Unification of Galileon Dualities

We study dualities of the general Galileon theory in d dimensions in terms of coordinate transformations on the coset space corresponding to the spontaneously broken Galileon group. The most general duality transformation is found to be determined uniquely up to four free parameters and under compositions these transformations form a group which can be identified with GL(2,R). This group represents a unified framework for all the up to now known Galileon dualities. We discuss a representation of this group on the Galileon theory space and using concrete examples we illustrate its applicability both on the classical and quantum level.


Introduction
The Galileons represent a theory of real scalar field φ with derivative interactions with interesting properties. It emerges in its simplest form as an effective theory of the Dvali-Gabadadze-Porrati model [1,2] as well as of the de Rham-Gabadadze-Tolley massive gravity theory [3] in the decoupling limit. Generalization of the Galileon Lagrangian was proposed by Nicolis, Rattazzi and Trincherini [4] as the long distance modifications of General relativity. In the seminal paper [4] also the complete classification of possible terms of the Galileon Lagrangian has been made and some of the physical consequences have been studied in detail, i.a. it was demonstrated that such theories exhibit the so-called Vainshtein
Let us note that the operator basis L n is not unique, we can choose also another set which differs from (2.4) by a total derivative and possible re-scaling. One of the many equivalent forms of the Lagrangian which can be obtained from (2.4

) by means of the integration by parts and simple algebra is
c n (∂φ · ∂φ) L der n−2 (2.7) Let us mention useful formulae (for derivation see e.g. [10]) and thus after integration and omitting the surface terms we get (2.11) For further convenience let us also write down explicitly the Feynman rule for n−point vertex V n (1, 2, . . . , n) = (−1) n d n (d − n + 1)!(n − 1)! σ∈Zn G(p σ(1) , p σ(2) , . . . , p σ(n−1) ) , (2.12) where we have introduced the Gram determinant G(p 1 , . . . , p n−1 ) G(p 1 , . . . , p n−1 ) = − 1 (d − n + 1)! ε p 1 ...p n−1 µn...µ d ε p 1 ,...,p n−1 νn...ν d d j=n η µ j ν j (2.13) and where the sum is over the cyclic permutations only 2 . Using this Feynman rules, one can in principle calculate any tree-level n−point amplitude in the pure Galileon theory, however, due to the complicated structure of the vertices it is not an easy task. The most economic way how to organize the rather lengthy and untransparent calculation is the machinery of the Berends-Giele like recursion relations 3 [18] which allows for an efficient computer algoritmization of the problem. In this way we can systematically calculate in principle any tree-level amplitude. What does it involve for n = 3, 4, 5 in the language of Feynman diagrams is depicted in Fig. 1. Note that crossing is tacitly assumed for these graphs which finally leads to four diagrams for 4-pt scattering and 26 for 5-pt scattering. In four dimension (in the theory with d 1 = 0) the calculation leads to the following results (we were also capable to calculate the 6-pt diagrams which involves 235 Feynman diagrams). Without the deeper understanding of the structure of the Galileon theory these results look suspiciously simple; 4 in fact it was our main motivation for starting to study this model more systematically. In what follows we shall i.a. show how to understand these results and how they can be obtained almost without calculation on a single sheet of paper.

Coset construction of the Galileon action
The Galilean symmetry is a prominent example of the so called non-uniform symmetry, i.e. a symmetry which does not commute with the space-time translations [20,21]. Indeed, denoting the infinitesimal translations and Galilean transformations of the Galileon field δ c φ and δ a,b φ respectively, we get [δ c , δ a,b ] φ = c · b = δ c·b,0 φ (3.3) Let us add to this transformations also the Lorentz rotations and boosts δ ω we get then [δ ω , δ a,b ] φ = −b · ω · x = δ 0,−b·ω φ = δ 0,ω·b φ (3.5) Therefore the infinitesimal transformations δ c , δ ω , δ a,b form a closed algebra with generators P a , J ab = −J ba , A and B a respectively. In terms of these generators δ c = −ic a P a (3.6) [J ab , J cd ] = i (η bc J ad + η ad J bc − η ac J bd − η bd J ac ) (3.9) which corresponds to the Galileon group GAL(d, 1) (see [17]). Within the Galileon theory this group is realized non-linearly on the fields φ and space-time coordinates x µ . Indeed, for the generators P a , A a B a we have (3.10) The generators A a B a are spontaneously broken, the order parameter can be identified with This corresponds to the symmetry breaking pattern GAL(d, 1) → ISO(d − 1, 1). Let us note that the above transformations are not completely independent in the sense of refs. [22,21] . Indeed, their localized forms with space-time dependent parameters a(x) and b µ (x) yield the same local transformation which corresponds to the local shift of the Galileon field φ. More precisely, writing a b (x) = x · b(x), we can identify δ a b (x),0 = δ 0,b(x) . (3.12) Physically this means that the local fluctuations of the order parameter which correspond to the Goldstone modes are not independent. As a result the particle spectrum does not contain the same number of Goldstone bosons as is the number of the broken generators (i.e. d + 1) but just one zero mass mode which can be identified with the Galileon field φ.
(see [21,23] for recent discussion of this issue). Construction of the low energy effective Lagrangian describing the dynamics of the Goldstone bosons corresponding to the spontaneous breakdown of the non-uniform symmetries is a generalization of the coset construction of Callan, Coleman, Wess and Zumino [24,25] and has been formulated by Volkov [26] and Ogievetsky [27]. Applied to the Galileon case, where the only linearly realized generators of the Galileon group are the Lorentz rotations and boosts J ab , the coset space is GAL(d, 1)/SO(d − 1, 1) the elements of which are the left cosets {gSO(d − 1, 1)} where g ∈ GAL(d, 1). The coordinates on this coset space can be chosen in a standard way by means of a unique choice of the representant U of each left coset. Such a representant can be written in terms of the coset coordinates x a , φ and L a as The general element of the galileon group g ∈ GAL(d, 1) acts on the cosets by means of the left multiplication and consequently the coset coordinates transform according to where h ≡ h(g, x, φ, L) ∈ SO(d − 1, 1) is the compensator arranging U to be of the form (3.13). As usual, the stability group, which is the Lorentz group SO(d − 1, 1) here, is realized linearly (φ transformed as a scalar and x and L are vectors), and the general element (3.14) of the Galileon group g ∈ GAL(d, 1) acts on U as follows As a result, for the general element of the Galileon group g ∈ GAL(d, 1) we have the following compensator Note that, the compensator does not depend on the coset coordinates (x, φ, L) and therefore treating φ and L a as space-time dependent fields, the compensator has no explicit or implicit x dependence. This simplifies the application of the general recipe [26,27] significantly, because the requirement of the invariance with respect to the local stability group can be replaced by much simpler requirement of global invariance. The basic object for the construction of the effective Lagrangian is the Maurer-Cartan form, which can be expressed in the coordinates x a , φ and L a as (3.18) where in the second line we have used the fact that A commutes with all the other generators. Using further we get finally The form ω c P is particularly simple. In the general case we get ω a P = e a µ (x)dx µ and e a µ plays a role of d−bein, intertwining the abstract group indices a, . . . with space-time indices µ, . . . and the flat metric η ab with the effective space-time metric g µν according to In our case e a µ = δ a µ , the space-time metric is therefore flat g µν = η µν (3.22) and the abstract group indices are identical with the space-time ones. This also ensures that the volume element d d x is invariant with respect to the non-linearly realized Galileon group. Note also that there is no term of the form ω ab J J ab on the right hand side of (3.20). This implies that the usual group covariant derivative is in our case identical with ordinary partial derivatives ∂ α . The forms ω c P , ω A and ω d B transform under a general element of the Galileon (3.14) group g ∈ GAL(d, 1) (cf. (3.17)) according to These forms span three irreducible representations of the stability group SO(d − 1, 1) (namely two vectors and one scalar) and can be therefore used separately as the basic building blocks for the construction of the effective Lagrangian. The general recipe requires to use this building block and their (covariant) derivatives to construct all the possible terms which are invariant with respect to local stability group. As we have mentioned above, in our case we make do with ordinary partial derivatives and the last requirement can be rephrased as the global SO(d − 1, 1) invariance when we identify the abstract group and space-time indices with help of the trivial d-bein δ a µ . Therefore, writing the most general invariant term of the Lagrangian is the Lorentz invariant combinations of the fields ∂ µ L ν and D µ φ, where and their derivatives. Apparently we have ended up with d + 1 Goldstone fields φ and L µ however this is not the final answer. In fact these fields are not independent. The standard possibility how to eliminate the unwanted degrees of freedom is to require an additional constraint [17,21] which is invariant with respect to the group GAL(d, 1) and which is known as the inverse Higgs constraint (IHC) [28]. Then the only remaining nontrivial building blocks are ∂ µ ∂ ν φ and its derivatives 5 and the general Lagrangian is The Galileon Lagrangian represents a different type of possible terms contributing to the invariant action, namely those which are not strictly invariant on the Lagrangian level, but are invariant only up to a total derivative. Such terms can be identified as the generalized Wess-Zumino-Witten (WZW) terms [29,30] , as was proved and discussed in detail in [31,32]. From the point of view of the coset construction, the WZW terms originate in the integrals of the closed invariant (d + 1)-forms 6 ω d+1 on GAL(d, 1)/SO(d−1, 1) (these correspond to the variation of the action) over the d+1 dimensional ball B d+1 the boundary of which is the compactified space-time S d = ∂B d+1 (3.27) 5 Another possibility how to treat the problem of additional degrees of freedom is based on the field where ψ µ are new fields. Then The invariant term M 2 DµφD µ φ, which was responsible for the kinetic term of the field φ in the original Lagrangian goes within the new parametrization in terms of φ and ψ µ into the mass term of the field ψ µ . This field then does not correspond more to the Goldstone boson and can be integrated out from the effective Lagrangian, provided we are interested in the dynamics of the field φ only. We end up again with the just one nontrivial building block ∂µ∂ν φ. See [21] for detailed discussion of this aspect of the spontaneously broken non-uniform symmetries. 6 More precisely we integrate the pull-back of the form ω d+1 with respect to the map B d+1 → In order to prevent these contributions to the action to degenerate into the strictly invariant Lagrangian terms discussed above it is necessary that the form ω d+1 is not an exterior derivative of the invariant d-form on GAL(d, 1)/SO(d − 1, 1). This means that ω d+1 has to be a nontrivial element of the cohomology H d+1 (GAL(d, 1)/SO(d − 1, 1), R) (see [33] and also [34] for a recent review on this topic). In the case of Galileon such forms can be constructed out of the covariant 1-forms ω µ P , ω A a ω µ B with indices contracted appropriately to get Lorentz invariant combinations. As was shown in [17], there are d + 1 such ω d+1 , namely where n = 1, 2, . . . , d + 1. These forms are closed and therefore Note that the d-forms β

Galileon duality as a coset coordinate transformation
The canonical coordinates (x, φ, L) on the coset space GAL(d, 1)/SO(d − 1, 1) which we have defined according to (3.13) are not the only possible ones. We can freely use any other set of coordinates connected with them by a general coordinate transformation of the form Not all such new coordinates are of any use, e.g. those transformations (4.1) which are not covariant with respect to the SO(d−1, 1) symmetry will hide this symmetry in the effective Lagrangian. Even if the covariance is respected, in the general case the resulting Lagrangian might be difficult to recognize as a Galileon theory. In this section we shall make a classification of those coordinate changes which preserve the general form of the Galileon action as a linear combination of the d + 1 terms discussed in the previous sections (though we allow for change of the couplings). Such a transformation of the coset coordinates can be then interpreted as a Galileon duality. It is obvious from (3.27) and (3.28) that, provided the forms ω µ P , ω A a ω µ B can be expressed in the primed coordinates as a (covariant) linear combination (with constant coefficients) of the primed forms ω µ P , ω A and ω µ B where the coordinate transformation corresponds to a duality transformation of the Galileon action. Indeed, provided 8 and, after imposing the IHC constraint 9 (3.25), the corresponding term in the action satisfies .
(4.5) This means that the coordinate transformation maps linear combination of the d + 1 basic building block of the Galileon action onto different linear combination of the same building blocks and the two apparently different Galileon theories are in fact dual to each other.
The conditions (4.3) constraint the form of the duality transformation (4.1) strongly. We have in the primed coordinates (here and in what follows the superscript at the symbol of partial derivative indicates the corresponding primed variable, e.g. ∂ (φ) ≡ ∂/∂φ ) and comparing the coefficients at ω ν P , ω A and ω ν B in the expressions for ω µ P and ω µ B with the corresponding right hand sides of (4.3) we get the following set of differential equations for ξ µ and Λ µ Integration of these equations is trivial, we get (up to the additive constats 10 ) Comparisson of coefficients in both expressions for ω A gives, after using the explicit form (4.8) of ξ µ and Λ µ , the following differential equations for f From the first equation it follows where the function F of two variables satisfies Integration of these equation is possible only if the integrability conditions are satisfied This constraints the possible values of the constants α IJ which means Imposing this constraint, the equations (4.10) transforms into the form which can be easily integrated (again up to the additive constant corresponding to trivial shift of φ) As a result we get the most general formulae 11 for the duality transformation of the coset coordinates and of the basic building blocs of the Galileon Lagrangian Imposing the IHC constraint (3.25) we get finally Let us note that the last formula of (4.18) (the transformation of ∂φ) is compatible with the first two as a result of the compatibility of the IHC constraint with the coordinate transformation mentioned above. We can also prove this easily by explicit calculation (see Appendix B). Let us finally write down the explicit formula for the duality in terms of the Galileon action. It is expressed by the identity Up to the remnants of the omitted additive constants, as discussed above. and the couplings of the two dual action are interrelated as where the matrix A nm (α) has the following form (4.23)

GL(2, R) group of the Galileon dualities
The duality transformations introduced in the previous section has natural GL(2, R) group structure under compositions. This is immediately seen from their action on the 1-forms ω A , ω µ P and ω µ B (cf. (4.17)) and on the coset coordinates x µ and L µ : the duality transformation is in one-to-one correspondence with the matrix and composition of two duality transformations corresponding to the matrices α and β is again a duality transformation described by matrix α · β. The condition det α = 0 ensures regularity 12 of the transformation of the coordinates on the coset space (4.16). A little bit less obvious is the group property for the duality transformation of φ. To demonstrate it let us rewrite (4.16) in the form Then composition of two dualities means However, as can be proved by direct calculation, and therefore as expected.
On the space D d+1 of the Galileon theories, which can be treated as a d+1 dimensional real space with elements corresponding to d + 1-tuples of the couplings d n , we have a linear representation of the duality group GL(2, R) by the matrices A nm (α) explicitly given by (4.23).
Let us now discuss some important special cases. The duality transformations corresponding to the one-parameter subgroup of matrices result in the following explicit transformation Fixing the parameter ζ = H 2 /4 the dual theory can be interpreted as an expansion of the original Galileon field about the de Sitter solution The fact, that the fluctuations φ about such a background are described by a dual Galileon Lagrangian has been established already in the seminal paper [4]. For the transformation of the couplings we get explicitly Another example concerns the following matrix It results in the duality transformation which can be rewritten in the more symmetric form as and which corresponds to the Legendre transformation. Duality properties of the Galileon theory with respect to this transformation has been discussed in detail in [9]. Explicit form for the dual couplings reads Let us now assume the diagonal matrix corresponding to the scaling transformation (∆ is the Galileon scaling dimension) for which the dual couplings simply scale according their dimension as More general scaling is also possible, namely for which x = λx , φ = λκφ , ∂φ = κ∂ φ (5.20) and in the dual theory d n (α S (λ, κ)) = κ n λ d−n+2 d n . (5.21) Let us assume now duality transformations induced by the matrices of the form 13 which represents a one-parameter subgroup and the coordinate and field transformation reads The rationale for the minus sign of the element αP B is that with this choice the infinitesimal form of this duality transformation is See Appendix A for bottom up construction of the finite duality transformation from the infinitesimal one.
Such a type of duality (with special value of the parameter θ) has been discussed in the papers [15,16,35] and its one-parametric group structure has been recognized in a very recent paper [36]. The couplings transform according to It is obvious, that any duality transformation can be obtained as a combination of the above elementary types of transformations. Indeed, for general matrix α we can write the following decomposition Let us give another simple example of such a type of decomposition. For instance, we have and therefore we can understand the one-parametric duality (5.24) as a Legendre transfor- which can be written in the symmetric form (cf. (5.14)) as As we will see in the following sections, the duality transformations (5.24) are the most interesting ones relevant from the point of view of physical applications. Let us briefly comment on some properties of its representation on the Galileon theory space (5.7). First, because the matrices A(θ) ≡ A nm (α D (θ)) are lower triangular matrices, any subspace D (k) d+1 ⊂ D d+1 spanned by the d + 1-tuples with first k couplings equal to zero (i.e. D (k) d+1 = {d|d n = 0 for n ≤ k}) is left invariant by A(θ). We can therefore restrict ourselves to some fixed D (k) d+1 in what follows 15 . Note also that α D (θ) is a one-parametric subgroup and thus the matrices A(θ) satisfy a differential equation The function ψ (x ) can be obtained by means of application of the dual transformation corresponding to the product of matrices in the second square brackets in (5.27), similarly for ψ(x). 15 For the physical applications it is natural to set d1 = 0 in order to avoid tadpoles and assume therefore the subspace D d+1 . and consequently we get for d This is a system of d − k nontrivial ordinary differential equations (note that the first of the equations (5.32) is trivial i.e. d k+1 can be taken as fixed 16 once for ever) describing the "running" of the couplings with the change of the duality parameter θ. Of course, the solutions are just d n (α D (θ)) given by (5.25) with d n , n > k as the initial conditions at θ = 0. Such a system have in general d − k − 1 functionally independent integrals of motion I k+3 , I k+4 , . . . , I d+1 which do not depend explicitly on θ. Once these are known, any other such an integral of motion can be then expressed as where f is some function. The set I k+3 , I k+4 , . . . , I d+1 represents therefore a basis of the α D (θ) duality invariants on the subspace D (k) d+1 of the Galileon theory space. The set of independent invariants I k+3 , I k+4 , . . . , I d+1 can be constructed by means of elimination of the initial conditions and θ the from the solution (5.25). This can be done as follows. Note that (5.25) for n = k + 2 and d n = 0 for n ≤ k reads and thus we have unique solution θ * for θ such that d k+2 (θ * ) = 0. According to the group property we can rewrite the solution of (5.32) in the form with new initial conditions d(θ * ). Inverting (5.35) we get the right hand side of which is θ independent. For d k+2 the equation (5.35) reads and thus we can easily eliminate θ − θ * solely in terms of d k+2 (θ). Inserting now this for the explicit θ − θ * dependence into (5.36) for n = k + 3, . . . , d + 1 we get the desired integrals of motion I l (d k+2 (θ), . . . , d d+1 (θ)). Their interpretation is clear, according to our construction I l represents a value of couplings d l in the theory dual with the original one such that in the dual theory the coupling d k+2 is zero. These integrals form the basis of the α D (θ) duality subgroup invariants on the Galileon theory subspace D (k) d+1 we started with.
Let us illustrate this general construction of α D (θ) duality invariants in the case of three and four dimensional Galileon theory. We will restrict ourselves to the theory subspaces D and according the general recipe, the only α D (θ) duality invariant is and we have two independent duality invariants I 4,5 = d 4,5 (θ * ) where θ * = 3d 3 , explicitly

Applications
Because the duality relates different Galileon theories, its main benefit is based on the possibility to solve a given problem in the simplest exemplar of the set of theories connected by duality and then to translate the result back to the apparently more complex original theory for which the problem has been formulated. In order to realize this approach effectively it is necessary to establish the transformation properties of various physically relevant quantities under the duality transformations. The main role play those which are invariants of the duality. As we have mentioned above, the most useful duality is the oneparametric subgroup α D (θ), which is (together with α L ) the only one for which the field and coordinate transformation is nontrivial. Therefore the invariants with respect to this subgroup are the most important ones. In this section we will discuss the above aspects of the duality using several examples both on classical and quantum levels.
In what follows we almost exclusively work in four dimensions with Minkowski metric and in the Galileon Lagrangian we set d 1 = 0 to avoid the tadpole and d 2 = 1/12 to get a canonical normalization of the kinetic term.

Classical static solution
As a warm up, we will illustrate the use of duality on a simple example of the static axialsymmetric solution of Galilean equations with an external source coupled to the Galileon field as The source T will be represented by an infinite "cosmic string" along the x 1 -axis with linear density σ > 0 Note that, for general external source, the part S int of the complete action violates duality. Therefore we cannot in general case simply argue that the duality transformation of the classical solution in the original theory is also a solution of the dual theory with the same external source. However, our source term is very special being local and therefore it modifies the equations of motion only on the set of points of zero measure. As we shall explicitly see, for such a source the duality works, which supports the conclusions made in the very recent paper [37]. Our axial-symmetric ansatz is thus where we have introduced the complex coordinates z and z: i.e.
In order to obtain the explicit form for the classical equations of motion we will start with the following useful formula [9] L der where we can easily work out the left hand side because the matrix η + w∂∂φ is block- Comparing this with the right hand side of (6.6) we get where we will set d 2 = 1/12 in the following. In our case T = 2σδ (2) (z, z) so the equation of motion becomes First we can easily solve the theory for d 3 = 0. The axial symmetric solution is (up to a constant term) In the case when d 3 = 0 we can first rewrite the equation of motion to where the prime means a derivative with respect to zz. By further integration over the disc with zz ≤ R 2 and using the Gauss theorem in two dimension we will arrive to which can be algebraically solve to 14) The final result can be obtained by elementary integration. We have two solutions, which is for d 3 > 0 defined only for R 2 > 12d 3 σ/π (for R 2 < 12d 3 σ/π this solution has an imaginary part) 17 . We will show in the following how this can be obtained using duality in a much simpler and pure algebraical way. The transformation of duality under the subgroup α D (θ) can be expressed in our coordinates as while the remaining coordinates x 0 and x 1 are left unchanged (cf. (5.24)). Let us assume that φ(z, z) is the solution of the theory (6.11) with d 3 = 0. The duality transformation of φ(z, z) is then given implicitly as We have therefore Let us note that for θ > 0 the transformation z → z θ double covers the complement of a circle z θ z θ < 4 σθ π ; inside of this circle φ θ is not defined. For θ < 0 this transformation double covers the whole complex plane, the circle zz = − σθ π is mapped to the point z θ = 0. The inversion of (6.17) which shall be inserted to the right hand side of φ θ (z θ , z θ ) is then Now the duality means that where S and S θ are the actions (without the external source term S int ) of the general Galileon theory and its α D (θ) dual respectively. In our case we take the former to be the general interacting theory (with d 3 = 0) and the latter we identify with its dual chosen in such a way that d 3 (θ) = 0. As we know from (5.40) such a theory can be obtained from the general one by duality transformation with θ = 3d 3 and thus for this value the eq.(6.16) is expected to represent the wanted solution of (6.10, 6.13). Let us now verify that it is indeed the case.
Using the duality transformation of the derivatives (c.f. the last equation of (5.24)) Inserting this in the left hand side of (6.13) we get where in the last line we used the explicit form of φ(zz). Therefore for θ = 3d 3 and expressing back zz in terms of z θ z θ we get which means that φ 3d 3 is a solution of the equation (6.13).

Hidden symetries
The Galileon duality often interrelates apparently very different theories. For instance, let us assume a Galileon theory with additional Z 2 symmetry which corresponds to the intrinsic parity, namely On the Lagrangian level this symmetry requires d n = 0 for all n odd. Under the general duality transformation such a Z 2 invariant theory might be mapped onto a dual with some d 2k−1 = 0 and therefore the Z 2 symmetry ceases to be manifest in the dual theory. In this section we will study the way how the symmetries of the original Lagrangian are realized on the dual one. Let us remind the definition of the dual action corresponding to the matrix α where φ α is the duality transformation of the field φ given by (cf. (4.18)) Here we have denoted 18 (∂φ) α ≡ ∂φ α /∂x α . Within this notation the group property of the duality transformations can be formally expressed as The inverse of the transformation (6.26) has the same form with the exchange and can be written symbolically as (6.28) Using this notation the formula (6.25) can be rewritten in the form Now any transformation of the general form Provided the original action is symmetric with respect to the transformation (6.30) we have using (6.29) (6.32) and the dual action is invariant with respect to (6.31).
Let us now give some explicit examples of these general formulae. The first example is the intrinsic parity transformation mentioned in the introduction to this section. In this case, the formula (6.31) simplifies considerably. Let us note that the intrinsic parity transformation (6.24) can be treated as a special case of the duality transformations (5.20) with a matrix Therefore φ P = φ α P and (6.31) has the form and the Z 2 symmetry is realized in the dual theory with action S α [φ] as a duality transformation associated with the matrix β P (α) = α −1 · α P · α = det (α) −1 α P P α BB + α P B α BP 2α P B α BB −2α BP α P P −α P P α BB − α P B α BP , (6. 35) or explicitly The transformation corresponding to the intrinsic parity is therefore realized in the dual theory non-linearly and non-locally as a simultaneous transformation of both space-time coordinates and fields. 19 Here F ∂φ is in fact not independent because it has to be compatible with the remaining two functions in such a way that F ∂φ (Y ) = ∂ φ In the same way we can find the dual realization of the Galileon symmetry (2.1). The general formula (6.31) reads in this case 38) or explicitly A dual Galileon transformation is therefore superposition of space-time translation and Galileon transformation with special values of parameters, as was recognized for the duality of the type (5.24) in [16].

Tree level amplitudes
As we have mentioned in Section 2, the tree level amplitudes up to the 5pt one have surprisingly simple structure though they are sums of a large number of nontrivial contributions stemming from individual Feynman graphs with different topologies. Therefore large cancelations between different contribution have to occur the reason of which is not transparent on the Lagrangian level. In this subsection we will show on an elementary level how these results can be understood better with the help of the duality. For general tree amplitudes we have where I and E represents number of internal and external lines respectively, V is number of vertices and α n is number of vertices with n legs; putting this together we get n (nα n − 2) = E − 2 . Here the sum is over the sequences {α n } d+1 n=3 which satisfy the condition (6.41) and the coefficients M {αn} (1, . . . , E) represent the sum of Feynman diagrams with α n vertices with n legs. In what follows we restrict ourselves to case d = 4, i.e. the sum in (6.42) is over the ordered triplets {α 3 , α 4 , α 5 }.
As we will see in Subsection 6.5, the tree-level S matrix is invariant with respect to the duality α D (θ), therefore the amplitudes have to satisfy the following condition As a last example we take E = 6, the computer calculation of which though possible gives rather lengthy and untransparent final output so it is difficult to reveal any regular structure hidden in it. As we will see also here the duality helps considerably. As we have illustrated above, the tree-level amplitudes are invariants of the subgroup α D (θ) but also their transformation properties with respect to the scalings α S (λ) and more generally α S (λ, κ) are transparent. Let us remind that, under the α S (λ) the couplings d n scale according its dimension (cf. (5.18) with ∆ = (d − 2)/2, which is the canonical dimension of the field φ) which just corresponds to the re-scaling of the units. Note that, for d even we can generalize the above scaling also to λ < 0. From the homogeneity of the tree 21 amplitudes Therefore not only that it is sufficient to know the amplitudes for some representant of the group orbit of α D (θ) in the theory subspace D (1) d+1 but we can also travel between different (but qualitatively similar) orbits using the formula (6.66). 21 Note that, at the loop level, we have additional dependence of the amplitudes on additional dimensionfull parameters, namely on the counterterm couplings as well as on the renormalization scale. 22 As we will see in the subsequent sections, the loop amplitudes have higher degree of homogeneity with respect to re-scaling the momenta.

Classification of the Galileon theories
As we have shown, some physical consequences of the Galileon theories are not directly visible from the Galileon Lagrangian. This concerns e.g. the cancelations of the various contributions to the tree-level amplitudes as well as the hidden Z 2 symmetry of the Galileon action discussed in the previous sections. However, as was seen in the latter case, such properties are usually shared by the theories which are connected by the group of duality transformations (or by some of its subgroup). It is therefore important to describe the equivalence classes of the Galileon theories with respect to the duality.
In what follows we will classify in this sense the Galileon theories in d = 3 and 4. We will restrict ourselves to the theory subspace D (1) d+1 with d 2 = 1/4 and 1/12 respectively and we will consider only the dualities corresponding the upper triangular matrices α which make sense also in the quantum case.

Galileons in d = 4
The properties of the theory which belongs to the theory subspace D • Every theory is dual to theory with d 3 = 0 • I 4 < 0, then theory is dual to just two theories with odd interactions where d 4 = 0 (this can be achieved by duality transformation corresponding to α D (θ ± ) for θ ± = 3d 3 ± √ −2I 4 ) • I 4 > 0, then there is no dual with d 4 = 0 • For (8I 4 ) 3 + (15I 5 ) 2 > 0 theory is dual to exactly one theory with d 5 = 0 • For (8I 4 ) 3 + (15I 5 ) 2 < 0 theory is dual to exactly three theories with d 5 = 0 The invariants I 4 and I 5 scale as d 4 and d 5 , namely I 4 (λ) = λ −6 I 4 , I 5 (λ) = λ −9 I 5 . The situation in three dimension is even simpler. There is only one invariant of the α D (θ) duality From the previous it follows readily Provided I 4 > 0 there is no α D (θ) dual with d 4 (θ) = 0 and naively there is no possibility to change the sign of I 4 by simple scaling because λ < 0 is not allowed in odd dimension for theory with d 2k−1 = 0. However, we can first remove d 3 by α (3d 3 /2) and only then scale with λ < 0 to arrange I 4 = 1, because there is no odd vertex in the dual theory. This leads to the following classification • I 4 = 0 -free theory To summarize, up to the above described α D (θ) and α S (λ) dualities there is only one non-trivial Galileon theory in three dimension the only nonzero amplitudes of which are the even ones. with d 2 = 1/12 fixed. The intersection of these surfices corresponds to the duals of a free theory. Both surfaces are invariant with respect to the scaling.

Duality of the S matrix
On the quantum level the most importatnt object is the S matrix. In this section we will discuss its properties with respect to the Galileon duality.
Let us first briefly remind the well known equivalence theorem which makes a statement about the invariance of the S matrix with respect to the field redefinitions (see e.g. [24]). The S matrix can be obtained by means of LSZ formulae from the generating functional Z[J] of the Green functions which can be expressed in terms of the functional integral.
In this formula we tacitly assume appropriate regularization which preserves the properties of the action with respect to the Galileon symmetry and duality transformations. The action can be expanded in powers of where S 0 [φ] is the Galileon Lagrangian (2.3). The higher order terms S n [φ] in the expansion (6.72) summed up in S CT [φ] represent the counterterms which are necessary in order to renormalize the UV divergences stemming form the n-loop graphs. The discussion of these counterterms we postpone to the Section 6.6, here we only stress that, because of the derivative structure of the Galileon interaction vertices, the counterterms S n [φ] have more derivatives per field than the basic action S 0 [φ], and that under our assumptions on the regularization the counterterms should respect the Galileon symmetry.
In the functional integral the field φ is a dummy variable and can be freely changed by means of the field redefinition φ → F[φ] according to Ignoring the Jacobian on the right hand side of (6.73) for a moment, the sufficient condition for the perturbative equivalence of the S matrices in the theories with actions S[φ] and S F [φ] is that the Fourier transforms of the Green functions of the operators φ(x) and F[φ](x) have the same one-particle poles at p 2 i = 0 up to a simple re-scaling of the residues. This is achieved provided F[0] = 0 and with Z F = 0. This requirement is respected by the Galileon duality transformation which are represented by the upper triangular matrices with α P P = 1. To prove this, it is sufficient to investigate the dualities corresponding to α D (θ) and α S (1, κ) separately because of the decomposition (5.26). In the former case we have and therefore 24 Z F = 1 + O( ) (6.77) while in the latter case we trivially 25 get Z F = κ. Thus the only obstruction which prevents us to make a statement on the equivalence of the on-shell S matrices in both theories also at the loop level is the possible nontrivial functional determinant on the right hand side of 23 In what follows we will often use this notation without further comments unless it shall lead to misinterpretation. 24 The O( ) part stems from the contributions of the terms bilinear and higher in the derivatives of the field φ on the right hand side of (6.76). These terms start to contribute to ZF only at the loop level. 25 The case of αS (λ, κ) is more complicated, because and the behaviour of the Green functions under the re-scaling of the momenta is governed by the renormalization group. This is the rationale for the constraint αP P = 1. However, at the tree level this simplifies to scaling respecting the canonical dimensions.
(6.73). Its actual value depends on the regularization. In what follows we will show that using dimensional regularization the functional determinant equals to one for the duality transformations α D (θ) (the case α S (1, κ) is of course trivial).
For the infinitesimal θ we can expand the functional determinant according to (6.76) as det δφ θ (x) δφ(y) = 1 + 2θTr (∂φ(x)∂) . (6.79) The trace can be further expressed in a standard way (introducing the operators X ≡ x, K ≡ −i∂ and their eigenvectors |x) and |k) respectively) as The first factor equals to zero for well behaved φ while the second one vanishes within the the dimensional regularization. We have thus For the finite transformation we can use the fact that the transformation forms a oneparametric group and thus  26 Let us note that without the knowledge that the Jacobian of this transformation is equal one, in a standard way, we can introduced the ghost fields which would reproduce the studied determinant. At the end, however, one would find that propagators of such ghosts are proportional to 1, and thus every integration over ghost loop with momentum l is of the type: which is true for the dimensional regularization.
The on-shell S matrices in theories with actions S F [φ] and S[φ] are therefore formally equivalent for the above duality transformations. This statement, however, must be taken with care. The first reason is that the counterterm part S CT [φ] of the action has not the form of the Galileon Lagrangian and transforms therefore highly nontrivially with respect to the duality (note that the duality transformation is in general non-local and involves infinite number of terms, cf. Appendix A). The second reason is that though we have formally established equivalence of the on-shell S matrices, the off-shell Green functions stay to be different in both theories. Indeed, we have in fact only proved that Green functions of operators φ(x) in original theory and those of operators F[φ](x) in the dual theory coincide. Therefore, the recursive construction of the counterterms in the dual theory starting with the dual basic action S 0 [F[φ]] will lead to counterterm action S F ]. On the other hand, the counterterms from S CT [F[φ]] will be sufficient to cancel the divergences of the on-shell amplitudes in the dual theory.

Counterterms
From the results of the previous section it seems possible to use the duality relations also at the quantum level. However, this is true only provided the quantum level makes sense. Starting with the basic (i.e. the tree-level) Galileon Lagrangian and chosing an appropriate regularization prescription which preserves the Galileon symmetry (in what follows we use exclusively a dimensional regularization), we can construct one-loop diagrams. Such diagrams will be divergent and will thus need to be renormalized by the counterterms. From a simple dimensional consideration it is clear (and it will be explicitly shown below) that it is not possible to create such counterterms using the basic tree-level Lagrangian. We will thus have to add qualitatively new terms in the action constrained in their form only by the Galileon symmetry. In fact at any order of the loop expansion an infinite tower of new counterterms is necessary. This is of course nothing new, such a mechanism is well studied in many different effective theories e.g. in the Chiral perturbation theory (ChPT) [38,39]. The problem of construction of higher order Lagrangians (e.g. next-toleading-order and next-to-next-to-leading order as is the nowadays status in ChPT) is the problem by itself. Here we will merely classify the order (i.e. the degree of homogeneity in the external momenta or the number of derivatives) of the graphs and the corresponding counterterms at the given loop level.
Let us start with the Weinberg formula [40] in d-dimension for the number of derivatives in the counterterm for a given graph with L loops and vertices V with d V derivatives 27 The number of external legs E and internal lines I is connected via where n V is the number of legs for the given vertex V . We can also simply extract number of loops Together with the previous relation this leads to and thus Let us now define an index of general vertex δ V as a surplus of the derivatives for the general vertex in comparison with the basic Lagrangian, namely for all the vertices of the basic Lagrangian δ V = 0). In terms of such a defined index we can rewrite the formula (6.92) in the form the right hand side of which defines the index δ Γ of a L-loop graph Γ built from the vertices with indices δ V . This formula is in fact an Galileon analog of the Weinberg formula for ChPT and represents thus the connection between the loop expansion and expansion in the diagram index δ Γ , which is the order of the diagram homogeneity in momenta (modulo logs) relative to tree-level diagrams constructed from the basic Lagrangian. Note that according to the formula (6.94) each loop contributes with an additional d + 2 term in the couterterm index δ CT . This means that the counterterms induced by the loops have δ CT > 0 and therefore (because for the vertices of the basic Lagrangian δ V = 0) they must be different form the terms of the basic Lagrangian. In other words the basic Galileon Lagrangian is not renormalized by loops. This proves what is often meant in the literature as the non-renormalization theorem [11,12,13].
Let us note that similarly to the Weinberg formula for the ChPT, the formula (6.94) itself cannot be used for the proof of the generalized renormalizability. Note that the restriction δ CT = N = const. constrains only the number of derivatives d according to but it does not constrain the number n of fields. In the case of ChPT the additional principle is a chiral symmetry which ensures that the infinite number of couterterms differing by the number of fields at each order combine into a finite number of chiral invariant operators. In our case we have only the Galileon symmetry at our disposal. As we have discussed above, it tells us only that the most general Galileon invariant Lagrangian is built from the building blocks ∂ µ 1 ∂ µ 2 . . . ∂ µ k φ where k ≥ 2, therefore the general couterterm with n legs satisfying (6.95) has the general form . Though for n fixed we have finite number of terms, the number n increases to infinity and already at the one loop level (where N = d + 2) we get infinite number of independent terms.

Examples of one-loop order duality
In the previous sections we have explicitly calculated the tree-level scattering amplitudes of the Galileon fields up to six particles. The non-trivial results start with the four-point scattering. In this section we will focus on this process and will study it at one-loop order. Of course, as mentioned above, such a full calculation would necessary need inclusion of so-far undefined Lagrangian L (4) CT , which would play a role of counterterms in this process. However, our main motivation is to explicitly show that the duality is not spoiled at the quantum level (i.e. by loop contributions) at least for the graphs with the vertices from the basic Galileon Lagrangian. We will thus first calculate dimesionally regularized individual contributions to 4-pt scattering at one-loop order in one Galileon theory and show that the final result is connected with other Galileon theory connected by duality.
In the Table 6.7 we summarize the one-loop diagrams to be calculated and their corresponding divergent parts in d = 4 dimension (the full results in d dimension for A 1−6 are summarized in Appendix C). Here we have used the standard Mandelstam variables for four-point scattering: where all momenta are ingoing and on-shell so that s + t + u = 0. The singularity in d = 4 dimension is given by Due to specific form of the of 3-pt vertex in the Galileon theory which can be rewritten in the form the contributions A 7 and A 8 (corresponding to graphs for which V 3 is one of the two vertices of a bubble) are zero also for general d. Indeed, with external momenta on shell, the only term of (6.99) which could contribute is schematically (p ext · l)(p ext − l) 2 where p ext is one of the external momenta and l is the loop momentum. Therefore the (p ext − l) 2 factor cancels one of the bubble propagators which thus degenerate in a massless tadpole and the latter is zero in dimensional regularization. Summing the diagrams together, we get that the divergent part of the amplitude for the 4-pt galileon-scattering at the one-loop order is Note that the degree of homogeneity in external momenta is in accord with the formula (6.94). As we have expected, the singular part (and in fact also the complete result (C.8), cf. Appendix C) depend on the α D (θ) duality invariant I 4 which illustrates the conclusions of Section 6.5 that α D (θ) dual theories produce the same S-matrices. This offers also another possibility how to use the duality relations similar to that we have discussed for the tree amplitudes in Section 6.3. Because I 4 is the coupling d 4 in the dual Galileon theory with new constants d i (θ * ) such as d 3 (θ * ) = 0 we can effectively eliminate 3pt vertices by passing to this dual theory. The only diagram which is left to calculate in such a dual theory is A 4 which simplifies the calculations considerably. Let us present another simple example of the one-loop calculation concerning the selfenergy correction for the Galileon field. The relevant graph is depicted in Fig.3  cf. also [41]. Therefore, on one hand, in Galileon theories with d 3 = 0 we need a counterterm L

and the
CT of the general form (6.96) to renormalize this divergence. The corresponding Feynman rule reads where µ is the dimensional regularization scale which is necessary to restore the canonical dimension of the loop integration and where C r (µ) is a linear combination of the finite parts of the counterterm couplings c (l) k 1 k 2 in (6.96) renormalized at scale µ. On the other hand, in the above mentioned dual theory with d 3 (θ * ) = 0 such a divergence does not occur. This is consequence of the fact that off-shell Green functions are not invariants with respect to the duality as discussed in Section 6.5.
Extreme example of such non-invariance of the counterterms is the case of the free theory and some of its α D (θ) duals. While the free theory does not need any counterterm of the above type, its dual always 28 does. However, as far as the S matrix is concerned, no counterterms are needed for the graphs with the vertices from the basic dual Lagrangian because these graphs have to combine into the trivial S matrix of the original free theory which is trivially divergence-free. Therefore, the contribution of the divergent part of (6.102) and analogical counterterms (which are needed to renormalize the divergent subgraphs in the Table 6.7) has to cancel in the final result. This is, however, not true for the finite part of the counterterms the couplings which are in principle independent. E.g. the renormalization of the bubble subgraph in the graphs A 3 in Table 6.7 brings about the contribution (for d → 4) (6.103) Therefore, the only possibility how to recover the free theory S matrix also in the dual theory with counterterms is to set at some scale all the renormalized counterterm coupling constants equal to zero. Becasue the couplings run with the renormalization scale, this might seem to be insufficient, because at another scale the finite parts of the counterterms are in general nonzero. However, because in the amplitude all the contributions of the divergent parts of the counterterms cancel, in the same way are also canceled all the 28 Note that any such dual has d3 = −θ/3 = 0.
contributions stemming from the changes of the counterterms couplings with change of the renormalization scale 29 .

Summary
In this paper we have studied the duality transformations of the general Galileon theories in d dimensions. According to the interpretation of the Galileon action as the generalized WZW term of the effective theory describing a Goldstone boson of the spontaneous breakdown of the Galileon group down to the d-dimensional Poincaré group we have studied the most general coordinate transformations on the corresponding coset space. The requirement that such a general transformation acts linearly on the basic building blocks of the Galileon Lagrangian (and therefore it represents a duality transformation) constraints the form of the transformation uniquely up to four free parameters. Under composition these duality transformations form a group which can be identified with GL(2, R). All the up to now known Galileon dualities can be identified as special elements (or one-parametric subgroups) of this duality group. We have also studied its action on the space of the Galileon theories and found a basis of the independent invariants of one of its most interesting one-parametric subgroups denoted α D (θ). This subgroup is represented in the space of fields as a field redefinition which can be understood both as a simultaneous space-time coordinates and field transformation or as a non-local change of the fields which includes infinite number of derivative dependent terms. We then have studied the applications of the duality group on concrete simple examples. We have shown that we can use it to generate classical solution of the interacting Galileon theory from the solution of the more simple one even when the Galileon is coupled to the local external source. As a second example we have demonstrated the usefulness of the Galileon duality for calculations of the tree on-shell scattering amplitudes and for finding the relations between the contributions of the apparently very different Feynman graphs with completely different topologies. We have also established the dual formulation of the additional symmetries of the Lagrangian and found that the Z 2 symmetry is realized non-linearly and non-locally in the dual theory.
As another example we have classified the equivalence classes (with respect to the duality subgroup α D (θ) combined with scaling) of the Galileon theories in three and four dimensions and found e.g. that there is up to the above dualities only one nontrivial interacting theory in three dimensions which exhibits the Z 2 symmetry. Then we have discussed the transformation properties of the S matrix on the loop level and established its formal invariance within the dimensional regularization, though only the tree-level part of the complete action with couterterms transforms nicely under the duality field redefinition and as we have discussed on a concrete example of the one-loop four-point on-shell amplitude, due to the counterterms the duality is not completely straightforward. It rather holds on the regularized level for the loop graphs with vertices form the basic tree-level Lagrangian. We have also touched the problem of the couterterms classification based on a generalization of the Weinberg formula and with help of the latter we discussed the non-renormalization theorem.
Note added: After this work was completed two works [37,36] closely connected with the topic studied in this paper appeared. Both these papers concern the properties of the one-parametric duality subgroup denoted as α D (θ) in our notation and partially overlap with our results.
A. Bottom up construction of the duality subgroup α D (θ) In this Appendix we get more elementary treatment of the Galileon duality corresponding to the subgroup α D (θ). In fact this was the way we had started to think about the Galileon duality.
Let us assume an infinitesimal field transformation where θ infinitesimal parameter. The infinitesimal change of φ can be also understood as an action of the following operator (which is defined on the space of the functionals F Therefore to the first order in θ the transformation (A.1) conserves the Galileon structure of the Lagrangian and merely shifts the coupling constants d n by δd n . Note that the transformation (A.1) with finite θ can be used to eliminate the cubic term from the interaction Lagrangian, however, the Galileon structure of the Lagrangian is spoiled with additional interaction terms which are generated by the transformations. The way how to eliminate the cubic term consistently without leaving the space of the Galileon theories is now clear. It suffices to construct the finite transformation by means of iteration of the infinitesimal one, i.e. to exponentialize it according to φ θ = exp (δ θ ) φ = exp θ∂φ · ∂φ δ δφ φ = φ + θ∂φ · ∂φ + 2θ 2 ∂φ · ∂∂φ · ∂φ + . . . It is not difficult to show that d n (θ) = d n (α D (θ)) where the right hand side is given by (5.25). From this construction it is clear that the transformatitons φ θ form a one parametric group.
C. Full form of 4-pt scattering amplitude and self-energy