CHY Theory for Several Fields

The Cachazo-He-Yuan (CHY) formula was originally proposed to describe on-shell scattering of particles from a single massless field. We present a method to modify it to include several interacting scalar fields, all possessing different masses and possibly off-shell momenta. The method is applied to Yukawa interactions between a number of scalar nucleons and pions, and to the $\f_1\f_2\f_3$ coupling of three different scalar fields. Composite models constructed from existing theories can be used to broaden the scope of the method. The modification is applied to describe Compton scattering from a massive particle, and to photon bremsstrahlung. It is also employed to generalize the disk function $Z$ and the sphere function $J$.


I. INTRODUCTION
Inspired by string theory in the zero Regge slope limit, Cachazo-He-Yuan (CHY) came up with a formula for the tree amplitude of several massless field theories, valid in any number of space-time dimension [1][2][3]. Like the string theory, the CHY formula is given by a multiple integral over the complex plane, possessing Möbius invariance. A set of scattering equations plays an important role in this formalism. Propagators are now closely related to the scattering functions in the scattering equations, and vertices are tied up with the global structure of the amplitude.
As originally conceived, the CHY formula describes the scattering of a massless scalar field with φ 3 coupling, the pure Yang-Mills theory, and Einstein's gravity. These are all massless fields with massless external momenta. Because the formula is applicable in any number of space-time dimension, dimensional reduction can be used to construct many other theories in lower dimensions [4,5].
As interesting and as novel as this new formulation is, it would not be helpful to high energy physics unless it can describe the Standard Model. That would require a number of additional developments not contained in the original formula. The formulation must be broadened to include off-shell amplitudes in order to coincide with field theory, and for loops to be computed. Non-zero masses must be included because most particles in nature are massive. The usual way to do that relies on the validity of the CHY formula in any number of dimension, so four-dimensional masses and off-shell momenta can be obtained from momenta in extra dimensions [65][66][67]. This works in some special cases, but not in general, because one cannot be assured that all correct propagators can be obtained this way. For example, the propagators of a massive φ 3 theory cannot be so obtained if the total number of external particles far exceeds the total number of extra dimensions. This is further explained in Appendix A.
There is however a different method to generate a massive φ 3 theory and/or off-shell momenta that guarantees correct propagators in any number of dimension, achieved simply by modifying the scattering equations [68,69]. The method has also been used to extend the CHY Yang-Mills theory off-shell [70].
To reproduce the Standard Model, massive fermions in the fundamental representation must be included. That proves to be quite difficult in the CHY formalism although some interesting progress has been made [71,72].
Nature contains many massive particles in a variety of interactions, so the CHY formula must also be generalized to describe them, on-shell and off-shell. In this article we propose a method to do so for scalar particles, again by modifying the scattering equations. We shall work out in detail the Yukawa coupling of several scalar nucleons and pions, all with different masses. In the special case of an infinite pion mass, this degenerates into the φ 4 theory. We also discuss how to couple three different scalar fields via φ 1 φ 2 φ 3 interaction.
In addition, a very useful method to construct composite models from existing theories will be discussed. In this way we can for example construct a model in which two nucleons of different masses couple to form a di-nucleon resonance, and the resonances can interact with one other via an exchange of pions. Beyond scalar particles we also take a first look at photons. Compton scattering and bremsstrahlung from a charged massive scalar particle can both be computed this way.
A disk function Z and a sphere function J have been introduced to connect massless field theory amplitudes to string amplitudes [60,73,74]. The modification of scattering functions mentioned above naturally leads to a possible modification of the Z and J functions. It would be interesting to see whether such a modification can be used to generate new string amplitudes and/or new effective field theories.
In Sec. II, the general setup of the method is described, together with illustrations taken from the two known cases: massless on-shell φ 3 scattering [3], and massive on-shell or off-shell scattering [68]. This method is then applied to the Yukawa coupling of one massive scalar nucleon and one massive pion in Sec. III, and to the Yukawa coupling of several massive scalar nucleons and pions in Sec. IV. Three field coupling φ 1 φ 2 φ 3 is taken up in Sec. V, and the construction of composite models is discussed in Sec. VI. Compton scattering and bremsstrahlung from charged scalar particles will be considered in Sec. VII and Appendix C. Modified Z and J functions will be discussed in Sec. VIII. Appendix A explains why extra dimensions cannot be simply used to obtain massive and/or off-shell amplitudes in four dimensions, and Appendix B proves a general covariant condition needed for the CHY formula to be extended.

A. Equation of motion
Consider a meromorphic function F (σ) defined by where σ i , a ij = a ji are arbitrary complex parameters subject to the constraints a ii = 0, and Under a Möbius transformation, and as a consequence of the constraint,f i (σ) can be shown to transform covariantly aŝ The residue of F (σ) at σ = σ i is Given an a ij , F (σ) = 0 for all σ if and only all its residues are zero, which is so if and only if the n parameters σ i are solutions of the n scattering equations f i = 0. In light of covariance, three σ i can be chosen arbitrarily, so only n−3 of these f i can be linearly independent. This linear dependence is encoded in the three sum rues Both Eq.(4) and Eq.(6) are known to be true for the original CHY massless on-shell φ 3 theory [1][2][3]. They are also true for the massive off-shell φ 3 theory [68]. That they are generally true for any a ij satisfying Eq.(2) is shown in Appendix B. It is this covariant property that allows the CHY formula Eq. (7) to be extended to cover many other field theories.
We shall refer to F (σ) = 0 as the equation of motion (EOM) because it generates all the scattering equations, and because it resembles the Klein-Gordon equation in ordinary field theory. In a field theory, the EOM is a differential equation true for all spacetime points x.
In the CHY theory, it is an algebraic equation true for all complex variables σ.

B. CHY amplitudes
Solutions of the scattering equations determine the n-particle tree amplitude A n . One way to express that is through the CHY formula where Γ is a clockwise contour encircling all f i = 0, with f i given by Eq. (5), and I n is chosen so that A n is invariant under the Möbius transformation Eq.(3). Möbius invariance permits a choice of three arbitrary gauge constants σ r , σ s , σ t in the integrand that does not affect the outcome of A n in Eq. (7).
Each f i factor plays the role of an inverse scalar propagator, and the I n factor plays the role of interaction vertices. For scalar theories which we shall concentrate mostly on in this article, where α = (α 1 α 2 · · · α n ) is a permutation of (12 · · · n) determined by the specific Feynman diagram under consideration. The quantity σ (p 1 p 2 ···p k ) stands for the product k i=1 σ p i p i+1 , with k + 1 ≡ 1, and σ ab = σ a − σ b . When spin-1 particles are involved, I n would be different.
We shall discuss a special case in Sec. VII.
It is implicitly assumed in Eq.(8) that every Feynman tree diagram is written as a planar diagram, with the external lines ordered clockwise and cyclically in the natural order 1, 2, · · · , n. The amplitude Eq. (7) is that of a single Feynman diagram, or that of a sum of several or all Feynman diagrams, all depending on what σ (α) is. See Sec. IIE for more discussions on that point.
The parameters a ij are so far arbitrary except for the constraint in Eq.(2). They will be chosen to reproduce the correct propagators in a given theory. When the external lines are ordered in the way described in the last paragraph, every propagator takes on the form , where k S = i∈S k i is the total external momentum of a set S of ≤ (n − 2) consecutive external lines, k i is the outgoing momentum of line i, and m 2 S is the (square) mass of the propagator. In a CHY scalar theory given by Eq. (7), the inverse propagators are related to the parameters a ij by the formula [68] Different theory specifies different m 2 S , and it is from this propagator equation that all a ij are determined. Once a ij are determined this way, we must still check that the symmetry condition a ij = a ji and the covariant condition Eq.(2) are satisfied. Otherwise Eq.(7) cannot be used. Moreover, we must make sure that Eq.(9) is self consistent in the following sense.
For any set S of consecutive external lines, let S be its complementary set, consisting of all the external lines not in S. Since S and S share the same propagator in a tree diagram, we must make sure that k 2 S − m 2 S = k 2 S − m 2 S . With momentum conservation, k S = −k S , so this is satisfied if and only if for every set S. We shall refer to that as the complementary condition.
The two cases in which a ij are known are the original CHY massless on-shell φ 3 theory [3], and the massive off-shell φ 3 theory [68]. They will be reviewed in the following subsections. After that, we shall elaborate on how the propagator condition Eq.(9) can be used to determine a ij for other theories, and then proceed to apply the method to a scalar Yukawa theory whose 'scalar nucleon' and 'pion' possess different masses. In the limit of an infinite pion mass, this becomes the φ 4 theory. This Yukawa theory can be generalized to include many kinds of nucleons with different masses, provided they interact with one another only through the exchange of pions. In these theories, a ij determined by the propagator condition can be made to obey the symmetry and the covariant conditions, as well as Eq.(10). We will also discuss a theory with three fields interacting according to φ 1 φ 2 φ 3 . The propagator condition once again determine its a ij for arbitrary masses of the three fields, but it turns out that such a a ij satisfies the required conditions only when the three masses are identical. Composite models and photons, as well as modified disk and sphere functions, will be discussed in later sections.

C. φ 3 theory, on-shell and massless
This is the original CHY theory [3]. With k 2 i = 0 and m S = 0, Eq.(10) is automatically satisfied. The choice of a ij for i = j is The symmetry condition a ij = a ji is obviously true. Owing to momentum conservation and the massless requirement, the covariant requirement Eq.(2) is also satisfied because n j=1 a ij = −k 2 i = 0. The propagator condition Eq.(9) with m 2 S = 0 also follows because i,j∈S D. φ 3 theory, off-shell and with mass m This situation was studied in [68]. The solution with a ii = 0 is The ρ ij = ρ ji term is responsible for off-shell extensions, and the µ ij = µ ji term is needed for a non-zero mass m. Specifically, for n > 4, The indices i ± 1 and i ± 2 could turn out to be non-positive, or larger than n. In that case they should be interpreted to have a value mod n.
For n ≤ 4, j could be equal to or less than two lines away on both sides of i. In that case Eq.(13) should be used to add up both sides to get the correct result for ρ ij and µ ij .
Since all the propagators have mass m 2 , Eq.(10) is satisfied. With this a ij , the covariant condition Eq.(2) is satisfied because n j=1 a ij = n j =i,j=1 To show that the propagator requirement Eq.(9) is also satisfied, let S = {p, p + 1, p + 2, · · · , q}, with |S| := q−p+1 ≤ n−2. The latter condition is necessary to ensure that there are at least two lines in S to merge to form a propagator somewhere. Since a ii = 0, i,j∈S For j > i, the only non-zero ρ ij 's are for j = i + 1 and j = i + 2. Hence and Thus the propagator condition Eq.(9) is satisfied with m 2 S = m 2 .

E. Connection between σ (α) and Feynman diagrams
Given a planar tree diagram such as Fig.1(a) whose external lines are ordered clockwise and cyclically according to (12 · · · n), there are several techniques [3,[7][8][9][10][11]19] to determine the appropriate σ (α) that yields the diagram. In this subsection we review how to do that following the approach of [11,19]. One of them would give the σ (α) we are after.
To find out which, we have to know the general relation between propagators and σ (α) .
For planar diagrams, propagators come from a merging of consecutive external lines. It turns out that consecutive external lines labelled by α u , α u+1 , · · · , α u+m would merge into a propagator if and only if they form a permutation of m consecutive numbers p, p+1, · · · , p+m.
When that happens we will put a square bracket around them, as in [α u α u+1 · · · α u+m ]. A Feynman tree diagram has n−3 propagators, so (α) must contain n−3 such compatible square brackets. Two brackets are compatible if they either do not overlap, or one is completely inside another.
If there is only one way to partition (α) into such compatible square brackets, then this σ (α) would give rise to one Feynman diagram. If there is more than one way to partition (α), then the amplitude would receive contribution from several Feynman diagrams, each corresponding to one such partition.
In summary, given a single diagram, the appropriate σ (α) is given by that flipping of external lines that produces a single compatible partition for (α). A different flipping may result in a sum of several diagrams, and σ (α) = σ (12···n) would give rise to a sum of all Feynman diagrams.
This rule works not only for the φ 3 theory, but also all the other theories to be discussed in the following sections.
It would be useful to know the origin of this rule [11,19], so that it can be generalized to the situation in Sec. VII. The (n − 3)-fold integration in Eq. (7) is taken over every σ i , except σ r , σ s , σ t . Contribution to the integral comes from simple poles in the factor 1/σ (α) , located when the σ i 's inside a square bracket coincide with one another. When the numbers inside a square bracket are permutations of a consecutive subset of (12 · · · n), a pole also occurs in the other factor 1/σ (12···n) of Eq. (8), but this other pole is compensated by a zero found in the factor 1/ i =r,s,t f i , leaving behind a propagator as the residue. In this way the (n − 3) integrations in Eq. (7) give rise to (n − 3) propagators that make up a Feynman diagram.

F. Beyond the φ 3 theory
If we decompose a ij in the form of Eq.(12), with ρ ij given by Eq.(13), then Eq. (14) and Eq. (16) show that the symmetry condition a ij = a ji , covariant condition Eq.(2), and the propagator condition Eq.(9) are already satisfied by the momentum part of a ij . That leaves only the mass part µ ij , which must satisfy where |S| is the number of lines in S. Since m 2 S depends on all the lines in S, µ ij must depend on the particle nature of i, j, and all the lines in between. To exhibit such a dependence explicitly, we shall denote µ ij by (a i a i+1 · · · a j ) when j > i, where a is a particle identification symbol: a i specifies the particle of line i, a j the particle of line j, and the remaining a's specify the particle identity of lines in between. The inequality j > i means that line j is downstream from line i when external lines are ordered cyclically in the clockwise direction.
It does not necessarily mean that the number j is larger than the number i because of the cyclic nature of the external lines.
Among the three terms of a ij in Eq.(12), the k i ·k j term is the simplest because it depends only on lines i and j. The ρ ij term is more complicated because it can depend on the lines between i and j, e.g., ρ i,i±2 = 1 2 k 2 i±1 , but it becomes zero when j and i are more than two lines apart. The last term µ ij is the most complicated because not only it depends on all lines between i and j, it may also be non-zero no matter how far apart lines i and j are.
Moreover, it also depends on the precise ordering and identity of the particles in between.
Since kinematics are universal, the k i ·k j and ρ ij terms are the same in all theories. Thus it is the µ ij term that tells theories apart. However, even for the same theory, different ordered sets of external lines may give rise to different µ ij 's. For example, in the Yukawa theory to be discussed in the next section, where solid and dotted lines represent a scalar nucleons and pions respectively, the amplitude in Fig.2(a) and the amplitude in Fig.2 may have different sets of µ ij . Diagrams (a1), (a2), (a3) are the tree diagrams contained in  Let us delve a bit more into the details on how Eq.(9) can be used to compute µ ij . If S consists of consecutive lines from p to q, then we will write the left hand side of Eq.(20) as a p a p+1 · · · a q . With that notation, Eq. (20) can be written as This allows µ ij to be computed by four combination of m 2 S using the formula if n − 2 ≥ |S| = q − p + 1 ≥ 4. For smaller |S|, the appropriate formulas are (a p a p+1 ) = a p a p+1 , (a p a p+1 a p+2 ) = a p a p+1 a p+2 − a p a p+1 − a p+1 a p+2 .
To use these formulas to determine µ ij , we must know what m 2 S is. That depends on the propagator, which in turn depends on the nature of interaction and the particle content of the external lines in S. More specifically, an interaction determines a set of 'topological requirements', which in turn fixes m 2 S . It will become clear that m 2 S depends only on the particle content of S, but never on their ordering, so a p a p+1 · · · a q is permutation invariant in the symbols. However, (a p a p+1 · · · a q ) does depend on how the lines are ordered.

G. Complementary, symmetry, and covariant conditions
It will be shown in this subsection that once the complementary condition Eq. (10) Each of the four terms in µ ij contains lines that are complementary to lines in one term of µ ji . For example, the complement of the first term in µ ij is the last term of µ ji . Using the complementary condition m 2 S = m 2 S , it follows that µ ij = µ ji .

Covariant condition
To how the covariant condition Eq.(19) from the propagator condition, it is important to note that Eq. (20) is valid only for |S| ≤ n − 2. With that caveat and using the symmetry condition for the last two terms, The Solving for I and I , the solution is In order for I, I to be non-negative integers, E must be an even integer ≥ 2. When E = 2, then we must have E ≥ 1. This topological requirement applies not only to the whole Feynman diagram, it also applies to a sub-diagram consisting of a propagator and all its corresponding external lines. It is the basis from which µ ij for the Yukawa theory is determined. If For |S| ≥ 4, µ ij can be written in the form Let us apply Eq. (7) to the scattering diagrams shown in Fig.3. By choosing the Möbius constant lines r, s, t to be 2, 3, 4, only f 1 and µ 1j enter into Eq. (7).
Recall that Eq. (21) can be used to compute µ ij only when j − i + 1 ≤ n − 2, because it requires at least two external lines to merge into a propagator. Thus for n = 4, Eq. (31) can be used only when j = i + 1. In particular, for i = 1, only µ 12 can be so calculated, though µ 14 can also be calculated using the symmetry condition µ 14 = µ 41 = µ 45 . That leaves µ 13 , which cannot be calculated from Eq. (21), but it can be calculated from Eq. (19) as µ 13 = −(µ 12 + µ 14 ).
Thus the inverse propagators are 2a 12 = 2k 1 ·k 2 + k 2 1 + k 2 2 − µ 12 = (k 1 + k 2 ) 2 − m 2 0 for Fig.3(a), Fig.3 Note also that we can get Fig.1(a) and Fig.1(b) together, also Fig.1(c) plus Fig.1(d) at It can be directly checked from these relations that the covariant requirement j =i µ ij = 0 is satisfied for i = 1, 3, 4. The three inverse propagators are as shown in the diagram. can also be chosen to yield a sum of several, or all, Feynman diagrams in the amplitude.
Since the method of choosing σ (α) has nothing to do with f i , it is expected to be valid for the Yukawa theory as well. This is indeed so, but with one caveat.
Recall the method consists of partitioning (α) into n−3 compatible square brackets, each representing a propagator in a Feynman diagram. For the Yukawa theory, this is still true as long as we avoid those partitions containing square brackets with only pion lines. We must exclude those because pions alone cannot merge into a propagator, which is why we assign an arbitrary constant M 2 to that case in Eq. (29). If it were not for the complementary condition which forces M 2 to be m 2 0 , even those diagrams need not be avoided because we could have set M 2 = ∞ to get rid of them at the end. As it is, M 2 = m 2 0 , but this is not the propagator mass of several pions, because such a set does not even produce a propagator.
To get things right, we must eliminate those phantom propagators by hand, by excluding the partitions containing pure pion square brackets. As long as that is followed, everything else is the same as in the φ 3 theory.
For example, the particles in the six external lines of Fig.4 are of the type (111001), with lines 4 and 5 being two consecutive pion lines. By setting σ (α) = σ (123456) , one would have included diagrams containing a two-pion phantom propagator and produced a wrong result, unless we exclude by hand partitions that have pure pion square brackets. However, if the fifth pion line were attached to the downstream side of the next nucleon, so that the particle content becomes (111010), then there are no consecutive pion set so σ (α) = σ (123456) for that kind of µ ij would produce the correct sum of all Feynman tree diagrams with those particle contents.

E. The φ 3 limit
By setting m 0 = m 1 = m we recover the φ 3 theory from the Yukawa theory. In that limit, it follows from Eq.(31) that µ i,i±1 = m 2 , µ i,i±2 = −m 2 , and µ ij = 0 for |j − i| > 2, which is what it should be according to Eq.(13).

F. The φ 4 theory
In the m 0 → ∞ limit, a multi-nucleon tree diagram such as Fig.5(a) becomes a φ 4 diagram such as Fig.5(b), provided the coupling constant is suitably scaled by a power of m 0 . However, it is important to remember that the m 0 → ∞ limit should be taken after the integral Eq.(7) is carried out.
For |S| = 4, only a few results are listed below for illustration, but clearly we can compute all the results for every |S|.

A. Topological rules for tree amplitudes
For easy reference, we shall refer to the three fields as having three different 'colors'.
Consider a connected tree diagram with E a (I a ) external (internal) lines of color a, and a total of n external lines. Such a diagram has n − 3 internal lines, and n − 2 vertices, hence Given E a , these rules lead to the solution In order for I a to be a non-negative integer, we must have E 1 , E 2 , E 3 to be either all odd, or all even numbers. We must also have E a + E b ≥ 2 for every pair a, b, thereby forbidding diagrams with only one color of external lines. These topological requirements are important for the determination of µ ij .    1 , E S 2 , E S 3 ] contains one even and two odd numbers, with the even number having color a, then S belongs to class S a . If they contain one odd and two even numbers, with the odd number having color a, then S belongs to class S a . In both cases, the propagator must have color a to satisfy the topological requirement, so With all the masses in Eq.(43) equal, the only difference between the φ 1 φ 2 φ 3 and the φ 3 amplitudes is that some Feynman diagrams allowed in φ 3 would not be allowed in φ 1 φ 2 φ 3 .
These are the diagrams violating the topological requirements discussed in Sec. VA.

VI. COMPOSITE THEORIES
Once a set of µ ij satisfying the complementary condition Eq.(10), the symmetry condition Eq. (18), and the covariant condition Eq. (19) is given, a Möbius invariant scalar amplitude can be computed from Eq.(7) and Eq. (8). Since these three conditions are all linear in µ ij , a linear combination of two such sets of µ ij also satisfies the three conditions. In this way we can construct many composite theories from two or more known theories by making linear combinations. In this section we shall illustrate how to do that with a simple example.

Clearly many other examples can be similarly constructed.
Let µ ij = αµ ij + βµ ij , where µ ij comes from a Yukawa theory, with nucleon mass m 1 and pion mass m 0 , and µ ij comes from another Yukawa theory, with nucleon mass m 1 and pion mass m 0 . α and β are arbitrary real parameters.  Fig.8(b) is the same diagram expressed in terms of the composite particles π, N 1 , N 2 , and N * , and the effective vertices N 1 N 2 N * and N * N * π. Dotted lines are composite pions π, and thin, medium, thick solid lines are the composite nucleons N 1 , N 2 and the resonance N * .
By adjusting the values of α, β, m 2 0 , m 2 1 , m 2 0 , m 2 1 , we can have any value for the compound masses m 2 0 , m 2 1 , m 2 2 , and M 2 . As a direct check, Table I shows the nine propagator masses in Fig.8(b), obtained from this formula using the total (ν , ν ) values read off from all the corresponding external lines in Fig.8(a). They agree with what Fig.8(b) shows.  Table I. Propagator masses m 2 S obtained from Eq.(44) and Fig.8(a) can be seen to be the same as those shown in Fig.8(b)

VII. SCALAR QED
The scattering function f i for scalar QED is identical to that of a Yukawa theory in Sec. III, provided we interpret m 1 ≡ m as the mass of the charged particle, and m 0 = 0 as the mass of photon. However, Feynman amplitudes in QED also contain non-trivial numerators describing photon polarizations as well as its derivative coupling to the charged particles. To implement that, the I n factor in Eq.(7) must be different from Eq. (8). For an amplitude with two charged legs and and n − 2 photon legs, we propose to use Ψ is the 2n × 2n antisymmetric matrix and Ψ ij ij is Ψ with the ith and jth columns and rows removed. The matrix elements are where i is the polarization vector of photon i. Particles 1 and n are the charged scalars, the others are photons.
The motivation of this proposal is explained in Appendix C. In the rest of this section, the proposal will be explicitly verified for Compton scattering (n = 4) and for the process with a single photon emission (n = 5), by comparing the result of Eq. (7) with the Feynman amplitude obtained from Feynman rules.
A. Compton scattering Up to normalization, the Feynman amplitude of Fig.9(a) is To be gauge invariant, we must also add in the cross diagram shown in Fig.9(b), which is the same as Fig.9(a) with k 2 and k 3 as well as 2 and 3 interchanged.
which agrees with the Feynman amplitude A q f in Eq. (48) up to a normalization factor. The correctness of the prescription Eq.(45) for n = 4 is thereby verified.
The cross diagram Fig.9(b) can be obtained either by interchanging 2 and 3 in Eq.(50), or by changing I n in Eq. (45) to N/σ (1324) . Up to normalization, the Feynman amplitude of Fig.10 is

B. Photon emission
Again for gauge invariance, we must add in five other cross diagrams obtained by permuting 2, 3, 4, but we will not consider them here. As before, choose σ r,s,t = σ 1,2,3 in Eq. (7) which up to a normalization factor is the same as the Feynman amplitude A F in Eq.(51), thereby verifying the correctness of Eq.(45) for n = 5.

VIII. DISK AND SPHERE INTEGRALS
A disk integral Z P (q 1 q 2 · · · q n ) = (α ) n−3 D(P ) dz 1 dz 2 · · · dz n vol(SL(2, R)) i<j |z ij | α k i ·k j z q 1 q 2 z q 2 q 3 · · · z q n−1 qn z qnq 1 (53) was introduced in [60] to connect a Yang-Mills field theory amplitude with an open string amplitude. Following [60], the integration variables σ i are now denoted as z i . α is the Regge slope, and k i are the light-like external momenta. The integration domain is the real line with the integration variables ordered according to a permutation P = {p 1 p 2 · · · p n } ∈ S n , The Z function depends on the permutation P , and also on another permutation Q = {q 1 q 2 · · · q n } ∈ S n . The integrand is invariant under a SL(2,R) transformation, which is the Möbius transformation with real coefficients.
The Z function possesses the following properties [73]: Z P (q 1 q 2 · · · q n ) = Z P (q 2 q 3 · · · q 1 ), Z P (q 1 q 2 · · · q n ) = (−) n Z P (q n q n−1 · · · q 1 ), k q 1 ·(k q 2 +k q 3 +· · ·+k q j )Z P (q 2 q 3 · · · q j q 1 q j+1 · · · q n ), whereB is the transpose of the set B, and A B is the shuffle product between set A and setB. In the zero Regge slope limit, Z P (Q) is essentially the scalar amplitude Eq.(7), with P = {12 · · · n} and Q = α. Eq.(59) should be replaced by the sum of a ij factors: In the zero Regge slope limit, it again reduces to the field theory amplitude defined by the same a ij .
With these similarities, it would be interesting to find out whether modified string amplitudes and/or new effective field theories can be sensibly constructed with the help of the modified Z function.
A similar remark applies to the modified sphere function, defined by i<j |z ij | 2α a ij (z p 1 p 2 z p 2 p 3 · · · z p n−1 pn z pnp 1 )(z q 1 q 2 z q 2 q 3 · · · z q n−1 qn z qnq 1 ) .
When the corresponding field theory is massless and on-shell, a ij = k i ·k j , the J function returns to the original one in [74].
Appendix A: Masses and off-shell momenta coming from extra dimensions Consider an n-particle scattering amplitude with external momentak i in a higher dimensional space, and supposek 2 i = 0 for all i. Decomposek i = (k i , k i ) into a four-dimensional component k i and an extra-dimensional component k i . Since k 2 i = −k 2 i may take on any value, a non-zero mass m and/or an off-shell four-momentum can be produced in real spacetime. We show in this Appendix that masses produced this way do not always yield the right propagators in a massive theory, though it might do so if only certain particles in the amplitude are massive. Similar arguments apply also to off-shell momenta.
Consider a theory whose particles carry a mass m. Label the external lines cyclically in the natural order i = 1, 2, · · · , n. For any p and any q ≤ n − 2, there is always an inverse This concludes the proof that we cannot obtain all propagators of all Feynman diagrams correctly if masses for a massive field theory come solely from the extra dimensions.

Appendix B: Covariance and sum rules
If a ij satisfies the constraints a ii = 0 and j a ij = 0, thenf i (σ) will transform covariantly as in Eq.(4), and the sum rules Eq.(6) will be satisfied.
For the sum rules, recall that f i =f i (σ i ) = j =i a ij /(σ i − σ j ), hence The last of the first equation is 0 because the summand is antisymmetric in i, j. The last term of the second equation is zero because j a ij = 0. The first term of the last expression of the last equation is zero because j a ij = 0, and the second term of that expression is zero because the summand is antisymmetric in i and j.
Appendix C: Numerator factor for QED The numerator factor N in Eq.(45) comes from a comparison between the Feynman rules for QED and those for gluon scattering. A gluon amplitude is a function of i · j , i ·k j , and k i ·k j . In the CHY theory [2,3], the numerator of the amplitude comes from the reduced Pfaffian of a 2n × 2n antisymmetric matrix of the form where for i = j, A ij = k i ·k j /σ ij , C ij = i ·k j /σ ij , and B ij = i · j /σ ij . A scalar QED amplitude with two charged scalars occupying lines 1 and n, such as Fig.9 or Fig.10, contains i · k j factors but no i · j nor k i ·k j factors in the numerator. Since Fig.11(c) is essentially the same as Fig.11(f), we may obtain the numerator factor N in such QED diagrams from Eq.(C1) by putting A = 0, and B = 0 except for the 1 · n /σ 1n and the n · 1 /σ n1 elements. This is then the N factor in Eq. (45).
There is however a caveat in using N . A number of gluons can merge to form a gluon propagator, but photons in a tree diagram cannot merge to form photon propagators. Hence those poles in N that lead to the phantom photon propagators must be excluded by hand.
Other than this, N has another imperfection. The reduced Pfaffian for the gluon matrix in Eq.(C1) is gauge invariant, but this is not so for the photon matrix in Eq. (46).
Moreover, this argument to produce N no longer works when two pairs of charged legs are present, for two different reasons. First, there would now be two i · j factors in the B block for the two charged pairs. If the factor B in Eq.(C1) contains two such pairs of i · j , then their product would appear somewhere in the reduced Pfaffian. In the case of gluon scattering, this product could come from the four-gluon vertices Fig.11(e), or terms such as Fig.11(b) or Fig.11(d). In the case of scalar QED, all these diagrams are absent, so this term would be extraneous and wrong. The second reason why that does not work is because k i ·k j factors would appear when the two pairs exchange a photon, so the A block in Eq.(C1) can no longer be dropped. However, Möbius invariance demands A to be a ij /σ ij when masses are present [70], not just k i ·k j /σ ij . Then mass factors µ ij /σ ij ∼ m/σ ij would also appear in A and in the numerator of the scattering amplitude. That is wrong and should not happen.
For both of these reasons, the N factors for scalar QED can no longer be obtained so simply from the gluon N factor when more than one pair of charged particles are present.