Extended solutions for the biadjoint scalar field

Biadjoint scalar field theories are increasingly important in the study of scattering amplitudes in various string and field theories. Recently, some first exact nonperturbative solutions of biadjoint scalar theory were presented, with a pure power-like form corresponding to isolated monopole-like objects located at the origin of space. In this paper, we find a novel family of extended solutions, involving non-trivial form factors that partially screen the divergent field at the origin. All previous solutions emerge as special cases.


Introduction
The study of (quantum) field theories remains a highly active research area, given that such theories describe the four fundamental forces in nature. It is widely believed that these forces may emerge as the low energy limit of an underlying framework, which may not necessarily be a field theory (e.g. it may be a string theory). It is thus interesting to elucidate common structures between theories, and also to investigate how different types of theory are related to each other. In this context, so-called biadjoint scalar theory has recently been increasingly studied, whose Lagrangian is given by Here the (un)primed indices are adjoint indices associated with two different Lie groups, with structure constants f abc andf a b c respectively, and y is a coupling constant. Although this theory does not seem to be directly physically applicable by itself, there is increasing evidence that it underlies the dynamics in more relevant theories. For example, perturbative scattering amplitudes in this theory can be used as building blocks for amplitudes in non-abelian gauge theories [1,2]. The latter are related to amplitudes in gravity theories by the double copy of refs. [3][4][5], thus providing a ladder of theories related by taking the field Φ aa of eq. (1), and replacing either of its adjoint indices with Lorentz (spacetime) indices. This same procedure can be applied to exact classical solutions as well as perturbative amplitudes [6][7][8][9][10][11][12][13], where again the biadjoint theory plays a crucial role.
Given the wide range of instances in which the biadjoint scalar theory appears, it is clearly worth studying this theory in its own right. All of the above examples of its use involve perturbative solutions of the equation of motion, which from eq. (1) is found to be Instead, one may consider exact nonlinear solutions, which may involve inverse powers of the coupling y. A number of solutions of this type were recently presented in ref. [38], consisting of singular point-like disturbances located at the origin. The first of these has spherical symmetry, and is applicable when the two Lie groups are the same as each other: Here r is the radial space coordinate 3 , and we define Further solutions are possible if one restricts both Lie groups to SU (2), which allows mixing between spacetime and adjoint indices, analogous to known non-perturbative solutions in nonabelian gauge theories [39][40][41][42][43][44]. In particular, ref. [38] presented a one-parameter family of solutions with an axial form in the internal space: where 0 ≤ k ≤ 2 if Φ aa ∈ R. Like eq. (3), this has a pure power-like behaviour in r. Indeed, this is the only possibility if an axial form is imposed [38], such that it tempting to think that the spectrum of nonperturbative solutions in biadjoint scalar theory is much simpler than that of non-abelian gauge theories. The aim of this paper, however, is to show that extended solutions do in fact exist. That is, it is possible to dress the power-like divergence in r with a non-trivial form factor, such that all previously found solutions emerge as special cases. The new solutions are still singular at the origin (a consequence of Derrick's theorem for scalar field theories [45], which prohibits finite energy). However, one solution in particular will exhibit an interesting screening behaviour, such that the strength of the divergence in energy is ameliorated. Our results will be useful in future studies of biadjoint scalar theory, including the issue of whether or not double copy-like relationships can be generalised beyond the perturbative sector.
The structure of the paper is as follows. In section 2, we present extended solutions for the case in which both Lie groups in the biadjoint scalar Lagrangian are the same. In section 3, we apply similar techniques to construct an extended solution for the case in which both Lie groups are SU (2). We discuss our results and conclude in section 4.

Extended solutions for common Lie groups
In this section, we consider the Lagrangian of eq. (1), but where both sets of structure constants are associated with the same Lie group. We may then make a similar ansatz to that made in ref. [38], namely Substituting this in eq. (2) yields a second-order non-linear differential equation forS(r): We now look for extended solutions in which the r −2 divergence of the field at the origin, of eq. (3), is dressed by a finite form factor. To this end, we may write where K(r) is finite for all r. Equation (7) then becomes We can study this further by defining so that eq. (9) becomes We may transform this into a more recognisable form by setting so that eq. (11) implies This is an Abel equation of the second kind. Unlike some equations of this type, there appears to be no analytic solution in terms of known functions. Instead, we may revert back to eq. (11) and analyse it using a method similar to that used by Wu and Yang [44] to construct nonperturbative solutions in pure Yang-Mills theory. First, one may turn eq. (11) into two coupled first-order differential equations as follows: This defines a vector field in the (K, ψ) plane, where the curves that are tangent to this vector field are the solutions we desire. We show a plot of these curves in figure 14, and solutions for K(ξ) that are finite for all values of r correspond to curves which are bounded in the plane. By inspection, and noting that the vector field is zero at (K, ψ) = (±1, 0), there are three possibilities: Figure 1: The integral curves of the vector field defined by (14). The stationary points (±1, 0) are marked in red.
Although a complete analytic solution for K(ξ) is not possible, we can examine its behaviour in asymptotic limits. For ξ → −∞ one may write Upon substituting this into eq. (11) and neglecting terms quadratic and higher in f (ξ), one finds the general solution The boundary condition f (ξ) → 0 as ξ → −∞ then requires B = 0. Similarly, for ξ → +∞ we may substitute K(ξ) = 1 + g(ξ) in eq. (11) and ignore terms quadratic and higher in g(ξ), leading to the general solution  Figure 2: (a) Numerical solution of (11) with asymptotic behaviour given by (17). We have chosen A = 1; other choices of A will lead to the same graph but translated along the ξ-axis. (b) The behaviour of K as a function of r.
where the first term gives the dominant behaviour. We conclude that or, translating back to the functionS(r), The two parameters A and A are not independent. As discussed above, all solutions for K(r) constitute a rescaling of a canonical solution. The scaling is fixed by choosing either A or A , such that the other parameter also becomes fixed. For the particular choice A = 1, we show a numerical solution for K(ξ) in figure 2.
The simple power-like solution of eq. (3) is reminiscent of a Coulomb solution in gauge theory, with a correspondingly divergent field at the origin 4 . The new solution found here consists of dressing the previous solution with a form factor, such that the divergence at r → 0 is partially screened. That is, the r −2 behaviour is softened to r −1 , and by choosing the rescaling parameter λ in K(r/λ), this screening can be made as smooth or abrupt as desired. The softening of the divergence in the field is also reflected in the energy of the solution. From eq. (1), the Hamiltonian density of the biadjoint scalar theory is found to be [38] which for the ansatz of eq. (6) evaluates to with N the dimension of the (common) Lie group. The energy is divergent due to the singularity at the origin, but can be evaluated by implementing a short-distance cutoff r 0 , corresponding to a charge radius: For the solution with the asymptotic behaviour of eq. (18), this gives (as r 0 → 0) which is a much softer divergence than the energy associated with eq. (3), that behaves as E ∼ r −3 0 [38].

Extended solutions for SU (2) ⊗ SU (2)
Having succeeded in constructing an extended solution for the case in which both gauge groups are identical, let us see if further such solutions are possible upon restricting each gauge group to be SU (2), for which one may make a similar ansatz to that used in ref. [38] Φ aa = 1 r 2 A(r)δ aa + B(r) x a x a r 2 + C(r) aa d x d r .
IntroducingĀ = A/y etc., eq. (2) then implies the coupled non-linear differential equations r 2Ā − 2rĀ + 2ĀB + 2Ā 2 + 2Ā + 2B = 0; To simplify this we make a further ansatz, namely we writē and then we look for special values of the constants c 1 , . . . , c 6 so that the three equations (24 -26) reduce to a single differential equation for f (r). We thus find two new types of solution that were not presented previously in ref. [38].

Multiple power-like solution
The first new solution is obtained upon choosinḡ in which caseB(r) satisfies the linear homogeneous differential equation whose general solution isB where b 1 and b 2 are arbitrary constants. The full solution for Φ aa is then Here the second term in the square brackets has a part which diverges more rapidly at the origin than the r −2 behaviour of the solutions of ref. [38], and a term which is softer. If one requires a finite energy upon integrating to infinity, however, one must set b 1 = 0. The energy, from eq. (21), is then found to be where again r 0 is a small-distance cutoff. For small r 0 , the energy diverges as E ∼ r −
Upon writingB one finds that K(r) satisfies eq. (9). Thus, an extended solution for Φ aa is given by In section 2 we found three solutions for K(r), which lead to the following cases: 1. K(r) = 1: in this case, eq. (38) reduces to By replacing we see that eq. (39) is the same as the solution of eq. (5), that was already presented in ref. [38].
2. K(r) = −1: in this case, eq. (38) reduces to which is the same as eq. (3) for the special case in which both gauge groups are SU (2), so that T A = 2.
3. The most general solution has K(r) given by the function of figure 2, with asymptotic limits given by eq. (18). This solution is new.
The extended solution no longer has the pure-axial property of eq. (5), consistent with the fact noted in ref. [38] that extended axial solutions cannot exist. The energy of the solution in case 3 above, in terms of the usual short-distance cutoff r 0 , is found to have the leading behaviour (as Note that the extended solution of section 2 (where again T A = 2) can also be obtained from eq. (38), by choosing c → ∞: Finally, we note that a complex solution is also possible. By choosing c 2 = −1 in eq. (36), this reduces to eq. (31), whose general solution is given by eq. (32). One then obtains where we have again set b 1 → 0 to ensure that the energy is bounded at spatial infinity.

Conclusion
Biadjoint scalar field theory occurs in a number of contexts, and plays an intriguing role in determining the dynamics of perturbative scattering amplitudes in gauge and gravity theories. The nonperturbative properties of this theory remain relatively unexplored, and in this paper we have presented a number of new solutions involving inverse powers of the coupling constant. Unlike the first solutions presented in ref. [38], the results of the present paper have an extended structure, indicating that the spectrum of nonperturbative solutions of biadjoint scalar theory is much richer than has previously been suggested. Interestingly, the solutions presented here include those in which the strong divergent behaviour of the field at the origin is partially screened.
There are a number of avenues for further work. Firstly, it may be possible to find more solutions of the biadjoint scalar theory, including non-static solutions. Secondly, one may imagine coupling the biadjoint scalar to a gauge field, as occurs in some applications of the double copy. Finally, the question of whether the nonperturbative solutions found here and in ref. [38] can themselves be copied to gauge theory or gravity deserves further attention.