---
author:
- |
    \
    Goethe-Universität Frankfurt am Main, Institut für Theoretische Physik,\
    Max-von-Laue-Stra<span>ß</span>e 1, D-60438 Frankfurt am Main, Germany\
    E-mail:
- |
    Owe Philipsen\
    Goethe-Universität Frankfurt am Main, Institut für Theoretische Physik,\
    Max-von-Laue-Stra<span>ß</span>e 1, D-60438 Frankfurt am Main, Germany\
    E-mail:
title: 'Definitions of a static SU(2) color triplet potential'
---

**Calculating the static potential in SU(2) gauge theory: basic principle**

The calculation of the singlet static potential is usually based on trial states\
$| \Phi^\textrm{\scriptsize sing} \rangle \equiv \bar{Q}(-r/2) U(-r/2;+r/2) Q(+r/2) | \Omega \rangle$, while for the triplet static potential typically\
$| \Phi^{\textrm{\scriptsize trip},a} \rangle \equiv \bar{Q}(-r/2) U(-r/2;s) \sigma^a U(s;+r/2) Q(+r/2) | \Omega \rangle$ is suggested or used (cf. e.g. @Brambilla:1999xf). Here $\pm r/2 \equiv (0,0,\pm r/2)$, $Q$ and $\bar{Q}$ are static quark/antiquark operators, $U$ are spatial parallel transporters (on a lattice products of links) and $\sigma^a$ denote Pauli matrices acting in color space. From the asymptotic behavior of the corresponding temporal correlation function the static potential $V_0^X(r)$, $X \in \{ \textrm{singlet}, \textrm{triplet}\}$ can be extracted.

**Lattice computations without gauge fixing**

On the lattice the singlet correlation function is proportional to Wilson loops,\
$\langle \Phi^\textrm{\scriptsize sing}(t_2) | \Phi^\textrm{\scriptsize sing}(t_1) \rangle \propto W(r,\Delta t)$, $\Delta t = t_2 - t_1$, from which the singlet potential can be determined (cf. the figure on page 1, blue dots). Since the triplet correlation function is not gauge invariant, one obtains $\langle \Phi^{\textrm{\scriptsize trip},a}(t_2) | \Phi^{\textrm{\scriptsize trip},a}(t_1) \rangle = 0$ and cannot determine a triplet potential.

**Lattice computations in temporal gauge**

Temporal gauge $A_0^g = 0$ in the continuum corresponds to temporal links $U_0^g(t,\mathbf{x}) = 1$ on a lattice. These links gauge transform according to $U_0(t,\mathbf{x}) \rightarrow U_0^g(t,\mathbf{x}) = g(t,\mathbf{x}) U_0(t,\mathbf{x}) g^\dagger(t+a,\mathbf{x})$, where $g(t,\mathbf{x}) \in \textrm{SU(2)}$. On a lattice with finite periodic temporal extension it is not possible to realize temporal gauge everywhere. There will be a slice of links, where $U_0 \neq 0$ (in the following wlog. $U_0^g(t=0,\mathbf{x}) \neq 1$, while $U_0^g(t=1 \ldots T-1,\mathbf{x}) = 1$; $T$ is the periodic temporal extension of the lattice). A possible choice for the corresponding gauge transformation $g(t,\mathbf{\mathbf{x}})$ is\
$g(t=2a,\mathbf{x}) = U_0(t=a,\mathbf{x})$,\
$g(t=3a,\mathbf{x}) = g(t=2a,\mathbf{x}) U_0(t=2a,\mathbf{x}) = U_0(t=a,\mathbf{x}) U_0(t=2a,\mathbf{x})$,\
$g(t=4a,\mathbf{x}) = g(t=3a,\mathbf{x}) U_0(t=3a,\mathbf{x}) = U_0(t=a,\mathbf{x}) U_0(t=2a,\mathbf{x}) U_0(t=3a,\mathbf{x})$, $\ldots$

*Non-perturbative computations (lattice), singlet potential:*

The trial states $| \Phi^\textrm{\scriptsize sing} \rangle$ are gauge invariant. Therefore, the result is identical to the result without gauge fixing (cf. the figure on page 1, blue dots).

Gauge transforming the temporal links to $U_0^g(t,\mathbf{x}) = 1$ and computing $$\begin{aligned}
\langle \Phi^\textrm{\scriptsize sing}(t_2) | \Phi^\textrm{\scriptsize sing}(t_1) \rangle \ \ = \ \ \Big\langle \textrm{Tr}\Big(U^g(t_1,-r/2;t_1,+r/2) U^g(t_2,+r/2;t_2,-r/2)\Big) \Big\rangle\end{aligned}$$ (here we assume $1 \leq t_1 < t_2 < T$, "case (A)") is equivalent to consider the manifestly gauge invariant observable $$\begin{aligned}
\nonumber & & \hspace{-0.7cm} \langle \Phi^\textrm{\scriptsize sing}(t_2) | \Phi^\textrm{\scriptsize sing}(t_1) \rangle \ \ = \ \ \Big\langle \textrm{Tr}\Big(U(t_1,-r/2;t_1,+r/2) \underbrace{g^\dagger(t_1,+r/2) g(t_2,+r/2)}_{U(t_1,+r/2;t_2,+r/2)} \\
%
 & & \hspace{0.675cm} U(t_2,+r/2;t_2,-r/2) \underbrace{g^\dagger(t_2,-r/2) g(t_1,-r/2)}_{U(t_2,-r/2;t_1,-r/2)}\Big) \Big\rangle \ \ = \ \ W(r,\Delta t)\end{aligned}$$ (cf. the figure on page 1). Similar considerations yield the same result for "case (B)", $0 = t_1 < t_2 < T$ or $1 \leq t_2 < t_1 < T$. This technique of transforming a non-gauge invariant observable into an equivalent manifestly gauge invariant observable will be helpful for interpreting the triplet potential.

A helpful theoretical tool to understand, which states are contained in a correlation function, is the transfer matrix formalism (cf. e.g. @Philipsen:2001ip [@Jahn:2004qr]). Without gauge fixing the transfer matrix is $\hat{T} = e^{-H a}$, $\hat{T} | \psi^{(n)} \rangle = \lambda^{(n)} | \psi^{(n)} \rangle$, $\lambda^{(n)} = e^{-E^{(n)} a}$ (lattice discretization errors neglected), where $E^{(n)}$ are the energies of gauge invariant states (e.g. the vacuum, glueballs). Similarly the transfer matrix in temporal gauge is $\hat{T}_0 = e^{-H_0 a}$, $\hat{T}_0 | \psi_0^{(n)} \rangle = \lambda_0^{(n)} | \psi_0^{(n)} \rangle$. In temporal gauge remaining gauge degrees of freedom are time-independent gauge transformations $g(\mathbf{x})$. One can show $[\hat{T}_0,g(\mathbf{x})] = 0$, i.e. eigenstates of $\hat{T}_0$ can be classified according to SU(2) color quantum numbers $(j(\mathbf{x}),m(\mathbf{x}))$ at each $\mathbf{x}$. $\lambda_0^{(n)} = e^{-E_0^{(n)} a}$, where $E_0^{(n)}$ are the energies of the gauge invariant states already mentioned as well as of additional non-gauge invariant states with $j(\mathbf{x}) \neq 0$. Such states can be interpreted as states containing static color charges (= static quarks)[^1]. One can derive $$\begin{aligned}
\langle \Phi^\textrm{\scriptsize sing}(t_2) | \Phi^\textrm{\scriptsize sing}(t_1) \rangle \ \ = \ \ \sum_k e^{-V_k^\textrm{\scriptsize sing}(r) \Delta t} \sum_m e^{-\mathcal{E}_m (T - \Delta t)} \sum_{\alpha,\beta} \Big|\langle k , \alpha \beta | \hat{U}_{\alpha \beta}(-r/2;+r/2) | m \rangle\Big|^2 ,\end{aligned}$$ where $\alpha \equiv m(-r/2) = \pm 1/2$ and $\beta \equiv m(+r/2) = \pm 1/2$ are color indices at $\pm r/2$. As expected this correlation function is suited to extract the common singlet potential $V_0^\textrm{\scriptsize sing}(r)$.

*Non-perturbative computations (lattice), triplet potential:*

Again one has to distinguish the two cases (A) and (B), which this time yield different results. When including the gauge fixing in the observable, one finds that $(s,t_1)$ and $(s,t_2)$, the spacetime positions of the "triplet generators" $\sigma^a$, are connected by an adjoint static propagator: $\textrm{Tr}(\sigma^a U(t_1,s;t_2,s) \sigma^b U(t_2,s;t_1,s))$. Within the transfer matrix formalism one can derive for case (A) $$\begin{aligned}
\nonumber & & \hspace{-0.7cm} \langle \Phi^{\textrm{\scriptsize trip},a}(t_2) | \Phi^{\textrm{\scriptsize trip},a}(t_1) \rangle \ \ = \\
%
 & & \hspace{0.675cm} \sum_{\alpha,\beta} \Big|\langle k , \alpha \beta , m(s)=a | \hat{U}_{\alpha \beta,a}(-r/2;s;+r/2) | m \rangle\Big|^2\end{aligned}$$ and for case (B) $$\begin{aligned}
\nonumber & & \hspace{-0.7cm} \langle \Phi^{\textrm{\scriptsize trip},a}(t_2) | \Phi^{\textrm{\scriptsize trip},a}(t_1) \rangle \ \ = \ \sum_k e^{-V^\textrm{\scriptsize sing}_k(r) \Delta t} \sum_m e^{-\mathcal{E}_m^{Q^\textrm{\scriptsize adj}} (T - \Delta t)} \\
%
 & & \hspace{0.675cm} \sum_{\alpha,\beta} \Big|\langle k , \alpha \beta | \hat{U}_{\alpha \beta,a}(-r/2;s;+r/2) | m , m(s)=a \rangle\Big|^2 .\end{aligned}$$ The conclusion is that one can either extract a three-quark potential (one quark at $+r/2$, one antiquark at $-r/2$, one adjoint quark at $s$) (case (A)) or the ordinary singlet potential (case (B)).

**Perturbative calculations in Lorenz gauge**

Most perturbative calculations of the static potential are carried out in Lorenz gauge $\partial_\mu A_\mu = 0$. The leading order result for trial states $| \Phi^\textrm{\scriptsize sing} \rangle$ is $V_0^\textrm{\scriptsize sing}(r) = -3 g^2 / 16 \pi r$, i.e. an attractive singlet potential. This result can be compared to the non-perturbative lattice result (in any gauge), since the trial state is gauge invariant. To perform a precise matching of lattice and perturbative static potentials, higher orders (NNLO or NNNLO) are required (cf. e.g. @Jansen:2011vv [@Bazavov:2012ka] for recent work on this topic), but nevertheless qualitative agreement is found (cf. the figure on page 1, blue dots and blue line). The leading order result for trial states $| \Phi^{\textrm{\scriptsize trip},a} \rangle$ is $V_0^{\textrm{\scriptsize trip}}(r) = +g^2 / 16 \pi r$, i.e. a repulsive triplet potential. Note, however, that in Lorenz gauge a transfer matrix does not exist, which renders a physical interpretation difficult. One can also calculate the gauge invariant triplet diagram obtained by using temporal gauge (cf. the figure on page 1, "triplet, case (A)"). Then one obtains\
$V_0^{Q \bar{Q} Q^\textrm{\scriptsize adj}}(r) = -9 g^2 / 16 \pi r$ (for $s = 0$), i.e. an attractive three-quark potential. Again qualitative agreement with the lattice result is found (cf. the figure on page 1, red dots and red line).

**Conclusions**

The singlet potential corresponds to a gauge invariant trial state\
$\bar{Q}(-r/2) U(-r/2;+r/2) Q(+r/2) | \Omega \rangle$. It is the same in any gauge and its interpretation as a static quark antiquark potential is clear.

The triplet potential corresponding to trial states\
$\bar{Q}(-r/2) U(-r/2;s) \sigma^a U(s;+r/2) Q(+r/2) | \Omega \rangle$ is different, when using different gauges: (1) without gauge fixing it cannot be calculated/computed; (2) in temporal gauge it corresponds to a three-quark potential and not to a potential between a quark and an antiquark in a color triplet state, i.e. the name "triplet potential" is misleading; (3) in Lorenz gauge a perturbative calculation yields a repulsive potential; since a transfer matrix does not exist, the physical interpretation is unclear.

**Acknowledgements**

We thank Felix Karbstein for discussions. M.W. acknowledges support by the Emmy Noether Programme of the DFG (German Research Foundation), grant WA 3000/1-1. This work was supported in part by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse.

<span>99</span>

N. Brambilla, A. Pineda, J. Soto and A. Vairo, Nucl. Phys. B <span>**566**</span>, 275 (2000) \[hep-ph/9907240\]. O. Philipsen, Nucl. Phys. B <span>**628**</span>, 167 (2002) \[hep-lat/0112047\]. O. Jahn and O. Philipsen, Phys. Rev. D <span>**70**</span>, 074504 (2004) \[hep-lat/0407042\]. K. Jansen <span>*et al.*</span> \[ETM Collaboration\], JHEP <span>**1201**</span>, 025 (2012) \[arXiv:1110.6859 \[hep-ph\]\]. A. Bazavov, N. Brambilla, X. Garcia i Tormo, P. Petreczky, J. Soto and A. Vairo, arXiv:1205.6155 \[hep-ph\].

[^1]: We use the following notation of energy eigenvalues $E_0^{(n)}$: (1) gauge invariant states, i.e. no static quarks: $\mathcal{E}_n$ ($j(\mathbf{x}) = 0$ for all $\mathbf{x}$); (2) a static quark/antiquark at $-r/2$ and at $+r/2$: $V^\textrm{\scriptsize sing}_n(r)$ ($j(-r/2) = j(+r/2) = 1/2$); (3) an adjoint static quark at $s$: $\mathcal{E}_n^{Q^\textrm{\scriptsize adj}}$ ($j(s) = 1$); (4) a static quark/antiquark at $-r/2$ and at $+r/2$, an adjoint static quark at $s$: $V^{Q \bar{Q} Q^\textrm{\scriptsize adj}}_n(r)$ ($j(-r/2) = j(+r/2) = 1/2$, $j(s) = 1$).

{'source': 'arXiv_out/1211'}