Non-perturbative quantum Galileon in the exact renormalization group
PPrepared for submission to JCAP
Non-perturbative quantum Galileonin the exact renormalization group
Christian F. Steinwachs
Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 3, 79104Freiburg, GermanyE-mail: [email protected]
Abstract.
We investigate the non-perturbative renormalization group flow of the scalarGalileon model in flat space. We discuss different expansion schemes of the Galileon trun-cation, including a heat-kernel based derivative expansion, a vertex expansion in momentumspace and a curvature expansion in terms of a covariant geometric formulation. We find thatthe Galileon symmetry prevents a quantum induced renormalization group running of theGalileon couplings. Consequently, the Galileon truncation only features a trivial Gaussianfixed point.
Keywords: quantum field theory in curved spacetime, renormalization group, scalar fieldtheory a r X i v : . [ h e p - t h ] J a n ontents The Galileon in flat space is a higher-derivative theory of a single scalar field which arisesfrom the decoupling limit of the Dvali-Gabadadze-Porrati model [1] and whose particularsymmetry structure ensures the second-order character of the field equations and the absenceof ghost-like excitations [2, 3]. Its generalization to the covariant Galileon in curved space,Horndeski’s scalar-tensor theory [4, 5], has many applications in cosmological models of theearly and late time acceleration of the universe, see e.g. [6–14].Various aspects of the classical and quantum properties of higher-derivative scalar ef-fective field theories including the Galileon, the Dirac-Born-Infeld model and the non-linearsigma model have been studied [15–17], including the perturbative renormalization of theGalileon [2, 18–27] and of the non-linear sigma model [28–31], as well as on-shell scatteringamplitudes in effective scalar field theories [32–40].In this article we study the non-perturbative renormalization group (RG) flow of theGalileon truncation in the framework of the exact renormalization group equation (ERGE);for a recent review on the ERGE see [41]. The scalar nature of the Galileon field also permitsto focus on the structural aspects of higher-derivative theories in the ERGE. In particular,it permits to investigate the dependence of the RG properties on the regulator choice and totest various optimization criteria without having to deal with the additional complicationsand off-shell ambiguities present in gauge theories, see e.g. [42–46].Truncations of scalar field theories based on derivative expansions up to order O ( ∂ ) havebeen studied previously in the in the context of the ERGE, see e.g. [46–57]. In [58], the RGflow of d -dimensional surfaces embedded in a d + 1 -dimensional space and the correspondenceto the Galileon, which emerges from this geometric formulation in the non-relativistic limit,has been studied.In contrast, in the present work, we directly investigate the non-perturbative RG flow ofthe Galileon truncation in different expansion schemes, including a derivative expansion up to O ( ∂ ) , a vertex expansion in momentum space up to O ( π ) , and a curvature expansion within– 1 – compact covariant geometric formulation in terms of an effective Galileon metric which re-sults from a resummation of infinitely many operators with a fixed number of derivatives perfield. We find that the non-renormalization property of the Galileon in the perturbative quan-tization [23, 26, 59–61] carries over to the non-perturbative RG flow. That is, the Galileonbeta functions do not receive any quantum-induced contributions and are only driven by theclassical (canonical) scaling dimensions of the Galileon couplings, which implies that the RGsystem only features a trivial Gaussian fixed point. Nevertheless, compared to the pertur-bative calculation of the one-loop divergences [18, 21, 22, 25, 26], a larger set of operators,which lead out of the Galileon truncation, is induced in the functional trace of the Wetterichequation. These terms correspond to power-law divergences and are absent in dimensionalregularized perturbative calculations.The article is structured as follows: In Sect. 2 we introduce the required formalism forthe non-perturbative quantization of the Galileon within the Exact Renormalization GroupEquation (ERGE). In Sect. 3 we calculate the functional trace in the Wetterich equationby three different expansion schemes, which are based on a O ( ∂ ) derivative expansion, avertex expansion in momentum space and a covariant geometric resummation techniques.We discuss the implications of our calculation in Sect. 4 and conclude in Sect. 5. The exact renormalization group (RG) is defined by the Wetterich equation [62–64] ˙Γ k = Tr (cid:32) ˙ R k Γ (2) k + R k (cid:33) . (2.1)The Euclidean effective averaged action (EAA) Γ k [ π ] is a functional of the Galileon field π and depends on the abstract RG scale k . It approaches the bare action Γ ∞ = S in the limit k → ∞ and the effective action Γ := Γ in the limit k → . A dot denotes differentiationwith respect to the logarithmic k -derivative ˙ X k := ∂ t X k := k∂ k X k . The presence of theregulator R k in (2.1) provides an infrared (IR) cutoff, while its scale derivative ˙ R k providesan ultraviolet (UV) cutoff such that the result of the trace is both IR and UV finite. Theone-loop structure of (2.1) is reflected by the functional trace. The Galileon truncation of theEAA in d = 4 dimensional flat space with Euclidean metric δ µν = diag (1 , , , reads Γ k [ π ] = (cid:88) i =2 Γ ( i ) k [ π ] , (2.2) Γ (2) k [ π ] = − c (cid:90) d x π(cid:15) µνρσ (cid:15) ανρσ π µα , (2.3) Γ (3) k [ π ] = c (cid:90) d x π(cid:15) µνρσ (cid:15) αβρσ π µα π νβ , (2.4) Γ (4) k [ π ] = − c (cid:90) d x π(cid:15) µνρσ (cid:15) αβγσ π µα π νβ π ργ , (2.5) Γ (5) k [ π ] = c (cid:90) d x π(cid:15) µνρσ (cid:15) αβγδ π µα π νβ π ργ π σδ . (2.6) The regulator enters the action in the path integral quadratic in the fields and hence acts as an effective k -dependent mass term R k ∝ k for fluctuations with p (cid:28) k while it satisfies R k ≈ for p (cid:29) k ( R k → in the limit k → ). – 2 –he symmetric tensor π µν is defined as π µν := ∂ µ ∂ ν π and ε µνρσ is the totally antisymmetricLevi-Civita symbol. The Galileon field and the partial derivatives have mass dimension [ π ] = [ ∂ µ ] = 1 , while the running couplings c i , i = 2 , . . . have mass dimension [ c i ] = − i − . (2.7)Since the Galileon action only involves derivative interactions, it is obviously invariant undershift symmetries π → π + λ with a constant λ . However, the action (2.2) is even invariantunder the more restrictive Galileon transformations with a constant vector v µ , π → π + λ + v µ x µ . (2.8)The invariance (2.8) leads to the particular structure of the Galileon interactions (2.3)-(2.6)and ensures that, despite the presence of higher derivative terms, the field equations are ofsecond order and no ghost-like excitations appear in the spectrum.Expanding the tensor contractions in (2.3)-(2.6) and integrating by parts, the Galileontruncation (2.2) is expressed in terms of a basis of invariants which does not contain self-contractions of derivatives acting on the same field (we use ∂ µ ...µ n = ∂ µ . . . ∂ µ n ), Γ k = 12 (cid:90) d x (cid:8) c ( ∂π ) + c ( ∂ µ π )( ∂ ν π )( ∂ µν π ) + c [( ∂ µ π )( ∂ ν π )( ∂ ρ π )( ∂ µνρ π )+3( ∂ µ π )( ∂ ν π )( ∂ ρµ π )( ∂ νρ π ) (cid:3) + c [( ∂ µ π )( ∂ ν π )( ∂ ρ π )( ∂ σ π )( ∂ µνρσ π )+12( ∂ µ π )( ∂ ν π )( ∂ ρ π )( ∂ µσ π )( ∂ σνρ π ) + 12( ∂ µ π )( ∂ ν π )( ∂ µρ π )( ∂ ρσ π )( ∂ σν π ) (cid:3)(cid:9) . (2.9)We perform the linear split of the Galileon field into background plus perturbation π ( x ) = ¯ π ( x ) + δπ ( x ) . (2.10)We identify the physical mean field (cid:104) π (cid:105) with ¯ π and omit the bar indicating a backgroundquantity. The scalar second-order fluctuation operator F ( ∂ ) is defined by the Hessian δ Γ k [ π ] δπ ( x ) δπ ( x (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) π = F ( ∂ ) δ ( x, x (cid:48) ) = − (cid:0) G − (cid:1) µν ∂ µ ∂ ν δ ( x, x (cid:48) ) . (2.11)The structure of the fluctuation operator in (2.11) suggests to identify (cid:0) G − (cid:1) µν with theinverse of an “effective Galileon metric” and is explicitly defined in terms of π (cid:0) G − (cid:1) µν := (cid:16) G − (cid:17) µν + (cid:16) G − (cid:17) µν + (cid:16) G − (cid:17) µν + (cid:16) G − (cid:17) µν , (2.12) (cid:16) G − (cid:17) µν = c ε µαρσ ε ναρσ = c g µν , (2.13) (cid:16) G − (cid:17) µν = − c ε µαρσ ε νβρσ π αβ = c [ ∂ µ ∂ ν π + g µν ∆ π ] , (2.14) (cid:16) G − (cid:17) µν = 32 c ε µαρσ ε νβγσ π αβ π ργ = 32 c [2( ∂ µ ∂ ρ π )( ∂ ρν π ) + 2( ∂ µν π )(∆ π )+ g µν (∆ π ) − g µν ( ∂ ρ ∂ σ π )( ∂ ρ ∂ σ π ) (cid:3) , (2.15) (cid:16) G − (cid:17) µν = − c ε µαρσ ε νβγκ π αβ π ργ π σκ = 2 c (cid:2) ∂ µ ∂ ν π )(∆ π ) + 6( ∂ µ ∂ ρ π )( ∂ ρ ∂ ν π )(∆ π )+6( ∂ µ ∂ ρ π )( ∂ ρ ∂ σ π )( ∂ σ ∂ ν π ) − ∂ µ ∂ ν π )( ∂ ρ ∂ σ π )( ∂ ρ ∂ σ π ) + g µν (∆ π ) − g µν ( ∂ ρ ∂ σ π )( ∂ σ ∂ λ π )( ∂ λ ∂ ρ π ) − g µν ( ∂ ρ ∂ σ )( ∂ ρ ∂ σ )(∆ π ) (cid:105) . (2.16) We suppress the subscript k that indicates the RG scale dependence. – 3 –he subscript of the tensor (cid:16) G − i ) (cid:17) µν indicates the number of background derivatives and ∆ := − δ µν ∂ µν defines the positive definite Laplacian. The number of background derivativesdefines the background derivative order (BDO) of a tensor. For example, we write (suppressingtensor indices) G − i ) = O (cid:0) ∂ i (cid:1) to indicate that the G − i ) have BDO i . The effective Galileonmetric G µν is defined as the inverse of (2.11) via the relation G µρ (cid:0) G − (cid:1) ρν = δ νµ . (2.17)The Galileon metric G µν is assumed to be positive definite to ensure its non-degeneracy det( G ) (cid:54) = 0 . (2.18)The metric G µν is not constant, but importantly its inverse (cid:0) G − (cid:1) µν is divergence-free ∂ µ (cid:0) G − (cid:1) µν = 0 . (2.19)Due to the particular structure of (cid:0) G − (cid:1) µν this also holds separately for the individual tensors (cid:16) G − i ) (cid:17) µν , i = 0 , , , . The regulated propagator is defined in terms of Γ (2) k and R k by G R := 1 F + R . (2.20)In the next section, we calculate the functional trace in the Wetterich equation in differentexpansion schemes. The Wetterich equation only provides an “exact” equation if the RG flow “closes”, that is, ifthe operators that are arise in the functional trace of the flow equation (2.1) are of the sameform as those present in the ansatz for the EAA. However, in general, the fluctuation operatorderived from the Hessian of the EAA induces operators in the functional trace which leadout of the truncation. Consistency of the Wetterich equation then requires to neglect theseoperators in order for the flow to close at least approximately within the projection onto thefinite truncation.Therefore an important practical question is connected to the principles according towhich the operators of a finite truncation should be chosen. In most cases, standard expansionschemes common to those in effective field theories (EFTs) provide a natural guiding principle.Any truncation is ultimately based on counting the number of derivatives, the number offields, the powers of mass parameters, or any other small parameters such as /N in thelarge N limit, or (cid:15) in the expansion around two-dimensional space d = 2 + (cid:15) , or a particularcombination of these. The most agnostic power counting scheme may be viewed to correspondto truncations ordered by operators with increasing canonical mass dimension. However, ingeneral, the choice of the best adapted expansion schemes depends on the underlying physicalproblem. In particular, in case the model under consideration has additional symmetries, moresophisticated combined expansion scheme may be more appropriate as e.g. in the context ofchiral perturbation theory [65, 66]. Positive definiteness requires that all eigenvalues of G µν must be positive. As the eigenvalues are in generalfunctions of the running couplings positive definiteness implies constraints on the c i . We suppress the k -label indicating the dependence eon the RG scale. – 4 – .1 Derivative expansion We denote the functional trace in the Wetterich equation (2.1) for the ansatz (2.2) by T := Tr (cid:32) ˙ R F + R (cid:33) . (3.1)We also define the regulated fluctuation operator F R = F + R and sort the individual termsin F R according to their BDO F R = P + Π , P (∆) = c ∆ + R (∆) , Π( ∂ ) = (cid:104) G − + G − + G − (cid:105) µν ∂ µν . (3.2)Following the classification scheme of [67], we choose a type I regulator R without specifyingany concrete profile function. We denote the BDO of the individual operators by F = O ( ∂ ) , P = O ( ∂ ) , Π = O ( ∂ ) , G − = O ( ∂ ) , G − = O ( ∂ ) , G − = O ( ∂ ) . (3.3)In view of the BDOs in (3.3), we can perform a systematic derivative expansion in powers ofthe perturbation Π by expanding the Greens function G R of F R up to the required order G R = 1 F R = 1 P (∆) − P (∆) Π( ∂ ) 1 P (∆) + 1 P (∆) Π( ∂ ) 1 P (∆) Π( ∂ ) 1 P (∆) + . . . (3.4)For a O ( ∂ ) truncation, we need to expand (3.4) up to terms involving four powers of Π . Inorder to explicitly evaluate the functional trace we need to arrange the resulting operatorsinto a standard form by commuting all functions of the Laplacian to the very right. This canbe accomplished iteratively by repeated use of the basic identity (cid:20) Π , P (cid:21) = 1 P [ P, Π] 1
P . (3.5)In view of
Π = O ( ∂ ) and the fact that each commutator [ P, Π] increases the BDO at leastby one, this expansion is guaranteed to be efficient. We need to evaluate commutators [ P, Π] which involve an arbitrary function P (∆) of ∆ . These commutators are again expanded upto the required BDO order r by using the general formula [ P, Π] = r (cid:88) n =1 n ! ( − n [Π , ∆] n P ( n ) (∆) . (3.6)Here, P ( n ) (∆) is the n th derivative of P (∆) with respect to the argument and all dependenceon ∆ has been ordered to the very right in (3.6). The n -fold commutator is defined iteratively [Π , ∆] n := [[Π , ∆] n − , ∆] , [Π , ∆] := [Π , ∆] , [Π , ∆] := Π . (3.7)Once the required BDO has been reached all factors of /P can freely be commuted to theright. The result of this algorithm is that the functional trace (3.1), truncated at O ( ∂ ) ,reduces to a sum of functional traces of the form Tr (cid:104) B µ ...µ j ∂ µ . . . ∂ µ j W ( m )( i,j,k,l ) (∆) (cid:105) , (3.8)– 5 –ith background-tensorial coefficients B µ ...µ j and ∆ -dependent functions W ( m )( i,j,k,(cid:96) ) (∆) , W ( m )( i,j,k,(cid:96) ) = (cid:16) P (1) (cid:17) i (cid:16) P (2) (cid:17) j (cid:16) P (3) (cid:17) k (cid:16) P (4) (cid:17) (cid:96) P − m ˙ R (3.9)In order to evaluate these functional traces explicitly, we perform the Mellin transform W (∆) = (cid:90) d s L − [ W ] ( s ) e − s ∆ . (3.10)In this way, the evaluation of the functional traces in (3.8) reduce to the evaluation of theUniversal Functional Traces (UFTs) [68] of the bare Laplacian ∆ , U µ ...µ j (∆ | s ) := ∂ µ . . . ∂ µ j e − s ∆ δ ( x, y ) (cid:12)(cid:12) y = x = 1(4 πs ) ∂ µ . . . ∂ µ j e − ( x − y )24 s (cid:12)(cid:12)(cid:12)(cid:12) y = x . (3.11)In the last equality of (3.11) we used the exact result for the heat-kernel of the Laplacian in d = 4 dimensional flat space. The chain of derivatives in (3.11) only leads to a combinatorialfactor times the totally symmetrized product of metric tensors normalized such that eachterm carries unit weight [ δ sym ] µ ...µ j := (2 j − δ ( µ µ . . . δ µ j − µ j ) . (3.12)Due to the structure of (3.11) only UFTs with an even number of indices j are non-zero, U µ ...µ j (∆ | s ) = ( − j (2 π ) (2 s ) − ( j +2) [ δ sym ] µ ...µ j . (3.13)We absorb the s -integral into the Q-functionals defined for an arbitrary function W (∆) , Q n [ W ] := 2 − n (2 π ) (cid:90) ∞ d s s − n L − [ W ] ( s ) . (3.14)Here, L − [ W ]( s ) is the inverse Laplace transform of W [ z ] and the Q -functional can be ex-pressed in terms of the original function W [ z ] by Q n [ W ] = 2 − n (2 π ) n ) (cid:90) ∞ d z z n − W ( z ) . (3.15)The algorithm (3.1)-(3.15) is similar to that proposed in [69] but differs in the expansion whichis based on a derivative expansion and not on an expansion of curvatures or background massdimension.Sometimes, it might happen that different expansion schemes accidentally fall togethersuch as in the context of gravity. In this case diffeomorphism invariance suggests a manifestlycovariant expansion which runs in powers of (derivatives of) curvature invariants. In view ofthe non-linear relation between the curvature and the metric (and its derivatives), a vertexexpansion in powers of the metric perturbations breaks this manifest covariance. As the metricfield is dimensionless (for coordinates having the physical dimension of length), a curvatureexpansion might also be viewed as a derivative expansion R ∼ ∂ g . However, in view of Note that the definition of the Q -functional is slightly different from the standard definition as it involvesan additional numerical factor of − n / π . – 6 –he diffeomorphism symmetry, a more appropriate viewpoint is to consider the expansion incurvatures as a covariant vertex expansion. A true covariant derivative expansion would thencount derivatives acting on curvatures and not on metric perturbations. In certain situations,even a resummation of terms is possible and adequate. In the aforementioned context of thecovariant perturbation theory, this corresponds to resum all operators with a fixed numberof curvatures but an arbitrary number of derivatives leading to non-local form factors [70–72]. Recently, such truncations have also been studied in the context of the ERGE [73, 74].Similarly, the Galileon in flat spacetime can be reformulated in geometrical terms [61] andthereby permits a resummation of an infinite number of operators with at least two powersof derivatives per field into curvature tensors, which are defined with respect to an effectiveGalileon metric as we will discuss in more detail in Sect. 3.3.Combining the expansions (3.4), (3.5) and (3.6) with (3.10), (3.13) and (3.14), the O ( ∂ ) derivative expansion of the functional trace (3.1) acquires the form T = 12 (cid:90) d x { C + C ( ∂ µν π )( ∂ µν π ) + C ( ∂ µνρ π )( ∂ µνρ π ) + C a ( ∂ µνρ π )( ∂ µν π )( ∂ ρ π )+ C b ( ∂ µ ∂ ν π )( ∂ ν ∂ ρ π )( ∂ ρ ∂ µ π ) + C ( ∂ µνρσ π )( ∂ µνρσ π )+ C a ( ∂ µνρ π )( ∂ ρ ∂ σ π )( ∂ µνσ π ) + C b ( ∂ µν π )( ∂ µνρσ π )( ∂ ρσ π )+ C c ( ∂ µνρσ π )( ∂ µνρ π )( ∂ σ π ) + C a ( ∂ µν π )( ∂ µν π )( ∂ ρσ π )( ∂ ρσ π )+ C b ( ∂ µ ∂ ν π )( ∂ ν ∂ ρ π )( ∂ ρ ∂ σ π )( ∂ σ ∂ µ π ) + C c ( ∂ µνρσ π )( ∂ µν π )( ∂ ρ π )( ∂ σ π )+ C d ( ∂ µνρ π )( ∂ µνσ π )( ∂ ρ π )( ∂ σ π ) + C e ( ∂ µνρ π )( ∂ µσ π )( ∂ σ ∂ ν π )( ∂ ρ π )+ C f ( ∂ µνρ π )( ∂ µν π )( ∂ ρσ π )( ∂ σ π ) } + t . d . + O (cid:0) ∂ (cid:1) . (3.16)In the final result (3.16), we have used the explicit expressions (2.14)-(2.16) for the G − i ) andintegrated by parts in order to reduce the invariants to a canonical basis. Total derivatives,denoted by “t.d.”, are neglected. The explicit results for the coefficients C - C f in terms of theGalileon coupling c - c and the Q-functional (3.15) are provided in Appendix A. Comparingthe result (3.16) with the ansatz (2.9), we find that only invariants with a higher number ofderivatives per field than originally present in (2.9) are induced in the functional trace (3.16),implying β c i = 0 , i = 2 , . . . , . (3.17)This result is a consequence of the Galileon symmetry (2.8) responsible for structure of theeffective inverse Galileon metric (2.12) and the operator Π defined in (3.2). Since the (cid:0) G − i (cid:1) µν only involve background tensor structures with two derivatives per field and the derivativeexpansion ultimately reduces to contractions among the (cid:0) G − i (cid:1) µν , projecting (3.16) to thebasis of Galileon invariants in (2.9) shows that there are no quantum induced contributionsto the Galileon beta functions. In this section, we extract the beta functions by a vertex expansion in momentum space. Thevertex expansion involves the scale-dependent n -point functions Γ ( n ) k ( x , . . . , x n ) := δ n Γ k δϕ ( x ) , δϕ ( x ) . . . δϕ ( x n ) . (3.18)– 7 – yG R ( x, y ) Figure 1 . Diagrammatic representation of the regulated Greens function (2.20).
Due to the scalar nature of the field ϕ ( x ) , the n -point functions (3.18) are totally symmetricunder exchange of the x i , i = 1 , . . . , n . Diagrammatically, the regulated Green’s function G R ( x, y ) defined in (2.20) is represented by a line.As shown in Fig. 2, the functional trace is diagrammatically represented by joining the twoends (identifying x = y ) of the line in Fig. 1, illustrating the one-loop structure of (2.1). ˙Γ k = Figure 2 . Diagrammatic representation of the Wetterich equation (2.1). The encircled cross corre-sponds to an insertion of the scale derivative of the regulator ⊗ = k∂ k R = ˙ R . Taking n functional derivatives of the Wetterich equation (2.1) with respect to the Galileonfield π defines the functional flow of the n -point functions ˙Γ ( n ) k . The flow equations of the ˙Γ ( n ) k are diagrammatically represented by insertions of n -point vertices Γ ( n ) k into the loop.In order to extract the beta functions for the Galileon truncation (2.9) from a vertexexpansion, we need to derive the flow of the first five n -point functions, which are diagram-matically represented in Fig. 3. In general, the one-loop structure of the flow equation impliesa hierarchy between the flow of the n -point functions, as ˙Γ ( n ) k involves all Γ ( m ) k with m ≤ n +2 .Evaluating the functional derivatives (3.18) at π = 0 and performing a Fourier transform, weobtain the momentum space expressions for the regulated propagator G R ( p ) = 1 / Γ (2) ( p , − p ) and the interaction vertices Γ ( n ) ( p , . . . , p n ) with n > . For the inner product between twofour-momentum vectors p iµ and p jµ we write ( p i · p j ) := p iµ p jν δ µν , p i := ( p i · p i ) . (3.19)Using the convention that all momenta are incoming and stripping off an overall momentum-conservating delta function δ ( (cid:80) ni =1 p iµ ) , we obtain G R ( p ) = 1 c p + R ( p ) , (3.20) Γ ( n ) k ( p , . . . , p n ) = c n ( n − n M ( p , . . . , p n ) , n > . (3.21)The, M ( p , . . . , p n ) are defined as the sum of the diagonal minors M ( ii ) ( p , . . . , ˆ p i , . . . p n ) ofthe n × n Gram matrix G ij := ( p i · p j ) . Despite the fact that we are working in d = 4 , That is, M ( p , . . . , p n ) = (cid:80) ni =1 M ( ii ) ( p , . . . , ˆ p i , . . . p n ) where M ( ii ) ( p , . . . , ˆ p i , . . . p n ) is the determinantof the matrix obtained by deleting the i th row and i th column of G ij . – 8 – a ) ˙Γ (1) k = − (2 a ) ˙Γ (2) k = − (2 b ) ˙Γ (4) k = (4 a ) − (4 b ) +3 (4 c ) +4 (4 d ) − (4 e ) ˙Γ (3) k = − (3 a ) +3 (3 b ) − (3 c ) ˙Γ (5) k = − (5 a ) +120 (5 b ) − (5 c ) − (5 d ) +10 (5 e ) +5 (5 f ) − (5 g ) Figure 3 . Diagrammatic representation of the RG flow ˙Γ ( n ) k of the n -point functions Γ ( n ) k . The blackdots represent insertions of n -point vertices Γ ( n ) k , the solid lines represent regulated propagators G R .The numerical coefficient in front of each diagram results from the functional differentiation of (2.1). in order to make the d -dependent numerical factors more transparent, we write the analyticexpressions of the ˙Γ ( n ) k corresponding to the one-loop diagrams in Fig. 3 for general d andreduce the expressions to d = 4 at the end. For example, the RG running of the two-pointfunctions reads ˙Γ (2) ( p, − p ) = (cid:90) d (cid:96) d (2 π ) d (cid:104) G ( (cid:96) ) Γ (3) ( (cid:96), p, − p − (cid:96) ) G R ( p + (cid:96) ) Γ (3) ( p + (cid:96), − p, − (cid:96) ) G R ( (cid:96) ) ˙ R ( (cid:96) ) (cid:105) − (cid:90) d (cid:96) d (2 π ) d (cid:104) G R ( (cid:96) ) Γ (4) ( (cid:96), p, − p, − (cid:96) ) G R ( (cid:96) ) ˙ R ( l ) (cid:105) . (3.22)We refrain from providing the analytic expressions for the momentum integrals of the diagramsthat enter the flow equation of the higher n -point functions, since the general structure andthe momentum flow is analogue to that in (3.22). Momentum conservation requires thatthe sum of all momenta at each vertex add up to zero and overall momentum conservation (cid:80) ni =1 p iµ = 0 permits to express the n external momenta p iµ , i = 1 , . . . , n through a linearcombination of n − external momenta.Looking at the structure of the Galileon truncation (2.2), the beta functions of theGalileon operators are obtained by extracting those parts of the ˙Γ ( n ) k that have n − powers of the external momenta. The vertices Γ ( n ) k only contribute positive powers of externalmomenta. Expanding the propagators around zero external momenta up to the required order,all contributions with a particular power of external momenta contributing to the ˙Γ ( n ) k canbe extracted. For example, the expansion of the propagator up to O (cid:0) p (cid:1) reads G R ( (cid:96) + p ) = G R ( (cid:96) ) + 2 G (1) R ( (cid:96) )( (cid:96) · p ) + (cid:104) G (2) R ( (cid:96) )( (cid:96) · p ) + G (1) R ( (cid:96) ) p (cid:105) + 13 (cid:104) G (3) R ( (cid:96) )( (cid:96) · p ) + 6 G (2) R ( (cid:96) )( (cid:96) · p ) p (cid:105) + 16 (cid:104) G (4) R ( (cid:96) )( (cid:96) · p ) + 12 G (3) R ( (cid:96) )( (cid:96) · p ) p + 3 G (2) R ( (cid:96) ) p (cid:105) + O ( p ) . (3.23)– 9 –he superscript denotes derivatives with respect to the argument G ( n ) R ( (cid:96) ) = ∂ n G R ( z ) /∂z n | z = (cid:96) .The tensor loop integrals are reduced to scalar ones by exploiting Lorentz invariance [75], (cid:90) d d (cid:96) f µ ...µ n ( p ) (cid:96) µ . . . (cid:96) µ n = Γ( d/ n Γ( n + d/ (cid:90) d d (cid:96) (cid:0) (cid:96) (cid:1) n f µ ...µ n ( p ) [ δ n sym ] µ ...µ n . (3.24)Here, [ δ n sym ] µ ...µ n is defined in (3.12) and the prefactor in (3.24) is determined by the trace [ δ n sym ] µ ...µ n δ µ µ . . . δ µ n − µ n = 2 n Γ( n + d/ d/ . (3.25)Transforming to polar loop coordinates and performing the angular integrals results in (cid:90) d d (cid:96) (2 π ) d = S d (2 π ) d (cid:90) d (cid:96) (cid:96) d − . (3.26)The surface of the unit sphere in d dimensions is defined by S d := 2 π d/ / Γ( d/ . Finally, wetransform the loop variable z = (cid:96) , d z = 2 (cid:96) d (cid:96) and use the definition (3.15) in order to absorbthe remaining loop integrals in the Q-functionals. In order to extract the beta functions ofthe Galileon couplings, we extract those contributions to the ˙Γ ( n ) k with n − powers ofexternal momenta and find ˙Γ ( n ) ( p , . . . , p n ) (cid:12)(cid:12)(cid:12) p n − = 0 . (3.27)In agreement with the result (3.16), this implies that all beta functions vanish β c i = 0 , i = 2 , , , . (3.28)The result (3.28) can again directly be traced back to the Galileon symmetry (2.8) and theone-loop structure of the Wetterich equation (2.1). The Galileon symmetry implies that each n -point vertex Γ ( n ) k carries n − powers of momenta. The one-loop structure implies thateach vertex Γ ( n ) k inserted in the one-loop diagrams in Fig. 3 will at least contribute n − powers of external momenta, since at most two legs can carry a loop momentum. In view ofthe expansion (3.23), each propagator can only further increase the power of external momentain a diagram. Hence, all n -point one-loop diagrams in Fig. 3 have at least n > n − powers of external momenta and therefore cannot contribute to the Galileon beta functions. The effective Galileon metric (2.17) defines the required geometric structure for an applicationof covariant heat kernel techniques for the fluctuation operator (2.11). Following the approachproposed in the context of the perturbative one-loop calculation [26], We define ∇ G µ as thetorsion-free covariant derivative compatible with G µν , [ ∇ G µ , ∇ G ν ] π = 0 , ∇ G ρ G µν = 0 , (3.29)The connection Γ ρµν ( G ) associated to ∇ G reads Γ ρµν ( G ) = 12 (cid:0) G − (cid:1) ρσ ( ∂ µ G σν + ∂ ν G σµ − ∂ σ G µν ) . (3.30) In addition, the -point and -point vertices entering the diagrams (4 c ) , (5 f ) and (5 g ) in Fig. 3 are triviallyzero in the Galileon truncation (2.2). – 10 –ndices are raised and lowered exclusively with G µν and (cid:0) G − (cid:1) µν , respectively. We define thepositive definite covariant Laplacian as ∆ G := − (cid:0) G − (cid:1) µν ∇ G µ ∇ G ν . (3.31)When the Laplacian ∆ G acts on scalars, it is related to the fluctuation operator (2.11) by F ( ∂ ) = − (cid:0) G − (cid:1) µν ∂ µ ∂ ν = ∆ G − (cid:0) G − (cid:1) µν Γ ρµν ( G ) ∇ G ρ . (3.32)In addition, we define the “bundle connection” acting on scalars Σ ρ := 12 (cid:0) G − (cid:1) µν Γ ρµν ( G ) = det( G ) / (cid:0) G − (cid:1) ρµ ∂ µ det( G ) − / . (3.33)Combining (3.32) with (3.33), the operator (3.31) can be written as F ( ∇ G ) = ∆ G − ρ ∇ G ρ . (3.34)In terms of D µ := ∇ G µ + G µν Σ ν , the operator (3.34) acquires minimal second order form F ( D ) = −D + E, (3.35)with −D = − (cid:0) G − (cid:1) µν D µ D µ and the endomorphism E := ∇ Gν Σ ν + Σ ν Σ ν . (3.36)For the scalar π , the bundle curvature vanishes due to the antisymmetry of the commutator, R µν π := [ D µ , D ν ] π = 0 . (3.37)Adopting a “spectrally adjusted” type III regulator [67], for which the argument is identifiedwith the minimal second-order operator (3.35), we obtain the regulated Greens function G R = 1 P ( −D + E ) , P ( −D + E ) := −D + E + R ( −D + E ) . (3.38)In terms of this geometric reformulation, an infinite number of operators are resummedinto curvatures of the effective Galileon metric G µν which have the schematic structure R ( G ) ∼ ∂ ( G − ∂ G ) + ( G − ∂G ) . Since G = O ( ∂ ) , it follows R µνρσ ( G ) = O (cid:0) ∂ (cid:1) such thatan expansion up to BDO O ( ∂ ) corresponds to an expansion up to O ( R ) . The functionaltrace (3.1) with the Greens function (3.38) of the covariant operator (3.35) reads T = 12 Tr (cid:16) G R ˙ R (cid:17) = 12 (cid:90) d x (cid:90) d s L − (cid:104) G R ˙ R (cid:105) ( s ) (cid:104) x | e − s ( −D + E ) | x (cid:105) = 12 (cid:90) d x √G (cid:90) d s (4 πs ) L − (cid:104) G R ˙ R (cid:105) ( s ) (cid:2) s a + s a + O (cid:0) s (cid:1)(cid:3) = 12 (cid:90) d x √G (cid:110) Q [ G R ˙ R ] + 2 Q (cid:104) G R ˙ R (cid:105) a + 4 Q (cid:104) G R ˙ R (cid:105) a + O (cid:0) R (cid:1)(cid:111) . (3.39)Here, a and a are the traced coincidence limits of the first two Schwinger-DeWitt coefficientsfor the minimal scalar second-order operator (3.35), a ( x, x ) = E − R ( G ) , (3.40) a ( x, x ) = 1180 [ R µνρσ ( G ) R µνρσ ( G ) − R µν ( G ) R µν ( G )] + 12 (cid:18) E − R ( G ) (cid:19) + t . d . (3.41)– 11 –imilar to the procedure in [25], the result of the functional trace in terms of the originalGalileon field is recovered by expanding (3.39) in powers of perturbations δ nπ (cid:0) G − (cid:1) µν evaluatedat zero mean field π . In view of the explicit expression (2.12) we find the structural relations δ π (cid:0) G − (cid:1) | π =0 ∼ ∂ δπ, (3.42) δ π (cid:0) G − (cid:1) | π =0 ∼ (cid:0) ∂ δπ (cid:1) (cid:0) ∂ δπ (cid:1) , (3.43) δ π (cid:0) G − (cid:1) | π =0 ∼ (cid:0) ∂ δπ (cid:1) (cid:0) ∂ δπ (cid:1) (cid:0) ∂ δπ (cid:1) , (3.44) δ nπ (cid:0) G − (cid:1) | π =0 = 0 , n > . (3.45)It is easy to see that the derivative structure of the invariants obtained by expanding theresummed geometric result in powers of δπ coincides with the structures obtained in thederivative expansion (3.16). In particular, for all invariants generated by this expansionthe number of derivatives per field is higher than for the invariants present in the Galileontruncation (2.9). For example, extracting the invariants that involving two powers of δπ leadsto the structural relations, δ π √G| π =0 ∼ (cid:2) δ π (cid:0) G − (cid:1) δ π (cid:0) G − (cid:1) + δ π (cid:0) G − (cid:1)(cid:3) π =0 ∼ (cid:0) ∂ δπ (cid:1) (cid:0) ∂ δπ (cid:1) , (3.46) δ π √G R ( G ) | π =0 ∼ (cid:2) δ π (cid:0) G − (cid:1) ∂ δ π (cid:0) G − (cid:1) + ∂ δ π (cid:0) G − (cid:1)(cid:3) π =0 ∼ (cid:0) ∂ δπ (cid:1) (cid:0) ∂ δπ (cid:1) , (3.47) δ π √G E ( G ) | π =0 ∼ (cid:2) δ π (cid:0) G − (cid:1) ∂ δ π (cid:0) G − (cid:1) + ∂ δ π (cid:0) G − (cid:1)(cid:3) π =0 ∼ (cid:0) ∂ δπ (cid:1) (cid:0) ∂ δπ (cid:1) , (3.48) δ π √G R ( G ) | π =0 ∼ ∂ (cid:0) δ π G − (cid:1) ∂ (cid:0) δ π G − (cid:1) | π =0 ∼ (cid:0) ∂ δπ (cid:1) (cid:0) ∂ δπ (cid:1) . (3.49) δ π √G E ( G ) | π =0 ∼ ∂ (cid:0) δ π G − (cid:1) ∂ (cid:0) δ π G − (cid:1) | π =0 ∼ (cid:0) ∂ δπ (cid:1) (cid:0) ∂ δπ (cid:1) . (3.50)Hence, no invariant ( δπ ) ∂ ( ∂π ) which is present in the ansatz (2.9) will be generated bythe re-expanding (3.39) in terms of the Galileon field. This pattern continues for invariantsinvolving higher powers of δπ and, in agreement with (3.17) and (3.28), implies β c i = 0 , i = 2 , , , . (3.51)Again the Galileon symmetry which leads to the particular structure of the inverse Galileonmetric is responsible for the structural relations (3.42)-(3.45), (3.46)-(3.50) and the vanishingbeta functions (3.51). According to (2.7), the Galileon couplings c i have negative mass dimension. Since the RGscale k has mass dimension [ k ] = 1 , we define the dimensionless Galileon couplings ˜ c i byrescaling the c i with the appropriate power of k , c i := k − i − ˜ c i . (4.1)The beta functions β ˜ c i := ˙˜ c i for the dimensional couplings ˜ c i are obtained by β c = ˙ c i = k∂ k c i = (cid:2) − i − c i + ˙˜ c i (cid:3) k − i − . (4.2) Each derivative w.r.t. π increases the number of fields by one. From the invariant line element √G onlystructures with two derivatives per field are generated (which follows from (3.42)-(3.45)), while additionalcurvature R ( G ) or endomorphism E ( G ) adds two additional derivatives. – 12 –ence, the Galileon RG system does not receive any quantum contributions and the onlycontribution to the dimensionless Galileon beta functions arise form the canonical mass scalingof the couplings c i , β ˜ c i = 3( i − c i . (4.3)A fixed point ˜ c ∗ i is defined by β ˜ c i (˜ c ∗ i ) = 0 . Consequently, the the Galileon RG system (4.3)only features the trivial Gaussian IR fixed point ˜ c ∗ i = 0 , i = 2 , , , . (4.4)This result is independent of the expansion scheme and choice of the regulator and a directconsequence of the Galileon symmetry (2.8). We have investigated the non-perturbative RG flow of the scalar Galileon in flat space withinthe ERGE. We have evaluated the functional trace in the Wetterich equation by variousexpansion schemes with different type of regulators, including on a heat-kernel techniquebased O ( ∂ ) derivative expansion, a vertex expansion in momentum space and a covariantformulation based on a geometric resummation of an infinite number of operators in terms ofcurvature tensors of an effective Galileon metric.Independently of the method and the type of regulator, we have found that the RG flowof the dimensionless Galileon couplings is solely driven by their classical mass dimension, thatis the beta functions do not receive any quantum-induced contributions. This leads to themain result that the only fixed point of the Galileon truncation is the trivial Gaussian IRfixed point.This result might be natural to expect from and is in agreement with the non-renorma-lization property of the Galileon found in the perturbative quantization [23, 26, 59–61]. How-ever, in contrast to the counterterms that arise in dimensional regulated perturbative calcula-tions, the functional trace in the Wetterich equation also induces operators which correspondto power-law divergences in perturbative one-loop calculations (which are annihilated by di-mensional regularization). Nevertheless, even these additional operators are not of the form ofthe operators present in the ansatz for the EAA (2.9) such that a projection of the functionaltrace to the Galileon truncation vanishes. As emphasized, the origin of this result is directlyconnected to the Galileon symmetry, which manifests itself in different ways in the derivativeexpansion, the momentum space vertex expansion and the geometric resummationIt would be interesting to investigate the non-perturbative RG flow of a general shift-symmetric higher-derivative scalar field theory which breaks the Galileon symmetry (2.8)within a derivative expansion of the ERGE . This would allow to compare the structure ofthe RG flow for less symmetric theory with the enhanced Galileon symmetry [76]. Besidebreaking the Galileon symmetry directly, a natural extension of the present work wouldbe the non-perturbative RG flow of the covariant Galileon in curved space. Due to thepresence of the gravitational interaction, the beta functions of the couplings in the resultinghigher-derivative scalar-tensor theory will receive quantum contributions and might featurea non-trivial interacting UV fixed point underlying the asymptotic safety scenario. In thiscontext, it would also be interesting to study the relation between the RG flow of this higher-derivative theories with that of scalar-tensor theories and geometric f ( R ) theories, whoserenormalization structure has been studied in perturbative one-loop calculations [77–83] andthe ERGE [84–96]. – 13 – cknowledgments I thank Roberto Percacci for many interesting conversations and constructive comments onthe first version of the draft. I also thank Omar Zanusso for helpful discussions.
A Derivative expansion: Coefficients
The explicit expressions for the coefficients in (3.16) were obtained by the
Mathematica tensor-algebra bundle xAct [97–100] and read C = Q (cid:104) W (1)(0 , , , (cid:105) , (A.1) C = 15 c Q (cid:104) W (3)(0 , , , (cid:105) , (A.2) C = − c Q (cid:104) W (4)(1 , , , (cid:105) + 60 c Q (cid:104) W (5)(2 , , , (cid:105) − c Q (cid:104) W (4)(0 , , , (cid:105) , (A.3) C = 15 c Q (cid:104) W (5)(2 , , , (cid:105) − c Q (cid:104) W (4)(0 , , , (cid:105) − c Q (cid:104) W (6)(3 , , , (cid:105) + 180 c Q (cid:104) W (5)(1 , , , (cid:105) − c Q (cid:104) W (4)(0 , , , (cid:105) + 720 c Q (cid:104) W (7)(4 , , , (cid:105) − c Q (cid:104) W (6)(2 , , , (cid:105) + 120 c Q (cid:104) W (5)(1 , , , (cid:105) − c Q (cid:104) W (6)(2 , , , (cid:105) + 180 c Q (cid:104) W (5)(0 , , , (cid:105) + 120 c Q (cid:104) W (5)(1 , , , (cid:105) − c Q (cid:104) W (4)(0 , , , (cid:105) , (A.4) C a = 120 c c Q (cid:104) W (3)(0 , , , (cid:105) + 510 c Q (cid:104) W (4)(0 , , , (cid:105) , (A.5) C b = 60 c c Q (cid:104) W (3)(0 , , , (cid:105) + 150 c Q (cid:104) W (4)(0 , , , (cid:105) , (A.6) C a = 93 c Q (cid:104) W (4)(0 , , , (cid:105) − c c Q (cid:104) W (4)(1 , , , (cid:105) − c Q (cid:104) W (5)(1 , , , (cid:105) + 720 c c Q (cid:104) W (5)(2 , , , (cid:105) − c c Q (cid:104) W (4)(0 , , , (cid:105) + 10760 c Q (cid:104) W (6)(2 , , , (cid:105) − c Q (cid:104) W (5)(0 , , , (cid:105) , (A.7) C b = 39 c Q (cid:104) W (4)(0 , , , (cid:105) − c c Q (cid:104) W (4)(1 , , , (cid:105) − c Q (cid:104) W (5)(1 , , , (cid:105) + 240 c c Q (cid:104) W (5)(2 , , , (cid:105) − c c Q (cid:104) W (4)(0 , , , (cid:105) + 5280 c Q (cid:104) W (6)(2 , , , (cid:105) − c Q (cid:104) W (5)(0 , , , (cid:105) , (A.8) C c = 30 c Q (cid:104) W (4)(0 , , , (cid:105) − c c Q (cid:104) W (4)(1 , , , (cid:105) − c Q (cid:104) W (5)(1 , , , (cid:105) + 480 c c Q (cid:104) W (5)(2 , , , (cid:105) − c c Q (cid:104) W (4)(0 , , , (cid:105) + 4600 c Q (cid:104) W (6)(2 , , , (cid:105) − c Q (cid:104) W (5)(0 , , , (cid:105) , (A.9) C a = 36 c Q (cid:104) W (3)(0 , , , (cid:105) + 594 c c Q (cid:104) W (4)(0 , , , (cid:105) + 2067 c Q (cid:104) W (5)(0 , , , (cid:105) , (A.10) C b = 24(3 c + 5 c c ) Q (cid:104) W (3)(0 , , , (cid:105) + 1098 c c Q (cid:104) W (4)(0 , , , (cid:105) + 2694 c Q (cid:104) W (5)(0 , , , (cid:105) (A.11) C c = 12(9 c + 10 c c ) Q (cid:104) W (3)(0 , , , (cid:105) + 2322 c c Q (cid:104) W (4)(0 , , , (cid:105) + 6756 c Q (cid:104) W (5)(0 , , , (cid:105) , (A.12)– 14 – d = 12(9 c + 10 c c ) Q (cid:104) W (3)(0 , , , (cid:105) + 2322 c c Q (cid:104) W (4)(0 , , , (cid:105) + 6756 c Q (cid:104) W (5)(0 , , , (cid:105) , (A.13) C e = 36(9 c + 10 c c ) Q (cid:104) W (3)(0 , , , (cid:105) + 5076 c c Q (cid:104) W (4)(0 , , , (cid:105) + 11448 c Q (cid:104) W (5)(0 , , , (cid:105) , (A.14) C f = 24(9 c + 10 c c ) Q (cid:104) W (3)(0 , , , (cid:105) + 4644 c c Q (cid:104) W (4)(0 , , , (cid:105) + 13512 c Q (cid:104) W (5)(0 , , , (cid:105) . (A.15) References [1] G. R. Dvali, G. Gabadadze and M. Porrati, , Phys. Lett.
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