Exact solutions and residual regulator dependence in functional renormalisation group flows
EExact solutions and residual regulator dependencein functional renormalisation group flows
Benjamin Knorr ∗ Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, ON N2L 2Y5, Canada
We construct exact solutions to the functional renormalisation group equation of the O(N) modeland the Gross-Neveu model at large N for < d < , without specifying the form of the regulator.This allows to investigate which quantities are independent of the choice of regulator without beingplagued by truncation artefacts. We find that only universal quantities, like critical exponents,and qualitative features, like the existence of a finite vacuum expectation value, are regulator-independent, whereas values of coupling constants are generically arbitrary. We also provide ageneral algorithm to construct a concrete operator basis for truncations in the derivative expansionand the Blaizot-Méndez-Wschebor scheme. I. INTRODUCTION
The large N limit is a useful and well-known expansionscheme in quantum field theory [1]. By introducing aninfinite number of a specific kind of field, one can controlthe leading order quantum effects, allowing for analyticexpressions for, e.g. , critical exponents characterising asecond order phase transition as a series in /N . In somecases, the (typically asymptotic) series can then be usedto make predictions for a finite, physical number of fields.As a downside, it is generally hard to prove statementsabout the accuracy of such an expansion, and indepen-dent, non-perturbative computations are desirable to ar-rive at trustworthy estimates of physical quantities.One such tool to perform non-perturbative calculationsis the functional renormalisation group ( FRG ) [2–8]. Inthis approach, one considers the so-called effective aver-age action Γ k , which depends on a fiducial momentumscale k . It interpolates between the classical action at amicroscopic scale Λ , and the standard effective action at k = 0 . The k -dependence is dictated by the Wetterichequation [2, 3], ˙Γ k = 12 Tr (cid:20)(cid:16) Γ (2) k + R k (cid:17) − ˙ R k (cid:21) . (1)In this, an overdot denotes the scale derivative k∂ k , thetrace includes a functional and an index trace as well asa minus sign for fermions, Γ (2) k is the second variationof the effective average action, and R k is a regulator,which essentially acts as a k -dependent mass term, mak-ing the right-hand side well-defined. In particular, in thelimit k → , the regulator vanishes. This equation de-fines a vector field on theory space, and the points wherethis vector field vanishes are called fixed points. Fixedpoints correspond physically to second order phase tran-sitions. The flow around such a fixed point is dictated bythe universal critical exponents, which are related to theeigenvalues of the stability matrix at the fixed point. ∗ Electronic address: [email protected]
For a general theory, the flow equation (1) typically hasto be solved in an approximation which is not closed, i.e. , in general one has to deal with systematic errorsin the form of truncation errors. As a consequence,estimates of physical quantities obtained in truncatedrenormalisation group flows will rather generically showa residual regulator dependence [5, 10–13]. It is thusof interest to understand what quantities are truly in-dependent of the choice of regulator in exact solutionsto give a good hint for what kinds of regulator depen-dencies one should study in truncated renormalisationgroup flows. We will exemplify this by considering thelarge N limit of some well-known theories: the O ( N ) model and the (partially bosonised) Gross-Neveu model.These models are relevant to condensed matter and highenergy physics, since they describe well-known second or-der phase transitions [1, 14–17]. As we will see, the large N limit allows for a non-perturbative discussion of the ex-istence and both universal and non-universal propertiesof the regular (Wilson-Fisher type) fixed point in thesemodels. As a by-product, we will discuss some structuralaspects of the flow equations in these models that carryover to finite N .This work is structured as follows. In section II wediscuss the O ( N ) model, focussing on the large N limit.We will first derive the simplified exact flow equation ob-tained in this limit in subsection II A. In subsection II B,we will construct an operator basis that parameterisesthe effective average action in the form of a derivativeexpansion, which reduces the functional flow equation toan infinite tower of first order ordinary differential equa-tions. With that in place, in subsection II C we showthat the derivative expansion is a perfect truncation inthe sense that the flow of any n -th order operator onlydepends on the solution of operators of lower or the sameorder. Since the effective potential plays a central role in See however [9] for recent developments in deriving general exactsolutions to the flow equation. Recently, non-regular solutions in this limit have been discussedin [18–21]. These solutions are based on a more subtle large N limit where naively sub-leading terms nevertheless contributedue to their divergence structure. a r X i v : . [ h e p - t h ] D ec the discussion of fixed points, we will investigate it indetail in subsection II D. Subsection II E is devoted to apartial resummation of the derivative expansion, whichhowever still represents a perfect truncation. The re-summation allows us to directly study momentum- andfield-dependent correlation functions. With the completefixed point solution, we are in the situation to discuss theregulator dependence in subsection II F. Finally, we dis-cuss some aspects of the model at finite N in subsectionII G. As a second example, we discuss the Gross-Neveumodel at large N in section III. We first present the ex-act flow equation in the large N limit in III A, then wediscuss the lowest order critical correlation functions andcritical exponents in III B and III C, respectively, finish-ing the section with a short discussion of the regulatordependence in III D. We close with a summary of themain results in section IV. II. O(N) MODELA. Exact flow equation
We will start our investigation by considering an O ( N ) -symmetric scalar field φ a in d Euclidean dimensions with < d < . Due to its simplicity and interesting appli-cations, it is potentially the best-studied model in thecontext of the FRG [22–38]. Let us derive the exact flowequation in the large N limit for the effective action Γ N →∞ [ φ ] = N (cid:90) d d x ( ∂ µ φ a ) ( ∂ µ φ a ) + N ¯Γ[ ρ ] , (2)where all non-trivial behaviour is carried by the secondterm ¯Γ . Here we introduced ρ = φ a φ a / . The regulatoris chosen to respect the O ( N ) symmetry, ∆ S k = N (cid:90) d d x φ a R k ( − ∂ ) φ a . (3)First we show that in the regular large N limit the effec-tive action is indeed of this form (2). The crucial observa-tion for the simplification in the large N limit is that forthe right-hand side of the flow equation to yield a factorof N , the trace over the O ( N ) bundle indices must be atrace over the corresponding field space identity. For thatreason, we only need the part of the two-point functionwhich is proportional to the identity: δ Γ N →∞ δφ a δφ b = N (cid:20) − ∂ + δ Γ δρ (cid:21) δ ab + . . . . (4) To improve the readability, we will suppress the index k on allquantities from hereon (except the regulator), and in a slightabuse of language refer to Γ k as the effective action. Since wewill not discuss the limit k → in this work, this should not leadto any confusion. Here, the dots indicate terms that, when traced over, donot give rise to a factor of N , i.e. , their bundle indices arecarried by φ and derivatives of it. Note that the identitycomponent of the inverse of this operator only involvesthe inverse of the identity component of the operator it-self. With this observation, we can immediately writedown the exact flow equation at large N: ˙¯Γ = 12 (cid:90) d d x (cid:90) d µ q ˙ R k ( q )( q − i ∂ ) + R k (( q − i ∂ ) ) + δ ¯Γ δρ . (5)In this we introduced the momentum integral measure (cid:90) d µ q = (cid:90) d d q (2 π ) d . (6)Two points deserve being mentioned with regards to (5):first, the full trace is already performed, and only a stan-dard momentum integral is left. Second, in contrast tostandard flows, only the first functional derivative of theaction appears in the propagator on the right-hand side.This is indeed at the heart of the simplifications in thelarge N limit, and allows to write down perfect trunca-tions where increasing the truncation order doesn’t alterthe flow equations of previously calculated orders.We still have to argue that the standard kinetic termused above is the only possibility for a term which doesn’tdepend on ρ only. In particular, one could imagine a field-or momentum-dependent wave function renormalisationfactor, also giving rise to a potentially non-vanishinganomalous dimension. However, since the flow equationdoesn’t give any contribution to the flow of the kineticterm, requiring locality and regularity at vanishing fieldimmediately fixes a trivial wave function renormalisationand a vanishing anomalous dimension. B. Spanning an operator basis
Even though the flow equation (5) is much simpler thanit is in the generic case, a direction solution is still in-volved. To make things tractable, we will now discussspanning the non-trivial part of the effective action, ¯Γ ,in a derivative expansion. For this, we need to specify aunique basis at every fixed number of derivatives wherewe took partial integrations into account. Since deriva-tives always have to come in pairs to form a Lorentzscalar, we will refer to the n -th order as all terms hav-ing n derivatives. The convergence properties of thisexpansion at finite N have been thoroughly discussed in[36, 37].At a given order n , a general operator in the derivativeexpansion of ¯Γ has the form X ( ρ ) ( ∂ µ · · · ρ ) ( ∂ ν · · · ρ ) · · · , (7)that is, it is a product of a ρ -dependent function anda series of derivatives acting on different ρ s so that for FIG. 1. The graph indicating the partial integration rulesand the basis for the first order of the derivative expansionfor the O ( N ) model at large N . The pictograph in the firstline correspond to an operator X ( ρ ) ∂ ρ , which is related tothe operator represented by the pictograph in the second line, Y ( ρ )( ∂ µ ρ )( ∂ µ ρ ) , by a partial integration. The blue dashedline indicates that X and Y are related by Y ( ρ ) = − X (cid:48) ( ρ ) . constant ρ , all interaction terms except for the potentialvanish. We will now describe a graphical procedure toobtain a unique basis at order n , and all partial integra-tion rules to write an arbitrary operator in this basis. Forthis, we draw a box for every ρ on which derivatives act,and a link connecting two boxes (or a box with itself)which corresponds to a pair of derivatives which is con-tracted, the end points marking the fields on which theyact. With this, the algorithm at order n is as follows:1. In the first row, draw all possible graphs with anynumber of boxes and exactly n links, where all linksstart and end on the same box, and no box canhave zero links attached to it. This corresponds toall operators of the form (7) where we only have ∂ as derivative operators, each acting on a single ρ .2. To arrive at the next row of the graph, employ thefollowing partial integration rules. For any graphof the first row, from the box with the most links,disconnect one of the links which starts and endson the box, and draw all distinct graphs where theloose link is connected to either one of the existingboxes, or a new box, with the following weights:(a) if the link is connected to an already existingbox, multiply with a factor of − ,(b) if the link is connected to a new box, multiplywith a factor of − x .If connecting the loose link to different boxes givesthe same new diagram, multiply the weight by thismultiplicity.3. Draw lines from every “parent” box of the givenlevel of the graph to its “children” as obtainedby the previous integration rules, indicating theweights.4. Perform the two previous steps for all other links,but don’t attach weights to the lines from the par-ents to these “illegitimate children”. This step en-sures that all distinct graphs on a given row aregenerated, since not all graphs of a given row need
20 links1 link2 links
FIG. 2. The graph indicating the partial integration rules andthe basis for the second order of the derivative expansion forthe O ( N ) model at large N . to have parents in the above sense. The first timethat new operators are generated in this way forthe O ( N ) model is at fourth order.5. Repeat the above steps until all links connect dif-ferent boxes.The bottom line of the graph represents the independentbasis at order n that we will use, and corresponds to thebasis where all ∂ operators are partially integrated. Towrite any element in this basis, follow the graph startingfrom this element down to the bottom row, multiplyingthe weights from row to row. Once the bottom row isreached, any factor of x corresponds to a ρ -derivative ofthe operators’ prefactor, that is the function X in (7). Toillustrate this procedure, we show the graphs for orders n = 1 , , in Figure 1, Figure 2 and Figure 3, the graphfor n = 4 can be found in the supplemental materialas it is too bulky to present it here. In these graphs,we only indicate multiplicities explicitly. Factors of − are indicated by olive, full lines while factors of − x areindicated by blue, dashed lines. Illegitimate relations areindicated by red, dotted lines.We can now span the effective action in terms of ourbasis in a derivative expansion: ¯Γ = (cid:90) d d x (cid:34) V ( ρ ) + Y ( ρ ) ( ∂ µ ρ ) ( ∂ µ ρ )+ W ( ρ ) ( ∂ µ ∂ ν ρ ) ( ∂ µ ∂ ν ρ ) + W ( ρ ) ( ∂ µ ρ ) ( ∂ µ ∂ ν ρ ) ( ∂ ν ρ )+ W ( ρ ) ( ∂ µ ρ ) ( ∂ µ ρ ) ( ∂ ν ρ ) ( ∂ ν ρ ) + O ( ∂ ) (cid:35) . (8)The potential V has no graphical representation. Thewave function renormalisation for the field ρ is repre-sented in the bottom line of Figure 1, and the functions W , , correspond to the bottom line of Figure 2, fromleft to right. The number of terms increases rapidly withthe order – for n = 3 , there are eight coupling functions,whereas for n = 4 , there are already 23 coupling func-tions.
22 22 2 2 2 40 links1 link2 links3 links
FIG. 3. The graph indicating the partial integration rules and the basis for the third order of the derivative expansion for the O ( N ) model at large N . C. Derivative expansion as a perfect truncation
In this section we will show that the derivative expan-sion in the large N limit is a perfect truncation, thatis, the flow of the n -th order coupling functions doesn’tdepend on the coupling functions of any higher order.In fact, we can show that for every coupling function X except the potential, the flow equation reads ˙ X ( ρ ) = − X (cid:48) ( ρ ) (cid:90) d µ q ˙ R k ( q )( q + R k ( q ) + V (cid:48) ( ρ )) + n X V (cid:48)(cid:48) ( ρ ) X ( ρ ) (cid:90) d µ q ˙ R k ( q )( q + R k ( q ) + V (cid:48) ( ρ )) + f X ( ρ ) , (9) where n X is the number of ρ s in the operator on whichderivatives act. Graphically, this is the number of boxesin the graph associated to the coupling function X . Fur-thermore, f X ( ρ ) is a function which doesn’t depend on X ,but depends on some operators of lower order, and candepend linearly on coupling functions of the same order,but not their derivatives. Let us point out that the flowequations are thus first order linear ordinary differentialequations.To show this form of the flow equation, we will use thebasis introduced in the previous section. For a generaloperator that appears in the action as X ( ρ ) ( D ρ ) · · · ( D n X ρ ) , (10)where the D i are collections of contracted derivatives inthe form of the basis of the previous section, the variationw.r.t. ρ reads δδρ (cid:90) d d x X ( ρ ) ( D ρ ) · · · ( D n X ρ )= X (cid:48) ( ρ ) ( D ρ ) · · · ( D n X ρ ) + n X (cid:88) l =1 (cid:104) ( − |D l | D l ( X ( ρ ) ( D ρ ) · · · ( D l − ρ ) ( D l +1 ρ ) · · · ( D n X ρ )) (cid:105) . (11)In this, |D l | is the number of derivatives contained inthe operator D l . Since this expression is still of the samederivative order, it cannot contribute to the flow of opera-tors of lower orders, and we can expand the flow equationto linear order in this expression to obtain its contribu- tion to the same order. The first term is still in thecorrect basis, so that it only contributes to the flow ofitself at that order, and is indeed the first term of (9).The additional minus sign comes from the expansion ofthe propagator. The second term gives a contribution − (cid:90) d d x (cid:90) d µ q ˙ R k ( q )( q + R k ( q ) + V (cid:48) ( ρ )) n X (cid:88) l =1 (cid:104) ( − |D l | D l ( X ( ρ ) ( D ρ ) · · · ( D l − ρ ) ( D l +1 ρ ) · · · ( D n X ρ )) (cid:105) (12)We have to perform a partial integration on D l to arriveagain in our chosen basis. Since the D l don’t include ∂ by construction, performing the partial integration di- rectly gives an expression in the correct basis. In partic-ular, we can easily isolate the term which contributes tothe flow of X itself: it is that term where all the deriva-tives of D l act on a single ρ . From this, we get a factor of − V (cid:48)(cid:48) ( ρ ) and another propagator from taking the deriva-tive of the squared propagator, and a factor of n X fromthe sum, so that we arrive at the second line of (9). Allother terms that X contributes to are different opera-tors of the same order, where it contributes linearly andwithout derivatives, or higher orders. This completes theproof of (9). D. Fixed point potential
We will now start to discuss the fixed point, that is thepoint where the flow of all dimensionless coupling func-tions vanishes. For this, we rescale the field and couplingfunctions by appropriate powers of k , e.g. , ρ = k d − r , V ( ρ ) = k d v ( k − d ρ ) . (13)Here, r is the dimensionless version of the field ρ and v the dimensionless potential. With this, at the fixed point we have to solve d v ( r ) − ( d − r v (cid:48) ( r ) = 12 (cid:90) d µ q ˙ R k ( q ) q + R k ( q ) + v (cid:48) ( r ) , (14)to arrive at the critical potential. In fact, it is easier tosolve the derivative of this equation for u ( r ) ≡ v (cid:48) ( r ) : u ( r ) − ( d − r u (cid:48) ( r )= − u (cid:48) ( r ) (cid:90) d µ q ˙ R k ( q )( q + R k ( q ) + u ( r )) . (15)The Wilson-Fisher type solution of this equation can begiven implicitly, r = 12( d − (cid:90) d µ q ˙ R k ( q )( q + R k ( q )) × F (cid:18) , − d − d (cid:12)(cid:12)(cid:12)(cid:12) − u ( r ) q + R k ( q ) (cid:19) . (16)The complete set of solutions of (15) for a specific choiceof regulator is discussed, e.g. , in [35].In fact, (16) establishes a bijective map between r ∈ ( −∞ , ∞ ) and u ∈ ( u −∞ , ∞ ) , even though a priori physi-cal values of r are non-negative. To see this, we calculate u (cid:48) ( r ) and show that it is positive. Taking the derivativeof (16) with respect to r and solving for u (cid:48) , we get u (cid:48) ( r ) = (cid:90) d µ q ˙ R k ( q )( q + R k ( q )) q + R k ( q ) u ( r ) F (cid:18) , − d − d (cid:12)(cid:12)(cid:12)(cid:12) − u ( r ) q + R k ( q ) (cid:19) −
11 + u ( r ) q + R k ( q ) − . (17)The right-hand side is positive as long as u ( r ) > − min q ( q + R k ( q )) ≡ u −∞ . (18)In particular, for any regulator u −∞ is negative, sincefrom general regulator criteria q + R k ( q ) > . It is easyto see that u = u −∞ corresponds to r = −∞ due to theproperties of the hypergeometric function in (16).Independent of the regulator, this establishes thatthere must be a finite r such that u vanishes, that isthe potential has a minimum. Inserting u ( r ) = 0 intothe implicit solution gives the explicit expression r = 12( d − (cid:90) d µ q ˙ R k ( q )( q + R k ( q )) , (19) Since the wave function renormalisation factor Z doesn’t renor-malise in the large N limit of the O ( N ) model, we don’t have toinclude it in the rescaling. This is different in the Gross-Neveumodel discussed below, where we assume that proper powers ofthis factor have been included in all dimensionless couplings. which is positive for any regulator, so that the minimumis always in the physical regime.By evaluating (16) at r = 0 , we get an implicit equa-tion which fixes u (0) . This indeed also gives an upperbound on u (0) since for the integral to vanish, the argu-ment of the hypergeometric function has to have a min-imum value, as it is positive for all arguments x largerthan some small negative value. The bound for u (0) isshown in Figure 4 for general dimensions d . In d = 3 ,the hypergeometric function vanishes at approximately . . This extremal value is exactly reached for theLitim cutoff [11], for which the hypergeometric functionis independent of the integration variable and thus itsargument must be its zero.The bijective relation between r and u will play a cen-tral role in the following discussion, since as it turns out, u is a much more natural variable than r .At this point it is also worthwhile to note the specialform of the fixed point equations for all the higher orderoperators. Translating (9) into dimensionless form, we - - - - - m a x u ( ) FIG. 4. The maximal allowed value of u (0) for dimensions With the results provided so far, in principle the fullfixed point effective action can be reconstructed, orderby order in the derivative expansion. As it turns out,this procedure can be optimised by resumming certainsubclasses of operators. This comes from the observa-tion that an operator corresponding to a diagram ele-ment with n boxes only contributes to diagram elementswith n or more boxes, that is, there is a further typeof hierarchy which gives again rise to a perfect trunca-tion. To prove this statement, we use momentum spacetechniques, and consider the n -th ρ -derivative of the flowequation, evaluated at constant ρ . Because in the large N limit, the propagator only involves the first derivativeof the effective action, the flow of the n -point functioninvolves correlators up to order n + 1 . In particular, theonly dependence on the higher order correlator is by atadpole diagram. Explicitly, this dependence is ˙Γ ( n ) ( p , . . . , p n | ρ ) ⊃ − (cid:90) d µ q ˙ R k ( q )( q + R k ( q ) + V (cid:48) ( ρ )) × Γ ( n +1) ( p , . . . , p n , − ( p + · · · + p n ) | ρ ) . (26)For better readability, we introduced a vertical bar to sep-arate the momentum arguments from the field argument.The dependence of the n -point functions Γ ( n ) on the mo-menta p i is fully symmetric. Using momentum conserva-tion for the last momentum argument of the ( n +1) -pointfunction, we find ˙Γ ( n ) ( p , . . . , p n ) | ρ ) ⊃ − 12 Γ ( n +1) ( p , . . . , p n , | ρ ) (cid:90) d µ q ˙ R k ( q )( q + R k ( q )+ V (cid:48) ( ρ )) . (27) The key observation is that the last momentum in the ( n + 1) -point function is zero, so that there is no contri-bution from operators with ( n + 1) boxes.Readers familiar with FRG techniques will realise thatthis partial resummation is a concrete realisation of theso-called Blaizot-Méndez-Wschebor ( BMW ) approxima-tion [22, 39–42], which resolves both momentum and fielddependence. In fact, the diagrammatic representationcan be used as a way to give a concrete action as a start-ing point to the BMW scheme, and motivates the originof the truncation rules employed there. Concretely, theresummed ansatz reads in our case ¯Γ = (cid:90) d d x (cid:34) V ( ρ ) + (cid:88) l ≥ Y l ( ρ ) ( ∂ µ · · · ∂ µ l ρ ) ( ∂ µ · · · ∂ µ l ρ )+ (cid:88) n ≥ n − (cid:88) l =1 W − ,n,l ( ρ ) ( ∂ µ · · · ∂ µ l ρ ) (cid:0) ∂ µ · · · ∂ µ l ∂ µ l +1 · · · ∂ µ n ρ (cid:1) ( ∂ µ l +1 · · · ∂ µ n ρ )+ (cid:88) n ≥ (cid:98) n (cid:99) (cid:88) l =1 (cid:98) n − j (cid:99) (cid:88) j = l W ∆ ,n,l,j ( ρ ) (cid:0) ∂ µ · · · ∂ µ l ∂ µ l +1 · · · ∂ µ l + j ρ (cid:1) ( ∂ µ l +1 · · · ∂ µ l + j ∂ µ l + j +1 · · · ∂ µ n ρ ) (cid:0) ∂ µ · · · ∂ µ l ∂ µ l + j +1 · · · ∂ µ n ρ (cid:1) + O (cid:0) ( ∂ρ ) (cid:1) (cid:35) . (28)The first line corresponds graphically to and boxes,the second line comprises the diagrams with three boxesarranged in a line, whereas the third line lists all oper-ators with three boxes arranged in a triangle. To ourknowledge, such an explicit expression for the effectiveaction giving rise to the BMW scheme hasn’t been putforward before.We will now iteratively study the flow equations forthe n -point functions to gain some structural insights.The zero-point function is identical to the potential and has been dealt with in the last subsection. The first non-trivial momentum dependence arises for the two-pointfunction. 1. Two-point function Taking the second ρ -derivative of the flow equation andevaluating it for constant field gives ˙¯Γ (0 , ( p | ρ ) = − δ ¯Γ δρ ( p ) δρ ( − p ) δρ (0) (cid:90) d µ q ˙ R k ( q )( q + R k ( q ) + V (cid:48) ( ρ )) + (cid:104) ¯Γ (0 , ( p | ρ ) (cid:105) (cid:90) d µ q ˙ R k ( q )( q + R k ( q ) + V (cid:48) ( ρ )) p + q ) + R k (( p + q ) ) + V (cid:48) ( ρ ) . (29) We emphasise that the two-point function of the composite ρ corresponds to a combination of parts of the two-, three- andfour-point functions of the elementary field φ . Similarly, higherorder composite correlators correspond to combinations of higherorder elementary correlators. Note that no loop momentum appears in the vertex func-tions. This is a general feature in the large N limit andonce again arises due to the fact that the exact flow equa-tion (5) depends on the first, not the second derivativeof the effective action. For that reason, the two inter-nal φ -legs of any vertex actually count as a single ρ -leg,with a momentum equal to minus all external legs due tomomentum conservation, and hence no loop momentum.As a consequence, we still only have to deal with (nowpartial) differential equations and not integro-differentialequations, as would be expected when resolving momen-tum dependencies.We will call the dimensionless version of the two-pointvertex γ . Furthermore we will introduce the integrals I n ( u ( r )) = (cid:90) d µ q ˙ R k ( q )( q + R k ( q ) + u ( r )) n , (30) I n,j ( p , . . . , p j | u ( r )) = (cid:90) d µ q ˙ R k ( q )( q + R k ( q ) + u ( r )) n × j (cid:89) l =1 1( p + ··· + p l + q ) + R k (( p + ··· + p l + q ) )+ u ( r ) . (31)Taking the above result that the tadpole contribution tothe flow doesn’t include any higher order correlators, wefind that the fixed point equation for γ reads (4 − d ) γ ( p | r ) − ( d − r γ (0 , ( p | r ) − p γ (1 , ( p | r )= − γ (0 , ( p | r ) I ( u ( r )) + γ ( p | r ) I , ( p | u ( r )) . (32)Note that we made it explicit in the argument of I , that because of Lorentz invariance, the integral must be afunction of the squared dimensionless momentum, whichwe denote by the same letter p . We can simplify theequation in two steps. First, we again use the fixed pointequation for the derivative of the potential to replacethe first integral on the right-hand side of the equationvia (15). Second, we use the bijective map from r to u and make a coordinate transformation to replace alloccurrences of r by u. This results in the fixed pointequation (4 − d ) γ ( p | u ) − p γ (1 , ( p | u ) − uγ (0 , ( p | u )= γ ( p | u ) I , ( p | u ) . (33)Taking care of the correct regular boundary conditions,this gives rise to the solution γ ( p | u ) = 1 (cid:82) d ω ω − d I , ( ω p | ω u ) . (34)The integral exists for all d ∈ (2 , . It can be checkedstraightforwardly by comparison with (17) and usingidentities for F that for p = 0 indeed γ (0 , u ) = u (cid:48) ( r ) ,as it should be. Expanding the solution in a Taylor seriesin x around , one can directly read off the (dimension-less) fixed point coefficient functions y l ( r ) of the ansatz(28). Explicitly, γ ( p | u ) = u (cid:48) ( r ) + 2 (cid:88) l ≥ y l ( r ) p l , (35)where both r and u (cid:48) ( r ) have to be replaced according to(16) and (17). We can also derive the large momentum limit at finite u . In this limit, γ ( p | u ) ∝ (cid:0) p (cid:1) − d , as p → ∞ . (36)Comparing to the standard behaviour of a two-pointfunction at large momentum, γ ( p | u ) ∝ (cid:0) p (cid:1) − η , (37)this suggests to define an anomalous dimension with re-spect to the field ρ with value η ρ = d − . (38)This is to be contrasted with the anomalous dimension ofthe fundamental field φ , which vanishes. Note also thatwhile the power law exponent is regulator-independent,the overall prefactor of the limit does depend on thechoice of the regulator. 2. Higher order correlation function For the higher order correlation functions, it is conve-nient to choose the inner product of distinct momenta asa basis for the momentum dependence, y ij = p iµ p µj , ≤ i < j ≤ n . (39)Any squared momentum can be replaced by a combina-tion of these by imposing momentum conservation, n (cid:88) l =1 p l = 0 , (40)where we have the convention that all momenta are in-going. This implies the replacement rules p i = − p iµ n (cid:88) l =1 l (cid:54) = i p µl = − i − (cid:88) l =1 y li − n (cid:88) l = i +1 y il . (41)Whenever squared momenta appear in the following,they are to be understood as a shorthand for their ex-pression in terms of the y ij .Following the same steps as for the two-point function,we get the fixed point equation for the three-point func-tion, (cid:2) (6 − d ) − y ij ∂ y ij − u ∂ u (cid:3) γ ( y , y , y | u )= γ ( y , y , y | u ) (cid:88) i =1 γ ( p i | u ) I , ( p i | u ) − γ ( p | u ) γ ( p | u ) γ ( p | u ) (cid:88) i,j =1 i (cid:54) = j I , ( p i , p j | u ) . (42)By requiring momentum conservation and again usingLorentz symmetry, it is clear that all terms are symmetricfunctions of the three squared momenta x i = ¯ p i , andcorrespondingly of the y ij . Structurally, we thus have (cid:2) (6 − d ) − y ij ∂ y ij − u ∂ u (cid:3) γ ( y , y , y | u )= γ ( y , y , y | u ) L ( y , y , y | u )+ C ( y , y , y | u ) . (43)The solution with the correct boundary condition at van-ishing momenta is γ ( y , y , y | u )= − (cid:90) d ω ω d − C ( ωy , ωy , ωy | ωu ) × exp (cid:20) − (cid:90) ω d ττ L ( τ y , τ y , τ y | τ u ) (cid:21) . (44)The solution can be mapped to the coupling functions W appearing in (28), but the expression is rather lengthyand provides no additional insight, so we refrain fromspelling it out.Indeed, this structure carries over in the same way forthe higher order correlation functions. The generalisationof the above for the n -point function, with n ≥ , reads (cid:2) (2 n − ( n − d ) − y ij ∂ y ij − u ∂ u (cid:3) γ n ( { y ij }| u )= γ n ( { y ij }| u ) L n ( { y ij }| u ) + C n ( { y ij }| u ) , (45)where L n = n (cid:88) i =1 γ ( p i | u ) I , ( p i | u ) , (46)comes from the self-energy type diagrams. All squaredmomenta should be replaced by combinations of the y ij ,and C n arises from all flow diagrams which involve onlycorrelators of order less than n , and thus are fixed al-ready. The corresponding solution to the equation is γ n ( { y ij }| u ) = − (cid:90) d ω ω d − ( n − − C n ( { ωy ij }| ωu ) × exp (cid:20) − (cid:90) ω d ττ L n ( { τ y ij }| τ u ) (cid:21) . (47)Since the flow diagrams entering C n can be calculatedeasily in an iterative way, this completes the calculationof the exact fixed point action for the O ( N ) model atlarge N . We emphasise that while the general idea of anexact solution in the large N limit has appeared in the lit-erature before [39], the systematic discussion of the fixedpoint correlation functions, filling in all intermediate de-tails, has been missing. F. Regulator dependence We will now discuss the regulator dependence of theexact solution and its properties. Clearly, as it must be, the existence of the fixed point and all critical exponentsare regulator-independent, see (25). The latter are phys-ically observable, at least in principle, and characterisethe second order phase transition. Further regulator-independent features are the existence of a positive vac-uum expectation value at r = r > , the monotonicityof the derivative of the potential, and the bound on thevalue of u (0) . These are however qualitative, not quan-titative features.The more interesting question is thus about quanti-tative regulator independence, e.g. , whether any combi-nation of coupling constants is regulator-independent aswell. If such a combination would be found, it wouldbe possible to assess the quality of a given truncation ina new way. On the other hand, if this is not the case,so that coupling constants are essentially free and com-pletely independent of each other, then any general com-parison of values for coupling constants obtained with dif-ferent regulators is meaningless. In particular, any strongregulator dependence of strictly non-universal quantitieswould not be a sign of a bad truncation whatsoever. Wewill find that indeed the latter option is the case: fixedpoint values of coupling constants are not universal.To make this more precise, let us study the coefficientsof the Taylor expansion of the critical potential aroundthe minimum. Taking n derivatives of the fixed pointequation and using Faà di Bruno’s formula gives (2 n − ( n − d ) v ( n ) ( r ) − ( d − rv ( n +1) ( r )= 12 n (cid:88) k =1 ( − k k ! I k +1 ( v (cid:48) ( r )) B n,k ( v (cid:48)(cid:48) ( r ) , . . . , v ( n − k +2) ( r )) . (48)Here, the B n,k denote the Bell polynomials. We nownotice that the only term on the right-hand side whichinvolves v ( n +1) ( r ) is the term with k = 1 . Evaluatingthis equation at the minimum r = r , we find that anydependence on v ( n +1) ( r ) drops out of the equation, andwe get a linear equation for v ( n ) ( r ) in terms of the lowerderivatives at the minimum. The first few derivativesread v ( r ) = 12 I (0) ,v (cid:48) ( r ) = 0 ,v (cid:48)(cid:48) ( r ) = 4 − d I (0) . (49)In general, one finds that v ( n ) ( r ) depends on the thresh-old integral I n +1 (0) . For n ≥ , the explicit dependenceon this integral reads v ( n ) ( r ) = 12 n ! d − n (cid:18) d − I (0) (cid:19) n I n +1 (0) + . . . , (50)where the dots indicate terms which only include thresh-old integrals with indices ≤ n . Since these threshold in-tegrals are in general independent of each other, we con-clude that all couplings are independent of each other,0 FIG. 5. The graph indicating the partial integration rulesand the basis for the first order of the derivative expansionof the O ( N ) model at finite N . In the convention of [37], thebottom row corresponds to the operators with prefactors Z and Y , from left to right. and no (finite) combination of couplings can be foundwhich is independent of the regulator choice.In a similar way, one can perform an expansion aroundinfinite field values, r = ∞ . Here, one finds that succes-sive expansion coefficients involve the threshold integralswith negative index, I − n (0) , which are in general in-dependent of each other and of those with positive in-dex. Thus there doesn’t even seem to be a combina-tion of terms from different expansions which combinesto a regulator-independent quantity. We conclude thatin general, nothing that doesn’t need to be regulator-independent is regulator-independent. G. Beyond large N /N expansion An obvious question to ask is whether we can repro-duce the results of the /N expansion. Unfortunately,this seems to be practically extremely hard, since alreadyat order /N , both the derivative expansion and its par-tial resummation fail to be perfect truncations. This caneasily be seen by considering the exact flow equation forthe potential – at order /N , it will always depend on the(field- and momentum-dependent) wave function renor-malisation at the same order, and not only on the large N solution. Likewise, the flow of the wave function renor-malisation will depend on the higher order correlatorsof the same order in /N . In a large N expansion, theonly simplification beyond the leading order is that theequations are linear, but in general they will form an infi-nite tower of integro-differential equations. This doesn’texclude that by clever rewriting, the corrections to thecritical exponents and the anomalous dimension cannot For this expansion to exist, we thus have to assume that theintegrals (cid:90) d µ q ˙ R k ( q ) (cid:0) q + R k ( q ) (cid:1) n , are finite for all positive n . This is the case if the (scale derivativeof the) regulator falls off faster than any polynomial, that is it isa Schwartz function on R + . be extracted, but we haven’t found a way to do so. Weconclude that to derive exact expressions for higher or-der terms of the large N expansion of critical quantities,standard large N methods [1, 43] are more efficient. 2. Finite N Let us now discuss some implications and benefits ofthe presented results for studies at finite N . From theanalysis of the O ( N ) model at varying N , it was foundthat the fixed point has a smooth dependence on N [37].We thus propose the following new way to close the flowequation at finite N . Instead of setting correlation func-tions that are not resolved to zero, the large N solutionmight be substituted instead. For large enough N , thisshould still be an excellent approximation, and improvethe convergence of any other truncation scheme. Thisgives a practically feasible implementation of the ideapresented in [44] to use information on higher order op-erators to stabilise lower order truncations.On a conceptual level, at finite N it is indeed moreuseful to employ a somewhat different basis than that ofthe large N limit, since we now have to deal with a bundleindex. It is most convenient to then let all derivativesact on fields φ a only instead of both φ a and ρ , since thatway we can avoid having to introduce two kinds of boxesand deal with more complicated integration rules. Thismodifies our algorithm to obtain the basis and partialintegration rules in the following way:• boxes now stand for fields φ a , and we have a secondkind of link, which we represent by a wavy link,indicating contraction of the bundle index,• only an even number of boxes can exist, since allfields must be contracted to form a scalar,• when adding a new element by partial integrationcorresponding to a weight of − x , we have to addtwo boxes which are linked by a wavy link, andthe derivative link connects to one of them; nofactors of two arise because of the normalisation ρ = φ a φ a / so that ( ∂ µ ρ ) = φ a ( ∂ µ φ a ) .We illustrate these changes in the graphs correspondingto the first two non-trivial orders in the derivative ex-pansion in Figure 5 and Figure 6. This choice of basiscoincides with the choice made in the recent investigationof the n = 2 truncation of the O ( N ) model at finite N [37], and we indicate the map to the prefactors of thatwork to our basis in the caption of the graphs. As hasbeen noted in [37], for the case N = 1 once again dif-ferent rules apply since the wavy links don’t representa contraction anymore and can be dropped. Then, anypair of completely unconnected boxes can be reabsorbedinto the prefactor coupling function, so that the num-ber of independent operators at any given order of thederivative expansion is lower for N = 1 than for N > .1 FIG. 6. The graph indicating the partial integration rules and the basis for the second order of the derivative expansion ofthe O ( N ) model at finite N . From left to right, the bottom row corresponds to the operators with prefactors W , W , W , W , W , W , W , W , W and W of the investigation done in [37]. We will close this section by commenting again on therelation of this basis to the BMW scheme [22, 39–42].At finite N and order n , this corresponds to the resum-mation of diagram elements where exactly n boxes areconnected by at least one derivative link. In contrastto the large N limit, this includes graphs with up to n boxes. III. GROSS-NEVEU MODELA. Exact flow equation As a second example for an exact fixed point solu-tion, we will discuss the partially bosonised Gross-Neveumodel. In this, we discuss N f Dirac fermions in a d γ -dimensional representation of the Clifford algebra, cou-pled to a Z -symmetric scalar field. For investigations atfinite N f in different dimensions and in both a condensedmatter and a high energy physics context, see [25, 29, 45–67]. The large N limit consists in taking the flavour num- ber to infinity, N f → ∞ , with the consequence that onlydiagrams with a complete fermion loop contribute to theflow in this limit. By assumption, the four-fermion cou-pling is set to zero in this model at the microscopic levelsince it has been transformed to a Yukawa coupling bythe Hubbard-Stratonovich transformation. In that way,the fermionic sector of the action doesn’t flow in the large N limit, but remains classical. Employing again localityand regularity arguments, this fixes the form of the actionto Γ N →∞ [ φ, Ψ] = d γ N f (cid:20)(cid:90) d d x (cid:2) i ¯Ψ /∂ Ψ + i ¯ gφ ¯ΨΨ (cid:3) + ¯Γ[ φ ] (cid:21) . (51)Both the fermionic wave function renormalisation andthe Yukawa interaction ¯ g are restricted to be constant bylocality and regularity. With the regulator choice ∆ S k = d γ N f (cid:90) d d x i ¯Ψ /∂ r k ( − ∂ )Ψ , (52)where r k is a dimensionless shape function, we can writedown the exact flow as ˙¯Γ = − (cid:90) d d x (cid:90) d µ q tr (cid:2) ( i /∂ − /q )(1 + r k (( q − i ∂ ) )) + i ¯ gφ (cid:3) − (cid:0) − /q ˙ r k ( q ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d γ N f , (53)where the remaining trace over the Dirac indices has tobe projected onto the term proportional to d γ N f . Tobring this into a more useful form, we first introduce thefermionic propagator at constant scalar field φ , G ( q | φ ) = (cid:2) − /q (1 + r k ( q )) + i ¯ gφ (cid:3) − = − /q (1 + r k ( q )) + i ¯ gφq (1 + r k ( q )) + 2¯ g ρ . (54) From the general flow (53) we can now extract the flowsof all correlation functions at constant field, similar tothe case of the O ( N ) model. Since the only couplingbetween the bosonic and the fermionic sector is the field-independent Yukawa coupling, only diagrams with three-point functions will appear in the flow. In that way, wecan write explicitly δ n ˙¯Γ δφ n ( p , . . . , p n | φ ) = − ( − i ¯ g ) n n ! (cid:90) d µ q tr G ( q | φ ) /q ˙ r k ( q ) G ( q | φ ) G ( q + p | φ ) · · · G ( q + p + · · · + p n − | φ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d γ N f . (55)A unit strength symmetrisation over the external mo- menta is implied on the right-hand side of the equation.2Let us point out two key differences to the O ( N ) model:first, since the flow only contributes to bosonic operatorsbut is driven purely by fermionic fluctuations, the result-ing fixed point equations for the correlation functions arelinear first order differential equations that can be inte-grated directly. Second, as is well-known, since the flowcontributes to all bosonic operators, we will have a non-trivial bosonic anomalous dimension η φ (cid:54) = 0 . B. Low order critical correlators We will now discuss the lowest order correlation func-tions and the spectrum. In the following we will occa-sionally use the identification q (1 + r k ( q )) = q + R k ( q ) , (56)to bring the flow equation into a form which resembles abosonic flow.Going somewhat out of order, we will first discuss thefixed point equation for the Yukawa coupling ¯ g . The rea-son for this is that it is dimensionful, so that requiringa fixed point in fact determines the bosonic anomalous dimension. For this we assume that ¯ g (cid:54) = 0 , which isreasonable since otherwise the flow is a constant, andwe are only left with the Gaussian fixed point. In gen-eral dimensions, the flow equation for the dimensionlessYukawa coupling g reads ˙ g = d − η φ g = 0 . (57)We conclude that for g (cid:54) = 0 , the bosonic anomalous di-mension is η φ = 4 − d . (58)Next, we will discuss the bosonic potential. Its flowequation reads ˙ v ( r ) = − dv ( r )+( d − η φ ) rv (cid:48) ( r ) − (cid:90) d µ q ˙ R k ( q ) q + R k ( q ) + 2 g r . (59)Here we used the rewriting (56). Using (58), the regularfixed point solution reads v ( r ) = − d (cid:90) d µ q ˙ R k ( q ) q + R k ( q ) F (cid:18) , − d − d (cid:12)(cid:12)(cid:12)(cid:12) − g rq + R k ( q ) (cid:19) . (60)For non-negative r the potential is monotonically increasing, so that the fixed point potential is in the symmetricregime. Let us now discuss the bosonic two-point function. At constant r , the flow equation reads ˙ γ ( p | r ) = ( − η φ ) γ ( p | r ) + ( d − η φ ) r γ (0 , ( p | r ) + 2 p γ (1 , ( p | r )+ 2 g (cid:90) d µ q ˙ r k ( q )[ q (1 + r k ( q )) + 2 g r ] [( p + q ) (1 + r k (( p + q ) )) + 2 g r ] × (cid:26) ( q + p · q )(1 + r k (( p + q ) ))( q (1 + r k ( q )) − g r ) − q (1 + r k ( q )) g r (cid:27) . (61)Denoting the integral (without the prefactor) as I ψ ( p | r ) , the solution to the equation with the correctboundary conditions reads γ ( p | r ) = 2 g d − I ψ (0 | − g (cid:90) d ω ω − d (cid:104) I ψ ( ωp | ωr ) − I ψ (0 | (cid:105) . (62)One can easily check that this reduces to v (cid:48) ( r ) + 2 rv (cid:48)(cid:48) ( r ) We stress that in this section, γ denotes the second variation ofthe effective action with respect to φ , in contrast to the discussionon the O ( N ) model, where the variation was with respect to ρ . Here we again used that by Lorentz invariance, the integral mustbe a function of p . at p = 0 , as it must for consistency, and that γ ( p | r ) ∝ (cid:0) p (cid:1) d − = (cid:0) p (cid:1) − ηφ , as p → ∞ , (63)which is consistent with the scaling expected from theanomalous dimension (58).It is worthwhile to discuss the equation for the anoma-lous dimension, which is related to the canonical nor-malisation of the scalar field. For this, we take the p -derivative of (61) and set p = r = 0 . Using the normali-sation condition γ (1 , (0 , 0) = 1 to fix the wave function3renormalisation, we find η φ g = 1 − d (cid:90) d µ q ˙ R k ( q ) q ( q + R k ( q )) + 32 (cid:90) d µ q ˙ R k ( q ) (cid:0) R (cid:48) k ( q ) + d q R (cid:48)(cid:48) k ( q ) (cid:1) ( q + R k ( q )) − d (cid:90) d µ q q ˙ R k ( q )(1 + R (cid:48) k ( q )) ( q + R k ( q )) . (64)While usually this equation fixes the anomalous dimen-sion, in our case it actually fixes the value of the Yukawacoupling g , since η φ is already fixed by (58). This equa-tion will be important in the next subsection to discussthe spectrum of the theory.The fixed point equations for the higher order corre-lation functions can be found in a similar way, and so-lutions have a similar form as (62). We note that forall correlators, including the two-point function, generalsolutions are only defined up to a multiple of the homoge-neous solution, but this is however uniquely fixed by theconsistency condition at vanishing momenta and regular-ity. We thus will not present the higher order correlatorsexplicitly. C. Critical exponents We are now prepared to discuss the critical exponentsof the fixed point. First, we will discuss the spectrumthat is related to variations of the potential, with van-ishing variation of the Yukawa coupling and anomalousdimension. With these conditions, we find − θδv ( r ) = − dδv ( r ) + ( d − η φ ) rδv (cid:48) ( r ) , (65)with the solution δv ( r ) = c r d − θ . (66)Here, c is a constant. In fact, these are the same per-turbations as for the O ( N ) model, with the replacement r → u , once again reinforcing that for the O ( N ) model u is the more natural variable. In conclusion, this partof the spectrum agrees with the spectrum of the O ( N ) model, θ = d − n , n ∈ N . (67)There is one additional critical exponent which corre-sponds to a finite perturbation of the Yukawa coupling.Taking the variation of (57), we get − θδg = 12 g δη φ , (68)where we used the fixed point value for the anomalousdimension. On the other hand, the variation of (64) alsorelates the variations of η φ and g via δ (cid:18) η φ g (cid:19) = 0 , (69) so that δη φ = 2 η φ g δg . (70)Combining (68) with (70), we find the final critical expo-nent θ η = − η φ = d − . (71)Notice that due to the linear nature of the fixed pointequations, it is clear that one can find solutions to theperturbation equations of all n -point correlation func-tions for all critical exponents. For example, for δg = 0 ,we find ( d − θ − δγ ( p | r ) = 2 r δγ (0 , ( p | r ) + 2 p δγ (1 , ( p | r ) . (72)The general solution to this equation reads δγ ( p | r ) = (cid:0) p (cid:1) d − − θ c n (cid:18) rp (cid:19) , (73)where c n is a free function. The set of critical exponents(67) then makes the first factor regular, and requiringthat the perturbation is well-defined for all momenta con-strains c n to be a polynomial of order at most ( n − .Since δγ (0 | r ) is related to δv ( r ) , there will be condi-tions on certain coefficients of this polynomial in relationto the normalisation constant c of the perturbations ofthe potential.It is interesting to point out explicitly that almost allcritical exponents of the O ( N ) model and the Gross-Neveu model at large N agree. The only exception isthe extra critical exponent θ η which is exclusive to thelatter model. This has the interesting consequence thatto uniquely characterise any universality class, the com-plete set of critical exponents has to be determined. Anyfinite or infinite subset is, in general, insufficient to spec-ify the universality class. D. Regulator dependence Once again we discuss the regulator dependence of thesolution. The existence of the fixed point and its spec-trum are independent of the regulator, as required. Bycontrast, the situation for the coupling constants is theexact same as for the O ( N ) model. Again taking theexample of the couplings making up the potential, theTaylor coefficients can be directly read off from the func-tional solution (60), since the hypergeometric functioncan be defined in terms of a power series around van-ishing argument. In complete similarity to the O ( N ) model, the n -th coupling constant is related to an in-dependent threshold integral, I n (0) . We conclude thatall remarks made earlier also apply to the Gross-Neveumodel, and indicate that indeed generically, fixed pointcouplings carry no regulator-independent information atall.4 IV. SUMMARY In this work we studied the O ( N ) model and the Gross-Neveu model within functional renormalisation in dimen-sions between two and four. In both models, we derivedthe complete non-perturbative fixed point effective ac-tion in the large N limit without specifying the regulatorfunction. That was possible because of the simplificationof the flow equation in the large N limit which allowsfor perfect truncations, where improving the truncationdoesn’t alter the renormalisation group equations of pre-vious orders.For the O ( N ) model, with regards to a large N ex-pansion, we have found indications that already at next-to-leading order, the perfect truncation property of thelarge N limit is lost. It thus seems difficult to derivethe /N corrections to critical quantities from the func-tional renormalisation group analytically. Nevertheless,the form of the effective action at infinite N might serveas an approximate closure of the flow equation at finite N , potentially improving the convergence of common ap-proximation schemes.The Gross-Neveu model is even simpler than the O ( N ) model in the large N limit, since the flow is exclusivelydriven by fermionic fluctuations contributing to the run-ning of bosonic operators. This results in linear partialdifferential equations for the correlation functions. No-tably, the spectrum of the Gross-Neveu model and the O ( N ) model is almost the same.A key result of the study of these models is thatonly observables and qualitative aspects are regulator-independent. Concretely, this concerns the existence ofthe fixed point and its spectrum, and qualitative resultslike the existence or absence of a non-trivial vacuum ex-pectation value. By contrast, generically fixed point cou- plings depend on the choice of the regulator even fornon-truncated solutions to the flow equation, and thusare essentially arbitrary. This is not entirely surprising,but having concrete calculations without systematic er-rors supporting this view is reassuring.To our knowledge, this is the first time that a completenon-perturbative fixed point action was derived. Thisserves as a showcase to test general properties of theeffective action independent of truncation errors. Theexplicit fixed point correlators represent a useful testingbed for numerical methods that aim to resolve field andmomentum dependencies.As a new technical result, we have put forward analgorithm to derive an operator basis at n -th order ofthe derivative expansion, which has an intuitive graph-ical representation, and that at the same time providesthe partial integration rules to map any operator of agiven order into the basis. Its application to higher or-ders, while cumbersome, is entirely straightforward, andthe algorithm can be implemented in a computer code toautomatise these computations.It would be interesting to derive the conformal data[68–70], such as the coefficients of the operator productexpansion, for these analytically solvable cases to gainsome knowledge about what techniques work best to ex-tract the data from a non-perturbative renormalisationgroup flow, but we leave this for future work. 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