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Monopoles and string tension in SU(2) QCD

Monopole and photon contributions to abelian Wilson loops are calculated using Monte-Carlo simulations of SU(2) QCD in the maximally abelian gauge. The string tension is well reproduced only by monopole contributions, whereas photons alone are responsible for the Coulomb coefficient of the abelian static potential.

In the case of QCD, there is a difficult problem. We have to find a color magnetic quantity in QCD. The 'tHooft idea of abelian projection of QCD [10] is very interesting in this respect. The abelian projection of QCD is to extract an abelian theory performing a partial gauge-fixing such that the maximal abelian torus group remains unbroken. After the abelian projection, SU(3) QCD can be regarded as a U(1) × U(1) abelian gauge theory with magnetic monopoles and electric charges. 't Hooft conjectured that the condensation of the abelian monopoles is the confinement mechanism in QCD [10].
There are, however, infinite ways of extracting such an abelian theory out of SU (3) QCD. It seems important to find a good gauge in which the conjecture is seen clearly to be realized. A gauge called maximally abelian (MA) gauge has been shown to be very interesting [11][12][13]. In the MA gauge, there are phenomena which may be called abelian dominance [12,14]. Especially the string tension in SU(2) QCD is well reproduced by Wilson loops composed of abelian link fields alone in the MA gauge. Moreover the monopole current k µ (s) can be defined similarly as in compact QED [5]. It is shown in the MA gauge that the abelian monopoles are dense and dynamical in the confinement phase, whereas they are dilute and static in the deconfinement phase [13].
Recently we have derived an effective U(1) monopole action in the MA gauge and in SU(2) QCD [15,16]. Entropy dominance over energy of the monopole loops, i.e., condensation of the monopole loops seems to occur in the confinement phase if extended monopoles [17] are considered [15,16]. After the abelian projection in the MA gauge, infrared behaviors of SU(2) QCD may be described by a compact-QED like U(1) theory with the running coupling constant instead of the bare one and with the monopole mass on a dual lattice.
If the monopoles alone are responsible for the confinement mechanism, the string tension which is a key quantity of confinement must be explained by monopole contributions alone. This is realized in compact QED [9]. The aim of this note is to show that the same thing happens also in SU(2) QCD by means of evaluating monopole and photon contributions to abelian Wilson loops. Preliminary results are reported by the present authors [15] and other group [18].

II. ABELIAN PROJECTION AND GAUGE INVARIANCE
We adopt the usual SU(2) Wilson action. The maximally abelian gauge is given [11] by performing a local gauge transformation V (s) such that After the gauge fixing is over, there still remains a U(1) symmetry. We can extract an abelian link gauge variable from the SU(2) ones as follows; where u(s,μ) is diagonal and A(s,μ) has off-diagonal components. It is easy to show that the above fields u(s,μ) and A(s,μ) behave under the residual U(1) transformation d(s) as an abelian gauge field and charged matters, respectively: Let us repeat that a U(1) invariant quantity written in terms of the abelian link variables u(s,μ) after an abelian projection can be rewritten in a SU(2) invariant (but complicated ) form using the original link variables U(s,μ) [14]. The gauge function V (s) which maximizes R is a functional of U(s,μ). First study the transformation property of V (s) under any SU (2) transformation W (s), fixing the U(1) ambiguity of V (s) in some way. Considering that a gauge-fixed quantity does not transform, we see where d(s) (∈ U(1)) ensures the form invariance of V W (s) and V (s) and is determined uniquely by V (s) and W (s). Using the definition (1) and the transformation property (5), for any SU(2) transformation W (s). Hence all U(1) invariant quantities composed of U(s,μ) (and u(s,μ)) are automatically SU(2) invariant after the abelian projection.
As an example, consider a U(1)-invariant 1 × 1 plaquette variable u P (s, µ, ν) composed of u(s,μ) alone. Using (1) and (2), we get where Under an arbitrary SU(2) transformation W (s), we see from (4), (5) and (6) B Namely, U ′ (s,μ) transforms like the corresponding original link field U(s,μ). The SU (2) invariance of u P (s, µ, ν) is seen also from this property. Moreover, it is seen how different the abelian plaquette variable is from the full one. u P (s, µ, ν) in a different abelian projection corresponds to a differently modified full plaquette variable composed of U ′ (s,μ).

III. MONOPOLE AND PHOTON CONTRIBUTIONS TO ABELIAN WILSON LOOPS
We show an abelian Wilson loop operator written in terms of u(s,μ) alone after the abelian projection is given by a product of monopole and photon contributions. Here we take into account only a simple Wilson loop, say, of size I × J. Then such an abelian Wilson loop operator is expressed as where J µ (s) is an external current taking ±1 along the Wilson loop and θ µ (s) is an angle variable defined from u(s,μ) as follows: Since J µ (s) is conserved, it is rewritten for such a simple Wilson loop in terms of an an- where ∂ ′ is a backward derivative on a lattice. M µν (s) takes ±1 on a surface with the Wilson loop boundary. Although we can choose any surface of such a type, we adopt a minimal surface here. We get where f µν (s) = ∂ µ θ ν (s) − ∂ ν θ µ (s) and ∂ µ is a forward derivative on a lattice. The gauge corresponds to a field strength and n µν (s) is an integer-valued plaquette variable denoting the Dirac string 1 . Since M µν (s) and n µν (s) are integers, the latter does not contribute to Eq. (13). Hence f µν (s) in Eq. (13) is replaced byf µν (s). Using a decomposition rule where a monopole current k µ (s) is defined as k µ (s) = (1/4π)ǫ µαβγ ∂ αfβγ (s) following DeGrand-Toussaint [5]. D(s) is the lattice Coulomb propagator. Sincef µν (s) corresponds to the field strength of the photon field, W 1 (W 2 ) is the photon (the monopole) contribution to the Wilson loop. To study the features of both contributions, we evaluate the expectation values W 1 and W 2 separately and compare them with those of W .

IV. SIMULATIONS AND EXTENDED MONOPOLES
The Monte-Carlo simulations were done on 24 4 lattice from β = 2.4 to β = 2.7. All measurements were done every 30 sweeps after a thermalization of 1500 sweeps. We took 50 configurations totally for measurements. The gauge-fixing criterion is the same as done in Ref. [19]. Using gauge-fixed configurations, we evaluated monopole currents and obtained the ensemble of monopole currents.
As shown in the previous note [15,16], type-2 extended monopole loops with b > b c ∼ Hence the effective lattice volume is reduced. We call the effective lattice as a renormalized lattice. We evaluate the averages of W using abelian link variables (called abelian) on the original lattice, of W 1 (photon part), and W 2 (monopole part) on each renormalized lattice, separately.

V. RESULTS
The results are shown in the following: 1. The monopole contributions to Wilson loops are obtained with relatively small errors.
Surprisingly enough, the Creutz ratios χ(I, J) of the monopole contributions having small errors are almost independent of the loop size for example as shown in Fig. 2.
This means that the monopole contributions are composed only of an area, a perimeter, and a constant terms without a Coulomb term.
2. Assuming the static potential is given by a linear + Coulomb + constant terms, we try to determine the potential using the least square fit. There are various ways, but we adopt a method similar to that [20] using the Creutz ratios. The assumption of the form of the static potential leads us to the Creutz ratio where χ 0 is the string tension and χ 1 corresponds to the Coulomb coefficient of the static potential. Using the fitted values of χ 0 and χ 1 , we can reproduce each static potential.
We plot their data in Fig. 3 (at β = 2.5) and in Fig. 4 (at β = 2.6). We find the monopole contributions are responsible for the linear-rising behaviors. When the 2 3 extended monopoles are used, we obtain almost the same results. The photon part contributes only to the short-ranged region. There seems to exist a small discrepancy between the abelian and the monopole + photon parts for R/a = 12, but finite-size effects are expected there. Moreover, the assumption of the form of the static potential looks too simple. Similar data are obtained for β = 2.4 and 2.7.
3. This is seen more clearly from the data of the string tensions which are determined from the static potentials. They are shown in Fig. 5. The full and the abelian string tensions have large errors but they are seen to be well reproduced by the monopoles alone for β ≤ 2.7 and the photon part has almost vanishing string tensions.
4. The string tensions from the monopoles and the photons of various sizes are plotted in Fig. 6. It is interesting to see that they are almost independent of the extendedness contrary to our preliminary data on a smaller lattice [21], although the monopole actions determined in [15,16] depend on the extendedness. Note that the extended monopoles of the type-2 are composed of the sum of the smallest monopoles. Hence even when the extended monopoles alone look to condense, the smallest monopoles also have some information of the condensation.

5.
We have derived also Coulomb coefficients from the static potentials as shown in Fig.   7. The monopole part has almost vanishing Coulomb coefficients which is in agreement with the constant behaviors of the Creutz rations of the monopole part as shown above.
The 1 3 photon part has large coefficients and they reproduce well the coefficients of the abelian static potentials.
6. We also measured the Coulomb coefficients of the photon parts on each renormalized lattice. The values depend on the extendedness. But the following fact is interesting.
The photon parts are evaluated on a renormalized lattice with b = na(β).
The scale parameter Λ determined is Λ ∼ 48Λ L which is quite near to the value Λ ∼ 42Λ L fixed from the monopole action [15].
In conclusion, our analyses done here strongly suggest that abelian monopoles are responsible for confinement in SU(2) QCD and condensation of the monopoles is the confinement mechanism if the abelian projection is done in the MA gauge. To extend our method to a finite-temperature system and also to SU(3) with or without dynamical quarks is very interesting. These studies are also in progress.
We wish to acknowledge Yoshimi Matsubara for useful discussions. This work is financially supported by JSPS Grant-in Aid for Scientific Research (c)(No.04640289).