1 Introduction

IFUP-TH 2007/18

Renormalization

Of Lorentz Violating Theories

Damiano Anselmi and Milenko Halat

Dipartimento di Fisica “Enrico Fermi”, Università di Pisa,

Largo Pontecorvo 3, I-56127 Pisa, Italy,

and INFN, Sezione di Pisa, Pisa, Italy

damiano.anselmi@df.unipi.it, milenko.halat@df.unipi.it

Abstract

We classify the unitary, renormalizable, Lorentz violating quantum field theories of interacting scalars and fermions, obtained improving the behavior of Feynman diagrams by means of higher space derivatives. Higher time derivatives are not generated by renormalization. Renormalizability is ensured by a “weighted power counting” criterion. The theories contain a dimensionful parameter , yet a set of models are classically invariant under a weighted scale transformation, which is anomalous at the quantum level. Formulas for the weighted trace anomaly are derived. The renormalization-group properties are studied.

## 1 Introduction

The set of power-counting renormalizable theories is considerably restricted by the assumptions of unitarity, locality, causality and Lorentz invariance. If we relax one or some of these assumptions we can enlarge the set of renormalizable theories. However, usually the enlargement is too wide. For example, there exist an infinite set of renormalizable nonunitary theories. Improving the behavior of propagators at large momenta with the help of higher-derivative kinetic terms [1] it is possible to define a renormalizable higher-derivative version of every theory, including gravity [2]. Relaxing locality can in principle make every theory renormalizable, smoothing away the small distance singularities that originate the UV divergences [3]. Unitarity violations due to higher derivatives can in some cases be traded for causality violations [4, 5].

The purpose of this paper is to investigate the issue of renormalizability in the presence of Lorentz violations, while preserving both locality and unitarity. The UV behavior of propagators is improved with the help of higher space derivatives. It is proved that, under certain conditions, renormalization does not turn on terms with higher time derivatives, thus preserving unitarity. Renormalizability follows from a modified power-counting criterion, which weights time and space differently. The set of consistent theories is still very restricted, yet considerably larger than the set of Lorentz invariant theories. Renormalizable models exist in arbitrary spacetime dimensions.

The quadratic terms that contain higher space derivatives, as well as certain vertices, are multiplied by inverse powers of a scale . Despite the presence of the dimensionful parameter certain models have a weighted scale invariance, which is anomalous at the quantum level. The weighted trace anomaly is worked out explicitly.

In this paper we concentrate on scalar and fermion theories, leaving the study of gauge theories and gravity to separate publications. Lorentz violating models with higher space derivatives might be useful to define the ultraviolet limit of theories that are otherwise nonrenormalizable, including quantum gravity, and allow to remove the divergences with a finite number of independent couplings. Other domains where the models of this paper might find applications are Lorentz violating extensions of the Standard Model [6], effective field theory [7], renormalization-group (RG) methods for the search of asymptotically safe fixed points [8], nonrelativistic quantum field theory for nuclear physics [9], condensed matter physics and the theory of critical phenomena [10]. Certain -models that fall in our class of renormalizable theories are useful to describe the critical behavior at Lifshitz points [11] and have been widely studied in that context [12], with a variety of applications to real physical systems. Effects of Lorentz and CPT violations on stability and microcausality have been studied [13], as well as the induction of Lorentz violations by the radiative corrections [14]. The renormalization of gauge theories containing Lorentz violating terms has been studied in [15]. For a recent review on astrophysical constraints on the Lorentz violation at high energy see ref. [16].

The paper is organized as follows. In section 2 we study the renormalizability of scalar theories, while in section 3 we include the fermions. In section 4 we analyze the divergent parts of Feynman diagrams and their subtractions. We prove the locality of counterterms and study the renormalization algorithm to all orders. The one-loop divergences are computed explicitly. In section 5 we analyze the renormalization structure and the renormalization group. In section 6 we study the energy-momentum tensor, the weighted scale invariance and the weighted trace anomaly. In section 7 we generalize our results to nonrelativistic theories. Section 8 contains the conclusions. In the appendices we collect more observations about the cancellation of subdivergences and the locality of counterterms, and some expressions of Euclidean propagators in coordinate space.

### Preliminaries.

We use the dimensional-regularization technique whenever possible. Since the analysis of divergences is the same in the Euclidean and Minkowskian frameworks, we write our formulas directly in the Euclidean framework, which is more explicit. Yet, with an abuse of language, we still speak of “Lorentz symmetry”, since no confusion is expected to arise.

We first consider models where the -dimensional spacetime manifold is split into the product of two submanifolds, a -dimensional submanifold , containing time and possibly some space coordinates, and a -dimensional space submanifold . Lorentz and rotational symmetries in the two submanifolds are assumed. This kind of splitting could be useful to describe specific physical situations (for example the presence of a non isotropic medium in condensed matter physics), but here it is mainly used as a starting point to illustrate our arguments in concrete examples. Indeed, most Lorentz violating theories contain a huge number of independent vertices, so it is convenient to begin with models where unnecessary complicacies are reduced to a minimum. The extension of our construction to the most general case, which is rather simple, will be described later. In the same spirit, a number of discrete symmetries, such as parity, time reversal, , etc., are often assumed.

To apply the dimensional-regularization technique, both submanifolds have to be continued independently. The total continued spacetime manifold is therefore split into the product , where and are complex and . Each momentum is split into “first” components , which live in , and “second” components , which live in : . The spacetime index is split into hatted and barred indices: . Notations such as , and refer to the same object, as well as , , . Frequently, Latin letters are used for the indices of the barred components of momenta. Finally, .

We say that is a weighted polynomial in and , of degree and weight , where is a multiple of , if is a polynomial of degree in . Clearly,

 Pk1,n(ˆp,¯¯¯p)Pk2,n(ˆp,¯¯¯p)=Pk1+k2,n(ˆp,¯¯¯p).

We say that is a homogeneous weighted polynomial in and , of degree and weight , if . It is straightforward to prove that a weighted polynomial of degree can be expressed as a linear combination of homogeneous weighted polynomials of degrees .

## 2 Renormalizability by weighted power counting

In this section we classify the renormalizable Lorentz violating scalar field theories that can be constructed with the help of quadratic terms containing higher space derivatives and prove that renormalization does not generate higher time derivatives.

Consider a generic scalar field theory with a propagator defined by the quadratic terms

 Lfree=12(ˆ∂φ)2+12Λ2n−2L(¯¯¯∂nφ)2, (2.1)

where is an energy scale. Up to total derivatives it is not necessary to specify how the derivatives contract among themselves. The of (2.1) should be understood as the highest power of that appears in the quadratic terms of the total lagrangian. Other quadratic terms of the form

 am2Λ2m−2L(¯¯¯∂mφ)2,m

could be present, or generated by renormalization. They are weighted monomials of degrees and weight . For the purposes of renormalization, it is convenient to consider such terms as “interactions” (two-leg vertices) and treat them perturbatively. Indeed, the counterterms depend polynomially on the parameters , because when the integral associated with a graph is differentiated a sufficient number of times with respect to the ’s it becomes overall convergent. The -polynomiality of counterterms generalizes the usual polynomiality in the masses. Thus we can assume that the propagator is defined by (2.1) and treat every other term as a vertex. Then the propagator is the inverse of a weighted homogeneous polynomial of degree and weight . The coefficient of the term must be positive, to have an action bounded from below in the Euclidean framework or, equivalently, an energy bounded from below in the Minkowskian framework.

Label the vertices that have -legs with indices , to distinguish different derivative structures. Each vertex of type defines a monomial in the momenta of the fields. Denote the weighted degree of such a monomial by . A vertex with derivatives , derivatives and -legs is symbolically written as

 [ˆ∂p1¯¯¯∂p2φN]α

and its weighted degree is

 δ(α)N=p1+p2n.

Consider a Feynman graph made of loops, external legs, internal legs and vertices of type . The integral associated with has the form

 IG(k)=∫dLˆDˆp(2π)LˆD∫dL¯¯¯¯D¯¯¯p(2π)L¯¯¯¯DI∏i=1P(i)−2,n(p,k)V∏j=1V(j)δj,n(p,k),

where are the loop momenta, are the external momenta, are the propagators, which have weighted degree , and are the vertices, with weighted degrees . The integral measure is a weighted measure of degree Ð. Performing a rescaling , accompanied by an analogous change of variables , it is straightforward to prove that is a weighted function of degree

 L\it\DH−2I+V∑j=1δj=L\it\DH−2I+∑(N,α)δ(α)Nv(α)N.

By the locality of counterterms, once the subdivergences of have been inductively subtracted away, the overall divergent part of is a weighted polynomial of degree

 ω(G)=L\it\dj−2I+∑(N,α)δ(α)Nv(α)N

in the external momenta, where đ. The usual relations

 L=I−V+1,E+2I=∑(N,α)Nv(α)N, (2.3)

allow us to write

 ω(G)=d(E)+∑(N,α)v(α)N[δ(α)N−d(N)], (2.4)

where

 d(X)≡\it\dj(1−X2)+X ; (2.5)

The theory is ) renormalizable, if it contains all vertices with , and only those: does not increase when the number of vertices increases; ) super-renormalizable, if it contains all vertices with , and only those: decreases when the number of vertices increases; ) strictly-renormalizable, if it contains all vertices with , and only those: does not depend on ; ) nonrenormalizable, if it contains some vertices with : increases when the number of those vertices increases.

The vertices with are called “weighted marginal”, those with are called “weighted relevant” and those with are called “weighted irrelevant”.

By locality, cannot be negative. Moreover, polynomiality demands that there must exist a bound on the number of legs that the vertices can contain. It is easy to show that these requirements are fulfilled if and only if

 \it\dj>2 (2.6)

and the bound is

 Nmax=[2\it\dj\it\dj−2], (2.7)

where denotes the integral part of . The existence of nontrivial interactions () requires , while the existence of nontrivial even interactions () requires .

To complete the proof of renormalizability, observe that when the weighted degree of divergence of a graph satisfies

 ω(G)≤d(E). (2.8)

The inequality (2.6) ensures also that decreases when the number of external legs increases. Finally, since the vertices that subtract the overall divergences of are of type with , it is straightforward to check that the lagrangian contains all needed vertices. Indeed, (2.8) coincides with the inequality satisfied by .

Now we prove that the renormalizable models just constructed are perturbatively unitary, in particular that no higher time derivatives are present, both in the kinetic part and in the vertices, and no higher time derivatives are generated by renormalization. Indeed, a lagrangian term with higher time derivatives would have for or (terms with need not be considered, since they cannot contain derivatives). This cannot happen in a renormalizable theory, because (2.6) and imply in general and for . In particular, true vertices () cannot contain any -derivative at all, because invariance under the reduced Lorentz and rotational symmetries of and exclude also terms containing an odd number of ’s or an odd number of ’s. Similar conclusions apply to the counterterms, because of (2.8). Therefore, renormalization does not turn on higher time derivatives, as promised.

### Weighted scale invariance.

The strictly renormalizable models have the . Their lagrangian has the form

 (2.9)

Here denotes a basis of lagrangian terms constructed with fields and -derivatives acting on them, contracted in all independent ways, and are dimensionless couplings.

In the physical spacetime dimension (the continuation to complex dimensions will be discussed later) the classical theories with lagrangians are invariant under the weighted dilatation

 ^x→^x e−Ω,¯x→¯x e−Ω/n,φ→φ eΩ(\it\dj/2−1), (2.10)

where is a constant parameter. Each lagrangian term scales with the factor đ, compensated by the scaling factor of the integration measure d of the action.

We call the models (2.9) homogeneous. Homogeneity is preserved by renormalization, namely there exists a subtraction scheme in which no lagrangian terms of weighted degrees smaller than are turned on by renormalization. This fact is evident using the dimensional-regularization technique. Indeed, when , the equality in (2.8) holds, so .

The weighted scale invariance (2.10) is anomalous at the quantum level. The weighted trace anomaly and its relation with the renormalization group are studied in section 6.

Nonhomogeneous theories are those that contain both weighted marginal and weighted relevant vertices. In these cases the weighted dilatation (2.10) is explicitly broken by the super-renormalizable vertices, and dynamically broken by the anomaly.

Let us analyze some explicit examples, starting from the homogeneous models.

### Homogeneous models.

We begin with the -theories. Setting in (2.7) we get

 103<\it\dj≤4. (2.11)

One solution with đ is the usual Lorentz-invariant -theory in four dimensions (, ). A simple Lorentz-violating solution is the model with described by the lagrangian

 L(2,4)=12(ˆ∂φ)2+12Λ2L(¯¯¯¯¯△φ)2+λ4!Λ2Lφ4. (2.12)

in six dimensions, with , . The -theories with are used to describe the critical behavior at Lifshitz points [11, 12].

It is clear that (2.11) admits infinitely many solutions for each value of đ. For example, given a solution, such as (2.12), infinitely many others are obtained multiplying and by a common integer factor. For đ we have the family of -dimensional theories

 L(2,2n)=12(ˆ∂φ)2+12Λ2(n−1)L(¯¯¯∂nφ)2+λ4!Λ2(n−1)Lφ4. (2.13)

In general, for every Lorentz-invariant renormalizable theory there exists an infinite family of Lorentz-violating renormalizable theories.

Let us now concentrate on four dimensions. The spacetime manifold can be split as , , , , . There is no nontrivial solution with . Indeed, (2.7) implies

 Nmax=[42−n],

so can only be 1, which gives back the Lorentz invariant -theory. For we get

 Nmax=[2(n+3)3−n].

The only nontrivial solution is , which implies and

 L(1,3)=12(ˆ∂φ)2+12Λ2L(¯¯¯¯¯△φ)2+λ66!Λ4Lφ4(¯¯¯∂φ)2+λ1010!Λ6Lφ10. (2.14)

For we get : every integer defines a nontrivial solution in this case. The simplest example is , . Listing all allowed vertices we get the theory

 L(2,2)=12(ˆ∂φ)2+12Λ2L(¯¯¯¯¯△φ)2+λ44!Λ2Lφ2(¯¯¯∂φ)2+λ66!Λ2Lφ6. (2.15)

This model belongs to a family of đ, -dimensional -theories, whose lagrangian is

 L(2,n)=12(ˆ∂φ)2+12Λ2(n−1)L(¯¯¯∂nφ)2+λ66!Λ2(n−1)Lφ6, (2.16)

when is odd and

 L(2,n)=12(ˆ∂φ)2+12Λ2(n−1)L(¯¯¯∂nφ)2+14!Λ2(n−1)L∑αλα[¯¯¯∂nφ4]α+λ66!Λ2(n−1)Lφ6, (2.17)

when is even. Observe that (2.16) includes the Lorentz-invariant -theory in three spacetime dimensions, which is the case .

For we get

 Nmax=[2(3n+1)n+1].

The solution with has . However, this solution is trivial, since its unique vertex would have just one -derivative. Instead, for every , is equal to . For example, the theory with is

 L(3,1)=12(ˆ∂φ)2+12Λ4L(¯¯¯∂¯¯¯¯¯△φ)2+λ′33!Λ3Lφ2¯¯¯¯¯△2φ+λ33!Λ3Lφ(¯¯¯¯¯△φ)2+λ44!Λ2Lφ2(¯¯¯∂φ)2+λ55!ΛLφ5,

which is clearly unstable. Imposing the symmetry we have the modified -theory

 Leven(3,1)=12(ˆ∂φ)2+12Λ4L(¯¯¯∂¯¯¯¯¯△φ)2+λ44!Λ2Lφ2(¯¯¯∂φ)2,

which is stable for . Finally, for we get again the Lorentz-invariant -theory.

### Nonhomogeneous models.

Nonhomogeneous theories can be obtained from the homogeneous ones adding all super-renormalizable terms, which are those that satisfy the strict inequality . For example, keeping the symmetry , the nonhomogeneous extension of (2.12) is just

 Lnh(2,4)=12(ˆ∂φ)2+a2(¯¯¯∂φ)2+m22φ2+12Λ2L(¯¯¯¯¯△φ)2+λ4!Λ2Lφ4

and the one of (2.15) is

 Lnh(2,2)=12(ˆ∂φ)2+a2(¯¯¯∂φ)2+m22φ2+12Λ2L(¯¯¯¯¯△φ)2+λ44!Λ2Lφ2(¯¯¯∂φ)2+λ′44!φ4+λ66!Λ2Lφ6.

### Splitting the spacetime manifold into the product of more submanifolds.

Instead of splitting the spacetime manifold into two submanifolds, we can split it into the product of more submanifolds, eventually one for each coordinate. This analysis covers the most general case. We still need to distinguish a -dimensional submanifold containing time from the -dimensional space submanifolds , , so we write

 Md=Mˆd⊗ℓ∏i=1M¯¯¯di.

Denote the space derivatives in the th space subsector with and assume that they have weights . Then the kinetic term of the lagrangian reads

 Lkin=12(ˆ∂φ)2+12φP2(¯¯¯∂i,ΛL)φ,

where is the most general weighted homogeneous polynomial of degree 2 in the spatial derivatives, , invariant under rotations in the subspaces . The -dependence is arranged so that has dimensionality 2. The previous analysis can be repeated straightforwardly. It is easy to verify that the weighted power-counting criterion works as before with

 \it\dj=ˆd+ℓ∑i=1¯¯¯dini.

### Edge renormalizability.

By edge renormalizable theories we mean theories where renormalization preserves the derivative structure of the lagrangian, but the powers of the fields are unrestricted. With scalars and fermions, such theories contain arbitrary functions of the fields and therefore infinitely many independent couplings. The notion of edge renormalizability is interesting in the perspective to study gravity. Indeed, Einstein gravity is an example of theory where all vertices have the same number of derivatives, but are nonpolynomial in the fluctuation around flat space. Yet, diffeomorphism invariance ensures that the number of invariants with a given dimensionality in units of mass is finite. Therefore, in quantum gravity a polynomial derivative structure is sufficient to reduce the arbitrariness to a finite set of independent couplings.

Edge renormalizable theories are those where does not decrease when increases, rather it is independent of . By formula (2.8) this means đ (), in which case is always equal to 2. Since đ, can be either or . The theories with contain higher time derivatives, so they are not unitary. Thus we must take . The homogeneous theory in four dimensions has lagrangian

 L=Lfree+LI, (2.18)

where

 Lfree=12(ˆ∂φ)2+12Λ4L(¯¯¯∂¯¯¯¯¯△φ)2

and

 LI=V1(φ)(ˆ∂φ)2+ V2(φ)[(∂iφ)2]3+V3(φ)¯¯¯¯¯△φ(∂iφ)2(∂jφ)2+V4(φ)(∂i∂jφ)(∂i∂j¯¯¯¯¯△φ) +V5(φ)¯¯¯¯¯△2φ(∂iφ)2+ V6(φ)(¯¯¯¯¯△φ)3+V7(φ)(∂i¯¯¯¯¯△φ)2+V8(φ)(∂i∂j∂kφ)2+V9(φ)¯¯¯¯¯△3φ, (2.19)

where the ’s are unspecified functions of with , .

The lagrangian of the most general nonhomogeneous theory is (2.18) with

 Lfree=12(ˆ∂φ)2−12φ⎛⎝a¯¯¯¯¯△+b¯¯¯¯¯△2Λ2L+¯¯¯¯¯△3Λ4L⎞⎠φ

and equal to (2.19) plus

 V10(φ)+V11(φ)¯¯¯¯¯△φ+V12(φ)¯¯¯¯¯△2φ+V13(φ)(¯¯¯¯¯△φ)2+V14(φ)[(∂iφ)2]2,

with , .

## 3 Inclusion of fermions

In this section we classify the models of interacting fermions and scalars. We start from pure fermionic theories, with quadratic lagrangian

 Lfree=¯¯¯¯ψˆ∂/ψ+1Λn−1L¯¯¯¯ψ¯¯¯∂/nψ,

where is the maximal number of -derivatives. The propagator

 −iˆp/+(−i)n¯¯p/nΛn−1Lˆp2+(¯¯p2)nΛ2n−2L,

is, in momentum space, a weighted function of degree 1. The loop-integral measure is, as usual, a weighted measure of degree đ. For the purposes of renormalization, the kinetic terms with fewer than -derivatives can be treated as vertices.

Label the vertices that have --legs by means of indices and denote their weighted degree with . Consider a diagram with external --legs, constructed with vertices of type . Once the subdivergences have been subtracted away, its overall divergence is a weighted polynomial of degree

 ω(G)=\it\dj−E(\it\dj−1)+∑(N,α)v(α)N[δ(α)N−\it\dj(1−N)−N]

in the external momenta. Renormalizability demands

 δ(α)N≤\it\dj(1−N)+N≡dF(N). (3.1)

Polynomiality demands

 \it\dj>1,

in which case the maximal number of external --legs is

 Nmax=[\it\dj\it\dj−1].

Pure fermionic homogeneous models have strictly renormalizable vertices, namely those with . Their lagrangian has the form

 L=¯¯¯¯ψˆ∂/ψ+1Λn−1L¯¯¯¯ψ¯¯¯∂/nψ+∑(N,α)λ(N,α)(N!)2Λ(n−1)(N−ˆd−Nˆd)L[¯¯¯∂ndF(N)¯¯¯¯ψNψN]α.

Here denotes a basis of lagrangian terms constructed with fields , fields and -derivatives, invariant under the reduced Lorentz symmetry. For simplicity, we can assume also invariance under parities in both portions of spacetime.

Let us concentrate on four spacetime dimensions. The Lorentz split gives , which admits infinitely many nontrivial solutions, beginning from . For example, the and theories read

 L(1,3) = ¯¯¯¯ψˆ∂/ψ+1Λ2L¯¯¯¯ψ¯¯¯¯¯Δ¯¯¯∂/ψ+∑αλαΛ2L[¯¯¯¯ψ2ψ2]α, L′(1,3) = ¯¯¯¯ψˆ∂/ψ+1Λ5L¯¯¯¯ψ ¯¯¯¯¯Δ3ψ+∑αλαΛ5L[¯¯¯∂3¯¯¯¯ψ2ψ2]α+∑αλ′αΛ5L[¯¯¯¯ψ3ψ3]α,

respectively. The Lorentz splits and do not admit nontrivial solutions, since in those cases.

Now we study the models containing coupled scalars and fermions. It is important to note that when different types of fields are involved, they must have the same . We classify the vertices with labels , where is the number of --legs, is the number of -legs and is an extra label that distinguishes vertices with different structures. Call the weighted degree of the -th vertex. Consider a diagram with external --legs, external -legs and vertices of type . Once the subdivergences have been subtracted away, the overall divergent part of a is a weighted polynomial of degree

 ω(G) = \it\dj−Eψ(\it\dj−1)−Eφ2(\it\dj−2) +∑(Nψ,Nφ,α)v(α)(Nψ,Nφ)[δ(α)(Nψ,Nφ)−\it\dj(1−Nψ−Nφ2)−Nψ−Nφ].

in the external momenta. Renormalizability demands

 δ(α)(Nψ,Nφ)≤\it\dj(1−Nψ−Nφ2)+Nψ+Nφ≡d(Nψ,Nφ).

Because is nonnegative, the numbers of fermionic and bosonic legs are bound by the inequality

 Nψ(\it\dj−1)+Nφ2(\it\dj−2)≤\it% \dj.

Polynomiality demands đ.

The homogeneous models have a lagrangian of the form

 L = ¯¯¯¯ψˆ∂/ψ+ηΛn−1L¯¯¯¯ψ¯¯¯∂/nψ+12(ˆ∂φ)2+12Λ2n−2L(¯¯¯∂nφ)2 +∑(Nψ,Nφ,α)λ(Nψ,Nφ,α)Nφ!(Nψ!)2Λ(n−1)(Nφ+Nψ+ˆd−ˆdNψ−ˆdNφ/2)L[¯¯¯∂nd(Nψ,Nφ)¯¯¯¯ψNψψNψφNφ]α.

In four dimensions the splitting has a unique nontrivial solution, which is the model (2.14) coupled to fermions. It has and its lagrangian reads

 L(1,3) = ¯¯¯¯ψˆ∂/ψ+ηΛL¯¯¯¯ψ¯¯¯¯¯Δψ+12(ˆ∂φ)2+12Λ2L(¯¯¯¯¯Δφ)2+λ22Λ2Lφ2(¯¯¯¯ψ↔¯¯¯∂/ψ)+λ′22Λ2Lφ2¯¯¯∂⋅(¯¯¯¯ψ¯¯¯γψ) +λ44!Λ3Lφ4¯¯¯¯ψψ+λ66!Λ4Lφ4(¯¯¯∂φ)2+λ1010!Λ6Lφ10.

The splitting admits infinitely many solutions. The simplest one is the theory with , symmetric under , that couples (2.15) to fermions:

 L(2,2)=¯¯¯¯ψˆ∂/ψ+ηΛL¯¯¯¯ψ¯¯¯¯¯Δψ+12(ˆ∂φ)2+12Λ2L(¯¯¯¯¯Δφ)2+λ22ΛLφ2¯¯¯¯ψψ+λ44!Λ2Lφ2(¯¯¯∂φ)2+λ66!Λ2Lφ6,

The splitting admits, again, infinitely many solutions.

## 4 Renormalization

In this section we study the structure of Feynman diagrams, their divergences and subdivergences, and the locality of counterterms. For definiteness, we work with scalar fields, but the conclusions are general.

### One-loop.

Consider the most general one-loop Feynman diagram , with external legs, internal legs and vertices of type and weighted degree . Collectively denote the external momenta by . The divergent part of can be calculated expanding the integral in powers of . We obtain a linear combination of contributions of the form

 I(I,n)μ1⋯μ2r|j1⋯j2sˆkν1⋯ˆkνu ¯¯¯ki1⋯¯¯¯kiv, (4.1)

where

 I(I,n)μ1⋯μ2r|j1⋯j2s=∫dˆDˆp(2π)ˆD∫d¯¯¯¯D¯¯¯p(2π)¯¯¯¯Dˆpμ1⋯