New Factorization Relations for Non-Linear Sigma Model Amplitudes

We obtain novel factorization identities for non-linear sigma model amplitudes using a new integrand in the CHY double-cover prescription. We find that it is possible write very compact relations using only longitudinal degrees of freedom. We discuss implications for on-shell recursion.


INTRODUCTION
Cachazo, He and Yuan invented in ref. [1] a new method for calculating S-matrix elements. This formalism has numerous applications and many interesting connections, see for instance refs. [2][3][4]. The CHY construction was formally proven by Dolan and Goddard in ref. [5].
The main ingredients are the n-point scattering equations where z a are auxiliary variables on the Riemann sphere and k a are momenta. In the CHY formalism one has to integrate over a contour containing the (n − 3)! independent solutions of the scattering equations.
As computations in the CHY formalism grow factorially in complexity with n, integration rules have been developed at tree [6][7][8][9] and loop level [10], so that analytical results for amplitudes can be derived without solving the scattering equations explicitly.
Recently, the CHY formalism was reformulated by one of us in the context of a double cover [11] (called the 'Λ-formalism' in refs. [11,12]). Here, the basic variables are elements of CP 2 , and not CP 1 as in the original CHY formalism. One advantage of the extra machinery is that amplitudes in the double-cover formulation naturally factorize into smaller CP 1 pieces, and this is a useful laboratory for deriving new amplitude identities.
We will start by reviewing the CHY formalism for the non-linear sigma model (NLSM) and provide an alternative formulation that employs a new integrand. Next, we will show how the double-cover formalism naturally factorizes this new CHY formulation in a surprising way.

A NEW CHY INTEGRAND
As explained in ref. [13], the flavor-ordered partial U (N ) non-linear sigma model amplitude in the scatter-ing equation framework is given by the contour integral where (α) = (α(1), ..., α(n)) denotes a partial ordering. The integrand is given by Here PT(α) and Pf ′ A are the Parke-Taylor (PT) factor and the reduced Pfaffian of matrix A, respectively. The n × n anti-symmetric matrix, A, is defined as, In general, (A) i1···ip j1···jp denotes the reduced matrix obtained by removing the rows, {i 1 , ..., i p }, and columns, {j 1 , ..., j p }, from A. Note that when the number of external particles n is odd, Pf ′ A = 0, and A n (α) vanishes.
Using eq. (5), we have With the choice {i, j} = {m, p}, this product of Pfaffians becomes a determinant, We will now discuss the following new matrix identities. On the support of the scattering equations and the massless condition, {S a = 0, k 2 a = 0}, we find when m = p = q det[(A) mp pq ] = 0 if n odd.
A proof of these identities will be provided in ref. [14].
Using the non-antisymmetric matrix, (A) ij jk , we define the objects (with i < j < k) Note that in eqs. (11) and (12) we have reduced the A matrix with the indices {i, j, k} associated with the Faddeev-Popov determinant. This gauge choice will be convenient later. We now have the following equality, when all particles are on-shell. When there are off-shell particles, the identity is true only if the number of particles is even. When the number of particles is odd and there are off-shell particles, one has A n (α) = 0 while A ′ n (α) = 0. Since the A matrix has co-rank 2 on the support of the scattering equations and the massless condition, {S a = 0, k 2 a = 0}, A (ij) n (α) vanishes trivially. However when there are off-shell particles the amplitude A (ij) n (α) is no longer zero. These observations will be crucial in obtaining the new factorization relations.

THE DOUBLE-COVER REPRESENTATION
In the double-cover version of the CHY construction, the n-point amplitude is given as a contour integral on the double-covered Riemann sphere with n punctures. The pairs (σ 1 , y 1 ), (σ 2 , y 2 ), . . . , (σ n , y n ) provide the new set of doubled variables restricted to the curves A translation table has been worked out in detail in ref. [11]. The double-cover formulation of the NLSM is given by the integral The Γ contour is defined by the 2n − 3 equations for d = {i, j, k, r} and a = 1, ..., n.
The integrand is given by (18) where (yσ) a ≡ y a + σ a . To obtain the kinematic matrix and the Parke-Taylor factors we need to do the following replacements Analogous to eq. (11), we can now write down a new form for the integrand where {i, j, k} are the same labels as in ∆(ijk) ∆(ijk|r). For more details on the double-cover prescription, see refs. [11,14,15].

FACTORIZATION
Let us start by considering the four-point amplitude, , with the gauge fixing (ijk|r) = (123|4). We will denote sums of cyclically-consecutive external momenta (modulo the total number of external momenta) by P i:j ≡ k i + k i+1 + . . . + k j−1 + k j ,. For expressions involving only two (not necessarily consecutive) momenta, we are using the shorthand notation P ij ≡ k i + k j . We focus on the configuration where the sets of punctures (σ 1 , σ 2 ) and (σ 3 , σ 4 ) are respectably on the upper and the lower sheet of the curves Expanding all elements in A ′ 4 (1, 2, 3, 4) around Λ = 0, we obtain (to leading order) where we have introduced the new fixed punctures σ P34 = σ P12 = 0. Since we want to arrive at factorization identity for non-linear sigma model amplitudes (inspired by previous work for Yang-Mills theories, see ref. [15]) we are now going to introduce polarizations associated with the punctures, σ P34 = σ P12 = 0, i.e. ǫ M 34 and ǫ M 12 . Thus, After separating the labels {1, 2} and {3, 4}, it is simple to rearrange the eq. (24) as a product of two reduced determinants, The new matrices in eq. (27) can be obtained from the A matrix by replacing the off-shell momenta, P 34 and P 12 , by their corresponding off-shell polarization vectors, where the A matrix in eq. (28) is the 3 × 3 matrix related with the punctures (σ 1 , σ 2 , σ P34 ), while the matrix in eq. (29) corresponds to the punctures (σ P12 , σ 3 , σ 4 ).
Using the measure, dµ Λ 4 = 1 2 2 dΛ Λ , we now perform the Λ integral and the amplitude becomes where the notation, P ǫ M i , means one must make the replacement, P i → 1 Following the integration rules in ref. [15], we also have the contribution (up to summing over mirrored configurations)

Thus, the final result is
Since the non-linear sigma model is a scalar theory it is an interesting proposition to consider longitudinal degrees of freedom only Doing so we arrive at the equation Surprisingly, it is possible to generalize this equation to higher point amplitudes. Here the overall sign of each contribution depends of the number of points of the subamplitudes. In ref. [14], we will give more details on this phenomenon.
In the previous expressions, the left hand side is gauge independent, but the precise form of the factorization identities on the right-hand side depends on a gauge choice. The three punctures which must be fixed in the smaller off-shell amplitudes are given by the set, ({All punctures in the amplitude} ∩ {1, 2, 3, 4}) ∪ {off − shell punctures} = {fixed punctures} (see refs. [11,15]).

NEW RELATIONS
We now wish to generalize the two new factorization relations given in eqs. (35) and (36). As will be shown in great detail elsewhere [14], using the double-cover prescription for a partial non-linear sigma model amplitude one is led to the following general formula where an n-point amplitude is factorized into a product of two (single-cover) lower-point amplitudes: (37) Here n is an even integer and we have used eq. (26). The above expression is valid using the Möbius and scale-invariance gauge choice (ijk|r) = (123|4).
From the decomposition obtained by the double-cover method in eq. (37), we are able to write down a new factorization relation, where only longitudinal degrees of freedom contribute, where eq. (33) was used. We checked this formula up to ten points.
Since the above factorization relation includes only longitudinal contributions, we can rewrite it in a more elegant form, involving only the A ′ q amplitudes. Using the definitions given in eqs. (11) and (12) and under the gauge fixing (ijk), with i < j < k, we have the following two identities [14] A Applying the identities eqs. (39) and (40), it is straightforward to obtain where the factorization formula has been written in terms of the generalized amplitude A ′ q . Other gauge choices will naturally with lead to alternative factorization formulas.
Finally, it is important to remark that after evaluating the residues, P 2 2i:2 (z) = 0, in eq. (43), one obtains extra non-physical contributions, which cancel out combining with terms associated with the residue at z = ∞. Therefore, the effective boundary contribution is just given by the sub-amplitudes with an odd number of particles . . , n, 1 z P 2 23 z=∞ .

CONCLUSIONS
We have proposed a new CHY integrand for the U (N ) non-linear sigma model. For this new integrand, the kinematic matrix, (A) ij jk , is no longer anti-symmetric. We have found two new factorization identities, eq. (37) and eq. (38). We have written the second factorization formula in an elegant way, which only involves the generalized amplitude, A ′ q . This formula turns out to be surprisingly compact (we have checked agreement of the soft-limit of this formula with ref. [17]). This has implications for BCFW recursion since the two new factorization formulas can be split among even and odd sub-amplitudes, for example A ′ 2q × A ′ 2m and A ′ 2q+1 × A ′ 2m+1 respectively. Using this we are able to give a physical meaning to the odd sub-amplitudes as boundary contributions under such recursions.
Despite similarities between the three-point amplitudes with the Feynman vertices obtained in ref. [26], the construction presented here is different. For example, the numerators found in eq. (32) are not reproduced by the Feynman rules found in ref. [26]. Understanding the relationship between the formalisms would be interesting.