Field Redefinitions and Infinite Field Anomalous Dimensions

Field redefinitions are commonly used to reduce the number of operators in the Lagrangian by removing redundant operators and transforming to a minimal operator basis. We give a general argument that such field redefinitions, while leaving the $S$-matrix invariant and consequently finite, lead not only to infinite Green's functions, but also to infinite field anomalous dimensions $\gamma_\phi$. These divergences cannot be removed by counterterms without reintroducing redundant operators.


Introduction
The S-matrix of quantum field theories is unchanged by field redefinitions [1][2][3][4], so that Lagrangians related by field redefinitions are equivalent, and give the same physical theory.While the S-matrix remains invariant under field redefinitions, Green's functions can (and do) change.Field redefinitions are often used to reduce the number of operators in the Lagrangian, and their couplings, to a minimal basis.In general, working in a minimal basis unavoidably leads to Green's functions and field anomalous dimensions which are infinite even after the addition of renormalization counterterms, even though the S-matrix is finite.A classic example of this phenomenon occurs with the penguin diagrams in the low-energy theory of weak interactions (see the discussion in [4, §6]), where Green's functions are infinite starting at one-loop order.Green's functions cannot be made finite by a simple rescaling of the fields -any attempt to make them finite reintroduces the redundant operators which were eliminated to obtain a minimal basis.This observation is relevant for theories of inflation, where one computes fluctuations from correlation functions of quantum fields.We also show that field anomalous dimensions are in general infinite starting at two-loop order when redundant operators are removed by field redefinitions.
Start with an EFT Lagrangian including all allowed operators which contribute to the action, i.e. all operators that are not total derivatives.This is equivalent to reducing the set of operators by using only integration-by-parts identities.The resulting set of operators is referred to as a Green's basis in the literature.Only a subset of operators in the Green's basis is independent under field redefinitions.The choice of independent operators is arbitrary, but the number of them is not.The independent operators (in some convention) are referred to as "physical" operators O i , and the remaining ones are referred to as "redundant" operators R j .The Green's basis has both sets of operators {O, R}.The Lagrangian coefficients of the physical operators are denoted by C i and of the redundant operators by D j .Field redefinitions can remove the redundant operators from the Lagrangian and modify the coefficients from {C, D} → {C, 0}.The resulting Lagrangian and coefficients will be referred to as being in the physical basis.
The Lagrangian in the Green's basis is renormalized in the MS scheme.The Lagrangian has counterterms which depend on {C, D}, and Green's functions and Smatrices computed with the renormalized Lagrangian are finite.The β-functions and field anomalous dimension when ǫ → 0 are finite, and depend on all the parameters in the Lagrangian.After a transformation to the physical basis, the β-functions and field anomalous dimensions have the form and depend only on the physical couplings.The β-functions are finite, but Green's functions and the field anomalous dimension γ φ are infinite, as was recently encountered in a specific case in ref. [5].Infinite Green's functions and field anomalous dimensions generically arise from field redefinitions.Start with the renormalized Lagrangian in the physical basis.Loop graphs computed with insertions of only the physical operators O can still lead to divergences which require counterterms with redundant operators R.These divergences induce non-zero values for the redundant coefficients D which are 1/ǫ k poles and generate β-functions for the redundant couplings: µ dD i /dµ = 0.These are, however, obscured because the theory is parametrized at the special point in theory space with D(μ) = 0. Nevertheless, the β-functions of the physical couplings and the field anomalous dimension depend on the counterterms of the redundant operators.An additional field redefinition is required to remove the counterterms of the redundant operators, and thereby transform the Lagrangian back to the physical basis, such that D(µ) = 0 for all µ, and the bare coupling of redundant operators vanishes, D b = 0.This field redefinition is infinite, since it removes counterterm coefficients of redundant operators.Since the S-matrix is invariant under field redefinitions, and remains finite, this means that any resulting β C i is finite.However, Green's functions are modified by this field redefinition, and become infinite, typically starting at one-loop order.Likewise, the field anomalous dimension has 1/ǫ poles starting at two-loop order when the scale dependence of the redundant couplings D i is ignored.
Infinite β-functions were found recently for the Standard Model Yukawa couplings at three-loop order [6][7][8].The origin of the divergence is due to an infinite µ-dependent flavor rotation, and has a different origin than the divergences studied in this paper.
We now demonstrate the above results with an explicit computation in the O(n) EFT to two-loop order.

Example
The example theory is the O(n) EFT to dimension six with Lagrangian where φ is an n-component real scalar field.The subscripts b refer to bare quantities.The dimension six terms are where we have divided the dimension-six operators into "physical" operators O 4,6 and "redundant" operators R 4,2 .The subscript denotes the number of fields in the operator.
The O(n) EFT has an expansion in a mass scale M, so the dimension-six coefficients C 4 , C 6 , D 4 , D 2 are order 1/M 2 , and terms of higher order in 1/M are neglected in eq.(2.1).The physical operator coefficients are denoted collectively by {C}, and the redundant operator coefficients by {D}.We include the dimension-two mass term (φ•φ) and dimension-four (φ • φ) 2 interaction in the physical operators, and m 2 and λ in the physical coefficients.One can make a field redefinition in eq.(2.1) to eliminate two of the dimension-six operators.Our choice in this paper is to eliminate R 4,2 and retain O 4,6 , so that the minimal basis of dimension-six operators is {O 4 , O 6 }.The choice of minimal operator basis is arbitrary, but the number of minimal operators is the same in any basis.All dimension-six operators {O 4 , O 6 , R 4 , R 2 } are included in the Green's basis.
The Lagrangian eq.(2.1) in the Green's basis can be renormalized in dimensional regularization in the MS scheme.The counterterms to two-loop order and dimensionsix are given in Appendix.A.1, and the β-functions and field anomalous dimension are given in Appendix.A.2.The field anomalous dimension and β-functions are all finite, and are functions of all the parameters in eq.(2.1).The 't Hooft consistency conditions [9] for the counterterms given in [10, §6] are satisfied, which implies that the β-functions and field anomalous dimensions are finite.
The field redefinition can be used to eliminate redundant operators in the Lagrangian.f and g are functions of the bare couplings of order 1/M 2 , and independent of µ, so the field-redefinition eq. ( 2.3) preserves µ-independence of the Lagrangian.The field redefinition with h a function of the bare couplings corresponds to a simple rescaling of the bare field and will modify Z φ while keeping Z of the couplings unchanged.Equations (2.3) and (2.4) comprise the most general field redefinition compatible with O(n) invariance to order 1/M 2 .There are two independent terms in eq. ( 2.3), so we can eliminate at most two operators from the Lagrangian, which have been chosen to be R 4 and R 2 .The general form for f and g must be compatible with the EFT power counting, so that the new Lagrangian retains the 1/M expansion.In addition, the dimensions of the terms must match in 4 − 2ǫ dimensions, where coupling constant dimensions can be fractional, e.g.λ b has dimension 2ǫ.The allowed redefinition is in terms of bare fields, or in terms of renormalized fields where a i are numbers, and the Lagrangian becomes In terms of renormalized couplings and fields, the Lagrangian is eq.(2.7) with and F i is the number of fields in O i , which determines the fractional part of the classical scaling dimension of the operator in 4 − 2ǫ dimensions.The field renormalization for φ is then Note that the kinetic term is no longer canonically normalized, not even at tree level.The additional rescaling transforms the Lagrangian to restoring canonical normalization of the kinetic energy term, and gives Z φ = Z φ .
Comparing with the original Lagrangian in eq.(2.1) gives the transformed coefficients ( C, D) and are the values of ( C, D) at In general, we have coefficients which are functions of the field redefinition parameters {a} and the original coefficients {C b }, {D b }.To go to the physical basis, the parameters {a} are chosen to set D i,b = 0 giving substituting the values for the parameters which set D = 0 back in eq.(2.12).The transformation is shown schematically in figure 1.
The original theory was parameterized by C and D. We can use eq.(2.13) to determine C, and parameterize the theory instead by C and D. We still need to retain D so that eq.(2.13) can be inverted to obtain the original couplings C and D from C and D. However, we emphasize that it is arbitrary to keep C and D. For example, in the O(n) case we consider, it also valid to parametrize the theory by C and C. In general, one has to retain as many total parameters as in the original theory.
The new renormalized couplings are given by the same functions F i of the original renormalized couplings obtained by dropping the 1/ǫ terms in eq.(2.13).In the O(n) example, the relations are The renormalization for the field in eq. ( 2.9) at The overall field redefinition that has been performed is a combination of eq.(2.5) and eq.(2.8), and is infinite, leading to infinite Green's functions.The only way to restore finite Green's functions is to undo the field redefinition and reintroduce the redundant operators.A simple rescaling of φ does not make the Green's functions finite, since the transformation eq.(2.5) is non-linear, and cannot be compensated for by a rescaling.
The new bare and and renormalized couplings are related by where f i is defined below eq.(2.7).The counterterms for C i to two-loop order are given in Appendix.B.
which completely removes all D 2 and D 4 dependence from the Lagrangian.This shows that when working in the physical basis, i.e. setting D 2 (µ) = 0 and D 4 (µ) = 0, we are implicitly making the infinite field redefinition eq.(2.20) and using the field φ.
The anomalous dimension of φ given in eq.(B.9) is infinite, because the logarithmic derivative of 1 + a 4 D 2 + a 5 D 4 does not vanish even at D 2,4 = 0 since the β-functions of D 2,4 do not vanish at that point, and a 4 and a 5 are infinite.
As noted before, it is equally valid to parametrize the theory by, for example, {C 6 , C 4 , C 4 , D 2 } instead of {C 6 , C 4 , D 4 , D 2 }.This affects only the form of Zφ, leaving Z of the physical couplings unchanged.Moreover, the field anomalous dimension (both the finite and infinite part) depends on the calculation, and not only on the final choice of physical basis.For example in the O(n) model, one can use the physical basis The field anomalous dimension computed in basis B 2 is different from that computed in basis B 1 and then converted to basis B 2 by an additional field redefinition for both the finite and infinite pieces.However the S-matrix computed in the two methods agrees.
Given that the field anomalous dimension is infinite starting at two-loop order, one can abandon attempts to renormalize the field, and simply use the bare field to compute S-matrix elements.In this case γ φ = 0.However, now the two point function φ(x)φ(y) is infinite, and the (infinite) wavefunction factor has to be included in the S-matrix computation to obtain a finite S-matrix, which makes the computation more involved.
Figure 2 shows the renormalized coupling constant space for the theory.(Figure 1 showed the bare coupling constant space.)In the space of renormalized couplings, we have two different flows.There is a flow due to field redefinitions, analogous to that in figure 1.Along these flow lines, the S-matrix is invariant.In addition, we have a flow due to a change in µ which also leaves the S-matrix invariant. 2Since the S-matrix is invariant under both flows, the two are compatible.A point on the field-redefinition curve at µ = µ 1 flows to some point on the field redefinition curve at µ = µ 2 .
Suppose we compute loop corrections in the Green's basis starting with a renormalized Lagrangian with D(µ) = 0 at µ = µ 1 .Even though the renormalized coupling vanishes, D(µ 1 ) = 0, loop corrections can generate counterterms for D(µ) which depend on the non-zero physical couplings C(µ).Thus RG evolution induces non-zero couplings D(µ) as µ evolves from µ 1 to µ 2 .This flow is shown by the orange arrow in figure 2. One then needs to do a field redefinition at µ 2 to make D(µ 2 ) = 0, shown by the blue arrow in figure 2. RG evolution in the EFT with only physical couplings is equivalent to a combination of RG evolution and field redefinitions in the Green's basis.Performing the field redefinition to all orders in the counterterms is the same as the transformation using bare couplings discussed earlier.During all the transformations, the S-matrix is invariant, and remains finite.The S-matrix is determined by the physical couplings (and vice-versa), so they remain finite as well, and the RG flow for the physical couplings is finite.The evolution of the physical couplings (the black dot in figure 2) is determined by the intersection of the field-redefinition invariance curve with the C axis; it does not depend on the starting point on the curve, i.e. the field-redefinition curves flow to other field-redefinition curves under a change in µ.As a result, the β-functions C(µ) are finite, and only depend on C(µ).The key point is that they do not depend on D.
In the above analysis, we have made use of (a) the invariance of the S-matrix under field redefinitions and (b) a one-to-one relation between the physical couplings and the S-matrix.These conditions do not apply to the quantum field φ, Green's functions are not invariant under field redefinitions, and the field anomalous dimension is generally infinite.The infinity arises due to the additional rescaling eq. ( 2.20) to remove D dependence in Z φ , or equivalently, dropping the D derivative in eq.(2.18).
If one wants to keep Green's functions and field anomalous dimensions finite, redundant operators cannot be ignored and one has to use the full Green's basis at all steps in the computations.

A Green's basis results
The counterterms, β-functions and field anomalous dimension of the O(n) theory to two-loop order in the Green's basis of eq.(2.1) are listed below, where in 4 − 2ǫ dimensions, with f i defined below eq.(2.7), and

A.1 Counterterms
The renormalization factors for the Lagrangian eq.(2.1) are: A.2 β-Functions and γ φ The β-functions and field anomalous dimensions computed from the counterterms are

B Physical basis results
The counterterms, β-functions, and field anomalous dimension in the physical basis with physical couplings C and redundant couplings D = 0 are listed below.Note that the wavefunction renormalization Zφ depends on the redundant couplings D, which parametrized the Lagrangian before the field redefinition.
) which are functions of the original couplings and a i .The choice a 1 = −1/2, a 2 = −1, a 3 = 1 gives D 4,b = 0, D 2,b = 0, so that the redundant operators are eliminated.The new bare couplings in the physical basis (C, D) are functions of the original bare couplings, Figure 1.Field redefinitions lead to a set of equivalent theories with the same (finite) Smatrix, shown by the green curves in the space of bare couplings.The coefficients C b and D b vary along the field redefinition curves as the parameters {a} in the field redefinition are varied.The coefficients C b are the values of C b when the green curve intersects the C b -axis and the redundant couplings D b vanish.The bare couplings are infinite.

1 Figure 2 .
Figure 2. Field redefinitions and RG flow lead to a set of equivalent theories with the same (finite) S-matrix, shown by the green and red curves respectively in the space of renormalized couplings.The coefficients C and D vary along the field redefinition curves as the parameters {a} in the field redefinition are varied.The coefficients C are the values of C when the green curve intersects the C-axis.The renormalized couplings are finite.Starting from vanishing redundant couplings D(µ 1 ) = 0 can still lead to non-zero redundant couplings D(µ 2 ) = 0 at a different value of µ.