Projectivity of Planar Zeros in Field and String Theory Amplitudes

We study the projective properties of planar zeros of tree-level scattering amplitudes in various theories. Whereas for pure scalar field theories we find that the planar zeros of the five-point amplitude do not enjoy projective invariance, coupling scalars to gauge fields gives rise to tree-level amplitudes whose planar zeros are determined by homogeneous polynomials in the stereographic coordinates labelling the direction of flight of the outgoing particles. In the case of pure gauge theories, this projective structure is generically destroyed if string corrections are taken into account. Scattering amplitudes of two scalars with graviton emission vanish exactly in the planar limit, whereas planar graviton amplitudes are zero for helicity violating configurations. These results are corrected by string effects, computed using the single-valued projection, which render the planar amplitude nonzero. Finally, we discuss how the structure of planar zeros can be derived from the soft limit behavior of the scattering amplitudes.


Introduction
Zeros in scattering amplitudes are useful devices to test interesting properties of the standard model. For example, the vanishing of the tree-level amplitude of certain processes involving the emission of a gauge boson is very sensitive to the form of the trilinear couplings. Thus, the detection of amplitude zeros were proposed as a way to constraint the existence of anomalous couplings in the standard model [1] (see [2] for reviews). Although these so-called type-I zeros are corrected by both loops and higher-order emissions, they manifest themselves in the existence of dips for a set of observables, a fact that has been confirmed by various experimental groups [3].
A second class of amplitude zeros appear for particular kinematic configurations in which all momenta are confined to a plane [4]. The phenomenological implications of these planar (or type-II) zeros has been recently studied in [5] in the context of a five parton amplitude, and it was shown how the planar zeros are determined by simple relations involving rapidity differences.
In a previous paper [6], we have studied the mathematical structure of planar zeros in gauge theories and gravity, focusing on the five-point amplitude for gluons and gravitons. There it was found that, once the outgoing momenta are expressed in terms of stereographic coordinates, the loci of planar zeros is determined by a cubic integer curve in the projective plane defined by these coordinates.
Although the analysis presented in [6] focused on the five-point scattering amplitude, the projective nature of the planar zeros in Yang-Mills theories is present for any multiplicity. To see this let us recall that, in a (super) Yang-Mills theory, the n-gluon tree-level amplitude can be written in the form [7] A n = (ig) n−2 σ∈S n−2 c σ A n 1, 2, σ(3 . . . , n) , To evaluate the amplitude in the planar limit, it is convenient to consider a center-of-mass frame where the incoming particles propagate along the z axis: (1, 0, 0, 1), while the momenta of the on-shell outgoing gluons can be parametrized in terms of stereographic coordinates according to p a = −ω a 1, ζ a + ζ a 1 + ζ a ζ a , i ζ a − ζ a 1 + ζ a ζ a , ζ a ζ a − 1 1 + ζ a ζ a with a = 3, . . . , n. (1.4) For the particular case of MHV amplitudes, the Parke-Taylor formula [8] gives the following expression for the color-ordered amplitudes A n 1 − , 2 − , σ(3 + , . . . , n + ) = i 12 4 12 2σ(3) . . . σ(n − 1)σ(n) σ(n) 1 . (1.5) Without loss of generality, we can consider planar scattering where all momenta lie on the plane y = 0, which means that the stereographic coordinates giving the direction of flight of the outgoing gluons are all real (ζ a = ζ a ). The relevant spinor inner products are computed to be 12 = √ s, 1σ(j) = −i √ 2s 1 4 |ζ σ(j) | ζ σ(j) (1.6) .
The aim of this paper is to further investigate these issues, focusing on the conditions under which the projective structure of the planar zeros is preserved. We will see that the equation determining them is invariant under a simultaneous rescaling of the outgoing stereographic coordinates for theories with gauge invariance, even when matter scalar fields are introduced.
The resulting projective curves are of the same type as the ones found for gluon scattering in Ref. [6]. Pure scalar theories, on the other hand, give rise to equations for the existence of planar zeros which are not homogeneous in the stereographic coordinates.
In the case of graviton scattering, we trace the vanishing of the planar five-graviton amplitude found in [6] to the fact that the amplitude becomes effectively three-dimensional in this limit. According to a general result of [9], odd-multiplicity three-dimensional gravitation amplitudes vanish due to helicity non-conservation. This conclusion is confirmed by a computation of the planar limit of the helicity-preserving six-graviton amplitude, which gives a nonzero result. We revisit the amplitudes for the scattering of scalars with graviton emission to find that they also vanish identically in the planar limit, similarly to what happens with the five-graviton amplitude computed in [6].
We also study the corrections to planar zeros associated with the ultraviolet completions of gauge theories and gravity provided by string theory. For Yang-Mills scattering, we compute the five gauge bosons disk amplitude using the methods developed in Ref. [10]. Expanding the generalized Euler integrals in powers of the inverse string tension, we find that the planar zero condition found in the field theory limit gets corrected by equations which fail to preserve the projective structure found in [6]. In the context of gravity amplitudes, we compute the five-graviton closed string amplitude on the sphere and its α expansion using the single-valued projection [12,13,14]. Unlike its field theory limit, the planar graviton string amplitude is generically nonzero, thus avoiding the consequences of the theorem proved in [9].
The plan of the paper is as follows. In Section 2 we analyze planar zeros in a cubic nongauge massless scalar theory transforming as bi-adjoints under a global symmetry group. We particularize our analysis to the case of a φ 3 theory, where we study the structure of the curves determining the planar zeros. We dedicate Section 3 to the study of planar zeros in a theory of scalars coupled to a gauge field. Having completed our presentation of the properties of planar zeros in gauge field theories, we proceed in Section 4 to the computation of the α corrections to the five-gluon amplitude in the planar limit.
In Section 5 we turn our attention to the planar zeros of gravitational scattering amplitudes, considering the collision of two scalars, both distinguishable and indistinguishable, with emission of a graviton. Here we also analyze the helicity-preserving six-graviton amplitude, which turns out to be nonzero in the planar limit. Section 6 is devoted to the study of the α corrections to the planar five graviton tree level amplitude. Finally, in Section 7 we discuss how the projective properties of planar zeros in theories with gauge invariance emerge from the structure of the amplitude in the soft limit. Our conclusions are summarized in Section 8. To avoid cluttering the main text with cumbersome expressions, some long equations have been deferred to the Appendix.

Pure scalar theories
We begin with the analysis of a pure scalar theory transforming in the biadjoint representation of a generic global symmetry group G × G, with action and study the tree-level, five point amplitude. A very economic way of obtaining this amplitude is by using the so-called zeroth-copy prescription [15], consisting in replacing kinematic numerators in the pure-gauge theory amplitude with a second copy of the color factors where c i , c i are the color factors of G and G respectively. With this, we find where we have defined the kinematic invariants 4) and the color factors are given by As in [6], we work in a center-of-mass reference frame in which the incoming particles have momenta given by (1.3), whereas the momenta of the three outgoing particles are parametrized using the stereographic coordinates as given in Eq. (1.4). In our convention all momenta enter the diagram. We consider planar scattering processes taking place on the plane y = 0, i.e. ζ a = ζ a for a = 3, 4, 5. In the case of the five-point amplitude, imposing energy-momentum conservation completely determines the energies of the outgoing particles: , .
Writing the kinematic invariants in (2.3) using our parametrization of the momenta, we arrive at the following equation for the planar five-point scalar amplitude , where P 10 (ζ 3 , ζ 4 , ζ 5 ) is a degree-ten polynomial whose coefficients depend on the color factors.
The explicit expression for this polynomial can be found in Eq. (A.1). The amplitude has collinear singularities at ζ a → 0, ∞ together with soft poles at ζ a ζ b → −1 (with a < b) where the energy of one of the outgoing particles tend to zero. In addition, the collinear limits ζ a → ζ b lead to a divergence of the energies of the outgoing particles.
Planar zeros are thus determined by the equation Inspecting Eq. (A.1), we find that, unlike the case of pure gauge theories studied in [6], this equation is not homogeneous in the stereographic coordinates, since it contains monomials of both degree 10 and 8. Thus, unlike the case of pure gauge theories studied in [6], planar zeros are no longer determined by a projective curve 1 . To study the corresponding geometric loci of planar zeros, we focus on the five-point amplitude for φ 3 theory, which can be retrieved from Eq.

Distinguishable scalars
To further explore this possibility, we study now the presence of planar zeros in scalar QCD (sQCD), in particular in the scattering of two distinct scalars with emission of a gluon in the final state. This process has been studied in Ref. [16]. We label momenta and color quantum numbers according to where + (p 5 ) indicates the polarization vector of the gluon. We slightly modify the conventions of Ref. [16], and consider all momenta as incoming. The amplitude takes the form A = g 3 C 1 n 1 s 24 s 35 + C 2 n 2 s 24 s 15 + C 3 n 3 s 24 + C 4 n 4 s 13 s 45 + C 5 n 5 s 13 s 25 + C 6 n 6 s 13 + C 7 n 7 s 13 s 24 , where now there are seven different color factors 3) We allow for the possibility of the two scalars transforming in different representations of the gauge group. The color factors satisfy four Jacobi identities These relations can be used to express the five-point amplitudes in terms of only three independent color factors, that we take C 1 , C 2 , and C 4 . Namely, Again, we work in the center-of-mass reference frame and use (1.4) to express outgoing momenta in terms of the stereographic coordinates. In the planar limit ζ a = ζ a we take the gluon polarization vector to be which indeed satisfies p 5 · ± (p 5 ) = 0. In the following, we specialize our analysis to a positive helicity gluon, ≡ + . With this, the numerators in the amplitude (3.2) take the following form in the planar limit , , As a nontrivial test of the previous equations, it can be checked that the amplitude satisfies the gauge Ward identity. Combining the numerators with the expressions for the kinematic invariants we arrive at the following form of the tree-level amplitude of distinct scalars with gluon emission in the limit of planar scattering: Similarly to what happens for pure gauge theories [6], planar zeros are determined by a homogeneous cubic polynomial An important difference with the pure gauge theory case is, however, that the polynomial factorizes. One of the factors, the trivial branch, is linear and independent of the color factors of the interacting particles, It is also independent of the direction of flight of the emitted gluon. The second, non-trivial branch is a quadratic equation whose coefficients depend on the three independent color factors.
Being also homogeneous in the color factors, the polynomial (3.9) defines an integer curve in the projective plane defined by the coordinates (ζ 3 , ζ 4 , ζ 5 ). It seems natural now to single out the direction of flight of the emitted gluon and study this curve in the patch centered around the point (0, 0, 1) using the coordinates Now, the trivial branch of planar zeros is determined by the straight line whereas the non-trivial quadratic curve takes the form 14) The quadratic curve (3.14) can be easily classified for a generic gauge group in terms of the three invariants (∆, δ, I) and the semiinvariant σ (see, for example, [17]) defined by Since δ ≤ 0, no ellipses are possible. It is also impossible to have δ = 0 with ∆ = 0, so parabolas are ruled out as well. Thus, the only possible class of curves are hyperbolas (∆ = 0, δ < 0), intersecting lines (∆ = 0, δ < 0), or parallel lines (∆ = δ = 0, σ < 0). Notice that this classification is valid for all gauge groups and all representations of the scalar fields.
As an illustrative example, we study the case of two scalars with charges e and e coupled to a photon. This correspond to having the U(1) generators giving the following values for the color factors In the patch centered around U = V = 0, the projective curve determining the planar zeros is given by the loci of planar zeros are hyperbolas with asymptotes along the coordinates axes and whose center is located at the point A particularly simple case arises when we consider that both scalars, though distinct, have the same electric charge, e = e . In this case the curve is given by U V = 1.

Indistinguishable scalars
The previous analysis of the scattering amplitude of distinct scalars coupled to a gauge field in an arbitrary representation illustrates how the derivative couplings required by gauge invariance are enough to restore the projective nature of planar zeros, that was absent in the pure scalar theories studied in Section 2. This is also the case when considering sQCD with a single scalar field in the adjoint representation of the gauge group. We consider again a five-point amplitude corresponding to the process [19] After all quartic couplings are resolved in terms of trivalent vertices, the 15 topologies contributing to this amplitude are the ones already encountered in both pure Yang-Mills theories and the scalar theories studied in Section 2. The amplitude takes the form where the color factors are the ones defined in (2.5), while the numerators are given by We have assumed again that the emitted gluon has positive helicity.
A long but straightforward evaluation of the amplitude in the planar limit gives the result The prefactor ζ 2 3 − ζ 3 ζ 4 + ζ 2 4 does not have real nontrivial zeros, corresponding to two complex straight lines in the (ζ 3 , ζ 4 ) plane. After multiplying by ζ 3 ζ 4 ζ 5 , which does not introduce any spurious physical zeros, we arrive at the cubic homogeneus equation Interestingly, the condition (3.25) for the existence of planar zeros in the scattering of two indistinguishable scalars with the emission of a gluon is identical to the one found for the fivegluon scattering amplitude in [6]. The reader is referred to this reference for the analysis of the curves for various gauge groups. where a = 1, . . . , (n − 3)! and Π a denotes the elements of S n−3 . Then, Eq. (4.1) can be written (1, σ 1 , n − 1, n) . . .
String corrections to field theory gauge amplitudes are obtained by expanding the integrals (4.2) in powers of α . The coefficients of the series are expressed in terms of kinematic invariants and multiple zeta values (MZV). Thus, the (n − 3)! × (n − 3)! matrix F has the following expansion in powers of the inverse string tension [18], with P 0 = I and M 1 = 0. At order α k , the matrix coefficient is a homogeneous function of degree k in the kinematic invariants s ij .
Let us particularize the analysis to the five-point function , where the matrix entries have the following expansion in powers of the string slope The coefficient A

Gravitational amplitudes
One of the results of Ref. [6] is that the planar, MHV five-point graviton amplitude is identically zero. This fact can be seen as a consequence of the theorem proved in [9], stating the vanishing of all helicity violating amplitudes in three dimensions. Indeed, at the level of the tree amplitude, the graviton couplings are of the form p i · ε k · p j with i, j = k, so imposing planarity decouples the graviton polarization normal to the plane. This renders the scattering effectively threedimensional and, as a consequence, the planar MHV amplitude is equal to zero.
In this section we are going to explore other gravitational amplitudes involving scalar particles minimally coupled to gravity. We begin with the scattering of two distinguishable scalars with graviton emission The tree-level amplitude was computed in Ref. [21] using the Feynman rules for a scalar theory coupled to gravity. Using the Sudakov decomposition, the amplitude has the tensor structure where K ⊥ ≡ k 1,⊥ + k 2,⊥ and the coefficients A i are rational functions of the Sudakov parameters α i , β i . The tensor structure of the amplitude shows again how, once the planar limit ζ i = ζ i is taken, the polarizations outside the interaction plane decouple and the amplitude becomes effectively three-dimensional. In this limit, the Sudakov parameters take the following form in terms of the stereographic coordinates: , , while the graviton polarization tensor is taken to be ε ± = ± ⊗ ± , with ± defined by (3.6).
Using the explicit expression for the coefficients in (5.3) given in [21], we find that the planar amplitude vanishes identically .
The gravitational amplitude (5.3) cannot be retrieved using the double-copy BCJ construction [22] from the gauge scattering amplitude of two distinct scalars with a gluon emission [16].
Despite this, the putative gravitational amplitude obtained from the double-copy of the gauge amplitude (3.2), with denominators satisfying color-kinematics duality, identically vanishes in the planar limit.
A similar result is obtained for the gravitational scattering of two indistinguishable scalars.
In this case, the amplitude can be obtained by double copy from the gauge scattering of adjoint identical scalars given in Eq. (3.22) using color-kinematics duality [19], where the numerators are the ones given in Eq. (3.23). In fact, the cancellation of this amplitude in the planar limit can be seen to happen by a mechanism similar to the one found in [6] for the pure gravitational case. Indeed, replacing the color factors by the corresponding numerators in the condition for the gauge planar zeros (3.25), we find the following condition for the existence of planar zeros n 7 ζ 2 3 ζ 4 − n 8 ζ 2 3 ζ 5 − n 6 ζ 3 ζ 2 4 + n 11 ζ 3 ζ 2 5 +(n 2 + n 6 − n 7 + n 8 − n 11 − n 13 )ζ 3 ζ 4 ζ 5 + n 13 ζ 2 4 ζ 5 − n 2 ζ 4 ζ 2 5 = 0. (5.8) In the planar limit (i.e., real stereographic coordinates), the relevant numerators have the following form Substituting these values in Eq. (5.8), we conclude that the condition for the existence of planar zeros is identically satisfied for any kinematic configuration. Since the numerators now are far more complicated than the ones for gluon scattering [6], the cancellation taking place is less trivial.
Since the scalar gravitational amplitudes studied above do not preserve helicity, the fact that they are zero in the planar limit is also a consequence of the vanishing of all helicityviolating supergravity amplitudes when reduced to three dimensions [9] (see also [23]). Indeed, the gauge amplitude for indistinguishable adjoint scalars (3.22) can be embedded in a N = 2 super Yang-Mills theory [19]. Thus, the corresponding double copy can be thought of as a scattering amplitude in N = 4 supergravity [24]. In the case of the gravitational scattering of two distinct scalars (5.1), on the other hand, the theory can be also embedded in a fourdimensional supergravity theory, such as the ones studied in [24]. Both amplitudes vanish in the planar limit, where the dynamics becomes effectively three-dimensional.
In the case of graviton MHV amplitudes, their vanishing in the planar limit follows from the explicit expression of the n-graviton amplitude [25] M ∈ R). Another scattering amplitude whose vanishing in the planar limit is not implied by the results of [9] is the six-graviton, helicity preserving amplitude M 6 (1 + , 2 − , 3 − , 4 − , 5 + , 6 + ).
Hence, the 6-graviton NMHV amplitude can be obtained by using the KLT formula in eqs.(??) and (??). The explicit dependence on the stereographic coordinates is quite messy to write it here; however, some arbitrary planar configurations can show numerically that the full amplitude, as expected, does not vanish. See Figure 1. Fig. 1. Arbitrary NMHV n = 6 graviton processes for which the momentum planar configuration lead to a non-vanishing amplitude.
The explicit expression for the amplitude M 6 (1 + , 2 − , 3 − , 4 − , 5 + , 6 + ) in the planar limit in terms of the stereographic coordinates is very cumbersome and will not be given here. However, it can be seen that this amplitude does not vanish. In Fig. 2 we have depicted two kinematic planar configurations for which a calculation of the tree-level amplitude gives a nonzero result.
The generalized KLT relations (6.1) can be recast in matrix form as [14] M where in the second identity we have changed the basis of the first-copy amplitudes to express them in terms of of the basis A n used in Eq. (4.4). Using now this same equation, we can express the string graviton amplitude in terms of field theory gauge amplitudes as The single-valued projection [11,12,13,14]  With this, the string amplitude takes the form Incidentally, dropping the term sv(F ) in the previous expression we retrieve the KLT expression of the field theory graviton amplitude.
We particularize our analysis to the five-point amplitude In order to get the closed string expression, we have to perform the rescaling α → α /4, as explained in [12]. Notice that the single-valued projection (6.6) eliminates many terms in the α -expansion of F . Plugging (6.9) into Eq. However, a first nonvanishing string correction survives the planar limit, This term is independent of the directions of the final states and is never zero. Using the expansion (4.5), it is possible to compute higher order corrections, whose coefficients are functions of the stereographic coordinates ζ a . We obtain the structure It is interesting to notice that the planar closed string amplitude (6.13) does not exhibit the soft poles at ζ a ζ b = −1 (with a < b), unlike the planar disk amplitude in Eq. (4.12). This reflects the peculiar relation between the soft and planar limits of amplitudes with gravitons, in both string and field theories. It would be worthwhile to clarify the interplay between the two limits using recent results for soft theorems in string theory [29,30].

Remarks on soft limits
We turn now to the problem of whether the mathematical structure of planar zeros can be fully captured in the soft limit. We begin with the gauge case analyzing the simple example of two distinguishable scalars studied in Section 3.1. In the limit in which the emitted positive (resp. negative) helicity gluon is soft, p 5 → 0, the leading behavior of the amplitude takes the form [

31]
where A 4 is the four-scalar tree level amplitude. In terms of the stereographic coordinates ζ a and taking the planar scattering limit, the soft amplitude reads The condition for the vanishing of the soft gauge theory amplitude in the planar limit is given by which reproduces the nontrivial loci of planar zeros for the full tree level amplitude discussed in Eq. (3.11). We notice, however, that in taking the soft limit we miss the trivial branch 2ζ 3 − ζ 4 = 0. In fact, this loci cannot be captured in the soft-gluon limit of the amplitude, since in the limit ω 5 → 0, which implies that 2ζ 3 − ζ 4 never vanishes. This shows that the trivial branch of planar zeros is not accesible from the soft limit of the amplitude. Therefore, not all planar zeros can be realized in the limit in which the gluon is taken to be soft. Notice, however, that this does not contradict the statements made in [6]. Indeed, any planar zero can be realized in the limit in which one of the particles is taken to be soft. However, once we decide which particle is soft, not all planar zeros can be realized in this regime, as we have seen in this case.
This being said, soft limits can be exploited to make a general analysis of planar zeros in the gauge case. We study the scattering of n charged particles in QED, parametrized by stereographic coordinates ζ i (i = 1, . . . , n), with the emission of a soft photon whose momenta we write in terms of the coordinate ζ n+1 , p a = ω a 1, ζ a + ζ a 1 + ζ a ζ a , i ζ a − ζ a 1 + ζ a ζ a , ζ a ζ a − 1 1 + ζ a ζ a , a = 1, . . . , n + 1.
The soft theorem for massless QED can be recast in terms of stereographic coordinates as [32] lim ω n+1 →0 + ω n+1 A n+1 (p 1 , . . . , p n+1 ) A n (p 1 , . . . , p n ), (7.7) where we have used the following form for the polarization vector of the photon A planar zero is now obtained by setting The projective nature of gauge planar zeros is also fragile with respect to the inclusion of string effects. We have seen how the α corrections to the five gluon amplitude introduces terms which do not share the projective structure of the field theory result.
The features of planar gravitational scattering differ in many aspects from those of gauge theories. Due to the peculiar features of three-dimensional gravity, odd-multiplicity amplitudes are zero in the planar limit while for even multiplicities they are only nonzero when helicity is conserved. We have checked this fact explicitly in various cases. String corrections to the field theory amplitude are generically nonvanishing in the planar limit, independently of their helicities and multiplicities, thus correcting the strong constraints imposed by the results of [9].
There are some intriguing elements in the interplay between planar zeros and soft limits in gauge theories that are worth exploring. Although planar zeros are expected to be corrected by quantum effects, the very fact that they are determined by the soft limit indicate that they might be of relevance for the infrared properties of the theory. In particular, it would be interesting to explore whether planar zeros are of any relevance for the asymptotic symmetries for theories like QED [32,33,34]. Due to Bose symmetry, the polynomial is invariant under permutations of its three variables ζ 3 , ζ 4 , and ζ 5 , provided this is supplemented with the corresponding permutation of S 3 acting on the color factors, as explained in [6].

A.2 The coefficients A
5 and A 5 of the α expansion (4.12) The coefficient A 5 of the α 2 correction to the five-gluon amplitude is a degree 10 polynomial in the stereographic coordinates, containing monomials of degree 8, 6, 4, and 2 as well