Renormalization group equations as 'decoupling' theorems

We propose a simple derivation of renormalization group equations and Callan-Symanzik equations as decoupling theorems of the structures underlying effective field theories.


INTRODUCTION
It is well known that any known quantum field theory could at best be an effective field theory (EFT) that include ingredients as many as possible for the description of certain phenomena. Various divergences (UV, IR and other types) arise in an EFT simply due to its simplification made at the scales widely separated from the dominating ones. In other words, an EFT is usually not a complete framework for accounting for the quantum fluctuations over any distances, it fails for certain modes underlying the 'effective ones'.
The conventional ways to deal with the divergences within EFT framework are renormalization (for UV divergences) and factorization (for IR and/or collinear divergences): The true non-calculable contributions from the underlying structures could be separated and put into some EFT ingredients (operators, couplings or matrix elements, etc.) that could be determined later. Such tricky procedures lead to the celebrated renormalization group equations (RGE). However, in the conventional treatment, an artificial regularization (as simple and temporary substitutes for the true underlying structures) must be employed in the course of calculations and the associated tricky procedures seem to obscure the physical rationality behind the treatments, though the final results should be independent of such regularization procedures.
In this presentation, I wish to give a rather simple (to some extent, less rigorous) derivation of RGE and the Callan-Symanzik equation, solely based on the simple picture that there are some elaborate details underlying the effective theories in use for certain purpose. In other words, the scale hierarchy substantiating the EFT construction is the only starting point. It is hoped that such simple lines of arguments could help the further exploration of the more fundamental implications of the EFT reconstructions as well as various factorization approaches that prevail in physics literature.

EFFECTIVE VERSUS UNDERLYING STRUCTURES
To proceed, we take the existence of a well-defined formulation of the theory underlying (UT) the EFT's as a natural fact or postulate. In fact, we could take the well defined formulation of an EFT as a result of reconstruction or projection procedure from the related sectors in the complete UT. The hierarchy between the scales in the formulation discriminates the parameters into effective and underlying ones, respectively. Such hierarchy should automatically facilitate a well defined expansion in terms of the ratios like Λ EF T /Λ UT (for UV underlying structures) or Λ UT /Λ EF T (for IR or other non-UV underlying structures), so that the resulting formulation is expressed in terms of the effective parameters and some possible 'agent' constants (only arise in the loop contributions) in the limit that the underlying details are apparently 'decoupled'. No unphysical divergences should appear in the course of such reconstruction or projection procedure.
For convenience, we introduce the symbolP EFT to assume all the elaborate procedures of the reconstruction or projection of an EFT out of a complete underlying theory, which should contain at least the following three operations: (1) projecting into the subspace of EFT; (2) averaging over the associated underlying dynamical processes (integrating out); (3) taking the decoupling limits with respect to the typical underlying constants, {σ}. With the help of this projection symbol, we could easily identify the technical origin of various divergences in EFT. To see this point, we employ the path integral formalism.
According to the above arguments, a well defined generating functional for an EFT should be obtained from the EFT projection on the generating functional for UT,P EFT .
with [g] being the 'effective' parameters. That means, the path integral should be performed in the presence of the underlying structures. If one perform the projection first before the path integral, an ill-defined EFT is resulted, Thus, the appearance of various divergences in an EFT implies that path integral and the EFT projection P EFT do not commute. In conventional approaches, the role of the sophisticated projection procedureP EFT is played by the R operation procedure in the well known BPZH program when there is only UV divergences. That is, R operation could be seen as one tricky realization of the projection operationP EFT (in the case of UV divergences alone), which could in turn be put into the forms with multiplicative renormalization,

Now let us consider a general vertex function
denoting the external momenta and the Lagrangian couplings (including masses) in an EFT and {σ} denoting the underlying parameters or constants. Now it is easy to see that such a vertex function must be a homogeneous function of all its dimensional arguments, that is where d ··· refers to the canonical mass dimension of any parameters involved. The corresponding equation in EFT could be obtained through the application of the projectionP EFT to both sides of Eq. (4), Here some constants {c} must appear as the agents of {σ} to maintain the dimension balance between {σ} and the EFT couplings and masses. Note that {c} only appear in the loop diagrams of EFT. The differential form for Eq.(4) reads Since d g g∂ g = −iΘ (the trace of the stress tensor), the alternative form of Eq.(6) reads, Obviously only dimensional constants contribute to the scaling behavior. Eq. (6) or Eq. (7) is just the most general UT version of the EFT scaling laws. They differ from naive EFT scaling laws only by the canonical contribution from the underlying structures ( d σ σ∂ σ ). This is just the origin of EFT scaling anomalies. To see this point, we first note the consequence of applyingP EFT to d σ σ∂ σ , Then, it is straightforward to see that, within EFT, dcc∂c has to be expanded into the insertion of appropriate EFT operators ([I Oi ], 'elementary' or composite) with appropriate coefficients (δ Oi ). Thus, we arrive at the following decoupling theorem, Note that each δ Oi must at least be a function of EFT couplings [g] and {c}. At present stage, we do not exclude nonlocal operators from the set of [O i ], which might be more relevant in the presence of IR or other non-UV divergences. Thus Eq. (9) is a rather general form of 'decoupling theorem' for any sort of underlying structures, UV or IR.
As the final step, it is easy to classify these operators into the kinetic operators (for the EFT fields [φ]), the coupling operators (with couplings [g]), and 'composite' ones, [O N ], that do not appear in the EFT lagrangian, Now with Eqs. (8,10) we can turn the primary decoupling theorem in Eq. (9) and the full scaling law in Eq. (6) into the following forms, Here Eqs. (11,12) are only true for the complete sum of all graphs (or up to a certain order). It is obvious that the canonical contributions from the underlying structures become the anomalies in terms of EFT parameters. δ Oi is just the anomalous dimension for the operator O i in EFT. Again the operators contributing to the scaling anomalies should contain the ones corresponding to IR or other non-UV singularities.

NOVEL PERSPECTIVE OF RGE AND CALLAN-SYMANZIK EQUATION
In this section we limit our attention to a special type of theories beset only with UV divergences: the ones without the scaling anomalies {ON } δ ON ([g]; {c})Î ON , i.e., the renormalizable theories in conventional terminology, and all the operators are now local. Then, Eqs. (11) and (12) become simpler, These equations just correspond to the usual RGE and Callan-Symanzik equation (CSE) for renormalizable theories. We could turn these equations into more familiar forms. For this purpose, we note that all the agent constants could be parametrized in terms of a single scaleμ and a series dimensionless ones (c 0 ). In the conventional programs, they are first predetermined through renormalization conditions, finally transformed into the physical parameters [1] or fixed somehow [2].

RGE and CSE as decoupling theorems
In Eq. (13) and (14) only the insertion of kinetic operators appears unfamiliar. To remove this unfamiliarity, let us note that φ δ φÎφ lead to the following consequences [3]: with n g;φ and n g;ψ being respectively the number of bosonic and fermionic field operators contained in the vertex with coupling g.
Then Eq. (13) and (14) take the following forms: with D g ≡δ g + d g . Eqs.(16,17) take the familiar forms of RGE and CSE. Here, all the constants (finite!) survive or arise from the 'decoupling' limit implied by the projection procedure. Thus, we could naturally interpret these equations as 'decoupling' theorems for the canonical scaling laws with underlying structures. The RGE and CSE for the generating functional read, with D denoting the spacetime dimension. We note that the operator trace anomalies [4] could be readily read from Eq. ( 19), The right hand side could be further simplified after using motion equations.

RGE and CSE for composite operators
For the vertex functions in the presence of composite operators, the only complication lies in the contribution from the related composite operators (more could show up than those in the vertex functions), that is the general cases in Eqs. (11,12), Here Underlying structures and the notion of renormalization As remarked above, the EFT parameters [g] and the UT agents {c} should be 'derived' from UT (they could not be derived from EFT!). Thus, in EFT, they have to be determined or fixed somehow through physical boundaries or data [1] or through sensible procedures [2]. Now to see the origin of the notion of renormalization, we solve the equation for scaling law, Eq.(17). This could be conveniently achieved through the introduction of 'running' parameters [ḡ(λ)] for [g] based on Coleman's bacteria analogue [5]. Then the solution of Eq.(17) can be found as the solution of the following equation, withḡ(=ḡ([g]; λ)) satisfying the following kind of equation, ]. Accordingly, one could define the 'renormalization' constants (finite again!) as z g ([g]; λ) ≡ḡ([g]; λ)/g. Thus renormalization is a notion in EFT associated with the rescaling, whose genuine origin is the contributions from underlying structures. The renormalization constants for operators (kinetic or composite) could be introduced in similar manner [3], including the cases with 'mixing' [6]. As a result, in the underlying theory point of view, the 'renormalization' constants are finite and could be introduced afterwards as byproducts, not as compulsory components. We suspect that this simple scenario might be helpful in more complicated EFT's, e.g., the Standard model, especially in its sectors with unstable fields and with flavor mixing.

APPELQUIST-CARAZZONE DECOUPLING THEOREM AND UNDERLYING STRUCTURES
Now let us discuss the decoupling theorem a la  from the underlying structures' perspective.

Decoupling and repartition
First let us note that, in the underlying theory perspective, the EFT parameters and the underlying parameters are grouped or partitioned into two separate sets by the reference scale that naturally appear in any physical processes (e.g., center energy in a scattering ) according to the relative magnitudes: the effective set [g] and the underlying set {σ}. When an EFT field 'becomes' too heavy to directly participate the EFT dynamics, it only induces a new partition between the effective and underlying parameters, with the union of the two sets kept 'conserved' in the course of decoupling: This repartition yields a new EFT that differs from the original one by a very massive field, hence a new set of agent constants {c} ′ is generated from this repartition. In terms of the scaling behavior, that means, or, equivalently, Then from the configuration of the parameters described in Eq. (27), the decoupling of a heavy EFT field could be formally be taken as the following two operations: (i) repartitioning the EFT and underlying parameters; (ii) taking the low energy limit with respect to the new underlying parameters, including the heavy EFT field's mass. This is the mathematical formulation of EFT field decoupling.
From Eq.(28) or (29), we see that, one could also work with the EFT containing M H , only adding some unnecessary technical complexities. This is also a well-known fact.
It is not difficult to see that our arguments above amounts to provide a relatively 'physical' rationale to the 'subtraction' solution of decoupling [8]. Of course different boundary conditions are needed across the threshold, which must lead to certain matching conditions for the 'running' parameters [9]. We hope our understanding could also be useful in the heavy quark effects in deeply inelastic scattering [10] and other important phenomenologies.

DISCUSSION AND SUMMARY
Of course, all the results presented here are not new. However, we still feel the way we derived these results seems more general and more natural. This underlying theory perspective might be of helps in deepening our understanding of the EFT methods. This is relevant to all the quantum theories, as any known theory is in fact an effective theory to some extents.
Of course, we did not touch the cases with concrete IR or other non-UV singularities. In such cases the underlying structures at large distances (hence often nonperturbative) must be incorporated (expanded) in a controllable way. Considering the complications in various factorization formulations, we refrain here from a naïve extrapolation of the scaling law. But the abstract operator form of the decoupling theorems (or scaling laws) in Eq.(9) (or Eq.(12)) is valid independent of such concrete details. The next step for deriving decoupling theorems or scaling laws in the case of non-UV singularities is to elaborate the contents of the operators and their anomalous dimensions that are responsible for non-UV singularities. We hope this line of investigations would lead to a different approach to the problem of non-UV singularities [11] and help to clarify the universal contents of factorization and/or its violation.
In summary, we derived the scaling laws in any EFT assuming the existence of nontrivial underlying structures with renormalization group equation and Callan-Symanzik equation being interpreted as 'decoupling' theorems of the underlying structures. The Appelquist-Carazzone theorem was briefly discussed in this underlying structures perspective.