Abelian dominance and adjoint sources in lattice QCD

TL;DR

This paper investigates how abelian dominance in lattice QCD extends from fundamental quarks to adjoint sources, providing theoretical derivations and numerical evidence supporting the reproduction of string tensions via abelian Wilson loops.

Contribution

It demonstrates that abelian dominance applies to higher representations in lattice QCD and clarifies differences between projection methods with supporting numerical data.

Findings

Fundamental and adjoint string tensions are reproduced by abelian Wilson loops.

Abelian dominance extends to higher representations beyond fundamental quarks.

Numerical evidence supports the extension of abelian dominance to adjoint sources.

Abstract

Certain properties of maximal abelian projection are derived which suggest that the fundamental and adjoint SU(2) string tensions are reproduced by singly and doubly charged abelian Wilson loops, respectively. Thus, abelian dominance, which has been observed for color sources (quarks) in the fundamental representation, can be extended to higher representations. Numerical evidence in support of this conjecture also for adjoint quarks is presented. The difference between maximal abelian and local projections is elucidated and the role of non-Wilson-like terms in the effective abelian action is discussed.

Tables

Table 1. TABLE I.: Summary of various Wilson loop operators considered in this work and the corresponding string tensions. Notation is as in the text: j 𝑗 j labels the SU(2) and n 𝑛 n the U(1) (abelian) representations; d 𝑑 d denotes diagonal rescaled SU(2) links and dSU(2) the expectations (not measurements) in the limit where the “diagonal approximation” is exact, corresponding to w 0 2 + w 3 2 superscript subscript 𝑤 0 2 superscript subscript 𝑤 3 2 w_{0}^{2}+w_{3}^{2} ≡ \equiv 1 at the operator level (equivalently, cos ⁡ ϕ ≡ 1 italic-ϕ 1 \cos\phi\equiv 1 ). These are to be contrasted with the results in MA and F12 projection abelian QCD (last two columns). Abelian dominance (AD) is tested with operators expressed in terms of abelian link variables. Thus, when possible, operators are presented in terms of abelian Wilson loops, with θ L ≡ subscript 𝜃 𝐿 absent \theta_{L}\equiv cos ⁡ ( ∑ i ∈ L θ i ) subscript 𝑖 𝐿 subscript 𝜃 𝑖 \cos(\sum_{i\in L}\theta_{i}) .

notation	interpretation	definition	dSU(2)	MA	F12
$W_{j=1/2}$	SU(2), fund.	$w_0$	$\sigma$	$\sigma$	$\sigma$
$W_{j=1/2}^d\equiv W_{n=1}$	U(1), n=1	$\cos {\theta }_L$	$\sigma$	$\approx \sigma \left(\text{AD}\right)$	$>\sigma$
$W_{j=1}$	SU(2), adj.	$\frac{4}{3}{w}_0^2-\frac{1}{3}$	$0$	$\frac{8}{3}\sigma$	$\frac{8}{3}\sigma$
$W_{j=1}^{\pm }$	$\text{adj.},m=\pm 1$	$\frac{2}{3}\left(w_0^2-w_3^2\right)$		$\approx \frac{8}{3}\sigma$	$\approx \frac{8}{3}\sigma$
$W_{j=1}^0$	$\text{adj.},m=0$	$\frac{2}{3}\left(w_0^2+w_3^2\right)-\frac{1}{3}$	$0$	$\approx \frac{8}{3}\sigma$	$\approx \frac{8}{3}\sigma$
$W_{j=1}^d$	adj., diag.	$\frac{2}{3}\cos \left(2{\theta }_L\right)+\frac{1}{3}$	$0$	$0$	$0$
$W_{n=2}$	$U\left(1\right)$, n=2	$\cos \left(2{\theta }_L\right)$		$\approx \frac{8}{3}\sigma \left(\text{AD}\right)$	$>\frac{8}{3}\sigma$

Equations

$$d=\mbox{diag}\Bigl{(}e^{i\alpha_{1}},e^{i\alpha_{2}},\dots e^{i\alpha_{N}}\Bigr{)}\ ,\quad\sum_{i}\alpha_{i}=0\ .$$

$$U=\left(\matrix{\cos\phi\;e^{i\theta}&\sin\phi\;e^{i\chi}\cr-\sin\phi\;e^{-i\chi}&\cos\phi\;e^{-i\theta}\cr}\right)=\left(\matrix{\cos\phi&\sin\phi\;e^{i\gamma}\cr-\sin\phi\;e^{-i\gamma}&\cos\phi\cr}\right)\left(\matrix{e^{i\theta}&0\cr 0&e^{-i\theta}\cr}\right)\ ,$$

$$\theta \equiv \arctan \left(\frac{u_3}{u_0}\right),$$

$${\theta }_{x,\mu }$$

$${\gamma }_{x,\mu }$$

$$\Omega \equiv \sum _{x,\mu }\mathrm{Tr}\left[{\mathrm{U}}_{\mu }^{†}(\mathrm{x}){\sigma }_3{\mathrm{U}}_{\mu }(\mathrm{x}){\sigma }_3\right]=\sum _{\mathrm{x},\mu }\cos (2{\varphi }_{\mathrm{x},\mu }).$$

$$S_P=\frac{1}{2}\text{mbox}Tr\left(U_1{U}_2{U}_3^{†}{U}_4^{†}\right)=S_{\theta }+S_{\chi }+S_{\theta \chi },$$

$$S_{\theta }$$

$$S_{\chi }$$

$$S_{\theta \chi }$$

$$\frac{1}{2{\pi }^2}{\int }_0^{\pi /2}d\varphi \cos \varphi \sin \varphi {\int }_{-\pi }^{\pi }d\theta {\int }_{-\pi }^{\pi }d\chi ,$$

$$w_0+i{w}_3$$

$$<W_{j=1/2}>$$

$$=$$

$$W_{n=1}=\mbox{Re}\left\{\prod_{i\in L}e^{i\theta_{i}}\right\}=\cos\Bigl{(}\sum_{i\in L}\theta_{i}\Bigr{)}\equiv\cos\bigl{(}\theta_{L}\bigr{)}\ ,$$

$$W^{d}_{j=1/2}={1\over 2}\mbox{Tr}\left\{\prod_{i\in L}\left(\matrix{e^{i\theta_{i}}&0\cr 0&e^{-i\theta_{i}}\cr}\right)\right\}=\cos\Bigl{(}\sum_{i\in L}\theta_{i}\Bigr{)}=W_{n=1}\ .$$

$${\sigma }_{j=1/2}={\sigma }_{n=1}^{MA}.$$

$${\sigma }_{j=1}^{abel,MA}$$

$$=$$

$$W_j\equiv \frac{1}{2j+1}\text{mbox}Tr\left\{\prod _{i\in L}{\mathcal{D}}_j[U_i]\right\},$$

$${\left({\mathcal{D}}_1[U]\right)}_{ab}$$

$$=$$

$$W_{j=1}=\frac{4(W_{j=1/2}{)}^2-1}{3}=\frac{4w_0^2-1}{3}.$$

$$W_{j=1}^0$$

$$W_{j=1}^{\pm }$$

$$<W_j(\mathcal{C})>$$

$$\approx$$

$$=$$

$$\Phi \to G\Phi G^{-1}.$$

$$({\cal D}_{1}[\Phi])\rightarrow\left(\matrix{\cos(2a)&\sin(2a)&0\cr-\sin(2a)&\cos(2a)&0\cr 0&0&1\cr}\right)({\cal D}_{1}[\Phi])\ .$$

$$j=\text{mbox}even$$

$$j=\text{mbox}odd$$

$$W_{j=1}^d=\frac{4(W_{j=1/2}^d{)}^2-1}{3}=\frac{4{\cos }^2({\theta }_L)-1}{3}=\frac{2}{3}\cos (2{\theta }_L)+\frac{1}{3}.$$

$$V(R,T{)}_{j=1}^{d,AP}=\log \left[\frac{W_{j=1}^{d,AP}(R,T-1)}{W_{j=1}^{d,AP}(R,T)}\right].$$

Full text

††preprint: NIKHEF 95-064

I Introduction

Understanding the confinement mechanism remains one of the most important topics in non-perturbative QCD. A minimalistic approach is to search for a subset of degrees of freedom that can lead to confinement. In that respect, considerable progress has been made in the case of compact quantum electrodynamics (CQED) where both analytical and numerical studies suggest that confinement can be understood to arise via monopole condensation in a dual superconductor picture [1, 2, 3]. The abelian projection of ’t Hooft [4] is an attempt to apply the same ideas to QCD by mapping SU(N) gauge theory onto its U(1)N-1 largest abelian (Cartan) subgroup. This mapping is effected by partial gauge fixing that leaves only a residual U(1)N-1 symmetry. In the original paper by ’t Hooft this is done by choosing a gauge so that an adjoint operator $X$ is diagonalized, $X\rightarrow VXV^{-1}$ = diag$[\exp(i\phi_{1}),\exp(i\phi_{2}),\dots\exp(i\phi_{N})]$, with the constraint $\sum_{i}\phi_{i}=0$. This gauge fixing is only partial since $V$ is determined modulo left multiplication by factors

[TABLE]

$\{d\}$ lives in the Cartan subgroup U(1)N-1. After the abelian projection, the diagonal components $-i[A^{\mu}]_{ii}$ of the (adjoint) gauge field transform under this residual gauge symmetry as $N$ abelian potentials (labelled “photons” by ’t Hooft), $-i[A^{\mu}]_{ii}$ $\rightarrow$ $-i[A^{\mu}]_{ii}$ $-\partial^{\mu}\alpha_{i}$, where $\sum_{i}[A^{\mu}]_{ii}=0$. The off-diagonal components $-i[A^{\mu}]_{ij}$, $i\neq j$, transform as $N(N-1)$ charged vector fields (labelled “gluons”) $-i[A^{\mu}]_{ij}$ $\rightarrow$ $-i\exp(i[\alpha_{i}-\alpha_{j}])[A^{\mu}]_{ij}$. Quark fields in the fundamental representation transform as singly charged with respect to the appropriate “photon”, $\psi_{i}$ $\rightarrow$ $\exp(i\alpha_{i})\psi_{i}$. The case of SU(2) is somehow special since the constraints imply that there is only one free phase $\alpha$ = $\alpha_{1}$ = $-\alpha_{2}$, with respect to which the off-diagonal “gluons” are doubly charged. Lattice QCD provides an ideal framework to carry out ’t Hooft’s programme [5]. On the lattice, an SU(2) link $U_{x,\mu}$ = $u_{0}+i\vec{\sigma}\cdot\vec{u}$, with $u_{0}^{2}+\vec{u}^{2}=1$, can be alternatively parametrized [7, 8] as $U^{11}_{x,\mu}$ = $\cos\phi_{x,\mu}\exp(i\theta_{x,\mu})$, $U^{12}_{x,\mu}$ = $\sin\phi_{x,\mu}\exp(i\chi_{x,\mu})$, $U^{21}_{x,\mu}$ = $-U^{12*}_{x,\mu}$, $U^{22}_{x,\mu}$ =$U^{11*}_{x,\mu}$, with $\phi\in[0,\pi/2]$, and $\chi,\theta\in(-\pi,\pi]$. The link $U_{x,\mu}$ can be decomposed as [6, 7]

[TABLE]

where the abelian phases $\theta$ are defined as

[TABLE]

and where $\cos\phi$ = $\sqrt{u_{1}^{2}+u_{2}^{2}}$. Under the residual U(1) symmetry $\theta$ and $\gamma=\chi+\theta$ transform like

[TABLE]

that is, like abelian gauge field and charge-two matter field (in the continuum), respectively. Notice that $\exp(i\theta)$ = $(u_{0}+iu_{3})/\sqrt{u_{0}^{2}+u_{3}^{2}}$ and therefore diag$[\exp(i\theta),\exp(-i\theta)]$ can be viewed as a rescaled diagonal SU(2) link. Besides photons, gluons and quarks the abelian projected theory also contains abelian monopole world lines in four dimensions ($d=4$) and monopole points (“instantons”) in $d=3$, which are identified as singularities in the gauge-fixing condition and are on the lattice extracted from the phases $\theta$ following the algorithm of DeGrand and Toussaint [3]. Having started from QCD, one is then in position to repeat the numerical studies that have been done in the case of compact QED. The abelian projection is gauge-dependent; the subset of degrees of freedom that can account for confinement may be different in different gauges [7, 8] and choosing a gauge becomes an art, although there is some evidence that this gauge dependence reflects short distance fluctuations and tends to go away in the infrared [10]. Most lattice studies of the abelian projection are performed using the so called maximal abelian [6] (MA) projection [9], corresponding in the continuum to $D^{\mu}_{0}A^{\mu}_{\pm}$ = $\partial^{\mu}A^{\mu}_{\pm}$ $-$ $ig[A^{\mu}_{0},A^{\mu}_{\pm}]$ = 0, where $A_{\pm}$ and $A_{0}$ are off diagonal and diagonal gluons, respectively. On the lattice, with SU(2) gauge group, this amounts to making the SU(2) links maximally-diagonal after the abelian projection. This projection has nice properties that seem to support ’t Hooft’s conjecture: the abelian monopole density in MA projection seems to scale [10, 11, 12, 13]; monopoles show a correlation with confinement in that they are dynamical in the confining phase and static above the finite temperature phase transition [5, 14]. A particularly interesting feature of MA projection is that abelian Wilson loops constructed from the diagonal photons $\theta$ reproduce the fundamental QCD string tension [14, 15, 16], a result named abelian dominance. This is an important result because it is the photons $\theta$ that contain the monopoles that are supposedly responsible for confinement. Further support to the dual superconductor picture is provided by measuring the string tension generated from monopoles alone and showing that it reproduces the full SU(2) string tension [15].

Abelian dominance may be interpreted as evidence that in maximal abelian (MA) projection the off-diagonal gluons $A_{\pm}$ can be neglected altogether, so that the Wilson loop is constructed simply from diagonal “photons”. However, this “diagonal approximation” [2, 17] is easily seen to be oversimplified by considering color sources (quarks) in SU(2) representations $j$ other than the fundamental ($j=1/2$). For $j=1$ (adjoint representation) the diagonal approximation forces the string tension to be zero not only in MA, but in any abelian projection, since the $m_{j}=0$ component of the adjoint source cannot couple to diagonal gluons and remains unconfined [18, 19]. This result is in disagreement with the fact that the adjoint potential shows for intermediate separations a Casimir scaling linear behavior [20, 21, 22, 23] and has been interpreted as evidence for the failure of the abelian projection mechanism [18, 19, 24]. Simulating QCD with adjoint color sources can therefore provide valuable insight into the dynamics of abelian projection.

In this work the maximal abelian projection is reviewed. It is shown that the dynamics of the theory in MA projection can be understood in terms of the following properties: (a) $<\cos\phi>$ is fixed to a value close to (but not exactly equal to) one (b) $\phi$ dependence factorizes (c) the effective action may be reasonably approximated as involving only $\theta$ fields. At $\beta=2.4$, where the calculations reported here are performed, $90\%$ of the SU(2) action is carried by a CQED type action $\tilde{\beta}\cos\theta_{P}$, where $\theta_{P}$ is an abelian plaquette constructed out of $\theta$ fields and $\tilde{\beta}=\beta(\cos\phi)^{4}$ is an effective coupling. There is also a ${\cal O}[10\%]$ correction that involves $\chi$-dependent terms and $\theta$-dependent terms coming from the expansion of the Fadeev-Popov determinant. Thus, to a first approximation the phases $\chi$ can be treated as random, and the phases $\phi$ as frozen. By integrating over $\chi$ and using $\phi$-factorization it is shown that abelian dominance for sources in the fundamental representation readily follows from this approximation. For adjoint sources the Wilson loop can be split [24] into contributions from the electrically (i.e., with respect to the residual U(1) symmetry) neutral ($m=0$) and electrically charged ($m=\pm 1$) components of the adjoint source. Is is shown that the “diagonal” approximation is not reproducing the MA projection results as can be most clearly seen by comparing their respective predictions for the electrically neutral adjoint Wilson loop $W^{0}_{j=1}$: the diagonal approximation predicts it should be $1/3$, while ignoring the $\chi$-dependent terms in the action predicts it should fluctuate around 0, albeit with a perimeter law falloff. This latter result simply means that to account for the confining behavior of the $m=0$ components of the adjoint source, fluctuations of the diagonal gluon field have to be taken into account as well. On the other hand, the above approximation (i.e. ignoring the small, $\chi$-dependent terms in the action) implies that the string tension from the electrically charged part of the adjoint Wilson loop $W^{\pm}_{j=1}$ must be given by doubly charged abelian Wilson loops $W_{n=2}$. It is numerically found that $W^{\pm}_{j=1}$ generates roughly the same string tension as the full adjoint Wilson loop $W_{j=1}$ = $W^{\pm}_{j=1}+W^{0}_{j=1}$, in accord with calculations in three dimensions [24]. This last result may then be used as phenomenological input, which, combined with the above analysis, suggests that doubly charged abelian Wilson loops should reproduce the full SU(2) string tension in MA projection. Numerical evidence in support of this result is presented. Thus, abelian dominance can be extended to quarks transforming in the adjoint representation. Doubly charged loops in CQED generate a string tension that Casimir-scales, and is therefore equal to $2^{2}=4$ times the one from singly charged loops [35]. Abelian dominance for adjoint sources on the other hand means that this ratio is in MA projection equal to the ratio of quadratic Casimirs $j(j+1)$ between adjoint ($j=1$) and fundamental $(j=1/2$) SU(2) representations, that is, $8/3$ $\approx$ 2.7 (in F12 projection is found to be less than 2). These results show that abelian QCD is certainly closer to CQED in MA projection than it is in local projections like F12 but still quite different, as is necessary due to the qualitative differences such as deconfinement phase transition, asymptotic freedom etc., between QCD (which abelian QCD in MA projection presumably reproduces) and CQED. In the context of the MA projection properties we mentioned above these non-Wilson type of $\theta$-dependent action terms originate in the expansion of the Fadeev-Popov determinant. Finally, the mechanism of charge screening (breaking of the adjoint flux tube at large quark-antiquark separations) is discussed in the framework of MA projection abelian QCD.

The structure of this article is as follows: in section II we discuss some properties of MA projection and then use them to derive abelian dominance for fundamental quarks (in section III) and for adjoint quarks (in section IV). A summary of our investigation is presented in section V.

II The Maximal Abelian projection

MA projection amounts to maximizing the quantity

[TABLE]

Following Ref. [7] the SU(2) plaquette action $S_{P}$ is decomposed in $S_{P}=S_{\theta}+S_{\chi}+S_{\theta\chi}$. $S_{\theta}$ contains $\theta$ but not $\chi$ fields, $S_{\chi}$ involves $\chi$ but not $\theta$ fields, and $S_{\theta\chi}$ involves interaction between $\theta$ and $\chi$ fields:

[TABLE]

where

[TABLE]

Here $\theta_{P}\equiv\theta_{1}+\theta_{2}-\theta_{3}-\theta_{4}$ and $\chi_{P}\equiv\chi_{1}-\chi_{2}+\chi_{3}-\chi_{4}$ are U(1)-gauge invariant plaquettes constructed from phases $\theta$ and $\chi$, respectively. The dots stand for 5 more terms involving U(1) invariant (c.f. Eq. (4)) combinations of two $\theta$ and two $\chi$ phases each. $S_{\theta}$ is proportional to the Wilson action for compact electrodynamics. The gauge fixing condition, Eq. (6), forces the $\phi$ phases to fluctuate around zero and thus one expects [7] that $S_{\chi}<S_{\theta\chi}<S_{\theta}$. Using 50 configurations on a $12^{4}$ lattice at $\beta=2.4$ we find that $<\cos\phi_{x,\mu}>$ is equal to 0.6665(1), 0.6711(1) and 0.9263(1) in unprojected, F12-projected, and MA-projected QCD, respectively. The value in the former two cases corresponds to $\phi$ being basically random, as can be seen using the group measure corresponding to link parametrization, Eq. (2)

[TABLE]

from which follows that $<\cos\phi>$ = $2/3$. In MA projection $<\cos\phi>$ is close to 1 and, accordingly, we find $S_{P}$ = 0.6298(1), $S_{\theta}$ = .5739(4) which means that they differ by $(S_{P}-S_{\theta})/S_{P}$ = 9%. Moreover we find that $<\prod_{P}\cos\phi><\cos\theta_{P}>$ =0.5586(5) which shows that the SU(2) plaquette to a good approximation factorizes in a product of $\cos\phi$ terms around the loop times an abelian plaquette constructed from the photons $\theta$. In F12 projection we find that $S_{\theta}$ = 0.0890(1) accounts for a small fraction of $S_{P}$ and, moreover, that factorization does not hold as $<\prod_{P}\cos\phi><\cos\theta_{P}>$ = 0.0565(1). In MA projection $\cos\phi$ behaves like a parameter (more appropriately, like a spin-glass coupling since the effective coupling is $\beta\cos\phi^{4}$); for example, $<\prod_{P}\cos\phi>$ = 0.7377(3), while $<\cos\phi>^{4}$ = 0.7362. This behavior is shown in Fig. 1 for the product $<\prod_{L}\cos\phi>$ around $T\times R$ rectangular Wilson loops. The solid curve corresponds to $0.9263^{{\cal P}}$ and the dotted curve to $(2/3)^{{\cal P}}$, where ${\cal P}=2(R+T)$ is the perimeter of the loop. It is clear that the product $<\prod_{L}\cos\phi>$ around Wilson loops falls with the perimeter of the loop. In F12 projection $<\cos\phi>$ is slightly larger than in the unprojected case because of the gauge condition (diagonalization of the plaquette in the (1,2) plane). For future reference we show the behavior of $<\prod_{L}\cos^{2}\phi>$ in Fig. 2. The solid curve corresponds to $0.9263^{2{\cal P}}$ and the dotted curve to $(1/2)^{{\cal P}}$, where $1/2$ is, from Eq. (11), the expectation value for randomly distributed $\cos^{2}\phi$.

III Abelian Dominance: fundamental sources

For an $T\times R$ Wilson loop in the fundamental representation, $W$ = $w_{0}+i\vec{\sigma}\cdot\vec{w}$, we can write a generalization of Eq. (8),

[TABLE]

The notation is as follows: $L$ stands for $R+T$, and $\theta_{L}$ is the abelian, singly charged, Wilson loop constructed from photons $\theta$. Similarly, $\chi_{L}$ is the charged matter field analog of Wilson loop and $\Omega_{n}[2m\theta,2(L-m)\chi]$ is an angle containing $2m$ $\theta$-phases and $2(L-m)$ $\chi$-phases. There can be an even-only number of either of them and therefore $m=1$ for the $1\times 1$ Wilson loop. The index $n$ denotes the specific combination of phases once $m$ is fixed. There are $\scriptstyle{2L\choose 2m}$ such combinations for given $L$ and $m$, so that the total number of terms is $2+\sum_{m=1}^{L-1}\scriptstyle{2L\choose 2m}$ = $2^{2L-1}$ (for example, in the case of the plaquette R=T=1, L=2, m=1, and the total number of terms is 8). Finally, $s_{n}$ is a sign factor for the given phase combination. Henceforth, $\cos\phi$ and $\sin\phi$ are treated as fixed parameters, that is, integration over $\phi$ using the MA projection condition, Eq. (6), amounts to fixing the magnitude of $\cos\phi$ with the residual effect of corrections to the action introduced by the Fadeev-Popov determinant [7]. Consider now the expectation value of the trace of the fundamental Wilson loop, $<W_{j=1/2}>$ = $<w_{0}>$. Ignoring the small $\chi$-dependent part in the action simply means that all terms in the operator involving angles containing $\chi$ variables vanish and therefore

[TABLE]

Here $\Delta S_{FP}$ is an effective action coming from the Fadeev-Popov determinant after $\phi$ integration. From Fig. 3 one sees that Eq. (14) is a good approximation in MA projection and, not surprisingly, very bad in F12 projection. Abelian dominance follows from Eq. (14): the string tensions from the SU(2) expectation value $<W_{j=1/2}>$ and the abelian expectation value $<\cos\theta_{L}>_{abel}$ (that is, the singly charged abelian Wilson loop) should be equal since the two differ by a perimeter term that disappears when forming Creutz ratios. Notice, for future reference, that $\cos\theta_{L}$ can be thought as constructed from either U(1) links $\exp(i\theta)$, Eq. (3), oriented along a path $L$

[TABLE]

or, from rescaled diagonal SU(2) links

[TABLE]

Abelian dominance is by now well established [14, 15, 16]. For completeness however, we test abelian dominance for the fundamental case in Fig. 4. The Creutz ratios from full SU(2) Wilson loops $W_{j=1/2}$ are compared to Creutz ratios from singly charged abelian Wilson loops $W^{AP}_{n=1}$ (equivalently, fundamental diagonal (rescaled) SU(2) Wilson loops $W^{d,AP}_{j=1/2}$) in maximal abelian (AP=MA) and field strength (AP=F12) projection. Calculations are performed at $\beta=2.4$, in four dimensions, where monopoles in MA projection seem to scale [11, 12] and fundamental abelian dominance has been already observed [26]. Results for SU(2) Creutz ratios in fundamental and adjoint representation come from 80 measurements on a $16^{4}$ lattice separated by 20 updates. An iterative smearing (with 20 iterations) is used for the space-like links, with parameter $c=2.5$ [27]. The abelian projection results come from two sets of measurements. The first comprises of 200 measurements on a $16^{4}$ lattice and the second of 350 measurements on a $12^{4}$ lattice. Measurements are separated by 60 updates. Error bars for the effective potential and Creutz ratios are obtained using the jackknife method. The MA condition, Eq. (6), is enforced iteratively using the overrelaxation algorithm of Ref. [28] with parameter $\omega=1.7$. Since the effective U(1) theory after projection fixing is not precisely known one cannot use multihit [29] variance reduction techniques. A straightforward modification of smearing was tried by creating smeared U(1) links after the abelian projection which did not prove successful. Without such techniques the range of $R,T$ values that results can be obtained becomes severely restricted, especially for the operators relevant for the adjoint source case (next section). No attempt is made to fit the Creutz ratios to some ansatz from which to extract the string tension. In MA projection abelian dominance is observed as the full SU(2) Creutz ratios are reproduced by the abelian ones at intermediate separations $R$

[TABLE]

Here, $n$ labels the U(1) representation (charge) in the abelian Wilson loop operator, $W_{n}=\cos(n\theta_{L})$. Fundamental quarks transform as singly charged with respect to the residual U(1), hence $n=1$ in this case. The fact that Creutz ratios show more noise in F12 projection than in MA projection reflects short distance fluctuations that plague local projections such as F12 but which are washed out in maximal abelian projection which is non-local [10]. F12 projection does not satisfy abelian dominance: the abelian string tension is higher than the SU(2) one [16] in commensurate with higher monopole density in local projections like F12 compared to MA projection [16, 10].

IV Abelian Dominance: adjoint sources

Consider now color sources (quarks) transforming in the adjoint representation of SU(2). At intermediate separations $R$ the potential between adjoint sources shows a linear behavior with a string tension that exhibits Casimir scaling, that is, scales like $j(j+1)$, where $j$ labels the SU(2) representation ($j=1/2$ for the fundamental, $j=1$ for the adjoint, etc.) Thus the adjoint string tension is roughly $8/3$ times the fundamental one. Eventually however, the adjoint potential is expected to saturate as it becomes energetically favorable for color-singlet bound states of adjoint source and glue (“gluelumps”) to screen the adjoint flux tube [2, 30, 31]. The Casimir scaling behavior is of perturbative (weak coupling) origin [34]: the one-gluon exchange contribution to the potential scales with the charge squared, hence the Casimir factor $j(j+1)$. There is substantial numerical evidence in favor of Casimir scaling in both three [22, 23] and four [20, 21] dimensions, for SU(N), as well as compact QED [35] (where $n$-ply charged Wilson loops develop a string tension that scales like $n^{2}$). Surprisingly enough, Casimir scaling seems to persist at large separations. In three dimensions it is still seen at $R$ values corresponding to almost twice the screening distance where gluelumps were expected to have saturated the adjoint potential [22]. Thus, at least for separations $R\leq 7a$ in $d=4$ at $\beta=2.4$ ($a\approx 0.12$ fm), where our measurements are made, adjoint abelian dominance requires abelian QCD in maximal abelian projection to generate a string tension

[TABLE]

where $8/3$ $\approx$ $2.7$ is the ratio of Casimirs between adjoint and fundamental SU(2) representations. Eq. (19) would be direct evidence for adjoint abelian dominance, whereas Eq. (21) would be indirect, since it involves comparison between abelian observables in MA projection and is based on the observed Casimir scaling of the adjoint string tension, Eq. (20), and the established abelian dominance for fundamental sources, Eq. (18). The critical issue is to decide from which abelian Wilson loops should $\sigma^{abel,MA}_{j=1}$ be extracted. We shall come to this question shortly. The adjoint string tension $\sigma_{j=1}$ is obtained from adjoint SU(2) Wilson loops, defined in a generic representation $j$ as [35]

[TABLE]

with ${\cal D}[U_{i}]$ the appropriate irreducible representation of the link $U_{i}$ $\in$ SU(2). The adjoint ($j=1$) representation of an SU(2) element $U$ with fundamental representation ${\cal D}_{1/2}[U]$ = $u_{0}+i\vec{\sigma}\cdot\vec{u}$, with $u_{0}^{2}+\vec{u}^{2}=1$, reads [36]

[TABLE]

where $a,b\in\{1,\dots N^{2}-1=3\}$. Using Eqs. (22,23) one has

[TABLE]

Writing $W$ = $w_{0}+i\vec{\sigma}\cdot\vec{w}$ for the fundamental Wilson loop, the trace of the adjoint Wilson loop can be split into the contribution of neutral ($m=0$) and charged ($m=\pm 1$) states [24]. From Eq. (23) we obtain

[TABLE]

This decomposition is projection-dependent as only the sum (26) $+$ (27) = (25) is SU(2)-invariant. Before applying the considerations of the previous section to the operators (26,27) let us discuss the abelian projection in the approximation where the Wilson loop of the full SU(2) theory is dominated by diagonal gauge field contributions [2, 17]

[TABLE]

Although Eq. (28) is sometimes referred to as the “abelian dominance” approximation [18], we prefer the term “diagonal approximation”. Notice that if there was a “perfectly abelian” projection, that is, one where the coset fields would be identically zero, $u_{2}^{2}+u_{1}^{2}\equiv 0$, then the above approximation would be exact. Abelian dominance would also be exact, since the full SU(2) Wilson loop operator would be identical to the one obtained from diagonal rescaled SU(2) links (equivalently, the singly charged abelian Wilson loop, Eq. (17)). Of course such a projection does not exist, but due to its very definition maximal abelian projection seems to be a close approximation to the diagonal SU(2) limit, since (in the language of section II) the diagonal approximation corresponds to $\cos\phi=1$. By just observing abelian dominance for fundamental quarks one cannot distinguish between the diagonal approximation and the approximations of section II since their difference affects only the perimeter term (c.f. Eq. (14)). The reason for distinguishing between them becomes clear when introducing color sources (quarks) in higher than the fundamental representation. In particular, consider a color source in the adjoint representation of SU(2)

[TABLE]

Under the residual abelian symmetry $G$ = diag$(\exp(ia),\exp(-ia))$ we write, using Eq. (23)

[TABLE]

Thus, the $m=\pm 1$ source components (corresponding to indices 1,2 in the above matrix) are electrically doubly charged (that is, they interact with the photons of the abelian projection) while the $m=0$ component (corresponding to index 3) is electrically neutral and does not interact with photons. In the diagonal approximation therefore, one expects from Eq. (26) that $<W^{0}_{j=1}>$ $\rightarrow$ $1/3$, corresponding to the $m=0$ contribution in Eq. (28). Even if $<W^{\pm}_{j=1}>$ does develop a string tension, the total adjoint Wilson loop should fluctuate around $1/3$ and, clearly, the adjoint string tension should be zero. Thus, approximation (28) imposes dramatic constraints to higher representation string tensions: for even representations $j$, such as the adjoint, the $m=0$ eigenvalue contributes a factor of $(2j+1)^{-1}$ in $<W_{j}({\cal C})>$ which does not allow for confining behavior. One therefore expects [2, 17]

[TABLE]

The adjoint string tension is of course expected to be zero at separations beyond the screening distance $R_{c}$. Still, though, the abelian projection mechanism should be able to generate the observed Casimir scaling adjoint string tension for $R<R_{c}$. Within the diagonal approximation, the abelian (i.e, constructed exclusively from $\theta$ links) operator that should generate the adjoint SU(2) string tension is the adjoint Wilson loop constructed from rescaled diagonal SU(2) links, diag$(\exp(i\theta),\exp(-i\theta))$ [18]

[TABLE]

However, the discussion above suggests that this operator can not lead to a confining potential. Indeed, in Ref. [18] it was shown that the SU(2) string tension in three dimensions from adjoint diagonal Wilson loops in MA projection is zero. This is not so much a test of the diagonal approximation, but rather, is a demonstration that Eq. (34) does follow from the diagonal approximation, whether the latter is justified or not. In order to make this point clear, we show the effective (time dependent) potential

[TABLE]

in four dimensions for maximal abelian projection (AP=MA) in Fig. 5 and field strength projection (AP=F12) in Fig. 6. The potential is consistent with zero in both projections; not only the string tension vanishes but the perimeter term as well. This verifies the expectation from Eqs. (28,26) that in a diagonal SU(2) theory the adjoint Wilson loop $W^{d,AP}_{j=1}$ $\rightarrow$ $1/3$, as is evident from Fig. 7.

Let us now use the techniques developed in the previous sections to address the adjoint Wilson loop. Using Eq. (13) we find

[TABLE]

The first issue is to understand why the neutral part of the Wilson loop does not fluctuate around $1/3$ as the diagonal approximation suggests but rather fluctuates around zero even in MA projection [24]. Consider therefore the operator $(w_{0}^{2}+w_{3}^{2})$. Performing the $\chi$ variable integration in the path integral by ignoring the small $\chi$-dependent part of the action allows only terms with no $\chi$ fields to survive. Since there is no $\theta$ dependence at this level we obtain

[TABLE]

Thusly, $W^{0}_{j=1}$ in MA projection fluctuates around [math] for any value of $\cos\phi$ not exactly one and not around $1/3$ as the diagonal approximation suggests. This shows that the adjoint diagonal Wilson loop is evidently unsuited for testing adjoint abelian dominance as is clearly seen in Fig. 7. Notice that in order to obtain the correct value around which $W^{0}_{j=1}$ fluctuates it is necessary to include $(\sin\phi)$ terms which, although small, are associated with large degeneracy factors. This is qualitatively different from the behavior of the fundamental Wilson loop, Eq. (14), and the behavior of the charged adjoint Wilson loop (Eq. (48) below) where only ($\cos\phi$) terms survive the free $\chi$ integration. Since the $\chi$-dependent part of the action is associated with ($\sin\phi$) factors, consistency requires that the correct behavior of $W^{0}_{j=1}$ cannot be predicted unless $\chi$-fluctuations are properly taken into account by using the $\chi$-dependent part of the action. Indeed, from Fig. 10 it appears that $W^{0}_{j=1}$ falls with an area law; Eq. (44) fails to describe $W^{0}_{j=1}$ except for very small loops. Physically, this amounts to the fact that $m=0$ source components interact either via the exchange of charged gluons (necessitating $<\sin\phi>\neq 0$) or via photons which see an $m=0$ source “dressed” with a virtual cloud of charged gluons. In our picture this latter mechanism amounts to bringing down from the exponential powers of the $\chi$ part of the action, which, coupled to $W^{0}_{j=1}$ will generate an assortment of perimeter and area terms involving $\theta_{L}$. The confining behavior $W^{0}_{j=1}$ depends on the interplay between these terms. We conclude that our picture cannot predict a simple abelian operator that generates the string tension corresponding to the neutral adjoint Wilson loop $W^{0}_{j=1}$.

The situation is different for the operator $(w_{0}^{2}-w_{3}^{2})$. Since there is no $n,n^{\prime}$ such that $\cos(\Omega_{n}+\Omega_{n}^{\prime})$ contains no $\chi$, the $\chi$ integration leaves simply

[TABLE]

This is the adjoint analog of Eq. (14) and is shown in Fig. 8 to be a very good approximation in MA projection (only). The $(\cos^{2}\phi)^{2L}$ factor is a perimeter term as can be seen in Fig. 2. The physical interpretation of this prediction is completely analogous to the fundamental case: the adjoint source components $m=\pm 1$ are electrically doubly charged; thus, abelian dominance in this case requires that the string tension from the doubly charged abelian Wilson loop $<\cos(2\theta_{L})>$ reproduces the corresponding SU(2) string tension. What is not clear yet is the relation between the SU(2) string tension from $W^{\pm}_{j=1}$ and the string tension from the gauge-invariant adjoint Wilson loop $W_{j=1}$. From Figs. 9,10 can be seen that $<w_{0}^{2}-w_{3}^{2}>$ $\approx$ $W_{j=1}$ $\approx$ $<2(w_{0}^{2}+w_{3}^{2})-1>$ in both F12 and MA projection. Thus, the full adjoint string tension and the string tension from the charged adjoint loop $W^{\pm}_{j=1}$ are roughly the same. For example, the (2,2) Creutz ratio from $<w_{0}^{2}-w_{3}^{2}>$ is $6\%$ higher than the gauge-invariant Creutz ratio, and the one from $<2(w_{0}^{2}+w_{3}^{2})-1>$ is $11\%$ lower. This is in agreement with the results in three dimensions [24]. We can only speculate about the origin of these results. A first argument is that it seems difficult to find string tensions $\sigma_{j=1}$, $\sigma^{\pm}_{j=1}$ and $\sigma^{0}_{j=1}$ such that $\exp(-\sigma_{j=1}\cdot A)$ = $a\,\exp(-\sigma^{\pm}_{j=1}\cdot A)$ $+$ $b\,\exp(-\sigma^{n}_{j=1}\cdot A)$ for a range of values for the area $A$ unless $\sigma_{j=1}$ = $\sigma^{\pm}_{j=1}$ = $\sigma^{0}_{j=1}$. Another argument is using the spectral decomposition: although gauge-dependent, $W^{\pm}_{j=1}$ is a correlator evolving in time with the gauge-invariant Hamiltonian. The gauge-dependence is manifest only in the ground state to vacuum matrix element of the operator, which does not affect the area law. If correct, this would explain why $\sigma_{j=1}$ = $\sigma^{\pm,AP}_{j=1}$ independent of the projection AP. At least what we can do is use the equality of the “full adjoint” and the “charged adjoint” string tensions as phenomenological input. Using $\sigma_{j=1}$ = $\sigma^{\pm}_{j=1}$ in combination with Fig. 8 and Eq. (48) suggests that the full adjoint SU(2) string tension is given by the doubly charged abelian Wilson loop, $W_{n=2}=\cos(2\theta_{L})$. In other words, the observed adjoint string tension arises from the interaction of diagonal gluons (photons) with the part of the adjoint source that is charged (doubly) with respect to the Cartan subgroup. If this is indeed verified by simulation it should be regarded as adjoint abelian dominance. Notice the difference from the fundamental representation. There, the Wilson loops from singly charged U(1) links and the one from diagonal rescaled SU(2) links are identical (c.f. Eqs. (16,17)). In the adjoint case, however, $<\cos(2\theta_{L})>$ and the diagonal adjoint Wilson loop, Eq. (36), are proportional up to a constant $1/3$. Thus, they can not simultaneously lead to an area law. This difference reflects the fact that fundamental abelian dominance is oblivious to the difference between MA projection and the diagonal approximation, whereas adjoint abelian dominance is not. If verified by simulations, adjoint abelian dominance is a non-trivial result for the following reason: it is well known that in compact electrodynamics Wilson loops constructed from U(1) links in representation $n$ (i.e., $n$-ply charged) lead to string tension scaling with the U(1) Casimir $n^{2}$ [1, 35]. Thus, for $\beta$ values such that the effective coupling is less than the critical CEQD coupling, $\beta(\cos\phi)^{4}<\beta_{c}=1$, and if we ignore the Fadeev-Popov determinant in the action, we expect

[TABLE]

while from Eq. (21) adjoint abelian dominance requires the relative strength to be $2.7$ rather than $4$. Thus, the non-Wilson like terms in the effective abelian action that come from the loop expansion of the Fadeev-Popov determinant, such as $\prod_{P}\cos^{2}\phi\cos(2\theta_{P})$ terms [7], need play a crucial role in bringing this number down from 4 to 2.7. Direct evidence for the presence of such terms in the effective action has been reported in Ref. [32] by using an inverse microcanonical demon method to generate the effective abelian action given an ensemble of MA-projected QCD configurations. In Fig. 12 we show results for the ratio

[TABLE]

of doubly to singly charged abelian Creutz ratios in maximal abelian (AP=MA) projection from 200 configurations on $16^{4}$ lattices and 350 configurations of $12^{4}$ lattices. For the $R$ values in Fig. 12 the ratio of adjoint to fundamental SU(2) Creutz ratios is 2.6027(3), 2.553(7) and 2.67(17) at $R$ = 1,2,3, respectively. At $R=1,2$ the singly charged Creutz ratios in MA projection underestimate the full SU(2) and therefore $\chi$ is seen in Fig. 12 to be lower than 4 but above 2.7. Certainly, more calculations are needed to verify that $\chi$ $\rightarrow$ 2.7 in MA projection. The corresponding ratios for in F12 projection are extremely noisy; only $R=1,2$ could be measured with values 2.111(1) and 1.93(50). Assuming a monotonic decrease with $R$ the results indicate that abelian QCD in F12 projection is rather different than compact QED. Wensley [37] has measured the string tension from doubly and singly charged abelian loops using monopoles alone. At $\beta$ = 2.5 he finds $\sigma_{j=1/2}$ = 0.033(1) for the unprojected theory, $\sigma^{MA}_{n=1}$ = 0.034(1) for the singly charged abelian loops and 0.093(1) for the doubly charged $\sigma^{MA}_{n=2}$, thusly yielding a ratio 2.8(3). The trend seen in Fig. 12 is in qualitative agreement with these results.

Direct numerical evidence for adjoint abelian dominance is shown in Fig. 11 where Creutz ratios from full SU(2) Wilson loops are compared to Creutz ratios from doubly charged abelian loops. As we have remarked earlier on, the lack of variance reduction and smearing techniques makes the calculation of doubly charged abelian Wilson loops very difficult. Thus, only results up to $R=3$ are shown. These results seem to support adjoint abelian dominance but certainly more calculations (demanding resources unavailable to us) are needed before definitive conclusions may be drawn. The F12 projection doubly charged Creutz ratios (not shown in the figure) behave similarly to the singly charged ones in Fig. 4: they are more difficult to measure but consistently higher than the unprojected SU(2) ones.

The last topic that should be discussed is the physics of charge screening in abelian projected QCD with adjoint sources. In the diagonal approximation picture screening comes about automatically since $W^{d}_{j=1}$ does not support a string tension, as we have already discussed. However, with respect to the conjecture that abelian dominance can be understood in the context of doubly charged abelian Wilson loops, there appears to be an important caveat: once the doubly charged abelain Wilson loops develop a string tension there seems to be no way that this abelian flux-tube can break, since the photons are neutral themselves and cannot couple with the charged sources to form electrically neutral states (the analog of “gluelumps”). Of course one may argue that the off-diagonal gluons $A_{\pm}$ which carry double charge can now form electrically neutral states with the charge-two sources thus screening the abelian potential obtained from doubly charged Wilson loops. It is more subtle though to address this question in the context of an effective abelian theory that results from the integration of the charged gluons ($\chi$). This theory (c.f. Eq. (48)) contains just photons (and monopoles). What happens is probably the following [38]: the loop expansion of the Fadeev-Popov determinant leads to terms in the effective $\theta$ action that contain doubly charged plaquettes of size $L\approx R_{c}$ where $R_{c}$ corresponds to the critical distance where gluelumps screen the adjoint sources in full SU(2). Due to the numerical difficulties the presence of these terms in the action has not been established in Ref. [32] but they are presumably there. When the Wilson loop is large enough to “accommodate” such terms one can use them (only a small number is necessary!) to “patch” the area of the Wilson loop [39]. As a result the Wilson loop does not fall like the area any more (as would be the case with only $1\times 1$ plaquettes in the action) and a crossover to perimeter law falloff occurs.

V summary

In abelian projected QCD monopoles, contained in the diagonal gluon (photon) fields $\theta$ are supposed to be responsible for confinement, by squeezing the abelian flux between electrically charged sources inside Abrikosov flux-tubes. Thus, a demonstration that abelian Wilson loops reproduce the full QCD string tension is crucial evidence in favor of this mechanism. The maximal abelian projection is the only projection in which this property of “abelian dominance” has been observed, and for quarks transforming in the fundamental representation. In this work a framework for describing the dynamics of abelian projected QCD in the maximal abelian projection is discussed. Specifically, it is shown that in maximal abelian QCD

1. (I)

Gluons are mostly diagonal: at $\beta=2.4$ $<\cos\phi>\simeq 0.93$. The magnitude of off-diagonal gluon fields $<\sin\phi>$ is small and does not fluctuate. Thus, $(\cos\phi)$ terms factorize in expectation values. 2. (II)

In commensurate with (I) the SU(2) action is by $90\%$ (at $\beta=2.4$) accounted for by a compact QED action with effective coupling $\beta(\cos\phi)^{4}$. Thus, the phases $\chi$ of off-diagonal gluons are basically random. 3. (III)

Using (I) and (II) to integrate over the (free) $\chi$ variables implies that

• (a)

fundamental SU(2) Wilson loop and singly charged abelian Wilson loop expectation values differ by a perimeter term, $(\cos\phi)^{2L}$. Thus, they generate the same string tension, which is the familiar abelian dominance for fundamental representation quarks.

• (b)

in analogy with [a], doubly charged abelian Wilson loops must generate the same string tension as charged (i.e., $m=\pm 1$) adjoint SU(2) Wilson loops. The perimeter term in this case is $(\cos^{2}\phi)^{2L}$.

• (c)

Adjoint Wilson loops constructed from rescaled SU(2) links should not be used for testing adjoint abelian dominance because they fluctuate around $1/3$, in any abelian projection, and therefore lead to a vanishing abelian potential.

• (d)

the neutral (i.e., $m=0$) adjoint Wilson loop fluctuates around 0, not around $1/3$ as ignoring the off-diagonal gluons would predict, albeit with a perimeter law falloff. However, $\chi$-integration cannot be carried out consistently unless the $\chi$-dependent part of the action is also included. Thus, the abelian operator that generates the string tension of the neutral adjoint Wilson loop cannot be predicted in a simple way. 4. (IV)

Numerically it is then found that charged Wilson loops have approximately the same, Casimir scaling, string tension as the gauge invariant full adjoint Wilson loop. Using this in conjunction with [b] suggests that doubly charged abelian Wilson loops generate the full SU(2) string tension. This is verified numerically. 5. (V)

The ratio of double to singly charged Wilson loops in MA projection is below 4:1 which would be the Casimir scaling limit if the effective U(1) theory after the abelian projection is simply compact QED. In MA projection the ratio tends towards the $2.7$ limit that fundamental and abelian dominance, together with Casimir scaling for the unprojected SU(2) theory, would require. In F12 projection the ratio lies below 2 which is an indirect indication that abelian QCD in F12 projection is less close to CQED than abelian QCD in MA projection.

Before definitive conclusions can be drawn, two points need to be clarified. Firstly, the relation between $W^{\pm}_{j=1}$ and $W_{j=1}$ must be understood. Although we have presented an argument suggesting they should generate the same Creutz ratios, we feel that before this issue is completely resolved it is not clear whether our results confirm Eq. (19) or only Eq. (48). Secondly, our results should certainly be confirmed at larger interquark separations, $5a\leq R\leq 7a$ at this $\beta=2.4$ value, where an non-vanishing, roughly Casimir scaling, adjoint string tension is observed. Assuming that they survive this test what is their implication for the abelian projection picture? Clearly, if one sticks to the notion that the off-diagonal gluons must play no role in the abelian projection dynamics then the abelian projection breaks down, as has been advocated in Refs. [18, 19, 24]. The interpretation emerging from this study is somehow different: off-diagonal gluons cannot be neglected, but still, abelian degrees of freedom, i.e., electrically charged source components and diagonal gluons (photons), “dominate”, in the sense that they generate the full SU(2) string tension, in both fundamental and adjoint representations. In that respect the MA projection appears to be uniquely successful: it effects what all these studies have set out to accomplish, namely, to identify a subset of degrees of freedom in terms of which the nonperturbative aspects of QCD can be understood. The success of the MA projection in reproducing the Casimir scaling behavior of the adjoint string tension provides further support to the idea that confinement arises due to monopoles contained in the abelian photon fields.

Acknowledgements.

The author wishes to thank Ken Yee, Maxim Chernodub, Jan Smit, Richard Woloshyn, and, especially, Mikhail Polikarpov for fruitful discussions and suggestions. This work is supported by Human Capital and Mobility Fellowship ERBCHBICT941430.