Recent advances in the classical Double Copy DC procedure have revealed a profound connection between gauge theories and Tduality invariant frameworks with Double Field Theory the classical DC of YangMills theory emerging as the first explicit example. Extending this procedure to higherderivative gauge theories predicts the existence of a HigherDerivative Double Theory HDDT which incorporates Weyl gravity along with bfield and dilaton contributions all in a Tduality invariant manner. In this work we show that combining both mappings leads to DFT a Tduality invariant model related to the bosonic string incorporating firstorder alpha corrections upon parameterization. Our results expand the potential applications of the DC program towards constructing perturbative alphacorrected Lagrangians while also opening up possibilities for reversing the map by considering the single and zeroth copies.
['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']
Quadratic Curvature Corrections in Double Field Theory via Double Copy
{'Quadratic Curvature Corrections in Double Field Theory via Double Copy': "Eric Lescano 1, ∗ and Jes'us A. Rodr'ıguez 2, †  \n1 Institute for Theoretical Physics (IFT), University of Wroclaw,  \npl. Maxa Borna 9, 50-204 Wroclaw, Poland  \n2  \nUniversidad de Buenos Aires, FCEyN, Departamento de F'ısica, Ciudad Universitaria, 1428 Buenos Aires, Argentina  \nRecent advances in the classical Double Copy (DC) procedure have revealed a profound connection between gauge theories and T-duality invariant frameworks, with Double Field Theory (DFT)-the classical DC of Yang-Mills theory-emerging as the first explicit example. Extending this procedure to higher-derivative gauge theories predicts the existence of a Higher-Derivative Double Theory (HDDT), which incorporates Weyl gravity along with b -field and dilaton contributions, all in a Tduality invariant manner. In this work, we show that combining both mappings leads to DFT+, a T-duality invariant model related to the bosonic string, incorporating first-order α ' corrections upon parameterization. Our results expand the potential applications of the DC program towards constructing perturbative α ' -corrected Lagrangians, while also opening up possibilities for reversing the map by considering the single and zeroth copies.", 'I. HIGHER-DERIVATIVE CORRECTIONS AND THE DOUBLE COPY MAP': "The NS-NS sector of the low energy limit of string theory exhibits a global O( n, n ) symmetry when the fields are independent of n spatial coordinates [1]. This continuous T-duality symmetry is exact to all orders in α ' [2], which motivates the study of higher-order corrections in theories with manifest duality invariance, such as Double Field Theory (DFT) [3, 4], a proposal to include T-duality as a fundamental symmetry of a field theory on a doubled space 1 . DFT reformulates supergravity in terms of O( D,D ) multiplets within a doubled geometrical framework, where D is the space-time dimension, and D ≥ n .  \nAlthough the geometric structure of DFT makes it difficult to directly construct invariant objects that are quadratic in curvatures, substantial progress has been made in obtaining higher-derivative corrections [8-17]. In [12], an exact mechanism was introduced through a generalization of the Green-Schwarz transformation, which requires an infinite tower of O( D,D )-covariant higher-derivative terms in the gauge-invariant action. Specifically, the first-order corrections reduce to the biparametric theory presented in [10], which interpolates between different low-energy limits of String Theory.  \nHowever, as stated in [18], DFT and similar frameworks cannot reproduce all of the higher-order terms in the α ' expansion of string theory, and hence developing new techniques to obtain higher-derivative terms remains a major challenge. It is within this context that the Double Copy (DC) construction [19-27] emerges as a promising approach to potentially provide new insights for systematically generating higher-derivative corrections.  \nThe quadratic and cubic contributions to perturbative DFT can be derived by applying a DC map to the quadratic and cubic Yang-Mills (YM) Lagrangian [28]. This connection between YM theory and perturbative DFT suggests that DFT is part of a broader class of T-duality invariant theories, all of which can be derived through the DC procedure applied to different gauge theories. More recent research [29, 30] delved into a new DC prescription applied to the higher-derivative gauge theory [31, 32]  \nL = a 1 κ ab D µ F µνa D ρ F ρ ν b + a 2 κ αβ D µ φ α D µ φ β , (1)  \nwhere µ, ν, · · · = 0 , . . . , D -1 are space-time indices, a, b, . . . , are indices in the adjoint representation of the gauge group and α, β . . . , are indices in some real representation of the same group. The fundamental fields of the theory are a gauge field A µ a , and a scalar field φ α . The coefficients a 1 , a 2 are real and depend on the space-time dimension through [30]  \na 1 = -2 ( D -3 D -2 ) , a 2 = a 1 2( D -1) . (2)  \nAfter applying the DC map on (1) at quadratic order in fields, the resulting theory contains Weyl gravity alongside b -field and dilaton contributions written in the form of a perturbative Higher-Derivative Double Theory (HDDT),  \n[ a 1 ( DF ) 2 + a 2 ( Dφ ) 2 DC --→ pert. HDDT ] (2) . (3)  \nIts field content consists of a generalized frame e µ ¯ µ ( x, ˜ x ), and a generalized dilaton Φ( x, ˜ x ), both obtained by identifying A µ a and φ α as described in [29, 30]. In particular, in [30] the authors considered interaction terms into Lagrangian (1) and demonstrated the existence of HDDT beyond the quadratic order.  \nIn this work we explore the possibility of using a classical, off-shell DC map to generate perturbative, fourderivative corrections to the bosonic string. To do this,  \nwe begin by considering DFT+ [10], a well-known Tduality invariant theory whose four-derivative corrections are related to the closed bosonic string theory upon parametrization, described by the action principle (11), up to field redefinitions. This action is invariant under the standard symmetries of DFT. When the strong constraint is imposed, the theory reduces to the standard NS-NS supergravity Lagrangian, along with the firstorder α ' corrections of the closed bosonic string. This reduced action corresponds to the result obtained by applying the DC prescription to a theory that includes YangMills (YM) together with (1), suggesting the relation  \n[ Y M + a 1 ( DF ) 2 + a 2 ( Dφ ) 2 DC --→ pert . DFT+ ] . (4)  \nThe findings presented in this study offer a starting point for several potential research directions, which will be further discussed in the final section of this work.", 'II. DFT + BEHIND DOUBLE COPY MAPS': "In this section, we construct the non-perturbative DFT+ theory (up to field redefinitions) by considering O( D,D ) multiplets in a double geometry, following the standard approach of incorporating higher-derivative terms into DFT.  \nThe dimension of the fundamental representation of O( D,D ) is 2 D , then the coordinates are defined in double space through X M = ( x µ , ˜ x µ ), with M,N,... indices in the fundamental representation of the duality group. The coordinates ˜ x µ are dual coordinates and are removed by  \nimposing the strong constraint,  \n∂ M ( ∂ M /star ) = ( ∂ M /star )( ∂ M /star ) = 0 , (5)  \nwhere /star denotes arbitrary generalized fields/parameters or products of them. The fundamental fields are a generalized frame E M A and a generalized dilaton d . These fields transform with respect to generalized diffeomorphisms and double Lorentz transformations as  \nδ ˆ ξ, Λ E M A = L ˆ ξ E M A + E M B Λ B A , (6)  \nδ ˆ ξ d = ˆ ξ N ∂ N d -1 2 ∂ M ˆ ξ M , (7)  \nwhere L ˆ ξ is the generalized Lie derivative. The generalized frame satisfies,  \nE MA H AB E NB = H MN , E MA η AB E NB = η MN , (8)  \nwhere η AB and H AB are double Lorentz invariant metrics, η MN is the O( D,D ) invariant metric, and H MN is known as the generalized metric. Using this metrics we can define the (flat) projectors  \nP AB = 1 2 ( η AB -H AB ) , P AB = 1 2 ( η AB + H AB ) ,  \nacting on arbitrary double-Lorentz vectors, lead to P A B V B = V A and P A B V B = V A .  \nAt this point it is necessary to introduce the generalized fluxes  \nF ABC = 3 ∂ [ A E M B E | M | C ] , (9)  \nwith the flat derivative defined as ∂ A = √ 2 E M A ∂ M . The action of DFT+ is given by  \nF A = √ 2 e 2 d ∂ M ( e -2 d E M A ) , (10)  \nS DFT+ = α ' ∫ d 2 D Xe -2 d ( R-R ( -) -R (+) + 1 2( D -2) R AB R AB -1 2( D -2)( D -1) R 2 ) , (11)  \n= 2 ∂ A F A + F A F A 1 F ABC F ABC 1 F ABC F ABC ,  \nR -6 -2 (12)  \nC C D C  \nR AB = ∂ A F B -∂ C F AB + F DA F BC -F F ABC , (13) R (+) = -1 2 [ ( ∂ A ∂ B F B CD ) F ACD +( ∂ A ∂ B F A CD ) F BCD +2( ∂ A F B CD ) F A CD F B +( ∂ A F ACD )( ∂ B F B CD ) +( ∂ A F B CD )( ∂ A F B CD ) + 2( ∂ A F B ) F B CD F ACD +( ∂ A F BCD ) F C CD F ABC -( ∂ A F BCD ) F C CD F ABC +2( ∂ A F A CD ) F B CD F B -4( ∂ A F B CD ) F A CE F BE D + 4 3 F E AC F BED F C CD F ABC + F B CD F A CD F B F A + F A CE F BED F A CG F BGD -F B CE F AED F A CG F BGD -F ABD F D CD F C CD F ABC ] . (14)  \nHere, R is the generalized scalar curvature and R AB is  \nthe generalized Ricci tensor. R (+) is a higher-derivative  \nwith  \ncombination [10] that can be constructed by extending to the heterotic duality group and applying the generalized Bergshoeff-de Roo identification [12], while R ( -) = R (+) [ P ↔ P ]. The four-derivative extra terms depending on R and R AB are contributions that are typically not considered in higher-derivative DFT, as they are proportional to the leading-order equations of motion and can therefore be absorbed through field redefinitions.  \nThe DFT+ action is, by construction, invariant under global O( D,D ) transformations as well as local generalized diffeomorphisms. It also exhibits invariance under double Lorentz transformations, which generalize the Green-Schwarz mechanism in the doubled space. This is realized through a deformation of the Lorentz transformation of the generalized frame, given by  \nδ (1) E M A = -F B CD ∂ A Λ CD E M B , δ (1) E M A = F A CD ∂ B Λ CD E M B . (15)  \nWhile one might be tempted to break Lorentz invariance and define S HDDT as consisting only of the fourderivative terms in (11), thereby recovering the double copy map (3) and restoring ordinary Lorentz invariance upon parametrization, this proposal for the HDDT action does not reduce to Weyl gravity in the pure gravitational limit. The inclusion of the generalized Ricci scalar is crucial due to higher-derivative field redefinitions at the supergravity level.  \nThe parametrization of the DFT+ degrees of freedom follows the same form as in standard DFT  \nE M A = 1 √ 2 ( -e µa -b ρµ e ρ a e µ a , e µa -b ρµ e ρ a e µ a ) , (16)  \ne -2 d = √ -˜ ge -2 ˜ φ , (17)  \nwhere e µa and e µa are two vielbein generating the same D -dimensional metric ˜ g µν . As mentioned previously, it is necessary to perform the following field redefinition,  \n˜ g µν = g µν -α ' 2 Ω ( -) µab Ω ( -) ν ab -α ' 2 Ω (+) µab Ω (+) ν ab , (18)  \nto eliminate the anomalous transformation of the metric under Lorentz transformations. Additionally, the dilaton must be redefined to ensure that the integration measure remains invariant, i.e., √ -˜ ge -2 ˜ φ = √ -ge -2 φ . In (18) we include the torsionful spin connection  \nΩ ± µab = w µab ± 1 2 H µab , (19)  \nwhere w µab is the spin connection, and H µab is the field strength of the Kalb-Ramond field.  \nThe low energy effective action that describes the previous theory once the duality group is broken is given by  \nS = ∫ d D x √ -ge -2 φ [ L (2) -α ' 4 ( Riem 2 -4 D -2 R µν R µν + 2 ( D -2)( D -1) R 2 + L (4) matter ( b, φ ) )] , (20)  \nwith L (2) the NS-NS sector of supergravity. The fourderive terms of this Lagrangian reproduces Weyl gravity in the pure gravitational limit (p.g.: b µν = φ = 0), and by setting α ' = -4 the action reduces to,  \nS | p.g. = ∫ d D x √ -g ( R + C µνρσ C µνρσ ) , (21)  \nwhere C µνρσ is the Weyl tensor. This is the theory obtained from applying the DC prescription on the gauge side of (4) and hence, the confirmation of the existence of a DC map between that gauge theory and DFT+.", 'III. DISCUSSION': "In this work, we explored the role of the Double Copy construction as a tool for deriving higher-derivative corrections in the context of Double Field Theory. Our analysis demonstrates that applying the DC prescription [28-30] to the Yang-Mills Lagrangian, along with higherderivative terms from gauge theory, leads to a perturba-  \ntive structure consistent with previous studies on higherderivative corrections, of the form DFT+HDDT.  \nWe further showed that the DFT+HDDT structure is connected to an exact formalism given by the DFT+ theory, which includes four-derivative corrections that are invariant under global O( D,D ) transformations and local generalized diffeomorphisms. These corrections align with the α ' expansion of the closed bosonic string when properly parameterized. Notably, the action contains contributions essential for maintaining Lorentz invariance and recovering Weyl gravity in the pure gravitational limit, underscoring the importance of higherderivative field redefinitions at the supergravity level.  \nBy employing the DC map, we successfully related the structure of gauge theory to T-duality-invariant theories, with a particular emphasis on the non-trivial effects of higher-derivative corrections. Now that the connection with DFT+ is established, further study of higher-order gauge theories becomes highly relevant, as they may connect to T-duality-invariant models through the DC.  \nOn the other hand, constructing a Conformal Field Theory (CDFT) in the double geometry, as outlined in  \n[40], using a higher-derivative DFT is limited by double Lorentz transformations. To the best of our knowledge, the only higher-derivative combination in the biparametric extension given in [10] that can be expressed in terms of the generalized metric is provided by the DFT- or HSZ theory [8, 9], whose parameterization does not lead to Riem 2 contributions. We do not rule out the possibility that CDFT could include additional noncovariant contributions, potentially incorporating noncovariant terms to resolve the lack of invariance within the generalized frame formalism (both in the bosonic or heterotic cases). We leave this issue for future investigation.  \nSeveral directions for future research arise from the results of this work. We briefly outline some of them below:  \n- · New Double Copy Realizations: The connection established in this work between DC maps and DFT+ provides a strong foundation for further exploring links between gauge theories and T-dualityinvariant frameworks. 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