In most equations in this paper, we suppress the arguments that denote the spacetime dependence of functions. As an example of this notation, the quadratic term in the action is written i2∫d4xd4yφ(x)Gno·int-1(x-y)φ(y)→i2φGno·int-1φ.(1)The notation Gno·int indicates the bare propagator. The classical action is S[φ]=i2φGno·int-1φ-i4!λφ4,iGno·int-1=-(□+m2).(2)For notational convenience, we use a scaled version of the physical coupling constant (λphys=iλ). The extra factor of i will be removed when rotating to Euclidean space to do numerical calculations.

Using the functional renormalization group method, we add to the action in (2) a nonlocal regulator term: Sκ[φ]=S[φ]+ΔSκ[φ],ΔSκ[φ]=-12R^κφ2.(3)The parameter κ has dimensions of momentum. The regulator function is chosen to have the following properties: limQ≪κR^κ(Q)∼κ2 and limQ≥κR^κ(Q)→0. The effect is therefore that for Q≪κ the regulator acts like a large mass term which suppresses quantum fluctuations with wavelengths 1/Q≫1/κ, but fluctuations with Q≫κ and wavelengths 1/Q≪1/κ are not affected by the presence of the regulator.

The n-point functions of the theory depend on the parameter κ. One obtains a hierarchy of coupled differential “flow” equations for the derivatives of the n-point functions with respect to κ. This hierarchy is automatically truncated when the effective action is, and there is therefore no need to introduce additional approximations. The set of truncated flow equations can be integrated from an ultraviolet scale κ=μ down to κ=0 where the regulator goes to zero and the desired quantum n-point functions are obtained. The parameter μ is the scale at which the bare masses and couplings are defined (we use μ instead of the traditional Λ because that letter will be used for a four-point kernel). One chooses μ large enough that when κ=μ the theory is classical. The two- and four-point functions are then known functions of the bare parameters, and these classical solutions can be used as initial conditions on the differential flow equations.

The 2PI generating functionals are calculated from the regulated action (3): Zκ[J,J2]=∫[dφ]exp{i(S[φ]+Jφ+12J2φ2-12R^κφ2)},(4)Wκ[J,J2]=-ilnZκ[J,J2],(5)δWκ[J,J2]δJ=⟨φ⟩≡ϕ,δWκ[J,J2]δJ2=12⟨φ2⟩=12(ϕ2+G).(6)The 2PI effective action is obtained by taking the double Legendre transform of the generating functional Wκ[J,J2] with respect to the sources J and J2 and taking ϕ and G as the independent variables: Γ^κ[ϕ,G]=Wκ-JδWκδJ-J2δWκδJ2=Wκ-Jϕ-12J2(ϕϕ+G).(7)After performing the Legendre transform, the functional arguments of the effective action ϕ and G are formally independent of the regulator function and the parameter κ, but the noninteracting propagator does depend on κ. We define iGno·int·κ-1=iGno·int-1-R^κ=-□-(m2+R^k).(8)Using this notation, the effective action Γ^κ[ϕ,G] can be written Γ^κ[ϕ,G]=Γno·int·κ[ϕ,G]+Γint[ϕ,G],Γ^no·int·κ[ϕ,G]=i2ϕGno·int·κ-1ϕ+i2TrlnG-1+i2TrGno·int·κ-1G,Γint[ϕ,G]=-i4!λϕ4-i4λϕ2G+Γ2[ϕ,G;λ],(9)where Γ2 contains all 2PI graphs with two and more loops. We define an effective action that corresponds to the original classical action at the scale μ: Γκ=Γ^κ-ΔSκ(ϕ).(10)

Throughout this paper, we use the notation Γ=-iΦ where both Γ and Φ carry the same subscripts or superscripts. For example, the interacting part of the action is written Γint[ϕ,G]=-iΦint[ϕ,G]. To make the equations look nicer we also define an imaginary regulator function Rκ=-iR^κ (the extra factor i will be removed when we rotate to Euclidean space). Using this notation, Eqs. (8)–(10) are rewritten Φκ=Φno·int·κ+Φint,(11)Φno·int·κ=-[12TrlnG-1+12Gno·int·κ-1G]-[12Gno·int·κ-1+12Rκ]ϕ2,(12)Gno·int·κ-1=Gno·int-1-Rκ.(13)We extremize the effective action by solving the variational equations of motion for the self-consistent one- and two-point functions. These self-consistent solutions will depend on the parameter κ and are therefore denoted ϕκ and Gκ. We will work throughout with the symmetric theory by setting ϕκ=0. In Fig. 1 we show the diagrams that contribute to Φint to four-loop order, and the names we will use for these diagrams.

We define a set of n-point kernels by taking functional derivatives of the effective action Λ(n,m)=2mδnδϕnδmδGmΦint.(14)Since we work with the symmetric theory, we consider only kernels with n=0 and we suppress the corresponding 0 index in the superscript. Substituting the self-consistent solutions into the definition of the kernels (14), we obtain Λκ(m)=Λ(m)|ϕ=oG=Gκ.(15)We introduce specific names for the kernels we will write repeatedly: Λ(1)=Σ,Λ(2)=Λ,Λ(3)=ϒ.(16)The kernels Σ, Λ and ϒ have two, four and six legs, respectively. The Fourier transformed functions are written Σ(P), Λ(P,K) and ϒ(P,K,Q). The diagrams that contribute to Λ and ϒ in the four-loop approximation are shown in Fig. 2. We note that the four-kernel contains two-loop diagrams that involve nasty overlapping divergences. When we use the RG method, any kernel that contains subdivergences is calculated from a flow equation. This is explained in detail in the next section.

Using the chain rule, and the fact that Φint and therefore Λ(m) does not explicitly depend on κ, we find ∂κΛκ(m)=12∂κGκΛκ(m+1).(17)In momentum space (restoring arguments), this equation has the form ∂κΛκ(m)(P1,P2,⋯Pm)=12∫dQ∂κGκ(Q)Λκ(m+1)(P1,P2,⋯Pm+1,Q).(18)Equation (18) is an infinite hierarchy of coupled integral equations for the n-point kernels. This structure is typical of continuum nonperturbative methods, for example Schwinger-Dyson equations and traditional (1PI) RG calculations. In our formalism however, the hierarchy truncates at the level of the action. This can be seen immediately from Eq. (18) since it is clear that when the effective action is truncated at any order in the skeleton expansion, the kernel on the right side of (18) is zero when the largest number of propagators that appears in any diagram in the effective action is less than m+1. As will be explained in Sec. III C, the hierarchy can be truncated at an even earlier level.

We show below how to rewrite the flow equations (18) in a more convenient form. The stationary condition is δΦκ[ϕ,G]δG|G=Gκ=0,(19)and using Eqs. (11)–(14) and (16), the variation produces a Dyson equation for the nonperturbative two-point function in terms of the two-kernel Σ: Gκ-1=Gno·int-1-Rκ-Σκ(ϕ,Gκ).(20)Using (20) we have ∂κGκ=-Gκ(∂κGκ-1)Gκ=Gκ(∂κ(Rκ+Σκ))Gκ.(21)The first two equations in the hierarchy (18) can now be rewritten using (16) and (21) as ∂κΣκ(P)=12∫dQ∂κ[Σκ(Q)+Rκ(Q)]Gκ2(Q)Λκ(P,Q),(22)∂κΛκ(P,K)=12∫dQ∂κ[Rκ(Q)+Σκ(Q)]Gκ2(Q)ϒκ(P,K,Q).(23)These equations are shown in Fig. 3. The grey boxes denote the kernels Σ, Λ and ϒ (the specific kernel is indicated by the number of legs attached to the box). The crosses indicate the insertion ∂κ(Σκ(Q)+Rκ(Q)).

Finally, we note that the flow equation for the two-kernel Σ can be rewritten in terms of the Bethe-Salpeter (BS) vertex. Iterating Eq. (22) we obtain ∂κΣκ(P)=12∫dQ∂κRκ(Q)Gκ2(Q)Mκ(P,Q),(24)with Mκ(P,K)=Λκ(P,K)+12∫dQΛκ(P,Q)Gκ2(Q)Mκ(Q,K).(25)

In the 2PI formalism, we can also define nonperturbative vertices in terms of the change in the effective action with respect to variations in the field evaluated at the stationary point. These are usually called ‘physical’ vertices. The physical four-point function, which we call V, can be written in terms of the BS vertex as V=λ+3(M-Λ).(26)We note that the vertex V involves a resummation in all three (s, t and u) channels, but using our shorthand notation which suppresses indices, the three channels combine to produce the factor (3) in Eq. (26). Details of the derivation of (26) are given in Refs. [31,72].

In order to do numerical calculations, we rotate to Euclidean space by defining the Euclidean variables: q0→iq4,dQ→idQE,Q2→-QE2,(27)m2→mE2,λ→-iλE,δλ→-iδλE,Gno·int-1→i(Gno·int-1)E,Σ→-iΣE⇒G-1→iGE-1=i(PE2+mE2+ΣE),Λ→iΛE,ϒκ→-iϒκE,M→iME,,V→iVE.Rκ=-iRκE.(28)The extra factors of i in the definitions of λE and RE remove the factors that were introduced in the definitions λphys=iλ [under Eq. (2)] and R^=iR [under Eq. (10)]. From this point forward we suppress the subscripts which indicate Euclidean space quantities. The flow equations (22) and (23), and the BS equation (25) have the same form in Euclidean space. The Euclidean space Dyson equation is G-1(P)=Gno·int-1(P)+Σ(P),(29)and the physical vertex becomes V=-λ+3(M-Λ).(30)The regulator function we use has the Euclidean momentum space form Rκ(Q)=Q2eQ2/κ2-1.(31)

Physical considerations dictate the renormalization conditions which are enforced on the nonperturbative quantum n-point functions that are obtained at the κ=0 end of the flow. We use standard renormalization conditions: G0-1(0)=m2,M0(0,0)=-λ.(32)The quantum n-point functions in (32) are obtained by solving the integro-differential flow equations (22), (23), starting from some set of initial conditions at κ=μ. The regulator function Rκ [Eq. (31)] is chosen so that at the scale κ=μ the theory is described by the classical action and the initial conditions for the flow equations can be taken from the bare masses and couplings of the original Lagrangian.

It is clear that one must know the values of the bare parameters from which to start the flow at the beginning of the calculation. A different choice of bare parameters will give different quantum n-point functions at the end of the flow, and therefore different renormalized masses and couplings. The procedure to figure out the values of the bare parameters that will satisfy the chosen renormalization conditions is called tuning. Starting from an initial guess for the bare parameters, we solve the flow equations, extract the renormalized parameters, adjust the bare parameters either up or down depending on the result, and solve the flow equations again. The calculation is repeated until the chosen renormaliation condition is satisfied to the desired accuracy.

Since RG procedure involves initializing the flow equations at an ultraviolet scale κ=μ, and also enforcing renormalization conditions at the κ=0 end of the flow, we must address the question of whether or not the initial and renormalization conditions can be defined consistently.

Consider an arbitrary n-point kernel of the form Λκ(m)(P1,P2…)=Λ˜κ(m)(P1,P2…)+C(P1,P2…)(33)where Λ˜(m)(P1,P2…) is the result obtained from integrating the corresponding flow equation, and C(P1,P2…) is a κ independent integration constant. In the limit κ→μ we require that the vertex function approaches a momentum independent constant: limκ→μΛκ(m)(P1,P2…)=Λμ(m)(P1,P2…)≡-λμ(34)comparing Eqs. (33) and (34) we have C(P1,P2…)=-λμ-Λ˜μ(m)(P1,P2…),(35)so that (33) becomes Λκ(m)(P1,P2…)=-λμ+Λ˜κ(m)(P1,P2…)-Λ˜μ(m)(P1,P2…).(36)

Now we look at the κ→0 end of the flow and determine the conditions under which we can enforce the renormalization condition Λ0(0,0,…)=-λ. We start by adding and subtracting two different terms to the original vertex, and grouping into square brackets the differences we will consider below. This gives Λκ(m)(P1,P2…)=-λ+[Λκ(m)(P1,P2…)-Λ0(m)(P1,P2…)]+[Λ0(m)(P1,P2…)-Λ0(m)(0,0,…)].(37)The renormalization condition is satisfied if the square brackets go to zero in the limit that κ and the momentum arguments go to zero. Setting κ=0 and using Eq. (36) we obtain []=(Λ˜0(m)(P1,P2…)-Λ˜0(m)(0,0…))-(Λ˜μ(m)(P1,P2…)-Λ˜μ(m)(0,0…)).(38)The second term in (38) is zero in the limit μ≫P, since this is (by construction) the limit in which loop contributions are suppressed and the momentum dependence of the vertex disappears. We have therefore shown that the renormalization condition will be consistent with the initial condition if Z=lim(P1,P2…)→0(Λ˜0(m)(P1,P2…)-Λ˜0(m)(0,0…))→0.(39)In the next section, we will show that this condition is satisfied if the truncation of the hierarchy in (18) is performed correctly.

We comment on the fact that the discussion above is misleading in one important way. It appears that the bare vertex λμ cancels exactly when we go from Eq. (36) to Eq. (37). If this were true, the initial condition and the renormalization condition would be unconnected to each other, and tuning would not be possible. The apparent cancellation occurs because the self-consistent nature of the set of coupled equations for kernels with different numbers of legs is not evident when we consider one kernel in isolation.

Finally, we note that renormalization conditions are typically defined on vertices constructed from resummed kernels [see Eq. (32)] and not the kernels themselves. However, the condition derived above (39) is still sufficient to guarantee the consistency of the procedure. The crucial point is that the kernels are two-particle-irreducible (see Fig. 2) and therefore when they are chained together to form a resummed BS vertex, no additional subdivergences are produced. It is easy to see how this works in our calculation, where we only need to enforce a renormalization condition on the four-point function (as will be explained in the next section). When we define the renormalization condition on the BS vertex instead of the kernel Λ, the result is only a shift in the final value of the bare coupling λμ that is produced by the tuning procedure.

Now we return to the issue of the truncation of the hierarchy of flow equations in Eq. (18). The kernels obtained from direct functional differentiation of the action using (15) will automatically satisfy the correct flow equations (18). As explained in Sec. III A, this is just an obvious application of the chain rule. Next we observe that if a given kernel obtained from functional differentiation satisfies the condition (39), then using the analysis of the previous section, we know it will also satisfy Λ0(m)(0,0⋯)=-λ and Λμ(m)(0,0⋯)=-λμ. The conclusion is that we do not need to solve its flow equation (since the result from solving the flow equation would be precisely equal to the expression obtained from the functional integration, with the addition of the appropriate constant). The smallest value of m for which (39) is satisfied is the “terminal” kernel which truncates the hierarchy of flow equations. After we have identified the terminal kernel, the set of flow equations for the kernels with 2×(1,2,3,…m-1) legs can then be solved self-consistently.

The final step is to show that the flow equation for each kernel can be initialized at the classical solution, which is just the corresponding bare coupling. This is just a consequence of the structure of the flow equations, in which the kernel with 2m legs is constructed by calculating a one-loop integral obtained by joining two legs of the kernel with 2(m+1) legs (see Fig. 3). If Λ0(m+1) is finite up to a momentum independent bare coupling constant, then clearly the result for Λ0(m) obtained from solving a flow equation of the form (18) will also be finite.

In order to identify the terminal kernel, we need to know under what circumstances the condition (39) will be satisfied by a given 2PI kernel Λκ(m). We start by looking at an example where it will not. We consider the self-energy diagram on the right side of Fig. 4 which gives Σ0(P)=-λ26∫dQ∫dLG0(L)G0(L+Q)G0(P+Q).(40)This sunset contribution to the self-energy is produced by the BBALL diagram in the effective action (see Fig. 1). The quantity Z in Eq. (39) now takes the form Z=-λ26∫dQ∫dLG0(L)G0(L+Q)[G0(P+Q)-G0(Q)],(41)=-λ26P2∫dQ∫dLG0(L)G0(L+Q)[G0′(Q)+⋯],(42)where in the last line we have expanded around P2=0. The prime denotes differentiation with respect to Q2 and the dots represent terms that are higher order in P2/Q2. It is clear that the divergent L integral is unaffected by the subtraction, and therefore we cannot conclude that Z→0 when P approaches zero.

Another example is the contribution to the four-kernel Λ from the last diagram in the first line of Fig. 2. If we route the momenta as shown in the right side of Fig. 4 and rescale momenta as described above, the divergence in the L2 integration is unaffected by the subtraction, and therefore the condition (39) is not satisfied.

In general, any loop that does not necessarily carry one of the external momenta is a “bad” loop. If a kernel does not have any bad loops, it satisfies (39) and its flow equation does not have to be solved. The smallest of these kernels is the terminal kernel. In the following two subsections, we explain how to apply these ideas to the 2PI calculation at the three- and four-loop level.

We initialize the flow of the two- and four-kernels, Σμ(P)=mμ2-m2,(43)Λμ(P,K)=-λμ,(44)and we take the propagator in the ultraviolet limit as Gμ-1(P)=P2+mμ2.

Schematically the numerical procedure can be described as follows: (1)

Choose values for the physical mass m and coupling λ. We use always m=1, which means we give all dimensionful quantities in mass units.

The differential equations are solved using a logarithmic scale by defining the variable t=lnκ/μ, in order to increase sensitivity to the small κ region where we approach the quantum theory. We use κmax=μ=100, κmin=10-2 and Nκ=50. We have checked that our results are insensitive to these choices. In addition, we have checked for possible dependence on the form of the regulator function by using a generalization of (31): Rκ(Q;z)=κ2(Q2/κ2)z(eQ2/κ2-1)z.(50)The original expression (31) corresponds to the choice of exponent z=1. Using z=1/2 and z=1/4 produces results that are virtually identical.

The four-dimensional momentum integrals are written in the imaginary time formalism as ∫dKf(k0,k→)=∑n∫d3k(2π)3f(mtn,k→),(51)with mt=2πT. Numerically we take Nt terms in the summation with β=1T=Ntat where the parameter at is the lattice spacing in the temporal direction.

The integrals over the three-momenta are done in spherical coordinates and using Gauss-Legendre integration. We use typically Nx=Nϕ=8 points for the integrations over the cosine of the polar angle and the azimuthal angle. The dependence on these angles is weak, and results are very stable when Nx and/or Nϕ are increased. To calculate the integral over the magnitude of the three-momenta, we define a spatial length scale analogous to the inverse temperature L=asNs where as is the spatial lattice spacing and Ns is the number of lattice points.

The numerical method replaces a continuous integration variable with infinite limits by a discrete sum over a finite number of terms. For numerical accuracy, we need generically that the upper limit of the sum is big and the step size is small. This means we require pmax∼1as≫1 and Δp∼1L=1Nsas≪1. The number of lattice points Ns is limited by memory and computation time, and therefore there is a limit on how small as can be taken while maintaining Nsas large. However, there is another more subtle issue that limits how small we can choose as. The theory has a Landau pole at a scale that decreases when λ increases. When λ becomes large, as must increase (pmax must decrease) so that the integrals are cut off in the ultraviolet at a scale below the Landau scale. However, decreasing the ultraviolet cutoff pmax will eventually cause important contributions from the momentum phase space to be missed. When λ has increased to the point that the Landau scale has moved down and dipped into the momentum regime over which the integrand is large, physically meaningful results cannot be obtained. In our calculation we have determined that the maximum coupling we can calculate is λ≈5.

Finally, we note that it is well known that scalar ϕ4 theory in four dimensions is noninteracting if it is considered as a fundamental theory valid for arbitrarily high momentum scales (quantum triviality), but the renormalized coupling is nonzero if the theory has an ultraviolet cutoff and an infrared regulator. In our calculation the mass m regulates the infrared and the lattice spacing parameter provides an ultraviolet cutoff.