UV consistency conditions for CHY integrands

We extend on-shell bootstrap methods from spacetime amplitudes to the worldsheet objects of the CHY formalism. We find that the integrands corresponding to tree-level non-linear sigma model, Yang-Mills and $(DF)^2$ theory are determined by demanding enhanced UV scaling under BCFW shifts. Back in spacetime, we also find that $(DF)^2$ theory is fixed by gauge invariance/UV scaling and simple locality assumptions.


I. INTRODUCTION
recently become apparent that a recursion may be set-up even with poles at infinity, as long as the asymptotic behavior is sufficiently tame and can be probed by other properties [24][25][26][27][28][29][30], while other UV and unitarity considerations may fix gravity loop integrands [31]. However, the above table demonstrates that the UV scaling may be considered more fundamental than the recursion itself, as no other input beyond locality is needed. Unfortunately conditions like UV scaling, or even simple gauge invariance, are quite non-transparent as they require numerous cancellations between different Feynman diagrams, but if they are completely constraining this suggests that scattering amplitudes may have radically different descriptions, which make other properties manifest, at the expense of the traditional locality and/or unitarity.
A step in this direction is given by the CHY formalism [32][33][34][35][36][37][38][39], which can be viewed as a transform from spacetime to a worldsheet, where local singularities corresponding to propagators are replaced by worldsheet singularities (σ i − σ j ) −1 , with the precise map encoded in the scattering equations (SE): Manifest locality and unitarity inherent to Feynman diagrams are lost, but gauge invariance and the double copy are more transparent in this formalism. This construction can be traced to an ambitwistor string theory [40,41], but it is not clear what (if any) principles determine the worldsheet objects I directly.
In this article we propose that simple scaling under BCFW shifts is sufficient to fully determine the integrands relevant for NLSM, YM, as well as DF 2 theory [42]. The only assumption needed is that these integrands are permutation invariant functions of dot products between momenta and polarization vectors (with no (e · e) factors for DF 2 theory, a necessary extra assumption), with the only singularities coming from products of σ ij = σ i − σ j , initially unrelated to the numerators. This result holds even without assuming the scattering equations, but we do require that shifting particles i and j also shifts the corresponding σ's: Under such shifts, we find that all three integrands scale as 1 : Next we can also consider what happens when we assume "worldsheet locality": demanding that any dot product x i ·y j is rescaled only by the appropriate worldsheet factor (σ ij ) −1 . This provides an ansatz that is very close to manifesting the correct scaling: for vector theories, no term scales worse than O(z −1 ), so only a minor improvement is required to obtain the correct scaling of O(z −2 ). Moreover, the correct scaling in most (but not all) shifts holds term by term in the expansion of the Yang-Mills reduced Pfaffian, while the cycle expansion of the non-reduced Pfaffian manifests correct scaling term by term. This suggests that the worldsheet is in fact a more natural home for the BCFW shift, getting us closer to a formalism which trades locality and/or unitarity for enhanced UV behavior.
The paper is organized as follows. In Section II we first briefly review the CHY construction (a full description can be found in refs. [32][33][34][35][36]). In Sections III-V we then present evidence that the integrands relevant for NLSM, YM and DF 2 theory are fully determined by UV scaling and various assumptions. Motivated by this result, we also extend such observations to the usual spacetime DF 2 amplitudes, which we find to be fixed by gauge invariance and similar UV conditions. We conclude with possible future directions in Section VI.

II. CHY REVIEW
The CHY formula expresses various scattering amplitudes as: where the scattering equations are given by: and the half-integrands I are functions of kinematics, polarization vectors (for vector theories), and worldsheet coordinates σ. Their specific expression dictates the particular theory 1 We note that some extra care is required when discussing permutation invariance and UV scaling for NLSM and YM, which are given by reduced Pfaffians. As will be discussed later, for these integrands, two rows/columns are deleted -we associate these deletions with two particles which behave differently from the others, since we are not using the scattering equations: the integrands are not permutation invariant in these particles, and scalings involving them can be slightly worse.
to be obtained, in a form manifesting the BCJ double-copy. There are four ingredients relevant to our discussion, necessary to build NLSM, YM and DF 2 theory. For all three theories, one of the integrands is the Parke-Taylor factor which encodes the ordering of the resulting amplitudes: The NLSM integrand is given by: where the reduced matrix A ab is obtained by removing rows and columns a and b from the n × n matrix A: In this paper we will focus on the simpler object The YM integrand is given by: where the reduced matrix Ψ ab is obtained by removing rows and columns a and b from the 2n × 2n matrix Ψ: with Despite this reduction, both integrands are permutation invariant on the support of the scattering equations. We will keep track of the labels a and b, and associate them with two particles which we single out as having a distinct scaling behavior under shifts. To avoid clutter, we will sometimes drop the extra label ab from objects under consideration.
The DF 2 integrand is given by: and is directly permutation invariant. Note that the only dot products appearing are of the form (e·p), a fact we use as an assumption.
Other theories such as gravity, Born-Infeld or the special Galileon may be obtained by mixing these ingredients: It is important to note that the integral (5) fully localizes on the delta functions, so in fact the amplitude is simply given by a sum over solutions to the scattering equations.
where J is some Jacobian factor resulting from solving the delta functions. In practice the scattering equations are non-trivial to solve (see [43][44][45][46] for developments), but they do not enter into our discussion.

A. BCFW scaling
We will use the following BCFW two particle shift [16,18]: We also rescale in the worldsheet coordinates: Under this shift we find the following scalings: for NLSM, Pf A ab : for YM, Pf Ψ ab : and for DF 2 : The σ shifts can be motivated by requiring the scattering equations corresponding to particles i and j to have an improved scaling under the combined shift [39]: Note that the other SE do not have this behavior. For instance, at 4 points under a [1, 2 shift, the following scattering equation becomes: While not mandatory, solving these shifted equations up to order O(z −2 ) further improves the scaling of the YM integrand to O(z −4 ).
It is interesting to note that the Laplace expansion of the reduced Pfaffian, manifests the right BCFW scaling term by term in all shifts except those involving the "reduced" particles. Similarly, the cycle expansion of the full Pfaffian (which sums to zero) [37,47], manifests both correct scaling and permutation invariance term by term: These special building blocks are given by: where the Ψ's are gauge invariant, either via linearized field strengths or via the scattering equations: Note that in this notation the integrand for DF 2 is precisely P 1111...1 .

B. Worldsheet locality
One property common to all integrands is what we will call worldsheet locality: the correspondence between worldsheet coordinates and spacetime dot products: With integrands given as a sum over products of such factors. This has an immediate effect on UV scaling: no factor may scale worse than O(z 0 ), while for vector theories, because of multilinearity in polarization vectors, no term can scale worse than O(z −1 ).
Quite surprisingly, we will see that even this property follows directly from demanding improved scaling, as our initial ansatze allow "non-local" factors: where k and l need not be related to i and j.

III. NON-LINEAR SIGMA MODEL
As mentioned before, we are looking at essentially √ I NLSM . Therefore our ansatz is defined by: • numerators are (n/2 − 1) dot products of p i ·p j • denominators are (n/2 − 1) factors of σ ij .
For instance, the four point ansatz is given by: Then after fixing some a and b, we impose the following scalings: Pf We can also perform the check at 6 points: this ansatz already has around 5000 terms, but imposing the correct scaling, with a = 5 and b = 6, we find a unique solution:

IV. YANG-MILLS
We are using the following assumptions: • numerators are multi-linear in all polarization vectors, with mass dimension [n − 2] • denominators are products of (n − 1) σ ij factors where again we note that the denominators are unrelated to the numerators. We then claim that the YM reduced Pfaffian of Ψ ab is uniquely fixed by the following scalings under BCFW shifts: which is easily checked at four points.
At five points the ansatz is already too large to verify the claim, but there is a simple assumption which reduces the complexity: that each σ i should appear at least once per term.
Then we are able to find a unique solution: the reduced five point Yang-Mills Pfaffian.

V. DF 2 THEORY
We are using the following assumptions: Then we claim that the DF 2 integrand is uniquely fixed by a BCFW scaling of O(z −2 ) under all two particle shifts. This is easily verified at four points, but at five points we again need to impose the presence of every σ i in each term, which allows us to check a somewhat weaker claim.
It is interesting to note that if we also allow ( It remains an open question whether this is still true at higher points, and whether there exist any constraints which select the linear combinations of the P i 1 i 2 ...ir that correspond to F 3 -type interactions [37].
A. Spacetime DF 2 In spacetime we are back to usual propagators, which in the case of DF 2 [42], a nonunitary theory, come in two types: gluon propagators, which are sum of consecutive momenta ( p i ) 4 , and scalar propagators, which can be sums of non-consecutive momenta ( p i ) 2 , since the scalar color structure allows non-planar interactions. Like the CHY integrand, we do not allow dot products (e·e), but only (e·p) and (p·p). A four point ansatz is given by: Imposing gauge invariance in all four particles: we find a unique result, the DF 2 amplitude. This is similar to the YM uniqueness from gauge invariance [14,15]. Finally, we also find that the BCFW scalings: O(z 0 ) for non-adjacent shifts , uniquely fix the DF 2 amplitudes, at least up to n = 5.

VI. CONCLUSION
We have shown that the constraining power of BCFW shifts is sufficient to fully determine the main ingredients of the CHY formalism, further demonstrating the surprising universality of enhanced UV scaling. While the claim for YM and DF 2 theory only holds if also assuming permutation invariance, it appears that this assumption can be replaced with another type of UV scaling: single particle shifts [i , introduced in [16]. These shifts are simply given by: At four points, we find that the usual two particle shifts, together with the single particle scalings: are enough to fully constrain the respective integrands, pointing to a description purely in terms of UV scalings.
The O(z −2 ) scaling also guarantees the lack of a pole at infinity, and so the integrands may be rebuilt from finite residues 2 . It is worth mentioning that the O(z −2 ) of the integrands also implies the existence of so-called "bonus relations" [23] between residues.
A related theory is the so-called "DF 2 +YM", relevant to the bosonic string, and which also has a CHY construction [48][49][50]. It would be interesting to investigate whether there exist (purely field theoretic) consistency conditions which determine this object as well.