1 Introduction

Planar Zeros in Gauge Theories and Gravity

Diego Medrano Jiménez111d.medrano@csic.es, Agustín Sabio Vera222a.sabio.vera@gmail.com and Miguel Á. Vázquez-Mozo333Miguel.Vazquez-Mozo@cern.ch

C/ Nicolás Cabrera 15, E-28049 Madrid, Spain

Instituto Universitario de Física Fundamental y Matemáticas (IUFFyM)

Plaza de la Merced s/n, E-37008 Salamanca, Spain

Abstract

Planar zeros are studied in the context of the five-point scattering amplitude for gauge bosons and gravitons. In the case of gauge theories, it is found that planar zeros are determined by an algebraic curve in the projective plane spanned by the three stereographic coordinates labelling the direction of the outgoing momenta. This curve depends on the values of six independent color structures. Considering the gauge group SU() with and fixed color indices, the class of curves obtained gets broader by increasing the rank of the group. For the five-graviton scattering, on the other hand, we show that the amplitude vanishes whenever the process is planar, without imposing further kinematic conditions. A rationale for this result is provided using color-kinematics duality.

## 1 Introduction

In the last decade, many studies have permitted a deeper understanding of the relationship between gravity and gauge theories from the point of view of scattering amplitudes (see [1] for a comprehensive review). One of the most interesting results is color-kinematics duality [2], which allows the construction of gravity amplitudes by replacing color factors by a second copy of the kinematic numerators. This double copy structure has a historic antecedent in the Kawai, Lewellen, and Tye (KLT) relations [3], showing how, at tree level, closed string amplitudes admit a decomposition in terms of products of open string amplitudes. Similar structures have been found in various other setups [4, 5]. It seems clear that, at the level of scattering amplitudes, there is a sense in which gravity can be considered the “square” of a gauge theory.

Given this double copy structure, a natural question to ask is how certain properties of gauge theory amplitudes translate into the gravitational side. One of these is the existence of radiation zeros [6]. This is a peculiar feature of certain scattering processes where one or more massless gauge bosons are radiated, consisting in the vanishing of the amplitude for certain phase space configurations. The phenomenon was first identified in processes involving gauge bosons trilinear couplings, in particular [7]. It has been experimentally observed both at the Tevatron [8] and LHC [9]. Their existence has been also studied in graviton photoproduction [10].

These so-called Type-I zeros appear for momentum configurations satisfying the constraint , where is the momentum of the gauge boson, and are the charge and momenta of the other particles, and is a numerical constant. In Ref. [11] it was realized that zeros in the amplitude may also occur when the spatial momenta of the particles involved in the process lie on the same plane. These Type-II or planar zeros have been identified in the processes [12] and [13], in both cases in the soft photon limit.

So far, the only study of planar zeros beyond the soft limit has been carried out in the interesting work [14], where the five parton amplitude in QCD was analyzed. Using the maximally helicity violating (MHV) formalism, planar zeros were found both for the and processes. In the case of the five-gluon amplitude for general color factors the planar zero condition depends on the color quantum numbers of incoming and outgoing gluons.

The present article has a double aim. One is to study the conditions for the emergence of planar zeros in the five-gluon amplitude. We show that planar zeros are a “projective” property of the amplitude, in the sense that they are preserved by a simultaneous rescaling of the stereographic coordinates labeling the flight directions of the three outgoing gluons. In terms of stereographic coordinates, we find that the existence of planar zeros is determined by a cubic algebraic curve whose integer coefficients are given in terms of the color factors. In the case of SU() gauge groups, we find that the casuistic of curves obtained for different color configurations gets broader as the rank increases, starting with the case of SU(2) where no physical zeros are found for external particles with well-defined color quantum numbers. Our second target consists in exploiting color-kinematics duality to study planar zeros in the gravitational case, where we find that the five-graviton amplitude vanishes whenever the process is planar. This can be understood applying the BCJ prescription to the equation determining the zeros in the gauge case. By replacing color factors with kinematic numerators satisfying color-kinematics duality, the condition for the planar zero is seen to be identically satisfied without further kinematic constraints.

The plan of the paper is as follows. In Section 2 we review the calculation of the five gluon amplitude using the MHV formalism. Section 3 is devoted to the conditions for planar zeros in the gauge case, while in Section 4 we study the transformation of the loci of planar zeros under permutations of the color labels of the external gluons. In Section 5 the graviton amplitude is obtained using color-kinematics duality and the condition for the existence of amplitude zeros is obtained. Finally, in Section 6 we summarize our conclusions.

## 2 The five-gluon amplitude

In this section we revisit the construction of the five-gluon amplitude

 g(p1,a1)+g(p2,a2)⟶g(p3,a3)+g(p4,a4)+g(p5,a5), (2.1)

where we take all momenta incoming. The tree level amplitude is computed in terms of 15 nonequivalent trivalent diagrams, leading to the expression

 A5 =g3(c1n1s12s45+c2n2s23s15+c3n3s34s12+c4n4s45s23+c5n5s15s34+c6n6s14s25+c7n7s13s25+c8n8s24s13 +c9n9s35s24+c10n10s14s35+c11n11s15s24+c12n12s12s35+c13n13s23s14+c14n14s25s34+c15n15s13s45), (2.2)

where we have introduced the kinematic invariants

 sij=(pi+pj)2=2pi⋅pj,i

The color factors in Eq. (2.2) are given by111Our conventions for the color factors differ from those in Refs. [1, 2].

 c1 =fa1a2bfba3cfca4a5,c2=fa1a5bfba4cfca3a2, c3 =fa3a4bfba5cfca1a2,c4=fa4a5bfba1cfca2a3, c5 =fa5a1bfba2cfca3a4,c6=fa1a4bfba3cfca5a2, c7 =fa1a3bfba4cfca5a2,c8=fa1a3bfba5cfca4a2, (2.4) c9 =fa3a5bfba1cfca2a4,c10=fa4a1bfba2cfca3a5, c11 =fa1a5bfba3cfca4a2,c12=fa3a5bfba4cfca1a2, c13 =fa1a4bfba5cfca3a2,c14=fa5a2bfba1cfca3a4, c15 =fa1a3bfba2cfca4a5,

and satisfy nine independent Jacobi identities

 c3−c5+c14 =0,c3−c1−c12=0, c4−c1+c15 =0,c4+c2−c13=0, c5+c2−c11 =0,c13−c6+c10=0, (2.5) c14−c7+c6 =0,c7−c8+c15=0, c8−c9−c11 =0,(c9−c10+c12=0).

On general grounds, the amplitude can be written in terms of color-ordered amplitudes as

 A5=g3∑σ∈S4c[1,σ(2,3,4,5)]A5[1,σ(2,3,4,5)], (2.6)

where the sum is over noncyclic permutations of the external legs. However, the set of color-ordered amplitudes is over complete, a fact expressed by the Kleiss-Kuijf relations [15]. In the case of the five-point amplitude, there are different ways of choosing a basis in the space of independent color structures [16]. We select one of these basis by fixing the incoming gluons (see Fig. 1), so the five-gluon amplitude in Eq. (2.2) can be re-expressed in terms of color ordered amplitudes according to

 A5=g3∑σ∈S3c[1,2,σ(3,4,5)]A5[1,2,σ(3,4,5)], (2.7)

where the subamplitudes are explictly given in terms of the numerators by

 A5[1,2,3,4,5] =n1s12s45−n2s23s15+n3s34s12+n4s45s23+n5s15s34, A5[1,2,3,5,4] =−n1s12s45−n13s23s14+n12s35s12−n4s45s23+n10s14s35, A5[1,2,4,3,5] =−n12s12s35−n11s24s15−n3s34s12+n9s35s24−n5s15s34, (2.8) A5[1,2,4,5,3] =n12s12s35−n8s24s13−n1s45s12−n9s35s24−n15s13s45, A5[1,2,5,3,4] =−n3s12s34−n6s25s14−n12s35s12+n14s34s25−n10s14s35, A5[1,2,5,4,3] =n3s12s34−n7s25s13+n1s12s45−n14s34s25+n15s13s45.

Going back to the expression for the color factors in Eq. (2.4), these partial amplitudes are respectively associated with the six color factors , , , , , and .

At this point we can exploit the generalized gauge freedom in the definition of the numerators to implement color-kinematics duality, so the numerators mimic the Jacobi identities (2.5). Solving the corresponding equations we can eliminate to finding the following solution for the numerators in terms of the basis of color-ordered amplitudes

 n1 =−n12=n15=s12s45A5[1,2,3,4,5], n2 =n3=n4=n5=n11=n13=n14=0, n6 =n7=n10=s14s35A5[1,2,3,5,4]+s14(s35+s45)A5[1,2,3,4,5], (2.9) n8 =n9=s14s35A5[1,2,3,5,4]+(s14s35+s14s45+s12s45)A5[1,2,3,4,5].

Color-kinematics duality is independent of the polarization of the gluons. Here we are going to use the MHV formalism and assign negative helicity to the incoming gluons. Using the Parke-Taylor formula [17] we have

 A5[1−,2−,σ(3+,4+,5+)]=i⟨12⟩4⟨12⟩⟨2σ(3)⟩⟨σ(3)σ(4)⟩⟨σ(4)σ(5)⟩⟨σ(5)1⟩, (2.10)

for any permutation of the last three indices. Expressing in addition the kinematic invariants in terms of spinors, , we arrive at the following expressions for the numerators

 n1 =−n12=n15=i⟨12⟩4[21][54]⟨23⟩⟨34⟩⟨51⟩, n6 =n7=n10=i⟨12⟩4[14][52]⟨23⟩⟨34⟩⟨51⟩, n8 =n9=i⟨12⟩4[24][51]⟨23⟩⟨34⟩⟨51⟩, (2.11) n2 =n3=n4=n5=n11=n13=n14=0.

With this result, the five-gluon amplitude can be written as

 A5 =−ig3⟨12⟩3(c2⟨23⟩⟨34⟩⟨45⟩⟨51⟩+c6⟨25⟩⟨53⟩⟨34⟩⟨41⟩+c7⟨25⟩⟨54⟩⟨43⟩⟨31⟩ +c8⟨24⟩⟨45⟩⟨53⟩⟨31⟩+c11⟨24⟩⟨43⟩⟨35⟩⟨51⟩+c13⟨23⟩⟨35⟩⟨54⟩⟨41⟩). (2.12)

Alternatively, this expression can be obtained from Eq. (2.7) by a direct application of the Parke-Taylor formula.

The spinor products appearing in the five-gluon amplitude can now be recast in terms of momenta. Working in the center-of-mass frame, the incoming momenta take the form

 p1=√s2(1,0,0,1),p2=√s2(1,0,0,−1). (2.13)

On the other hand, for the three outgoing gluons their spatial momenta are parametrized using stereographic coordinates (with ) according to

 pa=−ωa⎛⎝1,ζa+¯¯¯ζa1+ζa¯¯¯ζa,i¯¯¯ζa−ζa1+ζa¯¯¯ζa,ζa¯¯¯ζa−11+ζa¯¯¯ζa⎞⎠, (2.14)

where the global minus sign reflects that all momenta are taken entering the diagram. The stereographic coordinates are related to the rapidity and the azimuthal angle by

 ζa=eya+iϕa. (2.15)

## 3 Gauge planar zeros

We focus now on planar five-gluon scattering with general color quantum numbers. Since the incoming particles travel along the axis, without loss of generality we can take all momenta lying on the -plane. This means that and therefore has to be real and the outgoing momenta read

 pa=−ωa1+ζ2a(1+ζ2a,2ζa,0,ζ2a−1). (3.1)

Alternatively, the planarity condition implies that all emitted particles have azimuthal angles with either or .

Implementing momentum conservation gives three independent equations that determine the gluon energies in terms of the center-of-mass energy and the flight directions of the outgoing gluons labelled by ,

 ω3 =√s2(1+ζ23)(1+ζ4ζ5)(ζ3−ζ4)(ζ3−ζ5), ω4 =√s2(1+ζ24)(1+ζ3ζ5)(ζ4−ζ3)(ζ4−ζ5), (3.2) ω5 =√s2(1+ζ25)(1+ζ3ζ4)(ζ5−ζ3)(ζ5−ζ4).

Furthermore, the finite positive energy condition imposes constraints on the possible values of . In particular, let us remark that finite energy implies that for .

Using this parametrization, the amplitude (2.12) takes the form

 A5 =2ig3√s(ζ3−ζ4)(ζ3−ζ5)(ζ4−ζ5)(1+ζ3ζ4)(1+ζ3ζ5)(1+ζ4ζ5)[−c2ζ5−ζ3ζ3−c6ζ4−ζ5ζ5 +c7ζ3−ζ5ζ5−c8ζ3−ζ4ζ4+c11ζ5−ζ4ζ4+c13ζ4−ζ3ζ3]. (3.3)

In order to find the zeros of the amplitude, we notice that the finite energy condition implies that the prefactor can never vanish. As a consequence, we find the following equation depending on the color factors

 c2ζ5−ζ3ζ3+c6ζ4−ζ5ζ5−c7ζ3−ζ5ζ5 +c8ζ3−ζ4ζ4−c11ζ5−ζ4ζ4−c13ζ4−ζ3ζ3=0. (3.4)

The planar zero condition just derived is a homogeneous equation of vanishing degree. Since the amplitude (3.3) diverges whenever any of the vanishes, we can multiply the previous equation by without generating spurious solutions in the physical region. Taking projective coordinates

 (ζ3,ζ4,ζ5)=λ(1,U,V),λ,U,V≠0 (3.5)

the planar zeros of the five-gluon amplitude are determined by the loci defined by the following equation

 c7U−c8V−c6U2+c11V2+(c2+c6−c7+c8−c11−c13)UV+c13U2V−c2UV2=0. (3.6)

Moreover, this equation is homogeneous in the color factors and therefore independent of the normalization of the gauge group generators. Since there exists a normalization of the generators that makes all structure constants integer numbers [18], the planar zeros are determined by a cubic curve with integer coefficients.

In terms of the projective coordinates (3.5), the energies of the outgoing particles take the form

 ω3 =√s2(1+λ2)(1+λ2UV)λ2(1−U)(1−V), ω4 =√s2(1+λ2U2)(1+λ2V)λ2(U−1)(U−V), (3.7) ω5 =√s2(1+λ2V2)(1+λ2U)λ2(V−1)(V−U).

We have seen already that in order to keep the amplitude finite we have to exclude and from the physical region. Now, energy finiteness further demands that , , and . By requiring we find that, for example, the region , has to be considered unphysical as well. Indeed, if this is the case all three numerators in (3.7) are positive whereas the three denominators cannot have the same sign simultaneously. As a consequence, at least one of the energies has to be negative and the configuration is excluded. Studying the values of and in which the three energies are simultaneously positive for a given , we arrive at the physical regions shown in Fig. 2. Notice that the plot is symmetric under the exchange .

The conformal structure of the equation defining the planar zeros indicates that each solution of Eq. (3.6) can be realized in infinitely many physical setups, depending on the value of the parameter . Notice that the boundaries of the allowed regions depend on as well, so they move as this parameter varies, while the position of the zeros, being a projective invariant, remain fixed.

A particularly interesting regime is the soft limit, in which one or various of the emitted gluon energies tend to zero. From Eq. (3.7) we see that the points in the plane for which vanish are given by

 UV =−1λ2(ω3=0), V =−1λ2(ω4=0), (3.8) U =−1λ2(ω5=0),

which are indicated by the dashed lines in Fig. 2. On general grounds, a planar zero corresponding to a point of the cubic (3.6) can be physically captured in the soft limit provided there is a value of for which this point collides against any of the “soft” lines defining the boundaries of the physical region.

The first example to analyze is the case of incoming gluons in a singlet state for arbitrary gauge group, already studied in [14]. Using the fact that , we find

 c2=c6=−c7=c8=−c11=−c13=−fa3a4a5. (3.9)

 U+V+U2+V2−6UV+U2V+UV2=0. (3.10)

The associated algebraic curve is represented in Fig. 3. Comparing with Fig. 2 we see that for small enough there is indeed a large part of the curve lying on physically allowed regions. In particular, for there are solutions with arbitrarily large and .

We study next the loci defined by Eq. (3.6) for SU() gauge groups with different ranks and various color configurations:

### Su(2).

In the case of SU(2) it is easy to write a closed expression for the color factors

 c2 =δa3a4ϵa2a5a1−δa2a4ϵa3a5a1, c6 =δa5a3ϵa2a4a1−δa2a3ϵa5a4a1, c7 =δa1a4ϵa3a5a2−δa3a4ϵa1a5a2, c8 =δa2a5ϵa4a1a3−δa4a5ϵa2a1a3, (3.11) c11 =δa4a3ϵa2a5a1−δa2a3ϵa4a5a1, c13 =δa4a5ϵa1a2a3−δa1a5ϵa4a2a3,

where a convenient normalization of the gauge group generators has been chosen. In principle, the color factors can only take the values , and , since each term on the right-hand side of these equations is either or . However, the case is excluded. The reason is that due to the structure of indices of the Levi-Civita tensor, sharing the last two entries, they cannot have oposite signs. As a consequence, they cannot add up and we conclude that for SU(2) the color factors satisfy .

An exploration of the possible external color numbers show that there are no solutions containing physical points. We illustrate this with a few examples. Our first case has color structure , giving the same value for all color factors

 c2=c6=c7=c8=c11=c13=1. (3.12)

The resulting cubic equation completely factorizes as

 (U−1)(V−1)(U−V)=0. (3.13)

We see that the three solutions lie outside the physical region and as a consequence there are no planar zeros for this gauge configuration.

Next we try , which corresponds to color factors

 c2=c7=c8=c13=0,c6=−c11=1. (3.14)

In this case the equation for the zeros becomes quadratic and factorizes as

 (U−V)2=0. (3.15)

The geometric loci of zeros coincide again with the unphysical region corresponding to two particles in the final state with infinite energy.

As a last example, we take , which gives

 c2=c8=c11=c13=0,c6=c7=1. (3.16)

In this case the cubic again degenerates into a quadratic equation

 U(U−1)=0, (3.17)

which has no physical solutions.

To summarize, a scan of possible values of the external color numbers show that the only curves obtained in this case coincide with unphysical regions in the plot of Fig. 2, , or . The only possibility for planar zeros in this case is to consider external states without well-defined color numbers, such as the singlet case studied above.

### Su(3).

We work out a first example where we take color quantum numbers and color factors

 c2=−c7=c8=−c13=2,c6=−c11=−1. (3.18)

The planar zeros are given by the factorized cubic

 (U+V−2)(U+V−2UV)=0. (3.19)

This is a hyperbola together with its tangent at (see the left panel of Fig. 4). The loci has nonvanishing intersection with the physically allowed region in the plane for appropriate values of .

A different hyperbola is obtained for with

 c2=−c11=−1,c6=−4,c7=c8=0,c13=−2. (3.20)

The equation determining the zeros also factorizes in this case, giving

 (2U−V)(−2U+V+UV)=0. (3.21)

Again, we have a hyperbola and one of its tangents, this time at the origin. The curves are shown in the RHS panel of Fig. 4.

As in the SU(2) cases all examples explored for the gauge group SU(3) show factorization of the cubic equation. In this latter case, however, not only the type of curves is enlarged to include hyperbolas which were absent for SU(2), but the curves contain physical points. In addition, considering quantum numbers in a SU(2) subgroup of SU(3) generates the curves obtained for the former group.

### Su(5).

Enlarging the gauge group to SU(5) brings more general types of cubic algebraic curves. This is for example the case taking . The resulting color factors are

 (3.22)

Since and vanish, it results in the following quadratic equation determining the planar zeros

 U−2U2−2V+2UV+V2=0. (3.23)

Unlike the examples encountered for SU(2) and SU(3), this curve does not factorize and corresponds to the hyperbola shown in the LHS panel of Fig. 5.

A second interesting example is provided by . The corresponding color factors are

 (3.24)

The resulting equation for the zero

 U−2U2−2V+2UV+U2V=0 (3.25)

is the cubic curve shown in the RHS panel of Fig. 5.

As a last example we take , with color factors

 c2=−c7=−2,c6=c8=−c11=−c13=1. (3.26)

We get the cubic curve

 2U−U2−V−U2V−V2+2UV2=0, (3.27)

which, as shown in Fig. 6, contains a singular point at .

We see how SU(5) provides more general types of curves than the ones found for unitary groups of lower rank. We also have to take into account that SU(5) contains SU(3) and SU(2) subgroups. Using the standard generators (see, for example, [19]) these subgroups are respectively generated by and . Thus, setting the external indices in the subsets or we recover previous examples. For instance, gives the curve shown in the LHS panel of Fig. 4, whereas reproduces Eq. (3.13).

## 4 Planar zeros and color permutations

It is interesting to see how the zeros here investigated transform under permutations of the color quantum numbers of the external particles. We begin considering those permutations preserving the choice of amplitudes basis implied by Eq. (2.7). These are elements of permuting the color indices of the three outgoing gluons (see Fig. 1).

In order to find the action of these permutations on the geometric loci of planar zeros, we can see how the color factors (2.4) transform under permutations of the color indices. Here instead we use a more geometric approach and work with the homogenization of the cubic equation (3.6)

 c7Z2U−c8Z2V−c6ZU2+c11ZV2+(c2+c6−c7+c8−c11−c13)ZUV +c13U2V−c2UV2=0. (4.1)

The group acts passively by permutation of the coordinates . Let us discuss the geometrical meaning of these transformations. Equation (4.1) is defined in the whole projective plane, which is covered by the three affine patches centered at , , and . The group is generated by transformations interchanging the two coordinates within each patch, together with cyclic permutations of the three patches. This defines a coset decomposition of with respect to its cyclic subgroup generated by .

The physically allowed regions shown in Fig. 2 correspond to the patch centered at . It has been already pointed out that it is invariant under the interchange of the two coordinates . Moreover, the corresponding plots in the other two affine coordinate patches are identical to this one. This is easy to see from Eq. (3.2), where it is glaring that cyclic permutations of the three patches only interchange the energies of the three outgoing gluons. Thus, the positivity conditions remain algebraically the same in any of the three affine patches. The final conclusion is that only acts on the axes labels of the plot in Fig. 2. This is a passive version of the fact that the energies of the outgoing gluons are determined by momentum conservation alone and that the color structures play no role in it.

Applying a permutations of to Eq. (4.1), we find that the color factors transform under the (six-dimensional) regular representation of the group: in particular, if we write the group acts through the matrices

 (1)(2)(3) =⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝100000010000001000000100000010000001⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,(123)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝000100100000000001010000001000000010⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (132) =⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝010000000100000010100000000001001000⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,(12)(3)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝000010001000010000000001100000000100⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (4.2) (13)(2) =⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝001000000001100000000010000100010000⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,(1)(23)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝000001000010000100001000010000100000⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

Notice that the combination of color factors in the coefficient of the term itself transforms with the one-dimensional parity representation,

 σ(c2+c6−c7+c8−c11−c13)=(−1)π(σ)(c2+c6−c7+c8−c11−c13), (4.3)

where for even and odd permutations respectively. This is a consequence of the fact that is an invariant under the permutation group.

Applying the transformations given by the matrices (4.2) to the curve (3.10) obtained in the case of the scattering of two gluons in a singlet state,