Off-Shell CHY Amplitudes

The Cachazo-He-Yuan (CHY) formula for on-shell scattering amplitudes are extended off-shell. The off-shell amplitudes are M\"obius invariant, and have the same momentum poles as the on-shell amplitudes. The same technique is also used to obtain off-shell massive scalar and vector boson amplitudes.


I. INTRODUCTION
S-matrix theory was very popular in the late 1950's and early 1960's. It sought to deal more directly with physical observables, and to avoid ultraviolet divergences by staying away from local space-time interactions. Unfortunately, it never got too far because dynamics could not be fully introduced without a Lorentz-invariant interaction Lagrangian density.
This problem is now nicely circumvented by the Cachazo-He-Yuan (CHY) scattering theory [1][2][3][4][5][6], where local Lorentz invariance is supplemented by Möbius invariance of the scattering amplitude in an underlying complex plane. Since its inception, there has been many other papers discussing the properties of the scattering equations [7][8][9][10][11][12][13][14][15][16], calculations of the amplitude [17][18][19][20][21], its relation to string theory [22][23][24], the soft and collinear limits [25], and generalization to include massive and/or other particles [20,[26][27][28][29]. The CHY formula, in its original form, is a tree amplitude for massless particles. In order to implement unitarity, generalization to loop amplitudes [30][31][32][33][34][35][36][37] is required. To facilitate such a generalization and to understand better its connection with local quantum field theory, it is necessary to study the off-shell behavior of these scattering amplitudes. This is what we propose to do in this paper. In Sec. II, we will extend the CHY on-shell scalar amplitude off-shell to get the amputated Green's functions. We will also use the same technique to extend massless amplitudes to massive scalar amplitudes in Sec. III, on-shell and off-shell. The on-shell version agrees with the result obtained previously by Dolan and Goddard [20]. The same consideration also yields an off-shell extension of the CHY gauge amplitude, which is Möbius invariant, but the implication of such an extension requires more study as we shall discuss in Sec. IV.
Some of the illustrative details are contained in the three appendices.

II. OFF-SHELL MASSLESS SCALAR AMPLITUDE
Consider a set of scalar fields φ ia in which the first index is in the adjoint representation of some Lie algebra and the second index is in another. If they interact tri-linearly through f 's and g's being the structure constants in the Lie algebras, then the Green's function for n particle with momenta k i , i = 1 · · · n at the tree level will be a function of products of propagators 1 s i 1 i 2 ···im with 2 ≤ m ≤ n − 2 and s i 1 i 2 ···im ≡ (k i 1 + k i 2 + · · · + k im ) 2 . The coefficients will be a product C i of n − 2 f 's of the first Lie algebra and another product D a of n − 2 g's of the other. For some subsets of indices, they satisfy the Jacobi identities Because of this and because of f and g being totally antisymmetric, only (n − 2)! of the C's and (n − 2)! of the D's are independent. We can choose an independent set, such that C's are of the form f i 1 j 2 j 3 · · · f j n−2 j n−1 in and D's g a 1 b 2 b 3 · · · g b n−2 b n−1 an . The n-particle Green's function will be given as irrespective of whether k i are on-shell or not. Here C| is a vector formed from the independent set just mentioned and so is the vector |D . M is the (n − 2)! × (n − 2)! symmetric propagating matrix given by CHY formula when all the k 2 i = 0. Explicit expressions for n = 4, 5 were given earlier by Vaman and Yao [38]. In the next two sections, we shall explicitly solve for the modifications to the scattering functions f i in the CHY formulas such that M takes exactly the same form even when k 2 i = 0 for n = 4, 5. Generalization to any n will then be given in what follows.

A. Four particles
When all the particles are on shell, the amplitudes are given by [1][2][3][4][5][6] with and the scattering function f i is defined in (30). The integrals are to be evaluated at the We shall make changes for f 3 in eq.(4) to give correct M 1234,1234 , and M 1234, 1324 off-shell.
The other configurations will be given by the substitutions of eq.(6). To avoid confusion, we will usef 3 to denote the modified f 3 . The modifications we propose arê in which x 31 and x 32 are independent of σ's. This assumption of σ independence is predicated by our preference not having to solve a high order algebraic equation for the poles embedded in f 3 otherwise. As we shall see, with this assumption, Mobius covariance and energy momentum conservation for off-shell kinematics will lead to the determination of x ij .
However, we shall obtain x 31 and x 32 here directly by demanding that the Green's functions from eq.(4) should be the same as those given by the double-color field theory. We postpone to Appendix A to show that such modifications will abide by Mobius invariance, which allows us to set the three σ's to the values we gave. The pole is at and therefore These give M 1234, 1234 = 1 s 31 + s 32 + x 31 + x 32 Using the off-shell kinematics we obtain M 1234, 1234 = 1 To coincide with the field theory result M 1234, 1234 = 1 s 14 + 1 s 12 , we need With these, it is easy to obtain If we are to use these x ij forf 3 and assume that M 1324,1324 is given by eq.
We have not been able to find one single universalf 3 , which can produce all the results we want for all configurations for off-shell Green's functions. This color dependence of the off-shell scattering function is a new feature that will be further discussed in Sec. IIC.
The scattering functions f i are defined in (30). We shall obtain the other configurations The integrals are to be evaluated at the poles due to f 2 = 0 and f 4 = 0 simultaneously.
In this case, let us assume that to obtain the Green's function, the modifications arê in whichŝ ij = s ij + x ij and x ij = x ji are assumed to be independent of σ.
Upon using kinematics we arrive at Picking other pairs off i ,f j to evaluate M 12345, 12345 , we also obtain C. Any number of particles The CHY on-shell amplitude for n scalar particles is [1][2][3][4][5][6] where σ r , σ s , σ t are three Möbius constants which will be left arbitray, α = [α 1 α 2 · · · α n ] and β = [β 1 β 2 · · · β n ] are the two configurations of colors, with σ α = n a=1 σ αaα a+1 , σ β = n b=1 σ β b β b+1 , and n + 1 ≡ 1. The contour Γ encloses the (n − 3)! zeros of f i anti-clockwise, and the on-shell scattering functions are To get an off-shell amplitude, we assume the only change needed is to replace f i by an off-shell version given byf where x ij = x ji is assumed to be σ-independent. We also assume that the contour Γ is replaced byΓ to enclosef i = 0 (i = r, s, t) anti-clockwise. Since x ii do not appear, we may and will set them equal to zero. The rest of the parameters x ij = x ji are determined by requiring the off-shell amplitudes to be Möbius invariant, and to have propagators identical to those given by field theory.
Under a Möbius transformation σ j → (ασ j + β)/(γσ j + δ), with αδ − βγ = 1, the off-shell which, by using momentum conservation, implies n j=1 There are many solutions to this equation, so Möbius invariance alone is too general to fix the off-shell amplitude, and that is where the propagator requirement mentioned two paragraphs above comes in. First consider the case α = β = [123 · · · n]. Then any consecutive lines may form a propagator, with an inverse factor for some i and some ℓ. Here and after the line indices are understood to be mod n. For on-shell amplitudes, the inverse propagators r<t 2k r ·k t can be obtained by carrying out the integration of eq. (29). For off-shell amplitudes, the only change is to replace f i in eq. (29) byf i , namely, by replacing 2k r ·k t byŝ rt , hence the inverse propagator is r<tŝ rt . Equating this with eq.(34), we get or equivalently, In particular, if ℓ = 1, then and To obtain solutions for other x ij , subtract eq.(36) from the same equation with ℓ replaced by ℓ − 1 to get For ℓ = 2, it gives and The restriction n ≥ 5 comes about because the requirement that 2 = ℓ ≤ n − 3. In case n = 4, eqs. (40) and (41) are no longer valid. They would be replaced by a relation obtained from eqs. (33) and (37) to be This agrees with the result (13) obtained previously by direct calculation.
For ℓ ≥ 3, the solution can be obtained by subtracting (39) from the same equation with (i, ℓ) replaced by (i+1, ℓ−1) to get To summarize, the solutions are where m in the middle equation is the line between i and j. These solutions are symmetric in i and j, as they should be, and automatically satisfy the gauge-covariant condition eq.(33) because j =i More generally, if α = β = [α 1 α 2 α 3 · · · α n ], then the inverse propagators allowed would be s α i α i+1 α i+2 ···α i+ℓ , and the solution of x ij can be obtained from eq.(44) by a substitution to get where the colors are now indicated in the superscripts.
It is interesting to note that the on-shell scattering functions f i do not depend on the colors, but the off-shell functionsf i do.
D. Off-shell amplitude and off-shell extension of on-shell amplitudes Take the on-shell amplitude in eq. (29). There is a way to extend M α,β such that three of the particles are off-shell while the rest are on-shell. Let us call r, s, t constant lines and the others i = r, s, t variable lines. As long as all the variable lines are on-shell, i.e., k 2 i = 0 for all i = r, s, t, then the amplitude is Möbius invariant no matter whether k r , k s , k t are on-shell or not. In this way we can define a Möbius-invariant amplitude using (29) for up to three off-shell lines. We shall refer to this as the off-shell extension of the on-shell amplitude.
What we want to point out is that this off-shell extension is generally different from the off-shell amplitude considered above, by keeping all but at most three lines on-shell, because the off-shell extension amplitude may not satisfy the propagator requirement.
For example, suppose line r is off-shell and a variable line i is next to it in an amplitude with diagonal colors. The off-shell amplitude satisfies the propagator requirement and gives rise to a propagator 1/s ir , whereas the corresponding contribution from the off-shell extension amplitude is 1/2k i ·k r = 1/(s ir − k 2 r ).
In the case of diagonal colors, the only time an off-shell extension amplitude coincides with an off-shell amplitude is when there is only one off-shell line, say r, shielded on either side of it by the other two on-shell constant lines s and t. In this way no variable line can get next to the off-shell line to produce a different propagator.

III. OFF-SHELL MASSIVE SCALAR AMPLITUDE
We want to explore whether the amplitude of a double-color scalar theory with mass m can also be given by eq. (29), with f i replaced byf i of eq.(31), but with a different x ij than the massless case. The general solution is given below in this section, but to make it more concrete and easier to understand, explicit evaluations for n = 5 and n = 6 are given in Appendix C.
To be Möbius invariant the condition eq.(33) must be satisfied. For α = β = [123 · · · n], the inverse propagators are s i,i+1,i+2,···,i+ℓ − m 2 , so eq.(35) should be replaced by Setting ℓ = 1, we get Eq.(39) is still valid in the massive case, because subtraction cancels the m 2 terms. Setting ℓ = 2, we get Similarly, As in the massless case, n = 4 must be treated separately. There, we need to use eq.(33) to get For ℓ ≥ 3, the solution can be obtained by subtracting (39) from the same equation with (i, ℓ) replaced by (i+1, ℓ−1) to get The final solution is therefore It can easily be checked that the Möbius -invariant condition eq.(33) is also automatically satisfied. For general colors α and β, the solution can again be obtained from eq.(54) by a substitution as in the massless case.

IV. AN OFF-SHELL EXTENSION OF THE GAUGE AMPLITUDE
Similar to (29), the on-shell color-stripped n-gluon scattering amplitude is given by the CHY formula [1-6] to be with the reduced Pfaffian Pf ′ Ψ replacing the factor 1/σ β in the scalar theory. The reduced Pfaffian is related to the Pfaffian of Ψ ij ij by the matrix Ψ ij ij is obtained from the matrix Ψ by deleting its ith column and row and its jth column and row, and the antisymmetric matrix Ψ is made up of three n × n matrices A, B, C, The non-diagonal elements of these three sub-matrices are where ǫ a is the polarization of the ath gluon, satisfying ǫ a ·k a = 0. The diagonal elements of A and B are zero, and that of C is given by An off-shell extension of (55) can be obtained if we replace f i byf i obtained in the previous sections, the contour Γ byΓ enclosingf i = 0, and the elements of A in (58) by The row and column sums of A are still zero becausef i = 0 and because (32) is satisfied.
This off-shell extension is Möbius invariant and is independent of the choice of λ, ν as before, because all the conditions necessary to prove these properties for the on-shell amplitude have been preserved with the change.
These changes are the simplest extensions of the on-shell scattering formula to off-shell, but whether it is the amputated Green's function of an Yang-Mills field theory is not immediately clear. The reason is, off-shell Yang-Mills theory is gauge dependent, and in our extension gauges do not enter. It is possible that this extension determines a particular 'CHY gauge', or that the true off-shell extension is much more complicated than what is discussed in order to reflect the the freedom of gauge choice. It is even possible that fieldtheoretical off-shell expression is not Möbius invariant. Further study is required to know what is the truth.

V. CONCLUSION
The CHY scattering formulas for massless scalar particles are extended off-shell by changing 2k i ·k j in the scattering functionf i to (k i + k j ) 2 + x ij , where x ij = x ji is independent of σ. It can be determined uniquely from the requirement that off-shell amplitudes are Möbius invariant and have exactly the same invariant-momentum poles as the on-shell amplitudes.
The same requirements also allow us to extend the formula to massive scalar and vector amplitudes, on-shell and off-shell. A simple off-shell extension of the CHY gauge amplitude is also proposed, with many nice properties including Möbius invariance and the independence of λ and ν, but the true nature of this extension formula requires further study.
Appendix A: Möbius invariance of the n = 4 and n = 5 amplitudes One motivating and intriguing feature of the CHY formulas is that the on-shell amplitudes for scalar, gauge, and gravitational interactions are all invariant under Möbius transforma- In our extending the CHY formula to off-shell, it is very natural to ask if such invariance still holds. In fact, this is required of us, because if it were not so, then we would not have the freedom to fix three of the σ ′ s to the values we used in Sections IIA and IIB. Gladly, the answer is in the affirmative, although with some restrictions (eq.(6), eq. (17)). Let us begin with the case of four double-color scalars. The invariance of the off-shell Green's functions is intimately tied up with the transformation property of the scattering equations. Let us generalize these slightly before we fix three of the σ'ŝ 14 One can verify that they satisfy as a result of momentum conservation, and j =i which in turn means that only one of thef i is independent. Let us take this to bef 3 . Then, we find that under Möbius transformation The first term vanishes as indicated by eq.(A4), which makesf 3 Möbius covariant. Now, we also set σ 1 = 0, σ 2 = 1 and σ 4 → ∞. Using and we have The caveat here is that in order to have poles at the correct place, we have committed to the form off 3 as determined.
For the double-color five particle amplitudes, let us generalize the functions slightly tô It is easy to check that because of momentum conservation, similar to eq.(4-4), when we take which implies that only two of thef ′ i s are independent. We choose them to bef 3 andf 4 , which are Möbius covariant, in the sense that and we are led to .

(A13)
Appendix B: A diagonal element of the n = 5 amplitude In this note one of n = 5 scalar amplitudes is calculated. The others can be done in a similar fashion. For our purpose here, let us consider the most complicated case with diagonal colors, say with 1,3,5 as the constant lines.
Thus I 5 = I 5a + I 5b + I 5c consists of 5 terms, corresponding to five Feynman diagrams.