Special Galileon at one loop

We present a complete one-loop renormalization of the Special Galileon $S-$matrix. Especially we give a complete list of the higher derivative operators which are necessary for one-loop on-shell renormalization and prove the invariance of the one-loop on-shell effective action with respect to the Special Galileon symmetry. This enables us to enlarge the validity of the enhanced $O(p^3)$ soft behavior of the scattering amplitudes to the one-loop level. As an illustration we discuss explicitly the four-point and five-point one-loop scattering amplitudes and comment on some conjectures appearing in the existing literature.

1 Introduction: Why is the Special Galileon special?
The Galileons are remarkable derivatively coupled scalar effective field theories with plethora of very interesting properties both at the classical and the quantum level. In the literature, the (cubic) Galileon emerged at first time as the only interacting (zero helicity) mode in the decoupling limit of the Dvali-Gababadze-Poratti modified gravity model [1,2] and almost at the same time also in the similar limit of the massive gravity model [3]. Soon it has been recognized [4] that the Galileon in its generalized form might be a promising local modification of the General relativity at large scales with several appealing properties. In particular, regardless of the presence of highly nonlinear higher derivative couplings in the Lagrangian, the Galileon obeys the second order equation of motion which ensures the absence of the Ostrogradsky ghosts. It has been also shown, that near the massive sources the Galileon field is suppressed by the Vainshtein screening mechanism [5] and that the basic Lagrangian is stable with respect to the quantum corrections [6,7,8]. The seminal paper [4] initialized a boom of increasing interest in the Galileon theories concerning their formal properties , the possible generalizations, e.g. [9,10] as well as the cosmological applications. For a pedagogical reviews and for more comprehensive list existing literature see e.g. [11,12] In the general case, the basic flat space Galileon Lagrangian collects the most general terms built of the single scalar field and its derivatives (up to and including the second order ones, with n fields and 2n − 2 derivatives). These terms can be uniquely determined by the requirement of yielding the second order equations of motion and by invariance with respect to the polynomial shift symmetry of the first order, where a and b µ are real parameters. The latter property has an important consequence at the quantum level, namely the tree-level on-shell scattering amplitudes poses an enhanced soft behavior [13,14,15]. This means that the scattering amplitudes vanish as the second power of momentum when one of the external particles becomes soft, which is higher than one would naively expect from the simple counting of the derivatives in the Lagrangian. The general basic Galileon Lagrangian in D dimensions can be written in the form where d n are free real n−point couplings. For the theory to be well defined on the quantum level we demand d 1 = 0 (no tadpoles) and d 2 = (−1) D /2 (D − 1)! (canonical normalization of the kinetic term). The above mentioned enhanced soft behavior enables the full on-shell reconstructibility of the tree-level scattering amplitudes once the basic set of the seed amplitudes from the four-point up to the D + 1 point one is known [14]. On the other hand, due to the rich set of dualities of the Galileon Lagrangian [16,17,18,19], there is a many-to-one correspondence between the constants d n and the on-shell physics represented by the scattering amplitudes. Particular choice of the couplings d n can further increase the symmetry of the Lagrangian (1.2). Namely, we mean the choices  [20,21] δφ = θ 2 H µν α 2 x µ x ν ± ∂ µ φ∂ ν φ (1.5) where H µν is arbitrary fixed traceless 1 symmetric tensor and θ is an infinitesimal parameter. On the quantum level this symmetry is responsible for further enhancement of the soft behavior. In the above theories with β = 0 the tree-level amplitudes vanish as the third power of momentum in the single particle soft limit. The latter theories (1.3) and (1.4) with β = 0 are known as Special Galileons [13,22]. The Special Galileons are special in several aspects. First, their soft behavior is in a sense extremal, since as it was shown in [15], the O p 3 single soft limit is the highest possible within the single scalar effective theories with nontrivial power counting. To be more precise, let us assume a derivatively coupled single scalar effective field theory and for each elementary vertex V in the Lagrangian let us determine the ratio where D V is a number of derivatives and E V is a number of external legs of the vertex V . The soft behavior of such a theory can be characterized by the soft exponent σ which 1 For H µ µ = 0 the transformation (1.5) is a duality of the family, which transforms Lagrangian (1.2) and (1.4) with parametres (α, β) into the Lagrangian with parametres α, β ∓ θαH µ µ .
corresponds to the O (p σ ) behavior of the amplitudes in the single particle soft limit p → 0. Then for the general Galileon we get ρ V = 2 for each admissible vertex and the soft exponent is σ = 2. Provided we assume only theories with fixed ρ V = ρ ≥ 1, i.e. only with vertices with 2 + ρ (E V − 2) derivatives and with the soft limit characterized by soft exponent σ, then the theories with nontrivial enhanced soft behavior are those for which ρ ≤ σ since in the opposite case there is enough derivatives per field in each vertex to ensure the O (p σ ) soft limit automatically. However, as shown in [15], not all the nontrivial pairs (ρ, σ) are admissible, namely they have to satisfy the bounds ρ ≥ σ − 1 and σ ≤ 3. Note that the Special Galileons saturate both this bounds sitting in the very corner of the admissible (ρ, σ) region. This means that the Special Galileon together with the Dirac-Born-Infeld theory and the Non-linear sigma model belongs to the set of the exceptional scalar effective field theories for which there exist the Cachazo-He-Yuan representation [22]. The second special property of the Special Galileon is that it is (contrary to the general Galileon theories) potentially well defined as an perturbative effective quantum field theory -at least formally when we assume that the regularization and renormalization preserves all the symmetries. The full Lagrangian is then where L CT corresponds to the higher derivative counterterms needed for perturbative renormalization. Here we tacitly assume that L CT shares the symmetries of the basic Lagrangian L b . As was shown in [19], it is possible to organize the perturbative expansion according to the hierarchy of the vertices and according to the hierarchy of the corresponding Feynman graphs. This hierarchy is based on the assignment of an index i V to each vertex V (including those coming from L CT ), where Here D V and E V have the same meaning as above. Note that for the basic Lagrangian (1.2) we get i V = 0 and therefore this index measures the abundance of the number of derivatives relatively to the basic Lagrangian L b . The index of the counterterm vertices which are necessary in order to renormalize the one particle irreducible graph Γ is then given (for derivation of theis formula see [19]) where L Γ is number of loops and i V are the indices of the vertices of the graph Γ. The indices of the counterterms are thus related to the loop expansion. Therefore, in principle, the perturbative renormalization in the effective theory sense would be possible provided there is only finite number of vertices with fixed i CT . Then at each level of the hierarchy, given by the contributions of all the Feynman graphs Γ with i Γ CT fixed, there would be only finite number of unknown parameters.
In the case of the general Galileon, the only other constraint for the counterterms is the invariance with respect to the symmetry transformation (1.1). Then, however, there is an infinite number of possible uncorrelated vertices with given fixed i CT and therefore the formal effective theory looses its predictivity at each hierarchy level (see [19] for more detailed discussion).
On the other hand, the generalized polynomial shift symmetry (1.5) of the Special Galileon is strong enough to restrict the form of the counterterm Lagrangian with i CT fixed in such a way, that at each hierarchy level there is only finite number of unknown parameters in the S−matrix. The situation here is somewhat similar to the case of chiral perturbation theory [23,24] where the nonlinearly realized chiral symmetry allows only for finite number of low energy couplings at each order in the derivative expansion. This of course does not mean that there is only finite number of vertices at each hierarchy level. Rather the infinite number of possible vertices have correlated couplings and as a result they are combined into a finite number of operators invariant with respect to (1.5).
The third distinguished feature of the Special Galileon is the beautiful geometry behind the polynomial shift symmetry (1.5). The Special Galileon field can be understood as a scalar degree of freedom which describes fluctuations of a D−dimensional brane in a 2D−dimensional pseudo-Riemanian target space R 2,2D−2 treated as a Kāhler manifold 2 . The hidden Special Galileon symmetry (1.5) corresponds then to the nonlinearly realized subgroup of the symmetry group C D ⋊ U (1, D − 1) of the target space. This interpretation of the Special Galileon allows for simple construction and classification of the counterterm Lagrangian L CT (see [21] for more details) 3 . The basic building blocks for higher derivative Lagrangians are then the effective metric (the ± in the following formulas correspond to the two choices of the transformation prescription (1.5)) the extrinsic curvature tensor the Christoffell symbol the corresponding covariant derivative D µ and the covariant Levi-Civita tensor where g is the absolute value of the determinant of the effective metric g µν . Any diffeomorphism invariant built from g µν and its inverse g µν , K αµν , and their covariant derivatives 4 is automatically invariant with respect to (1.5). In order to construct the action, we need also the invariant measures, namely, for the plus sign in (1.5), the measures d D Z and d D Z

13)
2 This is the case of the "plus" branch of the Special Galileon. For the "minus" branch an appropriate analytic continuation of the parameter α is necessary. The target space is then R D,D which has no compatible complex structure. See for [21] more details. 3 Alternative way of the classification of the higher derivative Special Galileon Lagrangians was developed in [25,26] using the coset construction. 4 Thanks to the relations between these geometrical objects, the other building blocks, like e.g. the Riemann tensor and its descendants, are not needed without any loss of generality.
or, for the minus sign in (1.5), the measures d D Z + and d D Z − 14) and the canonical one As discussed in [21], the typical off-shell counterterm action which is invariant with respect to the generalized polynomial shift symmetry (1.5) is e.g. of the form according to the sign in (1.5) and (1.9). The functions L ZZ , · · · , L − are diffeomorphism invariants built from the above geometrical building blocks. However, this is not the whole story. As a consequence of the invariance of the independent measures d D Z + and d D Z we can costruct another invariant given as for the "plus" branch of the transformation (1.5) and for the "minus" branch. Note that these invariants has been missed in [21]. The existence of these invariants allows to forget the measures d D Z and d D Z or d D Z + and d D Z − since these can be constructed using the canonical measure (1.15) and the functions of the invariants σ, e.g.
The most general off shell counterterm action is then for the two branches of the Special Galileon where L CT is diffeomorphism invariant built from the scalar σ, the extrinsic curvature tensor K µνα , their covariant derivatives, the inverse metric g µν and Levi-Civita tensor E µναβ . Remarkably, the transformation (1.9) can be easily enlarged to the case of additional non-Galileon fields ψ µ 1 ... ν 1 ... . To get a minimal coupling of these fields with the Special Galileon, one simply constructs the diffeomorphism invariant Lagrangians of the non-Galileon fields on the general curved background and for the metric then substitutes the effective metric (1.9). The resulting Lagrangians are then automatically invariant under the simultaneous transformation (1.5) and the transformation of the ψ µ 1 ... ν 1 ... fields given schematically as and therefore can be added to the action given by the general formula (1.21). Note that the prescription (1.22) formally corresponds to the coordinate transformation For the fields, which carry also spinor indices, one needs additional geometrical structures, namely the vielbain m µ a and the spinor connection β a bµ which are expressed in terms of the second and third derivatives of the Galileon field φ (see [21] for the explicit formulas). The invariance of the resulting action with respect to the generalized polynomial shift symmetry (1.5) then guaranties under some additional assumptions 5 the enhanced O p 3 soft Galileon limits of the tree level scattering amplitudes.
Though the Special Galileon is well understood at the tree level, considerably less is known about its true quantum properties. In the literature, several one-loop calculations exists for the case of cubic Galileon [27,28,29,30,31] and for general Galileon [19,32], but systematic analysis with stress to the Special Galileon case is still missing. Namely, it is not known, whether the loops do or do not break the invariance with respect to the generalized polynomial shift symmetry (1.5) of the Special Galileon and whether the enhanced O p 3 soft behavior survives the quantum corrections. In this paper we initialize the studies in this direction and try to fill this gap partially by means of explicit calculation of the UV divergent part of the one-loop on-shell effective action using Dimensional Regularization (DR) and providing the complete one-loop renormalization of the S-matrix. We also calculate explicitly the lowest scattering amplitudes at one loop and discuss their compatibility with other approaches.
The paper is organized as follows. In the section 2 we briefly remind the basic facts concerning the functional approach to the S−matrix and its relation to the on-shell effective action. Section 3 is devoted to the issue of the quantum fluctuations in the classical Galileon background, we show here that the fluctuations can be described with action which is manifestly invariant with respect to the hidden Special Galileon symmetry. In the section 4 we calculate the infinite part of the one-loop on-shell effective action for the Special Galileon and give a complete classification of the counterterms in D = 4 for i Γ CT ≤ 6. In the section 5 we briefly discuss the extension of the soft theorem for Special Galileon to one loop. The section 6 is devoted to explicit examples of the scattering amplitudes. In section 7 we summarize and further discuss our results. The technicalities concerning the classification of the counterterms are presented in the appendix A 2 The generating functional of the S−matrix and the on-shell effective action For reader's convenience and in order to fix our notation let us first briefly review the interrelation of the S−matrix and the on-shell effective action (for original papers see [33,34,35]). The starting point for our calculation is the general formula for the perturbative S−matrix, which we treat here as the functional S [φ] of the external field φ and which is constructed is the interaction part of the action with UV counterterms included and the functional differential operator generates formally the chronological contractions of the perturbative Wick expansion of the S−matrix. From the functional S [φ] we can derive the operator S−matrix in the Dirac interaction picture inserting for φ the free field operators φ I and treating all the operator product as normally ordered 7 , namely Therefore it is sufficient to know S [φ] on shell, i.e. for the functional argument φ satisfying the free equation of motion φ = 0. The connected on-shell n−point scattering amplitudes A n (p 1 , . . . , p n ) can be obtained directly from the functional T [φ] by means of the functional differetiatial operation which is equivalent to the contraction of the operators φ I with the creation or annihilation operators in the initial or final state 8 Using Gaussian integration we can rewrite (2.1) to the form which formally includes the LSZ formulas is the complete action with UV counterterms included In order to obtain the loop expansion of Here and in what follows we often use the condesed notation where the dot means integration over the corresponding spacetime coordinates, e.g.
7 For this reason, the functional S [φ] is called the normal symbol of the S−matrix. 8 For simplicity, in the following formula we tacitly assume all the particles to be outgoing. and ξ is a new integration variable of the functional integral. Expanding now in powers of √ we get (up to an inessential overall constant) Let us now expand the functional T [φ] and the action S [φ cl ] in powers as are the one-loop counterterms and T tree and T 1−loop [φ] are the tree-and one-loop level connected S−matrices respectively. We can then identify Note that when understood as the functional of φ cl the one-loop connected S−matrix coincides with the one-loop effective action Γ 1−loop [φ cl ]. Note also that for φ on shell (i.e. for φ = 0) the field φ cl satisfies the classical equation of motion (EOM) Therefore provided we are interested only in the on-shell scattering amplitudes up to one loop order we need to know the one-loop on-shell effective action Γ 1−loop [φ cl ]. i.e. the one loop effective action with argument satisfying the classical equation of motion. This well known fact will be crucial for our following calculations.

Quantum fluctuations in the classical background
As discussed in the previous section, the key object for our further calculation is the oper- Taking the second functional derivative we derive for the fluctuation operator where we denoted , and therefore the quantum fluctuations in the quadratic approximation are described by the action In the case of Special Galileon we can sum up the right hand side of (3.3) and (3.4) in a closed form. Indeed, inserting (1.3) with β = 0 we get for the first branch of the Galileon Lagrangians 9 Using the general identity valid for any D × D matrix M µν we can rewrite the EOM in a compact form as where we denoted (3.8) 9 We have rescaled the coefficints d In the same way, using instead (1.4) for the second branch of the Special Galileon Lagrangians, we get δS Taking the partial derivative of (3.7) and (3.9) with respect to ∂ µ ∂ ν φ cl we get where ∂∂φ cl denotes the D ×D matrix with elements ∂ µ ∂ ν φ cl . This can be further simplified using the EOM (3.7) and (3.9) in the form 10 D (+) and similarly G µν Here g µν is the matrix inverse of the Special Galileon effective metric g µν = η µν ± ∂ µ ∂φ cl · ∂∂ ν φ cl /α 2 (see (1.9)) and g = |det g µν | is its determinant. The fluctuation action (3.5) is thus rewritten in the geometrical form 11 is real. Here we further suppose that D (+) + is positive as suggests the weak field limit D 11 Let us note, that in the case of general Galileon in D = 2 dimensions, the fluctuation action (3.5) can be also rewritten in the geometric form The exclusivity of the Special Galileons lies in the fact, that in this case the inverse metric Gµν is computable and equals to the explicitly known effective metric gµν . This ensures the invariance of the fluctuation action with respect to the hidden Special Galileon symmetry. This can be used for a formal proof of the invariance of the pure loop part of the on-shell effective action Γ 1−loop loop [φ cl ] with respect to the hidden Special Galileon symmetry. Indeed, up to an inessential constant 12 , where appropriate regularization compatible with the symmetry (e.g. the dimensional regularization will do the job as we will discussed later) is tacitly assumed. Thus under the transformations (1.5) and (3.16) where the Jacobian J = 1 − θTr H αβ ∂ α φ cl ∂ β can be shown to be equal to one (this holds within the dimensional regularization -see e.g. similar calculation in [19]), which proves the statement.
In what follows we will concentrate on the UV divergent part of Γ

The UV divergences at one loop
The most economic way how to obtain the UV divergent part of Γ 1−loop loop [φ cl ] within the dimensional regularization is to use the master formula stemming from the heat kernel expansion [36,37]. From (3.15) it follows that where D µ is the covariant derivative associated with the effective metric g µν . Therefore, up to an inessential constant, we can express Γ 1−loop loop [φ cl ] in terms of the heat kernel (here we use the Lorentzian signature, see e.g. [38]) using the well known formula for logarithm of the operator determinant At the coincident points x = y the heat kernel K (τ ; x, y) has the following asymptotic expansion for τ → 0 where a n (x)'s are the coincidence limits of the Seeley-DeWitt coefficients. The UV divergences of the effective action Γ 1−loop loop [φ cl ] are then connected with the lower limit of the integral in the formula (4.3) and can be cured by dimensional regularization writing D → D − 2ε. This gives the master formula valid for D even 13 where µ is the dimensional renormalization scale parameter. The Seeley-DeWitt coefficients a n (x) are calculable and explicitly known for n ≤ 10 in terms of the geometrical invariants (for a review and a complete list of references see [39]), e.g.
Here R αβµν , R µν and R are the Riemann curvature tensor, the Ricci tensor and the scalar curvature corresponding in our case to the effective metric g µν . All these objects can be expressed in terms of the extrinsic curvature tensor (1.10) and the inverse metric g µν , namely (see [21] for more details) As discussed in the introduction, due to their geometrical nature, all the Seeley-DeWitt coefficients are then automatically invariant with respect to the hidden Special Galileon symmetry and so is the divergent part of the one-loop on shell effective action (4.5).
In what follows we will concentrate on the four-dimensional case. Note that we can further simplify the above formulas dropping the last term in a 2 (x) which is a total derivative and also eliminating a total derivative corresponding to the four dimensional Gauss-Bonett term (4.8) The result then reads Note that the index of the vertices of (4.9) equals to (note that g µν contain exactly two derivatives per field while K σνβ has three derivatives per field and that RR ∼ K 4 ) 13 For D odd all the one loop infinities are removed automatically by means of the dimensional continuation.
This coincides with the index of the corresponding one-loop graphs (cf (1.8)). The dimensional regularization therefore respects the hierarchy of the counterterms, especially the basic Lagrangian is not renormalized. This is not true for other regularization schemes, e.g. for the momentum cutoff corresponding to the deformation of the integral in (4.3) by means of introducing lower limit of the integration of the order 1/Λ 2 . In such a case all the Seeley-DeWitt coefficients up to n = 2 (or n = D/2 for general D) contribute to Γ 1−loop loop [φ cl ] ∞ . Because we are concentrated on the S−matrix and on shell amplitudes, we can simplify (4.9) further using the classical EOM. Indeed, taking a derivative of (3.1), we get which can be rewritten with help of (3.4), (3.14) and (1.10) as The Ricci tensor given by eq. (4.7) then simplifies where the bold colon (and in general the bold dot in what follows) denotes the contraction of the adjacent indices with help of the inverse effective metric g κλ . Using the same shorthand notation we get for the scalar curvature where · denotes the trace with respect to the inverse metric g κλ . Finally we get or more explicitly The two operators on the right hand sides of the last two equations are therefore the only operators invariant with respect to the hidden Special Galileon symmetry which have to be inserted with infinite coefficients into the counterterm action S 1−loop

CT
. Interestingly, there are no other non-vanishing invariant operators of the form K 4 provided the EOM constraint (4.11) is satisfied 14 .
14 See appendix A for more details.
The one-loop counterterm action which is necessary to renormalize the infinities of the on-shell amplitudes up to and including the graphs with the graph index i Γ CT ≤ 6 (cf. (1.8)) therefore reads where we have denoted In the formula (4.16), Λ is the scale which controls the systematic derivative expansion, or more precisely the expansion in the graph index i Γ CT , and c i are dimensionless bare couplings to be specified later. The finite part of the counterterm action S 1−loop CT, finite has the general form (1.21) . However, provided the EOM (3.7) or (3.9) are satisfied, the invariant σ and their covariant derivatives do not correspond to the independent building blocs. Indeed, using the EOM in the form (3.7) we get from (1.18) for the "plus" branch and analogicaly for the "minus" branch. As a result, S 1−loop CT, finite can be written without loss of generality in the form 15 Note that the measure d 4 x √ g does not contribute to the index since it contains two derivatives per field. 16 Here complete means modulo integration by parts and use of EOM. 17 These two operators correspond to the quartic and quadratic divergences respectively when the momentum cutoff is used instead of dimensional regularization.

and we get two invariant operators with two covariant derivatives
However, as shown in the appendix A, the operators O 5 and O 6 vanish due to the relation which is valid as a consequence of EOM, and the operator O 7 can be rewritten (again using EOM and integration by parts) as a linear combination of the operators O 1 and O 2 plus a total derivative. The same is true for the apparently independent operator The parity odd operators with i O j ≤ 6 of the type where E µ 1 µ 2 µ 3 µ 4 is the contravariant Levi-Civita tensor (1.12), vanish on-shell due to the symmetry properties of the building blocks (see appendix A).
To summarize, there are only four independent on-shell counterterms, interestingly just those which already appeared in the Seeley-DeWitt coefficients (4.6) The bare parameters c i are expressed in terms of the finite couplings c r i (µ) renormalized at the scale µ according to 18 and their renormalization scale dependence is determined in terms of the coefficients β i as The natural values of c r i (µ) is of the order O (1). Therefore, in order to avoid incommensurable effects stemming from the counterterms and from the loops, we expect 2β j = O (1) and thus the natural value of the scale Λ is Λ ≃ 2 (4πα) 1/3 ≃ 4.6 × α 1/3 .

The scattering amplitudes at one loop and the soft theorem
Using the results of the previous section we can write the renormalized (i.e. finite) one-loop S−matrix in the form where we denoted as S 1−loop eff [φ cl ] the nonlocal one-loop effective action given by Note that φ cl is still determined by (2.7). Let us now change for a moment the definition of φ cl according to is the interaction part of the effective one-loop action S 1−loop eff . Then we can compare (5.1) and (5.4) with the tree-level formulas (2.11) and (2.7). Note that the latter correspond to amplitudes given as a sum of the tree graphs built from free propagators and vertices derived form the basic interaction action S int b . Therefore we can conclude, that the modified prescription (5.1) and (5.4)  As we have proved in the previous section, the effective action S 1−loop eff [φ cl ] is invariant 19 with respect to the hidden Special Galileon symmetry. Moreover, the effective action S 1−loop eff [φ cl ] contains only even verteices, especially there is no cubic vertex present. Therefore, using the general theorem connecting the symmetry of the action and the soft behavior of the tree-level scattering amplitudes [15], we can conclude that summing up all the tree graphs constructed with use of S 1−loop eff [φ cl ] we get amplitudes with enhanced O p 3 soft limit, i.e. the enhanced soft behavior is preserved also for the modified one-loop amplitudes.
Originally we were interested in the amplitudes given by graphs with index i Γ CT ≤ D + 2. These correspod to a subset of graphs contributing to the modified one-loop scattering amplitudes, which can be identified by counting the powers of the scale Λ introduced in (4. 16) and (4.20), and keeping the contributions up to and including the order O Λ −D−2 . Here we have to treat the vertices stemming form . Because the contributions of the graphs with different i Γ CT cannot cancell each other due to the different degree of homogeneity in momenta, this above soft theorem remains true also when we restrict ourselves to the contributions with i Γ CT ≤ D + 2 only. In this sense the soft theorem for the Special Galileon is valid also at the one-loop level.

Special Galileon amplitudes at one loop -explicit example
Let us illustrate the above considerations using explicit expressions for the one-loop four point amplitude in D = 4. For definiteness we will use the "plus" branch of the Special Galileon with the basic Lagrangian (1.3). The results for the "minus" branch can be obtained by appropriate analytic continuation in the parameter α.
The contribution i Γ = 0 corresponds to the basic Lagrangian and stemms from the fourpoint vertex 20 The corresponding off-shell vertex in the momentum representation is where G (p 1 , p 2 , p 3 ) is the Gramm determinant of the momenta p 1 , p 2 and p 3 . The on-shell amplitude reads then where s, t and u are the usual Mandelstamm variables and all the momenta are treated as outgoing. The O p 3 i soft behavior of the amplitude A i Γ =0 is manifest. The pure one-loop contributions corresponds to the bubble graphs with two vertices (6.2), explicitly and where q mn = l + 1 2 (p m + p n ). For further convenience, let us also introduce the notation The result of the loop integration has the form A bubble (p i , p j , p k , p l ) = P (s ij , s ik , s il ; ε) B (s ij ; ε) (6.8) where ε = 2 − d/2 , the function B (s ij ) is the scalar two-point function given as and P is the following polynomial Finally we get for the bubble The actual value of the constant c bubble depends on the details of the dimensional regularization scheme. We use the t'Hooft-Veltman scheme for the reduction of the tensor integrals, for which c bubble = −γ + ln 4π + 46 15 . (6.12) The complete one-loop contribution to the amplitude is then Let us now consider the contributions of the higher derivative counterterms. At the level i Γ = 2 we have where now the normal dot (and normal colon in what follows) means contraction of adjacent indices with flat metric η αβ and · denotes a trace with respect to the same flat metric. The effect of this Lagrangian is therefore twofold. The term quadratic in the fields contributes to the off-shell two point function of the field φ; for this contribution we get explicitly Note however, that the derivative of Σ i Γ =2 p 2 vanishes on shell and therefore there is no external leg renormalization. The quartic terms in (6.14) are responsible for the contact cotribution to the four point on-shell amplitude. We get and thus there is no i Γ = 2 contribution to the four-point amplitude. At the next level i Γ = 4 we have the following expansion of the Lagrangian Again, there is a contribution to the off-shell two point function, which, however, has no effect to the on shell four point amplitude. The contact contribution which stemms form the quartic term gives Finally, for the i Γ = 4 counterterm contribution we have the Lagrangian which yields the contact terms of the form The infinite part of A i Γ =6 CT then reads and cancels the infinite part of the of the loop contribution CT can be obtained from (6.20) by the replacement c i → c r i (µ). Finally we get 21 Here we used the identity (s 2 + t 2 + u 2 ) 3 = −12s 2 t 2 u 2 + 4 s 6 + t 6 + u 6 in order to be able to compare the results of the section 6 with existing literature.
where we abbreviated Note, that the O p 3 soft behavior is manifest for all the above components of the amplitude. In the case of the four-point amplitude it is somewhat trivial statement due to the special four-particle kinematics and due to the power counting of the individual contributions.

Summary and discussion
In this paper we have studied the issue of one-loop renormalization of the Special Galileon S−matrix. First we calculated the UV divergent part Γ 1−loop loop [φ cl ] ∞ of the one-loop on-shell effective action and proved its invariance with respect to the hidden Special Galileon symmetry for general space-time dimension D. The key ingredient of the proof was the fact, that we were able to express the action describing the quantum fluctuations in the on-shell classical background in terms of the geometric building blocks which were covariant with respect to the Special Galileon symmetry. We have further found appropriate prescription for the transformation of the fluctuating field with respect to the Special Galileon symmetry which ensures the invariance of the fluctuation action and, as a consequence, also the invariance of the complete one-loop on-shell effective action Γ 1−loop loop [φ cl ]. For D = 4 case we constructed the complete set of independent higher derivative counterterms relevant for the calculation of the scattering amplitudes, up to and including graphs with index i Γ = D Γ − 2E Γ + 2 = 6. The resulting counterterm action can be expressed in a manifestly invariant form with respect to the hidden Special Galileon symmetry, namely Here d 4 x √ g is invariant measure corresponding to the effective metric and the invariant operators O j are given in terms of the extrinsic curvature tensor and in terms of the inverse effective metric g µν as Here the bold dots and . . . mean contractions and trace with respect to g µν respectively. These operators form a complete basis of the on-shell counterterms relevant for the renormalization of the S−matrix at one loop and i Γ ≤ 6. We have identified the infinite parts of the bare coupling c j and established the running of the corresponding renormalized couplings c r j (µ) with the renormalization scale µ. Note, that the action S 1−loop CT generates only even vertices (especially there is no cubic vertex present). As we have proved, also the nonlocal part of the i Γ ≤ 6 on-shell effective action S 1−loop eff stemming form the loops is invariant with respect to the hidden Special Galileon symmetry and contains only even nonlocal vertices. Using the general theorem about the relation between generalized polynomial shift symmetries and the soft behavior of the scattering amplitudes we proved that the enhanced O p 3 soft limit of the amplitudes is preserved also at one-loop and i Γ ≤ 6.
Let us stress, that the proof of the manifestly invariant form of S 1−loop CT and of the nonlocal one-loop effective action S 1−loop eff heavily depends on the fact that only the on-shell configurations satisfying the classical equation of motion derived form the basic Lagrangians are relevant for the calculation of the on-shell scattering amplitudes. Also, this classical equation of motion were used for the elimination of the redundant operators and reduction of the basis of the counterterms. Therefore, though the counterterm action S 1−loop CT guaranties the finiteness of the on-shell scattering amplitudes, it is not sufficient to make also the off-shell Green functions finite.
In order to renormalize also the one-loop off-shell Green functions, the simple explicit calculation of the UV divergent part of the effective action is not possible. In general we can expect that additional counterterms will be needed, and that the simple structure of the higher order Lagrangians will be lost. For instance, the most general counterterm action invariant with respect to the hidden Special Galileon symmetry should be of the form (1.21), with additional invariant odd in the field which vanish on-shell, namely 22 Naively, at the level i Γ ≤ 6 we might expect the invariant action of the form where the diffeomorphism invariant operators O j are now constructed as contractions (with respect to the effective metric g µν ) of the extrinsic curvature tensor, its covariant derivatives and covariant derivatives of the scalar σ. Schematically with n j +m j +r j k j ≤ 4 (see e.g. the operators O 5 , . . . , O 8 discussed in section 4 and appendix A). To list a complete basis of such off-shell independent operators is beyond the scope of this paper. However, even if we were able to classify the operators O j of the above type, this is not the whole story. Note that the constants c j in (7.6) can be freely replaced with arbitrary functions c j (σ) without changing the index of the corresponding counterterms. Thus, the presence of the off-shell invariant σ obscures the complete classification of the offshell counterterm action S CT . The situation is somewhat similar to the three-flavor chiral perturbation theory with additional U (1) pseudoscalar corresponding to η ′ (cf. [23,24]).
In that case, the invariant Lagrangians is determined up to arbitrary potentials which are functions of the η ′ field. Note also, that because the off-shell invariant σ is odd in the field, the off-shell basis should contain also the operators giving rise to the odd off-shell vertices. However, the contributions of such vertices have to cancel each other in the on-shell amplitudes.
Recently, a complementary classification of the higher order Special Galileon Lagrangians in D = 4 was developed using the coset construction based on the Special Galileon symmetry algebra [25,26]. The basic building blocks of this constructions are the invariant measure d 4 x det (E), the covariant derivative ∇ a ξ a of the Goldstone field ξ a corresponding to the linear part of the general Galileon symmetry (1.1) and the covariant derivative ∇ c σ ab of the Goldstone field σ ab corresponding to the hidden Special Galileon symmetry 23 (1.5). The Latin indices are then contracted with the flat metric η ab to build the invariants. Though the direct comparison of these building blocks with our approach is difficult, we have found the following correspondence with our geometrical objects valid up to denoted higher orders in the field φ where is the vielbein for the inverse effective metric g µν = η ab m µ a m ν b mentioned in the introduction (in a particular gauge, see [21] for more details). We can thus conclude, that on shell and up to the higher orders mentioned above det (E) = √ g, ∇ a ξ a = 0, ∇ c σ ab = K cab . (7.10) Therefore the building blocks for the on-shell higher derivative action are the same as in our geometrical approach, at least for the vertices necessary for the calculation of the 4pt, 5pt and 6pt scattering amplitudes at i Γ ≤ 6. This especially means, that there is no cubic and 23 The authors restrict themselves to the "plus" branch and their α parameter corresponds to our 1/α. quintic vertex relevant for the on-shell amplitudes. Though such vertices might be present in the off-shell higher derivative action, e.g. in the Lagrangian introduced in [26] as their contribution have to vanish on shell. As a consequence, the leading order O(p 10 ) 5pt amplitude proportional to a 1 as well as the a 2 1 contribution to the 4pt amplitude have to vanish on shell.
As an illustration of the one-loop renormalization of the Special Galileon we have calculated the four-particle amplitude up to and including the contributions with i Γ = 6. The result reads where c r j (µ) and k r j (µ) are (linear combinations of) the couplings 24 renormalized at the scale µ. The running of these couplings with µ given by (4.32) and (4.31) ensures the manifest renormalization scale independence of A i Γ ≤6

4
. This amplitude satisfies also manifestly the soft theorem with O p 3 soft behavior. Let us note that the polynomial part of A i Γ ≤6 4 has been determined independently in [26] from the Lagrangian constructed using the coset formalism with the same result up to a redefinition of the couplings. Alternative determination of the polynomial part was presented in [41] using the soft bootstrap with the conclusion, that consistency of the soft BCFW recursion for the tree-level six-particle amplitude forces the constant k r 2 (µ) to vanish. This conclusion was supported by the KLT double copy construction of the amplitude. However, even if we set k r 2 (µ) = 0 at some scale, due to the running of k r 2 (µ) such a term is inevitably generated by the loop corrections at other scale µ ′ . The next five-particle amplitude with i Γ ≤ 6 vanishes, since there are no odd vertices in the one-loop on-shell effective action S 1−loop eff A i Γ ≤6 5 = 0 (7.13) Note, that in [41] it was constructed 5pt amplitude with i Γ = 6 consistent with the O p 3 soft behavior, namely Though the soft bootstrap probe does not exclude such an amplitude, it was established that it cannot be obtained by the KLT double copy construction. In our approach, such an amplitude is excluded, since it cannot stem from any operator invariant with respect to the Special Galileon symmetry. Our results therefore support the conjecture, that symmetry based definition of Special Galileon is in tension with the KLT double copy construction at the higher orders as claimed in [26]. However, we did not confirm the violation of the interrelation between the Special Galileon symmetry and the O p 3 soft behavior of the amplitudes, since we proved that at least for i Γ ≤ 6 there does not exist any cubic vertex relevant for the on-shell amplitudes.

A Classification of the one loop on shell counterterms
In this appendix we prove some statements concerning the elimination of redundant on shell counterterms. Let us first remind the Codazzi equation for the extrinsic curvature D µ K a νρ − D ν K a µρ = 0 (A.1) which is valid in our case due to the geometrical interpretation of the Special Galileon field as an effective theory of the D−dimensional brane in a flat 2D−dimensional target space [21]. This can be rewritten as a symmetry relation for the covariant derivative of the tensor K αµν = g αβ m β a K a µν , where m β a is the D−bein for the induced metric g µν , as 25 As a consequence, where the last equation holds on shell as a consequence of (4.11). Therefore the operators vanish on shell. Similarly, with help of (A.2) and (4.11) we get on shell (D · D) K αβρ = g µν D µ D ν K αβρ = g µν D µ D β K ανρ = g µν [D µ , D β ] K ανρ + D β g µν D µ K ανρ = g µν [D µ , D β ] K ανρ = g µν R σ αβµ K σνρ + R σ νβµ K ασρ + R σ ρβµ K ανσ = g µν g σκ (R καβµ K σνρ + R κνβµ K ασρ + R κρβµ K ανσ ) . (A.5) Now, using the Gauss equation for the Riemann tensor R καβµ = K κβ · K αµ −K κµ ·K αβ , (A.6) and after some algebra, using the total symmetry of K αµρ and (4.11) we have (D · D) K αβρ = 2g µν K νρ · K β · K αµ −K ρ : K · K αβ − K β : K · K αρ − K α : K · K ρβ . (A.7) Finally we get for the operator O 7 where O 1 and O 8 are defined as O 1 = K : K · K : K = R µν R µν O 8 = g κρ g γβ g δα g µν K κγδ K νρ · K β · K αµ . (A.9) As far as the operator O 8 term is concerned, let us remind, that R καβµ R καβµ = 2 K : K · K : K − 2g κρ g γβ g δα g µν K κγδ K νρ · K β · K αµ 25 Note that g αβ and m µ a are covariantly constant.
Therefore up to a total derivative corresponding to the Gauss-Bonnet term G (cf. (A.14) Let us now concentrate to the parity odd operators where n j + m j ≤ 4 and where Note that, due to the symmetry of K αµν , within the tensor (D n j K m j ) µ 1 µ 2 µ 3 µ 4 each K can have at least one uncontracted index. Also g µν K αµν = 0 on shell, so that only indices within different K's can be contracted. There are no i Γ = 2 and i Γ = 4 such operators, since there are not enough contractions to satisfy he above requirements. For instance for i Γ = 4 the following possible operator with n j = 0, m j = 2 vanishes O i Γ =4 9 = (K µ 1 µ 2 · K µ 3 µ 4 ) E µ 1 µ 2 µ 3 µ 4 = 0, since we have to contract the extrinsic curvature tensors K αµν once, however the result of this contraction is symmetric in µ 1 µ 2 and µ 3 µ 4 . Similarly, because D · K µν = 0 on shell, the possible operator with n j = 1, m j = 1 O i Γ =4 10 = D µ 1 K µ 2 µ 3 µ 4 E µ 1 µ 2 µ 3 µ 4 = 0.
At the next level 26 i Γ = 6 we have seemingly two operators, namely which however both vanish due to the symmetry (cyclic symmetry and µ 1 µ 2 and µ 3 µ 4 symmetry respectively) of the four times contracted extrinsic curvature tensors. In the case n j = 1, m j = 3 we need three contractions within the D K 3 building block, so we have only the following possibility O i Γ =6 13 = g αβ (D µ 1 K αµ 2 ) · K µ 3 · K µ 4 β E µ 1 µ 2 µ 3 µ 4 26 Note that the operators with nj + 3mj odd cannot be constructed, therefore there are no iΓ = 5 parity odd operators. which however vanish due to (A.2). For n j = 2 and m j = 2 we need two contractions within D 2 K 2 , the only possibility with one uncontracted index per K is then (modulo integration by parts) but this operator vanishes using (A.2). Finally, in the case n j = 3 and m j = 1 only one contraction is needed within D 3 K, however because K αµν is totally symmetric and traceless, the subsequent contraction with E µ 1 µ 2 µ 3 µ 4 vanishes. To summarize, there is no parity odd on-shell operator up to and including i Γ = 6.