Results from the 4PI effective action in 2- and 3-dimensions

Abstract

We consider a symmetric scalar theory with quartic coupling and solve the equations of motion from the 4PI effective action in 2- and 3-dimensions using an iterative numerical lattice method. For coupling less than 10 (in dimensionless units) good convergence is obtained in less than 10 iterations. We use lattice size up to 16 in 2-dimensions and 10 in 3-dimensions and demonstrate the convergence of the results with increasing lattice size. The self-consistent solutions for the 2-point and 4-point functions agree well with the perturbative ones when the coupling is small and deviate when the coupling is large.

Eur. Phys. J. C (2013) 73:2399DOI 10.1140/epjc/s10052-013-2399-5

Regular Article - Theoretical Physics

Results from the 4PI effective action in 2- and 3-dimensions

M.E. Carrington1,2,a, Wei-Jie Fu1,2,b

1Department of Physics, Brandon University, Brandon, Manitoba, R7A 6A9, Canada

2Winnipeg Institute for Theoretical Physics, Winnipeg, Manitoba, CanadaReceived: 4 January 2013 / Revised: 22 March 2013 / Published online: 20 April 2013 Springer-Verlag Berlin Heidelberg and Societ Italiana di Fisica 2013

Abstract We consider a symmetric scalar theory with quartic coupling and solve the equations of motion from the 4PI effective action in 2- and 3-dimensions using an iterative numerical lattice method. For coupling less than 10 (in dimensionless units) good convergence is obtained in less than 10 iterations. We use lattice size up to 16 in 2-dimensions and 10 in 3-dimensions and demonstrate the convergence of the results with increasing lattice size. The self-consistent solutions for the 2-point and 4-point functions agree well with the perturbative ones when the coupling is small and deviate when the coupling is large.

1 Introduction

The resummation of certain classes of Feynman diagrams to innite loop order is a powerful method in quantum eld theory. A well known example is the hard thermal loop theory [1], developed in the context of thermal eld theory, which resums all loop corrections which are of the same order as tree diagrams, when external momenta are soft.

In recent years, another kind of resummation approach, known as two-particle irreducible (2PI) effective action theory, has attracted a lot of attention. In the 2PI formalism, the effective action is expressed as a functional of the nonperturbative propagator [26], which is determined through a self-consistent stationary equation after the effective action is expanded to a certain order in the loop or 1/N expansion. This self-consistent equation of motion resums certain classes of diagrams to innite order. The classes that are resummed are determined by the set of skeleton diagrams that are included in the truncated effective action. Numerical studies indicate that the 2PI effective action formalism is very successful in describing equilibrium thermodynamics, and also the quantum dynamics of far from equilibrium

a e-mail: mailto:[email protected]

Web End [email protected]

b e-mail: [email protected]

of quantum elds. The entropy of the quark-gluon plasma obtained from the 2PI formalism shows very good agreement with lattice data for temperatures above twice the transition temperature [7, 8]. The poor convergence problem usually encountered in high-temperature resummed perturbation theory with bosonic elds is also solved in the 2PI effective action theory [9]. Furthermore, it has been shown that non-equilibrium dynamics with subsequent late-time thermalization can be well described in the 2PI formalism (see [1013] and references therein). The 2PI effective action has also been combined with the exact renormalization group to provide efcient non-perturbative approximation schemes [14]. The shear viscosity in the O(N) model has been computed using the 2PI formalism [15].

The 2PI effective action theory has its own drawbacks and limitations. When the effective action is expanded to only 2-loops, the 2PI effective action is complete. However, when the expansion is beyond 2-loops, one must use a higher order effective theory to obtain a complete description [16]. Higher order effective theories are dened in terms of self-consistently determined n-point functions for n > 2. It has been shown that the 2PI effective action is insufcient to calculate transport coefcients for high temperature gauge theories [17, 18], but that higher order nPI effective actions can be used [19].

The 4PI effective action for scalar eld theories is derived in Ref. [20] using Legendre transformations. The method of successive Legendre transforms is used in [16, 21]. A new method has been developed to calculate the 5-loop 5PI and 6-loop 6PI effective action for scalar eld theories [22, 23]. The 3PI and 4PI effective actions have been used to obtain a set of integral equations from which the leading order and next-to-leading order contributions to the viscosity can be calculated [24, 25].

A lot of effort has been devoted to numerical computations in 2PI effective theories. For higher order nPI theories numerical calculations are extremely difcult and little progress has been made (see, however, [26]).

Page 2 of 9 Eur. Phys. J. C (2013) 73:2399

This paper is organized as follows. In Sect. 2 we dene our notation, in Sect. 3 we present results from our numerical calculations in 2D and 3D, and in Sect. 4 we give our conclusions.

2 General formalism

We consider the following Lagrangian:1

L =

1

2[parenleftbig] m22[parenrightbig]

Connected and proper Green functions are denoted V cj and Vj , respectively, where the subscript j indicates the number of legs. They are dened thus:

V cj = [angbracketleftbig]j[angbracketrightbig]c = (i)j+1

j W

Rj1

, (5)

jj 1PI = i

jj [parenleftbig]W[R1] R1[parenrightbig]. (6)

The equations that relate the connected and proper vertices are obtained from their denitions using the chain rule. We organize the calculation of the effective action using the method of subsequent Legendre transforms [16, 21]. This method involves starting from an expression for the 2PI effective action and exploiting the fact that the source terms R3 and R4 can be combined with the corresponding bare vertex by dening a modied interaction vertex. The result is

[, G, V3, V4] =

Vj = i

i

4!

4. (1)

The classical action is

S[] = S0[] + Sint[],

S0[] =

1

2

[integraldisplay] ddx ddy (x)[bracketleftbig]iG1

0 (x y)[bracketrightbig](y),

Sint[] =

(2)

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 non-interacting part of the classical action is written

S0[] =

1

2

i

4!

[integraldisplay] ddx4(x).

i 2G102 +

i2Tr ln G1 +

i 2TrG10G

+int[, G, V3, V4], iint[, G, V3, V4]

=

(7)

4!

4 +

4 [parenleftbig]2G[parenrightbig] + 2[, G, V3, V4],

[integraldisplay] ddx ddy(x)

[bracketleftbig]iG10(x y)[bracketrightbig](y)

i 2G102,

(3)

G10 = i

2Scl[] 2

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]=0 =

i[parenleftbig] + m2[parenrightbig].

where 2[, G, V3, V4] represents diagrams with two and

more loops. In this paper we consider only the self-consistent 2- and 4-point functions in the symmetric phase. These are obtained by solving simultaneously the equations of motion:

[, G, V3, V4]

G

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]=0,G= G,V3=0,V4= V4 =

The effective action is obtained from the Legendre transformation of the connected generating functional:

Z[R1, R2, R3, R4]

=

[integraldisplay]

[d] Exp

0,

(8)

From now on we drop the subscript on the 4-point vertex function and write V := V4 and V := V4.

The variables m, , G and V in Sect. 2 should all carry a subscript B to indicate that they are bare quantities. These bare quantities (with subscript B) are related to the renormalized ones by the following relations:

m2 = Zm2B m2, = Z2B , GB = ZG, VB = Z2V,

ZG10B = G10 + G10, G10 = i[parenleftbig]Z + m2[parenrightbig], Z = Z 1.

(9)

In order to simplify the notation we have not introduced a subscript R for renormalized quantities and we have sup-

[, G, V3, V4]

V4

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]=0,G= G,V3=0,V4= V4 =

0.

[bracketleftbigg]i [parenleftbigg]Scl[] + R1 +

1

2R22

R44[parenrightbigg][bracketrightbigg],

W[R1, R2, R3, R4] = i LnZ[R1, R2, R3, R4], [, G, V3, V4]

= W R1

+

1 3!

R33 +

1 4!

(4)

WR1 R2

WR2 R3

WR3 R4

W R4 .

1The coupling constant is imaginary. Using this denition the lines and crosses in Feynman diagrams are propagators and proper vertex functions, as dened in Eq. (6), and the diagrams in Figs. 13 do not carry signs or extra factors of i. Numerical calculations are done in Euclidean space and the corresponding vertex is dened in Eq. (14).

Eur. Phys. J. C (2013) 73:2399 Page 3 of 9

Fig. 1 The effective action at 4-Loops including counter-terms. The open circle is a bare vertex, the circle with a cross denotes a counter-term, and the vertex V is indicated by a solid dot

Fig. 2 Self-consistent equation of motion for the proper 4-point vertex

Fig. 3 The self-energy obtained from Eqs. (10) and (12)

pressed the subscript B everywhere except in Eq. (9). All quantities in Sect. 2 are bare, and all quantities in the following sections are renormalized.

We divide the functional 2[G, V ] into two pieces: terms

without counter-terms or bare vertices, and terms that do contain either counter-terms or bare vertices. We denote these two pieces iint[G, V ] and i0[G, V ], respectively. Using this notation we write the effective action in

Eq. (7) as:

i [G, V ] =

1

2Tr ln G1

[integraldisplay] dQV (Pa, Pc, Q)G(Q)

G(Q + Pa + Pc)V (Pb, Pd, Q), Vt(Pa, Pb, Pc) = Vs(Pa, Pc, Pb),

Vu(Pa, Pb, Pc) = Vs(Pa, Pb, Pd), Pd = (Pa + Pb + Pc).

(11)

1

2TrG10G + 0[G, V ]

+int[G, V ],

(10)

0 =

1

2TrG10G +

18( + )G2 +

1 4!

( + )G4V.

The equation of motion for the 2-point function is obtained from the variational equation /G = 0 (see

Eq. (8)). It has the form of a Dyson equation where the self-energy is proportional to the functional derivative of the terms in the effective action with two and more loops:

G1 = G10 , = 2

2

The functional 0 has the same form for n 4 and all orders

in the loop expansion and int contains a set of skeleton diagrams which are determined by the orders of the Legendre transform and loop expansion. The sum 2 = 0 + int is shown to 4-loops in Fig. 1.2 In all diagrams, bare 4-vertices are represented by white circles, counter-terms are circles with crosses in them, and solid dots are the vertex V .

The equation of motion for V is obtained from the variational equation /V = 0 (see Eq. (8)). Using the 4-Loops

4PI effective action gives the result in Fig. 2 and Eq. (11). We use throughout the notation dQ = ddq/(2)d.

2All gures are drawn using jaxodraw [27].

V (Pa, Pb, Pc) = + + Vs(Pa, Pb, Pc)

+ Vt(Pa, Pb, Pc) + Vu(Pa, Pb, Pc),

Vs(Pa, Pb, Pc) =

1

2

G . (12)

The result is shown in the rst line of Fig. 3. The diagrams can be rearranged by substituting the V equation of motion into the vertex on the left side of the sixth diagram. This substitution cancels the 3-loop diagram and produces the result in the second line of the gure and Eq. (13).

(p) = i[parenleftbig]ZP

2 m2[parenrightbig] +

1

2( + )

[integraldisplay] dQ G(Q)

16( + )

[integraldisplay] dQ

+ [integraldisplay] dK V (P, Q, K)G(Q)G(K)G(Q + K + P ).(13)

Page 4 of 9 Eur. Phys. J. C (2013) 73:2399

Numerical calculations will be done in Euclidean space and therefore we redene the variables:

q0 = iqE, Z = ZE, m2 = m2E, G = iGE, = iE, = iE, = iE, V = iVE.

(14)

The Dyson equation in Euclidean space is (see Eq. (12)):

G1E(P ) = G10E(P ) + E(P ). (15) All variables from here on are Euclidean and we suppress the subscript E. The coupling constant and 4-point vertex have dimension 4d so that in 2D m2 and in 3D m.

We work in mass units, in which all dimensional quantities are scaled by the mass.

3 Numerical results

3.1 Perturbative theory

We start by looking at the perturbative theory. For the 4-point function at the 1-loop level the diagrams we need are

obtained from Fig. 2 with lines and proper vertices replaced by bare ones. In less than 4-dimensions there are no ultraviolet divergences and from Eq. (11) we nd

V (Pa, Pb, Pc)

= +

2 2

(2 d/2)

(4)d/2

[integraldisplay]

1 [braceleftbigg] 1

[m2 + x(1 x)(Pa + Pc)2]2d/2

1

+ [m2 + x(1 x)(Pa + Pb)2]2d/2

[bracerightbigg]. (16)

To obtain the 2-point function at 2-Loops we need to calculate the tadpole and sunset diagrams in Eq. (13) with lines and proper vertices replaced by bare ones. We use dimensional regularization and dene = 1 d/2. The tad-

pole diagram is momentum independent in any number of dimensions. In 3D it is nite using dimensional regularization and in 2D the divergent part is easily obtained as /(8 ). The sunset contribution to the self-energy is

1

+ [m2 + x(1 x)(Pb + Pc)2]2d/2

[integraldisplay]

1

0 dx

2 dims

sunset (P ) =

2 6(4)2

[integraldisplay]

1

0 dy

1

[y + (1 y)x(1 x)]m2 + y(1 y)x(1 x)P 2

.

[integraldisplay]

1

[integraldisplay]

1

(17)

3 dims

sunset (P ) =

2 6

( ) (4)32

dx(x(1 x))1/2+

dyy 1/2

0 [P 2y(1 y) + m2(1 y +

y x(1x) )]

.

0

The integral is nite in 2D. In 3D the divergence is momentum independent. We can write the divergent part of the self-energy as

div =

8

1 d2

1

212(4)2 d3, (18)

where the Kronecker deltas indicate which pieces contribute in 2D and 3D. Thus we nd that in 2D the tadpole has a momentum independent divergence and the sunset diagram is nite, while in 3D the situation is reversed and the sunset has a momentum independent divergence but the tadpole is nite. Using PauliVillars regularization the tadpole has a momentum independent divergence in 2D or 3D and the sunset has a momentum independent divergence in 3D only. In all cases, the counter-term m2 completely removes the divergence and we can set Z = 0. We use the renormal

ization condition (0) = 0 to determine the mass counter-

term. Since the tadpole diagram is independent of the ex-

ternal momentum (in any dimension), this renormalization condition completely removes the entire tadpole contribution, and we can just drop the diagram. In both 2D and 3D the 4-vertex does not UV-renormalize and the natural choice is to set = 0, so that is dened as the limit of the 4-point

function at large external momenta.

3.2 Non-perturbative calculation

The diagrams produced by expanding the nPI equation of motion are not the same as those produced by the perturbative expansion, some diagrams appear with different symmetry factors, and some diagrams are missing altogether. In less than 4-dimensions, however, the only fundamental divergences are the tadpole and sunset diagrams, and each insertion of a bare self-energy is accompanied by the mass counter-term that makes it nite. Iteration does not create new sub-divergences and therefore iterations amount to inserting renormalized self-energies, without introducing new

Eur. Phys. J. C (2013) 73:2399 Page 5 of 9

divergences. Therefore one can also renormalize the nonperturbative theory using only a mass counter-term. In order to compare the non-perturbative results with the perturbative ones, we use the same renormalization conditions.

To obtain non-perturbative results we solve the self-consistent equation of motion for the 2- and 4-point functions using a numerical lattice method. We use an Nd symmetric lattice with periodic boundary conditions. The size of the lattice is limited by the calculation time and memory constraints. In 2D we use N up to 16 and in 3D N up to 10.

The lattice spacing is a and we choose a = 2/(Nm). In

Euclidean space, each momentum component is discretized:

pi =

N

2 , (19)

and the periodic boundary conditions take the form ni +N =

ni for all momentum components. The lattice momenta given by Eq. (19) form a Brillouin zone. On the lattice, the equation of motion for the 4-point vertex (Eq. (11)) is transformed into

2aN ni, ni =

N

2 + 1, . . . ,

V (Pa, Pb, Pc) = +

1

2

1 (aN)d [summationdisplay]

Q

V (Pa, Pc, Q)G(Q)G(Q + Pa + Pc)V (Pb, Pd, Q)

+V (Pa, Pb, Q)G(Q)G(Q + Pa + Pb)V (Pc, Pd, Q)

+V (Pa, Pd, Q)G(Q)G(Q + Pa + Pd)V (Pb, Pc, Q)[bracketrightbig], (20)

and the 2-loop self-energy (Eq. (13)) is

(P ) = ZP 2 + m2 +

16( + )

1 (aN)2d [summationdisplay]

Q

[summationdisplay]

K

V (P, Q, K)G(Q)G(K)G(Q + K + P ).

(21)

We start from an initial 4-point vertex and propagator which we chose to be the bare vertex and free propagator. Then we use Eqs. (15), (20), and (21) and simultaneously search for self-consistent solutions using an iterative procedure. In order to make the iterations converge more quickly, we adopt the following formula to update the vertex and propagator at every iteration [9]:

Vupdate = Vnew + (1 )V, (22) Gupdate = Gnew + (1 )G, (23)

where is the convergence factor, and we choose = 0.8.

In all of our calculations, the full momentum dependence of the vertex and self-energy is taken into account. In order to produce gures, we must x some momentum components to obtain a 2-dimensional representation of the results. For the 4-point function, when we consider the dependence on either the number of iterations or the coupling, we choose all momentum components equal to zero. We also study the momentum dependence of the 4-point function as a function of (pa)x and {(pa)x, (pb)x} with all other mo

mentum components set to zero, where (pa)x and (pb)x are the x-components of the momentum of the rst and second legs. For the 2-point function, because of the renormalization condition, we consider (px = 2, 0, 0) as a function of

the number of iterations and coupling, and also (px, 0, 0) at xed coupling.

It is interesting to compare the results we obtain from the non-perturbative calculation with the corresponding perturbative ones. The continuum perturbative solution can be obtained from Eqs. (16) and (17). In order to check our equations, we also do the perturbative calculation on the lattice by solving Eqs. (20) and (21) with the self-consistent vertex and propagator replaced by the bare ones. We work in 3D and use N = 6, 8, 10, 12, and 30, the results are shown in

Fig. 4. For very large N the lattice calculation converges to the continuum limit, as it should. In this paper we do not go beyond N = 10, and although it is clear that larger Ns are

desirable, the gure shows that the calculation converges in the sense that N = 10 is closer to N = 8 than N = 8 is to

N = 6. Later in this section we discuss convergence further.

In Fig. 5 we show the 4-point function and self-energy as a function of the number of iterations. We choose = 5

(in mass units), and N = 16 in 2D and N = 8 in 3D. The

rst two iterations are not included so that the evolution can be seen more clearly. In both the 2D and 3D cases, self-consistent convergent solutions are obtained quickly after several iterations. The number of iterations that is needed increases slightly as increases, but it is easy to obtain converged solutions for 10.

Figure 6 shows the 2D and 3D 4-point vertex and self-energy as functions of the coupling strength , calculated from 4PI effective action and perturbation theory. In all cases the non-perturbative results agree well with the perturbative ones when the coupling strength is small. When the coupling constant becomes large the 4PI results differ signicantly from the perturbative ones, indicating the importance of a non-perturbative method in the strong coupling

Page 6 of 9 Eur. Phys. J. C (2013) 73:2399

Fig. 4 The perturbative vertex and self-energy from the lattice calculation with N = 6, 8, 10,

12 and 30, and in the continuum limit

Fig. 5 The 4-point vertex and self-energy as a function of the number of iterations for = 5,

and N = 16 for 2D and N = 8

for 3D

regime. In 2D, the results for the 4PI vertex are almost independent of the lattice number N. The self-energy depends more strongly on the lattice size but converges well when N is increased to 16 (the curve corresponding to N = 16 al

most coincides with that corresponding to N = 12). In 3D,

calculations can only be performed up to N = 10 because

the three independent external momenta of the 4-point vertex consume lot of computational resource. Convergence is good for the 4-point vertex, but the results for the self-energy have a stronger dependence on the lattice number.

In order to investigate whether convergence is obtained with N = 10, we look at the ratios of the vertex and self en

ergy calculated with N = 8 and N = 10, and with N = 6 and

N = 8. The results are shown in Fig. 7. The fact that these

ratios approach 1 indicates that our results are converging.

Figure 8 shows the dependence of the 4-point vertex on the rst momentum component in 2D and 3D, choosing all

momentum components other than (pa)x zero and using the coupling strength equal to 1 and 5. We choose N = 16, 12

and 8 for the 2D calculations, and N = 10, 8, and 6 for the

3D ones. The difference between the non-perturbative vertex and the perturbative one is greater when the coupling constant is larger, as expected.The momentum dependence is produced by the 1-loop diagram and at large momentum the 4-point vertex scales as V Cpd4, as expected.

The results obtained by increasing the lattice number converge in both 2D and 3D.

In Fig. 9 we show the dependence of the self-energy on px with all other momentum components zero. The momentum dependence comes from the sunset diagram and at large momentum the self-energy scales like C C /p2 in 2D and

C + C log(p) in 3D, as expected. The difference between

the non-perturbative self-energy and the perturbative one is greater when the coupling constant is larger. Convergence

Eur. Phys. J. C (2013) 73:2399 Page 7 of 9

Fig. 6 Comparison of the 2D and 3D 4-point vertex and self-energy as functions of the coupling strength . The 4PI calculations are done in 2D with N = 16, 12, and 8, and in 3D

with N = 10, 8, and 6. The top

left panel is the 2D 4-vertex, top right is the 2D self-energy, bottom left is the 3D 4-vertex, bottom right is the 3D self-energy. In each graph the perturbative result is the dotted line which joins the round markers (red) (Color gure online)

Fig. 7 Non-perturbative results for VN=8/VN=10, VN=6/VN=8,

N=8/N=10 and

N=6/N=8 in 3D as functions

of the coupling strength

with increasing lattice size is not as good as for the vertex, and not as good in 3D as in 2D. However, the analysis in Fig. 7 indicates that our results are converging.

It is interesting to compare the 4PI 2-point function with the 2PI version, which is obtained from Eq. (21) with the self-consistent vertex replaced by the bare one. For the values of chosen in Fig. 9 the 2PI result is almost identical to the perturbative one, and it is only for very large values of that one can see the difference. We illustrate this point in Fig. 10 where we show the perturbative, 2PI and 4PI self-energies for = 5 and 50, and N = 8.

In Fig. 11 we give contour plots of the 2D and 3D 4-point vertex which show the dependence of the vertex on the two momentum components (pa)x and (pb)x with all others

chosen to be zero. The vertex has a minimum at the origin of the coordinates, and the gradient varies with direction.

4 Summary and outlook

We have solved the integral equations which determine the self-energy and vertex functions in 2D and 3D at zero temperature using a numerical lattice method. All results agree with the perturbative ones when the coupling is small but deviate signicantly when the coupling strength increases. In 2D the 4PI calculations with lattice number N = 16 are

convergent and the non-perturbative 4-point vertex and self-energy show similar asymptotic behaviors at large momen-

Page 8 of 9 Eur. Phys. J. C (2013) 73:2399

Fig. 8 Dependence of the 2D and 3D 4-point vertex on (pa)x with all other external momenta components set to zero. Top left panel is 2D with = 1, top

right is 2D with = 5, bottom

left is 3D with = 1, and

bottom right is 3D with = 5.

The 4PI calculations are done in 2D with N = 16, 12, and 8, and

in 3D with N = 10, 8, and 6.

The perturbative result is the dotted line which joins the round markers (red) (Color gure online)

Fig. 9 Dependence of the 2D and 3D self-energy on the px with all other momentum components set to zero. Top left panel is 2D with = 1, top

right is 2D with = 5, bottom

left is 3D with = 1, and

bottom right is 3D with = 5.

The 4PI calculations are done in 2D with N = 16, 12, and 8, and

in 3D with N = 10, 8, and 6.

The perturbative result is the dotted line which joins the round markers (red) (Color gure online)

tum as the perturbative ones. In 3D the 4PI calculations with N = 10 are reasonably well convergent, especially for the

4-point vertex. To obtain more accurate results at large momenta in 3D we should extend our calculations to larger lattice number. This requires a different numerical method and

increased computing power, and work on this is currently in progress.

We comment that zero temperature is the simplest situation numerically, but not the one in which it is expected that nPI methods will have a substantial advantage over pertur-

Eur. Phys. J. C (2013) 73:2399 Page 9 of 9

Fig. 10 Comparison of the 3D perturbative, 2PI and 4PI self-energies for = 5 and

= 50, with N = 8

Fig. 11 Contour plot of the 4-point vertex in 2D and 3D as a function of (pa)x and (pb)x with all other momenta components set to zero for = 5

bation theory, which is known to break down at high temperatures. Our calculation makes use of the symmetries of the 2- and 4-point functions, namely the fact that they are symmetric under the interchange of legs, and the interchange of co-ordinate axes. At nite temperature, the number of symmetries will be reduced and the memory requirements will be correspondingly larger.

Our numerical calculations demonstrate that 4PI calculations are both interesting and feasible, and motivates further work on more physically interesting problems.

Acknowledgements This work was supported by the Natural and Sciences and Engineering Research Council of Canada. WJF is supported in part by the National Natural Science Foundation of China under contract No. 11005138.

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