Functional renormalization group and 2PI effective action formalism
FFunctional renormalization group and 2PI effective action formalism
Jean-Paul Blaizot a , Jan M. Pawlowski b , Urko Reinosa c a Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, 91191 Gif-sur-Yvette Cedex, France b Institut für Theoretische Physik, University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany c Centre de Physique Théorique (CPHT), CNRS, Ecole Polytechnique,Institut Polytechnique de Paris, Route de Saclay, F-91128 Palaiseau, France.
Abstract
We combine two non-perturbative approaches, one based on the two-particle-irreducible (2PI) action, theother on the functional renormalization group (fRG), in an effort to develop new non-perturbative approxi-mations for the field theoretical description of strongly coupled systems. In particular, we exploit the exact2PI relations between the two-point and four-point functions in order to truncate the infinite hierarchyof equations of the functional renormalization group. The truncation is “exact” in two ways. First, thesolution of the resulting flow equation is independent of the choice of the regulator. Second, this solutioncoincides with that of the 2PI equations for the two-point and the four-point functions, for any selectionof two-skeleton diagrams characterizing a so-called Φ -derivable approximation. The transformation of theequations of the 2PI formalism into flow equations offers new ways to solve these equations in practice, andprovides new insight on certain aspects of their renormalization. It also opens the possibility to developapproximation schemes going beyond the strict Φ -derivable ones, as well as new truncation schemes for thefRG hierarchy. Keywords: key words, key words, key words PACS codes here, in the form:
1. Introduction
There is an obvious need to develop non-perturbative methods in quantum field theory in order to dealwith systems, both in-equilibrium and out-of-equilibrium, that are strongly coupled. A natural startingpoint for discussing such methods is the generating functional of connected Green’s functions W [ J ] , orequivalently, the one-particle-irreducible (1PI) effective action Γ[ φ ] , a functional of the field expectationvalue. This functional can be obtained as the Legendre transform of W [ J ] with respect to a source J .Alternatively, it can be derived from more general functionals that involve Legendre transforms with respectto additional sources. Such functionals are commonly referred to as n -particle-irreducible effective actions Email addresses: [email protected] (Jean-Paul Blaizot), [email protected] (Jan M.Pawlowski), [email protected] (Urko Reinosa)
Preprint submitted to Annals of Physics March 1, 2021 a r X i v : . [ h e p - t h ] F e b n PI in short) and depend not only on the field expectation value, but also on the propagator, and possiblyon a number of dressed n -point vertices. This paper focuses on the 2PI effective action, denoted Γ[ φ, G ] ,which depends both on the field expectation value φ and on the propagator G . It coincides with the 1PIeffective action Γ[ φ ] once G is fixed to its stationary value determined by the condition δ Γ[ φ, G ] /δG = 0 .When φ also assumes its stationary value, the 1PI and 2PI functionals coincide with the free energy of thesystem.Our goal in this paper is to explore the connections between techniques based on the 2PI formalism,and the functional renormalization group (fRG). The fRG is usually formulated in terms of the 1PI effectiveaction, [1, 2, 3]. It has led to many applications in different areas (for reviews see [4, 5, 6, 7, 8, 9, 10, 11,12, 13, 14]. One of its major advantages is to allow for the formulation of approximations at the level of theeffective action itself, a prominent example being the local potential approximation [1]. A drawback of themethod is, however, that the flow for the effective action yields naturally an infinite hierarchy of equationsfor the n -point functions, analog to the Dyson-Schwinger hierarchy, whose solution requires in practice sometruncation. Such truncations are usually justified on the basis of physical considerations, or the ease oftheir implementation, which may introduce uncontrollable elements. One example of such uncontrollableelements relates to the presence of a regulator that controls the flow of the various n -point functions. Withina given truncation, final values of these n -point functions may depend on the choice of this regulator, whichintroduces uncertainties that are not easy to control a priori. The 2PI formalism has been developed initially in the context of the non-relativistic many-body problem[18, 19], and formulated in [20] in terms of a 2PI effective action more suited to applications to relativisticfield theories. An essential element of the 2PI effective action is a specific functional of the single-particlepropagator, Φ[ G ] , often referred to as the Luttinger-Ward (LW) functional, which is the sum of all two-particle irreducible skeleton diagrams evaluated with the full propagator G . The LW functional is also thegenerating functional for the self-energy, and other 2PI n -point functions that are obtained as functionalderivatives of Φ[ G ] with respect to G . Selecting a specific class of skeleton diagrams in Φ[ G ] yields so-called Φ -derivable approximations which have special symmetry conserving properties [21, 22] (see however thediscussion in [23], and references therein). In the context of relativistic field theories, the 2PI formalismhas been applied to the study of systems both in-equilibrium (see e.g. [24]) and out-of-equilibrium (see e.g.[25] and references therein), including the calculation of transport coefficients (see e.g. [26, 27, 28]). Onemay also mention a recent application in the more formal context of the SYK model [29]. There has alsobeen much effort to extend their application to gauge theories (see e.g. [30, 31, 32, 33]). The 2PI formalismleads to non-linear equations for the self-consistent propagator that are often difficult to solve, and may haveunphysical solutions that can annihilate with physical ones in some particular cases (see e.g. [34, 35]). In the One possibility is to try to select appropriate (optimal) regulators that minimize the dependence upon small variations,see e.g. [15, 16, 17]. ϕ theory in four dimensions. We shall in particular analyze further the truncation of the fRGflow equations that has been proposed in Ref. [39], and that exploits the relation between the four-pointfunction and the two-point function in the 2PI formalism. The truncation that we use is based on an exactrelation. It leads on the one hand to an exact reformulation of the flow equations for the two-point andfour-point functions, and on the other hand to an exact reformulation of the 2PI equations in terms of flowequations. Approximations are formulated in terms of the selection of skeletons in the LW functional, inthe spirit of Φ -derivable approximations. Thus, aside from providing a possible truncation of the usual flowequations of the fRG, conversely, the fRG provides an alternative way to solve the 2PI equations. It alsogreatly clarifies some issues concerning the renormalization of Φ -derivable approximations. Ref. [39] waslimited to the case of a super-renomalizable theory ( ϕ theory in three dimensions). Here we consider themore general case of renormalizable theories on the example of ϕ scalar theory in four dimensions. Thisallows us to analyze more fully the specificities of the renormalization of Φ -derivable approximations fromthe point of view of the exact renormalization group.We emphasize that our use of 2PI relations is distinct from that developed in [40, 41] or [42, 43]. Inthese works, one writes flow equations for the 2PI effective action, which in turn translates into a hierarchyof equations for the 2PI n -point functions. What we use in this paper is rather a hybrid scheme, where weclose the 1PI hierarchy of flow equations by using the 2PI relation between the two-point and four-pointfunctions [39]. While the latter relation can be seen as part of the 2PI hierarchy, we do not focus on the 2PI n -point functions, as done for instance in [44] (see also [45, 46, 47], and similarly [48]), but rather on differentobjects, identified as loop truncations of the four-point function. These objects emerge naturally in the flowformulation of Φ -derivable approximations and turn out to play an important role in their renormalization,whether this is achieved through the flow equations or via the more standard diagrammatic approach. Aswe shall see, the renormalization of Φ -derivable approximations via the flow equations brings both insightand flexibility. This flexibility can be exploited to extend standard Φ -derivable approximations beyond theirstandard diagrammatic formulations.The outline of the paper is as follows. In Sect. 2, we provide a short summary of both the functionalrenormalization group based on the 1PI effective action, and the 2PI formalism. We show how the centralequations of the 2PI formalism, the gap equation that defines the two-point function self-consistently, aswell as the equation that relates this two-point function to the free-energy, can be transformed into flowequations: one that relates the self-energy to the four-point function, and another that relates the free-energy The discussion could also be extended to other systems, in particular fermionic ones where truncations at the level offour-point functions are often employed.
3o the two-point function. These equations are identical to the flow equations for the two- and zero-pointfunctions that can be deduced from the fRG. In Sect. 3, we show how the relation between the two-pointfunction and the four-point function of the 2PI formalism allows us to close the infinite fRG hierarchyof equations for the 1PI n -point functions at the level of the four-point function. At this point we haveobtained a possible calculation scheme where a given Φ -derivable approximation is exactly reformulatedin terms of flow equations. To this point, all the relevant equations depend on an ultraviolet cutoff. Thefollowing sections will be concerned with the elimination of this cutoff dependence through renormalization.In Sect. 4 we develop further the analysis of the flow equation for the four-point functions, in preparationfor the renormalization proper, which is addressed in the following two sections. In particular, we emphasizethe role of auxiliary four-point functions truncated at a given loop order corresponding to the loop orderof the Φ -derivable approximation considered. In Sect. 5, we show that the equations derived in Sect. 4are finite and can be made independent of the bare parameters. One can then extend the strategy ofthe standard 1PI flow equations to renormalize the 2PI approximation, without having to introduce thecounterterms of the diagrammatic approach. In Sect. 6, we use these same flow equations to make contactwith the diagrammatic approach. The flow equations clarify how the divergences are distributed amongthe various n -point functions involved in a given Φ -derivable approximation, and provide clear prescriptionsfor the explicit determination of the counterterms. Overall, the flow equations bring new insights on thewhole renormalization procedure of the diagrammatic approach. In Sect. 7, we take a general view onwhat has been achieved in previous sections and emphasize features that could potentially be generalizedto other approaches, beyond the Φ -derivable framework. We also propose yet a third, more practical flowreformulation of Φ -derivable approximations that combines the benefits of the flow equations derived inSecs. 3 and 4. In the last section we discuss possible extensions of Φ -derivable approximations, as well asnew possible truncations of the fRG. Appendix A illustrates with the simple two-loop example many of theconcepts developed in the main text. The remaining appendices gather technical material that complementsthe developments of the main text. In particular, we show in Appendix B how, for certain generic collectionsof diagrams, it is possible to hide any reference to the quartic coupling (of the considered ϕ theory) usingthe exact four-point function. This property is crucial to the strategy followed in the present work and itmay have applications elsewhere.
2. A connection between the 2PI formalism and the functional renormalization group
We consider in this paper a scalar field theory with Euclidean action S [ ϕ ] = (cid:90) d d x (cid:26)
12 ( ∂ϕ ( x )) + m ϕ ( x ) + λ b ϕ ( x ) (cid:27) , (1)4n d = 4 dimensions. When discussing systems at finite temperature, the integration over the (imaginary)time is restricted to the interval [0 , β = 1 /T ] . For clarity, we put a subscript b on the mass and the couplingconstant, and refer to m b and λ b as to bare parameters, although that terminology takes its full (andstandard) meaning only when we discuss renormalization issues, which will come later. Our goal in this section is to relate two non-perturbative formulations of the same cut off theory, leavingaside the issues of ultraviolet divergences and renormalization which will be addressed in later sections.Thus, all the momentum integrals that we shall introduce will be assumed to be evaluated with an ultravioletregulator characterized by a cutoff scale Λ uv . For instance, this can be a sharp cutoff, in which case all theloop integrals are limited to momenta smaller than Λ uv . Except in a few cases where explicit calculationsare done, this ultraviolet regulator will be left implicit. The one-particle-irreducible (1PI) effective action Γ[ φ ] is obtained from the action S [ ϕ ] through a Leg-endre transform, with φ denoting the expectation value of the field in the presence of an external source: weintroduce a source J ( x ) coupled to the field ϕ ( x ) , and write the generating functional of connected Green’sfunctions as W [ J ] = ln (cid:90) D ϕ e − S [ ϕ ]+ (cid:82) x J ( x ) ϕ ( x ) , δW [ J ] δJ ( x ) = (cid:104) ϕ ( x ) (cid:105) ≡ φ ( x ) . (2)The Legendre transform of W [ J ] yields Γ[ φ ]Γ[ φ ] + W [ J ] = (cid:90) x J ( x ) φ ( x ) , δ Γ[ φ ] δφ ( x ) = J ( x ) . (3)In thermal equilibrium, the effective action Γ[ φ ] is the free energy for a given expectation value φ of the field(to within a factor /β ).The 1PI n -point functions are obtained from Γ[ φ ] by functional differentiation with respect to φ ( x ) .More precisely, we define Γ ( n ) ( x , . . . , x n ; φ ) ≡ δ n Γ δφ ( x ) · · · δφ ( x n ) , (4)where we leave explicit the functional dependence on the background field φ . By construction, the n -pointfunctions are invariant under any permutation of their arguments, a property known as crossing symmetry We keep the notation simple for the field ϕ , which here is also to be understood as the bare field. We shall switch to therenormalized field when discussing renormalization in later sections (see e.g. Sect. 6.1). Throughout this paper we use a shorthand notation for the integrations over spatial and momentum integrals, viz. (cid:90) x ≡ (cid:90) d d x , (cid:90) q ≡ (cid:90) d d q (2 π ) d . Γ ( n ) ( p , . . . , p n ; φ ) ≡ (cid:90) x · · · (cid:90) x n e i (cid:80) nj =1 p j x j Γ ( n ) ( x , . . . , x n ; φ ) . (5)Our choice of a common sign for all the exponential factors corresponds to a convention of all incoming orall outgoing momenta. For the diagrammatic representation to be used in this work, we choose incomingmomenta. It is easily seen that Γ ( n ) ( p , . . . , p n ; φ ) = δ n Γ δφ ( − p ) · · · δφ ( − p n ) , (6)which makes the crossing symmetry explicit in terms of the momentum variables.In the case of a constant background field φ ( x ) = φ , the n -point functions Γ ( n ) ( x , . . . , x n ; φ ) are invariantunder translations of the coordinates, and it is convenient to factor out of the definition of their Fouriertransform Γ ( n ) ( p , . . . , p n ; φ ) the δ -function that expresses the conservation of the total momentum: Γ ( n ) ( p , . . . , p n ; φ ) → δ ( d ) ( p + · · · + p n ) Γ ( n ) ( p , . . . , p n ; φ ) . (7)We shall refer to the function Γ ( n ) ( p , . . . , p n ) in the right-hand side of (7) as to the “reduced” Fouriertransform. Note that we shall use the same notation for both the reduced and the full Fourier transforms,unless confusion may arise, in which case we shall specify which one is used. It should be stressed thatthis reduced Fourier transform is actually a function of n − independent variables. The reason for usinga redundant notation is that it makes it simpler to track down the crossing symmetry of Γ ( n ) ( p , . . . , p n ) . Finally, we mention that a similar factorization of the momentum conserving delta-function applies in asense to the effective action itself. Indeed, for a constant background configuration, the effective action isproportional to the space time volume (cid:82) x δ ( d ) (0) (recall our convention for δ -functions in momentumspace) Γ[ φ ] → δ ( d ) (0) Γ( φ ) , (8)where Γ( φ ) in the right-hand side defines the so-called effective potential. We absorb a factor of (2 π ) d in the definition of the functional derivative δ/δϕ ( p ) in momentum space. We shall absorb asimilar factor in the definition of the δ -function in momentum space. One could decide to denote the reduced Fourier transform as a function of n − variables among p , . . . p n , for instance as Γ ( n ) ( p , . . . , p n − ) . In this case, however, crossing symmetry, which just amounts to permutation invariance of Γ ( n ) ( p , . . . , p n ) ,translates into permutation invariance of Γ ( n ) ( p , . . . , p n − ) plus invariance under the substitution of any momentum p i among p , . . . , p n − by − (cid:80) n − j =1 p j . S [ ϕ ] a non-localregulator term ∆ S κ [ ϕ ] of the form [14] ∆ S κ [ ϕ ] = 12 (cid:90) x (cid:90) y R κ ( x − y ) ϕ ( y ) ϕ ( x ) = 12 (cid:90) q R κ ( q ) ϕ ( q ) ϕ ( − q ) , (9)where the parameter κ runs continuously from a high momentum scale Λ (to be specified below) down to0 where the original theory is recovered. The regulator ∆ S κ modifies the free propagator. Its role is thatof a mass term that suppresses the fluctuations with momenta lower than κ , while leaving unaffected thosewith momenta greater than κ . This is usually achieved with a smooth cutoff function R κ ( q ) such that, forsmall momenta q , R κ ( q (cid:28) κ ) (cid:39) κ , while at large momenta, R κ ( q ) goes sufficiently rapidly to 0 as q > ∼ κ so that ∂ κ R κ ( q ) can play the role of an ultraviolet cutoff in the flow equations. A convenient choice for theregulator, to which we shall occasionally refer to, is one which substitutes in the propagator the scale κ forthe momentum q when q ≤ κ , and vanishes for q > κ , that is, R κ ( q ) = ( κ − q ) θ ( κ − q ) , [15]. We shall alsouse at some places a sharp cutoff that completely eliminates all contributions to loop integrals coming frommomenta below the scale κ : in this case the loop momenta run strictly from κ to Λ uv .We may consider the addition of the regulator term as a (non local) continous “deformation” of theoriginal ϕ theory. The generating functional of 1PI n -point functions in this deformed theory, Γ κ [ φ ] , obeys the following flow equation [1, 2, 3] ∂ κ Γ κ [ φ ] = 12 (cid:90) x (cid:90) y ∂ κ R κ ( x − y ) G κ ( y, x ; φ ) , (10)where G κ ( x, y ; φ ) denotes the full propagator in the deformed theory, related to the two-point function Γ (2) κ ( z, y ; φ ) by (cid:90) z G κ ( x, z ; φ ) (cid:104) Γ (2) κ ( z, y ; φ ) + R κ ( z − y ) (cid:105) = δ ( d ) ( x − y ) . (11)In Fourier space, this becomes ∂ κ Γ κ [ φ ] = 12 (cid:90) q ∂ κ R κ ( q ) G κ ( q, − q ; φ ) , (12)with (cid:90) r G κ ( p, − r ; φ ) (cid:104) Γ (2) κ ( r, − q ; φ ) + δ ( d ) ( r − q ) R κ ( r ) (cid:105) = δ ( d ) ( p − q ) , (13) The later is defined as in Eqs. (2)-(3) with S replaced by S + ∆ S κ and with the additional convenient subtraction of (cid:82) x (cid:82) y R κ ( x − y ) φ ( y ) φ ( x ) . igure 1: The two diagrams contributing to the flow of the two-point function, Eq. (15). The internal lines represent dressedpropagators, G κ . The circled cross represents an insertion of ∂ κ R k . The vertices denoted by grey dots are respectively thethree-point functions Γ (3) κ (left) and the four-point function Γ (4) κ (right). Note that Γ (3) κ vanishes when φ vanishes, leaving thenthe right diagram as the only contribution to the flow of the two-point function. and where G κ ( p, q ) and Γ (2) κ ( p, q ) denote the full Fourier transforms (see above). In the case of a constantbackground, Eq. (12) retains its form if we replace Γ κ [ φ ] by the effective potential (8) and G κ ( q, − q ) by thereduced propagator (7). This is because the same volume factor δ ( d ) (0) = (cid:82) x can be factored out on bothsides of the equation. In contrast, Eq. (13) becomes G − κ ( q, − q ; φ ) = Γ (2) κ ( q, − q ; φ ) + R κ ( q ) , (14)with G κ ( q, − q ; φ ) and Γ (2) κ ( q, − q ; φ ) the reduced Fourier transforms. In what follows, and unlike our choiceof notation for the higher reduced n -point functions, we denote G κ ( q, − q ; φ ) and Γ (2) κ ( q, − q ; φ ) simplyas G κ ( q ; φ ) and Γ (2) κ ( q ; φ ) , the crossing symmetry implying in this case that G κ ( q ; φ ) = G κ ( − q ; φ ) and Γ (2) κ ( − q ; φ ) = Γ (2) κ ( q ; φ ) .By taking two derivatives of Eq. (10) with respect to φ , exploiting Eq. (11), and Fourier transformingwhile restricting to a constant background field, one obtains the flow equation for the two-point function: ∂ κ Γ (2) κ ( p ; φ ) = (cid:90) q ∂ κ R κ ( q ) G κ ( q, φ ) ×× (cid:26) Γ (3) κ ( q, p, − p − q ; φ ) G κ ( q + p, φ )Γ (3) κ ( p + q, − p, − q ; φ ) −
12 Γ (4) κ ( q, − q, p, − p ; φ ) (cid:27) , (15)where it is understood that here all the n -point functions are the reduced ones. A diagrammatic illustrationof this equation is given in Fig. 1. Flow equations for other n -point function are obtained similarly by takingfurther functional derivatives of Γ[ φ ] .In this paper, we shall be considering only systems with φ = 0 . External background fields will beintroduced occasionally only as intermediate tools to derive equations of motion. When φ = 0 , the equation Alternatively, one could also start from Eq. (12), take two field derivatives in the momentum representation while exploitingEq. (13), and then restrict oneself to a constant background field configuration. Γ (3) κ ( p, q, − p − q ; φ = 0) = 0 . We shall also use thesimplified notation Γ ( n ) ( p , . . . , p n ; φ = 0) = Γ ( n ) ( p , . . . , p n ) and similarly G κ ( p ; φ = 0) = G κ ( p ) . The 2PI formalism involves an additional Legendre transform with respect to a source K ( x, y ) coupledto the bilinear ϕ ( x ) ϕ ( y ) . One defines the functional W [ J, K ] = ln (cid:90) D ϕ e − S [ ϕ ]+ (cid:82) x J ( x ) ϕ ( x )+ (cid:82) x (cid:82) y K ( x,y ) ϕ ( x ) ϕ ( y ) , (16)together with the fields φ ( x ) and G ( x, y ) conjugated to the sources J ( x ) and K ( x, y ) : δW [ J, K ] δJ ( x ) = (cid:104) ϕ ( x ) (cid:105) ≡ φ ( x ) , δW [ J, K ] δK ( x, y ) = (cid:104) ϕ ( x ) ϕ ( y ) (cid:105) ≡ G ( x, y ) + φ ( x ) φ ( y ) . (17)The Legendre transform of W [ J, K ] yields the 2PI effective action Γ[ φ, G ]Γ[ φ, G ] + W [ J, K ] = (cid:90) x J ( x ) φ ( x ) + (cid:90) x (cid:90) y K ( x, y ) (cid:2) G ( x, y ) + φ ( x ) φ ( y ) (cid:3) , (18)with δ Γ[ φ, G ] δφ ( x ) = J ( x ) + 2 (cid:90) y K ( x, y ) φ ( y ) , δ Γ[ φ, G ] δG ( x, y ) = K ( x, y ) . (19)The relation between the 1PI effective action Γ[ φ ] and the 2PI effective action Γ[ φ, G ] is easily unveiled bynoticing that W [ J ] as defined in Eq. (2) is obtained from W [ J, K ] in the limit K → . Taking the samelimit in Eqs. (18)-(19) and comparing to Eq. (3) one finds that Γ[ φ ] and Γ[ φ, G ] coincide when G is chosento obey its equation of motion: Γ[ φ ] = Γ[ φ, G ] , δ Γ[ φ, G ] δG ( x, y ) = 0 . (20)We mention that functional derivatives with respect to G ( x, y ) need to be taken in a symmetrical sense: δδG ( x, y ) → (cid:20) δδG ( x, y ) + δδG ( y, x ) (cid:21) . (21)Indeed, it follows from Eq. (17) that G is crossing symmetric for any choice of sources, and, therefore, thevariations of the propagator need to satisfy this constraint. This does not play any crucial role for thederivative that appears in Eq. (20) but becomes crucial once one considers higher derivatives of Γ[ φ, G ] with9espect to G . Just as the 1PI effective action can be expressed in terms of 1PI diagrams, the 2PI effective action admitsan expansion in terms of 2PI diagrams [18, 49, 20]. In the case of a vanishing field expectation value φ = 0 , Γ[ G ] takes the form Γ[ G ] = 12 Tr log G − + 12 Tr G − G + Φ[ G ] , (22)where Tr O denotes the trace of the operator O that can be expressed either as an integral over space-timecoordinates or as an integral over 4-momenta: Tr O ≡ (cid:90) x O ( x, x ) = (cid:90) p O ( p, − p ) . (23)The functional Φ[ G ] in Eq. (22), often referred to as the Luttinger-Ward (LW) functional, plays a centralrole in the 2PI formalism. It is built as the sum of the “two-particle-irreducible” (2PI) diagrams of ordinaryperturbation theory, with no external lines, but evaluated with the full propagator G rather than the freepropagator G . These diagrams, sometimes also called “two-line-irreducible” diagrams, are diagrams thatcannot be split apart by cutting two lines. In the absence of external lines, this notion coincides with that of“skeleton” diagrams (or two-skeletons), that is diagrams in which one cannot isolate self-energy insertions.In the absence of any external field, the case treated in this paper, Φ[ G ] is a functional of the full propagator G . The skeletons diagrams contributing to Φ up to order four-loop are displayed in Fig. 2. One defines Figure 2: The three skeleton diagrams that contribute to Φ[ G ] at order four-loop. The black dots represent bare vertices λ b . similarly skeleton diagrams with external lines as obtained by taking successive functional derivatives of Φ[ G ] with respect to G . For two or more such derivatives, the corresponding skeleton diagrams are notcompletely two-particle-irreducible: they are 2PI only with respect to cuts of two lines that leave the two Similarly, because K ( x, y ) only enters W [ J, K ] in the combination K ( x, y ) ϕ ( x ) ϕ ( y ) , we can restrict ourself to symmetricalsources such that K ( x, y ) = K ( y, x ) . In fact, this is a necessary requirement if one wants to ensure the invertibility of theLegendre transform. At finite temperature, integration over 4-momenta is replaced by integration over 3-momenta and summation over Mat-subara frequencies ω n = 2 πnT . δ/δG on the same side of the cut.Let us emphasize that G ( x, y ) in Eq. (22) is a priori a general symmetric function of the two positionvariables x and y . In contrast, the equation of motion for G given in Eq. (20) defines a specific value for G that may be restricted by symmetry. For instance, in equilibrium and in the presence of a constant φ (suchas the vanishing background case considered here), the propagator determined by the equation of motionis translation invariant: G ( x, y ) = G ( x − y, ≡ G ( x − y ) . For many purposes, it is enough to restrict thefunctional Γ[ G ] to such propagators. In this case, it is readily seen that a trivial volume factor δ ( d ) (0) factorizes from all terms in Eq. (22). By dropping this factor, one is left with the reduced effective action Γ[ G ] = 12 (cid:90) p log G − ( p ) + 12 (cid:90) p G − ( p ) G ( p ) + Φ[ G ] , (24)in terms of a reduced Luttinger-Ward functional. Here, G − ( p ) stands for the inverse free propagator, G − ( p ) = p + m . The full propagator is obtained by extremizing the functional Γ[ G ] . This yields aself-consistent Dyson equation, commonly called a “gap equation” G − ( p ) = G − ( p ) + Σ( p ) , (25)where the self-energy functional Σ[ G ] is obtained from Φ[ G ] via a functional derivative, viz. Σ( p ) = 2 δ Φ[ G ] δG ( p ) . (26)The skeleton diagrams contributing to Σ up to order three-loop are displayed in Fig. 3. By construction,these diagrams do not contain subdiagrams that can be identified as self-energy insertions. Solving the gapequation (25) represents a major task in the 2PI formalism. After substituting the solution of this equationinto Eq. (22), one obtains the free energy as the sum of all the Feynman diagrams of ordinary perturbationtheory (i.e., irreducible and reducible and evaluated with propagators G ). A further differentiation of Φ[ G ] with respect to G yields the two-particle-irreducible kernel I ( q, p ) = 2 δ Σ( p ) δG ( q ) = 4 δ Φ δG ( q ) δG ( p ) = I ( p, q ) . (27) Below, we will discuss examples where the most general G ( x, y ) needs to be used. We recall that our convention for the functional derivative in momentum space includes an implicit factor (2 π ) d . The counting of loops depends on the quantity that one is looking at. Thus for instance a four-loop contribution to Φ generates three-loop contributions to Σ . Note however that the functional derivative does not change the dependence on thecoupling constant, and both the four-loop contribution to Φ and the three-loop contributions to Σ are of order λ . Note that when substituting G for G in Φ , one ends up with an over-counting of the perturbative diagrams. This over-counting is precisely corrected by the first two terms in Eq. (22) (see e.g. [37]). However, in contrast to what happens for Φ ,the substitution G → G in skeletons with external lines does not generate any over-counting. igure 3: The skeleton diagrams that contribute to Σ[ G ] up to order three-loop. These are obtained from the skeletons of Φ[ G ] shown in Fig. 2 by taking a functional derivative with respect to G , Eq. (26). This kernel can be used to construct the full four-point function Γ (4) ( q, p ) , via a Bethe-Salpeter (BS) equation Γ (4) ( q, p ) = I ( q, p ) − (cid:90) r Γ (4) ( q, r ) G ( r ) I ( r, p )= I ( q, p ) − (cid:90) r I ( q, r ) G ( r ) Γ (4) ( r, p ) . (28)This equation allows the calculation of the four-point function at specific values of the external momenta,namely Γ (4) ( q, p ) ≡ Γ (4) ( q, − q, p, − p ) , where it is here understood that Γ (4) ( q, − q, p, − p ) denotes the reducedFourier transform. As follows from Eq. (21), the functional derivative with respect to the propagator G ( q ) needs to be taken in a symmetrical sense I ( q, p ) = 2 δ Σ( p ) δG ( q ) −→ δ Σ( p ) δG ( q ) + δ Σ( p ) δG ( − q ) , (29)reflecting the crossing symmetry G ( − q ) = G ( q ) . The diagrammatic interpretation of this operation is illus-trated in Fig. 4, and the skeletons contributing to I up to two-loop order are given in Fig. 6 below. Thekernel I ( q, p ) is the two-line irreducible contribution to Γ (4) ( q, p ) in one particular channel. Borrowing fromthe particle physics terminology, we refer to this particular channel as the s -channel, with momenta p, − p entering and q, − q leaving. The other two channels are the t and u channels.It may not be immediately obvious why we need to consider the four-point function when calculating thefree energy and the self energy according to Eqs. (22) and (25), since these are functionals of the propagator.However, we shall see later that the four-point function and the corresponding BS equation play an importantrole in the renormalization, as they serve in particular to eliminate subdivergences both in the self-energyand in the free energy. They also emerge naturally when reformulating the 2PI formalism in terms of flowequations as we show below.Not unrelated to this, the four-point function appears when one looks at how the solution of the gap Recall that we use a convention of “incoming” momenta: a momentum p is counted as + p if it enters a vertex, and − p ifit leaves it. igure 4: The symmetric derivative (see text) of Σ( p ) (left diagram) generates the two contributions to I ( q, p ) displayed onthe right. These two contributions, of order λ , correspond to the t and the u channels respectively. equation, i.e. the self-consistent propagator G or the self-energy Σ , depends on parameters, in particular onthe mass m b . To see that, let us take the derivative of the self-energy, Eq. (26), with respect to m . Weobtain ∂ Σ( p ) ∂m = 2 (cid:90) q ∂G ( q ) ∂m δ Φ[ G ] δG ( q ) δG ( p ) = 12 (cid:90) q ∂G ( q ) ∂m I ( q, p ) . (30)Observe next that, according to Eq. (25), G − = p + m + Σ[ G ] , so that ∂G ( q ) ∂m = − G ( q ) ∂G − ( q ) ∂m = − G ( q ) (cid:18) ∂ Σ( q ) ∂m (cid:19) . (31)It follows that Eq. (30) can be written as (cid:90) q ∂ Σ( q ) ∂m (cid:18) δ ( d ) ( q − p ) + 12 G ( q ) I ( q, p ) (cid:19) = − (cid:90) q G ( q ) I ( q, p ) . (32)This equation can be read as a (continuous) matrix equation. Let us then define the matrix M ( q, p ) ≡ Figure 5: Graphical illustration of Eq. (33) for the two-loop skeleton. The derivative with respect to m b is denoted by aslash. When acting on a self-consistent propagator, this derivative generates the infinite tower of one-loop kernels I that areresummed by the BS equation, Eq. (28). δ ( d ) ( q − p ) + G ( q ) I ( q, p ) . It is easily checked, using Eq. (28), that the inverse matrix is given by13 − ( q, p ) = δ ( d ) ( q − p ) − G ( q ) Γ (4) ( q, p ) . Multiplying now both sides of Eq. (32) by this inverse matrixand using Eq. (28) once more, one arrives finally at ∂ Σ( p ) ∂m = − (cid:90) q G ( q ) Γ (4) ( q, p ) (33)where, as anticipated, the four-point function explicitly appears. Note that this result has taken into accountthe fact that the dressed propagator depends on m b both explicitly and implicitly, through its dependenceon the self-energy, itself a functional of the propagator. It is this self-consistency which is at the origin of theappearance of the four-point function in the final result (see also Fig. 5). The flow equation (42) discussedin the next subsection can be seen as a generalization of this equation (33).If all skeletons are kept in Φ[ G ] , all the functional relations written above are exact relations in the cutofftheory. But, of course, the main interest of the LW functional is to lend itself to particular approximations.The so-called “ Φ -derivable” approximations [22] consist in selecting a class of skeletons in Φ[ G ] and calculat-ing Σ and Γ (4) from the equations above. It is easy to verify that all the relations mentioned above remainvalid then, whichever group of skeletons is selected. It will be convenient in the foregoing discussion torefer to a systematic expansion of the skeleton diagrams. One such expansion consists in selecting skeletonswith an increasing number of loops . Since we shall constantly refer to this loop expansion in the presentdiscussion, we introduce a specific notation for the various objects that appear in this expansion. We denoteby Φ L the contribution to the LW functional of all the skeletons that contains up to L loops. For instance,the four-loop approximation to Φ[ G ] is the following functional of G : Φ L =4 [ G ] = λ b (cid:18)(cid:90) q G ( q ) (cid:19) − λ (cid:90) p (cid:90) q (cid:90) r G ( p ) G ( q ) G ( r ) G ( r + q + p )+ λ (cid:90) p (cid:90) q (cid:90) r (cid:90) k G ( p ) G ( q ) G ( r ) G ( k ) G ( r + q + p ) G ( k + q + p ) . (34)We shall call Φ ( l ) the l -loop contribution to Φ , with the factor λ l − left out, that is, we write Φ L = λ b Φ (2) − λ Φ (3) + · · · + λ L − ( − ) L Φ ( L ) . (35)Similar expansions will be used for other objects, such as the self-energy Σ ( Σ L , Σ ( l ) ), the irreducible kernel Another systematic expansion is that in number of components of the field, in the case of a scalar theory with O(N)symmetry [50]. Most of the results of this paper extend to the corresponding /N expansion. A l -loop diagram that contributes to Φ contains l − vertices. It is tacitly assumed that the counting in loops is here madewith respect to a given propagator G . Of course, when such a propagator is replaced by the solution of the gap equation, eachof the finite loop contributions that we are considering contains infinitely many “perturbative” loops with respect to the freepropagator G . ( I L , I ( l ) ) or the four-point function Γ (4) ( Γ (4) L , Γ (4 ,l ) ), viz. Γ (4) L = λ b − λ Γ (4 , + · · · + λ L +1b ( − ) L Γ (4 ,L ) . (36)The skeletons corresponding to Φ L =4 [ G ] are shown in Fig. 2, and those which contribute to Σ L =3 [ G ] and to I L =2 [ G ] in this approximation are displayed respectively in Figs. 3 and 6. Figure 6: The skeleton diagrams that contribute to the kernel I ( q, p ) up to order two-loop, i.e., the contributions to I ( q, p ) .These are obtained from the skeletons of Φ [ G ] by taking two functional derivatives with respect to G , which amounts to cuttingtwo lines in the diagrams of Φ , that are subsequently labelled respectively p and q . These diagrams are two-line irreducible inthe s channel, defined by the external lines that carry momentum p (or equivalently q ). Only the t channel contributions ofthe one-loop and two-loop diagrams are shown (the u channel diagrams are obtained from the t channel ones by exchangingthe q lines, as in Fig. 4). It is useful for the foregoing discussion to observe how the successive loop contributions to Γ (4) ( q, p ) build up as one takes successive skeletons into account in the loop expansion. At a given order in the loopexpansion, the diagrams of Γ (4 ,l ) ( q, p ) that are two-line-irreducible in the s -channel, are included into thekernel I ( l ) , while those which are reducible are obtained from iterations in the Bethe-Salpeter equation ofcontributions I ( l (cid:48) ) with l (cid:48) < l . An illustration is provided in Fig. 7 which displays the three contributionsto Γ (4 ,l =1) ( q, p ) . This illustration makes apparent a well-known feature of the Φ -derivable approximations:by focussing on a specific channel, one looses the crossing symmetry which is present in each order ofperturbation theory. Let us stress that, within the approximation defined by the replacement Φ → Φ L , thesolution to the BS equation (with I → I L − ), which we shall denote throughout Γ (4)Φ L , contains of course Γ (4) L but also infinitely many higher order contributions that make it distinct from Γ (4) L . In particular, whilethe various channels are always treated in a symmetrical way in Γ (4) L , this is not so for Γ (4)Φ L . Both Γ (4)Φ L andthe Γ (4) L (cid:48) with L (cid:48) < L − play a role in the foregoing analysis. We return now to the functional renormalization group, and observe that all the functional relationsdiscussed in the previous subsection hold for the deformed theory, with the obvious modifications that areneeded in order to take into account the presence of the regulator in the free propagator. Consider inparticular the gap equation in the deformed theory: G − κ ( p ) = p + m + Σ κ ( p ) + R κ ( p ) , (37)15 igure 7: The diagrams that contribute to the four-point function Γ (4) ( p, q ) at order one-loop. The first diagram, whichis reducible in the s channel, is obtained by iterating once the four-point vertex (the lowest order contribution to I ) in theBethe-Salpeter equation. The other two diagrams are two-line irreducible in the s channel (but reducible in the t or the u channel), and correspond respectively to the one-loop contributions of the t and u channels to the kernel I ( l =1) ( p, q ) . where the κ dependence of the self-energy comes entirely from its dependence on the propagator G κ , i.e., Σ κ ≡ Σ[ G κ ] . For κ = 0 , since the regulator then vanishes, this equation reduces trivially to the gap equationof the original theory, Eq. (25). Now, by taking a derivative of Σ κ with respect to κ and using the extensionof Eq. (26) to the deformed theory, we obtain ∂ κ Σ κ ( p ) = 2 (cid:90) q ∂ κ G κ ( q ) δ Φ[ G ] δG ( q ) δG ( p ) (cid:12)(cid:12)(cid:12)(cid:12) G κ = 12 (cid:90) q ∂ κ G κ ( q ) I κ ( q, p ) , (38)where the irreducible kernel I κ ( q, p ) appears: this is given by the same skeletons as in the original theory,with the propagators replaced by G κ . Again, G κ is the sole source of κ dependence of the kernel, i.e., I κ = I [ G κ ] . A graphical illustration of this equation, analogous to Eq. (30), is presented in Fig. 8. The
5. Renormalized Φ -derivable approximations from the flow equations Up until now, we have been dealing with Φ -derivable approximations within the theory with cutoff.We have seen that such approximations could be reformulated in terms of flow equations, and that, foran appropriate choice of the initial conditions, the solution of these flow equations reproduces the resultsobtained by solving directly the gap equation and the BS equation. The initial conditions involve explicitlythe bare parameters, i.e. the parameters of the lagrangian, as indicated in Eq. (50), and the solutiondepends on the ultraviolet cutoff Λ uv . We now address the question of whether and how we can remove thedependence on both the ultraviolet cutoff and the bare parameters by renormalization. In order to so so, itis useful to recall how this is achieved within the standard formulation of 1PI flow equations. We then return to the functional renormalization group (see Sect. 2.1) and recall some general features ofthe usual flow equations for the 1PI n -point functions Γ ( n ) κ . These equations are obtained by taking successivefunctional derivatives of Eq. (12), with examples given in Eq. (15) for the two-point function Γ (2) κ ( p ) , andEq. (49) for the four-point function Γ (4) κ ( q, p ) . There are three important properties of these flow equationsthat need to be underlined. The first one is that they make no reference to the bare parameters, but areexpressed instead solely in terms of the n -point functions, which we may identify with the renormalized n -point functions (see Sect. 6.2 below). The second property is that the flows (i.e., the derivatives ∂ κ Γ ( n ) κ )are all given by one-loop integrals which are cut off by ∂ κ R κ . This means that the ultraviolet cutoff Λ uv can be removed from the loops without altering the solution in any significant way. Finally, the n -pointfunctions have simple behaviors for κ = Λ (cid:29) Λ phys , where Λ phys is the scale of relevant physical momenta(we are typically interested in n -point functions Γ ( n ) ( p i ) with external momenta p i (cid:46) Λ phys ). In the regime Λ (cid:29) Λ phys , the external momenta p i contribute to Γ ( n ) ( p i ) via propagators that always feature a scale much29arger than Λ phys , of the order of Λ or higher. In this regime, it makes then sense to approximate the n -point functions by their Taylor expansion in powers of the p i . In the case of the renormalized n -pointfunctions, which are our focus here, the corresponding coefficients are given by simple power counting: Γ (2) κ =Λ ( p ) ∼ m + Z Λ p , Γ (4) κ =Λ ( p, q ) ∼ λ Λ , Γ (2 n ≥ κ =Λ ( p i ) ≈ , Λ (cid:29) Λ phys , (72)where m ∼ Λ , Z Λ ∼ ln Λ and λ Λ ∼ ln Λ , and we have neglected contributions that are suppressed bypowers of Λ phys / Λ .It is interesting to see how this general behavior of the n -point functions indicated in Eq. (72) emergesas a self-consistent solution of the flow equations. Let us indeed assume that the leading behaviors of the n -point functions are given by Γ (2) κ ( p ) ∼ m κ + Z κ p , Γ (4) κ ∼ λ κ , Γ ( n> κ ∼ κ − n , (73)with m κ ∼ κ , Z κ ∼ ln κ and λ κ ∼ ln κ , where the notation f κ ∼ ln κ , or equivalently κ∂ κ f κ ∼ κ , is meantto indicate that the growth of f κ at large κ is slower than any power of κ . As a specific illustration of theargument, let us consider the flow equation for the four-point function, Eq. (49). The loop momentum r inthis equation is bounded by κ , because ∂ κ R κ ( r ) plays the role of an ultraviolet cutoff at the scale κ (e.g. κ∂ κ R κ ( r ) ∼ κ θ ( κ − r ) ). Then, by plugging the expected behaviors (73) into Eq. (49), using the fact thatthe regulated propagators G κ ( r ) at large κ behave as κ − and taking into account the four dimensionalphase space integration ∼ κ , one easily verifies that the leading order behaviors of the integrals in Eq. (49)are indeed ∼ κ , in agreement with the initial assumption that Γ (4) κ ∼ ln κ . This argument can be easilyextended to arbitrary n -point functions, thereby verifying Eqs. (73). Thus, when κ = Λ becomes large, onlythe two and four-point functions are relevant, and their values are determined by the parameters m , and λ Λ in addition to Z Λ : the effective action Γ Λ [ φ ] eventually takes the form of the classical action of Eq. (1),with suitably defined parameters. We may furthermore argue that when κ (cid:29) Λ phys (and p, q (cid:46) Λ phys ) the dependence of the four-point This is because, the total momentum of a given propagator is either less than Λ in which case the propagator features alarge mass of the order of Λ as given by the regulator R Λ ( q < Λ) ∼ Λ , or it is larger than Λ in which case it provides itselfthe large momentum scale. Subleading corrections to these coefficients involve terms controlled by m/ Λ where m denotes the mass. We stress alsothat the possibility of Taylor expanding in powers of p i applies both to renormalized and to unrenormalized n -point functions.The important difference in the case of the renormalized fluctuations is that Λ is the only large scale and the coefficients ofthe Taylor expansion are controlled by power counting with respect to this scale. In the case of unrenormalized fluctuations,in contrast, the coefficients of the Taylor expansion are functions of both Λ and Λ uv , and for Λ uv (cid:29) Λ (cid:29) Λ phys , one does nothave a simple power counting rule. The general structure outlined above depends crucially on the presence of a derivative of the regulator in all the loopintegrals. Indeed, for regulators with a sufficiently rapid decay or even compact support for momenta p (cid:46) κ all flows areinfrared and ultraviolet finite irrespective of the -perturbative- renormalizability of the theory at hand. For example, the current φ -theory in d = 6 dimensions is not renormalizable but features finite flows: the non-renormalizability of the theory reflectsitself in infinitely many correlation functions flowing with positive power laws as κ → ∞ . Γ (4) κ ( p, q ) on the external momenta p and q is negligible. Recall that κ plays a role similar to theexternal momenta in the loop integrals. When κ (cid:46) Λ phys , the effect of changing κ is negligible, there isessentially no flow: the external momenta play the role of the infrared regulator, and hide the effect of thevariation of κ : . When κ (cid:29) Λ phys , the opposite situation prevails: the n -point functions become independentof momenta. In other words, the flow is important only when κ is of the order of the external momenta (orthe mass). In the regime where Λ phys (cid:28) κ , the momentum dependence of the n -point functions shows upas small power corrections in p/κ which are obtained by expanding the loop integrals in powers of p/κ , thepresence of the regulator eliminating potential infrared divergences.One concludes from this discussion that if Λ is chosen large enough, Λ phys (cid:28) Λ (cid:28) Λ uv , then theappropriate initial conditions on the flows of the various n -point functions may be taken as indicated inEq. (72). The values of the parameters m , λ Λ and Z Λ can be chosen so that m κ , λ κ and Z κ take specifiedvalues at κ = 0 , which correspond to the renormalized parameters (see renormalization conditions (82)below). Note that the bare parameters, m b and λ b , do not enter at all in the discussion: they have beeneliminated in favor of the parameters m , λ Λ and Z Λ that specify the initial conditions of the flows. Similarly,the ultraviolet cutoff drops out because the flow equations are finite, and fluctuations are integrated overthe finite range of momenta [0 , Λ] . The independence of the renormalized parameters on the specific valueof Λ is ensured by the flow equations. These govern the variations of the initial parameters m , λ Λ and Z Λ under a change of Λ in a way compatible with power counting, as expressed in Eqs. (72).In the rest of this section, we shall verify that these features of the 1PI flow equations, in particularthe finiteness of the flow equations and the absence of reference to the bare parameters, hold, with someadjustments, for the flow equations for Φ -derivable approximations, which are summarized in Sect. 4.3. We first examine the dependence of the flow equations on the bare parameters. The flow equations (68)-(70) that we have derived in Sects. 3 and 4 are nearly independent of these parameters. If fact this is so forEq. (68) written in terms of Γ (2) L − ,κ rather than in terms of Σ L − ,κ . Since G κ ( r ) = (cid:2) Γ (2) L − ,κ ( r ) + R κ ( r ) (cid:3) − ,the bare mass non longer appears, as it would if one would relate the propagator to the self-energy, G κ ( r ) = (cid:2) r + m + Σ L − ,κ ( r ) + R κ ( r ) (cid:3) − .As for the Γ (4) L,κ ’s that appear in the right-hand side of Eqs. (65) they are still to be seen as cut-off regulatedFeynman diagrams with vertices given by λ b , as specified by the operator [ _ ] [ L ] (see after Eq. (53)), whichis indeed based on the expansion (36) in powers of λ b . To eliminate all reference to the bare coupling, werewrite the strict l -loop contribution λ l +1b ( − ) l Γ (4 ,l ) in Eq. (36) as ∆Γ (4) κ,l = Γ (4) L = l,κ − Γ (4) L = l − ,κ for l ≥ and ∆Γ (4)0 ,κ = Γ (4) L =0 ,κ . We then replace in the right-hand side of Eqs. (65) any occurence of Γ (4) L,κ by the following31xpansion Γ (4) L,κ = Γ (4) L =0 ,κ (cid:124) (cid:123)(cid:122) (cid:125) ∆Γ (4) L =0 ,κ + (cid:104) Γ (4) L =1 ,κ − Γ (4) L =0 ,κ (cid:124) (cid:123)(cid:122) (cid:125) ∆Γ (4) L =1 ,κ (cid:105) + · · · + (cid:104) Γ (4) L,κ − Γ (4) L − ,κ (cid:124) (cid:123)(cid:122) (cid:125) ∆Γ (4) L,κ (cid:105) , (74)where each consecutive term counts one extra loop, and we redefine the expansion operator [ _ ] L with respectto this formal loop expansion. For greater clarity, we shall denote by { _ } L the expansion operator [ _ ] L in formulas where this substitution is made. Of course, as far as the bare n -point functions are concerned,using either [ _ ] L or { _ } L makes no difference. However, by using { _ } L in equations such as Eqs. (60) and(65), we obtain equations that make no reference to the bare coupling, and which therefore describe notonly the bare n -point functions, but also the renormalized ones, the choice between one type of solution andthe other being essentially determined by the initial conditions, as we discuss below. Now that we have eliminated any reference to the bare parameters from the flow equations, let us showthat they are finite. Before we proceed to this analysis a few words of caution are required however.For one thing, the present model features an ultraviolet Landau pole Λ L . Strictly speaking, this preventsus from taking the limit Λ uv → ∞ . However, at sufficiently weak coupling, the scale Λ L is so large thatone can consider values of the cut off that are large with respect to the physical scales, without hitting theLandau pole (see the discussion after Eq. (A.6) in Appendix A). It makes then sense to ask whether the flowequations are essentially insensitive to the cut-off in this range of cut-off values. This is what will be meantin what follows by “checking the finiteness of the flow equations”, although we shall still use the short-handnotation Λ uv → ∞ in place of Λ L (cid:29) Λ uv (cid:29) Λ phys .The cut-off insensitivity of the flow equations will be granted in part by the presence of ∂ κ R κ whichregulates some of the loop integrals in the UV. Some other loops, however, will not feature this naturalregulator and proving their insensitivity to the cut-off will require a finer analysis based on power counting.We shall assume that, in estimating superficial degrees of divergence, G κ ( p ) and Γ (4)Φ L ,κ ( p, q ) count respec-tively as − and . This natural assumption follows from the application of Weinberg’s theorem on thecorresponding perturbative contributions (see Appendix D). Such perturbative power counting needs to betaken with a small pinch of salt, however, as the resummations entailed by the Φ -derivable approximationscan potentially alter the asymptotic behavior of the n -point functions. In the considered cut-off range, weexpect these modifications to be marginal, such that a superficial degree of divergence that is found to benegative based on the perturbative counting ( δ ≤ − ), will remain so once resummation effects are consid-ered. In other words, the perturbative power counting is sufficient to conclude on the finiteness of the flow32quations. With all these precautions taken, let us now analyze the finiteness of the flow equations. The flow of thetwo-point function is given by Eq. (66), and is identical to that obtained in the 1PI approach: it containsindeed a derivative of the regulator in its right-hand side, so that the flow is finite if Γ (4) κ ( q, p ) is finite.Thus, in this equation, we can ignore the cutoff Λ uv that controls the loop momentum without affecting thesolution in any significant way.The situation is a priori different for the flow equation for Γ (4) κ ( q, p ) , given by Eq. (46). The loopintegrals in the right-hand side of this equation are not cut off by a factor ∂ κ R κ ( q ) . Furthermore it containsconvolutions of n -point functions whose external momenta are not limited to the range p i (cid:46) Λ phys . Thus,even though we are a priori interested in Γ (4) κ ( p, q ) for p and q in the physical range, p, q (cid:46) Λ phys , in orderto calculate the integrals in the right-hand side of Eq. (46), we need the function Γ (4) κ ( p, q ) with at least oneof its arguments allowed to run freely to infinite values. The same remark applies to the irreducible kernel I κ ( p, q ) , or to the self-energy Σ κ ( r ) that enters the propagators in Eq. (46). We need therefore to examinemore closely whether Eq. (46) remains finite as Λ uv → ∞ , by using power counting.Let us then consider the first integral in the right-hand side of Eq. (46), which we rewrite here forconvenience (cid:90) r Γ (4)Φ L ,κ ( p, r ) G κ ( r ) ∂ κ G κ ( r ) Γ (4)Φ L ,κ ( r, q ) . (75)To analyze the convergence of this integral, we need to determine how much ∂ κ G κ contributes to the powercounting. This is easily done by combining the relation G − κ ( r ) = Γ (2) L − ,κ ( r ) + R κ ( r ) and Eq. (66), to obtain ∂ κ G κ ( r ) = − ∂ κ R κ ( r ) G κ ( r ) + 12 (cid:90) s ∂ κ R κ ( s ) G κ ( s ) Γ (4)Φ L ,κ ( s, r ) G κ ( r ) . (76)Since ∂ κ R κ ( r ) is strongly suppressed at large r , the dominant asymptotic behavior of ∂ κ G κ ( r ) is given by thesecond term in Eq. (76), which is a finite integral. It follows that ∂ κ G κ ( r ) contributes as G κ ( r ) , i.e. as − tothe power counting. The superficial degree of divergence of the integral (75) is then × − − − ,where we counted for the integration measure, × for the two four-point functions, − for the propagator,and − for the flow of the propagator. The integral is then finite. As for the other integrals in Eq. (46), theyare finite thanks to the fact that ∂ κ I κ ( p, q ) contributes as − to the power counting rules. This propertyis intimately related to the s-channel two-particle irreducibility of I κ ( p, q ) , as we discuss in Appendix D. We cannot exclude a situation where the resummation modifies the asymptotic behavior of the propagator in such a waythat certain loops that are marginally divergent with a perturbative counting ( δ = 0 ) become marginally convergent with thenon-perturbative counting. In fact, this assumption has been implicitly made in [53] when checking the finiteness of certainloop integrals. Here, we shall not make this assumption because, first, it is not clear whether the resummation tends to improvethe convergence (this could in fact depend on the considered truncation), and, second, because the obtained convergence ispossibly quite slow and therefore does not necessarily ensure a strong cut off insensitivity before hitting the Landau pole. Infact, as stated above, such considerations are not necessary here because as we shall see, the perturbative counting is sufficient. ∂ κ I κ in Eq. (46) have a superficial degree of divergence equal to − × − − , where we counted for the integration measure, for the four-point function, × ( − for the two propagators and − for ∂ κ I κ . The same goes for the two-loop integral in Eq. (46)since both its sub-loop integrals and the two-loop integral itself have superficial degrees of divergence equalto − . One concludes therefore that all the integrals involved in Eq. (46) are finite by power counting.Equations (59) and (65) are easier to handle. Because they involve diagrams with no explicit four-pointsubdiagrams (assuming the Γ (4) L ’s to be finite), the only possible divergence is an overall divergence. However,it is easily seen that the superficial degree of divergence of any integral appearing in ∂ κ I κ is δ = − . Thisis because the diagrams of ∂ κ I κ are nothing but those of I κ with one of the propagators G κ replaced by ∂ κ G κ . This implies that δ ∂ I = δ I − − − , where we used that δ I = 0 and that δ ∂ κ G κ = − .The same argument applies to ∂ κ Γ (4) L,κ .One conclude that all the integrals that are involved in the flow equations for Φ -derivable approximationsare finite. We may therefore safely eliminate the ultraviolet cutoff. Starting from the L -loop Φ -derivable approximation in the bare theory, we have derived a system ofcoupled finite flow equations for Γ (2) L − ,κ , Γ (4)Φ L ,κ and the Γ (4) L (cid:48) ,κ (for ≤ L (cid:48) ≤ L − ), that make no referenceto the bare parameters. These equations can be written in the same concise form as in Sect. 4.3: ∂ κ Γ (2) L − ,κ ( p ) = F (2) L − ,κ (cid:2) Γ (2) L − ,κ , Γ (4)Φ L ,κ (cid:3) ( p ) . (77) ∂ κ Γ (4)Φ L ,κ ( p, q ) = F (4)Φ L ,κ (cid:2) Γ (2) L − ,κ , Γ (4)Φ L ,κ , Γ (4) L − ,κ , . . . , Γ (4) L =0 ,κ (cid:3) ( p, q ) (78) ∂ κ Γ (4) L (cid:48) ,κ ( p i ) = F (4) L (cid:48) ,κ (cid:2) Γ (2) L − ,κ , Γ (4) L − ,κ , . . . , Γ (4) L =0 ,κ (cid:3) ( p i ) (0 ≤ L (cid:48) ≤ L − . (79)These equations have the same form as Eqs. (68)-(70). The main difference is that here, thanks to the useof the { _ } L expansion operator, the flows do not make any reference to the bare parameters. Moreover,since the flows are finite, and we restrict κ to values smaller than Λ (cid:28) Λ uv , the ultraviolet cuttoff Λ uv doesnot play any major role and can be dropped.One can write the solution of these equations generically as in Eq. (71) Γ ( n ) κ ( p i ) = Γ ( n )Λ ( p i ) + (cid:90) κ Λ d κ (cid:48) F ( n ) κ (cid:48) ( p i ) . (80)Assuming the flow to be initialized at the scale Λ (cid:29) Λ phys , the initial conditions, that is the values of the34 ( n )Λ ( p i ) ’s, are essentially constant, with only a very weak dependence on the momenta as discussed earlier(see Eqs. (72)). They can be chosen in the form Γ (2) L − ,κ =Λ ( p ) ∼ m + Z Λ p , Γ (4)Φ L ,κ =Λ ( p, q ) ∼ λ Λ , Γ (4) L (cid:48) ,κ =Λ ( p, q, r ) = λ L (cid:48) , Λ , (0 ≤ L (cid:48) ≤ L − , (81)for Λ (cid:29) Λ phys . The need for different initial conditions for the Γ (4) L,κ ’s, which also differ from the initialcondition for Γ (4)Φ L ,κ , reflects the fact that these are different approximations to the exact four-point function.In particular even though Γ (4) L,κ and Γ (4)Φ L ,κ should coincide in the exact case (correspoding to L → ∞ ), theyhave no reason to coincide for any finite L . The best we can do is to adjust the initial conditions such thatthese functions coincide at a given value of κ and a given value of the external momenta, see below. Contactwith the original formulation of Φ -derivable approximations is made by fixing the values of the parameters Z Λ , m Λ and λ Λ via the “renormalization conditions” Γ (2) L − ,κ =0 ( p = 0) = m , dΓ (2) L − ,κ =0 d p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p =0 = 1 , Γ (4)Φ L ,κ =0 ( p i = 0) = λ , (82)whereas the λ L, Λ ’s are fixed via the “consistency” conditions Γ (4) L (cid:48) ,κ =0 ( p i = 0) = Γ (4)Φ L ,κ =0 ( p i = 0) , (0 ≤ L (cid:48) ≤ L − . (83)This last condition ensures that Γ (4) L,κ and Γ (4)Φ L ,κ coincide as L → ∞ , a condition that is not necessarily metif one fixes the finite parts of the λ L, Λ ’s in an arbitrary way. A convenient feature that results from this choice of renormalization conditions, is that the renormalizedsolution for Γ (4) L,κ is a polynomial of order L + 1 in the renormalized coupling λ , whose coefficients do notdepend on L . That is Γ (4) L,κ = λ − λ ˜Γ (4 , κ + · · · + λ L +1 ( − ) L ˜Γ (4 ,L ) . (84)This results is established in Appendix E and will be exploited in the next section. Since the initial condi-tions λ L, Λ are nothing but the asymptotic form of the Γ (4) L,κ =Λ ’s for Λ (cid:29) p i , it follows that the λ L, Λ ’s havea similar polynomial structure.In summary, we have shown in this section that the flow equations for Φ -derivable approximations, thatare summarized in Set. 4.3, enjoy essentially the same properties as the usual flow equations for the 1PI Note that these consistency conditions have nothing to do with those introduced in [44]. Our choice follows the generalstrategy described in [38]. In a sense, consistency conditions are implicitly used also in perturbative calculations where, for instance, the four-pointfunction at any loop order is required to equal λ at the renormalization point. -point functions. In particular, they are finite, and can be made independent of the bare parameters. Therenormalization process can then be carried out in very much the same way as for the 1PI flow equations.The strength of this approach resides in that it gives direct access to the renormalized n -point functions,without having to deal with the ultraviolet divergences and the counterterms that are omnipresent in thediagrammatic approach, the topic of the next section.The derivation of the flow equations in Sect. 3 was initially motivated by the search for a truncation ofthe 1PI equations at the level of the four-point function. The obvious four-point function that appears inthis context is that which solves the Bethe-Salpeter equation. However, as was discussed at length in Sect. 4,other four-point functions naturally appear when calculating the flow of the irreducible kernel of the BSequation. As we saw in the present section, these functions Γ (4) L play an important role in the renormalization,and they need to be properly initialized. We may be concerned by the fact that the number of parametersthat need to be specified to fix the initial conditions seem to exceed the usual number of available parametersin a renormalizable field theory, and furthermore this number grows with the loop order of the consideredapproximation. However, as we have seen, this is not the case, since the excess number of parameters isbalanced by an equal number of consistency conditions. As we shall see in the next section, the very sameissue arises when we examine renormalization from the diagrammatic point of view.
6. Renormalized Φ -derivable approximations from the diagrammatic approach The approach based on flow equations, which was shown in the previous section to be so effective in thecalculation of renormalized n -point functions, proves also to be quite helpful in analyzing the renormalizationof Φ -derivable approximations from the diagrammatic point of view. In the previous section, we wrote theflow equations in the generic form Γ ( n ) κ ( p i ) = Γ ( n )Λ ( p i ) + (cid:90) κ Λ d κ (cid:48) F ( n ) κ (cid:48) ( p i ) (cid:12)(cid:12)(cid:12) Λ uv . (85)We argued that since the flows F ( n ) are finite, and Λ is also kept finite, we could ignore Λ uv . In this section,the same flow equations will be used, but we shall let Λ → ∞ (as in Sect. 4), keeping Λ uv fixed. We shallthen examine the divergences that arise when Λ uv → ∞ and see how these can be absorbed in countertermspresent in the initial conditions Γ ( n )Λ ( p i ) . Before we see how this works in detail, it is useful to recall howthe renormalization problem is formulated in the diagrammatic approach of Φ -derivable approximations.36 .1. Diagrammatic renormalization of Φ -derivable approximations: setting-up the problem Following the standard techniques of perturbative renormalization, one rewrites the action (1) in termsof a rescaled field, ϕ b = Z / ϕ , S [ ϕ ] = (cid:90) d d x (cid:26) Z ∂ϕ ( x )) + m + δm ϕ ( x ) + λ + δλ ϕ ( x ) (cid:27) , (86)where Zm ≡ m + δm , Z λ b ≡ λ + δλ, Z = 1 + δZ, (87)are conventionally split into renormalized parameters λ , m , and counterterms δλ , δm and δZ . To therescaling of the field corresponds a rescaling of the n -point functions: Γ ( n )b = Z − n/ Γ ( n ) . In particular,the propagator rescales as G b = ZG and, to within an inessential additive constant term, the 2PI effectiveaction (24) becomes Γ[ G ] = 12 (cid:90) p log G − ( p ) + 12 (cid:90) p ZG − ( p ) G ( p ) + Φ[ G ] , (88)with ZG − ( p ) = Zp + m + δm . The functional Φ[ G ] is made of the same 2PI diagrams as before butwith the coupling λ b replaced by λ + δλ . The corresponding gap equation is now G − = ZG − + Σ[ G ] , (89)where the self-energy functional Σ[ G ] is related to Φ[ G ] as before, see Eq. (26). To take a definite example,at -loop order in Φ , the gap equation reads G − ( p ) = Zp + m + δm + λ + δλ (cid:90) q G ( q ) − ( λ + δλ ) (cid:90) q (cid:90) r G ( q ) G ( r ) G ( r + q + p )+ ( λ + δλ ) (cid:90) q (cid:90) r (cid:90) k G ( q ) G ( r ) G ( k ) G ( r + q + p ) G ( k + q + p ) . (90)If one were doing perturbation theory (e.g. calculate contributions to the inverse propagator from theequation above with, in the right-hand side, G substituted by G ) one would follow the standard route. Thecounterterms δZ , δm , δλ contain the parts of the bare parameters that diverge as Λ uv → ∞ . They are The reason is quite simple: a given diagram of Φ[ G ] has no external leg, so the number I of is internal lines is twice thenumber V of its vertices. The rescaling of the propagator, G b = ZG , generates a factor Z I = Z V that can be combined withthe prefactor λ V b into Z V λ V b = ( λ + δλ ) V . It is sometimes convenient to reabsorb the one-loop term (cid:82) ( δZp + δm ) G ( p ) / in the definition of Φ[ G ] , in which case the2PI effective action and the gap equation can be written with Z = 1 and G − = p + m . The relation between the self-energyfunctional and Φ[ G ] remains unchanged. Again, the notation Λ uv → ∞ is to be understood as Λ Landau (cid:29) Λ uv (cid:29) Λ phys . m , λ and the cutoff Λ uv , and are determined as formal series in powers of λ , with their divergentpart chosen so as to absorb the ultraviolet divergences order by order in the expansion in λ . Their finiteparts are fixed by the same renormalization conditions as in Eq. (82), namely dΓ (2) ( p )d p (cid:12)(cid:12)(cid:12)(cid:12) p =0 = 1 , Γ (2) ( p = 0) = m , Γ (4) (0 , ,
0) = λ. (91)This method, rooted in perturbation theory, can be extended to the Φ -derivable approximations [36, 37,38]. For the specific 4-loop example considered above, this implies solving the full non-linear gap equation,Eq. (90), with the self-consistent propagator G in its right-hand side. The main difficulty that one has toface in this approach is the proper identification and treatment of the divergences and subdivergences in thegap equation. In particular, the various occurrences of the coupling counterterm δλ , because they absorbdistinct subdivergences, need to be treated on different footings. This is where the flow equations bringa significant clarification by providing a precise map of how these divergences are distributed among therelevant n -point functions and how they can be absorbed in counterterms. This is what we discuss in thenext two subsections. As was shown in Sects. 3 and 4.3, the bare diagrammatic expansion is a solution to the flow equations(68-70) with initial conditions (50) taken at a scale Λ well above the ultraviolet cutoff. Taking Λ → ∞ , onecan write the solutions of these equations formally as follows Γ (2) L − ,κ =0 ( p ) = p + m + (cid:90) ∞ d κ F (2) L − ,κ (cid:2) Γ (2) L − ,κ , Γ (4)Φ L ,κ (cid:3) ( p ) (cid:12)(cid:12)(cid:12) Λ uv , (92) Γ (4)Φ L ,κ =0 ( p, q ) = λ b + (cid:90) ∞ d κ F (4)Φ L ,κ (cid:2) Γ (2) L − ,κ , Γ (4)Φ L ,κ , Γ (4) L − ,κ , . . . , Γ (4) L =0 ,κ (cid:3) ( p, q ) (cid:12)(cid:12)(cid:12) Λ uv , (93) Γ (4) L (cid:48) ,κ =0 ( p i ) = λ b + (cid:90) ∞ d κ F (4) L (cid:48) ,κ (cid:2) Γ (2) L − ,κ , Γ (4) L − ,κ , . . . , Γ (4) L =0 ,κ (cid:3) ( p i ) (cid:12)(cid:12)(cid:12) Λ uv , (0 ≤ L (cid:48) ≤ L − . (94)In the above equations, the notation | Λ uv reminds one that the cutoff is kept in the integrals entering theflows F ( n ) , and the n -point functions there should be viewed as the bare n -point functions.At this point it is easy to make contact with the renormalized theory. To do so, we first move to therescaled theory, which is easily done since the flow equations are invariant under this operation, as is easily We consider only the massive case in this paper. Γ (2) L − ,κ =0 ( p ) = Zp + m + δm + (cid:90) ∞ d κ F (2) L − ,κ (cid:2) Γ (2) L − ,κ , Γ (4)Φ L ,κ (cid:3) ( p ) (cid:12)(cid:12)(cid:12) Λ uv , (95) Γ (4)Φ L ,κ =0 ( p, q ) = λ + δλ + (cid:90) ∞ d κ F (4)Φ L ,κ (cid:2) Γ (2) L − ,κ , Γ (4)Φ L ,κ , Γ (4) L − ,κ , . . . , Γ (4) L =0 ,κ (cid:3) ( p, q ) (cid:12)(cid:12)(cid:12) Λ uv , (96) Γ (4) L (cid:48) ,κ =0 ( p i ) = λ + δλ + (cid:90) ∞ d κ F (4) L (cid:48) ,κ (cid:2) Γ (2) L − ,κ , Γ (4) L − ,κ , . . . , Γ (4) L =0 ,κ (cid:3) ( p i ) (cid:12)(cid:12)(cid:12) Λ uv , (0 ≤ L (cid:48) ≤ L − . (97)A first important observation is that these equations clearly separate the subdivergences and the overalldivergences of a given n -point function. Indeed, since overall Λ uv -divergences are associated to the regimewhere all loop momenta of a given diagram contributing to Γ ( n ) grow large, they cannot depend on κ . Itfollows that the F ( n ) | Λ uv ’s in the above equations, which are nothing but the κ -derivatives of the correspond-ing n -point functions, are only sensitive to the subdivergences of these n -point functions, while the overalldivergences occur due to the explicit unbounded κ -integrals in Eqs. (95)-(97). A second observation is that,because the explicit loops that enter the F ( n ) κ | Λ uv ’s are convergent for Λ uv (cid:29) κ , the subdivergences neverappear explicitly but only as hidden divergences (including overall divergences) of the various n -point func-tions that enter as arguments of the F ( n ) κ | Λ uv ’s. It results from these observations that overall divergencesencompass actually all the possible divergences that can appear in the n -point functions (a well knownfeature, also in perturbation theory). It is a strength of the flow formulation to remove all divergences atonce by removing only overall divergences, since the counterterms never appear explicitly in F ( n ) κ | Λ uv , butonly as tree-level contributions in Eqs. (95)-(97) via the initial conditions. Thus, by fixing these countert-erms in order to satisfy renormalization conditions at κ = 0 , one handles simultaneously both the overalldivergences and the subdivergences present in the n -point functions. This consitutes in fact an essentialingredient towards a formal proof of renormalization in the diagrammatic formulation.The previous argument needs to be slightly amended to take into account the fact that Γ (4)Φ L and the Γ (4) L ’sare different approximations to the four-point function, and there is no reason for their overall divergencesto be absorbed by the same counterterm δλ . Such an observation was already made in the previous section(see Eq. (81)). To account for this possibility, we modify the initial conditions and set Γ (2) L − ,κ =0 ( p ) = Zp + m + δm + (cid:90) ∞ d κ F (2) L − ,κ (cid:2) Γ (2) L − ,κ , Γ (4)Φ L ,κ (cid:3) ( p ) (cid:12)(cid:12)(cid:12) Λ uv , (98) Γ (4)Φ L ,κ =0 ( p, q ) = λ + δ ˜ λ + (cid:90) ∞ d κ F (4)Φ L ,κ (cid:2) Γ (2) L − ,κ , Γ (4)Φ L ,κ , Γ (4) L − ,κ , . . . , Γ (4) L =0 ,κ (cid:3) ( p, q ) (cid:12)(cid:12)(cid:12) Λ uv , (99) Γ (4) L (cid:48) ,κ =0 ( p i ) = λ + δλ L (cid:48) + (cid:90) ∞ d κ F (4) L (cid:48) ,κ (cid:2) Γ (2) L − ,κ , Γ (4) L − ,κ , . . . , Γ (4) L =0 ,κ (cid:3) ( p i ) (cid:12)(cid:12)(cid:12) Λ uv , (0 ≤ L (cid:48) ≤ L − . (100)As shown by these equations, the counterterms determine the n -point functions at the large initial scale39 → ∞ . Upon imposing the same renormalization conditions as in the previous section (see Eqs. (91)), weextract the following formal expressions: δm = − (cid:90) ∞ d κ F (2) L − ,κ (cid:2) Γ (2) L − ,κ , Γ (4)Φ L ,κ (cid:3) (0) (cid:12)(cid:12)(cid:12) Λ uv , (101) δZ = − (cid:90) ∞ d κ dd p F (2) L − ,κ (cid:2) Γ (2) L − ,κ , Γ (4)Φ L ,κ (cid:3) ( p ) (cid:12)(cid:12)(cid:12) Λ uv (cid:12)(cid:12)(cid:12)(cid:12) p =0 , (102) δ ˜ λ = − (cid:90) ∞ d κ F (4)Φ L ,κ (cid:2) Γ (2) L − ,κ , Γ (4)Φ L ,κ , Γ (4) L − ,κ , . . . , Γ (4) L =0 ,κ (cid:3) (0 , (cid:12)(cid:12)(cid:12) Λ uv , (103) δλ L (cid:48) = − (cid:90) ∞ d κ F (4) L (cid:48) ,κ (cid:2) Γ (2) L − ,κ , Γ (4) L − ,κ , . . . , Γ (4) L =0 ,κ (cid:3) ( p i = 0) (cid:12)(cid:12)(cid:12) Λ uv , (0 ≤ L (cid:48) ≤ L − . (104)While these expressions may be of limited practical use, they summarize the general structure of the coun-terterms that are needed to cope with all the divergences of the L -loop Φ -derivable approximation. Theyexhibit the basic n -point functions on which to impose the renormalization conditions necessary to thecomplete determination of these counterterms, namely Γ (2) L − , Γ (4)Φ L and the Γ (4) L (cid:48) ’s. In the next subsection,we shall turn these formal considerations into a more explicit construction by integrating exactly the flowequations (98)-(100) in terms of the diagrams. This will also allow us to derive diagrammatic expressionsfor the counterterms (101)-(104), thus providing an explicit and synthetic procedure for the renormalizationof Φ -derivable approximations in their diagrammatic formulation. We begin with an observation concerning the general structure of the solution, and in particular on theway it depends on the parameters of the initial conditions. Let us consider the original 2PI effective action,Eq. (24 ), truncated at order L -loop, and write it as follows Γ L [ G ] = 12 (cid:90) p log G − ( p ) + 12 (cid:90) p ( zp + y ) G ( p ) + Φ L [ G ] , (105)with Φ L [ G ] = u Φ (2) [ G ] − v Φ (3) [ G ] + · · · + v L − ( − ) L Φ ( L ) . (106)Here, u , v , y and z are arbitrary constants and Φ ( l ) [ G ] is built with the same cut off integrals as in Eq. (35).It is then easily verified, by following the same steps as in Sects. 3 and 4, that Γ (2) L − ,κ ≡ zp + y + Σ L − ,κ and Γ (4)Φ L ,κ , defined from the LW functional Φ L [ G κ ] just given, obey the same flow equations as those derivedin Sects. 3 and 4. The Γ (4) L,κ ’s now involve v as coupling constant rather than λ b , and correspondingly (see40q. (36)) Γ (4) L,κ = v − v Γ (4 , κ + · · · + v L +1 ( − ) L Γ (4 ,L ) κ . (107)This result is in fact trivial: first, y and z disappear entirely from the flow equations once these are written interms of Γ (2) L − ,κ rather than Σ L − ,κ (see Eq. (66)); second, the term involving the coefficient u in Eq. (106)drops entirely from the flow equation since it gives a constant contribution to I L − ,κ which disappears in ∂ κ I L − ,κ ; finally, v is a mere renaming of the bare coupling.Clearly, the solution to the flow equations that corresponds to Eqs. (106) and (107) obeys the initialconditions Γ (2) L − ,κ = ∞ ( p ) = zp + y , Γ (4)Φ L ,κ = ∞ = u , Γ (4) L,κ = ∞ ( p ) = v . (108)This type of initial conditions accommodates the solution (95)-(97), upon the choice z = Z , y = m + δm , u = v = λ + δλ , which corresponds to a simple rescaling of the n -point functions. But it cannot correspondto the renormalized solution (98)-(100) because, although u can be chosen independently of v , as in Eq. (99),all the Γ (4) L,κ ’s are initialized at the same value, independent of L .We know, however, that the different δλ L ’s in Eq. (100) are finite order truncations of a unique series δλ in powers of λ : δλ = (cid:88) l ≥ λ l +1 δλ ( l ) , δλ L = (cid:88) ≤ l ≤ L λ l +1 δλ ( l ) . (109)This is because the counterterms are essentially nothing but the initial conditions for the renormalizedsolution at the scale Λ (cid:29) Λ uv . We have seen in the previous section that the renormalized Γ (4) L,κ ’s arepolynomials in λ with L -independent coefficients (see Eq. (84)). It follows that the same property holds forthe δλ L ’s, which is precisely what Eq. (109) states.Let us then consider the extension of Φ[ G ] in Eq. (106) to arbitrary order in v , and define Γ (2) κ ≡ zp + y + Σ κ and Γ (4) κ from the (exact) gap and BS equations. It is easily seen that these functions satisfythe flow equations (66) and (67), with ∂ κ I κ given by the exact relation (59), while Γ (4) κ = v − v Γ (4 , κ + · · · + v L +1 ( − ) L Γ (4 ,L ) κ + · · · (110)also obeys the exact equation (64). At this point we substitute v → λ + δλ and expand the exact equations(59) and (64) in a loop expansion in power of λ , using wherever needed the expansion operator { _ } L definedafter Eq. (74), as well as the expansion (109) of δλ . By doing so, one easily verifies that Γ (2) L − ,κ and Γ Φ L ,κ
41s derived from Φ L [ G ] = u Φ (2) [ G ] + (cid:104) − ( λ + δλ ) Φ (3) [ G ] + · · · + ( λ + δλ ) L − ( − ) L Φ ( L ) (cid:105) L = u Φ (2) [ G ] − (cid:2) ( λ + δλ ) (cid:3) L − Φ (3) [ G ] + · · · + (cid:2) ( λ + δλ ) L − (cid:3) L =0 ( − ) L Φ ( L ) , (111)as well as Γ (4) L,κ defined as Γ (4) L,κ = (cid:104) λ + δλ − ( λ + δλ ) Γ (4 , κ + · · · + ( λ + δλ ) L +1 ( − ) L Γ (4 ,L ) κ (cid:105) L = (cid:2) λ + δλ (cid:3) L − (cid:2) ( λ + δλ ) (cid:3) L − Γ (4 , κ + · · · + (cid:2) ( λ + δλ ) L +1 (cid:3) L =0 ( − ) L Γ (4 ,L ) κ (112)where [ _ ] L refers now the loop expansion in powers of λ , obey the same flow equations as those in Sect. 5.2.The solution just obtained satisfies now the initial conditions in Eqs. (95)-(97). It corresponds to thefollowing expression for the LW functional Φ[ G ] = ( λ + δ ˜ λ ) Φ (2) [ G ] + (cid:88) ≤ l ≤ L (cid:2) ( λ + δλ ) l − (cid:3) L − l ( − ) l Φ ( l ) [ G ] , (113)where the counterterms δ ˜ λ and δλ L are explicitly indicated. These are determined from the renormalizationconditions imposed respectively on Γ (4)Φ L , as given by the BS equation, and Γ (4) L , as given by Γ (4) L = (cid:88) ≤ l ≤ L (cid:2) ( λ + δλ ) l +1 (cid:3) L − l ( − ) l Γ (4 ,l ) . (114) The preceding analysis has taught us how the counterterms δλ L , δ ˜ λ , δm and δZ should be implementedin the diagrammatic formulation of Φ -derivable approximations. In this subsection we turn to their explicitdetermination. δλ L and renormalization of the Γ (4) L ’s The renormalized expression (114) is similar to that obtained in perturbation theory, the only differencebeing that the self-consistent propagator G is involved in the calculation of the integrals Γ (4 ,l ) instead of theperturbative one, G . So the renormalization works along the same lines as in perturbation theory, whichwe shall follow to determine the first contributions to δλ .Using the expansion (109) of δλ , one can recast Eq. (114) in the form Γ (4) L = (cid:88) ≤ l ≤ L λ l +1 ( − ) l ˜Γ (4 ,l ) , (115)42ith ˜Γ (4 , = 1 , (116) ˜Γ (4 , = Γ (4 , − δλ (1) , (117) ˜Γ (4 , = Γ (4 , + δλ (2) − δλ (1) Γ (4 , , (118) . . . Higher orders feature the same triangular structure. For a given ˜Γ (4 ,l ) , δλ ( l ) appears as a tree-level con-tribution whereas the δλ ( l (cid:48)
12 ˜ I L − , (134) ˜Γ (4 , L = δ ˜ λ (2) + 12 ( λ + δ ˜ λ (1) ) G ˜ I L − + 14 ˜ I L − G ˜ I L − (135) . . . The counterterm contribution δ ˜ λ (1) renormalizes the first rung (which has only an overall divergence sinceits subdivergences have been removed by the renormalization of the Γ (4) L ’s). This same counterterm allowsto remove a subdivergence in the two-rung contribution, as it is clear from (135). There only remains anoverall divergence that can be absorbed in δ ˜ λ (2) . The same logic extends in higher rung contributions.The counterterms δλ and δ ˜ λ have different structures and renormalize their respective four-point func-tions in different ways. Thus, as all the ladders are resummed in Γ (4)Φ L , the same counterterm (132) endsup renormalizing the subdivergences and the overall divergences of the rung expansion, as we have just dis-cussed. In contrast, in the case of a given Γ (4) L , different expansions of the same counterterm δλ renormalizesthe overall divergences and the subdivergences. We should note however that all these countertems are notcompletely independent since, as discussed at the end of Sect. 2.2, at a given loop order the diagrams thatcontribute to Γ (4) L can be separated into contributions to I and diagrams resulting from the iteration of theBS equation (see e.g. Fig. 13). Thus, when restricted at a given loop order, the counterterm δ ˜ λ L obviouslycoincides with δλ L , the difference between the two quantities being of order λ L +2 .Returning now to the general form of the BS equation, Eq. (28), we use the equation for the counterterm46 igure 13: The contributions to Γ (4 , that involve iterations of I in the BS equation up to order λ . δ ˜ λ in the form of Eq. (124) and subtract it from the BS equation. One obtains Γ (4)Φ L ( q, p ) − λ = ˜ I L − ( q, p ) − ˜ I L − (0 , − (cid:90) r Γ (4)Φ L ( q, r ) G ( r ) I L − ( r, p ) + 12 (cid:90) r Γ (4)Φ L (0 , r ) G ( r ) I L − ( r, I L − ( q, p ) − ˜ I L − (0 , − (cid:90) r Γ (4)Φ L ( q, r ) G ( r )[˜ I L − ( r, p ) − ˜ I L − ( r, (cid:90) r Γ (4)Φ L (0 , r ) G ( r )[˜ I L − ( r, − ˜ I L − ( r, q )] . (136)In going from the first to the second line, we have added and subtracted the same quantity but written intwo equivalent ways (that follows from the ladder structure of the diagrams resummed by the BS equation,as well as the symmetry under the exchanges of the arguments): (cid:90) r Γ (4)Φ L ( q, r ) G ( r ) I L − ( r,
0) = 12 (cid:90) r I L − ( q, r ) G ( r )Γ (4)Φ L ( r, . (137)The benefit of the second equality in Eq. (136) is that it provides an explicitly finite equation for Γ (4)Φ L ( q, p ) .Indeed ˜ I L − ( r, q ) has no subdivergences thanks to the renormalization of the Γ (4) L ’s, and the remainingoverall divergence cancels in the difference ˜ I L − ( r, q ) − ˜ I L − ( r, . Moreover, as we argue in Appendix D,due to the two-particule irreductibility of ˜ I L − ( r, q ) , the leading asymptotic behavior at large r does notdepend on q . It therefore cancels in the difference ˜ I L − ( r, q ) − ˜ I L − ( r, , ensuring the convergence of theintegrals in the second equality of Eq. (136). δZ and δm and renormalization of G We come now to the final steps involved in the renormalization of the gap equation. From Eq. (113),one derives the gap equation in the form G − ( p ) = Zp + m + δm + ( λ + δ ˜ λ ) δ Φ (2) δG ( p ) + (cid:88) ≤ l ≤ L (cid:2) ( λ + δλ ) l − (cid:3) L − l ( − ) l δ Φ ( l ) [ G ] δG ( p ) . (138)47mposing the renormalization conditions, we find δm = − ( λ + δ ˜ λ ) δ Φ (2) δG (0) − (cid:88) ≤ l ≤ L (cid:2) ( λ + δλ ) l − (cid:3) L − l ( − ) l δ Φ ( l ) [ G ] δG (0) , (139)and δZ = − (cid:88) ≤ l ≤ L (cid:2) ( λ + δλ ) l − (cid:3) L − l ( − ) l dd p δ Φ ( l ) [ G ] δG ( p ) (cid:12)(cid:12)(cid:12)(cid:12) p =0 , (140)where we have used that δ Φ (2) /δG ( p ) does not depend on p . Again, we know from the general considera-tions of Sect. 6.2 that, together with the previously determined δλ and δ ˜ λ , the counterterms δZ and δm renormalize the gap equation. We can use their expressions above to rewrite the gap equation as G − ( p ) = p + m + L (cid:88) l =3 (cid:2) ( λ + δλ ) l − (cid:3) L − l ( − ) l (cid:34) δ Φ ( l ) [ G ] δG ( p ) − δ Φ ( l ) [ G ] δG (0) − p dd p δ Φ ( l ) [ G ] δG ( p ) (cid:12)(cid:12)(cid:12)(cid:12) p =0 (cid:35) . (141)This equation still depends on the counterterm δλ needed to absorb the divergences associated to the Γ (4) L ’s(see Eq. (114)). Note, however, that the value of δ ˜ λ is not used at all. This is because the term inwhich it would appear in the gap equation is momentum independent, and it is cancelled by an identicalcontribution to δm that has the opposite sign. This is why, for instance, the renormalization of thegap equation in the vacuum can be done independently from that of the BS equation [54, 55]. At finitetemperature, this would remain true in a thermal scheme where renormalization conditions are imposed atfinite temperature. However, in this case the renormalized mass becomes temperature dependent and theanalysis of this temperature dependence relates to the BS equation and its renormalization [56]. The BS isalso mandatory when one does not use a thermal scheme (see the discussion in Appendix A).Similar considerations can be used to renormalize the free-energy density, to within an additive globalcounterterm. Let us summarize the above findings by providing a synthetic renormalization procedure for Φ -derivableapproximations. Consider the L -loop Φ -derivable approximation. In a first step, one replaces the bare L -loop truncation by Γ[ G ] = 12 (cid:90) p log G − ( p ) + 12 (cid:90) p ( Zp + m + δm ) G ( p )+ ( λ + δ ˜ λ )Φ (2) + (cid:88) ≤ l ≤ L (cid:2) ( λ + δλ ) l − (cid:3) L − l ( − ) l Φ ( l ) [ G ] . (142)48econd, the counterterms δZ , δm , δ ˜ λ and δλ are given by δZ = − (cid:88) ≤ l ≤ L (cid:2) ( λ + δλ ) l − (cid:3) L − l ( − ) l dd p δ Φ ( l ) [ G ] δG ( p ) (cid:12)(cid:12)(cid:12)(cid:12) p =0 , (143) δm = − ( λ + δ ˜ λ ) δ Φ (2) δG (0) − (cid:88) ≤ l ≤ L (cid:2) ( λ + δλ ) l − (cid:3) L − l ( − ) l δ Φ ( l ) [ G ] δG (0) , (144) δ ˜ λ = − ˜ I L − (0 , − (cid:82) r Γ (4)Φ L (0 , r ) G ( r )( λ + ˜ I L − ( r, − (cid:82) r Γ (4)Φ L (0 , r ) G ( r ) . (145) δλ = λ Γ (4 , ( p i = 0) + λ (cid:104) − Γ (4 , ( p i = 0) + 2 (cid:0) Γ (4 , ( p i = 0) (cid:1) (cid:105) + . . . (146)where the higher terms in δλ can be determined systematically by renormalizing the skeleton expansion ofthe four-point function, as explained above, and where ˜ I L − ( q, p ) ≡ (cid:88) ≤ l ≤ L (cid:2) ( λ + δλ ) l − (cid:3) L − l ( − ) l δ Φ ( l ) [ G ] δG ( q ) δG ( p ) . (147)These counterterms renormalize not only the gap and Bethe-Salpeter equations that one derives from (142),but also the free-energy density up to an overall shift.The equations (142)-(147) provide a concise summary of the renormalization of Φ -derivable approxima-tions, to all orders, in their diagrammatic formulation.
7. General remarks and practical implementations
We have seen in the previous section that the flow equations (77)-(79) allow us to understand therenormalization of Φ -derivable approximations in their original diagrammatic formulation. This is becausethe diagrammatic formulation can be based on the same flow equations that were used in the previous section,but keeping the ultraviolet cutoff and choosing the initial scale Λ = ∞ , as indicated at the beginning of thissection, see Eq. (85).As described above, this exact rewriting of the diagrammatic formulation provides a clear map of howdivergences are distributed among the relevant n -point functions and shows that their cancellations can beachieved by removing solely overall divergences. This is done by adjusting the counterterms in the initialconditions in Eqs. (98-100) such that the renormalization and consistency conditions are met at κ = 0 . Byimposing these conditions on the relevant n -point functions written in terms of the diagrams (and not as aformal solution to the flow equations), we can obtain the explicit expressions for the counterterms.In fact, for a given set of renormalization and consistency conditions, the renormalized solution definedby Eq. (80) and the renormalized solution defined by Eq. (85), with the appropriate modifications referred49o above, are one and the same renormalized solution. To make this clearer, let us rewrite Eq. (85) here: Γ ( n ) κ ( p i ) = Γ ( n )Λ ( p i ) + (cid:90) κ Λ d κ (cid:48) F ( n ) κ (cid:48) ( p i ) (cid:12)(cid:12)(cid:12) Λ uv . (148)On may read this equation backwards, that is as an equation specifying the n -point functions at scale Λ interms of their values at scale κ = 0 , i.e., in terms of their renormalized values. What was done in Sect. 5was to choose an initial scale large compared to the physical momenta, Λ (cid:29) Λ phys , but small compared tothe ultraviolet cutoff. In fact, as long as Λ is kept finite, the ultraviolet cutoff is not needed, and can justbe dropped. One gets then Γ ( n )Λ ( p i ) = Γ ( n ) κ =0 ( p i ) + (cid:90) Λ0 d κ (cid:48) F ( n ) κ (cid:48) ( p i ) . (149)The flow equation ensures that Γ ( n ) κ =0 ( p i ) , the renormalized n -point functions at scale κ = 0 , remain constantas we change Λ : the change of Γ ( n ) κ =0 ( p i ) is compensated by the change of the integration boundary. Whatwas done in the previous section was to rewrite the same solution with a different treatment of the initialconditions. Because we let Λ → ∞ , we need to keep Λ uv finite to avoid divergences. One gets then Γ ( n )Λ →∞ ( p i ) = Γ ( n ) κ =0 ( p i ) + (cid:90) ∞ d κ (cid:48) F ( n ) κ (cid:48) ( p i ) (cid:12)(cid:12)(cid:12) Λ uv . (150)Divergences appear as one lets Λ uv → ∞ . What we have seen in the previous section is that these diver-gences can be handled by counterterms in the initial values Γ ( n )Λ →∞ ( p i ) . These counterterms eliminate globaldivergences, subdivergences being effectively taken care of by the coupled flow equations. The initial con-ditions Γ ( n )Λ →∞ ( p i ) depend on Λ uv in a similar way as Γ ( n )Λ ( p i ) depend on Λ , this being dominated by powercounting. In a way, we could view the flow approach based on Eq. (149) as a (powerful) regularisation, whichavoids the needs to determine the counterterms of the more conventional regularizations of diagrammaticapproaches, such as summarized in Eq. (150). Because we are dealing with the same solution, it is clearthat the functions that need to be initialized in this “flow regularization” are nothing but those functionsthat need to be renormalized in the cutoff regularization.Our analysis has been based on a specific treatment of the flow of the irreducible kernel which, we believe,is that which best reveals the general structure of the approximations that we are considering. However,in practical applications, some hybrid alternative could be advantageous. For instance, since we have anexplicit form for the renormalized Γ (4) L,κ ’s, Eq. (114), we can use it directly in Eq. (60) with the expansionoperator [ _ ] L replaced by { _ } L or reinterpreted as referring to the loop expansion in powers of λ . One isthen left with the flow equations for the two-point function Γ (2) L − ,κ and for the four-point function Γ (4)Φ L,κ .The practical advantage of such an hybrid scheme is that computing directly the renormalized Γ (4) L,κ ’s could50e more economical than solving the tower of their flow equations. Yet another possibility is to couplethe flow equation for Γ (2) L − ,κ to the renormalized BS equation for Γ (4)Φ L ,κ , obtained from a straightforwardextension of Eq. (136) in the presence of R κ .
8. Improving on Φ -derivable approximations using their fRG reformulation In the previous sections, we have shown that the reformulation of Φ -derivable approximations in terms offlow equations provides much insight into their renormalization. The resulting flow equations form a finite setof equations, realizing a specific truncation of the fRG equations, and we have indicated several strategiesthat can be implemented to solve these equations in practice. In this section, we exploit the flexibilityof this reformulation and show how it can be used to construct approximations that extend Φ -derivableapproximations beyond their standard diagrammatic derivation. As an example, we derive approximationsthat are crossing-symmetric. We also provide possible generalizations of the present truncation of the fRGequations. Let us first recall that the standard 1PI flow equations are explicitly crossing symmetric, and remainso even in the presence of approximations. This is perhaps not completely obvious at the level of Eq. (49)because this equation is restricted to a particular momentum configuration. However, one may derive amore general version of the flow equation, valid for an arbitrary configuration of external momenta, viz. ∂ κ Γ (4) κ ( p , p , p , p ) = − (cid:90) r ∂ κ R κ ( r ) G κ ( r ) Γ (6) κ ( r, − r, p , p , p , p )+ (cid:90) r Γ (4) κ ( p , p , r, − p − p − r ) G κ ( r + p + p ) G κ ( r ) ∂ κ R κ ( r ) Γ (4) κ ( r, − p − p − r, p , p )+ (cid:90) r Γ (4) κ ( p , p , r, − p − p − r ) G κ ( r + p + p ) G κ ( r ) ∂ κ R κ ( r ) Γ (4) κ ( r, − p − p − r, p , p )+ (cid:90) r Γ (4) κ ( p , p , r, − p − p − r ) G κ ( r + p + p ) G κ ( r ) ∂ κ R κ ( r ) Γ (4) κ ( r, − p − p − r, p , p ) . (151)Equation (151) is now explicitly crossing symmetric, ensuring in a trivial way that the crossing symmetryof the n -point functions is preserved along the flow. As we have already mentioned, the same property doesnot hold for the flow formulation of the 2PI equations that is discussed in the main text (see the discussionat the end of Sect. 3.1). However, we shall see that the flow formulation allows for a simple extension of thisequation that preserves crossing symmetry through approximations.In the present discussion, we do not immediately restrict ourselves to translational invariant system andassume the propagator to be a function of two momentum variables, G ( p , p ) . By taking two derivatives51f the Luttinger-Ward functional with respect to G ( p, q ) , one obtains the kernel I ( p , p ; p , p ) = 4 δ Φ[ G ] δG ( p , p ) δG ( p , p ) . (152)This can be used to obtain the complete four-point function from the Bethe-Salpeter equation Γ (4) ( p , p ; p , p ) = I ( p , p ; p , p ) − (cid:90) q,k,r,s Γ (4) ( p , p ; q, r ) G ( − q, − k ) G ( − r, − s ) I ( − k, − s ; p , p )= I ( p , p ; p , p ) − (cid:90) q,k,r,s I ( p , p ; q, r ) G ( − q, − k ) G ( − r, − s )Γ (4) ( − k, − s ; p , p ) . (153)Here it is understood that G , I and Γ (4) are full Fourier transforms (see the discussion around Eq. (7),where full and reduced Fourier transforms are introduced). Restricting now the propagators to be those of atranslation invariant system, G ( p, q ) = δ ( d ) ( p + q ) G ( p ) , and extracting similar δ -functions from I and Γ (4) ,we arrive at the following equation for the reduced Fourier transforms (we keep the same notation for thereduced and full Fourier transforms, as done throughout the paper) Γ (4) ( p , p ; p , p ) = I ( p , p ; p , p ) − (cid:90) q Γ (4) ( p , p ; q, − p − p − q ) G ( q ) G ( p + p + q ) I ( − q, p + p + q ; p , p )= I ( p , p ; p , p ) − (cid:90) q I ( p , p ; q, − p − p − q ) G ( q ) G ( p + p + q )Γ (4) ( − q, p + p + q ; p , p ) (154)where it is understood that p = − ( p + p + p ) and we have used G ( − q ) = G ( q ) . In the case where p = p = − p and p = q = − p , we recover Eq. (28). We can proceed identically in the presence of a regulator κ . Then, after taking a κ -derivative and slightly We absorb two factors of (2 π ) d in the definition of δ/δG ( p, q ) . In order to verify that the kernels coincide, we recall that the functional Φ in (27) is not exactly the same as the one in(152). The former is the restriction of the latter to propagators of the form G ( p, q ) = δ ( d ) ( p + q ) G ( p ) and there is also a volumefactor V that has been factored out. For the sake of clarity, we momentarily denote the reduced functional φ [ g ] = Φ[ G = gδ ] .Now δφδG ( p ) = 1 V (cid:90) q,r δ ( δ ( d ) ( q + r ) G ( q )) δG ( p ) δ Φ δG ( q, r ) (cid:12)(cid:12)(cid:12)(cid:12) G ( q,r )= δ ( d ) ( q + r ) G ( q ) = 1 V (cid:90) q,r δ ( d ) ( q + r ) δ ( d ) ( p − q ) δ Φ δG ( q, r ) (cid:12)(cid:12)(cid:12)(cid:12) G ( q,r )= δ ( d ) ( q + r ) G ( q ) = 1 V δ Φ δG ( p, − p ) . (155)and similarly δ φ/δg ( p ) δg ( q ) = (1 /V ) δ φ/δG ( p, − p ) δG ( q, − q ) . If we explicitly factor out the momentum conservation deltafrom δ φ/δG ( p, − p ) δG ( q, − q ) while keeping the same notation for the reduced object, that is δ φ/δG ( p, − p ) δG ( q, − q ) → δ ( d ) ( p − p + q − q ) δ φ/δG ( p, − p ) δG ( q, − q ) we arrive at δ φ/δg ( p ) δg ( q ) = δ φ/δG ( p, − p ) δG ( q, − q ) and so the kernels identifyindeed. ∂ κ Γ (4) κ ( p , p ; p , p ) = ∂ κ I κ ( p , p ; p , p ) − (cid:90) q Γ (4) κ ( p , p ; − q, q − p − p ) ∂ κ [ G κ ( q ) G κ ( p + p − q )] Γ (4) κ ( q, p + p − q, p , p ) − (cid:90) q ∂ κ I κ ( p , p , − q, q − p − p ) G κ ( q ) G κ ( p + p − q ) Γ (4) κ ( q, p + p − q, p , p ) − (cid:90) q Γ (4) κ ( p , p , − q, q − p − p ) G κ ( q ) G κ ( p + p − q ) ∂ κ I κ ( q, p + p − p , p )+ 14 (cid:90) q (cid:90) k Γ (4) κ ( p , p ; − q, p + p − q ) G κ ( q ) G κ ( p + p − q ) ∂ κ × I κ ( q, p + p − q ; k, p + p − k ) G κ ( k ) G κ ( p + p − k ) Γ (4) κ ( − k, k − p − p ; p , p ) . (156)This equation is quite different from (49). In particular, even though its solution in the absence of approxi-mations obeys crossing symmetry, this symmetry is not manifest in the equation itself, and it generally getslost in the presence of approximations. It is in this sense that the flow equation (46), which is only a par-ticular case of Eq. (156), is not compatible with crossing symmetry. This of course relates to the discussionat the end of Sect. 2.2 of how the different channels are non-equivalently resummed by the Bethe-Salpeterequation.To cope with this issue, we write a symmetrize version of Eq. (156), viz. ∂ κ Γ (4) κ ( p , p , p , p ) = 13 (cid:34) ∂ κ I κ ( p , p ; p , p ) − (cid:90) q Γ (4) κ ( p , p ; − q, q − p − p ) ∂ κ [ G κ ( q ) G κ ( p + p − q )] Γ (4) κ ( q, p + p − q, p , p ) − (cid:90) q ∂ κ I κ ( p , p , − q, q − p − p ) G κ ( q ) G κ ( p + p − q ) Γ (4) κ ( q, p + p − q, p , p ) − (cid:90) q Γ (4) κ ( p , p , − q, q − p − p ) G κ ( q ) G κ ( p + p − q ) ∂ κ I κ ( q, p + p − p , p )+ 112 (cid:90) q (cid:90) k Γ (4) κ ( p , p ; − q, p + p − q ) G κ ( q ) G κ ( p + p − q ) × ∂ κ I κ ( q, p + p − q ; k, p + p − k ) G κ ( k ) G κ ( p + p − k ) Γ (4) κ ( − k, k − p − p ; p , p )+ ( p ↔ p ) + ( p ↔ p ) (cid:35) . (157)This equation is, by construction, crossing-symmetric. It is obeyed by the exact four-point function. Whencoupled to Eq. (42), and upon truncations of the Φ[ G ] functional, it generates a new expansion scheme for53oth the two- and the four-point function, where crossing symmetry is explicitly implemented. We can even go a step further. We have illustrated above that subdivergent contributions that appear inthe flow of I κ could always be written in terms of the loop approximations Γ (4) κ,L to the four-point function.These approximations are always crossing symmetric. However they differ from the four-point function thatenters Eq. (42). We can solve this issue by replacing any occurrence of Γ (4) κ,L by the function Γ (4) κ that solvesEq. (157). For instance, at four-loop order in Φ , Eq. (57) would be replaced by ∂ κ I κ ( q, p ) = − (cid:90) r Γ (4) κ ( r, q, p ) ∂ κ G κ ( r ) G κ ( r + q + p ) Γ (4) κ ( r, q, p ) + ( q → − q )+ 2 (cid:90) s ∂ κ G κ ( s ) (cid:90) r Γ (4) κ ( r, q, p ) G κ ( r ) G κ ( r + q + p ) × Γ (4) κ ( − r, − s, − p ) G κ ( r + s + p ) Γ (4) κ ( r + p + q, s, − q ) + ( q → − q ) . (158)By combining Eqs. (42), (157) and (158), we have now an “improved” four-loop Φ -derivable truncation where not only crossing symmetry is manifest, but also, where only one version of the four-point functionruns along the flow. This is a simplification since it eliminates the need to deal with various versions of thefour-point function.Of course, in doing what we do here, we depart from the strict correspondence with the diagrammaticapproach. In particular the solution of the above flow equations is likely sensitive to the choice of regulator.So there is a trade-of between extending Φ -derivable approximations in a crossing symmetric way, as wepropose to do here, and the reintroduction of a regulator dependence. We mention, however, that, since thenew scheme is still based on a skeleton expansion, we expect this regulator dependence to be controlled, atleast formally, by the number of loops kept in the expansion. The analysis of the present paper has revealed the role of various skeleton expansions, beyond the skeletonexpansion of the Luttinger-Ward functional at the core of Φ -derivable approximations. These other skeletonexpansions suggest other possibilities to truncate the fRG hierarchy which we now briefly review.One possibility is to use directly the two-skeleton expansion of Γ (4) L . Thus one may replace the exactflow equation (42) by ∂ κ Σ κ ( p ) = − (cid:90) q ∂ κ R κ ( q ) G κ ( q ) Γ (4) L,κ ( q, p ) , (159) The inclusion of channels beyond the standard Φ -derivable approximation has been discussed in Ref. [57] within a particularapproximation in the context of magnetic catalysis. The present discussion makes these considerations more systematic. Of course, as one increases the truncation order of Φ[ G ] new terms appear in (158) but they can alway be determined in asystematic way. Γ (4) L,κ ( q, p ) . These Γ (4) L,κ ( q, p ) can be obtained as renormalized diagramsor as solutions of the flow equations (65).Yet another truncation consists in using the expansion of the six-point function in terms of four-skeletons,as discussed at the end of Appendix B. We write ∂ κ Σ κ ( p ) = − (cid:90) q ∂ κ R κ ( q ) G κ ( q ) Γ (4) κ ( q, p ) (160)with ∂ κ Γ (4) κ ( p , p , p , p ) = − (cid:90) r ∂ κ R κ ( r ) G κ ( r ) Γ (6) L,κ ( r, − r, p , p , p , p )+ (cid:90) r Γ (4) κ ( p , p , r, − p − p − r ) G κ ( r + p + p ) G κ ( r ) ∂ κ R κ ( r ) Γ (4) κ ( r, − p − p − r, p , p )+ (cid:90) r Γ (4) κ ( p , p , r, − p − p − r ) G κ ( r + p + p ) G κ ( r ) ∂ κ R κ ( r ) Γ (4) κ ( r, − p − p − r, p , p )+ (cid:90) r Γ (4) κ ( p , p , r, − p − p − r ) G κ ( r + p + p ) G κ ( r ) ∂ κ R κ ( r ) Γ (4) κ ( r, − p − p − r, p , p ) (161)with Γ (6) L,κ an explicitly finite expression in terms of G κ and Γ (4) κ . Again, this truncation is systematicallyimprovable by adding more terms in the expansion of Γ (6) κ in terms of four-skeletons.
9. Conclusions
In the present work we have studied a particular truncation of the flow equations for the 1PI effectiveaction for a scalar ϕ theory in four dimensions in the symmetric phase, extending the fRG-2PI approachinitiated in [39]. This truncation exploits the relation that exists between the four-point function and the two-point function in the 2PI formalism based on the Luttinger-Ward functional. It provides an exact formulationof the 2PI equations in terms of flow equations. The solution of these flow equations remains independent ofthe choice of the regulator that controls the flows within the so-called Φ -derivable approximations based on aselection of a finite set of skeletons contributing to the LW functional. This suggests specific approximationschemes in which skeletons are ordered according to their number of loops, and one selects skeletons up toa given number of loops. Once such an approximation is made, the two-point and four-point functions thatare obtained are of course only approximate. However the independence of the solution of the flow equationson the choice of the regulator persists.One important benefit of writing the 2PI equations in terms of flow equations is that the functionalRG provides much insight on the renormalization, which constitutes a major part of the present paper.55e have shown that the flow equations that govern a given Φ -derivable approximation are finite, thatis, they are independent of any ultraviolet cutoff that may need to be introduced at intermediate stages.Furthermore, they can be made independent of the parameters of the bare theory. It follows that therenormalization can be achieved by following essentially the same route as in the standard 1PI fRG, i.e.without introducing counterterms: the choice of counterterms is replaced by a suitable choice of initialconditions. The flow equations also clarify the mechanisms of elimination of divergences, in particular theytake care automatically of the subdivergences, and resolve the subtle issue of hidden subdivergences thatcomplicates the diagrammatic approach to renormalization. All in all, the flow equations provide a completeunderstanding of the all-order renormalization of Φ -derivable approximations, and clarify a number of issuesconcerning higher orders that were left unsettled in previous studies.The formulation of the 2PI equations in terms of flow equations not only provides an efficient tool tounderstand the formal aspects of the renormalization of Φ -derivable approximations, it also leads to variouspractical implementations. It offers furthermore a flexibility that allows us to extend the Φ -derivable approx-imations beyond the strict diagrammatic domain where they are usually defined. We have provided examplesof such extensions, as well as suggestions for new possible truncations of the fRG equations. Finally, wenote that the strategy that has been developed in this paper could also be tried with other non-perturbativeapproaches defined in terms of diagrams, such as higher ( n PI) effective actions or Dyson-Schwinger equations.
Acknowledgements
We thank N. Wink for discussions. This work is supported by EMMI, the BMBFgrant 05P18VHFCA. It is part of and supported by the DFG Collaborative Research Centre SFB 1225(ISOQUANT) as well as by the DFG under Germany’s Excellence Strategy EXC - 2181/1 - 390900948(the Heidelberg Excellence Cluster STRUCTURES). JPB is thankful for the warm hospitality of the ITPHeidelberg which he visited as a Jensen professor during early stages of this work. UR wishes to thank boththe ITP Heidelberg and the ECT* for support during the initial stages of this work.
Appendix A. Illustrative example: the -loop approximation In this appendix, we illustrate some of the formal developments of the main text with the simplest nontrivial example, that of the two-loop Φ -skeleton (left diagram in Fig. 2). The self-energy is momentumindependent and so is the four-point function Γ (4) ( p, q ) = Γ (4) (0 , , with I ( p, q ) = λ b . To alleviate thenotation, we set λ = Γ (4) (0 , , and call m the solution of the gap equation, to be identified with thephysical (or renormalized) mass. Appendix A.1. General considerations
We start by considering the gap equation, the BS equation, and their solutions. This will allow us inparticular to recall well known features of the ϕ field theory in four dimensions.56n the two-loop approximation, the gap equation and the BS equation take simple forms. The gapequation reads m = m + Σ( m ) = m + λ b I ( m ) , I ( m ) ≡ (cid:90) q< Λ uv q + m , (A.1)and the BS equation (28) can be written as λ = 1 λ b + 12 J ( m ) , J ( m ) ≡ (cid:90) q< Λ uv q + m ) . (A.2)These two equations provide the values of the physical mass m , and the four-point function λ , as a functionof λ b , m b and the ultraviolet cutoff Λ uv . At weak coupling, and for Λ uv (cid:29) m , we have approximately m − m ≈ λ b Λ / (32 π ) , λ ≈ λ b + 116 π (cid:18) − m (cid:19) . (A.3)The last relation shows that, at fixed λ b , λ is a decreasing function of Λ uv (eventually going to zero as Λ uv goes to infinity). Alternatively, Eqs. (A.3) specify how the parameters of the Lagrangian, λ b and m b , needto change when one varies Λ uv , so as to maintain λ and m at their physical values. Consider for instancethe derivative of λ b with respect to Λ uv , at fixed λ . One gets Λ uv ∂∂ Λ uv (cid:18) λ b (cid:19) ≈ − π . (A.4)The negative sign is indicative of the presence of a so-called “Landau pole”: λ b blows up for a finite value Λ L of Λ uv . To see that, we integrate Eq. (A.4) and get, with Λ some reference scale, λ b (Λ uv ) = λ b (Λ )1 − λ b (Λ )32 π log Λ uv Λ . (A.5)This shows that λ b (Λ uv ) blows up when Λ uv = Λ L , with Λ L = Λ exp (cid:18) π λ b (Λ ) (cid:19) . (A.6)As well known, the presence of this Landau pole renders delicate the discussions concerning the renormal-ization, and in particular the limit of infinite cutoff Λ uv → ∞ . We always assume in this paper that wework in a regime of sufficiently weak coupling (that is λ b (Λ ) (cid:28) ), so that we can allow Λ uv to becomelarge while staying far below Λ L , i.e., Λ (cid:28) Λ uv (cid:28) Λ L . Note that the BS equation captures only 1/3 of the one-loop beta-function, as only a single channel out of three is beingtaken into account. ppendix A.2. Flow equations Consider now the analogs of Eqs. (A.1) and (A.2) obtained when a regulator is added to all the propa-gators, as in Eq. (37). These read respectively m κ = m + λ b (cid:90) q< Λ uv G κ ( q ) , G κ ( q ) = 1 q + m κ + R κ ( q ) , (A.7)and λ κ = 1 λ b + 12 (cid:90) q< Λ uv G κ ( q ) , (A.8)where we have set λ κ = Γ (4) κ (0 , . The presence of the regulator does not alter significantly the generalstructure of the equations. In particular, for κ = 0 one recovers trivially the solution written above.We may however proceed as in Sect. 2.3 and obtain m κ and λ κ as solutions of flow equations. The flowequation for m κ is obtained simply by taking the derivative of Eq. (A.7). One gets ∂ κ m κ = − λ b (cid:90) q< Λ uv G κ ( q ) (cid:0) ∂ κ m κ + ∂ κ R κ (cid:1) , (A.9)or, using Eq. (A.8), ∂ κ m κ = − λ κ (cid:90) q< Λ uv ( ∂ κ R κ ) G κ ( q ) . (A.10)This is indeed the expected flow equation for the two-point function (see Eq. (42)). Note that the flow ofthe mass and that of the coupling constant are naturally coupled. The flow equation for λ κ , can be obtainedsimilarly by differentiating Eq. (A.8) with respect to κ . It reads ∂ κ (cid:18) λ κ (cid:19) = 12 ∂ κ (cid:90) q< Λ uv G κ ( q ) , ∂ κ λ κ = − λ κ ∂ κ (cid:90) q< Λ uv G κ ( q ) . (A.11)The writing on the right corresponds indeed to Eq. (46), where only the first line contributes since ∂ κ I κ = 0 in the present case.It remains to determine the initial conditions for these flow equations (see the discussion in Sect. 4.3).We want to choose these at a large value κ = Λ , in such a way as to recover the standard 2PI results at κ = 0 .This easily done since we know the explicit solution, given in Eqs. (A.7) and (A.8) above: it is sufficientto analyze the behavior of this solution as κ = Λ becomes large. If we use a sharp infrared regulator, theanswer is immediate: when Λ = Λ uv , Σ Λ = 0 so that m Λ = m b , and λ Λ = λ b . For a smooth regulator, e.g.58 κ ( q ) = ( κ − q ) θ ( κ − q ) , and Λ > Λ uv , one gets instead λ Λ = 1 λ b + 164 π Λ (Λ + M ) , Σ Λ = λ b π Λ Λ + M , (A.12)where m = m + Σ Λ . The dependence on Λ reflects the fact that, with a smooth regulator, the fluctuations(carrying momenta below the ultraviolet cutoff Λ uv ) are only gradually suppressed as Λ grows; they areentirely suppressed only as Λ → ∞ : when Λ / Λ uv → ∞ , λ Λ → λ b and Σ Λ / Λ → , the deviations fromthese limits being powers of Λ uv / Λ . Appendix A.3. Renormalization and hidden sub-divergences
We now examine the renormalization with counterterms, as outlined in Sect. 6. In the 2-loop approxi-mation, there is no field renormalization ( Z = 1 ) and therefore we have simply λ b = λ + δλ, m = m + δm . (A.13)With the self-energy given in Eq. (A.1), the gap equation reads m = m + λ b I ( m ) = m + δm + λ b I ( m ) , (A.14)where we have used Eq. (91) to identify the renormalized mass m with the solution of the gap equation. Theone-loop self-energy is divergent when Λ uv → ∞ . This divergence may be absorbed in the mass counterterm δm = − λ b I ( m ) . (A.15)However, by doing this simple subtraction, one ignores the fact that there are subdivergences hidden in I ( m ) , and failing to eliminate those properly may lead to difficulties. This is the case in particular infinite temperature calculations. Such subdivergences appear explicitly when one performs a perturbativeanalysis, i.e., expand the self-consistent propagator in powers of the coupling, and they are related to therenormalization of the coupling constant (see Fig. A.14 for an explicit example at two-loop perturbativeorder).We show now that such subdivergences are eliminated when one simultaneously renormalize the self-energy and the BS equation. For better illustration, we consider a calculation at finite temperature andwrite the relevant integrals as follows I ( M ) = I ( M ) + I T ( M ) ,J ( M ) = J ( M ) + J T ( M ) (A.16)59 igure A.14: The leading order contribution to the self-energy (left) and it first perturbative correction (right). The subdiagramisolated by the dashed line is divergent. This divergence corresponds to a subdivergence of the leading order self-energy whenthe latter is evaluated with a self-consistent propagator. Such a subdivergence is eliminated by a renormalization of the couplingconstant. where I ( M ) and I T ( M ) denote respectively the zero temperature and the finite temperature contributionsto I ( M ) , and similarly for J and J T ( J = − ∂I/∂M ). At one-loop order this separation is not ambiguous.We denote by M the solution of the finite temperature gap equation, the renormalized mass m being givenby the solution of this equation at zero temperature. Similarly, we call Γ (4) ( M ) the solution of the BSequation, with the renormalized coupling given by λ = Γ (4) ( m ) at T = 0 . We have M = m + δm + 12 ( λ + δλ ) I ( M ) , (4) ( M ) = 1 λ + δλ + 12 J ( M ) . (A.17)At this point, one may impose the renormalization conditions (at T = 0 ) and determine the counterterms.We get, δλ = λ J ( m ) λ − λ J ( m ) , δm = − λ + δλ I ( m ) . (A.18)Using this expression for the mass counterterm, we rewrite the gap equation as follows M − m = 12 ( λ + δλ ) [ I ( M ) − I ( m )] . (A.19)Now, we have I ( M ) = I ( m ) + (cid:0) M − m (cid:1) ∂I ∂M (cid:12)(cid:12)(cid:12)(cid:12) M = m + C ( M, m )= I ( m ) − (cid:0) M − m (cid:1) J ( m ) + C ( M, m ) (A.20)where C ( M, m ) is a finite quantity, with C ( M, M ) = I T ( M ) . Note how the quadratic divergence dropsin the difference I ( M ) − I ( m ) , leaving a logarithmic divergence that cancels against that of J ( m ) (thedivergent intermediate loop in Fig. A.14). By using the explicit expression of δλ in terms of J ( m ) given60bove, Eq. (A.18), one finally obtains the following simple form of the gap equation M − m = λ C ( M, m ) . (A.21)At zero temperature, M = m is obviously solution (since then C ( m, m ) = 0 ). At finite temperature theequation remains finite. Observe that the elimination of the subdivergences has been crucial to obtain thissimple result. This was achieved by considering the “internal structure” of the self-consistent propagators inorder to exhibit the hidden subdivergences. As we shall see in the next section, handling subdivergences ismade much easier with the flow equations.Before moving to the next section, we mention that one could also impose the renormalization conditionsdirectly at finite temperature. In this case δm = − λ + δλ I ( m ) (A.22)and the gap equation becomes M = m which is the simple statement that the renormalized mass is chosento coincide with the solution of the gap equation. Here, it may seem that the four-point function playsno role in the gap equation. This is however an illusion since, in this scheme, the renormalized mass istemperature dependent, and, when inquiring how it depends on the temperature the four-point functionreemerges. Indeed, since the bare mass m = m + δm is temperature independent, on has m d T = d m d T − λ + δλ I d T , (A.23)where we used that λ b = λ + δλ is also temperature independent. Now the temperature dependence of I iseither explicit, through I T , or implicit through m . Then, one finds m d T − λ + δλ (cid:20) − J ( m ) d m d T + ∂I∂T (cid:21) , (A.24)where we used that ∂I/∂m = − J ( m ) . Solving for d m / d T , one find eventually d m d T = Γ (4) ( m )2 ∂I∂T , (A.25)which involves the four-point function, as announced.61 ppendix A.4. Renormalization with flow equations In order to illustrate the developments in Sects. 5 and 6 we write the flow equations for m κ and λ κ asfollows (here, for simplicity, we switch back to the zero temperature case) ∂ κ m κ = F (2) κ ( m κ , λ κ ) , ∂ κ λ κ = F (4) κ ( m κ , λ κ ) , (A.26)and recall the important properties that the flows F ( n ) κ are finite. It is convenient however, for integratingthe equations, to keep an ultraviolet cutoff. We shall then be able to illustrate the strategies followedrespectively in Sects. 6 and 5.In the first case, that of Sect. 6, one writes the solution of the flow equations formally as m κ = m + (cid:90) κ ∞ d κ (cid:48) F (2) κ (cid:48) ( m κ (cid:48) , λ κ (cid:48) ) (cid:12)(cid:12)(cid:12) Λ uv , λ κ = λ b + (cid:90) κ ∞ d κ (cid:48) F (4) κ (cid:48) ( m κ (cid:48) , λ κ (cid:48) ) (cid:12)(cid:12)(cid:12) Λ uv , (A.27)where we have made explicit the dependence of the flows on the ultraviolet cutoff. The initial condition isset at the scale Λ (cid:29) Λ uv , actually Λ → ∞ , and we used m →∞ = m and λ Λ →∞ = λ b . By replacing thebare parameters by their expressions (A.13) in terms of renormalized ones and counterterms, one recoversthe formal expressions (101) and (103) of the counterterms (note that δλ here is the same as what is called δ ˜ λ in Sect. 6): δm = − (cid:90) ∞ d κ (cid:48) F (2) κ (cid:48) ( m κ (cid:48) , λ κ (cid:48) ) (cid:12)(cid:12)(cid:12) Λ uv , δλ = − (cid:90) ∞ d κ (cid:48) F (4) κ (cid:48) ( m κ (cid:48) , λ κ (cid:48) ) (cid:12)(cid:12)(cid:12) Λ uv . (A.28)Explicit expressions are easily obtained by integrating the flow equation Eq. (A.11) for λ κ between the scale Λ and the scale κ . One gets λ κ − λ Λ = (cid:90) κ Λ d κ (cid:48) (cid:90) q< Λ uv G κ (cid:48) ( q ) ∂ κ (cid:48) G κ (cid:48) ( q ) , (A.29)or, letting Λ → ∞ , and setting λ κ =0 = λ , λ − λ b = (cid:90) ∞ d κ (cid:48) (cid:90) q< Λ uv G κ (cid:48) ( q ) ∂ κ (cid:48) G κ (cid:48) ( q ) = 12 (cid:90) q< Λ uv G ( q ) , (A.30)where we have used G Λ →∞ = 0 . The expression of the counterterms, already given in Eq. (A.18), followsby replacing λ b by λ + δλ . One may proceed similarly for the mass. By using directly the solution given inEq. (A.7), one gets m κ − m λ b = 12 (cid:90) q< Λ uv [ G κ ( q ) − G Λ ( q )] . (A.31)62etting Λ → ∞ , using m →∞ = m = m + δm , and setting m κ =0 = m , one immediately recovers theexpression (A.18) of the mass counterterm. In fact, it is interesting to rewrite this equation (A.31) in termsof the renormalized parameters m κ λ = m λ + 12 (cid:90) q< Λ uv (cid:2) G κ ( q ) − G ( q ) + ( m κ − m ) G ( q ) (cid:3) . (A.32)One recognizes in this integral the pattern of the elimination of divergences already discussed at the endof Sect. Appendix A.3, with the last term proportional to G cancelling the logarithmic divergence in thedifference G κ − G , and leaving a potential quadratic divergence that disappears in the difference. This simpleexample illustrates the efficiency of the flow equation in dealing with (hidden) subdivergences.We now return to the flow equations (A.26), and integrate them from 0 to Λ , and Λ is now kept finite.For λ κ the result is written in Eq. (A.29), which we can rewrite as follows λ κ − λ Λ = 12 (cid:90) q< Λ uv (cid:2) G κ ( q ) − G ( q ) (cid:3) . (A.33)The integral is now finite, which results from the fact that the integration over the scale parameter κ islimited to the finite interval [0 , Λ] . The limit Λ uv → ∞ can then be trivially taken. One sees that λ Λ isthe quantity that is related by the flow to the renormalized coupling λ = λ κ =0 . In other word this is theproper initial condition on the flow, when this is initialized at Λ < Λ uv . Note how λ κ remains independentof Λ : as one varies Λ , one changes the amount of fluctuations that are integrated out (in the integral inthe right-hand side), and that change is exactly compensated by a change in the value of λ Λ . If Λ is largeenough, the flow of λ Λ is simple, Λ ∂ Λ λ Λ (cid:39) − / (32 π ) . This specifies how λ Λ has to change as one modifiesthe initial scale Λ so as to leave invariant the flow of λ κ for κ < Λ , and in particular the renormalizedcoupling λ = λ κ =0 . Similar considerations apply of course for the dependence of the bare parameters or thecounterterms on Λ uv , as given for instance by Eqs. (A.3) for the bare parameters, or Eqs. (A.18) for thecountertermsSimilar manipulations can be done for the mass. One obtains easily the following equation, m κ λ Λ = m λ Λ + 12 (cid:90) q< Λ uv (cid:2) G κ ( q ) − G Λ ( q ) + ( m κ − m ) G ( q ) (cid:3) , (A.34)in terms of the parameters at scale Λ . Here again we can set Λ uv → ∞ . By choosing λ Λ and m Λ , therenormalized parameters at the scale Λ , as initial conditions for the coupled flow equations for Γ (4) κ and m κ ,one eliminates the need to determine the counterterms of the standard renormalization procedure. Renor-malization here amounts to appropriate subtractions at the scale Λ , as shown for instance in Eqs. (A.29) and(A.34). Once such subtractions are made at the initial scale, the coupled equations take care automaticallyof all potential four-point subdivergences. 63 ppendix B. Four point insertions in ∂ κ I κ In this section we generalize the analysis presented in Sect. 4 to arbitrary diagrams contributing to theflow of I κ . The diagrams contributing to I κ are obtained from the skeleton diagrams of Φ by opening twolines, and the derivative ∂ κ entails opening one extra line, thereby making the diagrams of ∂ κ I κ effectivelythose of a six-point function, δ I /δG . Examples of diagrams contributing to Φ and I , and exhibiting varioustopologies, are given below (Figs. B.15 and B.16) at five and six-loop orders for Φ . Our goal in thisAppendix is to analyze the four-point insertions in the diagrams contributing to ∂ κ I κ and show how thesecan be resummed into complete four-point functions. At the end of this Appendix, we mention possiblegeneralizations of this analysis to other classes of diagrams as well as to other n -point functions. Figure B.15: Some five and six-loop diagrams that contribute to Φ , with various topologies. Let us first specify what we mean by a four-point insertion, which we shall refer to simply as a four-insertion. Consider a diagram D contributing to ∂ κ I κ , noted ∂ I for short. We draw this, as in Fig. B.16,as a diagram contributing to I , with one line carrying a slash, which is the line to be opened to yield thecorresponding diagram of δ I /δG . The external lines of the diagrams are labelled by a number, specifyingthe channel in which I is irreducible (see Appendix C for more details). Thus, for instance, the diagramsin Fig. B.16 are irreducible in the channel (12;34). The external lines are considered parts of the diagram.A four-insertion is any 1PI four-point subgraph of D , possibly including some of the external lines of D ,that is obtained by cutting up to four internal lines of D , such that, after the cut, the diagram splits intotwo disconnected pieces. Including the external lines in the process allows us in particular to consider asinsertions the tree-level vertices (as well as insertions containing vertices attached to these external lines).The four-insertions that isolate a single tree-level vertex will be called trivial. A diagram in which onlytrivial four-insertions can be identified will be said to be irreducible. Examples are displayed in Fig. B.16.Given a diagram D , it is straightforward to make the list of all its four-insertions. Elements of this listthat have the same topology, but involve distinct elements of D , are to be considered as distinct. Amongthe four-insertions of D some can be imbedded into larger four-insertions (see Fig. B.16 for an example).We call maximal insertion a four-insertion which is not itself a four-insertion in a larger one. We shallargue in this appendix that these maximal insertions can be identified without ambiguity in any diagram D contributing to ∂ I . It is then possible to substitute them by trivial ones, i.e., by tree-level vertices. In64 Figure B.16: Some diagrams that contribute to I , and that can be deduced from those of Φ in Fig. B.15 by functionaldifferentiation. The lines that carry a slash are the lines that are opened when differentiating I , leading effectively to diagramsfor six-point functions. The dotted lines isolate four-insertions. The middle bubble in the left diagram is an insertion embeddedin the larger insertion made by the two most right bubbles which constitute a maximal insertion for ∂ I . The diagram on theright is irreducible, i.e., it contains only trivial four-insertions. so doing, one transforms D into an irreducible diagram, in the sense defined above, which we shall refer toas a four-skeleton . Examples of four-skeletons contributing to ∂ I are given in Fig. B.17. That the maximalfour-insertions can be identified unambiguously means that a given diagram of ∂ I has a unique four-skeleton.It follows that one can generate all the diagrams of ∂ I by replacing the vertices of the four-skeletons by thecomplete four-point function, which is our ultimate goal (see Sect. 4). This replacement does not changethe overall irreducibility properties of ∂ I : a four-skeleton of ∂ I is irreducible in the channel (12; 34) (it contains no four-insertion with lines 1,2 or 3,4 as external legs), and this property subsists after thereplacement of the trivial vertices by full four-point functions. It is also clear that all the diagrams of ∂ I can be generated in this way, since each diagram of ∂ I has a unique skeleton. Furthermore, the argumentthat we shall develop shortly, showing that maximal four-insertions cannot have common elements, indicatesthat symmetry factors factorize into symmetry factors attached to the skeletons, and those of the individualfour-point functions that sit at the vertices of the four-skeletons. This insures that each diagram is properlycounted. An explicit verification by direct evaluation of the symmetry factors will be given elsewhere [58]. Figure B.17: Some of the irreducible diagrams that contribute to ∂ I that are deduced from the three diagrams of Fig. B.15 bythe procedure indicated in the text. These diagrams contain only trivial four-insertions. The left diagram is clearly one-linereducible: it is split into two parts when the line that joins the two vertices is cut. The issues that we are considering in this appendix bear similarities with those addressed at the beginningof this paper about 2PI, or more generally n PI, approximations. Among typical such issues, let us recall,for instance, that vacuum diagrams do not, in general, have unique two-skeletons, and the substitution ofthe bare propagators by dressed ones in the Luttinger-Ward functional generates an over-counting of the65riginal diagrams. By contrast, the two-skeletons of the two-point function (or higher n -point functions)can be identified without ambiguity and it is then possible to generate, without double counting, all thediagrams of the two-point function by substituting in its two-skeletons the bare propagators with the fullones. However, the notion of maximal four-insertion, and the fact that we consider external lines in thedefinition of the four-insertions, are important elements that make the present analysis deviate somewhatfrom more standard diagrammatic analysis in n PI formalisms (see e.g. [49]). Note for instance that theanalysis of this appendix will apply to ∂ I but not to I . For one thing, with our definition, each diagram of I is a (maximal) four-insertion, so that I has a unique skeleton, the tree-level vertex. But the replacement ofthis vertex by a full four-point function would not be legitimate since it would not preserve the irreducibilityof I . It is easy to see that no such constraint remains on the four-insertions of ∂ I .The proof that maximal four-insertions in the diagrams of ∂ I can be non ambiguously identified relies onthe property that, in such diagrams, any two maximal four-insertions cannot have any element in common,that is, they do not overlap. In order to establish the latter property we shall consider first the case of atwo-point function for which such overlaps are possible, and we shall identify how two insertions M and M (not necessarily maximal) can overlap. Then we shall return to the diagrams of ∂ I and, by using areductio ad absurdum argument, we will show that no overlap of two four-insertions can occur there if weassume M and M to be maximal.Consider then a diagram D , in which two insertions M and M have been identified. The insertion M (a similar discussion can be made for M ) has been obtained by cutting a quadruplet of lines in thediagram, including possibly some of the external lines of D . We shall call these four lines the external legsof the insertion: these are composed, as just said, of cut internal lines of D and possibly some externallines of D . In M , there could also exist sets of internal lines such that, once cut, the insertion splits intotwo disconnected parts. We shall call connecting lines such internal lines. To make things more concrete a b c a b c M M C Figure B.18: Example of two overlapping (non maximal) four-insertions M and M , in a diagram D with two external lines( n = 2 ). consider for instance the diagram of Fig. B.18, that contributes to a two-point function ( n = 2 ). In Fig. B.18,66 abc c cba Figure B.19: Example of two overlapping maximal four-insertions in a diagram with n = 2 . The two diagrams represent twodistinct four-insertions that have all their vertices, and all their lines but two in common. M is isolated from D by cutting the lines a , b , c . Together with the external line labelled 1, these linesconstitute the external legs of M . By cutting the lines a , b , c one splits M into two disconnectedpieces, one of them containing the vertex attached to the line 1, the other part being a six point functionwith a , b , c and a , b , c as external legs. The lines a , b , c are connecting lines of M . Similarly, theinsertion called M , has a , b , c and 2 as external legs, and a , b , c as connecting lines. M and M share a common subgraph, the six-point function labelled C , already identified as part of M . The structurethat we see emerging on this simple example is that depicted more generally in Fig. B.20 below. It revealsone possible type of overlap between four-insertions which, however, are here non maximal.There is another possibility of overlap, which is illustrated in Fig. B.19. The two displayed four-insertionsthere are now maximal, and overlapping. They share all their vertices and their lines, except the line a ,opened in the left one and c opened in the right one. There are no connecting lines. The two lines a and c play a special role: they are not lines of C , the set of common elements, but they join two vertices thatbelong to C . For this reason, such lines will be called returning lines . Each of these lines belongs to one ofthe four-insertions but not to the other. Typically, opening such a line in an n -point function generates a n + 2 point function. Thus, returning lines may appear as parts of the external legs of a four-insertion, butthey play no role in isolating the insertion from the rest of the diagram. For a more general discussion onhow connecting and returning lines describe the insertion of a graph into another graph, see [58].With these elements in mind, we now return to the diagram displayed in Fig. B.20, which is a part of abigger diagran D . This exhibits two overlapping insertions M and M and has n external lines (we keephere n ≥ as a free parameter). We denote by C the part common to M and M . We assume that M contains m > lines connecting C and M / C , where M / C denotes the complementary part of C in M , and similarly for M . In order to isolate M in D , one has to cut a number of lines: these include theconnecting lines of M , as well as possibly internal lines of D . That is, the lines whose number is denotedby n C may contain external legs of D as well as cut internal lines of D that connect a vertex of C to the restof the diagram. These n C lines are common to both M and M . Similarly, the n M / C external lines of M / C is the set of all the vertices of M that are not in C together with the lines attached to them. The m connectinglines are not counted as part of the external lines of M / C . / C may contain external lines of D and cut internal lines of D . The total number of external lines of thefour-insertion M , n M , is therefore n M = n M / C + n C + m . The same reasoning holds for M so thatwe have n M = n M / C + n C + m = n M = n M / C + n C + m , (B.1)where we have used the fact that both M and M are four-insertions, so that n M = n M = 4 . M / C M / CC m m n C n M / C n M / C Figure B.20: Generic configuration with an overlap of two maximal vertex insertions.
At this point, we note that M ∪ M , the graph that contains all the elements of M and M , can beviewed as a 1PI subgraph of D with n M / C + n M / C + n C external legs. This number is necessary greaterthan 2 since we started from a diagram D that does not contain any self-energy insertion and our assumptionthat D has at least four external legs exclude the case n = 2 . Since it is not possible to isolate three-pointfunctions in D , we must have then n M / C + n M / C + n C ≥ . The case n M / C + n M / C + n C = 4 is excludedbecause it would imply that both M and M are insertions of M ∪ M , which is impossible since M and M are maximal. We conclude therefore that n M / C + n M / C + n C > which, with the help of Eq. (B.1)we can rewrite as m + m + n C < . (B.2)But m ≥ and m ≥ since M and M are 1PI subgraphs, so that m + m + n C ≥ , in contradictionwith Eq. (B.2). We conclude that two four-insertions of D cannot share a common subgraph C with thestructure exhibited in Fig. B.20.The acute reader may have noticed that we did not take into account the possibility that returning linescould be attached to C . That is, we have ignored the possibility that the dotted box on the right of Fig. B.20cuts one or various lines that start at a vertex of C and return to a vertex of C while not being cut by thedashed box on the left. One could also consider similar returning lines that are cut by the left box but notby the right one. In fact it is easy to argue that such cases cannot occur here. Assume indeed that there68s, for instance, one returning line contributing to M . It would contribute to two external lines of M ,meaning that the insertion that has been isolated from D before opening the returning line is a two-pointfunction. But this is excluded by our assumptions that D is 2PI and n ≥ .Now, even though this possibility is excluded in the present case, it is nevertheless interesting to push thereasoning by keeping open the possibility of returning lines, in particular in view of potential generalizationsof the present analysis [58]. To do so, let us assume that in addition to the lines indicated in Fig. B.20 thereare r returning lines in M and r returning lines in M . It is easy to see that the mere effect of theirpresence is to change m → m + 2 r and similarly for m . The inequality (B.2) gets modified by the samesubstitution and leads to the same contradiction, and hence the same conclusion.We need now examine the two possibilities that are not covered by the generic case that we have justconsidered and which assumed that both m and m were different from zero. The first situation is that inwhich the two four-insertions M and M share all their vertices. This corresponds to m = m = 0 . If weignore possible returning lines, it is clear that M = M which is just the trivial case of overlap between afour-insertion and itself. With the possibility of returning lines taken into account, Eq. (B.1) gets replacedby n C + 2 r = n C + 2 r , so that either r = r = 1 and n C = 2 or r = r = 2 and n C = 0 . In thefirst case, M ∪ M is a two-point function and in the second case a zero-point function, but this is justimpossible since we excluded these possibilities.The second situation is that where the common part C shares all its vertices with say M , but wherethere are still connecting lines within M . This correspond to the case m (cid:54) = 0 , m = 0 (clearly, the case m = 0 , m (cid:54) = 0 can be treated similarly). If we ignore possible returning lines, it is clear that M ⊂ M which is not possible since we assumed M to be maximal. If we include the possibility of returning lines,following the same reasoning as above, we arrive at m + 2 r + 2 r + n C < . But since m ≥ and at leastone of the r i is different from zero, we have m + 2 r + 2 r + n C ≥ which again leads to a contradiction.This concludes our proof that in the diagrams contributing to ∂ I there is no ambiguity in isolating four-vertex insertions, and consequently in identifying the four-skeletons. This proof extends in fact to a widerclass of diagrams. Clearly, it applies to any connected two-skeleton diagram that has a number n ≥ ofexternal lines (this is why we kept n ≥ as a free parameter in the previous discussion), and by extension,to any disconnected diagram made of such pieces. One example is provided by the diagrams that enterthe 2PI n -point functions δ m I /δG m , with m > ( m = 1 corresponding to ∂ I ). These correspond to thesum of n -point diagrams that are 2PI with respect to the cuts that leave the legs originating from a givenderivative δ/δG on the same side of the cut. Their connected parts involve skeleton graphs with more thanfour legs. Another case that fits the picture is Γ ( p ) for p ≥ once written in terms of two-skeleton diagrams(which it is always possible to do unambiguously), and also δ m Γ ( p ) /δG m . Note that both δ Γ (4) /δG and δ I /δG , enter the flow equations that are considered in Sect. 4, once convoluted with ∂ κ G κ . Of course thesubstitution of the bare vertices by the full four-point function requires that there is no restriction on the69ype of four-insertions that can appear (such as the restriction of irreducibility already mentioned for I ), aproperty that needs to be checked for any infinite class of diagrams that one wants to analyse.The previous discussion concerned four-point insertions in connected two-skeleton diagrams with n ≥ external lines (or disconnected combination of those), the case for which we have shown that the four-pointinsertions do not overlap. In a diagram with more than six external lines, we may encounter overlappingmaximal insertions with more than six external lines. To see that, we shall extend the discussion of thegeneric situation depicted in Fig. B.20 to the case of a diagram with n ≥ external lines built on four-skeletons, with two overlapping insertions M and M . For simplicity we restrict the discussion to thecase where the considered insertions have 6 external lines, and the assumption of a four-skeleton eliminatesreturning lines. The same reasoning as that leading to Eq. (B.1) yields now n M = n M / C + n C + m = n M = n M / C + n C + m = 6 . (B.3)Moreover, the union of M and M is a 1PI ( n M / C + n M / C + n C ) -point vertex function whose number Figure B.21: Example of overlapping six-vertex insertion within a -point function. of legs cannot be less than since we have assumed that D is a four-skeleton with more than six legs. Thisnumber of legs must be even, so it cannot be 5. It cannot be either since this would mean that M ∪ M is a six-point vertex that contains the maximal six-point vertices M and M . It follows that n C + n M / C + n M / C > . (B.4)Combining this with (B.3) together with the fact that m i ≥ (since M i is 1PI), one arrives at ≤ m + m + n C < (B.5)There is indeed room for overlap, with for instance the solution m = m = 2 and n C = 0 . An illustration is70iven in Fig. B.21 showing a contribution to the eight-point function with two overlapping maximal six-pointvertex insertions. Appendix C. A simple relation between Γ (4) L,κ and I L,κ
Following an analysis similar to that in Ref. [49], we present here a simple relation between the fourpoint function Γ (4) and its irreducible parts in the three independent channels. We consider here the generalsituation where the propagator is a function of two positions or two momenta, and we write it simply as G ij with G ij = G ( x i , x j ) or G ij = G ( p i , p j ) . The kernel I is then a function of four variables I ≡ δ Φ[ G ] δG δG , (C.1)where the semi-colon indicates the channel in which I is irreducible. The diagrams contributing to I cannot be split into two disconnected pieces containing respectively the pairs (1 , and (3 , by cuttingtwo of its internal lines: they are irreducible in the channel (12; 34) . I can be used to construct thefour-point function Γ (4)1234 from the Bethe-Salpeter equation Γ (4)1234 = I − I ij G ik G jl Γ (4) kl = I −
12 Γ (4)12 ij G ik G jl I kl ; a . (C.2)Note that Γ (4)1234 is crossing symmetric, in contrast to the kernel I , which only obeys the followingproperties I = I as follows from its definition, and I = I = I as follows from thesymmetry G ij = G ji . But, for instance, it is not invariant under the exchange of and .We shall write the BS equation in the following way Γ (4)1234 = I + ∆Γ (4)12;34 , (C.3)separating the contributions of the diagrams that are irreducible in the channel (12; 34) from those whichare not. One can write similar relations for the other two channels: Γ (4)1234 = I + ∆Γ (4)13;24 , (C.4) Γ (4)1234 = I + ∆Γ (4)14;23 . (C.5)In fact, when only quartic interactions are present and the field expectation value is assumed to vanish,any diagram contributing to Γ (4)1234 can be reducible in only one channel. The decompositions above are Indeed, consider a diagram that admits a cut that leaves (1 , and (3 , on each side. Then the diagram writes necessarily I , which are irreducible in the channel (12; 34) , there arediagrams that are also irreducible in the other two channels. We call ¯ I the sum of the diagrams that areirreducible in all channels (including the elementary vertex). Now, the diagrams that contribute to I are either fully irreducible, or reducible in either the channel (13; 24) or (14; 23) . The latter contributionsare respectively ∆Γ (4)13;24 and ∆Γ (4)14;23 , so that I = ¯ I + ∆Γ (4)13;24 + ∆Γ (4)14;23 . (C.6)It follows that Γ (4)1234 = I + ∆Γ (4)12;34 (C.7) = ¯ I + ∆Γ (4)13;24 + ∆Γ (4)14;23 + ∆Γ (4)12;34 . (C.8)From this relation and (C.3)-(C.5), a simple calculation yields the following identity Γ (4)1234 = 12 (cid:104) I + I + I − ¯ I (cid:105) . (C.9)This identity, valid at any loop order, allows one to construct Γ (4) L from I L . It also shows that ¯ I iscrossing-symmetric as we anticipated by our choice of notation. Appendix D. ∂ κ I κ and power counting Let us briefly recall how Weinberg’s theorem can be used to determine the large momentum behavior ofa given n -point function Γ ( n ) ( p , . . . , p n ) as some (not necessarily all) of its external momenta grow large.This involves the identification of subgraphs attached to the large external momenta, and such that allinternal lines of these subgraphs carry large momenta. Because of this, one can expand the propagatorsin powers of the small scales present in the subgraph (such as the mass or the regulator), or the (small)external momenta. The leading asymptotic behavior is then determined by power counting applied to thesubgraph in question. This leading behavior does not depend on the small scales, which however remainpresent in the rest of the diagram and may contribute as a multiplicative factor.More concretely, consider for instance a diagram contributing to Γ (4) κ ( p, q ) and suppose that p is “large”,i.e., p (cid:29) q, κ . It is easily seen that the subgraphs attached to p that have a maximal superficial degree of X ij G ik G jl Y kl . Now, if we assume that there is a second possible cut that leaves (1 , and (2 , on each side, then,because and cannot be on the same side of the cut, and similarly for and , the only possibility is that the diagram writes X ii (cid:48) G i (cid:48) j (cid:48) X (cid:48) jj (cid:48) G ik G jl Y kk (cid:48) G k (cid:48) l (cid:48) Y (cid:48) ll (cid:48) which involves three-point functions X , X (cid:48) , Y and Y (cid:48) . However, in a quartic theorywith vanishing field expectation value, three-point functions vanish and there are no such diagrams. p (by power counting). Similarresults hold in the regimes q (cid:29) p, κ and p, q (cid:29) κ . Returning to the case p (cid:29) q, κ , we note that a typicaldiagram contributing to Γ (4) κ ( p, q ) that possesses four-point subgraphs attached to p will yield contributionsof the form Γ (4) κ ( p, q ) ∼ ln p × r κ ( q ) , as discussed above . It follows in particular that for p (cid:29) q, κ , ∂ κ Γ (4) κ ( p, q ) ∼ ln p × ∂ κ r κ ( q ) . In other words, ∂ κ Γ (4) κ ( p, q ) also counts as in the power counting for p (cid:29) q, κ . This result extends to the regimes q (cid:29) p, κ and p, q (cid:29) κ .The situation is different for ∂ κ I κ ( p, q ) . Indeed, although I κ ( p, q ) may also contain subgraphs con-tributing to the four-point function, its s-channel two-particle irreducibility implies that the only four-pointsubgraph that one can attach to the external legs carrying the momentum p is the diagram itself. It followsthat here r κ ( q ) = 1 , and the logarithmic asymptotic behavior is independent of κ . As a result, ∂ κ I κ ( p, q ) is suppressed by at least one unit (in fact two) as compared to ∂ κ Γ (4) κ ( p, q ) and finally counts as − in thepower counting. Similar remarks apply to the regimes q (cid:29) p, κ and p, q (cid:29) κ .The same result can be obtained directly from the flow equation (47). The two-particle irreducibility of δ I /δG means that the leading asymptotic behavior of ∂ κ I κ ( p, q ) comes from the regime where all momentain the integral are large and is thus given by the superficial degree of divergence of that integral. Using that ∂ κ G κ counts as − and δ I /δG as − (since I counts as but δ/δG kills one integral and one propagator),we find the superficial degree of divergence δ ∂ I = 4 − − − , in agreement with the result above. Appendix E. Renormalized loop skeleton expansion
In this section, we show that the renormalized solution Γ (4) L,κ of the flow equations (65) with initialconditions such that the conditions (82) and (83) are fulfilled has the polynomial form (84). To this purposewe write Γ (4) L,κ as in Eq. (74) and show that ∆Γ (4) L,κ is proportional to λ L +1 .We start by considering ∆Γ (4) L =0 ,κ ( p i ) = Γ (4) L =0 ,κ ( p i ) , without any prejudice on its original diagrammaticstructure. Since its flow vanishes, see Eq. (61), the value of Γ (4) L =0 ,κ ( p i ) does not depend on κ . Furthermore,since Γ (4) L =0 , Λ ( p i ) does not depend on p i , Γ (4) L =0 ,κ ( p i ) cannot depend on p i either. The constant value of Γ (4) L =0 ,κ =0 ( p i = 0) is fixed from the conditions (82) and (83): Γ (4) L =0 ,κ ( p i ) = λ . (E.1)Let us now consider a generic ∆Γ (4) L,κ ( p i ) . We proceed recursively assuming that ∆Γ (4) L (cid:48) ,κ ( p i ) has been shownto be proportional to λ L (cid:48) +1 for L (cid:48) < L , and show that the property extends to L (cid:48) = L . Writing Γ (4) L,κ ( p i ) = λ L, Λ + (cid:90) κ Λ dκ (cid:48) ∂ κ (cid:48) Γ (4) L,κ (cid:48) ( p i ) , (E.2) By ‘ ln p ’ we mean a function that grows logarithmically. This could include powers of logarithms. λ L, Λ = λ − (cid:90) dκ (cid:48) ∂ κ (cid:48) Γ (4) L,κ (cid:48) ( p i = 0) , (E.3)so that Γ (4) L,κ ( p i ) = λ − (cid:90) dκ (cid:48) ∂ κ (cid:48) Γ (4) L,κ (cid:48) ( p i = 0) + (cid:90) κ Λ dκ (cid:48) ∂ κ (cid:48) Γ (4) L,κ (cid:48) ( p i ) . (E.4)By subtracting a similar equation with L replaced by L − , we obtain ∆Γ (4) L,κ ( p i ) = − (cid:90) dκ (cid:48) ∂ κ (cid:48) ∆Γ (4) L,κ (cid:48) ( p i = 0) + (cid:90) κ Λ dκ (cid:48) ∂ κ (cid:48) ∆Γ (4) L,κ (cid:48) ( p i ) . (E.5)We return now to Eq. (65), and we perform the substitution indicated in Eq. (74). We note then thateach term on the right-hand side of Eq. (65) now involves ∆Γ (4 L (cid:48) ,κ with L (cid:48) < L . We call ¯ L the number of loopsin a given term of Eq. (65); for instance the first term has ¯ L = 1 loop, the second term ¯ L = 2 loops and so on.From the definition of the operator [ _ ] { L } given after Eq. (74), we are instructed to keep in the expansionof terms of order ¯ L , all the terms that are such that L = ¯ L + (cid:80) L (cid:48) . Using our recurrence assumption, eachterm scales with λ as λ (cid:80) L (cid:48) + V , with V the number of explicit vertices of the considered term. It is easilyseen that V = ¯ L + 1 , from which it follows finally that ∂ κ ∆Γ (4) L,κ is proportional to λ (cid:80) L (cid:48) +¯ L +1 = λ L +1 . Inview of Eq. (E.5), this conclusion extends to ∆Γ (4) L,κ itself.
Appendix F. Remarks on the 2PI n -point functions In this Appendix we analyze the flow equations for the 2PI n -point functions. The first two equations,that for the self-energy Σ κ ( p ) and that for the irreducible kernel I κ ( p, q ) , have already been given in themain text, Eqs. (38) and (47) respectively. It is convenient to rewrite the equation for the self-energy interms of the two-point function, i.e., ∂ κ Γ (2) κ ( p ) = 12 (cid:90) q ∂ κ G κ ( q ) I κ ( q, p ) , (F.1)with G − κ ( q ) = Γ (2) L − ,κ ( q ) + R κ ( q ) and where I κ needs to be seen at this stage as a sum of skeleton diagramsin the bare theory. To remove this reference to the bare theory, we may obtain I κ from the integration ofa flow equation. This is easily obtained by noticing that the κ dependence of I κ ( q, p ) originates solely fromthe propagator G κ . We get (see Eq. (47)) ∂ κ I κ ( q, p ) = (cid:90) r ∂ κ G κ ( r ) δ I ( q, p ) δG ( r ) (cid:12)(cid:12)(cid:12)(cid:12) G = G κ . (F.2)74gain, since δ I /δG is to be seen as a sum of skeleton diagrams in the bare theory, we repeat the previousstep and obtain δ I /δG from the integration of a flow equation ∂ κ δ I ( q, p ) δG ( r ) (cid:12)(cid:12)(cid:12)(cid:12) G = G κ = (cid:90) r ∂ κ G κ ( r ) δ I ( q, p ) δG ( r ) δG ( r ) (cid:12)(cid:12)(cid:12)(cid:12) G = G κ . (F.3)We can continue this procedure until we reach δ m +1 I ( q, p ) /δG ( r m +1 ) · · · δG ( r ) = 0 with m equal to themaximal number of propagators in the diagrams of I , in the considered Φ -derivable approximation. In thiscase, the flow of δ m I ( q, p ) /δG ( r m ) · · · δG ( r ) vanishes and the tower of flow equations terminates One thenarrives at a reformulation of Φ -derivable approximations as a system of flow equations for the quantities Γ (2) κ ( p ) , I κ ( q, p ) and δ k I ( q, p ) /δG ( r k ) · · · δG ( r ) | G = G κ , with ≤ k ≤ m . These equations are (F.1), (F.2)and ∂ κ δ k I ( q, p ) δG ( r k ) · · · δG ( r ) (cid:12)(cid:12)(cid:12)(cid:12) G = G κ = (cid:90) r k +1 ∂ κ G κ ( r k +1 ) δ k +1 I ( q, p ) δG ( r k +1 ) · · · δG ( r ) (cid:12)(cid:12)(cid:12)(cid:12) G = G κ , ≤ k ≤ m . (F.4)As we have seen, these equations rely on the fact that the Luttinger-Ward functional Φ has no explicitdependence on κ , all the dependence on κ being carried by the propagator. p p p pq q p pr q q r rq qr s sp p Figure F.22: The various 2PI n -point functions that appear in the three-loop approximation to Φ . From left to right: Φ , Σ , I , J and K . As an illustration, consider the three-loop approximation for Φ (see Fig. F.22). The irreducible kernel I κ obeys ∂ κ I κ ( q, p ) = (cid:90) r ∂ κ G κ ( r ) J κ ( p, q, r ) , J κ ( p, q, r ) ≡ δ I κ ( q, p ) δG κ ( r ) . (F.5)where J κ ( p, q, r ) is a six-point function with a tree structure: a propagator connecting two vertices. Afurther functional derivative yields an eight-point function K κ ( q, p, r, s ) ∂ κ J κ ( q, p, r ) = (cid:90) s ∂ κ G κ ( s ) K κ ( q, p, r, s ) , K κ ( q, p, r, s ) ≡ δ J κ ( q, p, r ) δG κ ( s ) , (F.6)composed of two disconnected vertices. Clearly, in the three-loop Φ -derivable approximation, K κ ( q, p, r, s ) For instance, in the L -loop approximation, the diagrams that contribute to Φ contain up to L − propagators and thesequence of 2PI n -point functions that can be constructed terminates with 2PI (4 L − -point functions.
75s the last 2PI n -point function that can be constructed and we have ∂ κ K κ = 0 .While at a given loop order, the sequence of 2PI n -point functions is finite and provides therefore apossible practical scheme to determine ∂ κ I κ , there are two features that make this procedure somewhatunsatisfactory, as we now explain. The first one is easily dealt with, while the second one points to aconceptual issue that motivated the strategy adopted in Sect. 4.Let us consider the first issue. From the 2PI nature of the derivatives δ k I ( q, p ) /δG ( r k ) · · · δG ( r ) , itfollows that there are no four-point functions attached directly to the two legs associated to a given derivative δ/δG ( r i ) . From Weinberg theorem, it follows that δ k I ( q, p ) /δG ( r k ) · · · δG ( r ) counts as a strictly negativecontribution in the power counting of the integral in Eq. (F.4). Since ∂ κ G κ ( r k +1 ) counts as − , the integralis finite. In contrast, Eq. (F.2) is not finite by power couting since the superficial degree of divergence is δ = 4 − . There is however a simple solution to this problem. All one needs to do is replaceEqs. (F.1) and (F.2) by ∂ κ Γ (2) κ ( p ) = 12 (cid:90) q ∂ κ R κ ( q ) Γ (4) κ ( q, p ) , (F.7)and ∂ κ Γ (4) κ ( p, q ) = ∂ κ I κ ( p, q ) − (cid:90) r Γ (4) κ ( p, r ) ∂ κ G κ ( r ) Γ (4) κ ( r, q ) − (cid:90) r ∂ κ I κ ( p, r ) G κ ( r ) Γ (4) κ ( r, q ) − (cid:90) r Γ (4) κ ( p, r ) G κ ( r ) ∂ κ I κ ( r, q )+ 14 (cid:90) r (cid:90) s Γ (4) κ ( p, r ) G κ ( r ) ∂ κ I κ ( r, s ) G κ ( s ) Γ (4) κ ( s, q ) . (F.8)These equations have been shown to be finite by power counting in the main text. We then arrive ata system of finite flow equations, Eqs. (F.7), (F.8) and (F.4) for the functions Γ (2) κ ( p ) , Γ (4) κ ( q, p ) and δ k I ( q, p ) /δG ( r k ) · · · δG ( r ) , with ≤ k ≤ m . We note that I κ does not appear in this set since it en-ters the equations only through ∂ κ I κ , and its flow equation never needs to be integrated.The second unsatisfactory feature concerns the initialization of the system of equations. Because Γ (2) κ and Γ (4) κ are 1PI functions, their initial conditions are simple and given in Eq. (72). On the other hand, thefunctions δ k I ( q, p ) /δG ( r k ) · · · δG ( r ) ’s are not 1PI functions. They can contain in particular disconnectedpieces involving delta functions in momentum space. The developments in Appendix B allow us to clarifythe structure of the initial conditions to a large extent. In fact, the functions δ k I ( q, p ) /δG ( r k ) · · · δG ( r ) enjoy the same properties as δ I /δG ( r ) : they are expressible as skeleton diagrams in which tree-level verticesare replaced by the exact Γ (4) . In a given Φ -derivable approximation, one truncates at a given loop order, in76hich case the exact Γ (4) needs to be replaced by Γ (4) L , as appropriate. Among the skeleton diagrams, thereare diagrams where all propagators have been cut and there remain only the tree-level vertices multipliedby appropriate delta functions. After replacing the tree-level vertices by four-point functions, one obtainsproducts of Γ (4) L ’s (expanded to the relevant loop order) multiplied by delta functions, which survive when κ is taken large an lead to products of the initial conditions λ L, Λ (again, expanded to the relevant looporder). The other diagrams, involve, in addition, some Γ (4) L ’s connected by propagators. These diagrams aresuppressed at large κ because, once expressed in terms of the Γ (4) L ’s, all their possible loops have a negativesuperficial degree of divergence.To make things more concrete, let us take a few examples. At three-loop order, the tower of flow equationsinvolves δ I /δG and δ I /δG whose diagrammatic contributions are shown as the last two diagrams of Fig. 16and are obviously expressed in terms of Γ (4) L =0 . In δ I /δG , the two Γ (4) L =0 are connected by a propagatorwhich suppresses the contribution at large κ . One may then initialize δ I /δG to . On the other hand, nopropagator appears in δ I /δG together with the two vertices, so this quantity has a non-trivial initialization: δ I ( q, p ) δG ( r ) δG ( r ) (cid:12)(cid:12)(cid:12)(cid:12) G = G Λ = − λ , Λ δ ( p + q + r + r ) (F.9)(recall that λ , Λ is simply λ with our choice of renormalization conditions). At four-loop order, the towerof flow equations involves δ I /δG , δ I /δG , δ I /δG , δ I /δG . A simple analysis reveals that δ I /δG and δ I /δG do not contain contributions involving only disconnected four-point functions. These objects aretherefore suppressed at large κ and need to be initialized to . On the other hand δ I /δG and δ I /δG contain such contributions and thus require a non-trivial initialization. We have for instance δ I ( q, p ) δG ( r ) δG ( r ) = − (cid:104)(cid:2) Γ (4) L =1 ( p, q, r ) (cid:3) (cid:105) δ ( p + q + r + r ) (F.10)so the initial condition is δ I ( q, p ) δG ( r ) δG ( r ) (cid:12)(cid:12)(cid:12)(cid:12) G = G Λ = − (cid:104) λ , Λ (cid:105) δ ( p + q + r + r ) . (F.11)One can similarly deduce the initial condition for δ I ( q, p ) /δG ( r ) δG ( r ) δG ( q ) δG ( p ) | G = G κ . We note thatthe four-point functions Γ (4) L that were found to play a major role in our main discussion, also appear herein the determination of the initial conditions for some of the flow equations of the 2PI n -point functions.Therefore, these functions cannot be ignored, and need to be properly treated.Let us finally mention that while the hierarchy of equations (F.1), (F.2) and (F.4) is identical to thatdiscussed in Ref. [44], our analysis deviates from that presented in that reference. According to Ref. [44], If I is made of diagrams up to L − loops, δ k I ( q, p ) /δG ( r k ) · · · δG ( r ) should be seen as made of diagrams up to L − − k loops, provided one counts the possible delta functions that may appear, as negative loops. n -point function that fulfills certain “consistency conditions”,as defined in [44], that ensure that the corresponding flow equation can be integrated exactly in terms ofdiagrams. These consistency conditions, which are not the same as those discussed in the main text, seeEq. (83), express the regularity of the 2PI n -point functions X ( n ) ( p i ) in the zero-momentum limit, that is ∆ X ( n ) ( p i ) ≡ X ( n ) ( p i ) − X ( n ) ( p i = 0) should approach as p i → . According to the authors of [44] suchlimits may lead to a × ∞ indetermination due to the possible presence of subdivergences. We find notrace of such problematic limits. For one thing, the present theory has a Landau pole which requires thepresence of an explicit UV cut-off. This ensures that ∆ X ( n ) ( p i ) approaches as p i → . It could happenthat this limit is approached only for extremely tiny values of p i if subintegrals remain strongly sensitive to Λ uv . However, our finite flow equations all generate the appropriate subtractions at the scale Λ that ensurethat no strong dependence on Λ uv is present and ∆ X ( n ) ( p i ) approaches smoothly in a reasonable rangeof p i near 0.In our approach, we have introduced a hierarchy of flow equations beyond the equation for ∂ κ I κ , butthese are written in terms of the Γ (4) L,κ ’s rather than the 2PI n -point functions. As explained in the maintext, the main purpose of this hierarchy is to clarify the renormalization, see Sects. 5 and 6. As discussedin Sect. 7, for practical applications, we can directly evaluate ∂ κ I κ in terms of the diagrammatically renor-malized Γ (4) L,κ ’s.
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