Static Sources, Center Symmetry, and String Breaking in QCD

A static-source observable is meaningful only after its representation, renormalization, matter content, and limiting procedure are fixed. Rectangular Wilson loops extract zero-temperature static energies; Polyakov loops probe the free-energy response to a source at finite temperature; center symmetry makes either quantity an exact order parameter only in theories whose dynamical matter does not screen the corresponding center charge.

Required background. What Confinement Means in QCD with Dynamical Quarks supplies the operational confinement definitions; Wilson Lines and Loops supplies parallel transport, path ordering, and gauge transformation laws.

Helpful background. Breaking Higher-Form Symmetry and Diagnosing Phases supplies the one-form symmetry interpretation.

Wilson loops and the static-energy limit

For a closed contour $C$ and a representation $R$ of dimension $d_R$,

$$W_R(C)=\frac{1}{d_R}\operatorname{tr}_R \mathcal P\exp\!\left(ig\oint_C A_\mu^aT_R^a\,dx^\mu\right).$$

The trace closes the gauge parallel transporter, making the operator gauge invariant. For a Euclidean rectangle of spatial width $r$ and temporal extent $T$, transfer-matrix evolution gives

$$\langle W_R(r,T)\rangle_{\rm ren} =\sum_n |c_n(r)|^2e^{-E_n(r)T}.$$

Provided $c_0(r)\neq0$, the exact ground-state energy follows from either equivalent large-$T$ limit,

$$E_0(r)=-\lim_{T\to\infty}\partial_T\ln\langle W_R(r,T)\rangle_{\rm ren} =-\lim_{T\to\infty}\frac{1}{T}\ln\langle W_R(r,T)\rangle_{\rm ren}.$$

The second equality assumes that factors subexponential in $T$ have been separated. A linearly divergent static self-energy adds a representation- and scheme-dependent constant to $E_0$; energy differences, forces $dE_0/dr$, and properly matched thresholds remove that ambiguity.

If a large rectangle in pure gauge theory obeys

$$\ln\langle W_R(r,T)\rangle_{\rm ren} =-\sigma_k rT-\mu_R(2r+2T)-\Gamma_{\rm cusp}(C)+\cdots,$$

then taking $T\to\infty$ first gives

$$E_0(r)=\sigma_k r+2\mu_R+o(1).$$

The area coefficient $\sigma_k$ is the string tension in the $N_c$-ality sector $k$. The perimeter term contains static-line renormalization, and a contour with sharp corners has additional cusp renormalization. These terms cannot be read as a second string tension. Wilson’s lattice strong-coupling expansion supplies the canonical controlled area-law example Wilson 1974, pp. 2445–2452.

The limit order is part of the definition: project with $T\to\infty$ at fixed $r$, then study large $r$. Sending both sides of the rectangle to infinity along an unspecified path can combine ground-state projection with the infrared limit and obscure which energy is being measured.

Center charge and screening

In pure $SU(N_c)$ Yang–Mills theory, the electric $\mathbb Z_{N_c}$ one-form symmetry acts on a Wilson loop by

$$W_R(C)\longmapsto z^{k_R}W_R(C), \qquad z\in\mathbb Z_{N_c},$$

where $k_R$ is the representation’s $N_c$-ality. Adjoint gluons have $k=0$, so gluon screening can change $R$ without changing $k_R$. Consequently, an adjoint string can break into gluelumps even in pure gauge theory, whereas a nonzero-$k$ source cannot be screened by gluons alone. The generalized-symmetry statement and its line-operator hypotheses are given by Gaiotto et al. 2015, §§2.1–2.2.

Dynamical matter changes the group of genuine unscreened lines. Fundamental quarks carry $k=1$ and can terminate a fundamental Wilson line, so they explicitly break the pure-gauge center one-form symmetry. More generally, if the matter $N_c$-alities generate a subgroup $K\subset\mathbb Z_{N_c}$, only the quotient of center charges not screenable by $K$ can label stable strings. This representation test should precede any claim based on an area law.

String breaking as a level-mixing problem

With dynamical fundamental quarks, use at least two trial sectors: a string-like state $|S\rangle$ joining the static sources and a two-meson state $|M\bar M\rangle$. A minimal effective Hamiltonian is

$$H(r)= \begin{pmatrix} V_S(r) & x(r)\\ x(r)^* & 2M_{Q\bar q}^{\rm stat} \end{pmatrix}.$$

Its eigenvalues are

$$E_\pm(r)=\frac{V_S(r)+2M_{Q\bar q}^{\rm stat}}{2} \pm\sqrt{\frac{\bigl[V_S(r)-2M_{Q\bar q}^{\rm stat}\bigr]^2}{4}+|x(r)|^2}.$$

For $V_S(r)\ll2M_{Q\bar q}^{\rm stat}$, $E_-$ is string-like. When the unmixed levels cross, $x\neq0$ produces an avoided crossing; for $V_S(r)\gg2M_{Q\bar q}^{\rm stat}$, $E_-$ approaches the two-meson threshold. This is string breaking. The same additive static self-energy occurs in $V_S$ and the threshold, so the crossing condition is scheme independent.

A Wilson loop may have $|\langle0|W|M\bar M\rangle|\ll|\langle0|W|S\rangle|$. At finite $T$, its effective energy can therefore follow the excited string-like level long after the exact ground state has become screened. A variational basis containing both sectors repairs the interpretation; the numerical construction and continuum extrapolation belong to Wilson and Polyakov Loops, Static Energies, and Screening Diagnostics.

Polyakov loops at finite temperature

At inverse temperature $\beta$, the traced Polyakov loop is

$$P(\mathbf x)=\frac{1}{N_c}\operatorname{tr}\, \mathcal P\exp\!\left(ig\int_0^\beta A_0(\tau,\mathbf x)\,d\tau\right).$$

After multiplicative renormalization, its expectation value is conventionally expressed as

$$\langle P\rangle_{\rm ren}=e^{-\beta F_Q},$$

where $F_Q$ is the excess free energy of one static fundamental source, with a scheme-dependent additive constant. In pure gauge theory, $P$ is center charged. An unbroken center symmetry enforces $\langle P\rangle=0$ and hence an infinite isolated-source free energy, while spontaneous center breaking permits a nonzero expectation value. At finite volume the exact symmetric expectation remains zero unless a source or sector selection is used; the thermodynamic limit precedes removal of that source.

With dynamical fundamental quarks, a static source can bind a light antiquark, so $F_Q$ is finite and center symmetry is explicitly broken. The Polyakov loop remains a useful renormalized response observable but is no longer an exact order parameter. Moreover, it is a finite-temperature observable: it should not be substituted without argument for the zero-temperature Wilson-loop limit.

Diagnostic checklist

Before interpreting a line observable, record:

1. the gauge group’s global form and the probe representation;
2. all dynamical matter representations and the screening subgroup they generate;
3. whether the contour is spatial, temporal, or thermal;
4. the operator and cusp renormalization prescription;
5. the order of the $T$, $r$, volume, continuum, and thermodynamic limits;
6. the operator basis used to resolve string and screened states.

These entries decide whether an observed area law is an exact asymptotic diagnostic, a controlled-regime result, or an intermediate-distance feature.

Common pitfalls

Reading a finite-$T$ plateau as the ground state. A poor overlap can delay the screened state by an exponentially large Euclidean time. Check a correlation matrix with both flux-tube and two-hadron operators.

Ignoring perimeter and cusp terms. Bare loop values depend strongly on the regulator. Extract a force, a matched difference, or a consistently renormalized energy before assigning physical meaning.

Using the Polyakov loop as a universal order parameter. Its center charge is decisive only when the center symmetry is exact. With dynamical fundamental quarks it is a response observable, not a binary confinement criterion.

Exercise

Suppose $V_S(r)=\sigma r+C$, the two-meson threshold is $E_M$, and the mixing $x$ is a real constant. Find the minimum level splitting and show the large-$r$ limit of the lower eigenvalue.

Solution

The unmixed levels cross at $r_b=(E_M-C)/\sigma$. At that point,

$$E_+(r_b)-E_-(r_b)=2|x|.$$

For $\sigma r+C\gg E_M$,

$$E_-(r)=E_M-\frac{x^2}{\sigma r+C-E_M}+O(r^{-2}),$$

so the ground-state energy saturates at the screened threshold even though the upper eigenvalue remains string-like.

References

• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, no. 2 (2015): 172. DOI.
• Greensite, Jeff. An Introduction to the Confinement Problem. Lecture Notes in Physics 821. Berlin: Springer, 2011, chs. 4–6. DOI.
• Wilson, Kenneth G. “Confinement of Quarks.” Physical Review D 10 (1974): 2445–2459. DOI.