Renormalization group methods and the 2PI effective action
We consider a symmetric scalar theory with quartic coupling in 4-dimensions and compare the standard 2PI calculation with a modified version which uses an exact renormalization group method. The set of integral differential equations that are obtained from the exact renormalization group method truncate naturally, without the introduction of additional approximations. The results of the two methods agree well, which shows that the exact renormalization group can be used at the level of the 2PI effective action to obtain finite results without the use of counter-terms. The method therefore offers a promising starting point to study the renormalization of higher order PI theories.
There is much interest in the study of non-perturbative systems, which cannot be solved by expanding in some small parameter. Two formalisms that have been proposed to address non-perturbative problems are -particle irreducible (PI) effective theories Jackiw1974 (); Norton1975 (), and the exact renormalization group (RG) Wetterich1993 (); Ellwanger1993 (); Tetradis1994 (); Morris1994 ().
The 2PI formalism has been used to study finite temperature systems (see for example Blaizot1999 (); Berges2005a ()), non-equilibrium dynamics and subsequent late-time thermalization (see Berges2001 () and references therein), and transport coefficients Aarts2004 (); Carrington2006 (). Beyond 2-loop order the 2PI effective action is not complete Berges-hierarchy () and one must use a higher order PI theory. The 4PI effective action for scalar field theories is derived in Ref. Norton1975 () using Legendre transformations. The method of successive Legendre transforms is used in Berges-hierarchy (); Carrington2004 (). A new method has been developed to calculate the 5-loop 5PI and 6-loop 6PI effective action for scalar field theories Guo2011 (); Guo2012 (). The 3PI and 4PI effective actions have been used to obtain the integral equations from which the leading order and next-to-leading order contributions to conductivity and shear viscosity can be calculated Carrington2009 (). However, the integral equations that are produced by higher order theories are difficult to solve, and new methods must be developed.
One problem is the size of the phase space that is involved, but this can be significantly reduced using symmetry constraints Fu2013 (); Fu2014 (). Another problem is the renormalization of higher order theories in 4-dimensions. The renormalization of the 2PI effective theory was only understood through the labours of multiple authors over a period of many years vanHees2002 (); Blaizot2003 (); Serreau2005 (); Serreau2010 (). Using a diagrammatic approach, it was shown that renormalization requires introducing a set of vertex counter-terms, which obey different renormalization conditions and approach each other in the limit that the order of the approximation is taken to infinity. For higher order theories, the integral equations are too complicated to analyse in this way.
The exact renormalization group has been applied to a variety of problems (for reviews see Morris-98 (); Bagnuls2001 (); Berges2002 (); Pawlowski2005 (); Rosten2007 ()) and has lead to insight into the nature of renormalizability. A regulator function is introduced which depends on the continuous parameter whose role is to suppress fluctuations with momenta while leaving those with momentum unaffected. The regulated action is equal to the standard action when is taken to zero, which corresponds to removing the cutoff and including all fluctuations. On the other hand, when we can associate the regulated action with the classical action. The regulated action thus interpolates between the classical action and the full quantum action, as the parameter is lowered to zero. The exact RG equations describe the evolution of the system from the scale of large , where the solutions are known, to the scale , where the solutions are desired. As always, physical quantities should be independent of the regularization scheme, which means in this case independent of the choice of regulator function.
The formal relationship between the RG method and 2PI theories has been studied in Dupuis1 (); Blaizot-2PIa (); Blaizot-2PIb (); Dupuis2 (), and the connection with higher PI theories was developed in Carrington-BS (). One of the difficulties with the standard RG flow equations obtained from the 1PI effective action is that they take the form of an infinite coupled hierarchy of functional differential equations, and an additional approximation is needed to truncate this hierarchy BMW1 (); BMW2 (). An PI effective theory also produces a infinite hierarchy of coupled integral equations, but in this case the hierarchy truncates automatically when the effective action is truncated at some order in the approximation (for example, a loop or expansion), and the truncation respects gauge invariance, to the order of the approximation Smit2003 (); Zaraket2004 (). One advantage of the method we develop in this paper is that the RG flow equations that are obtained from the 2PI effective action also truncate naturally, and therefore do not require the introduction of additional approximations.
In this paper we look at a specific 2PI calculation and show that it can be done in a different way, using a RG method. We start by calculating the 2-point and 4-point functions using the standard 2PI method, which was done previously in Ref. Berges2005a (). Then we do the calculation in a different way, without introducing counter-terms, using a regulated 2PI effective action and solving the resulting flow equations. The results of the two methods agree well, which shows that the RG method can be used at the level of the 2PI effective action to obtain finite results without the use of counter-terms.
This paper is organized as follows.
In section II we present our version of the 2PI calculation. In sections II.1 and II.2 we define the action and -point functions. The renormalizability of the 2PI theory is discussed in section II.3. In II.4 we explain our numerical method, much of which will also be used in the RG calculation, which is presented in section III. In section III.1 we discuss the RG formalism, and in the following two sections, III.2 and III.3, we define the regulated action and obtain the general flow equations. In section III.4 we give the specific form of the flow equations when the effective action is truncated at order , and in section III.5 we derive the boundary conditions on the flow. In section III.6 we explain how the RG equations truncate. In section III.7 we discuss the connection between the RG calculation and the standard 2PI one. Some details of the numerical method are given in III.8. In section IV we present our numerical results from both calculations. In section V we compare the two methods from the point of view of computational difficulty, and present our conclusions.
In most equations in this paper we suppress the arguments that denote the space-time dependence of functions. As an example of this notation, the quadratic term in the action is written:
We use the notation for the bare propagator because we reserve for the propagator in the RG calculation in the limit that the regulator goes to zero.
Ii The 2PI Effective Theory
ii.1 2PI effective Action
The classical action is
For notational convenience we use a scaled version of the physical coupling constant (). The extra factor of will be removed when rotating to Eucledian space to do numerical calculations. We consider the 2PI theory and construct the generating functional using 1- and 2-point sources
Taking functional derivatives with respect to sources we obtain
The 2PI effective action is obtained by taking the double Legendre transform of the generating functional with respect to the sources and and taking and as the independent variables:
We write the result as a function of renormalized variables without introducing additional subscripts:
where contains all 2PI vacuum graphs whose vertices are given by the terms cubic or quartic in in the expanded expression of . Throughout this paper we use the notation where both and carry the same subscripts or superscripts. For example, for the 2PI effective action we write , for the interacting part of the 2PI effective action we have , etc.
The stationary condition is
The solution is an implicit function of the field. We define the resummed action
The minimum of the resummed action () satisfies the condition
In this paper we consider only the symmetric theory, which means we take .
ii.2 2pi -point functions
We can obtain -point functions which obey the symmetries of the original theory by taking functional derivatives of the resummed action
These resummed -point functions obey integral equations with kernels of the form
We introduce specific names for the kernels we will need:
Both of the kernels denoted correspond to 2-point functions, and the kernels are 4-point functions. Using this notation the stationary condition in Eq. (7) can be written:
Thus we have a self-consistent equation for the propagator which has the form of a Dyson equation.
In both of these expressions we have dropped terms that contain kernels with an odd number of derivatives, since they will be zero in the symmetric theory. In addition, some terms have been dropped using the stationary condition (7). The first equation looks like (13), but for an arbitrary truncation and therefore . In the second equation, the three terms in the round bracket represent the , and channels of the 1-loop seagull diagram with one vertex and the other vertex given by:
where again we have dropped terms that are zero in the symmetric theory. This equation can be rewritten
where the vertex is defined as
In summary we have defined the following vertices:
(1) Resummed vertices and (14)
In the exact (untruncated) theory
We now impose the stationarity condition (9) and set . We Fourier transform to momentum space and write the vertices as functions of their momentum arguments. (We do not introduce new notation to indicate that the function changes. For example, we should write ), but we suppress the bar.) The vertices in the symmetric theory and in momentum space are written:
ii.3 2PI renormalization
The 1PI effective action is renormalized by introducing three counter-terms in the Lagrangian (denoted , and ) which modify the bare parameters of the original theory (and the wave function normalization) and are determined by three renormalization conditions. To renormalize the 2PI theory we need multiple counter-terms, which we denote , and . Counter-terms differentiated by different subscripts come from the same term in the Lagrangian, but correspond to different orders in the approximation that is used to truncate the effective action. All counter-terms are determined from only three renormalization conditions. We describe the procedure below.
One starts by adding counter-terms to each local, mass dimension 4 operator in the effective action
In addition, one includes the usual counter-terms in the skeleton expansion of the effective action, to the approximation order. For example, to order we have (see Fig. 1):
Primes are used for counter-terms which have partners in (23). We define
The coupling counter-terms in are chosen to cancel divergences in the integrals in the 4-kernels, and the coupling counter-terms in cancel the remaining divergences in the resummed 4-point vertices. The 4-kernels are divergent (for example where is the quantity that is made finite by ), but this is not a problem since the 4-kernels are not directly related to physical quantities. The 2-point functions contain coupling constant counter-terms that must be obtained self-consistently from the appropriate BS equation. The vertices and their corresponding counter-terms are listed in Table 1.
|c-term — vertex||c-term — vertex||c-term — vertex|
|2-pt fcns||, , —||, , —|
In the truncated theory, the different -point functions in (22) are not the same, and similarly the counter-terms which are differentiated by subscripts are not the same. Renormalizability requires only that the untruncated (exact) theory contains one mass counter-term, one wave-function renormalization counter-term and one coupling constant counter-term, which produce one renormalized -point function (20) and one renormalized 4-point function (21). The counter-terms introduced in (23 - 25) must therefore satisfy , and when the order of the approximation is taken to infinity. To obtain this result, the counter-terms which carry different subscripts must be determined from the same renormalization condition, which means that the two 2-point functions must satisfy the same renormalization conditions, and the three 4-point functions must satisfy one renormalization condition
where the notation , , etc., indicates that all momentum components of each leg are set to zero.
ii.4 2PI Numerical Method
In this paper we truncate the effective action so that it includes all terms of order in the skeleton expansion. To this order , and , and therefore we consider only , and , and we drop the superscripts. The only counter-terms we will need are , , and , and therefore we drop the counter-term subscripts as well. Furthermore, at order the kernel does not contain any non-global divergences that require a coupling counter-term, which means that the division of the counter-term into two pieces is not necessary, and therefore we use only .
Equations (11, 12, 13, 19) determine the self-energy, propagator, 4-kernel and BS vertex. We rotate to Eucledian space, discretize, and solve the resulting set of equations using an iterative relaxation method. These three steps are described in the following three sub-sections. More details can be found in Refs. Fu2013 (); Fu2014 ()
ii.4.1 Eucledian space equations
where we have used .
The propagator and the BS vertex are given by:
We rotate to Eucledian space and define the Eucledian variables:
From now on we suppress the subscripts indicating Eucledian space. The counter-terms are determined from the renormalization conditions (26) which are written in Eucledian space
In order to do the numerical calculation, we restrict to a box in co-ordinate space of finite volume . Fourier transforming to momentum space one obtains discrete frequencies and momenta. This can be written
The parameters and are the lattice spacing in the temporal and spatial directions. Indices which fall outside of the range are wrapped inside using periodic boundary conditions. This is done using the function
where Mod[,] is an integer function that gives the remainder on division of by so that Mod[,] (for example, Mod[17,17]=0 and Mod[23,17]=6).
After discretization the bare propagator has the form
To simplify the notation we represent the arguments of a function of discrete variables using one boldface character, for example,
We also write the four summations which correspond to one discretized 4-momentum integral as one summation, for example,
The scalar theory in 4-dimensions is non-interacting if it is considered as a fundamental theory valid for arbitrarily high momentum scales (quantum triviality), but the renormalized coupling is non-zero if the theory has an ultra-violet cutoff and an infra-red regulator. In our calculation the lattice spacing parameters and provide an ultra-violet cutoff for the and momentum integrals, and the mass regulates the momentum integrals in the infra-red.
ii.4.3 The relaxation method
We solve the system of equations (42 - 49) using an iterative relaxation method. We use an index in round brackets to indicate the iteration number of a given quantity. In the first step of the calculation, the counter-terms are determined at fixed temperature . We will verify numerically that corresponds to the zero temperature limit, and we refer to it from here on as zero temperature. We will study the temperature dependence of the -point functions by decreasing , using the counter-terms obtained at zero temperature. From this point on, we scale all dimensionful variables by , or equivalently we set the renormalized mass to one and express all quantities in mass units.
First we describe the general method to find the counter-terms by imposing the renormaliztion conditions at . The zeroth iteration of the propagator is the bare propagator. The kernel at any iteration order is obtained from (45, 46) using propagators at the corresponding iteration order. The BS vertex at zeroth iteration order is defined to be at zeroth order. Thus we have
Starting from these zeroth order solutions we iterate to find self-consistent solutions
Iterations are terminated when the relative maximum difference between the th iteration and the th, for any quantity, at any point in the phase space, is less than .
If we simply ignore the vertex counter-term, the equations that determine the 2-point function are independent of those that determine the 4-vertex. This means that if renormalization were not necessary, one could calculate the propagator independently, and use this result in the BS equation to calculate the vertex . Once the counter-terms have been determined at zero temperature, calculations at different finite temperatures are much simpler because of the fact that when the counter-terms are known, the equations that give the 2-point function are decoupled from the vertex equations. This means that one can solve the equations at any finite temperature using a simpler procedure. Symbolically:
If the number of iterations it takes to obtain convergence of the self-energy equation is and the number it takes to converge the BS equation is , the first (zero temperature) procedure requires iterations, and the second (finite temperature) calculation takes iterations. Typically and therefore, after the renormalization is performed, subsequent calculations at different finite temperatures are much quicker.
The coupling of the equations for the 2-point and 4-point functions that we have described above is a general feature of the zero temperature calculation at arbitrary approximation order. However, when the effective action is truncated at order as in this paper, the zero temperature calculation can be done in a simpler way. The basic reason is that the vertex counter-term contributes to the self-energy only through the momentum independent tadpole diagram, and therefore one can proceed as follows.
set and drop the tadpole contribution in the self-energy in (43)
use in (46) to get and define
iterate the BS equation (44) starting from and obtain
calculate the tadpole term in the self-energy (43) using and
use the renormalization condition (47) to get
the full mass counter-term is the sum
The total number of steps is .
We will do the numerical calculations using the renormalized parameters and . The renormalization is done with , , and , and finite temperature calculations are done with . We present our results in section IV, together with the results from the RG calculation, which is described in the next section. All calculations are done using fast Fourier transforms, to improve performance.
Iii Exact Renormalization Group Calculation
iii.1 The RG formalism
Using the functional renormalization group method, we add to the action in (2) a non-local regulator term
The bare mass and coupling are defined at an ultra-violet scale which must be specified (we use instead of the traditional because that letter has already been used for the 4-point kernels). The parameter has dimensions of momentum and the regulator is chosen to have the following properties: when , , and when , . The effect is therefore that (1) for the regulator is a large mass term which suppresses quantum fluctuations with wavelengths and; (2) fluctuations with and wavelengths