Radiation of scalar modes and the classical double copy
The double copy procedure relates gauge and gravity theories through color-kinematics replacements, and holds for both scattering amplitudes and in classical contexts. Moreover, it has been shown that there is a web of theories whose scattering amplitudes are related through operations that exchange color and kinematic factors. In this paper, we generalize and extend this procedure by showing that the classical perturbative double copy of pions corresponds to special Galileons. We consider point-particles coupled to the relevant scalar fields, and find the leading and next to leading order radiation amplitudes. By considering couplings motivated by those that would arise from extracting the longitudinal modes of the gauge and gravity theories, we are able to map the non-linear sigma model radiation to that of the special Galileon. We also construct the single copy by mapping the bi-adjoint scalar radiation to the non-linear sigma model radiation through generalized color-kinematics replacements.
- I Introduction
- II Bi-adjoint scalar radiation
- III Non-linear sigma model radiation
- IV Single copy: bi-adjoint scalar to non-linear sigma model
- V Special Galileon radiation
- VI Double copy: non-linear sigma model to special Galileon
- VII Discussion and outlook
- VIII Acknowledgments
A surprising relationship between gauge theories and gravity has been shown to exist not only for scattering amplitudes but also in more general cases. In the scattering amplitudes context, an example of this relationship is the BCJ (Bern, Carrasco, Johannson) double copy Bern et al. (2008, 2010a, 2010b) which consists of applying color-kinematics replacements to the Yang-Mills scattering amplitudes in order to obtain the scattering amplitudes of a gravitational theory involving a graviton, a dilaton, and a two-form field. In this case, there is a duality between the color factors and the kinematic factors, since both can satisfy the same algebra. In the cases where the double copy maps observables other than scattering amplitudes, the idea of performing color-kinematics replacements persists, but the existence of an algebra satisfied by the analogue of the kinematic factors has been scarcely explored. An example of these duality satisfying kinematic factors in the classical context was introduced in Shen (2018). Some of the new cases consist of a classical realization of the double copy and follow two main directions: exact results Monteiro et al. (2014); Luna et al. (2015, 2016); Ridgway and Wise (2016); Carrillo-GonzÃ¡lez et al. (2018); Bahjat-Abbas et al. (2017); Lee (2018); Ilderton (2018); Berman et al. (2018), and perturbative results Saotome and Akhoury (2013); Neill and Rothstein (2013); Luna et al. (2017); Goldberger and Ridgway (2017a); Goldberger et al. (2017); Goldberger and Ridgway (2017b); Goldberger et al. (2018); Chester (2018); Shen (2018); Plefka et al. (2018). While the construction of the double copy in the case of amplitudes relies heavily on the cubic structure (or rather the ability to write the theory in a form in which there is a cubic interaction), in the classical double copy this technical requirement is not always obvious. In the case of exact results, one starts from the gravitational side with a solution in the form of a Kerr-Schild metric and applies the corresponding color-kinematics replacements to obtain the single copy, i.e.., the gauge theory analogue. Applying these replacements one more time leads to the bi-adjoint scalar analogue, the zeroth copy. In the perturbative case, one can take the opposite direction and start from the bi-adjoint scalar, then apply the corresponding replacements and obtain Yang-Mills theory, perform this one more time and obtain the gravitational theory. Other surprising examples of the double copy have been discovered in different contexts—see for example Anastasiou et al. (2014, 2017); Borsten and Duff (2015); Cardoso et al. (2017, 2016); Chu (2017); Mizera and Skrzypek (2018). In this paper, we will focus on the perturbative implementation of the classical double copy.
Such a procedure applies more broadly than between gauge and gravitational theories. For example, by considering a “dimensional reduction” of the gauge and gravity theories one can obtain the scattering amplitudes of the non-linear sigma model (NLSM) and the special Galileon, respectively Cachazo et al. (2015); Cheung et al. (2017a). The relation between these scalar theories and the gauge and gravity theories can also be explained from another point of view: if we consider massive Yang-Mills and massive gravitational fields, the corresponding longitudinal modes are described by the non-linear sigma model and the (special111One can choose the parameters of massive gravity such that the resulting scalar field theory in the decoupling limit is the special Galileon. Cachazo et al. (2015); Cheung et al. (2015); Hinterbichler and Joyce (2015); Novotny (2017) ) Galileon respectively Hinterbichler (2012); de Rham (2014). This suggests the possibility of a broader relationship between these sets of theories. Indeed, it has been shown that there is a web of relationships between their scattering amplitudes Cachazo et al. (2015); Cheung and Shen (2017); Cheung et al. (2017b, a); Zhou and Feng (2018); Bollmann and Ferro (2018), see Fig.1.
In this paper, we begin by analyzing the existence of a classical perturbative double copy for the NLSM radiation. The setup consists of point-particles weakly coupled to pions which evolve consistently with the NLSM field and whose deviations from their initial trajectories and color degrees of freedom are small. We assume that the NLSM coupling to the point-particles is invariant under the unbroken symmetry. This coupling is motivated by the fact that the NLSM can arise as the longitudinal mode of a massive Yang-Mills field which gives rise to pion couplings invariant under the unbroken symmetry. Similarly, we assume that the special Galileon couples through a conformal transformation which is motivated by the coupling that would arise in the decoupling limit of massive gravity for the Galileons. In addition to the double copy relation between these theories, it is also expected that one can perform a color-kinematics replacement from the bi-adjoint scalar and obtain the NLSM radiation as in the Yang-Mills case. Given this, we will also consider the zeroth copy case where the point-particles couple to the bi-adjoint scalar field, thus spanning the entire RHS of Fig. 1.
The observable that we want to map between theories is the radiation amplitude at spatial infinity . For example, the on-shell radiation amplitude for the bi-adjoint scalar is defined as
where the on-shell current gives the flux of energy-momentum, color and angular momentum at spatial infinity, and is defined by the equations of motion , with coupling constant . Similar definitions hold for the non-linear sigma model and the special Galileon. In 4d, the probability of emission of a scalar can be written as Goldberger (2007)
As the observation time grows, , the differential radiated power is given by
The final goal is to be able to map the scalar radiation power emitted by a set of point-particles among different theories. In order to do so, we will only need to map between on-shell currents.
i.1 Summary of results
In this section, we summarize the procedure for obtaining the classical double copy for the radiation of scalar modes. We show that by applying a special set of color-kinematics replacements, it is possible to transform the radiation field generated by point-particles interacting through a bi-adjoint scalar field to the one in which these particles interact through a non-linear sigma model field. Similarly, one can act on the NLSM radiation field to obtain the equivalent object for the double copy, i.e.., the special Galileon radiation. We consider the case where the impact parameters of the particles are large, and thus the particle number is conserved, since no particles are created or annihilated. The large separation of the particles accounts for the consistency of the perturbative calculation, a point that will be made more precise in the body of the paper. A crucial fact for the existence of the double copy is that the couplings of the scalar fields to the point-particles have the same coupling strength as the self-interactions of such fields. This is similar to the case of Yang-Mills and gravity.
For each theory, the point-particles carry different degrees of freedom depending on the couplings being considered. In the bi-adjoint scalar field case, the point-particles carry two color charges, and , each in the adjoint representation of the groups and . In the case of the NLSM corresponding to the symmetry breaking pattern (with the diagonal subgroup), we will consider a coupling to the point-particles that is manifestly invariant under the unbroken symmetries. This means that the coupling will involve the “covariant derivative” of the Goldstone modes, , which in our case will couple to the color dipole moment of the point-particles. Manifest invariance under is sufficient to ensure invariance under the full group Weinberg (1996). Finally, the special Galileon coupling we will be using follows from a conformal transformation of the point-particle action. This transformation is motivated by the one implemented in massive gravity to remove the kinetic mixing between the helicity-2 modes and the longitudinal mode before taking the decoupling limit, in that case .
Because we have different degrees of freedom carrying a color index in the bi-adjoint scalar and the non-linear sigma model, we will also need a replacement rule to map one to the other. Thus, for the single copy we need not only the usual color-kinematics replacements, which schematically are of the form
but also the color-color replacements
where stands for the collection of momenta involved in the process. At second order in the couplings, the single copy color-color replacements are given by Eq.(IV.2) and the color-kinematics by Eq.(IV.3). These replacements map the on-shell current Eq.(II.20) into the on-shell current Eq.(III.25). At quartic order, the color-color replacements are given by Eq.(IV.5) and the color-kinematics by Eq.(IV.7), these give a map between the on-shell currents in Eq.(II.23) and Eq.(III.29).
For the double copy case instead we can simply perform a color-kinematics replacement of the form
The replacement rules at second order are found in Eq.(VI.2), and the ones at quartic order in Eq.(VI.4). These replacements create a map between on-shell currents: Eq.(III.25) maps onto Eq.(V.15), and Eq.(III.29) maps onto Eq.(V.17). There are four main features that is worth highlighting about these replacements:
Coupling constants: the coupling constants in the three different theories are mapped into each other as follows:
Thus, a result obtained in the biadjoint case with a precision of will be mapped onto an equivalent result for NLSM and special Galileon at order and respectively. For the sake of brevity, from now on we will denote this level of precision as .
Color charges: The color charges and dipole moments are mapped as
where represents different momentum factors, depending on the specific color structure. This can be compared to the Yang-Mills-gravity case where the replacement is . In this case, we are mapping between scalar theories so no new structure with a Lorentz index appears uncontracted.
Three-point vertex: Color factors which involve only one structure constant are mapped to zero,
In the gravitational double copy the color factor of the Yang-Mills three-point function, , is mapped to the color-stripped Yang-Mills three-point vertex. This is motivated by the BCJ double copy where one replaces the Yang-Mills color factor by a second copy of the Yang-Mills kinematic factor in order to obtain a gravitational amplitude. In the present case, the NLSM does not have a cubic vertex and thus the above color structure is mapped to zero.
Color-kinematics duality for the double copy: The replacement rules that take the NLSM four-point amplitude color factor to the NLSM four point amplitude, i.e..
maps color factors satisfying the Jacobi identity
to kinematic factors that satisfy another Jacobi identity
This provides a new example of the color-kinematics duality at the classical level. The analogue case for the gravitational double copy was studied in Shen (2018).
In the following, we carry out the perturbative calculation for each theory in detail. In section II, we compute the bi-adjoint scalar radiation and in section III that for the non-linear sigma model case. In section IV we then explain the color-kinematics and color-color replacements that transform the bi-adjoint scalar result into the NLSM one. We continue in section V with the calculation of the special Galileon radiation and in section VI derive the color-kinematics replacements that lead to the double copy. We conclude in section VII.
Ii Bi-adjoint scalar radiation
In this section, we compute the radiation field produced by color charges coupled through the bi-adjoint scalar field. The result was first computed in Goldberger et al. (2017) and extended to order in Shen (2018). In the following, we show these results for completeness while clarifying some technical details of the calculation.
The bi-adjoint scalar field transforms in the adjoint representation of the group and has cubic interactions, described by the Lagrangian
Our goal is to compute perturbatively the scalar radiation field generated by a set of color charges coming from infinity, which will evolve consistently together with the field they generate. The point-particles carry color charges also transforming in the adjoint representation of and move along the worldlines , where is the coordinate along the worldline, and labels the individual particles. These point-particles are coupled to the scalar field in the following way:
where the einbein is a Lagrange multiplier that ensures invariance under reparametrizations of , and and are color charges transforming in the adjoint representations of and respectively. For the purpose of this paper, the specific Lagrangian realization giving rise to the color charges is not relevant and thus is not considered here, but a discussion regarding this can be found in Goldberger et al. (2017). The total color currents are
Here, and are the Noether currents derived from due to the invariance under and and read
while the leading order currents produced by the point-particles are given by
where is the velocity of the point-particle carrying color charge or . The next to leading order contributions to these currents include finite size effects. By varying the action and considering current conservation, we obtain the equations of motion for the coordinates and the color charges
where is the momentum of the particle and .
ii.1 Perturbative solutions
The equation motion for the bi-adjoint scalar field can be written as
where the source current is
This allows us to compute the radiation field at in terms of the Fourier transform of the source:
The initial configuration consists of charged particles that are moving with constant velocity at . Thus, the initial conditions for the color-charged point-particles are:
where are the (spacelike) impact parameters.
In what follows, we compute the solutions perturbatively in powers of the coupling strength. The actual dimensionless parameter that controls the expansion is a combination of the coupling strength and kinematic factors, given by Goldberger and Ridgway (2017a)
where is the energy of the point-particle and is its impact parameter. In this expression we have neglected the phase space volume. Notice that the perturbation parameter is inversely proportional to the impact parameter. This is consistent with our set up of particles that are far apart from each other and which only experience small deviations as they interact through the scalar field. Indeed, the fact that ensures that the deviations are small compared to the impact parameter. We can now find the field
Notice that on-shell () the field vanishes unless but is timelike, therefore there is no radiation at this order, as we should expect, since static point-particles do not radiate.
We now proceed to obtain the next order perturbation for the deviations of the point-particle trajectories and color charges. These are obtained by considering
where . The source at is given by
After using our previous results, the source current becomes
with , and
The above result for the source current heavily relies on the use of the delta functions in . This will be the case for all the final results that we present. One can think of this perturbative solution in terms of Feynman diagrams. At second order in the coupling, the contributions to the bi-adjoint current are given by the graphs in Fig. 3. The first term in the parentheses in Eq.(II.19) corresponds to the graph on the left-hand side of Fig. 3. This graph only shows the case of the scalar field radiated by particle , but we should also include the case where it is radiated from particle . This is taken into account by the sum over point-particles. The last term of Eq.(II.19) comes from the graph on the right-hand side which corresponds to the self-interactions of the field.
As we will see in the next section, the NLSM self-interactions will only contribute at next to leading order in perturbations. Hence, in order to construct a satisfactory copy we will compute the source current for the radiation field for the bi-adjoint scalar at . The source at this order is given by
which corresponds to the graphs in Fig.4.
The term in curly brackets contains the deflections of the point-particle coordinates and color charges at next to leading order, which are given by
Notice that the momentum involved in the propagators that appear in these calculations corresponds to the momentum exchanged with the point-particle. After using the fact that
with given by Eq.(II.20), the first term, which comes from the field self-interactions, reads
where is the straightforward generalization of , namely
Notice that the current should be symmetric under interchange of particles. This symmetry is not manifest in Eq. (II.28), but it is realized by the sum over the particle indices and .
Iii Non-linear sigma model radiation
Consider now the non-linear sigma model (NLSM) based on the simple compact Lie group ; that is, the model corresponding to the symmetry breaking , where . The leading order effective Lagrangian is given by
where , and is an element of the group . We will use the exponential parametrization
where are the Goldstone boson fields and are the generators of . Given that the pattern is a simple generalization of the one describing QCD pions, in what follows we will often refer to the NLSM fields simply as pions. Since all quantities with a color index will transform in the adjoint representation, we’ll find it convenient to follow the conventions that are often adopted in the amplitudes literature (see e.g. Kampf et al. (2013a, b); Chen and Du (2014); Du and Fu (2016)). Hence, our generators satisfy the following relations:
With this parametrization, the strength of self-interactions is determined by the coupling . In terms of the Goldstone fields, the Lagrangian can be rewritten as
where we have defined
In this case, we want to consider a coupling to the point-particle that preserves the unbroken symmetry , which means that it involves the pion covariant derivative . Consider a coupling to a dipole moment localized on the worldline:
where the pion covariant derivative in the exponential parametrization is
From this coupling, we can read off the current generated by the color charges. Up to next to next leading order this current is
We obtain the equation of motion that determines the evolution of the point-particle coordinates by varying the point-particle action . At next to leading order in the coupling this yields
Similarly, we obtain the equations of motion determining the evolution of the dipole moment from the conservation of the total color current
Here, is the Noether current derived from and reads
Given this, implies that the dipole evolves according to
The above equation is found after performing a Fourier transformation and using the equation of motion obtained from varying that reads