1. Introduction
There are many systems of physical interest that are strongly coupled and must be described with non-perturbative methods. Schwinger-Dyson (SD) equations are often used, but one problem with this approach is that the hierarchy of coupled SD equations needs to be truncated, and different truncations have been proposed. The
n-particle-irreducible effective action is an alternative non-perturbative method. The action is written as a functional of dressed vertex functions, which are calculated self consistently by applying the variational principle [
1,
2]. A fundamental advantage of
nPI is that the method provides a systematic expansion with the truncation occurring at the level of the action. Gauge invariance may be violated by the truncation [
3,
4], and various proposals to minimize gauge dependence have been discussed in [
5,
6,
7,
8,
9]. We are primarily interested in the renormalization of
nPI theories. The 2PI effective theory can be renormalized using a counterterm approach [
10,
11,
12,
13,
14], but the method requires several sets of vertex counterterms and cannot be extended to the 4PI theory. It is known that higher order
nPI formulations (
) are necessary in some situations. Transport coefficients in gauge theories (even at leading order) cannot be calculated using a 2PI formulation [
15,
16], and numerical calculations have shown that, for a symmetric scalar
theory, 4PI vertex corrections are large in three dimensions [
17], and for sufficiently large coupling the 2PI approximation breaks down at the 4 loop level in four dimensions [
18,
19].
In this paper we work with a symmetric scalar theory, to avoid some of the complications of gauge theories, and focus on the problem of renormalizability. We use the renormalization group (RG) method that was introduced in [
20]. Using this method, no counterterms are needed and the divergences are absorbed into the bare parameters of the Lagrangian, the structure of which is fixed and totally independent of the order of the approximation. In this sense, the RG method is designed to be used at any order in the
nPI approximation (and at any loop order).
2. Notation
We introduce a notation that suppresses the arguments that give the space-time dependence of functions. For example, the term in the action that is quadratic in the fields is written:
where
is the bare propagator. The classical action is
For notational convenience we use a scaled coupling constant (), and the factor of i that is introduced here will be removed when we rotate to Euclidean space for numerical calculations.
To use the functional renormalization group method, we add a non-local regulator function to the action [
21]
The scale denoted
has dimensions of momentum. The regulator function satisfies
and
so that for
the regulator plays the role of a large mass term which suppresses quantum fluctuations with wavelengths
, while in the opposite limit fluctuations with wavelengths
are unaffected. The regulated action (
3) can be used to obtain the 2PI generating functionals:
To obtain the 2PI effective action, we take the double Legendre transform of the generating functional
with respect to the sources
J and
, with
and
G now taken as the independent variables. The resulting effective action
can be written
where
means the set of all 2PI graphs with two and more loops and we have defined
. We have subtracted the regulator term so that the effective action corresponds to the classical action at the ultraviolet scale
. To simplify the notation we will write
where both
and
have the same subscripts, and we define an imaginary regulator function
(the extra factor
i will be removed when we change to Euclidean space variables).
The effective action is extremized by solving the variational equations of motion for the self consistent 1 and 2 point functions. These self consistent
dependent solutions are denoted
and
, but since we work with the symmetric theory we will set
. We calculate
n-point kernels by functionally differentiating the effective action
To simplify the notation we use special names for certain kernels:
3. Flow Equations
and
do not depend explicitly on
and therefore we can use the chain rule to obtain
In momentum space the equation becomes
We will show below that this infinite hierarchy of coupled integral equations for the
n-point kernels truncates at the level of the action. The flow equations can be rewritten in a more useful form using the stationary condition
which gives by a straightforward calculation
The first two equations in the hierarchy (8) now take the form
By iterating Equation (
11), we can reformulate the flow equation for the 2 point function
so that the kernel contains a Bethe-Salpeter (BS) vertex:
with
A different class of non-perturbative vertices can be defined by considering variations of the effective action with respect to the field. The 4 point function that is obtained in this way is related to the BS vertex as The vertex V contains terms from all three (s, t and u) channels, and the shorthand notation which suppresses indices combines the three channels to give the factor (3) in Equation (3).
We rotate to Euclidean space for the numerical calculation, and to simplify the notation we do not introduce subscripts to denote Euclidean space quantities. The flow Equations (
11) and (12) and the BS Equation (
14) have the same form in Euclidean space. The Dyson equation has the form
and the equation for the physical vertex in Euclidean space is
The regulator function becomes
At the 4 loop level, the hierarchy of flow equations can be truncated at the level of the second equation (this is explained below). The
n-point functions for the quantum theory can be obtained by starting from initial conditions defined at
and solving the integro-differential flow Equations (
11) and (12). We choose the regulator function
so that the theory is described by the classical action at the ultraviolet scale
. The initial conditions are therefore obtained from the bare masses and couplings of the Lagrangian. The values of the bare parameters are unknown, but the values of the renormalized parameters are specified by the renormalization conditions
that are enforced by choice on the
n-point functions that will be obtained at the quantum end of the flow. The method is to start from an initial guess for the bare parameters, solve the flow equations, extract the renormalized parameters, and then adjust the bare parameters (either up or down depending on the result). We then resolve the flow equations and repeat the procedure, continuing until the renormalization conditions are satisfied (to some numerically specified accuracy).
It can be shown [
19] that consistency between the initial conditions and the renormalization conditions requires
If the hierarchy in (
8) is truncated correctly, the condition (
17) will be satisfied. This statement is proved by showing that if a given kernel obtained from functional differentiation satisfies the condition (
17), it will also satisfy
and
[
19]. The result is that the flow equation for this kernel does not have to be solved. We therefore need to find the smallest value of
m for which (
17) is satisfied, and then solve self consistently the set of flow equations for the kernels with
legs.
It is straightforward to show that any kernel that contains a diagram with a loop that is not forced by the structure of the diagram to carry one of the external momenta, will not satisfy (
17), and the flow equation for this kernel must be solved [
19]. If the effective action is truncated at the 3 loop level, the self energy will include the sunset diagram which will not satisfy (
17). On the other hand, the kernel
has the tree graph and two 1 loop contributions that always carry external momenta, which means that
does not have to be flowed but can be simply substituted into the
flow equation. We have only to replace the tree vertex with the bare vertex (
) to satisfy the initial condition. At the 4 loop level the kernel
does not satisfy (
17), but the 6-leg kernel
does, and can be substituted directly into the
flow equation. There is no bare 6-vertex in the Lagrangian and therefore the integration constant is set to zero. The result is that at the 4 loop level we must solve the
and
flow equations self consistently.
4. Numerical Method
We start the flow of the 2 and 4 kernels from the initial conditions
and the propagator in the ultraviolet limit is
. We replace
with the variable
so that we approach the quantum theory more slowly. We use
,
and
and we have tested the insensitivity of our results to these choices. We have also used a generalized form of (
15) to verify that our results are not dependent on the form of the regulator. The renormalized mass and coupling are obtained from the quantum functions
and are then compared with the values specified in the renormalization conditions, adjusted, and tuned, by repeating the procedure until the renormalization conditions are satisfied to specified accuracy.
The 4-dimensional momentum integrals are written
with
. There are
terms in the summation with
and
is the lattice spacing in the temporal direction. We use spherical coordinates and Gauss-Legendre integration to do the integrals over the 3-momenta.
5. Results and Discussion
We use
points for the integrations over the cosine of the polar angle and the azimuthal angle, and we have checked that all results are stable when we increase the number of grid points in these dimensions. The momentum space grid spacing is
where
is the spatial lattice spacing and
is the number of lattice points for the momentum magnitude. The UV momentum cutoffs are
and
. We use
so that
. The numerics are stable if results are unchanged when
decreases while
is held fixed, and we have checked that this is true if
. To test the renormalization we increase
while holding
fixed. In
Figure 1 we show
versus
. For purposes of comparison we also show a calculation that is done incorrectly, by working at 3 loop level and replacing one of the vertices in the 4 kernel with a bare vertex. We have checked that dependence on the renormalization scale is very small.
To evaluate the 2, 3 and 4 loop approximations in the context of a physical quantity, we calculate the pressure, which can be obtained from the effective action using
where
V is the 3-volume. To 4 loop order, the contributions to the pressure are
There is a temperature independent divergence that can be subtracted off with a ‘cosmological constant’ renormalization, by setting the vacuum pressure to zero:
. The arrow on the right side of (
21) indicates that we have dropped a temperature independent constant which would have been removed by this shift. The term
is the non-interacting (
) pressure and since we want to compare
to the non-interacting expression, we define
. In
Figure 2 we show our results for the pressure as a function of the coupling at the 2, 3 and 4 loop orders of approximation.