One-loop effective actions and higher spins

The idea we advocate in this paper is that the one-loop effective action of a free (massive) field theory coupled to external sources (via conserved currents) contains complete information about the classical dynamics of such sources. We show many explicit examples of this fact for (scalar and fermion) free field theories in various dimensions $d=3,4,5,6$ coupled to (bosonic, completely symmetric) sources with a number of spins. In some cases we also provide compact formulas for any dimension. This paper is devoted to two-point correlators, so the one-loop effective action we construct contains only the quadratic terms and the relevant equations of motion for the sources we obtain are the linearized ones.


Introduction. Higher spins are everywhere
The idea we wish to support in this paper is that the one-loop effective action of a free (massive) field theory coupled to external sources (via conserved currents) contains complete information about the possible classical dynamics of the sources. We exhibit several examples of this fact for (scalar and fermion) free field theories in various dimensions d = 3, 4, 5, 6 coupled to (bosonic) sources with a large number of spins. In some cases we also provide compact formulas for any dimension. In this paper we concentrate on two-point correlators, so the one-loop effective action we construct contains only the quadratic part.
Consequently the equations of motion for the sources we obtain are the linearized ones. We postpone to a future work the analysis of one-point and three-point correlators. But our thesis is that the dynamics generated by the one-loop effective action (OLEA) contains all the information we need to reconstruct complete interacting equations of motion. This paper is a follow-up of [1], which contains a few (mostly parity odd) 3d examples of what has just been said. The present paper is more general and systematic, not limited to 3d, and devoted especially to the parity even sector.
As we have just mentioned, the crucial issue here is the calculation of the two-point functions of free massive field theories coupled to external sources. We do it via Feynman diagrams. This is in principle a simple calculation, and to carry it out we resort to a method introduced by Davydychev and collaborators, [2]. However, as we will see, although we can derive from it very general formulas they are expressed in terms of hypergeometric functions and derivatives thereof and not easily 'readable'. For this reason it is often very useful to expand such results near their IR and UV fixed points. These expansions in powers of the mass m in odd dimensions, and m and log m in even dimensions, allows us to single out the dynamics of the sources and will be referred to as tomography. In other words what we do is to describe RG trajectories of two-point current correlators that pass through those of free massive theories, but we focus in particular on their IR and UV expansions, where the physical content is more easily recognizable.
Such IR and UV expansions are necessary also for another reason: one has to check that the IR and UV limits of the one-loop effective action are well defined. We find in fact divergent and non-conserved term in the limit m → ∞. These terms are local and can be subtracted. We subtract as well the IR finite terms, which are also local. This is to make the OLEA well defined and scheme independent. The results obtained in this way, in particular in the even parity sector, transferred to the OLEA, allow us to find the linearized Fronsdal eom's (see [3,4,5]) for all the source fields we have considered, in the nonlocal form introduced by Francia and Sagnotti, [6]. In 3d we consider also the odd parity sector, and confirm the connection with Pope and Townsend's generalizations of Chern-Simons theory, already pointed out in [1].
In this paper, for the purpose of comparison, we also analyze massless free theories beside the corresponding massive ones. The difference between the two is that the latter allow us to control not only the UV but also the IR, while in the former only the UV is regularized. This explains the difference in the results. In general in the massless case we do not get all the information we can extract from the massive theory and many results are scheme dependent. Briefly stated, at least for the purpose of this paper, to make sure we get a complete information we must use massive models.
The subject of this paper is inspired by the idea of exploring theories with infinite many fields, [7], in particular string theories and Vasiliev-type higher spin theories, [8]. As shown in the body of the paper higher spin fields appear naturally in the one-loop effective action of the simplest free theories in any dimension and it is possible to make contact with the literature on classical higher spin theories, [9,10]. Other sources of inspiration have been [11,12,14,13,15,16]. The idea of exploring the one-loop effective action is far from new: the list of works which may have some overlap with our paper includes [21,23,22,24,25,26,27,28], but is likely to be incomplete. ¿From a technical point of view this paper continues the line of research started with [32,33,34,35] with more powerful techniques (a new Mathematica code).
The paper is organized as follows. In the next section we introduce the massive scalar and fermion model and define the relevant OLEA's. Section 3 is meant to explain the motivation for this research by means of simple concrete examples. We also introduce the issue of higher spin Fronsdal eom and their various forms. In section 4 we introduce a new representation of higher spin eom's in momentum space and their general form, which is independent on the dimension of space-time. Section 5 contains a short summary of Davydychev's method to compute one-loop Feynman diagrams. In section 6 to 10 we analyze the one-loop scalar and fermion model two-point functions and their IR and UV expansion (tomography) in 3, 5, 4 and 6 dimensions, respectively. In section 11 we produce the formulas for two-point correlators of spin 1, 2, 3 currents in any dimensions. Section 12 is devoted to the conclusion. Appendix A contains the demonstration of a result used in section 4 and Appendix B the analysis of the massless scalar and fermion models.

Free field theory models
In this paper we limit ourselves to two type of models, the free scalar and free fermion, although it is not hard to extend the analysis to other models. By the first we mean the complex scalar theory defined by the Lagrangian in any dimension. On shell the current is conserved. We can couple it to a gauge field via the action term´d d x A µ (x)J µ (x). The scalar-scalar-gluon vertex with momenta p, p ′ , k, respectively, (p incoming and p ′ , k outgoing), and the propagator are, respectively, But, of course we can define infinite many completely symmetric (on shell) conserved currents, of which (2.2) is only the simplest example: They couple minimally to external spin s fields, a µ 1 ...µs . The on-shell current conservation implies (to the lowest order) invariance under the gauge transformations δa µ 1 ...µs = ∂ (µ 1 Λ µ 2 ...µs) (2.5) where round brackets stand for symmetrization. In the case s = 2 the conserved current is the energy-momentum tensor and the external source is the metric fluctuation h µν , where g µν = η µν + h µν . In this case the action is the integral of (2.1) multiplied by √ g. The vertex for an incoming scalar with momentum p and outgoing scalar with momentum p ′ and an outgoing spin-s field with momentum k is The free fermion model is represented by a Dirac fermion coupled to a gauge field. The action is where A µ = A a µ (x)T a and T a are the generators of a gauge algebra in a given representation determined by ψ. We will use the antihermitean convention, so [T a , T b ] = f abc T c , and the normalization tr(T a T b ) = −δ ab .
The current J a µ (x) = iψγ µ T a ψ (2.8) is (classically) covariantly conserved on shell as a consequence of the gauge invariance of (2.7) (DJ) a = (∂ µ δ ac + f abc A bµ )J c µ = 0 (2.9) The next example involves the coupling to gravity The corresponding energy momentum tensor is covariantly conserved on shell as a consequence of the diffeomorphism invariance of the action, If we expand the metric around the flat spacetime, g µν (x) = η µν + h µν (x), then, contrary to spin-1 case, interaction is not linear in the gauge field, which is h µν . However, for the purposes of this paper, only linear term matters, and it is given by coupling the flat space energy-momentum tensor to the metric fluctuation h µν . Again we can couple the fermions to more general fields. Consider the free action 14) and the spin three conserved current Using the equation of motion one can prove that Therefore, the spin three current (2.15) is conserved on shell and its tracelessness is softly broken by the mass term. Similarly to the gauge field and the metric, we can couple the fermion ψ to a new external source b µνλ by adding to (2.14) the term Due to the (on shell) current conservation this coupling is invariant (to lowest order) under the infinitesimal gauge transformations In the limit m → 0 we have also invariance under the local transformations which are usually referred to as (generalized) Weyl transformations and which induce the tracelessness of J µνλ in any couple of indices. We can construct on-shell conserved currents for any spin s, but their form is more complicated than in the scalar case. The explicit expressions can be found in [1].
We notice that to lowest order in the external sources the relevant action, in all cases above, takes the form of the free action + a linear interaction term such as (2.18). We make the identification a µ = A µ , a µν ∼h µν , a µνλ ∼b µνλ , with the obvious exception of the non-Abelian field in (2.7). The latter will be the only case in which we consider non-Abelian external sources. 1

Generating functions and effective actions
In both scalar and fermion cases, the generating function for the external source a µ 1 ...µs is In particular a µ = A µ , a µν = 1 4 h µν and J (2) µν = 2T µν with a µνλ = b µνλ . The full one-loop 1-pt correlator for J µ 1 ...µs is The full one-loop conservation law for the energy-momentum tensor is A similar covariant conservation should be written also for the other currents, but for s > 2 we will content ourselves with the lowest nontrivial order in which the conservation law reduces to Warning. One must be careful when applying the previous formulas for generating functions. If the expression 0|T J (s) µ 11 ...µ 1s (x 1 ) · · · J (s) µ n1 ...µns (x n )|0 in (2.21) is meant to denote the n-th point-function calculated by using Feynman diagrams, a factor i n is already included in the diagram themselves and so it should be dropped in (2.21). When the current is the energy-momentum tensor an additional precaution is necessary: the factor i n−1 n! must be replaced by i n−1 2 n n! . The factor 1 2 n is motivated by the fact that when we expand the action the factor δS δg µν g=η = 1 2 T µν . Another consequence of this fact will be that the presence of vertices with one graviton in Feynman diagrams will correspond to insertions of the operator 1 2 T µν in correlation functions.
Our purpose in this paper is to compute the effective action for the external source fields at the quadratic order. As a consequence the first task is to compute the two-point functions In the sequel we compute them by using the Feynman diagram technique. For all twopoint functions the only relevant diagram is the bubble diagram with one spin s line of ingoing momentum k and one with the same outgoing momentum and one scalar or fermion circulating in the internal loop. For instance the 2pt function for the current J a in the fermion model is while for the e.m. tensor it is where symmetrization of indices (µ, ν) and (λ, ρ) and the factor 1 4 is introduced accordingly to the above warning.

An appetizer in 3d
In [1] we calculated in particular the two-point function of the current J a in the fermion model as well as its IR and UV limit. In the parity violating part we found a well-known result: when Fourier antitransformed and inserted in the generating function of the OLEA (2.21) it gives rise to the linearized version of the gauge CS action in 3d (which is in fact conformal invariant). In [1] we did the same for the two-point correlator of the e.m. tensor for the fermion model, and proceeding the same way we found the linearized version of the gravity CS action. Something that was also known before, [22]. Repeating the same thing for the spin 3 current above we found instead a previously unknown result: the UV limit in particular leads to a linearized action that corresponds to a spin 3 CS generalization postulated long ago by Pope and Townsend, [17,18,19,20].
These were the results found in the parity odd part (in [1] we were mostly interested in the latter). But the even parity parts of the two-point correlators have perhaps even more interesting interpretations, so let us briefly analyze the parity even parts of the linearized effective actions obtained from 2-point current correlators in the free massive Dirac fermion quantum field theory in 3d in [1].

Spin one and two -parity even sectors
The UV limit of the two-point function of the J a currents are nonlocal conformal correlators, according to expectations, see [14]. The same is true for the e.m. tensor two-point function. But now let us focus on the IR limits. According to [1], for the J a current two-point function, for large m we havẽ This term is local. Fourier anti-transforming it and inserting it into (2.21) it gives rise to the action which is the lowest term in the expansion of the YM action where g Y M ∼ |m|. Now let us go to the IR limit of the even part of the 2pt e.m. tensor correlator. Eq.(3.36) of [1] says This is a local expression multiplied by |m|. In fact Fourier anti-transforming it and inserting it into (2.19) it gives rise to the action which is the linearized Einstein-Hilbert action: where κ ∼ 1 |m| . These results for spin-1 and -2 are known have been known for a long time, see for instance [11]. Now, we ask the same question for the 2pt correlator of the 3-current (sec. 3.3). What action, if any, does it represent for the external source field?

Linearized equations for spin 3 in parity even sector
Before presenting our results in 3d, let us briefly review the status of the linearized equations for the massless spin 3 field described by the completely symmetric field ϕ µνλ . Historically the first formulation of equations for the unconstrained free massless spin 3 field was given by Fronsdal [3] where underlined indices mean the sum over the minimum number of terms necessary to completely symmetrize the expression in µ, ν and λ, i.e. for instance and where a prime ′ means that the tensor is traced over a pair of indices. In some formulas we shall use shorter notation in which all indexes are suppressed. Under the gauge variation (2.19), δϕ µνλ = ∂ µ Λ νλ + perm., the Fronsdal kinetic tensor transforms as δF µνλ = 3∂ µ ∂ ν ∂ λ Λ ′ . It follows that the Fronsdal equation is invariant only on restricted gauge transformations satisfying Λ ′ = 0 (this requirement holds for all higher spins). Also, the Fronsdal tensor is not divergence-free, ∂ · F = 0, so one cannot directly couple the spin 3 field to a conserved (i.e., divergence-free) current using the Fronsdal equation. As we construct effective actions and corresponding equations for the higher spin fields by (minimally) coupling to conserved currents, it is obvious that Fronsdal's formalism is not suited for our purposes.
The formulation appropriate for our purposes was proposed in [6], and analyzed in more detail in [29] (for a review, see [30]). It was shown that there is a one parameter class of equations for unconstrained spin 3 field, which are order 2 in derivatives, fully gauge invariant, and ready to be coupled to the external conserved current. These equations are most elegantly expressed by using gauge invariant linearized spin 3 Riemann tensor defined by The spin 3 equations are parametrized by real number a and given by where spin 3 Ricci tensors are defined by while their divergences are defined by 2 What is the difference between equations with different a? First of all, it can be shown that regardless the value of a, the free field equation (3.9)-(3.10) is equivalent to Fronsdal equation (3.7). They start to differ when interactions are introduced. Note that equations (for any a) are non-local. From the purely mathematical side, the equation for a = 0 plays a special role because it is the least singular on-shell 3 , and because of this it was originally promoted in [6]. However, it was later shown in [29] that equations with different parameters a propagate different set of excitations when coupled to a conserved external current J µνλ , In particular, it was shown that only equation with a = 1/2 propagates spin 3 massless excitations and nothing else, if one does not introduce additional constraints on ϕ or J. For a = 1/2 the tensor A can be also written as Let us emphasize that this by itself does not mean that the equation with a = 1/2 is the "right one" to be used for the consistent coupling to the dynamical matter.
The non-locality of equations (3.9)-(3.9) can be 'cured' by multiplying with r with r large enough. It is obvious that the equation with a = 0 is special in that r = 1 already does the job, while for a = 0 one needs r = 2. In this way one cures non-locality, but the price paid is that equations become higher-derivative (order 4 for a = 0 and order 6 for a = 0). This opens up an additional question when one considers coupling to the conserved current J: should we do this as in (3.13), or should we couple the current in the local way, with r large enough?
The moral of the above analysis is that, due to several reasons, there is a large degeneracy in formulating equations of motion for the free massless spin 3 field, and it is not obvious that all formulations can be used as a basis for constructing consistent interacting quantized theories. It would be advantageous to know which formulation(s) are more promising, before embarking into such enterprise. We shall now argue that the induced action method may give us a hint.
In section 3.2.4 of [1] it was shown that the parity even part of the spin 3 two-point current correlator for a massive Dirac fermion in 3d is given by where τ b and τ ′ b are form factors presented in [1], and are projectors which guarantee conservation. From (2.22) it follows that the linearized effective equation in momentum space for the background spin 3 field minimally coupled to a conserved current in free QFT with massive Dirac field in 3d, is given by The form factors contain branch-cuts, which means that this equation is strongly non-local. The fact that there are two independent conserved structures present in (3.16), and so in (3.18), is directly connected with the one-parameter degeneracy introduced in (3.10).
In the IR region (|k 2 |/m 2 < 4) the form factors are analytic, as expected, and the equation is weakly nonlocal (infinite sum of local terms) when expanded around |k|/m = 0. Using the expansions of form factors from [1], we obtain that the leading term in the IR is given by Observe that this is the lowest derivative conserved local expression, which is unique. Now, plugging (3.19) into (3.18) and Fourier antitransforming, we obtain for the linearized induced equation in the coordinate space where G is the conserved symmetric local tensor linear in ϕ, which is 4th-order in derivatives. As there is a unique such tensor, we can conclude (without doing any calculations) that it must be proportional to G(0), with G(0) defined in (3.9)-(3.10). Explicitly written, The result (3.20)-(3.22) is, in some sense, natural. First of all, it is the lowest derivative linear local parity invariant equation satisfying unrestricted gauge invariance and conservation condition. Also, the equation is of the same form as in spin 1 case, and we can identify the tensor F as spin 3 Maxwell tensor, while G appears to be spin 3 Riemann tensor (it is the lowest derivative local conserved gauge invariant parity even rank-3 tensor). 4 Let us connect our result with the known constructions, reviewed above. It is obvious that our result (3.20)-(3.22) is the same as (3.15) with a = 0 and r = 1, i.e., we have obtained a local version of the equation proposed in [6]. As we already mentioned, this equation does not propagate only spin 3 massless excitations, unless the conserved spin 3 current of the Dirac theory has some special properties which takes care of the redundant modes. This is the question we plan to investigate in the future.
Let us now briefly comment the UV limit (m/|k| → 0). After subtracting IR divergent terms (for a full explanation of this issue, see below) form factors in the UV limit tend to constants, which gives rise to a non-local correlator. However one of the subleading terms gives a combination of the two conserved quantities A : k 2 π µ 1 ν 1 π µ 2 ν 2 π µ 3 ν 3 B : k 2 π µ 1 µ 2 π µ 3 ν 1 π ν 2 ν 3 (3.23) which is not the same combination as the one present in IR limit (3.19). So, the corresponding induced linearized equation is also different.
A priori, one could freely linearly combine terms A and B and construct one parameter candidate equations for the free spin 3 field. For example, A by itself gives the following equation By combining with the traced equation, it can be shown that it is equivalent to the Fronsdal equation. The same can be shown for generic linear combination of A and B. There is though the special case, the combination 4B − 3A, which is traceless, for which the equation is In conclusion, we see that our simple analysis, based solely on the classification of possible conserved structures, recovers the Francia-Sagnotti analysis and gives an efficient method for analyzing higher spin actions. But, we emphasize that the induced action method, out of many possibilities, picks particular equations which are already coupled to particular external currents.
Comment. The previous results are limited to 3d and to the lowest spins. They are nevertheless enough to stir our interest and motivate a more in depth analysis. It is also clear enough that equations in the coordinate space are not always the best fit to generalizations to higher spins. Writing down the actions and equations of motion in the explicit form used so far becomes rapidly unwieldy with increasing spins and dimensions. Fortunately a language much sleeker than this and the formalism used so far in higher spin theories is at hand. We simply must go to momentum space and use the projector (3.17). Before plunging into the analysis of the results for 2pt correlators coming from Feynman diagrams, we'd better prepare the ground with a general analysis of their expected structure.

Universal EOM and conserved structures for spin s.
Our starting point is the 2-pt functions of symmetric conserved currents. We expect them to be conserved too, i.e. we expect to find 0 if we contract any index with the external momentum k. We exclude the presence of anomalies. In fact we will come across also some non-conservations, but they can be fixed by subtracting local counterterms. This aspect of our analysis is interesting in itself, but we will illustrate it later on in any detail. For the time being we ignore this fact and suppose that all 2-pt functions we deal with are conserved.
This said, the form of the conserved structures is universal, in the sense that is does not depend on the dimension d of spacetime. For spin s they can be easily constructed by means of the projector (3.17) and polarization vectors n 1 , n 2 : n 1µ , n 2ν .
For spin s let us write down the structures: where n·π (k) ·m = n µ π µν m ν . There are ⌊s/2⌋ independent such terms. Let us setẼ where a l are arbitrary constants. The explicit conserved structures are obtained by differentiating s times E (s) with respect to n 1 and s times with respect to n 2 . One obtains in this way conserved tensorsẼ Conservation is a consequence of the transversality property k µ π µν = 0 (4.5) This is the most general conserved structure for spin s (for a proof, see Appendix A). By Fourier anti-transforming and inserting into (2.21), one can construct the effective action corresponding to (4.6) multiplied by k 2 for the spin s field B µ 1 ...µs,ν 1 ...νs as follows where E(∂) is the formal Fourier transform ofẼ(k), i.e. the same expression with k µ replaced by −i∂ µ . The eom is of course After canonically normalization, it depends on ⌊s/2⌋ − 1 arbitrary constants. This is the most general linearized eom for a completely symmetric spin s field. ¿FromẼ (s) (k) we can obtain the most general traceless combination, by taking the trace of (4.6) and imposing it to vanish. This can be done by differentiating the implicit expressions (4.1),...,(4.3),... with respect to ∂ ∂n 1µ ∂ ∂n µ 1 . The resulting equation is the recurrence relation Setting a 0 = 1 the solution is Replacing this in (4.6) we obtain a traceless conserved structure. In turn this gives rise to a traceless eom.

Eom's from conserved structures
Any conserved structure (4.4) in coordinate space is in general a non-local differential operator. To each there corresponds a quadratic Lagrangian and a linearized eom. For the EOM it is enough to differentiate s times with respect to n ν 2 and saturate the exposed indices with the spin s tensor field a ν 1 ...νs , multiply by k 2 , set the result to zero and then differentiate also s times wrt n µ 1 . For the Lagrangian one saturates the lhs of the EOM with a µ 1 ...µs and divide by 2.
Therefore we can represent the eom symbolically as In the following instead of contracting the n 2 indices with the field a, we will always leave n 2 free and operate only on n 1 . The operation will be essentially tracing two n 1 indices. For instance tracing (n 1 ·π (k) ·n 1 ) over n 1 gives d − 1.
Let us consider the spin 3 case. In this compact notation, the most general eom will be Taking the trace over n 1 gives Thus, unless 6a + (d + 1)b = 0, i.e. for generic coefficients a and b, the second piece of (4.12) vanishes on shell and we can simply drop it. Therefore the relevant eom for spin 3 is i.e. (3.24).
Now we wish to prove that this is general, that is, for any spin s, for generic coefficients, the eom can be reduced to the form The strategy consists in taking the trace of (4.11) wrt to n 1 the maximum number of times and replacing the results in (4.11). For instance, for spin 4 we have to trace twice. Tracing p times (4.11) we get is not easy to compute, but these coefficients are generically non-vanishing. It is however possible to infer the important property that For s = 2n after n tracings, i.e. p = n, we arrive at Now let us consider p = n − 1. Using (4.16) and (4.18) we arrive at So, generically, Now we proceed by induction. Suppose after q traces, i.e. p = n − q + 1, we have Then, at level p = n − q, we remain with from which the conclusion (4.15) follows. For s = 2n + 1, we start from p = n and we can repeat the induction procedure arriving at the same conclusion (4.15).
The next task is to recover the Fronsdal equation from (4.15).
To this end we take the trace of (4.15), i.e. apply to it ∂ ∂n µ . This is easily seen to give and, in general, Using this we can easily calculate all the traces of (4.15). The end result is for even s, and for odd s. These two equations have to be understood as follows: any solution that satisfies (4.15) also satisfies either (4.24) or (4.25). Therefore we can replace these two eqs. into (4.15). The viceversa is not true in general: i.e. if a solution satisfies (4.24) or (4.25), it may not satisfy (4.15). For the time being we assume that Eq.(4.24) and (4.25) imply that in (4.15) we can make the replacement (n 2 ·k) 2 = k 2 (n 2 ·n 2 ) (see the comment below). The result of this substitution is: The first line gives the spin s Fronsdal operator. Therefore (4.26) identifies the spin s nonlocal Fronsdal equation. The compensator takes the form

Conserved odd parity structures
It is easy to obtain also all the odd parity structures. The spin 1 odd parity conserved Lorentz structure (linear in n 1 ·n 2 ·k) can only bẽ C (1) 0 (k·n 1 ·n 2 ) = (n 1 ǫn 2 ·k), (n 1 ǫn 2 ·k) = ǫ µνλ n µ 1 n ν 2 k λ (4.28) It is easy to realize that, for higher spin, the epsilon tensor can only appear in the form (n 1 ǫn 2 , k) in every single term, thus it can be factored out. What remains is an even spin structure of one order less. So the most general odd conserved Lorentz structure will be a combination ofC The odd parity action is supposed to be local (and higher derivative) Therefore the odd eom is The tracelessness condition (for spin s > 1) implies a recursion relation for the coefficients c l : Setting c 0 = 1 the solution is: A comment on the non-local Fronsdal equation In the previous derivations of eqs.(4.26) and (4.33), we have simplified a few steps by disregarding a number of alternatives. First, we have stated that several passages are generic, that is they do not hold in some very specific cases, leaving out in this way several (probably pathological) possibilities. Moreover, we have disregarded solutions that satisfy (4.24) or (4.25), but not (4.15). Therefore our conclusions concerning eqs.(4.26) and (4.33) are generic. They do not address more subtle questions, in particular the one pointed out in [29,30]: the non-locality of the Fronsdal equation contains a large freedom, so an important issue is to select the form of the equation that gives rise to the correct propagator for the higher spin field, and not all non-local equations which give rise to the Fronsdal equation upon gauge fixing also give the correct propagator 5 . We cannot say, on the basis of our previous derivation, that our non-local Fronsdal equations have the property of generating the correct propagator, but we can verify this a posteriori, by analyzing the effective actions we obtain for the massive scalar and fermion models in various dimensions. We will return to this issue in the concluding section.

The general method
In this section we illustrate the method to compute the 2-pt functions with Feynman diagrams. On first reading one can skip this section and go directly to the results in the next one. The integrals we have to compute in this paper are like the ones in (2.27) and (2.28), that is of the general form where, eventually, q 1 = 0, q 2 = −k. We will use the method invented by [2] to reduce the tensor integral to a sum of scalar ones stands for the complete symmetrization of the objects inside the curly brackets, for example The basic integral is now the scalar onẽ For instance, the bubble integral for the s = 1 current in the scalar model The integralĨ (2) (d; α, β; k, m) can be cast into the form of a hypergeometric series This representation is valid for large m compared to k. When m is small compared to k another representation is availablẽ In the sequel we consider also massless models. The relevant results can be obtained from the massive models by taking the m → 0 limit. But they can also be obtained by setting m = 0 from the very beginning. In such a case the basic integral is

Guidelines for the calculations
We will now set out to do explicit calculations and derive results for two-point functions in the scalar and fermion model in different dimensions. The method just outlined is the most convenient for our purposes, but it is nevertheless one out of many. In fact, even within it there are different possibilities or schemes. We expect that our results may depend on such schemes, but also to find a criterion to extract the scheme independent part. In most cases this is conservation and finiteness. In particular, by suitably choosing the scheme we will be able, for instance, to obtain both finiteness and conservation for spin 1 current in any dimension in the fermion model. The same is not as easy for higher spin currents.
In generic spin current correlators and, therefore, in the corresponding one-loop effective actions, we will find, beside non-conserved terms, also terms that diverge in the IR limit m → ∞. Fortunately these terms are finite in number and easy to identify by expanding the OLEA near the IR and the UV. Not only, all the nonconserved and all IR divergent terms are local. It is thus possible to subtract all the terms that diverge in the IR, which include, in particular, all the nonconserved ones and recover both conservation and finiteness in the IR.
In this process a particular attention has to be paid to the terms of order 0 in m, in even dimensions. In some cases they are local and conserved, and appear both in the IR and the UV. Even in this case we follow the attitude of subtracting the IR term from the corresponding UV one, on the assumption that physical information is contained in the difference between the UV and the IR, not in their absolute values.
Finally it should be added that the resulting IR and UV expansions are both convergent.
The calculations in the sequel are mainly carried out using a new Mathematica code [36].
To somewhat abbreviate the following formulas, at times we use the compact notation where a is some constant. The symbol k used in the above formula and in the sequel deserves an explanation: k saturated with n 1 , n 2 represents the vector k µ , while in the other cases it represents the modulus |k|. Finally, contrary to ( [1]), the latter is k ≡ |k| = √ k 2 .

3d scalar effective field action tomography
In this section we start the analysis of the two-point functions of spin higher than 1 currents. Before reporting on the general spin s case we would like to analyze in detail a few low spin cases. It is in in general possible to obtain compact expressions of the one-loop effective actions. However expanding it in powers of m near the IR and UV limits (an operation we call tomography) provides the most interesting information.
It is possible to use the parameter m to cut to slices the two-point function of currents of any spin. Let us consider the case of a massive scalar model (msm) in 3d. The basic formulas are (2.1,2.2,2.4,2.6) and (5.4) together with the analogous ones for higher spins, with d = 3.

3d msm: spin 1 current
This case is well known and simple, but it is excellent for pedagogical purposes. The exact 2-pt correlator for s = 1 is We can expand (6.1) in power of k m (IR) or of m k (UV). In the IR case we find .... ....
while the even powers of m vanish. The first is a (non-conserved and divergent in the IR limit) local term ∼ η µν , which must be subtracted away. The other terms are all conserved and proportional to the conserved structure The UV expansion is instead .... ....
In fact we have O(m 2n ) = 0 for n ≥ 2. The only nonvanishing terms with even powers of m are O(m 0 ), O(m 2 ). For these terms see the comment below. Except (6.7) the other terms are conserved and proportional to (6.5). The terms proportional to (6.5) are all non-local in the UV, and local in the IR, in particular (6.3) is local and corresponds to the YM action in 3d, see (3.1).
The two nonconserved terms are (6.2) in the IR and (6.7) in the UV. The first is local and the second is nonlocal, but their divergence is the same and local: This means that we can cancel it by subtracting a local term, ∼ m´d 3 x tr(A 2 ). This amounts to subtracting the IR contribution (which is local) from the UV one. Indeed we get So the term of order m in the UV and IR conjure up to reform again the same conserved structure as all the other terms. Taking the UV and IR limits splits apart this conserved structure. The conclusion is that, up to a local term we can view the effective action as a sum of infinite many terms, all proportional to n 1 · π (k) · n 2 with coefficients proportional to various monomials of m and k. In compact form: 6.2 3d msm: e.m. tensor We have to consider  . But putting together the analogous non-conserved terms in the UV and IR (that is subtracting the local IR terms from the (nonlocal) UV ones) we recover conservation.
So we find a result analogous to the 1-current. Up to local terms the effective action is a sum of infinite many terms, all proportional to the same conserved structure (6.22) with coefficients proportional to various monomials of m and k. They form a convergent series both in the IR and in the UV. In compact form: It should be noticed that the massless model case gives the result: This is conserved but not traceless, which is not surprising because a scalar massless model in d ≥ 3 is not conformal invariant. Eq.(6.21) is conserved. It does not coincide with the linearized Einstein-Hilbert action (in particular it is nonlocal), but this is simply a nonlocal version of the same, in the same sense as we have already seen for spin 3 and higher in section 3.

3d msm: higher spin currents
This scheme repeats itself for higher spin currents. For spin 4 there are 4 non-conserved terms in the IR and 4 in the UV, while the others are conserved or 0. Subtracting the IR non-conserved terms from the corresponding UV ones all the nonvanishing terms turn out to be proportional to the conserved structure: All terms with even powers of m vanish, except m 0 , m 2 , m 4 , m 6 , m 8 . For spin 5 there are 5 non-conserved terms in the IR and 5 in the UV, while the others are conserved or 0. Subtracting the IR non-conserved terms from the corresponding UV ones all the nonvanishing terms turn out to be proportional to the conserved structure: All terms with even powers of m vanish, except m 0 , m 2 , m 4 , m 6 , m 8 , m 10 . Comment 1. As we have seen above any conserved structure is connected to a (nonlocal) higher spin field equation of motion. In particular eqs.(6.3) and (6.21) are conserved structures which represent the linearized YM and EH actions, respectively, the second one in a nonlocal version. Eq.(6.36) is non-local and gives rise to a variant of the nonlocal Fronsdal equation discussed in sec.3. It is clear that any two-point correlator structure can be uniquely related to a given (linearized) equation of motion. The structure of the 2pt-functions conform to the general discussion in sec.4. This will be confirmed by the forthcoming analysis.
It is remarkable that the conserved structures that appear in the above expansions are always the same for any fixed 2pt correlator. As we will see this is not the case for the effective field action originating from a fermion model. Comment 2. The nonvanishing even m power terms are a finite number in all cases. They come from the fact that the UV expansion of coth −1 contains the factor − iπ 2 . This is the reason why they are a finite number and do not contain the factor i π like the others. The factor − iπ 2 comes from the logarithmic cut of coth −1 and it is determined by the choice of the Riemann sheet. So it is scheme dependent.
It is interesting to compare the O(m 0 ) results with the massless model case, obtained via (5.8 (k, n 1 , n 2 ) (6.44) These correlators are nonlocal and coincide with the O U V (m 0 ) terms evaluated above 6 . To be precise there is an indeterminacy in their sign due to the branch point at k = 0 originated from the choice of sign of the square root √ k 2 . This indeterminacy is present also in the m → 0 limit of the massive model and it is related to the choice of Riemann sheet mentioned above. As a consequence of it, in this paper we do not worry about the sign in front of the Maxwell and EH kinetic terms that appear in the effective actions. We postpone to a future work the task of finding a physically consistent prescription that eliminates this indeterminacy.

3d fermion effective field theory action tomography
We consider now the same analysis for the massive fermion model (mfm). The starting point are eqs.(2.7,2.10,2.27,2.28) and the like for higher spins (see also [1]).

3d mfm: spin 1 current
This case is rather simple. It takes a very compact form and is conserved without any subtraction. Expanding, the term corresponds to the linearized CS action (here ǫ (k·n 1 ·n 2 ) means ǫ µνρ k µ n ν 1 n ρ 2 ), and the term in the IR corresponds to the linearized YM action. 6 Appendix B contains a complete analysis of two-point functions for massless scalar and fermion models.

3d mfm: e.m. tensor -even part
For the e.m. tensor we have in the IR (all formulas below have to be multiplied by the factor 1 16 ) .... ....
The even powers vanish. The O(m 3 ) term is not conserved, while the other terms are all conserved and proportional to different combinations of the two conserved structures.
In the UV we have O(m 3 ) : −2 im 3 3πk 4 k 2 (n 2 ·n 2 ) − 2 (k·n 2 ) 2 (k·n 1 ) 2 + 2k 2 (n 1 ·n 2 ) (k·n 2 ) (k·n 1 ) +k 2 (n 1 ·n 1 ) (k·n 2 ) 2 . But putting together the analogous non-conserved term in the UV and IR (that is subtracting the local IR term from the (nonlocal) UV one) we recover conservation: 1 (k, n 1 , n 2 ) (7.14) Eq.(7.5) is the linearized and local version of the EH equation of motion (see sec.3). The other are non-local versions of the same (except (7.10). Actually, according to our general philosophy the term O IR (m), which is divergent in the IR limit, must be subtracted. It will therefore appear in the place of the vanishing term (7.9) with inverted sign. Once again up to local terms the effective action is a sum of infinite many terms, which form a convergent series both in the IR and in the UV, all of them proportional to various combinations of the conserved structures with coefficients proportional to various monomials of m and k. In compact form:

O(m 2 ) is not conserved, but
is. In summary, after subtracting O IR (m 2 ) the odd 2-pt correlator is: The term (7.17) and, in a scaling limit, also (7.19), give rise to the linearized CS action as discussed in [1].

3d mfm: spin 3, even part
This was already discussed in [1], so we report here only the final results. One must subtract the local terms O(m 5 ), O(m 3 ) in the IR, which are not conserved. After which the effective action becomes The O IR (m) term is conserved and has to be subtracted from it. The interpretation of these conserved structures in terms of massless Fronsdal eom has been discussed above. At each order they are different combinations of two conserved structures n 1 ·π (k) ·n 2 3 and (n 1 ·π (k) ·n 1 )(n 1 ·π (k) ·n 2 )(n 2 ·π (k) ·n 2 ) (7.24) but it is actually easy to prove that all these combinations give rise to the same eom (after taking the trace of the resulting equation and re-inserting it). The only condition is that the coefficient of the first structure be nonvanishing.

Tomography in 5d
There is no substantial difference between 3d and 5d. We start from the same formulas as in 3d and change only the dimension. For obvious reasons of readability we limit ourselves to the even parity part and the lowest spins, although the generalization is at hand. The analog of eq.(6.1) is This is not conserved, but the divergence is local. Expanding in powers of m like in 3d, we get in the IR . . . : . . .
The term O(m) is conserved but divergent in the IR limit. Therefore, according to our recipe, it must be subtracted and will appear with opposite sign in the UV list, where the corresponding term is missing. This term yields the Maxwell (or linearized YM) action and EOM, with a coupling ∼ m.
The terms corresponding to odd powers of m vanish. In the UV we have O(m 5 ) : − im 5 15π 2 k 4 k 2 (n 1 ·n 1 ) (k·n 2 ) 2 + 4k 2 (n 1 ·n 2 ) (k·n 1 ) (k·n 2 ) +k 2 (n 2 ·n 2 ) (k·n 1 ) 2 − 3k 4 (k·n 1 ) 2 (k·n 2 ) 2     (k, n 1 , n 2 )) (8.33) The term (8.32) corresponds to the spin 3 Fronsdal EOM. As we see from these examples the scheme for 5d is similar to 3d. Once again the term O IR (m) must be subtracted, although conserved, because it is divergent in the IR; as a consequence it will appear with opposite sign in the UV list, where the corresponding term is missing.

5d mfm: spin 1 current
The analog of eq.(7.1) (for the even part) is which is conserved. Expanding in powers of m like in 3d, all coefficients have of course the same conserved structure. In the IR all even m-power coefficient vanish and, for instance, which (with reversed sign) corresponds to the Maxwell action. In the UV we have instead,

O(m 5 ) is nonlocal and non-conserved, but
The remaining terms are conserved. In particular O U V (m 2 ) corresponds to the linearized EH action. The terms O IR (m), O IR (m 3 ) are conserved but divergent in the IR limit. So they must be subtracted and will appear in the UV list with opposite sign. It is curious that the fermionic model in 5d does not reproduce exactly the spin 3 Fronsdal operator. In fact the term (8.54) has the right form but lacks the essential k 2 n 1 ·π (k) ·n 2 3 part. This has to be considered a combinatorial coincidence. The terms O IR (m), O IR (m 3 ) are conserved but divergent in the IR limit. So they must be subtracted and will appear in the UV list with opposite sign.
Comment. The structure of the 2pt functions in 5d essentially repeats the scheme of 3d. The m-power expansions both in the IR and in the UV are similar: in the IR there are non conserved local terms, while in the UV there are non-conserved nonlocal terms. Subtracting the former from the latter one obtains conserved structures (and a finite IR limit). All the other terms are conserved and have analogous types of structures in both the fermionic and the scalar model.

Tomography in 4d
Even dimensional models present an additional problem concerning their regularization. For odd d works by itself as a complete regulator in carrying out the integrals generated by the Feynman diagrams. This is not anymore true for even d. The way out is well-known, we will set d = 4 + ε. Another difference we will come across with, which is related to this, is the appearance of log terms in the form factors. We will again expand the two-point functions in powers of m near the IR and UV limits.
In almost all the two-point correlators and, therefore, in all the one-loop effective actions, we will find non-conserved terms and terms that diverge in the IR m → ∞, like in the odd dimensional case, but we will find also ε-divergent terms. Our general attitude is to recover both conservation and finiteness in the IR. This is possible because all the nonconserved and all divergent terms in the IR, as well as all ε-divergent terms, are local. We will therefore subtract all the terms that diverge in the IR and in ε. They include, in particular, all the nonconserved ones.
There remains however an ambiguity. Beside divergent and/or nonconserved terms, in the case of m 0 we meet also finite contributions, both in the IR and in the UV. Also for these terms we subtract the IR from the UV contribution, on the assumption that it is this difference that contains the physical information.

4d msm: spin 1 current
The full formula for the 2pt correlator is expressed in terms of hypergeometric functions and parameter derivatives thereof, and we dispense with writing it down explicitly here, see however sec.11. We will focus on the power of m expansions. As just mentioned, we have to consider also log(m) and 1 ε factors. In the IR we find .... ....
These coefficients are conserved except O(m 2 ). All the odd powers of m vanish.
All odd powers of m vanish. The even powers are conserved except (9.7). Subtracting from the latter the analogous (local) non-conserved term in the IR we find a conserved term The O(log(m)) term is divergent in the IR, and the O(m 0 ) is divergent in the ε → 0 limit. Luckily they are local and can be subtracted with the following result: This term corresponds to the linearized Maxwell action with an energy dependent coupling.
The first two terms are not conserved, the logarithmic term is conserved but divergent in the IR, the m 0 term is divergent in the limit ε → 0. They all must be subtracted. The remaining terms are conserved.

4d Fermion Model
We consider now the same analysis for the fermion massive model. We start again from eqs.(2.7,2.10,2.27,2.28) and the like for higher spins.

4d mfm: Spin 1 current
The full formula for the 2pt correlator is similar to the scalar case and expressed in terms of parameter derivatives of hypergeometric functions, see sec.11. A full expression in terms of simple functions can be found in Appendix C. The m-power expansion in the IR is as follows .... ....
All odd powers of m vanish. The above terms are all conserved except (9.31) and (9.33). O(m 2 ) and O(log(m)) are divergent in the IR and O(m 0 ) is divergent in ε.

Tomography in 6d
10.1 6d Scalar Model The basic formulas are again (2.1,2.2,2.4,2.6 and (5.4) together with the analogous ones for higher spins, with d = 6 + ε. For the full two-point correlator formulas see next section.
Here we limit ourselves to IR and UV expansions.

6d msm: spin 1 current
Like in 4d, we have to consider also log(m) and 1 ε factors. In the IR the nonvanishing terms are .... ....
These coefficients are conserved except O(m 4 ). All the odd powers of m vanish.

6d Fermion Model
We consider now the same analysis for the fermion massive model. We start again from eqs.(2.7,2.10,2.27,2.28) and the like for higher spins and set d = 6 + ε.
10.2.1 6d mfm: spin 1 current We will limit ourselves to the power of m expansions in the IR. The terms proportional to m 4 , m 2 , m 0 and log(m) are local, nonconserved and/or divergent. Thus they must be subtracted. Therefore the first nonvanishing term in the IR is: .... ....
In the UV, after subtracting the local terms we find:

6d mfm: spin 2 current
In this subsection all results must be multiplied by a factor of 1 16 In the IR the odd powers of m vanish. The terms proportional to m 6 , m 4 , m 2 , m 0 and log(m) are local, nonconserved and/or divergent. Thus they must be subtracted. In the UV all the odd powers of m vanish. After subtracting the above local terms we have and

Spin s current two-point correlators in any dimension
In this section we derive general formulas for the two-point correlators for spin s = 1, 2 and 3 in any dimension. The procedure is slightly different from the one used so far. In the previous sections we fixed the dimension of space-time, that is we set d = 3, 4 + ε, 5, 6 + ε in the scalar integrals (see sec.5). In this section we leave the parameter d free and evaluate specific cases at the end. The two procedures often lead to different intermediate results.
Of course the results of physical interest must coincide.
In the following we focus on the massive case, while the massless computations are deferred to Appendix B.

Fermion model
In this subsection we compute the even part of two point correlator for a fermion in d dimensions for spin s = 1, 2, 3 where the Feynman vertices are

Fermion model -massive case
Let us compute the even part of two point correlator for a massive fermion in d dimensions for spin 1. The one-loop contribution is Tr (1) is the trace of the identity operator on the vector space on which the Dirac matrices act. Since we are working with the lowest dimensional complex spinors in each dimension we have Tr(1) = 2 ⌊ d 2 ⌋ . In the odd dimensional cases d can be simply replaced by the values 3, 5, ..., so this factor is an overall numerical factor. In the even dimensional cases we have to replace d by 2 + ε, 4 + ε, ..., so the same factor contains an ε dependence in addition to the overall numerical factor. The ε dependence cannot change the divergent pole part in dimensional regularization but will only affect the finite local part. However when subtracting the infinite and finite IR terms from the effective action this dependence disappears. So we will ignore it.
Warning. In evaluating the scalar integralĨ (1) (k), which is our basic quantity, the results below have been obtained by choosing a reference value 4 for Tr(1). This value is appropriate only for d = 4, 5, but must be corrected for the other dimensions: for d = 2, 3 the results must be divided by 2, for d = 6, 7 they must be multiplied by 2, and so on. By Davydychev tensor reduction procedure it is possible to rewrite such an amplitude in terms of scalar integrals as Because of dimensional reasons the superficial degree of divergence ofĨ (1) andĨ (2) are always such that deg(Ĩ (1) ) = deg(Ĩ (2) ) + 2 = d − 2. One can check the Ward identity k 2Ĩ (2) (k) +Ĩ (1) (k) = 0, (11.5) which implies we can rewrite the amplitude (11.3) as (11.6) to finite order polynomials according to the formulas z n so that one can easily check the 1/ε part is just So in general the divergent part (for ε → 0) is a polynomial in z = k 2 4m 2 of degree d/2 − 1, where the constant term is missing because the front factor of highest dimension is m d−2 . According to Weinberg's theorem this corresponds to the degree of divergence d − 4 of the two point functions, which is therefore lower than the expected one d − 2 because of gauge invariance. The above divergent terms are local and appear both in the IR and the UV limit, as we have seen many times above. Some of them are divergent for m → ∞ and must be subtracted to guarantee finiteness of the IR limit (or decoupling of infinite mass modes). Others are of order m 0 . The reason why we subtract them from the effective action is, according to our attitude, because the physical information is contained in the difference between the IR and UV limits (rather than in their absolute value).
As for the finite part we cannot give a closed formula for generic d, but thanks to the formulae we can recognize the IR behavior is analytic in z, which is to be expected as m acts as an IR regulator. More explicitly, we get the following behaviors . . . . . .

So the dominating term is
whereas the term with highest power of momentum and dimensionless constant is ∼ So, in coordinate space the following terms are dominating for large m whereas all the others are suppressed by negative powers of m. For d > 4 those terms would give a non-decoupling of IR dynamics from high-energy physics, but we can notice they are the same as the local counterterms appearing in the divergent part. 7 So they have to be subtracted, as we have done many times before. In (11.12) there are also terms of order m 0 . They have to be subtracted from the effective action for the same reason explained above: the physical meaning is contained in the difference between the UV and the IR. In d = 2 no pole shows up. In fact we have dimension the UV expansions for the finite part are: As it is to be expected the leading behavior is ∼ m d−2 z d/2−1 log(−z) corresponding to the UV behavior of the divergent part. The presence of logarithms is to be interpreted as the consequence of running of parameters. In ordinary (interacting) gauge theories, once these logarithms are reabsorbed in the running parameters, the remaining polynomial behavior can itself be subtracted by proper counterterms leading to a well-behaved amplitude in the UV.

d odd
Let us discuss now the odd dimensional case. In odd dimensions there is no divergent part in the ǫ-expansion. For the IR we get the expansion

Conclusion
We have seen a large number of examples that the one-loop effective action of a free massive model coupled to external sources contains complete information about the (classical) equations of motion of the sources. In this paper we have considered only the two-point functions and so the relevant information involve the linearized equations of motion. Moreover we have considered only completely symmetric bosonic external sources. Within these limitations we have produced overwhelming evidence that our previous statement is correct. We have considered both a free scalar model and a free fermion model in different dimensions, and shown that in all cases the two-point functions of conserved currents are built out of the differential operators which define the linearized (Fronsdal) equations of motion of the fields that couple to the currents. There is no doubt that such free field theories know about the dynamics of the fields that can couple to them (via a conserved current) 9 . At this stage a specification is in order. Our intent in this paper was to show the universal appearance of non-local Fronsdal (as well as Maxwell and EH) linearized eom in the one-loop effective actions of a free scalar and boson field coupled to external currents, while postponing other subtler questions to future research. In particular we did not tackle the problem raised by [29,30], concerning the form of the Fronsdal equation that guarantees the right propagator for the relevant higher spin field. In order to do that one must first of all specify to what equations one refers to, for we have seen that in the IR and UV limits of the OLEA's the conserved structures very often are different, and different from the various tomographic sections, although for spin s they are all characterized by the presence of the leading (4.1) term (the scalar model is in this sense a particular, though less interesting, case, because the conserved structures are always the same for given spin). This part of our research is work in progress, see [31].
The results of this paper opens a new research territory. Beside the just mentioned problem, we would like to know whether the above results extend to other external sources, fermionic fields as well as not completely symmetric fields. The next question is interaction, which requires analyzing three-point functions. In this context interactions have been considered for the simplest cases (spin 1 and 2) in 3d in [1]. From three-point functions one expects to find information about the consistency of the (field or fields) interaction with the source field symmetry. For instance, for spin 1 with gauge symmetry, for spin 2 with diffeomorphisms. For higher spins we do not know, in general, neither the interaction nor the full form of the symmetry transformations. But knowing the three-point functions may be the key to constructing both. There are anyhow some exceptions to our ignorance in this field (higher spin theories in 3d, or Vasiliev's higher spin theory in AdS 4 , or string field theory). One can hopefully use this knowledge to test the approach suggested here.
If our conjecture is correct, that is if the analysis of three-point correlators in theories coupled to external sources confirms their consistency with the dynamics of the latter, as we believe, an obvious question comes next: what does this mean? The correspondence between one free field theory and higher (or low) spin theories is not a type of duality we are familiar with, like AdS/CFT. First of all it concerns models in the same dimension. Secondly, from one free theory we retrieve knowledge about (infinite) many theories. So the correspondence would be one to (infinite) many. And this is clearly not satisfactory. The results of this paper points rather toward the possibility of a correspondence between theories with infinite many fields. If, say, a starting free (or interacting) theory knows about the dynamics of other fields, why shouldn't the latter be included with the initial one in a unique theory? Arguing this way one is led to a (for the time being, generic) concept of involutive theory: a theory is involutive if it includes all the fields it is able to couple with (in the OLEA) while preserving a fundamental symmetry.
A good playground to test this concept could be string field theory (SFT). Such a theory is formulated in terms of a basic string field Φ. The latter, in the field theory regime, is a superposition of Fock space states, each with a coefficient given by a suitable ordinary spacetime field. Restricting ourselves to bosonic SFT, the action formulated by Witten is well known, and is given by the formula below with Ψ replaced by Φ. Analyzing it in the spirit of this paper amounts to studying the theory where the first piece is the free SFT and the second is the simplest interaction with the source term (Ψ is the source string field). The first piece is invariant under the BRST transformation δΦ = QΛ. The second term carries the invariance under δΨ = QΛ provided that Φ is on shell, i.e. QΦ = 0. This mimics what we have done previously for simple field theory models.