The Schwinger Model, Screening, and Bosonization

The Schwinger model—one massless Dirac fermion coupled to a $U(1)$ gauge field in $1+1$ dimensions—is exactly reducible to a free massive scalar. With $D_\mu=\partial_\mu-ieA_\mu$ and the current normalization below, the gauge-invariant spectrum contains one neutral boson of mass

$$m_\gamma=\frac{e}{\sqrt{\pi}}.$$

The same mode makes the static potential between external probes saturate on the infinite line. This exact massless result does not transfer unchanged to a massive fermion, several flavors, compact spatial topology, a different electric-flux sector, or a statement about asymptotic dynamical charges.

Required background. The QED action, charges, and observables fixes gauge redundancy and probe couplings, while regulated Jacobians and measure variation supplies the axial anomaly. Helpful background. Quantum currents and improvements supplies renormalized current identities.

Massless Schwinger model and conventions

Use the site’s $1+1$-dimensional Lorentzian extension, choose $e>0$, and take

$$\mathcal L =\bar\psi\,i\gamma^\mu D_\mu\psi -\frac14F_{\mu\nu}F^{\mu\nu}, \qquad D_\mu=\partial_\mu-ieA_\mu .$$

Then the interaction is $+eA_\mu j^\mu$, with

$$j^\mu=\bar\psi\gamma^\mu\psi, \qquad j_5^\mu=\bar\psi\gamma^\mu\gamma_*\psi.$$

The coupling has mass dimension one, $[e]=1$, so the model can generate a mass proportional to $e$ without dimensional transmutation. The classical axial current is anomalous. In the stated orientation,

$$\partial_\mu j_5^\mu =-\frac{e}{2\pi} \epsilon^{\mu\nu}F_{\mu\nu} =-\frac{e}{\pi}E, \qquad E=F_{01}.$$

Changing the sign of $\epsilon^{01}$ or of the charge in $D_\mu$ changes intermediate signs but not $m_\gamma^2=e^2/\pi$. The gauge-invariant result follows from the anomaly and current algebra, as in Schwinger’s original solution Schwinger 1962, pp. 2425–2429.

Bosonization dictionary

For one massless Dirac fermion, normalize a real scalar $\phi$ so that

$$j^\mu =\frac{1}{\sqrt{\pi}} \epsilon^{\mu\nu}\partial_\nu\phi, \qquad j_5^\mu =-\frac{1}{\sqrt{\pi}}\partial^\mu\phi .$$

The relative sign follows from $j_5^\mu=-\epsilon^{\mu\nu}j_\nu$. Indeed, the vector dictionary gives $(j^0,j^1)=(\partial_x\phi,-\partial_t\phi)/\sqrt\pi$, so the axial components are $(j_5^0,j_5^1)=(-\partial_t\phi,\partial_x\phi)/\sqrt\pi$. Reversing the scalar field reverses both dictionaries; it cannot change their relative sign.

The current relation is an operator statement with a specified regulator and normal ordering Coleman 1975, pp. 2088–2092. It preserves the current two-point function and equal-time algebra; it does not identify the elementary fermion operator with $\phi$ itself. Fermion vertex operators also require Klein factors, normal ordering, and boundary-sector data.

The bosonized Lorentzian Lagrangian is

$$\mathcal L_{\mathrm{bos}} =\frac12\partial_\mu\phi\,\partial^\mu\phi +\frac{e}{\sqrt{\pi}} A_\mu\epsilon^{\mu\nu}\partial_\nu\phi -\frac14F_{\mu\nu}F^{\mu\nu}.$$

After integration by parts, the interaction is $(e/\sqrt{\pi})E\phi$ up to a boundary term. That boundary term is harmless for decaying fields on the infinite line but must be retained or replaced by explicit sector data on a circle or interval.

The transformation and its required checks appear in the chapter’s dual-variable map. The regime comparison separates this exact bosonization result from large-$N$ gap saddles.

Eliminating the electric field

In $1+1$ dimensions the Maxwell term is

$$-\frac14F_{\mu\nu}F^{\mu\nu} =\frac12E^2.$$

Gauss’s law allows a spatially constant integration constant $E_0$ labeling background flux. In the zero-flux sector on the infinite line,

$$E=-\frac{e}{\sqrt{\pi}}\phi.$$

Substituting into the first-order electric-field form gives

$$\mathcal L_{\mathrm{eff}} =\frac12\partial_\mu\phi\,\partial^\mu\phi -\frac12\frac{e^2}{\pi}\phi^2.$$

Hence

$$\left( \Box+\frac{e^2}{\pi} \right)\phi=0, \qquad m_\gamma^2=\frac{e^2}{\pi}.$$

The notation $m_\gamma$ is conventional: in $1+1$ dimensions there is no transverse photon polarization. The physical excitation is one neutral massive boson created by gauge-invariant operators such as $E$ or the vector current. There is no isolated charged fermion in the asymptotic spectrum.

Before eliminating $E$, variation of the scalar gives $\Box\phi=eE/\sqrt\pi$. Thus the same dictionary reproduces the anomaly:

$$\partial_\mu j_5^\mu =-\frac{1}{\sqrt{\pi}}\Box\phi =-\frac{e}{\pi}E,$$

with the sign fixed by the field equation and the declared chirality matrix. This also agrees with the real-time current identity in the course’s Schwinger-model calculation after translating its unit-charge gauge field as $A_{\mathrm{course}}=eA$. The axial-current identity, anomaly, mass, Green function, and signed force are independently checked in the chapter’s benchmark.

Static external probes

Introduce nondynamical probe charges $+q$ and $-q$ at $x=0$ and $x=R$ on the infinite line. Linear response is governed by the one-dimensional Yukawa Green function

$$G_m(x) =\frac{e^{-m\lvert x\rvert}}{2m}, \qquad (-\partial_x^2+m^2)G_m(x)=\delta(x).$$

Normalizing the coincident neutral pair to $V(0)=0$, the total screened-pair field energy is

$$V(R) =q^2\left[G_m(0)-G_m(R)\right] =\frac{q^2}{2m} \left(1-e^{-mR}\right), \qquad m=\frac{e}{\sqrt{\pi}}.$$

Thus

$$V(R\to\infty)=\frac{q^2}{2m}, \qquad F_R(R)=-\frac{\mathrm dV}{\mathrm dR} =-\frac{q^2}{2}e^{-mR}, \qquad \lvert F_R(R)\rvert=\frac{q^2}{2}e^{-mR}.$$

The force is attractive and decays exponentially, while the total field energy saturates at the two separated screened-charge self-energies. If those separated self-energies were subtracted instead, the interaction energy would be $-q^2G_m(R)$ and would approach zero from below. This calculation assumes a noncompact infinite line, the zero-background-flux vacuum, static nondynamical sources, and the massless one-flavor theory. It is not a statement that a gauge-charged local fermion appears as an asymptotic particle.

What changes away from the exact massless case

Fermion mass

After an anomalous axial rotation that places theta in the fermion mass phase, a fermion mass bosonizes to a normal-ordered cosine,

$$m_f\bar\psi\psi \longleftrightarrow -C(\mu)m_f :\!\cos(2\sqrt{\pi}\phi+\theta)\!:\,_{\mu},$$

where $C(\mu)$ depends on the normal-ordering convention. The effective theory is then interacting rather than a free massive scalar. Vacuum branches, probe string tension, and screening depend on $m_f/e$, $\theta$, and the probe charge $q/e$ Coleman, Jackiw, and Susskind 1975, pp. 267–275.

In an alternative convention, theta appears as a shift of the quadratic electric-flux term rather than in the cosine. The two descriptions agree only after the anomalous rotation, background flux, and operator phases are transformed together.

Several flavors

For $N_f$ massless flavors of equal charge, only the flavor-singlet boson couples to the gauge field and obtains a mass proportional to $e\sqrt{N_f/\pi}$. The remaining $N_f-1$ bosonic combinations are massless before additional interactions or fermion masses are introduced. Therefore “the Schwinger model is gapped” is true for the one-flavor massless model, not for every multiflavor generalization.

Boundaries and flux sectors

On a circle, Gauss’s law leaves a quantized or otherwise globally specified electric zero mode, depending on the gauge group and allowed charges. A theta term shifts the canonical flux center. In the standard massless one-flavor theory, an anomalous chiral—or equivalently bosonic—shift removes theta from the energy spectrum; theta-dependent vacuum energy requires a fermion mass or another obstruction to that shift. On an interval, edge charges and boundary conditions enter the bosonization dictionary. Eliminating $E$ with $E_0=0$ before specifying these data can discard physical sectors.

Common pitfalls

Calling mγ a Higgs mass. No scalar condensate breaks the gauge redundancy. The gauge-invariant mass arises from the anomaly and vacuum polarization, and the physical state is a neutral boson.

Using screening without naming the probe. The saturating potential is for nondynamical external charges in the stated massless, infinite-line, zero-flux problem. Dynamical spectrum and external-probe response are different observables.

Keeping the free scalar after adding mf. The mass term produces a cosine, so the massive Schwinger model is interacting. Its theta dependence and probe tension cannot be read from $m=e/\sqrt{\pi}$ alone.

Exercises

1. Starting from the bosonized Lagrangian, integrate by parts and eliminate $E$ in the zero-flux sector.

Solution

With decaying boundary data,

$$\frac{e}{\sqrt{\pi}} A_\mu\epsilon^{\mu\nu}\partial_\nu\phi =\frac{e}{\sqrt{\pi}}E\phi +\text{total derivative}.$$

The electric-field part is

$$\frac12E^2+\frac{e}{\sqrt{\pi}}E\phi =\frac12 \left( E+\frac{e}{\sqrt{\pi}}\phi \right)^2 -\frac12\frac{e^2}{\pi}\phi^2.$$

Eliminating the square gives the free scalar of mass $e/\sqrt{\pi}$.

2. Derive the static probe potential from the Yukawa Green function.

Solution

For sources $q_1=q$ at $0$ and $q_2=-q$ at $R$, the quadratic field energy is

$$\frac12\sum_{i,j}q_iq_jG_m(x_i-x_j) =q^2G_m(0)-q^2G_m(R).$$

Using $G_m(x)=e^{-m\lvert x\rvert}/(2m)$ gives

$$V(R)=\frac{q^2}{2m}(1-e^{-mR}).$$

The finite limit as $R\to\infty$ is screening; a linearly growing potential would instead signal an unscreened constant electric field between the probes.

3. Retain a constant electric-flux sector by writing $E=E_0-e\phi/\sqrt\pi$. In the noncompact zero-mode treatment, show that $E_0$ shifts the vacuum center but neither its minimum energy nor the excitation mass. Why must compact zero-mode identifications be treated separately?

Solution

Gauss’s law gives

$$\partial_x\left(E+\frac{e}{\sqrt\pi}\phi\right)=0, \qquad E=E_0-\frac{e}{\sqrt\pi}\phi.$$

The electric contribution to the Hamiltonian is therefore

$$\frac12E^2 =\frac12\left(E_0-\frac{e}{\sqrt\pi}\phi\right)^2.$$

Equivalently, the Lorentzian effective Lagrangian contains the negative of this potential. It is centered at

$$\phi_0=\frac{\sqrt\pi}{e}E_0.$$

Writing $\phi=\phi_0+\delta\phi$ gives

$$\mathcal L_{\mathrm{eff}} =\frac12(\partial\delta\phi)^2 -\frac12\frac{e^2}{\pi}(\delta\phi)^2.$$

Thus the sector shifts the vacuum center but not the minimum energy or the curvature $m_\gamma^2=e^2/\pi$. On a compact circle, however, the bosonic zero mode and electric flux can have global identifications. One must quantize those sectors and impose the fermion boundary conditions before deciding whether the shift is an allowed change of variables.

4. Distinguish the total pair energy $V(R)=q^2[G_m(0)-G_m(R)]$ from the self-energy-subtracted interaction energy, and compute the signed force.

Solution

$V(R)$ is normalized to vanish when the opposite charges coincide. At infinite separation it retains the two screened-charge self-energies and tends to $q^2G_m(0)=q^2/(2m)$. Subtracting that limiting value gives

$$V_{\mathrm{int}}(R)=-q^2G_m(R) =-\frac{q^2}{2m}e^{-mR},$$

which approaches zero from below. Both conventions give the same force on increasing separation,

$$F_R=-\frac{\mathrm dV}{\mathrm dR} =-\frac{q^2}{2}e^{-mR}.$$

The negative sign denotes attraction; its magnitude is the positive expression often quoted.

Continue

Compare the exact bosonized mass with the auxiliary-field masses on Gap Equations, Dimensional Transmutation, and Physical Mass.

References

• Coleman, Sidney. “Quantum Sine-Gordon Equation as the Massive Thirring Model.” Physical Review D 11 (1975): 2088–2097. DOI.
• Coleman, Sidney, R. Jackiw, and Leonard Susskind. “Charge Shielding and Quark Confinement in the Massive Schwinger Model.” Annals of Physics 93 (1975): 267–275. DOI.
• Schwinger, Julian. “Gauge Invariance and Mass. II.” Physical Review 128 (1962): 2425–2429. DOI.