Monopoles and Confinement in Three-Dimensional Gauge Theory

The previous pages studied instantons in the two-dimensional $O(3)$ sigma model. Those instantons were finite-action maps between spheres, and their effects were invisible in ordinary perturbation theory. This page turns to a gauge-theory version of the same lesson: in three Euclidean dimensions, monopoles are instantons.

The result is one of the cleanest analytic mechanisms for confinement. A weakly coupled theory can have massive charged particles, an apparently harmless unbroken $U(1)$ gauge field, and yet an external probe with odd center charge cannot be isolated in the infrared. The reason is not a perturbative photon diagram. It is a gas of monopole instantons. Their long-range Coulomb fields form a plasma, the plasma Debye-screens the dual magnetic field, and the unscreened Wilson loop acquires an area law.

The model is not four-dimensional QCD. It is better than that for our purposes: it is a controlled laboratory in which one can see, in formulas, how topological defects disorder a gauge field and produce a mass gap. The moral will survive far beyond the model: confinement is naturally tied to magnetic disorder, not merely to large perturbative corrections.

Required background. Theta angle and instanton corrections introduces dilute instanton sums, fugacity, and the plasma language used here.

Helpful background. Confinement and screening in two-dimensional QED distinguishes an area law from screening and string breaking, while dimensional transmutation and mass gaps reviews nonperturbatively generated scales. The mechanism has three layers: broken $SU(2)$ supplies a nonsingular monopole core and a weakly coupled long-distance photon; compactness requires monopole and anti-monopole events; and their grand canonical gas maps to a sine-Gordon theory for a compact dual photon. The resulting mass gap is generated dynamically, and its Wilson-loop area law is not ordinary Yukawa screening.

The SU(2) model and the monopole instanton

Normalization. We work in three Euclidean dimensions with coordinates $x^\nu$, $\nu=1,2,3$, and

$$\epsilon_{123}=+1.$$

For a compact Abelian gauge field normalized so that the minimal Wilson loop is $\exp(i\oint A)$, write

$$S_{\rm Max}[A]={1\over4e^2}\int d^3x\,F_{\mu\nu}F_{\mu\nu} ={1\over2e^2}\int d^3x\,B_\mu B_\mu,$$

where

$$B_\mu={1\over2}\epsilon_{\mu\nu\rho}F_{\nu\rho}.$$

In three dimensions $e^2$ has dimensions of mass. A unit monopole has magnetic flux

$$\int_{S^2} B_\mu\,dS_\mu=2\pi.$$

Precise numerical factors below depend on this normalization and on the microscopic regularization of the monopole core. The qualitative statements — monopole fugacity, dual-photon mass, and Wilson-loop area law — do not.

For the broken $SU(2)$ ultraviolet completion, we rescale the microscopic fields so that the action has an overall factor $1/g_3^2$. Then

$$[g_3^2]=[v]=1, \qquad m_W\sim v.$$

We normalize the surviving Abelian field so that a fundamental probe has charge one. The massive adjoint $W$ bosons then have charges $\pm2$, a unit monopole has flux $2\pi$, and $e^2$ is proportional to $g_3^2$. These charge conventions will matter when we discuss which Wilson loops can be screened.

A convenient ultraviolet completion of compact QED in three dimensions is the three-dimensional Georgi–Glashow model: an $SU(2)$ gauge theory with an adjoint scalar. In Euclidean form one may write, schematically,

$$S_E={1\over g_3^2}\int d^3x\, \left[ {1\over4}(F_{\mu\nu}^a)^2 +{1\over2}(D_\mu\phi^a)^2 +V(\phi^a\phi^a) \right].$$

In the rescaled-field convention, one possible potential is

$$V(\phi^2) ={1\over4}{\lambda\over g_3^2}(\phi^2-v^2)^2,$$

so $\lambda/g_3^2$ is dimensionless. Other normalizations merely reshuffle the parameters in the monopole core.

The potential is chosen so that

$$\phi^a\phi^a\to v^2$$

far from the origin. A vacuum expectation value of an adjoint field breaks

$$SU(2)\longrightarrow U(1).$$

The two charged gauge bosons $W_\mu^\pm$ become massive, $m_W\sim v$, while the Abelian photon remains massless in perturbation theory. At distances much larger than the $W$-boson Compton wavelength, the perturbative effective theory looks like compact $U(1)$ gauge theory in three dimensions. The semiclassical regime is

$${v\over g_3^2}\gg1,$$

which makes the monopole core weakly coupled and its action large.

The nonperturbative object is the finite-action monopole configuration. At spatial infinity the scalar field defines a map

$$S^2_\infty\longrightarrow SU(2)/U(1)\simeq S^2,$$

and such maps are classified by

$$\pi_2(S^2)=\mathbb Z.$$

The charge-one configuration can be represented by the hedgehog ansatz

$$\phi^a(x)=v\,h(r){x^a\over r}, \qquad A_\mu^a(x)={1-k(r)\over r^2}\epsilon_{\mu ab}x^b, \qquad r=\sqrt{x_\mu x_\mu}.$$

The profile functions satisfy

$$h(0)=0, \qquad k(0)=1, \qquad h(\infty)=1, \qquad k(\infty)=0.$$

The core is nonsingular because the non-Abelian fields and the Higgs field smooth out the Dirac monopole singularity. Far outside the core, only the unbroken Abelian magnetic field remains,

$$B_\mu\sim {x_\mu\over 2r^3},$$

for the unit flux convention above.

A monopole instanton in three Euclidean dimensions. The adjoint scalar approaches a hedgehog configuration $\phi^a\propto x^a/r$ at large radius, while the long-distance Abelian magnetic field carries quantized flux through $S^2_\infty$.

Why call this an instanton? In three Euclidean dimensions it is localized in all three coordinates. If one later interprets one coordinate as imaginary time, the monopole is an event: it changes the magnetic flux of the two-dimensional spatial slice. Equivalently, the topological current

$$j_m^\mu={1\over2\pi}B^\mu$$

would be conserved in noncompact Maxwell theory,

$$\partial_\mu j_m^\mu=0,$$

but monopole events violate this conservation law by integer amounts:

$$\partial_\mu B_\mu=2\pi\sum_i q_i\delta^{(3)}(x-x_i), \qquad q_i\in\mathbb Z.$$

This is the gauge-theory analogue of an instanton changing a winding number.

The action of a single monopole has the form

$$S_0={4\pi v\over g_3^2}\,C\!\left({\lambda\over g_3^2}\right),$$

where $C$ is an order-one function determined by the scalar self-coupling and by convention choices. In the BPS limit $C=1$ in the common normalization. What matters most is the exponential suppression. Since a fugacity is a weight per unit three-volume, write it dimensionally as

$$\zeta=\mathcal A(v,g_3^2,\lambda)e^{-S_0}, \qquad [\mathcal A]=[\zeta]=3,$$

where the prefactor $\mathcal A$ contains zero-mode Jacobians, the one-loop determinant, and powers of the core scales. The exponential is universal at leading semiclassical order; the prefactor is not simply a dimensionless constant.

From isolated monopoles to a plasma

A single monopole is exponentially suppressed at weak coupling, but the vacuum volume is large. The path integral must sum over any number of monopoles and anti-monopoles. In the dilute approximation, the grand canonical partition function has the Coulomb-gas form

$$Z_{\rm mon} = \sum_{N_+,N_-\ge0}{\zeta^{N_++N_-}\over N_+!N_-!} \int \prod_{i=1}^{N_++N_-}d^3x_i\, \exp\left[-S_{\rm Coul}(\{x_i,q_i\})\right],$$

where $q_i=+1$ for a monopole and $q_i=-1$ for an anti-monopole. With the flux convention above,

$$S_{\rm Coul} ={2\pi^2\over e^2}\sum_{i\ne j}q_iq_jG(x_i-x_j), \qquad -\partial^2G(x)=\delta^{(3)}(x), \qquad G(x)={1\over4\pi |x|}.$$

Like charges repel and opposite charges attract. The important point is that this is a three-dimensional Coulomb gas. A dilute gas of positive and negative charges does not remain a collection of isolated dipoles. It forms a plasma.

The monopole sector is a grand canonical gas of magnetic charges $q_i=\pm1$, with dimension-three fugacity $\zeta=\mathcal A e^{-S_0}$. The equivalent dual description is a compact scalar $\sigma$ with vertex insertions $e^{iq_i\sigma(x_i)}$; summing over monopoles produces a cosine potential.

The analogy with an ordinary electric plasma is literal. A test magnetic charge polarizes the monopole gas around it. The screening cloud cuts off the long-range magnetic Coulomb field. In the original gauge-field language, this means the photon is no longer massless. In the dual language, it means the dual photon is pinned by a periodic potential.

After powers of $v$ and $g_3^2$ from the determinant are restored, the useful semiclassical slogan is

$$m^2\sim (\text{microscopic mass})^2 e^{-\operatorname{const}\,v/g_3^2}.$$

This is the right physical scaling: the mass gap is nonanalytic at weak coupling. No finite order of perturbation theory can produce it.

Dual photon and sine-Gordon action

The dual formulation turns the plasma statement into a one-line calculation. Start from compact Maxwell theory without monopoles. Introduce $B_\mu$ as an independent field and impose the Bianchi identity using a periodic Lagrange multiplier $\sigma\sim\sigma+2\pi$:

$$Z_0=\int \mathcal DB\,\mathcal D\sigma\, \exp\left[ -\int d^3x\,{B_\mu B_\mu\over2e^2} +{i\over2\pi}\int d^3x\,\sigma\,\partial_\mu B_\mu \right].$$

Integrating over $\sigma$ enforces $\partial_\mu B_\mu=0$, so locally $B_\mu$ is the curl of a gauge field. Integrating over $B_\mu$ instead gives the free dual photon action

$$S_{\rm dual,0}[\sigma] ={e^2\over8\pi^2}\int d^3x\,(\partial_\mu\sigma)^2.$$

A monopole of charge $q$ at $x_i$ changes the Bianchi identity by

$$\partial_\mu B_\mu(x)=2\pi q\delta^{(3)}(x-x_i).$$

In the dual path integral this insertion contributes the vertex operator

$$e^{iq\sigma(x_i)}.$$

Summing independently over monopoles and anti-monopoles therefore gives

$$\sum_{N_+,N_-\ge0} {1\over N_+!N_-!} \left(\zeta\int d^3x\,e^{i\sigma(x)}\right)^{N_+} \left(\zeta\int d^3x\,e^{-i\sigma(x)}\right)^{N_-} = \exp\left[2\zeta\int d^3x\,\cos\sigma(x)\right].$$

Thus the monopole plasma is equivalent to the three-dimensional sine-Gordon theory

$$\boxed{ S_{\rm dual}[\sigma] = \int d^3x\, \left[ {e^2\over8\pi^2}(\partial_\mu\sigma)^2 +2\zeta(1-\cos\sigma) \right]. }$$

The additive constant has been chosen so that the vacuum energy of the cosine term vanishes at $\sigma=0$.

Expanding near a minimum,

$$2\zeta(1-\cos\sigma)=\zeta\sigma^2+O(\sigma^4),$$

so the dual photon has mass

$$\boxed{ m_\gamma^2={8\pi^2\zeta\over e^2} }$$

in this normalization. Since $\zeta=\mathcal A e^{-S_0}$,

$$m_\gamma^2={8\pi^2\mathcal A\over e^2}e^{-S_0}, \qquad m_\gamma\sim e^{-S_0/2}\times \text{power of microscopic scales}.$$

Monopoles generate a periodic potential for the compact dual photon $\sigma$. Expanding around any minimum gives a mass $m_\gamma$, which is nonperturbative in the microscopic coupling.

This mass is sometimes called a photon mass, but the phrase needs care. Gauge invariance is not broken. There is no Higgs field giving the photon a perturbative Proca mass. The physical statement is that gauge-invariant magnetic correlations decay exponentially:

$$\langle B_\mu(x)B_\nu(0)\rangle_c\sim e^{-m_\gamma |x|}$$

up to powers of $|x|$. The mass gap is produced by magnetic instantons.

Wilson loops and the area law

The diagnostic for confinement is the Wilson loop

$$W(C)=\exp\left(i\oint_C A_\mu dx^\mu\right).$$

With the normalization fixed above, this is the charge-one loop associated with a fundamental $SU(2)$ probe.

For a large rectangular loop of spatial size $R$ and Euclidean time size $T_E$,

$$\langle W(R,T_E)\rangle\sim e^{-V(R)T_E}.$$

If the loop obeys an area law,

$$\langle W(C)\rangle\sim e^{-T_{\rm string}\operatorname{Area}(C)},$$

then

$$V(R)=T_{\rm string}R$$

for large $R$. This is confinement: separating a pair of external electric charges costs energy proportional to their separation.

In the dual-photon description, the Wilson loop imposes a discontinuity of the compact scalar across any surface $\Sigma$ whose boundary is $C$:

$$\partial\Sigma=C, \qquad \Delta\sigma=2\pi \quad\text{across }\Sigma.$$

The path integral is therefore dominated, for a large loop, by a sine-Gordon domain wall sitting on the minimal surface spanned by the loop. The wall tension is the string tension.

For the unscreened charge-one probe, a Wilson loop forces the dual photon to jump by $2\pi$ across a spanning surface $\Sigma$. The sine-Gordon action assigns a finite tension to this sheet, producing an area law $\langle W(C)\rangle\sim e^{-T_{\rm string}\operatorname{Area}(\Sigma)}$. Charge-two strings can eventually break on the heavy $W$ bosons.

For the charge-one probe in the normalization of this page, the exact numerical coefficient in $T_{\rm string}$ follows from the low-energy sine-Gordon action, while changes of charge convention rescale it. Its parametric form is

$$T_{\rm string}\sim e^2m_\gamma.$$

This is a useful formula to remember. The gauge coupling $e^2$ supplies the stiffness of the dual photon, and $m_\gamma$ supplies the inverse thickness of the wall. Their product has dimension two, as a string tension should in $2+1$ dimensions.

A useful normalization check is the exact sine-Gordon wall tension in this low-energy action. Write the kinetic term as

$${1\over2}K(\partial\sigma)^2, \qquad K={e^2\over4\pi^2},$$

and the potential as

$$V(\sigma)=2\zeta(1-\cos\sigma).$$

The static wall connecting neighboring vacua obeys the first-integral equation

$${1\over2}K(\sigma')^2=V(\sigma),$$

so its tension is

$$T_{\rm wall}=\int_0^{2\pi}d\sigma\,\sqrt{2K V(\sigma)} =8\sqrt{2K\zeta} =8K m_\gamma.$$

Thus, in this normalization,

$$T_{\rm wall}={2e^2\over\pi^2}m_\gamma.$$

The coefficient above is exact within this leading dilute-gas sine-Gordon action. Higher harmonics, determinant corrections, and a different normalization of the minimal electric probe change it. The relation $T\propto e^2m_\gamma$ is the robust semiclassical lesson.

Which electric charges remain confined?

The domain wall above describes the minimal charge-one Wilson loop. The ultraviolet completion also contains dynamical $W$ bosons of Abelian charge $2$. They can screen an external even charge: a charge-two string can break by producing a $W^+W^-$ pair once its energy is large enough. Parametrically, string breaking becomes favorable near

$$R_{\rm break}\sim {2m_W\over T_{\rm string}}.$$

An odd external charge cannot be completely screened by particles of charge $2$. It can only be shifted by an even amount and therefore retains the charge-one string. Thus the asymptotic area law in the full $SU(2)$ model applies to Wilson loops with nontrivial $\mathbb Z_2$ center charge. Even-charge loops may display an intermediate area law in the low-energy Abelian theory, but eventually cross over to string breaking and perimeter behavior.

This derivation also explains why simply giving a photon a mass is not the whole story. A massive noncompact photon would screen fields, but screening alone need not produce a Wilson-loop area law for electric charges. Compactness and the periodicity of the dual photon are essential: the Wilson loop is a topological defect for $\sigma$, and the defect must be filled by a domain wall.

Abelian confinement and the lesson for QCD

The three-dimensional model gives a controlled example of Abelian confinement. The low-energy gauge group is $U(1)$, and the asymptotically confined probes are the odd-charge, center-nontrivial external charges that cannot be screened by the massive $W$ bosons. The mechanism is magnetic disorder:

$$\text{monopole instanton gas} \quad\Longrightarrow\quad \text{dual-photon mass} \quad\Longrightarrow\quad \text{Wilson-loop area law}.$$

This is often called the Polyakov confinement mechanism.

In four Euclidean dimensions the same monopole solution is not localized in all spacetime directions. It traces out a worldline. A dilute collection of monopole loops would contribute schematically as

$$Z_{\rm loops}\sim \sum_{\text{closed loops }\Gamma} \exp[-M_{\rm mon}\operatorname{Length}(\Gamma)]\times(\text{fluctuation factors}).$$

In a regulated loop ensemble, the number of random closed loops can grow exponentially with length. This motivates a competition:

$$\text{action cost per unit length} \quad\text{versus}\quad \text{entropy per unit length}.$$

If the entropy gain exceeds the action cost, monopole loops proliferate. This is a heuristic magnetic-condensate picture behind the dual-superconductor analogy for confinement; unlike the three-dimensional dilute plasma, it is not a controlled derivation of confinement in ordinary four-dimensional Yang–Mills theory.

In three Euclidean dimensions a monopole is an instanton event. In four Euclidean dimensions it becomes a worldline. The displayed action–entropy competition is a heuristic loop-ensemble picture, not the controlled three-dimensional dilute-plasma derivation.

For non-Abelian QCD, the situation is subtler. There is no weakly coupled adjoint Higgs field in ordinary QCD that leaves a clean compact $U(1)$ theory with semiclassical monopoles. Perturbation theory explains asymptotic freedom, but not confinement by itself. Nonperturbatively, the expected infrared physics involves a mass gap and an area law for appropriate Wilson loops, but deriving this directly in continuum four-dimensional Yang–Mills theory remains much harder than in the three-dimensional Abelianized model.

The value of the three-dimensional model is conceptual and technical. It shows explicitly that topological defects can produce confinement in a weakly coupled calculation. It also teaches the right infrared variables: Wilson loops, compactness, magnetic disorder, dual fields, and defect condensation.

Example: Debye screening from the dual action

The sine-Gordon action immediately reproduces plasma screening. For small fluctuations around a vacuum, use

$$S_{\rm dual}\approx\int d^3x\, \left[ {e^2\over8\pi^2}(\partial\sigma)^2+ \zeta\sigma^2 \right].$$

The dual propagator is therefore

$$\langle \sigma(p)\sigma(-p)\rangle ={4\pi^2/e^2\over p^2+m_\gamma^2}, \qquad m_\gamma^2={8\pi^2\zeta\over e^2}.$$

A test magnetic charge couples as $e^{iq\sigma}$. At quadratic order, the interaction between two test magnetic charges is governed by the massive Green function

$$G_m(r)={e^{-m_\gamma r}\over4\pi r}.$$

Thus the monopole plasma converts the long-range Coulomb potential into a Yukawa potential. This is Debye screening. The same mass scale controls the thickness of the confining electric flux tube.

Summary

In three Euclidean dimensions, the monopole of a broken $SU(2)$ gauge theory is a finite-action instanton. At long distances, the theory looks like compact $U(1)$ gauge theory, and monopole and anti-monopole events must be summed in the path integral.

The dilute monopole gas is a Coulomb plasma. Dualizing the Abelian photon maps this plasma to a sine-Gordon theory for a compact scalar $\sigma$:

$$S_{\rm dual}=\int d^3x\, \left[ {e^2\over8\pi^2}(\partial\sigma)^2+2\zeta(1-\cos\sigma) \right].$$

The cosine term gives the dual photon a nonperturbative mass,

$$m_\gamma^2\propto {\zeta\over e^2} \propto {\mathcal A\over e^2}e^{-S_0},$$

and the Wilson loop becomes a domain-wall problem for the dual photon. The result is an area law,

$$\langle W(C)\rangle\sim e^{-T_{\rm string}\operatorname{Area}(C)}, \qquad T_{\rm string}\sim e^2m_\gamma.$$

This is confinement by monopole instantons. In the microscopic adjoint-Higgs model, charge-two $W$ bosons screen even external charges, while odd center charge remains confined. The mechanism is not a proof of QCD confinement, but it is one of the sharpest examples of how nonperturbative topology turns a seemingly weakly coupled gauge field into a theory with a mass gap and an area law for unscreened probes.

Common pitfalls

Treating a three-dimensional monopole as a propagating particle. In three Euclidean dimensions it is an instanton event. In four Euclidean dimensions the analogous object is a worldline, and the physics becomes a question about loop proliferation.

Calling the dual-photon mass a perturbative Proca mass. Gauge invariance is unbroken; the mass gap arises from compactness and monopole events.

Equating a mass gap with electric confinement. Debye screening explains the dual mass, but the Wilson-loop area law follows because the loop forces a topological discontinuity of the compact dual scalar.

Ignoring screening by the heavy $W$ bosons. In fundamental-charge units the $W$ bosons have charge $2$. They can break even-charge strings, while an odd charge retains a nontrivial $\mathbb Z_2$ center charge and remains confined.

Promoting the four-dimensional analogy to a derivation. The controlled calculation is Abelianized and three-dimensional. Ordinary four-dimensional QCD has related magnetic-disorder ideas, but not the same weak-coupling derivation in flat space without additional structure.

Exercises

Exercise 1: Dimensions of the coupling and fugacity

Show by dimensional analysis that in three dimensions $[e^2]=1$ and $[\zeta]=3$, and check that

$$m_\gamma^2={8\pi^2\zeta\over e^2}$$

has the correct dimension.

Solution

The Maxwell action is

$$S={1\over4e^2}\int d^3x\,F_{\mu\nu}F_{\mu\nu}.$$

With the Wilson loop $\exp(i\oint A)$ dimensionless, $[A]+[x]=0$, so $[A]=1$ in mass units. Then $[F]=[\partial A]=2$, and $[F^2]=4$. Since $[d^3x]=-3$, the factor $1/e^2$ must have dimension $-1$, hence

$$[e^2]=1.$$

The fugacity $\zeta$ appears in the action as

$$\int d^3x\,\zeta\cos\sigma,$$

and $\sigma$ is dimensionless because it is compact. Therefore

$$[\zeta]=3.$$

Thus

$$\left[{\zeta\over e^2}\right]=3-1=2,$$

which is the dimension of a mass squared. The expression for $m_\gamma^2$ is dimensionally consistent.

Exercise 2: Dual-photon mass

Starting from

$$S_{\rm dual}[\sigma]=\int d^3x\, \left[ {e^2\over8\pi^2}(\partial\sigma)^2+2\zeta(1-\cos\sigma) \right],$$

derive the small-fluctuation mass of $\sigma$ around $\sigma=0$.

Solution

Expand the cosine:

$$1-\cos\sigma={\sigma^2\over2}+O(\sigma^4).$$

Then

$$2\zeta(1-\cos\sigma)=\zeta\sigma^2+O(\sigma^4).$$

The quadratic action is

$$S^{(2)}=\int d^3x\, \left[ {e^2\over8\pi^2}(\partial\sigma)^2+\zeta\sigma^2 \right].$$

Write it in the canonical quadratic form

$$S^{(2)}={1\over2}\int d^3x\, K\left[(\partial\sigma)^2+m_\gamma^2\sigma^2\right],$$

with

$$K={e^2\over4\pi^2}.$$

Matching the mass term gives

$${1\over2}Km_\gamma^2=\zeta,$$

so

$$m_\gamma^2={2\zeta\over K}={8\pi^2\zeta\over e^2}.$$

Exercise 3: Parametric string tension

Use the sine-Gordon action to estimate the parametric dependence of the Wilson-loop string tension. Explain why

$$T_{\rm string}\sim e^2m_\gamma.$$

Solution

A Wilson loop forces the compact field $\sigma$ to jump by $2\pi$ across a surface. The dominant configuration for a large loop is a one-dimensional domain wall transverse to that surface.

The kinetic stiffness of the dual photon is

$$K={e^2\over4\pi^2},$$

because

$${e^2\over8\pi^2}(\partial\sigma)^2={1\over2}K(\partial\sigma)^2.$$

The wall thickness is set by the inverse dual-photon mass,

$$\ell_{\rm wall}\sim {1\over m_\gamma}.$$

Across the wall the field changes by an order-one amount, $\Delta\sigma=2\pi$, so the gradient is of order $m_\gamma$. The kinetic contribution per unit area is therefore

$$T_{\rm kin}\sim K(m_\gamma)^2\ell_{\rm wall} \sim K m_\gamma.$$

Since $K\sim e^2$, this gives

$$T_{\rm string}\sim e^2m_\gamma.$$

The potential contribution gives the same scaling by the equations of motion. Numerical factors depend on the normalization of the minimal Wilson loop and the precise sine-Gordon potential.

Exercise 4: Linear potential and string breaking

A large rectangular Wilson loop of spatial size $R$ and Euclidean time size $T_E$ satisfies

$$\langle W(R,T_E)\rangle\sim e^{-T_{\rm string}RT_E}.$$

Show that the corresponding static potential is linear. Then explain why a charge-two probe in the microscopic $SU(2)$ model does not remain confined at arbitrarily large separation, whereas a charge-one probe does.

Solution

The static potential is defined by the large-time behavior

$$\langle W(R,T_E)\rangle\sim e^{-V(R)T_E}$$

as $T_E\to\infty$. Comparing with the area-law form gives

$$V(R)T_E=T_{\rm string}RT_E,$$

so

$$V(R)=T_{\rm string}R.$$

Thus the energy required to separate external electric charges grows linearly with separation. This is confinement.

This last conclusion assumes that the external charge cannot be screened. In the normalization of this page, a dynamical $W$ boson has charge $2$. Pair creation can therefore screen a charge-two probe and break its string once

$$T_{\rm string}R\gtrsim 2m_W.$$

A charge-one probe cannot be neutralized by any number of charge-two $W$ bosons. It retains odd $\mathbb Z_2$ center charge and its string remains asymptotically stable.

Exercise 5: Exact sine-Gordon wall tension

For the sine-Gordon action

$$S=\int d^3x\,\left[{1\over2}K(\partial\sigma)^2+2\zeta(1-\cos\sigma)\right],$$

compute the tension of a one-dimensional wall satisfying $\sigma(-\infty)=0$ and $\sigma(+\infty)=2\pi$. Express the answer in terms of $K$ and $m_\gamma^2=2\zeta/K$.

Solution

For a static wall depending on one coordinate $z$, the tension is

$$T=\int dz\,\left[{1\over2}K(\sigma')^2+V(\sigma)\right], \qquad V(\sigma)=2\zeta(1-\cos\sigma).$$

The equation of motion has the first integral

$${1\over2}K(\sigma')^2-V(\sigma)=0,$$

because both $\sigma'$ and $V$ vanish at $z=\pm\infty$. Therefore

$$T=\int dz\,2V(\sigma) =\int_0^{2\pi}d\sigma\,{2V(\sigma)\over\sigma'}.$$

Using

$$\sigma'=\sqrt{{2V(\sigma)\over K}},$$

gives

$$T=\int_0^{2\pi}d\sigma\,\sqrt{2K V(\sigma)}.$$

Since

$$V(\sigma)=4\zeta\sin^2{\sigma\over2},$$

we find

$$T=\sqrt{8K\zeta}\int_0^{2\pi}d\sigma\,\sin{\sigma\over2} =4\sqrt{8K\zeta}=8\sqrt{2K\zeta}.$$

Using $m_\gamma^2=2\zeta/K$ gives

$$T=8K m_\gamma.$$

For $K=e^2/(4\pi^2)$ this becomes $T=2e^2m_\gamma/\pi^2$. Different Wilson-loop normalizations can multiply this wall tension by charge-dependent factors.

References

• T. Banks, R. Myerson, and J. Kogut, “Phase Transitions in Abelian Lattice Gauge Theories,” Nuclear Physics B 129 (1977), 493–510, doi:10.1016/0550-3213(77)90129-8.
• G. ’t Hooft, “Magnetic Monopoles in Unified Gauge Theories,” Nuclear Physics B 79 (1974), 276–284, doi:10.1016/0550-3213(74)90486-6.
• A. M. Polyakov, “Compact Gauge Fields and the Infrared Catastrophe,” Physics Letters B 59 (1975), 82–84, doi:10.1016/0370-2693(75)90162-8.
• A. M. Polyakov, “Quark Confinement and Topology of Gauge Groups,” Nuclear Physics B 120 (1977), 429–458, doi:10.1016/0550-3213(77)90086-4.

Further reading

• S. Coleman, Aspects of Symmetry: Selected Erice Lectures, Cambridge University Press, Cambridge, 1985, especially “The Magnetic Monopole Fifty Years Later.”
• A. M. Polyakov, Gauge Fields and Strings, Contemporary Concepts in Physics, vol. 3, Harwood Academic Publishers, Chur, 1987.
• M. Srednicki, Quantum Field Theory, Cambridge University Press, Cambridge, 2007.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 5th ed., Oxford University Press, Oxford, 2021.