Effective operators in SUSY, superfield constraints and searches for a UV completion

We discuss the role of a class of higher dimensional operators in 4D N=1 supersymmetric effective theories. The Lagrangian in such theories is an expansion in momenta below the scale of"new physics"($\Lambda$) and contains the effective operators generated by integrating out the"heavy states"above $\Lambda$ present in the UV complete theory. We go beyond the"traditional"leading order in this momentum expansion (in $\partial/\Lambda$). Keeping manifest supersymmetry and using superfield {\it constraints} we show that the corresponding higher dimensional (derivative) operators in the sectors of chiral, linear and vector superfields of a Lagrangian can be"unfolded"into second-order operators. The"unfolded"formulation has only polynomial interactions and additional massive superfields, some of which are ghost-like if the effective operators were {\it quadratic} in fields. Using this formulation, the UV theory emerges naturally and fixes the (otherwise unknown) coefficient and sign of the initial (higher derivative) operators. Integrating the massive fields of the"unfolded"formulation generates an effective theory with only polynomial effective interactions relevant for phenomenology. We also provide several examples of"unfolding"of theories with higher derivative {\it interactions} in the gauge or matter sectors that are actually ghost-free. We then illustrate how our method can be applied even when including {\it all orders} in the momentum expansion, by using an infinite set of superfield constraints and an iterative procedure, with similar results.


Introduction
Effective field theories are our main tool for studying physics at high scales, like the physics beyond the Standard Model. There are two reasons for this. One reason is the absence of a fundamental theory (UV completion). The second reason is that these theories are convenient for practical purposes: we do not need them if we have the full theory and are able to compute everything in such case; but effective theories make calculations much easier by focusing on the relevant parameters for the physics at the (momentum) scale investigated. Then shorter distance physics can be ignored together with all particles too heavy to be produced at this scale. Eliminating (i.e. integrating out) these particles simplifies the calculation. The result is a non-renormalizable theory (even if initial theory was renormalizable), in which all nontrivial effects of heavy particles appear in operators with dimensions d > 4 [1,2]. In the full theory, these effects are included in the non-local interactions obtained by integrating out "heavy particles". But in the effective theories one replaces non-local interactions with virtual particles exchange by a set of local interactions such as to give the same low energy physics.
The high energy behaviour is affected, so the effective theory is only valid at momenta below the mass of the "heavy particles". This mass is the effective theory cut-off (Λ).
The Lagrangian of the effective theory then contains just local interactions, obtained from an expansion in momenta below this scale, i.e. in powers of ∂/Λ, up to some finite order.
Keeping all orders in momenta leads to a non-local theory equivalent to the original full theory 1 . The effective Lagrangian is analytic in ∂/Λ in the region relevant to the low energy theory and it can be dealt with in any finite order in the momentum expansion. This is a local Lagrangian that one is using. This picture is so familiar that it is usually implicitly assumed, but we reminded it to make clear our set up. For a review see [2].
In this work we would like to investigate such effective theories beyond the first leading term of the momentum expansion. This is relevant when the momentum is closer to Λ. Using this picture we also try to infer a UV theory at scales above the "heavy particles" mass. By "UV theory" we mean a two-derivative ghost-free theory with only polynomial interactions in superfields. If this theory is renormalizable we refer to it as a UV completion 2 .
Let us formulate the above picture and our goals in a more precise way. In 4D SUSY effective theories the study is often strongly restricted to Kahler potentials K and superpotentials W that depend only on the superfields Φ i . But one often encounters cases when K and W depend on more general arguments such as the superderivatives D α acting on the superfields i.e. K = K(Φ i , Φ † j , DΦ † k , DΦ l , D 2 Φ † m , D 2 Φ n , ...) and W = W (Φ i , D 2 Φ † i , ...). In component fields, this action contains powers of ∂ µ , ✷, etc. These account for the momentum expansion ∂/Λ giving the effective theory mentioned. Λ is here related to Kahler curvature.
For the two-point Green functions (propagators) the presence of an expansion in powers of ∂/Λ (giving higher dimensional derivative operators) can lead to additional poles; some of these are ghosts, (super)fields of negative kinetic terms. There is nothing "pathological" about their presence here. They are just artefacts of the effective approach that are eliminated by the field equations or non-linear field redefinitions, to leave a ghost-free effective theory.
As mentioned, the corresponding (derivative) operators are a common presence in the low energy limit of the UV theory, after integrating out the "heavy particles". They are thus related to the UV completion of the effective action. Such operators are also present in the interactions terms. When scalar fields in these interactions develop vev's, these terms can contribute to the two-point Green functions, with similar effects (additional poles and ghosts). These operators are studied below.
Let us detail how such operators emerge when classically integrating out massive states. 1 There is nothing wrong in using this non-local theory for calculations [2]. For the differences between this non-local and effective local theories, see [2]; we shall meet such an example in Section 5 and the Appendix. 2 An example of a UV theory is 4D N=1 supergravity (UV incomplete).

Consider a simple (UV complete) Lagrangian
For large Λ, the superfield χ has a large mass, so it can be integrated out using its equation of motion −D 2 χ † + Λχ + mΦ 0 = 0. This has an iterative solution Thus χ is an infinite series in (∂/Λ). This solution is used back in L to give where we replaced D 2 D 2 → −16✷ and Z = 1/(1 + m 2 /Λ 2 ). So a simple decoupling of a massive state generated the ✷-operators. Actually, due to eq.(2), eq.(3) contains an infinite series in momentum expansion (∂/Λ), from the initial renormalizable theory. One usually truncates this series to a low(est) order, as in (3). Then ✷-operators are often eliminated by simply using the leading order equations of motion 3 (for the non-SUSY case, see [3,4]).
Such operators are also generated dynamically by compactification, as loop counterterms.
They can be generated by bulk (gauge) interactions to give the F-term below that contributes to a one-loop running of the 4D effective gauge coupling in orbifolds [5,6,7,8,9,10,11] δL where 1/R is the compactification scale and W α is the gauge field strength. Regarding the D-term, it is generated by superpotential (Yukawa) interactions localised at the 4D fixed points of an orbifold (it can be a Higgs mass counterterm in such orbifolds [12,13]). So models with extra dimensions contain such effective operators 4 at one-loop. Such operators can also be present in other compactifications (Randal-Sundrum, etc). In conclusion, these operators are common and are related to the UV regime.
The main goal of this work is to clarify two problems for general effective theories: 1): to obtain a better understanding of the higher order terms in momentum expansion, and then "remove" these higher derivative operators from the effective Lagrangian. To do this we show that one can reformulate ("unfold") such a Lagrangian into a second-order Lagrangian i.e. without higher dimensional (derivative) operators. This is what we mean by "unfolding".
This result is interesting since a two-derivative formulation of a theory is easier to handle and the "unfolded" formulation is a first step towards identifying a (ghost-free) UV theory.
2): to identify a two-derivative ghost-free UV theory of the effective theory with such operators. It is of strong interest to find a UV quantum consistent theory leading in the infrared to these (effective) operators and to fix in this way their (otherwise unknown) coefficient and sign, in agreement with constraints derived from analyticity and causality [14]. The "unfolded" formulation will help us to achieve this. For a related discussion see [15,16,17,18].
We show how to "remove" the ✷-operators acting on chiral, vector or linear superfields, in a consistent way, while preserving manifest supersymmetry (in a superfield language). We show how (effective) theories that contain such operators can be "unfolded" into second order theories with only polynomial interactions and with additional massive states, sometimes with negative kinetic terms (ghosts). For this, one eliminates these operators by suitable superfield constraints. For example, in the chiral sector such constraints replace each Here m is a small arbitrary scale of the theory that enforces the constraint. All superderivatives are thus eliminated, to find a second-order theory. The method can be iterated to higher orders in D α , Dα and can also be applied to non-derivative effective operators. Subsequent integration of these massive states leads to a theory with effective polynomial operators only and this formulation can be used for phenomenology.
We then show how the "unfolded" formulation helps us identify a UV theory of the initial effective theory. In the above models the initial effective operator was quadratic in fields.
We also study other models with higher derivative interactions that are ghost-free: a chiral superfield model with such operators, a model of Dirac gaugino masses and a supersymmetric version of the Euler-Heisenberg Lagrangian. We "unfold" them and then find their UV theory.
In each of these Sections we use the "unfolded" formulation to identify a UV theory (or UV completion) that generates at low energy the effective operators considered. Section 2 also contains the unfolding of a higher-derivative ghost-free chiral superfield model. Section 4 contains ghost-free examples leading to Dirac gaugino masses and to the Euler-Heisenberg gauge Lagrangian. Section 5 comments on how to treat more general cases. The Appendix presents the "unfolding" of a theory with an (known) infinite series of superderivatives. Using an infinite set of superfield constraints it is shown that even in this case there exists an "unfolded" version with only polynomial (d=4) interactions and an infinite set of extra (massive) superfields. Truncating the "unfolded" theory to a number of such fields is equivalent to the truncation to a corresponding power in (∂/Λ) of the momentum expansion.
2 Effective operators in the chiral sector 2.1 "Unfolding" the effective operators Let us first consider the case the ✷-operators, eqs.(3), (4) in the matter sector [15]. Consider where m is a real, small but arbitrary mass scale of the theory. This is a superfield constraint that we add to L, using a Lagrange multiplier superfield 5 Φ 3 . Then The field equation for Φ 3 recovers the constraint. Also, this constraint is implemented with a coefficient 1/Λ because it must be removed when Λ → ∞, while m is an unphysical parameter that restores the mass dimension (at the end of the calculation we take m → 0). L becomes The (hermitian) matrix k ij of the kinetic (D-)terms Φ † i k ij Φ j has det k ij = ρ m 4 /(16Λ 4 ). Its real eigenvalues 6 control the nature of the superfields Φ 1,2,3 : positive (negative) eigenvalues correspond to particle-like (ghost-like) superfields, respectively. We therefore have: a) If ρ = 1 then we have two negative and one positive eigenvalue, the latter corresponding to the original particle-like degree of freedom. Two superghosts are present, one due to the operator ✷, the second because auxiliary F 1 of Φ 1 became dynamical, thus an extra d.o.f. is present which, by supersymmetry, demands the presence of an extra (ghost) superfield. 5 This Lagrange multiplier method is similar to the one used in [31] in a case without higher derivatives. 6 The matrix is k11 = 1, k12 = σ m/(4Λ) = k * 21 , k13 = m/Λ = k * 31 , k22 = −ρ m 2 /(16Λ), k23 = 0 = k32 = k33. b) If ρ = −1, one has one negative (1 ghost) and two positive (2 particles) eigenvalues 7 .
c) If ρ = 0 then one eigenvalue is 0, one is positive and one is negative. Thus we have one ghost and one particle superfields. All these eigenvalues are the roots of The exact expressions can be obtained. According to our discussion we have The notation ν 3 ∼ −ρ means ν 3 has the sign of (−ρ) and is 0 if ρ = 0. This covers all cases discussed above. For a complete analysis, we also bring L to canonical form using a transformation to Φ ′ i = u ij Φ j with diag(ν 1 , ν 2 , ν 3 ) = u k u † and u ij unitary. With the notation z kj = u * k2 u * j3 m 2 /(4Λ) and after rescaling The initial L was "unfolded" into a second order theory, with extra superfieldsΦ 2,3 (of mass ∼ Λ, see later) and at least one of them having "wrong"-sign kinetic term (superghost). None of the auxiliary fields is dynamical anymore. The effective operators are not present anymore and all interactions are polynomial, up to O(1/Λ 3 ).
Note that in the arguments of W in the last two equations and also inside the square bracket of the F-terms there are additional terms O(m/Λ) that we did not write since we now set m → 0 because we do not have a constraint anymore. As a side remark, notice that the F i are the auxiliaries of superfieldsΦ i . This allows V = 0 with broken global SUSY. 7 We cannot have 3 negative eigenvalues, since we had one positive value to begin with (the initial particle) The effect of the original higher dimensional operators was then to introduce ghost superfields, of large mass (of order Λ) as shown by the F-terms in the last two equations 8 . Using the equations of motion one can integrate out the massive ghost superfields. For example in eq.(13), one uses the equation of motion for Φ 2 where the derivative of W is wrt its shown argument. This gives One then expands Using this and expanding W ( where W ≡ W (Φ 1 ) and σ = ±1. Thus effective operators re-emerge but are now polynomial in fields. L eff is classically equivalent to starting L in eq.(5) for ρ = 0. Even if W ′ may contain a linear dependence onΦ 1 , no ghost can be generated in the D-term due to the suppression 1/Λ 2 (relative to dominantΦ † 1Φ 1 ). Eq. (17) can now be used for phenomenology. For eq.(12) integrating the superghostsΦ 2 ,Φ 3 of mass ∼ Λ is done similarly to find which is equivalent to eq.(5) for σ = 0. This is the formulation that can be used for phenomenology.
In both examples there are no ghost superfields as asymptotic (final) states in The method can be extended to more general cases and higher orders, etc. An alternative to our approach that leads to results identical to those in eqs. (17), (18), is to use in eq. (5) non-linear field redefinitions to "remove" the derivative operators [18].

A ghost-free UV theory in the matter sector
The above result indicates how a UV theory of the starting Lagrangian eq.(5) can be realised, that is ghost free and, in this case, also renormalizable (UV complete). If we ignore the form of the dimensionless constants and O(1/Λ 3 ) terms, Lagrangian (8) contains only dimensionfour operators. The only problem is that it has ghosts, so it is not UV complete. This is seen after its kinetic terms are diagonalised, leading to results (11) to (13) in which ghost superfields emerge. Their presence is induced by the kinetic mixing in (8) that is dominant in the D-term, due to the absence of diagonal kinetic terms for Φ 2 and Φ 3 . This indicates that a ghost free theory should thus "UV complete" Lagrangian (8) by the addition of diagonal kinetic terms with suitable values for real ζ, η. Let us examine the impact of these terms. The new L becomes with ξ real 9 . Let us first show that this UV completed theory, recovers at low energy the Lagrangian in eq.(5). The equations of motion for Φ 2,3 are Using this back in L gives, up to O(1/m 3 ) terms This L is identical to the original Lagrangian of eq.(5) provided that We thus need only one extra parameter η to find a UV theory. 9 ξ is just a dimensionless parameter (ξ → m/Λ in Section 2). In the UV theory ξ (m) becomes physical.
with ζ as in eq.(23). Let us determine the values of the new coefficient η so thatν 1,2, all roots are positive if one respects simultaneously the following conditions: In this last case, the solution for η exists and depends on the exact values of ρ and ξ. i.e. no Φ † 1 ✷Φ 1 in original L, there always exists a negative root. As a result, if an L contains only the effective operator d 2 θΦ 1 ✷Φ 1 there is no ghost-free, UV theory.
From the above cases we select those when all roots are positive.
The coefficientsũ ij andz jk are calculable and depend on ρ, σ and ξ. L provides one possible UV theory 10 of the original Lagrangian eqs. (5), (8), is free of ghosts (under the above assumptions for ρ, σ, ξ), and recovers the initial effective Lagrangian.
This method of identifying the UV theory can be extended to more complicated K and W .

"Unfolding" a higher-derivative ghost-free chiral theory
In this section we study a different theory with higher-derivatives that is known to be ghostfree, since it does not introduce additional degrees of freedom (poles in the propagator) [19,20,21,22]. Its Lagrangian is where the sign of ρ is not yet fixed. This effective action has interactions of the form |∂φ| 4 and higher-order algebric terms for the chiral auxiliary field of the form |F | 4 , but no dynamics for F is generated. With the help of four chiral Lagrange (super)field multipliers Σ 1,2,3,4 one can rewrite (26) as Eliminating Σ 1,2,3,4 recovers the previous Lagrangian while m 1,2,3 and ǫ are included for dimensional reasons. Further In this "unfolded" formulation, the parameters m 1,2,3 and ǫ are not physical, but they become so in the UV theory (see later). This form of L has no higher derivative terms anymore. Given the high dimension of the initial derivative operator, the terms in eq.(28) that correspond to the initial operator, although polynomial, have mass dimension larger than four.
Despite its appearance, the Lagrangian in eq.(28) has no ghosts since it is equivalent to the original (ghost-free) Lagrangian in eq. (26). Indeed, for vanishing vev 11 Φ = 0, it is obvious Σ 2,3 have no dynamics and therefore only enforce constraints on L. One of the constraints is that Σ 1 is a composite field, so the apparent "off-diagonal" ghost-like kinetic term of Σ 1 is actually a higher-order polynomial operator. 11 If this is not the case, the analysis is more complicated.
A UV theory of eq.(28) is found similar to the previous examples by adding the missing kinetic terms The resulting UV Lagrangian is manifestly ghost-free for ǫ < 1 4 and ρ > 0, which is the appropriate sign coming from general considerations [14]. Notice that in the UV Lagrangian of eq.(29) all parameters are physical, so the theory has now several mass scales. By integrating out the massive fields Σ i one finds the corresponding effective action.
One can ask if this UV theory also generates additional effective operators beyond the original one in (26) of dimensions lower or equal to its dimension. The field equations of the massive fields determine their solution: where the dots stand for terms that contribute to operators of dimension higher than eight.
Substituting this solution back in L UV of eq.(29) one finds a low energy, effective Lagrangian below We thus recovered our initial operator, the last term above, suppressed by Λ. The other two operators are suppressed by another mass scale m 1 ǫ . One can in principle arrange that Λ ≪ m 1 /ǫ by a suitable choice for (dimensionless) ǫ. In this case our operator of interest in eq.(26) is the leading one generated at low energy.
As seen above, consistency of the UV theory we found demands that ρ = 1.

Effective operators for a linear multiplet
Another important supersymmetric multiplet, notably in string models, is a real linear multiplet L [32], containing a real scalar field c, a fermion ψ and an antisymmetric tensor b µν .
The multiplet satisfies the constraints with a solution where Z α is a chiral spinor superfield. The superspace expansion in the rigid case is The massless action for a linear multiplet has the gauge symmetry if W α is a gauge superfield strength satisfying D α W α = DαWα. The free action of a massive linear multiplet is [33]

"Unfolding" effective operators in the linear multiplet sector
In applications one can encounter an effective operator of the type shown below acting in the linear multiplet sector where ρ = ±1 was introduced to allow either sign for the last term. L 0 can be written as by using the identity The above effective operator can be "unfolded" by using our experience so far 12 . The result can be guessed directly, by starting with with a constraint L ′ = D α Z ′ α + DαZ ′α ; here L ′ is a massive linear multiplet. a is a real dimensionless numerical coefficient to be identified shortly. The above Lagrangian has one standard linear multiplet (L) and an additional, ghost linear multiplet since the determinant of the kinetic terms matrix is negative ∝ −a 2 < 0. One can eliminate the massive linear multiplet L ′ via its equation of motion that is actually obtained from that for the unconstrained The effective Lagrangian after eliminating L ′ becomes This is identical to eq.(37) provided that a 2 = ρ/2 which has a solution for ρ = +1 only.
With this value, L of eq.(39) is an "unfolded" version of initial L 0 of eq.(37) since it has no higher dimensional operators, but it contains an additional ghost-like superfield.

Ghost-free UV theory for the linear multiplet case
A natural UV theory of the effective Lagrangian above is to add to eq.(39) a kinetic term for the massive linear multiplet L ′ L is ghost-free (if a 2 < 1) and recovers the Lagrangian L 0 plus higher derivative operators.
Other UV theories are possible. Consider for example the coupling of the massless linear multiplet L to a massive vector multiplet V 12 According to our method the constraint L ′ = D α Z ′ α +DαZ ′α should be imposed via a Lagrange multiplier superfield. We impose this constraint directly when using the eq of motion of Z ′ instead of (constrained) L ′ .
The field equation of the massive vector multiplet leads to which, when inserted back into eq.(43), leads to The above analysis can be extended to cases when such effective operators are present in the gauge sector. Without restriction to generality, we consider an Abelian case with 13 where we used that D 2 ǫ γα = −2D γ D α . Then As before, we introduce an auxiliary (real) superfield V ′ which enables us to remove the higher dimensional (derivative) operator via a constraint 13 We denote the gauge field strength by W α , not to be confused with the superpotential W .
Here m is a small arbitrary scale of the theory which is set to 0 at the end of the calculation.
We implement the constraint using a Lagrangian multiplier which is a real superfield Σ, as shown below: Using the Lagrangian in eq.(50), the eq of motion for Σ reproduces the constraint eq.(49).
V ′ can be eliminated (integrated out exactly) since its equations of motion are algebric, so L becomes: We obtained a second-order theory with renormalizable interactions (by power counting) with the original vector superfield V and an additional massive one Σ. The new field Σ is a ghost (vector) superfield since the determinant of the kinetic terms is negative. The gauge kinetic mixing can be diagonalised by a rotation and an appropriate rescaling. Finally, one can also integrate out Σ using the above L to obtain an effective operator of second order, as done for the matter sector, eqs. (17), (18). The result depends on the structure assumed for K (for example one can consider the simplest case Φ † e V Φ, etc).

A ghost-free UV completion in the gauge sector
Can we can find in this case a simple, ghost-free (renormalizable) UV theory of L in eq.(46), (51)? From the matter sector we know that simply adding a positive kinetic term, in this case for Σ in eq.(51), is the way to proceed. We thus add δL together with L of eq.(51) can be brought to a diagonal basis after a suitable rotation applied to V, Σ. Then one chooses values for δ so that there are no ghost vector superfields in L + δL. This new Lagrangian gives a UV theory that is also renormalizable (UV completion).

Application: an example generating Dirac gaugino masses
As discussed, not all theories with L having powers of superderivatives generate ghosts, if these are present in interactions. We construct another example in the following. Start with a model where Λ ′ is large, comparable to Λ. We have a massive gauge field V ′ of field strength W ′ α , a field X charged under it, a gauge kinetic term for V and an interaction term with Φ neutral under V, V ′ . If Φ has a scalar component with non-zero vev, then we would have kinetic mixing W W ′ and this would induce the presence of a ghost (for W ′ ). However, assume that the scalar component of Φ has a vanishing vev, ensured by a suitable choice of the superpotential

W (Φ). Then the field equation for
Using this in L, we obtain an effective Lagrangian up to additional terms (not shown) suppressed by extra powers Λ ′ . If we identify the chiral superfield X with the spurion of supersymmetry breaking, then L eff contains a Dirac mass term [34,35] This is seen by considering the fermionic component of Φ and the gaugino λ in W α giving a mass term for λ ψ where ψ is the Weyl fermion of Φ. Finally, a UV theory of Lagrangian (53) is obtained by adding there an F-term We obtain in this way a two-derivative, ghost free Lagrangian that generates in the low energy the Dirac gaugino mass term.

Supersymmetric Euler-Heisenberg Lagrangian
Another interesting case is that of a supersymmetric generalisation of the Euler-Heisenberg Lagrangian which is a higher-derivative gauge theory that can be ghost-free 14 . It is given by where ρ = ±1 accounts for possible signs of this operator. L contains the gauge field term One must have ρ > 0 according to the constraints discussed in [14]. Similar to the chiral superfields example discussed earlier, L can be re-formulated as ("unfolded" into) a secondorder theory. The idea is to introduce a constraint that replaces a pair W α W α of the D-term by a chiral superfield so this term becomes quadratic (in S) and thus "removes" the extra derivatives. We thus introduce two additional chiral superfields Φ and S, according to where Φ is the Lagrange multiplier (chiral superfield) that implements the constraint and M is an arbitrary mass scale. Notice that, up to total derivatives, there is a continuous shift symmetry Φ → Φ + iα, where α is a real parameter. After integrating out S and Φ one recovers L of (57) provided that ρ/Λ 4 = ǫ 2 . It is interesting that this second-order formulation of the starting L is actually consistent with the condition ρ > 0 [14].
The situation is however more subtle because the chiral operator W α W α is constrained by W α W α = 1 2 D 2 Ω, where the real superfield Ω is the Chern-Simons superfield 15 [37,38].
Accordingly, the field S in eq.(59) is not an independent chiral superfield, but the field strength of a three-form superfield S = − 1 4 D 2 U , where U is a dimensionless superfield 16 .
Let us briefly review some details regarding the real three-form multiplet U . This is defined in superspace by (60) 14 Here we "unfold" this operator at the classical level. However this operator can also be generated by loop corrections in a standard gauge theory. For studies of this operator in supergravity, see e.g. [36]. 15 In the Abelian case, it is given explicitly by Ω = − 1 2 D α Wα +DαWα + V D α Wα . 16 This is similar to the dilaton case, for example.
The difference between a three-form multiplet U and a regular vector superfield V is the replacement of the vector potential V m by the three-form C npq . To find the appropriate kinetic terms, the analog of the chiral field strength superfield W α for a vector multiplet is the chiral superfield [39,37] with F defined by F = 1 4! ǫ mnpq F mnpq , where F mnpq is the four-form field strength. While the massless three-form multiplet U has only two propagating bosonic (s) and two fermionic (λ) degrees of freedom, a massive three-form multiplet has four bosonic and fermionic degrees of freedom. Notice also that a massless three-form multiplet has the gauge invariance U → U −L, where L is a linear multiplet. This symmetry is broken by a mass term, similarly to the case of a standard vector multiplet V . References [39,37] contain detailed explanations about the three-form multiplet.
Returning to eq.(59), there is a dual formulation of this equation, which can be found by starting from the master Lagrangian where Q is a real superfield and L a (dimensionless) linear multiplet. The equation of motion for L gives M Q = Φ + Φ † , which when used into eq.(62) recovers the action of eq.(59).
Finding the UV theory of the Euler-Heisenberg Lagrangian is now done by adding the missing kinetic term in the chiral formulation The shift symmetric kinetic term for Φ is equivalent to a standard canonical one in rigid SUSY, but is different in the supergravity version. The dual UV Lagrangian is found by introducing a vector multiplet as above and eliminating it out via its equation of motion.
The result is and contains a massive three-form multiplet. The action in eq.(65) is fully gauge invariant.
In particular, under U (1) gauge transformations and in the unitary gauge L = 0, the relevant transformations are This discussion neglected subtleties related to the existence of boundary terms in the action.
While they are important to obtain a fully consistent Lagrangian (see e.g. [40]), they are not relevant for the above discussion.
It is interesting to note that the Lagrangian in eq.(65) (with the gauge field set to zero) is that of the chaotic inflationary model described in the last reference in [40].

More general cases
The method of superfield constraints that we introduced can be generalised to more arbitrary K, W which have as arguments chiral functions Note the dependence on the superderivatives of many fields Φ j . The dots stand for powers of such superderivatives that may also be present. All higher dimensional terms are suppressed by appropriate powers of a high scale (the Kahler curvature tensor). This action is difficult to compute and investigate in component fields. However, one can introduce constraints ..., and similar for the other, higher order superderivatives that may be present, by using an iterative procedure (as done in the Appendix). These constraints are then added to the original L as F-terms of type with Σ j the Lagrange multipliers chiral superfields whose eqs of motion recover the constraints. As in previous cases, the coefficient in front of the integral is useful to ensure the constraint is vanishing in the limit Λ → ∞ and that the kinetic mixing is under control.
Similar considerations apply for the vector (gauge) or linear multiplet sectors. In this way one obtains a K and W that only depend on the superfields χ k = {Φ i ,Φ j , Σ l } but not on the superderivatives. With K = K(χ k , χ † k ) and W (χ k ), the Grassmann integrals in L can then be performed as for the non-linear sigma-model by using a (Grassmann space) Taylor expansion, to find the component fields action [41]. This will be a second order theory, but with additional fields (that can then be eliminated by the field equations, if massive enough).
The "unfolding" method of superfield constraints applied to higher order terms in momentum expansion of the Lagrangian can also be applied to other general cases. One may be interested in cases when the momentum is closer to the effective cutoff, when even higher order terms (beyond those considered here) are relevant. In the Appendix we present the extreme case of including all orders in such an expansion and apply our method. To illustrate it, we simplify the analysis and use instead a known UV-complete (renormalizable) theory, to a given number of superfields is equivalent to truncating the initial Lagrangian to a given power of ∂/Λ. In this "truncated" theory, one can integrate out these massive fields to obtain a Lagrangian with new effective operators polynomial in superfields, as in eq.(17), (18).
While this example is very simple, it shows that our method can be extended to higher or all orders in momentum expansion by using an appropriate set of superfield constraints and their iteration.

Conclusions
Effective field theories contain a series in momentum expansion that includes a special class of higher dimensional operators. These are generated by classical integration of massive states even in renormalizable, UV complete theories (and also by quantum corrections of compactification). The result of such integration is in general truncated to a given order in momentum expansion ∂/Λ (Λ is the effective cutoff). This leads to an effective theory with higher derivative operators that can induce the presence of (super)fields of negative kinetic terms (ghosts). Contrary to a common perception, there is nothing "pathological" about their presence here. They are just artefacts of the expansion in ∂/Λ obtained after integrating out massive states (of mass ∼ Λ) of the UV theory. Their presence and that of the corresponding (derivative) operators is not problematic and we showed how to treat them. Finally, keeping all orders in the momentum expansion gives a non-local theory equivalent (classically) to the initial, fundamental theory that generated the effective operators in the low energy.
In a manifestly supersymmetric approach and using superfield constraints, we showed how such effective operators acting on chiral, vector and linear superfields can be "unfolded" into "standard" operators, polynomial in superfields, in the order of truncation in ∂/Λ considered. To see this, consider the case of chiral superfields. In such case one replaced the superderivatives D 2 Φ † 0 by new chiral superfields mχ = D 2 Φ † 0 where m is a small, arbitrary mass introduced for dimensional reasons. This chiral superfield constraint was enforced with a Lagrange multiplier chiral superfield and in this way all superderivatives are eliminated.
The method can be applied and iterated to higher orders in D, D (and also to non-derivative operators). After this "unfolding" there are no superderivatives left but only terms polynomial in superfields and additional superfields (some of which are ghost-like) of a mass of order Λ. These can be integrated out to obtain a ghost-free low energy effective action that is polynomial in superfields and corresponding to the order in ∂/Λ considered. The action so obtained can then be used for phenomenology. This procedure can be repeated to higher orders in ∂/Λ for improved accuracy. The method was then applied to cases when superderivative operators act in the gauge and linear multiplets sectors, with similar results.
The "unfolding" method of superfields constraints helps one identify (two-derivative and ghost-free) UV theories that generate at low-energy the effective operators considered. We applied this method to the case when ✷-operators acted on chiral, vector or linear superfields, and using their "unfolded" formulation we identified the associated UV theory (not necessarily unique). In these examples the initial effective operators were quadratic in fields or gauge fields strengths.
Further examples were provided of ghost-free effective theories with higher dimensional (derivative) interactions: a chiral superfield model with such an interaction operator, a model generating at low-energy Dirac-gaugino masses and an effective model that is a supersymmetric version of the Euler-Heisenberg Lagrangian. Each model was "unfolded" into a secondorder theory for which we subsequently identified a UV formulation (i.e. with two-derivatives only and ghost-free).
We also showed how our method can be extended to any K and W as arbitrary functions of chiral (functions) arguments. Finally, the Appendix provided a special case showing how our method can be used to all orders in momentum expansion. This is relevant for momenta closer to the effective cutoff. This was possible by using an iteration procedure and an infinite set of superfield constraints, with similar conclusions as in the examples quadratic in fields.

A Appendix: "Unfolding" effective operators to all orders
In this Appendix we show that the "unfolding" method of using superfield constraints to eliminate higher powers of the superderivatives can be extended to all orders in the momentum expansion of an effective theory.
The plan of this section is as follows. In Section A.1, we choose a simple model for which the UV completion is known, integrate a massive state χ exactly (to all orders), to generate an infinite series in momentum expansion and a non-local theory, see eqs.

A.1 Effective operators from a renormalizable theory: all orders analysis
Consider a simple renormalizable, UV complete Lagrangian of eq.(1) which has a (iterative) solution This solution is used back in L to integrate χ. Due to technical difficulties, one always truncates such expansion to a fixed (low) order. One finds for example after a rescaling Φ 0 → Φ 0 /(1 + ξ 2 ) 1/2 was made and where Note the presence of the ✷ operators in the F-and D-terms investigated in the text, also leading to the presence of the extra ghost superfields.
But we can go beyond the approximation of truncated series. One finds the exact result where n → ∞; in this limit there is no ghost, but truncating L to n finite generates ghosts (artefacts). One ignores the terms ✷ n * since integrating out χ of mass of order Λ we effectively integrate momenta that are below this mass scale. Further, the F-term can be replaced, up to a total space-time integral as follows: To see this expand (✷ * + 1) −1 and use repeated integration by parts. One then rescales Φ → Z 1/2 Φ, Eq.(A-8) is an exact result of integrating out the massive field χ to all orders in ∂/Λ. The global effect is a wavefunction renormalization, albeit in operatorial sense. The presence of ✷ inside W generates (ghost-free) derivative interactions. The derivative acting in the (denominator of the) bilinear F-term brings ghost superfields (when Taylor expanded and truncated, see Section 2). Its structure is non-local and follows from the sum of the whole series, as expected from the discussion in Introduction. Note that this term is equivalent to In the familiar approximation ✷ * ≪ 1 one finds L eff of eq.(A-4).

A.2 "Unfolding" the effective operators to all orders
For further insight into the momentum expansion of an effective theory, let us examine the result of eq.(A-8) by using the "unfolding" procedure. We would like to see if such "unfolded" version still exists to all orders in momentum, i.e. we do not "truncate" the Lagrangian after integrating χ in (A-1). To this end, we use as many constraints as needed to enforce the solution in eq.(A-3). Introduce Define new Lagrange multipliers superfields Σ i i=1,2...., that enforce constraints (A-10), so we have an equal number of Σ i and Φ i (with i = 0). Then integrating χ to all orders in momentum gives the Lagrangian below -12) or equivalently 18 This is an "unfolding" of Lagrangian (A-7) or (A-8) to all orders in (∂/Λ) giving the result of 17 We used in (A-10) the scale m for dimensional reasons. Using instead the scale Λ does not work since then the eigenvectors of the matrix of kinetic terms would not be normalizable to unity (unitarity violation). 18 Instead of multiplying the constraint by ξ one can alternatively demand Σ → 0 fast enough at large Λ.
integrating out (exactly) the massive state χ. Lagrangian (A-13) contains only renormalizable operators by power counting but also an infinite number of superfields 19 (n → ∞). Truncating the number of superfields to finite n is equivalent to a truncation of the momentum expansion of the solution χ and of the Lagrangian in eq.(A-6). Note the polynomial form of L.
In both cases λ = ±ξ, the fields Φ j are arbitrary up to the constraint n−1 j=1 ξ j Φ j = 0; one can choose any two fields to implement this constraint with the remaining fields set to 0.
The Lagrangian in the diagonal basis becomes, after a rescalingΦ z →Φ z / |λ z |: Φ n+1 , ...,Φ 2n are n the ghost superfields. From the F-term bilinears and ignoring contributions from W , it can be shown that one state is light (original particle Φ 0 ) while the other (particle and ghost-like) superfields are massive (mass of order Λ).
All superderivatives generated after the integration of massive superfield χ were eliminated. The above Lagrangian has only interactions polynomial in superfields and all operators of d > 4 were eliminated via superfield constraints 20 . The downside is the presence of an infinite set of fields, all massive, beyond initial Φ 0 . These can be integrated out as we did in the "truncated" case. This description is classically equivalent to L of eq.(A-1), (A-8) and may be useful in applications. This method can also be applied to more complicated cases. 20 A similar example exists for the effective Akulov-Volkov action for the goldstino; this action can be completely expressed in terms of superfields, in a interaction-free theory L = d 4 θG † G + d 2 θf G + h.c., f = 0, endowed with the constraint G 2 = 0 where G is the goldstino superfield [42,43,44,45,46,47].