Symmetry improvement of 3PI effective actions for $\mathrm{O}(N)$ scalar field theory

N-particle irreducible effective actions (nPIEA) are a powerful tool for extracting nonperturbative and nonequilibrium physics from quantum field theories. Unfortunately, practical truncations of nPIEA can unphysically violate symmetries. Pilaftsis and Teresi (PT) addressed this by introducing a “symmetry improvement” scheme in the context of the 2PIEA for an O (2) scalar theory, ensuring that the Goldstone boson is massless in the broken symmetry phase [A. Pilaftsis and D. Teresi, Nucl. Phys. B874, 594 (2013)]. We extend this idea by introducing a symmetry improved 3PIEA for $\mathrm{O}(N)$ theories, for which the basic variables are the one-, two- and three-point correlation functions. This requires the imposition of a Ward identity involving the three-point function. We find that the method leads to an infinity of physically distinct schemes, though a field theoretic analogue of d’Alembert’s principle is used to single out a unique scheme. The standard equivalence hierarchy of nPIEA no longer holds with symmetry improvement, and we investigate the difference between the symmetry improved 3PIEA and 2PIEA. We present renormalized equations of motion and counterterms for two- and three-loop truncations of the effective action, though we leave their numerical solution to future work. We solve the Hartree-Fock approximation and find that our method achieves a middle ground between the unimproved 2PIEA and PT methods. The phase transition predicted by our method is weakly first order and the Goldstone theorem is satisfied, while the PT method correctly predicts a second-order phase transition. In contrast, the unimproved 2PIEA predicts a strong first-order transition with large violations of the Goldstone theorem. We also show that, in contrast to PT, the two-loop truncation of the symmetry improved 3PIEA does not predict the correct Higgs decay rate, although the three-loop truncation does, at least to leading order. These results suggest that symmetry improvement should not be applied to $n$PIEA truncated to $<n$ loops. We also show that symmetry improvement schemes are compatible with the Coleman-Mermin-Wagner theorem, giving a check on the consistency of the formalism.

Figure 1

From left to right: the Hartree-Fock self-energy diagram, an example daisy or ring diagram, an example super-daisy graph. The whole class of super-daisy diagrams is obtained from iterating insertions of Hartree-Fock graphs in all possible ways.

Figure 2

Two- and three-loop diagrams contributing to ${\mathrm{\Gamma }}_3^{(3)}[\phi ,\mathrm{\Delta },V]$. We label these ${\mathrm{\Phi }}_1$ through ${\mathrm{\Phi }}_5$ from left to right, respectively, and their explicit forms are given in (2.21) through (2.25). Solid circles represent the resummed vertices $V$ and the crossed circles represent the bare vertices $V_0$ and $W$. The dashed lines represent the resummed propagators $\mathrm{\Delta }$. Note that these diagrams are called “EIGHT,” “EGG,” “MERCEDES,” “HAIR,” and “BBALL,” respectively, in the nomenclature of [4, 5].

Figure 3

Equation of motion for the 3PI vertex function $V$ up to one-loop order (2.26). Note that the bubble graph (last term) is implicitly symmetrized over external momenta and $\mathrm{O}(N)$ indices.

Figure 4

Sunset self-energy graph.

Figure 5

Defining equation for the auxiliary vertex functions ${\overline{V}}^{\mu }$ and $V_N^{\mu }$, obtained by taking the corresponding equations of motion for $\overline{V}$ and $V_N$, dropping the triangle graphs, and making the replacements $\overline{V}\to {\overline{V}}^{\mu }$, $V_N\to V_N^{\mu }$, and ${\mathrm{\Delta }}_{G/H}\to {\mathrm{\Delta }}_{G/H}^{\mu }$. The filled triangle represents the auxiliary vertices and the lines represent the auxiliary propagators. Crossed circles are bare vertices as before.

Figure 6

Solution for the auxiliary vertex function in terms of a six-point kernel ${\mathcal{K}}_{abcdef}$ which is represented by the gray box (the indices run from top to bottom down the left side, then the right). The vertical black bar in the kernel equation of motion represents symmetrization of the external lines (with a factor of $1/3!$).

Figure 7

Feynman graph topologies appearing in the self-energy function ${\mathrm{\Sigma }}_G^0(p)$ in (4.33).

Figure 8

Expectation value of the scalar field $v=⟨\varphi ⟩$ as a function of temperature $T$ computed in the Hartree-Fock approximation using the unimproved 2PIEA (solid black), the Pilaftsis and Teresi symmetry improved 2PIEA (dash dotted blue) and our symmetry improved 3PIEA (solid green). In the symmetric phase (dashed black) all methods agree. The vertical grey line at $T\approx 131.5\mathrm{MeV}$ corresponds to the critical temperature which is the same in all methods.

Figure 9

The Higgs mass $m_H$ as a function of temperature $T$ computed in the Hartree-Fock approximation using the unimproved 2PIEA (solid black), the Pilaftsis and Teresi symmetry improved 2PIEA (dash dotted blue) and our symmetry improved 3PIEA (solid green). In the symmetric phase (dashed black) all methods agree. The vertical grey line at $T\approx 131.5\mathrm{MeV}$ corresponds to the critical temperature which is the same in all methods.

Figure 10

The Goldstone mass $m_G$ as a function of temperature $T$ computed in the Hartree-Fock approximation using the unimproved 2PIEA (solid black), the Pilaftsis and Teresi symmetry improved 2PIEA (dash dotted blue) and our symmetry improved 3PIEA (solid green). In the symmetric phase (dashed black) all methods agree. The vertical grey line at $T\approx 131.5\mathrm{MeV}$ corresponds to the critical temperature which is the same in all methods.

Figure 11

Tree level of decay of the Higgs ($H$) to two Goldstone bosons $a=1,\cdots ,N-1$.

Figure 12

The self-energy diagram from (4.22) which, due to the inconsistent truncation of the Ward identity, gives the incorrect absorptive part to the Higgs propagator in the two-loop truncated symmetry improved 3PIEA. $a=1,\cdots ,N-1$ labels Goldstone boson lines and $H$ labels the Higgs boson line.

Figure 13

One-loop contribution to the Higgs self-energy. The tadpole and sunset graphs are from the Higgs self-energy ${\mathrm{\Sigma }}_G$, while the four remaining terms come from vertex corrections via the Ward identity (3.20). The momentum incoming from the lower Goldstone leg is zero, and the crossed vertex represents a factor of $v$.

Figure 14

A contribution to the six-point kernel ${\mathcal{K}}_{abcdef}$ resulting from (left to right) a derangement, two stabilizers and another derangement. The dashed boxes surround the permutations (to aid visualization only). Only stabilizers lead to divergent loops.

Figure 15

Solution for the auxiliary vertex $V_{abc}^{\mu }$ in terms of the four-point kernel which sums all iterated bubble insertions.