentials V n (n = 0, 1, 2) in the parityeven QQ system, and first and second gluonic-excitation energies ∆E n ≡ V n − V 0 (n = 1, 2) for both SU(3) and Abelian cases in Fig. 3.
As shown in Fig. 3 (a), all the SU(3) and Abelian potentials have approximately the same linear slope (the string tension) in long distances, which is also observed for the third-excited potential. This indicates universal Abelian dominance for the quark confinement force of the excited-state QQ potentials as well as the ground state potential.
The gluonic excitation energies are defined by the relative difference between the ground-state and excitedstate potentials, ∆E n (r) ≡ V n (r)−V 0 (r). Therefore their absolute values are physically meaningful, while all potentials have ambiguity of an overall constant shift. For the gluonic excitation energies, we expect cancellation of systematic errors on V n , especially for the Abelian part. From Fig. 3(b), the SU(3) gluonic-excitation energies ∆E n (r) seem to be roughly approximated with the Ansatz a n /r + E th n , and the best fit parameters are a 1 = 0.56(2), E th 1 = 0.97(1)GeV for ∆E 1 (r), and a 2 = 0.39(1), E th 2 = 1.977(7)GeV for ∆E 2 (r). On the other hand, the Abelian-projected gluonic-excitation energies ∆E Abel (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- 3. The short-distance 1/r-like behavior is significantly reduced in the Abelian-projected gluonic-excitation energies ∆E Abel n (r).
From the first two features, we conjecture that there is some threshold between 1GeV and 2GeV for the appli-cable excitation-energy region of Abelian dominance.
Here, Abelian dominance holds for nonperturbative properties such as confinement, spontaneous chiralsymmetry breaking, and instantons, but does not hold for perturbative QCD. Then, as an interesting conjecture, we expect that the first gluonic-excitation energy of about 1GeV in long distances is nonperturbative, since it exhibits Abelian dominance.
On the other hand, the higher gluonic-excitation energies, which do not show Abelian dominance, might have perturbative ingredients, which would obey ∆E Abel pQCD ≃ 1 4 ∆E pQCD , according to the gluon-number reduction through Abelianization. Finally, let us consider the significant reduction of the short-distance 1/r-like behavior in the Abelian-projected gluonic-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 ≃ 1 4 a n , which agrees with the gluon-number reduction through Abelianization, as is also seen in the perturbative one-gluon exchange. Then, as an interesting possibility, the shortdistance 1/r-like behavior might originate from perturbative QCD, instead of nonperturbative QCD. In any case, this finding would be a key to understand the shortdistance 1/r behavior in the excited SU(3) potentials for the static QQ system.

V. SUMMARY AND CONCLUSION
In this paper, we have presented the first study of the Abelian-projected gluonic-excitation energies in the static QQ system in SU(3) lattice QCD at the quenched level. Using smeared link variables on the lattice, we have examined four low-lying parity-even QQ potentials. We have found universal Abelian dominance for the quark confinement force also in the excited-state QQ potentials.
As a remarkable fact, we have found Abelian dominance in the first gluonic-excitation energy of about 1GeV in long distances in the maximally Abelian gauge, although it is an excitation phenomenon in QCD. In contrast, no Abelian dominance has been observed in the second and higher gluonic-excitation energies. From these two findings, we have conjectured that there is some threshold for the applicable excitation-energy region of Abelian dominance between 1GeV and 2GeV.
In addition, we have found that the short-distance 1/r behavior in gluonic excitation energies is significantly reduced by the Abelian projection. This finding might be a key to understand the short-distance 1/r behavior in the excited SU(3) potentials for the static QQ system.
As a future work, it is interesting to perform the similar study for the baryonic 3Q system. It is also meaningful to examine Abelian projection for parity-odd excitedstate QQ potentials, using asymmetric sample states as in Ref. [12].
It is also interesting to investigate the long-distance behavior of the gluonic excitation energies in Abelianprojected QCD and to compare with the stringy mode of the QQ flux tube. In SU(3) lattice QCD, the stringy modes grow up and appear in longer distances than 2 fm [12]. Since Abelian-projected QCD also exhibits quark confinement and the flux-tube formation [8], the stringy modes are expected also in the Abelian part in longer distances.