2PI and nPI Effective Actions

The two-particle-irreducible effective action treats the mean field and full propagator as independent variables. Its stationarity equation is a self-consistent Dyson equation whose self-energy comes from opening one line in a vacuum skeleton. This construction prevents diagrammatic double counting only when the skeleton functional, symmetry factors, counterterms, and any separately solved vertex equations are kept consistent. A finite-loop 2PI truncation can conserve global currents while still violating crossing or gauge Ward identities at a higher level.

Required background. The 1PI effective action supplies the first Legendre transform, while closure, symmetry, and branch selection supplies the truncation record. Helpful background. Diagrammatics and symmetry factors supplies the skeleton counting.

Double Legendre transform

For a real Euclidean scalar, introduce local and symmetric bilocal sources:

$$Z[J,K] =\int\mathcal D\varphi\, \exp\!\left[ -S_E[\varphi] +J_x\varphi_x +\frac12\varphi_xK_{xy}\varphi_y \right], \qquad W=\ln Z.$$

Repeated indices include integration. Define

$$\phi_x=\frac{\delta W}{\delta J_x}, \qquad \frac{\delta W}{\delta K_{xy}} =\frac12 \left( \phi_x\phi_y+G_{xy} \right).$$

The double Legendre transform is

$$\Gamma[\phi,G] =-W+J_x\phi_x +\frac12K_{xy} \left( \phi_x\phi_y+G_{xy} \right).$$

Eliminating the sources gives the exact representation

$$\begin{aligned} \Gamma[\phi,G] ={}& S_E[\phi] +\frac12\operatorname{Tr}\ln G^{-1} \\ &+\frac12\operatorname{Tr} \left[ D^{-1}(\phi)G-1 \right] +\Phi_2[\phi,G] +\Gamma_{\mathrm{ct}}[\phi,G], \end{aligned}$$

where

$$D^{-1}_{xy}(\phi) =\frac{\delta^2S_E[\phi]} {\delta\phi_x\delta\phi_y}.$$

$\Phi_2$ is the sum of vacuum diagrams that remain connected after any two internal lines are cut, built with full $G$ and the interaction vertices of the shifted action. Cornwall, Jackiw, and Tomboulis derive this form and its stationarity equations Cornwall, Jackiw, and Tomboulis 1974, §§ II–III.

Stationarity and the factor of two

At vanishing bilocal source,

$$\frac{\delta\Gamma}{\delta G_{xy}}=0.$$

Using

$$\frac{\delta}{\delta G} \frac12\operatorname{Tr}\ln G^{-1} =-\frac12G^{-1},$$

the stationary equation becomes

$$G^{-1} =D^{-1}+\Sigma, \qquad \Sigma_{xy} =2\frac{\delta(\Phi_2+\Gamma_{\mathrm{ct}})} {\delta G_{yx}}.$$

The factor of two follows from the symmetric bilocal source and the definition of $G$. Minkowski conventions distribute factors of $i$ differently; mixing a Euclidean $\Phi_2$ with a Minkowski self-energy rule changes signs.

Hartree plus sunset with checked skeleton factors

For

$$S_E[\varphi] =\int_x \left[ \frac12(\partial\varphi)^2 +\frac{m^2}{2}\varphi^2 +\frac{\lambda}{4!}\varphi^4 \right]$$

in the symmetric phase, the first two 2PI skeletons are

$$\Phi_2[G] =\frac{\lambda}{8}\int_xG(x,x)^2 -\frac{\lambda^2}{48} \int_{x,y}G(x,y)^4 +O(\lambda^3).$$

The double-bubble factor is $3/4!=1/8$: three Wick pairings close the four legs at one vertex. For the basketball, $4!$ contractions join two vertices, the expansion contributes $1/2!$, and the two $1/4!$ vertex factors give $4!/[2(4!)^2]=1/48$. The minus sign is the second connected cumulant in the Euclidean expansion.

Opening one full line gives

$$\Sigma(x,y) =\frac{\lambda}{2}G(x,x)\delta(x-y) -\frac{\lambda^2}{6}G(x,y)^3 +\Sigma_{\mathrm{ct}}(x,y) +O(\lambda^3).$$

The first term is the local Hartree self-energy; the second is the nonlocal sunset. Expanding the self-consistent $G$ in powers of $\lambda$ reproduces ordinary diagrams with the appropriate insertions. One must not add separate tadpole or sunset self-energies on top of this equation: they are already generated by differentiating $\Phi_2$.

What nPI adds

Legendre transforming sources for cubic, quartic, or higher composites produces 3PI, 4PI, and general nPI actions with self-consistent vertices. This can align propagator and vertex equations at a chosen loop order and reduce dependence on an external vertex ansatz.

It does not eliminate truncation:

• a finite-loop nPI action omits higher skeletons;
• different nPI levels become equivalent only through order-dependent statements that must be demonstrated;
• renormalization requires all counterterms allowed by the truncated variational structure;
• gauge identities can require vertex relations beyond those enforced by stationarity.

The shared closure and validation map locates the skeleton truncation. The functional-method validation comparison records stationarity, double-counting, symmetry, and loop-order tests.

Conserving does not mean fully symmetric

A $\Phi$-derivable approximation that respects a global continuous symmetry yields conservation laws for expectation values under suitable evolution. This is Baym’s conserving-approximation result Baym 1962, §§ II–IV.

At finite truncation, however, a vertex obtained by differentiating the stationary propagator need not equal a separately truncated variational vertex. Crossing symmetry and Ward identities for higher functions can be violated. In gauge theories, naive finite-loop 2PI truncations can produce gauge-parameter dependence at the same practical size as the claimed signal. “Conserving” must therefore name the conserved quantity and truncation order.

Common pitfalls

Missing the factor of two. It originates in the symmetric bilocal source. Omitting it halves every self-energy obtained from $\Phi_2$.

Adding a skeleton twice. Once a vacuum skeleton is in $\Phi_2$, opening its lines generates the associated self-energy. Adding that self-energy independently double counts it.

Equating stationarity with the global minimum. Nonlinear stationary equations can have several extrema. Branch and stability tests remain necessary.

Exercises

1. Differentiate the displayed $\Phi_2[G]$ and reproduce both self-energy coefficients.

Solution

For the double bubble,

$$2\frac{\delta}{\delta G(y,x)} \frac{\lambda}{8}\int_zG(z,z)^2 =\frac{\lambda}{2}G(x,x)\delta(x-y).$$

For the basketball, differentiating $G^4$ gives $4G^3$, so

$$2\left(-\frac{\lambda^2}{48}\right)4G(x,y)^3 =-\frac{\lambda^2}{6}G(x,y)^3.$$

The transpose in the derivative is immaterial for a symmetric scalar propagator.

2. Explain why solving a separate Schwinger–Dyson vertex equation can spoil the 2PI counting.

Solution

The stationary propagator already resums insertions generated by the selected vacuum skeletons. A separately truncated vertex equation can reinsert a diagram already implicit in those full lines or omit the functional derivative relation required by the 2PI action. A consistent hybrid must derive both equations from one higher nPI action or document and subtract the overlap diagram by diagram.

Continue

Bethe–Salpeter and Faddeev Bound-State Equations differentiates the self-energy to build a symmetry-consistent kernel. Functional-Method Validation and Error Control separates loop-order spread from stationary-solver error.

References

• Baym, Gordon. “Self-Consistent Approximations in Many-Body Systems.” Physical Review 127 (1962): 1391–1401. DOI.
• Cornwall, John M., R. Jackiw, and E. Tomboulis. “Effective Action for Composite Operators.” Physical Review D 10 (1974): 2428–2445. DOI.