$\Lambda$ Scattering Equations

The CHY representation of scattering amplitudes is based on integrals over the moduli space of a punctured sphere. We replace the punctured sphere by a double-cover version. The resulting scattering equations depend on a parameter $\Lambda$ controlling the opening of a branch cut. The new representation of scattering amplitudes possesses an enhanced redundancy which can be used to fix, modulo branches, the location of four punctures while promoting $\Lambda$ to a variable. Via residue theorems we show how CHY formulas break up into sums of products of smaller (off-shell) ones times a propagator. This leads to a powerful way of evaluating CHY integrals of generic rational functions, which we call the $\Lambda$ algorithm.

Denoting the position of n punctures on a sphere by {z 1 , z 2 , . . . z n } and using P SL(2, C) to fix three of them, say z 1 , z 2 , z 3 , there is a rational map from C n−3 → C n−3 which is a function of the space of kinematic invariants for the scattering of n massless particles (k 2 a = 0), s ab = k a · k b , , for a ∈ {1, 2, . . . , n}, (1.1) with a ∈ {4, 5, . . . , n}.
In section 2 we will give a short review about this ideas. The representation (1.2) of scattering amplitudes makes many properties manifest. Some of them are gauge invariance, soft limits, BCJ relations and the existence of KLT formulas [17,[45][46][47][48][49][50][51][52][53][54][55]. The drawback is that integrals of the form (1.2) require the solution of polynomial equations whose degree increases with the number of particles.
In this paper we reformulate the formula for scattering amplitudes in terms of a double cover of the punctured sphere. More precisely, we consider a sphere as a curve in CP 2 defined by where Λ is a non zero constant parameter. Clearly, the curve is invariant under a simultaneous scaling of the coordinates (y, σ, Λ). The new formulation is schematically given by M n = 1 Vol(P SL(2, C)) n a=1 (y a dy a ) n a=1 (y 2 a − σ 2 a + Λ 2 ) n b=1 dσ b H(σ, y, Λ) E b (σ, y, Λ) . (1.4) Integration over both residues of the curve implements the sums over choices of branches.
In the double cover description each puncture is specified by a pair of complex numbers (σ a , y a ). The value of y a indicates the branch where the puncture is located. The new form of the components of the map E a is E a (σ, y, Λ) = n b=1,b =a 1 2 This form is very easy to motivate and it is done in section 3. The differential form being integrated in the double cover version of the formula for M n is invariant under the global rescaling inherited from CP 2 . This C * group can be promoted to a full redundancy of the description introducing the scale measure where the Λ factor is proportional to the square root of the discriminant of the quadratic curve in (1.3), ∆ = 4Λ 2 . The C * action is non-trivial on the puncture locations, this means that one can combine the new C * action with the P SL(2, C) group of the sphere and use it to fix the σ coordinate of four punctures. Doing so leaves Λ as an integration variable to be fixed by the scattering equations. This is done in section 4. In section 5 we show that the global residue theorem can be used to replace one of the components of the map, say E n , by Λ. As it turns out the residue theorem only picks up poles at Λ = 0 and at Λ = ∞ and both are identical. At Λ = 0 the branch cut connecting the two branches of the double cover closes and the integrals separates into sectors. Each sector is determined by the distribution of the punctures between the two branches. The amplitude then becomes schematically (1.7) where the sum is over possible distributions of punctures and M off−shell refers to amplitudes where one particle, corresponding to the puncture created by the closing of the branch cut is off-shell. We apply this procedure to more general integrals over the moduli space which appear as parts of physical amplitudes in their CHY representation. In these more general cases, when the integrand has at most double poles on the boundary of the moduli space M n,0 then the propagator is a standard Feynman propagator. An example of an integral with at most double poles is d n z 1 E 1 (z)E 2 (z) · · · E n (z) 1 (123 · · · n) 2 (1. 8) where (123 · · · n) := (z 1 − z 2 )(z 2 − z 3 ) · · · (z n − z 1 ). (1.9) This integral is known to give the sum over all Feynman diagrams computing a partial amplitude in a cubic scalar theory in the bi-adjoint representation of U (N ) × U (M ). Iterating the procedure gives rise to a novel set of diagrams where the buliding blocks are four-particle amplitudes and propagators. When the integrand has higher order poles on the moduli space one finds generalized propagators which are made from higher powers of kinematic invariant. One example, explicitly compute in section 6, is a six-particle integral d 6 z 1 E 1 (z)E 2 (z) · · · E 6 (z) 1 (1234) 2 (56) 2 (1. 10) with (56) = (z 5 − z 6 )(z 6 − z 5 ) consistent with the definition (1.9). This integral has poles of the form 1/s 3 56 . A very familiar way of understanding this process is by analogy with the Stukelberg procedure for taking massless limits of massive vector bosons [56]. The mass parameter is played by the kinematic invariant controlling the factorization limit while the Stukelberg field is played by the Λ parameter. All this process is shown in section 6, where we formulate a new algorithm and in section 7 we give three non trivial examples.
In section 8 we compare our method with the rules given in [18] by Baadsgaard et al. We also generalize the new algorithm to non trivial numerators and we give a simple example.
Finally, we end in section 9 with discussions.

Preliminaries
In this section we review the basic CHY construction [1][2][3] and show some examples that motivate the double-cover construction.

CHY Construction
Consider the scattering of n massless particles. The scattering data is determined in terms of a set of n momentum vectors {k µ 1 , k µ 2 , . . . , k µ n } and n wave functions { µ 1 , µ 2 , . . . , µ n }. Here we take the wave functions to be polarization vectors as higher spin wave functions, e.g. for gravitons, can be constructed using tensor products. In a slightly different terminology from the original CHY construction, one introduces n rational functions of the puncture locations, z a , defined by [1,39] It is easy to show that three linear combinations vanish n a=1 z m a E a = 0 for m ∈ {0, 1, 2}. (2.2) Recalling that different configurations of punctures on a CP 1 are to be identified if they differ by a P SL(2, C) transformation. This means that the location of three punctures can be fixed. It is possible to show that for any rational function H(z) which transforms as that computes one of the local residues at a zero of the map C n−3 → C n−3 is independent of the choice of fixed punctures {z i , z j , z k } and of equations eliminated {E p , E q , E r } to construct the map. In this formula |ijk| stands for the Vandermonde determinant of z i , z j , z k . One way to see that this is the case is to realize that the generators of P SL(2, C) are Treating the P SL(2, C) as a redundance of the integral and using a gauge fixing procedure one can check that the Fadeev-Popov determinant is indeed (2.6)

Examples
The CHY representation of many theories are known (or are conjectured). In this subsection we review some of them in order to motivate the constructions in this paper. Let us start with Einstein gravity [1,2]. The integrand H is computed as the reduced determinant of a matrix a 2n × 2n antisymmetric matrix where A, B and C are n × n matrices. The first two matrices have components while the third is given by This matrix depends on the momenta k µ a and on polarization vectors µ a . The diagonal components of the C matrix can be written in a manifestly P SL(2, C) covariant way by choosing a momentum vector, say k n if a = n, and eliminating it using momentum conservation . (2.10) The integrand is given by where Ψ ij ij is the (n − 2) × (n − 2) matrix obtained from Ψ by removing the rows (i, j) and the columns (i, j).
The second example is that of the scattering of gluons in a U (N ) Yang-Mills theory [1,2]. The coefficient of the trace Tr(T a 1 T a 2 · · · T an ) is computed by the integrand Pf Ψ, (2.12) where Pf Ψ = (z i − z j ) −1 PfΨ ij ij and (123 · · · n) = (z 1 − z 2 )(z 2 − z 3 ) · · · (z n − z 1 ). The third example is that of a scalar theory in the bi-adjoint representation of U (N )×U (Ñ ) [2]. The coefficient of the trace Tr(T a 1 T a 2 · · · T an )Tr(T a w(1)T a w(2) · · ·T a w(n) ) with w some permutation of labels, is given by . (2.13) The last two examples are also purely scalar theories but with derivative interactions [3,11].
The fourth example is a special Galileon theory (sGal) that possesses more nonlinearly realized symmetries than a generic Galileon. Amplitudes in this theory are computed using where det A = (z i − z j ) −2 detA ij ij . The fifth and final example is the U (N ) non-linear sigma model. The term proportional to the trace Tr(T a 1 T a 2 · · · T an ) is computed by In order to illustrate the kind of integrals we are interested in performing let us consider det A for four particles, This means that the integrands of the Galileon and NLSM are

Singularities on M 0,n
The examples given above make it clear that a variety of integrands H(z) can appear. One way to classify them is by the kind of singularities they have as different boundaries on the moduli space of a punctured sphere are approached. The largest codimension singularities are when two punctures approach each other. Consider for example the integrands for four particles [2,3,11] Clearly, the first integrand has double poles as any two consecutive punctures approach each other z a → z a+1 and no other poles. The second integrand has a triple poles when z 1 → z 2 and when z 3 → z 4 and simple poles when z 2 → z 3 and when z 4 → z 1 . Finally, the last integrand only has fourth order poles when z 1 → z 2 and when z 3 → z 4 . It is easy to show that the order of the pole is related to the order of the propagator associated to the coincident punctures. If the integrand as a (m + 1) th order pole when z a → z b then the integral has a pole of the form 1/s m ab . In the rest of this paper we develop a double cover formulation which is tailored for exploiting the behavior of integrands near boundaries of the moduli space. This method not only becomes a powerful tool in the evaluation of integrals but it also makes physical properties manifest such as crossing and factorization.

Double-Cover Formulation
We consider a sphere as a curve in CP 2 defined by [14] (3.1) We call this curve Σ and it can be interpreted as two sheets joined by a branch cut. We take σ as the coordinate on a sheet and y as the variable determining the branch. Λ is taken to be a constant parameter that controls the opening of the branch cut joining the branch points σ = −Λ and σ = Λ. The location of n-punctures on Σ is given by n pairs {(σ a , y a )}. We would like to find formulation of the maps E a defining the scattering equations for this curve. Clearly, E a must have a simple pole when puncture a coincides with puncture b. On Σ, it is not enough to have σ a → σ b but we also need y a → y b , i.e., they must be on the same branch. When σ a → σ b one can have either y a → y b or y a → −y b . So we need a projector, P (a) b , that gives one in the former and zero in the latter. One choice is This turns out to be the correct choice and one has One important condition the equations have to satisfy is that they must be covariant under the exchange of σ and y (with Λ → iΛ) which is a symmetry of the curve Σ. It is easy to check that on the support of y 2 b = σ 2 b − Λ 2 , the function y a E a is invariant.
Having found the new version of the maps E a which give rise to the scattering equations, the next step is to translate the rational function H(z) which defines the theory under consideration. All such functions can be decomposed as linear combinations of functions of the form [17] H(z) = 1 (α(1)α(2) · · · α(n))(γ(1)γ(2) · · · γ(n)) f (r ijkl ), (3.4) where (α(1)α(2) · · · α(n)) and (γ(1)γ(2) · · · γ(n)) are Parke-Taylor factors with ordering α and γ (see (1.9) for the Parke-Taylor factor definition [57]). f is a rational function of r ijkl which are general cross ratios, i.e., where we have introduced a convenient shorthand notation In order to map H(z) to H(σ, y), we define any combinations of factors of the form as a chain (a 1 a 2 · · · a m−1 a m ) of length m. Chains are taken to have lengths 2 ≤ m ≤ n. A chain of length 2 is given by (a 1 a 2 ) = (z a 1 − z a 2 )(z a 2 − z a 1 ). (3.8) It is straightforward to check Now we propose to use the following replacement into the chain so as to construct the integrand H(σ, y), Note that while the left hand side is antisymmetric in the a and b labels the right hand side is not and hence the notation τ a:b . This fact becomes irrelevant when the substitution is made into chains and hence the importance of the appearance of them in the integrands. So, we complete the map H(z) → H(σ, y) by H(σ, y) = (τ α(1):α(2) · · · τ α(n):α(1) )(τ γ(1):γ(2) · · · τ γ(n):γ(1) )f τ i:k τ k:j τ j:l τ l:i τ i:j τ j:l τ l:k τ k:i . (3.11) In addition one can check, in a simple way, the chain property (a 1 a 2 · · · a m−1 a m ) = (−1) m (a m a m−1 · · · a 2 a 1 ), (3.12) (τ a 1 :a 2 · · · τ a m−1 :am τ am:a 1 ) = (−1) m (τ am:a m−1 τ a m−1 :a m−2 · · · τ a 2 :a 1 τ a 1 :am ), (3.13) and the inverse map works in the same way, τ a:b → 1 z ab . Moreover, it is straightforward to check the scattering equations can be written as where we have denotedτ a:b andẼ a as It is not obvious how chains appear in integrands that are computed using the Pfaffian or the determinant of the matrices Ψ of A.

Redundancies
Next we move to the discussion of the redundancies and how to gauge fix them. This subsection is only the first part of the discussion in which we show how to perform the standard gauge fixings. In the second part, presented in section 4, we perform a different gauge fixing which allow us to use residue theorems to break up contour integrals into integrals with smaller number of punctures.
We consider the following integral where Λ is a non-zero constant parameter and H(σ, y) is a general rational function as in (3.11). The factor Vol(P SL(2, C)) in the integral is there only as a reminder that the integral has a redundancy that has to be gauge fixed. The P SL(2, C) action is generated by the vectors (on the support of the algebraic curve y 2 where ∂ a ≡ ∂/∂σ a and they satisfy the algebra The covariance of the E a maps under these transformations imply that there are three linear combinations among them 1 n a=1 y a E a = 0, n a=1 σ a y a E a = 0, n a=1 y 2 a E a = 0. (3.19) In order to define local residues in (3.16), one must remove three of the elements of the map (σ 1 , σ 2 , . . . , σ n ) → (E 1 , E 2 , . . . , E n ) from C n → C n . This is welcome as one can use the P SL(2, C) group to fix the location of three σ a variables. Using the standard Fadeev-Popov procedure one has where the Fadeev-Popov determinants are given by likewise for |i, j, k| and Γ is the integration cycle defined by the solutions of the 2n − 3 equations The 2 3 factor appears when the P SL(2, C) symmetry is fixed and the (Z 2 ) 3 symmetry Note that the values of σ i , σ j and σ k have been fixed but their branches do not, i.e. y i , y j and y k can still take any of the solutions to

Promoting Λ to variable
In the previous prescription, (3.20), Λ is a constant parameter. In this section we show how to introduce Λ as a variable.
It is straightforward to check that the integral in (3.16) is invariant by the scale transformation (σ a , y a , Λ) → ρ(σ a , y a , Λ), ρ ∈ C * and a = 1, . . . , n, (3.24) Note that the P SL(2, C) measure is also invariant by the scale transformation in (3.24). In order to promote the Λ parameter to a variable we introduce the scale invariant measure dΛ Λ . Thus, the new measure is also scale and P SL(2, C) invariant, i.e GL(2, C) invariant. Clearly, the generators of this GL(2, C) symmetry are given by the elements {L 0 , L −1 , L 1 } and the scale generator Its algebra is given by on the support of the algebraic curve y 2 a = σ 2 a − Λ 2 . Now, note that the denominator in (3.25) can be written as the following determinant This determinant is just the Fadeev-Popov determinant for the gauge fixing of the three punctures (σ i , σ j , σ k ) and the branch cut variable Λ. Finally, we can rewrite the (3.16) prescription as Fixing the {E p , E q , E r } scattering equations , the (σ i , σ j , σ k ) punctures and the Λ branch cut variable one obtains which is the same expression as in (3.20).

Equivalence with the CHY Construction
The idea of this section is to show how the (3.16) prescription is in fact equivalent to the original CHY approach.
Let us define a map from the double-cover version of the sphere into a single cover of CP 1 . This should take us back to the original CHY construction. Such a map is very well known and it is given by where Λ = {0, ∞} is a constant and z a are the coordinates on CP 1 (CHY coordinates).
The first observation is that if all the punctures are located on the same branch, say the upper sheet, i.e. y a = + σ 2 a − Λ 2 , then In this expression it is easy to see that the lack of antisymmetry in the labels translate into an overall factor in the z a variables. Also it is simple to show which means that 1 2 This is indeed the natural differential form on Σ with simple poles at (σ a , y a ) = (σ b , y b ) and at σ a = ∞ with residues 1 and −1 respectively. Therefore, it is straightforward to conclude where H(z) is as in (3.4). Carrying out the integration over the y a variables on the contour given by the solutions y a = + σ 2 a − Λ 2 and performing the map (3.32), then (3.16) becomes where the 1 2 n factor comes from the integral Computing the integral over all possible configurations, this means the 2 n way of choosing (y 1 = ± σ 2 1 − Λ 2 , ..., y n = ± σ 2 n − Λ 2 ), and performing the map (3.32), one obtains .
This result agrees with the original CHY formula.

New Gauge Fixing
In this section we find that by using the full GL(2) redundancy one can gauge fix the location of four punctures, modulo branches. Thus, promoting Λ to a variable to be fixed by the scattering equations, one has the possibility of using a global residue theorem [14] that leads to a new diagrammatic expansion of general amplitudes. Moreover, the residue theorem allows the analytic evaluation of integrals with rational functions whose answers have non-local poles and thus are hard to obtain by other means.

New Gauge Fixing
Let us start by reviewing the generations of the GL(2, C) redundancy, as we did in (3.17) in section 3, Since all four vectors act on σ's one can use them to fix four of the punctures' σ. For simplicity of notation let us assume that they are σ 1 , σ 2 , σ 3 and σ 4 . The Fadeev-Popov jacobian, ∆ FP , is now In addition to this one still has to remove three elements from the map This procedure is not affected by the new gauge choice and the formula used in (3.20) is still valid. Putting all together and removing, without loss of generality, the scattering equations E 1 , E 2 and E 3 we arrive at the new formula where Γ is the integration cycle defines as in (3.22), given by the equations for the 2n − 3 variables (Λ, σ 5 , . . . , σ n , y 1 , y 2 , . . . , y n ). It is interesting to note that the opening of the branch cut connecting the two branches (sheets) becomes a function of the kinematic invariants k a ·k b . This means that as we move in the space of kinematic invariants the branch cut also moves. This is what makes factorization and crossing natural properties to address using this formulation.

Residue Theorem and Diagrammatic Expansion
The equations obtained at the end of the previous section are polynomial equations of increasing degree as the number of particles increases. In fact, the equations (4.4) lead to higher degree polynomials than the original CHY scattering equations. This seems to be an obstacle. However, using a residue theorem we will effectively replace one of the E a = 0 equations by the equation Λ = 0. This might come as a surprise as closing the cut is intuitively related to a factorization limit. Instead, what we will see is that once the cut closes a new puncture appears that represents an off-shell particle. The sum over solutions to the equations y 2 b = σ 2 b − Λ 2 give rise to y b = ±σ b and determine the branch location of the b th -puncture. For a given distribution of particles, say a subset U (L) is on the upper (lower) branch, the equation E a that was eliminated gives rise to the propagator 1/P 2 U where P U is the sum over the momenta of all external particles on the upper branch. In this way, the integral given in (4.3) becomes a sum of products of contour integrals with smaller number of particles. By iterating the process we find a diagrammatic description. The most important outcome is that at each step in the iteration process the degree of the scattering equation is lowered and analytic evaluations become possible.

Residue Theorem
Following the similar ideas as in section 3.2 les us consider the general integral 2) where γ is the contour defined by the equations and H(σ, y) n c=1 y 2 c /Λ is the integrand. Clearly there are more integration variables than contours, nevertheless, when the GL(2, C) symmetry is fixed then the number of integration variables becomes equal to the contour cycles.
Note that (5.2) depends over the Λ variable by the expression defines the integration cycle. Naively, using the global residue theorem, it is straightforward to see that the previous expression can be written as Nevertheless, in order to apply the global residue theorem one must also verify if the point at infinity is a pole. One says that the (5.2) integral has a pole at infinity if and only if [14] deg d 1 + · · · + d 2n is sum over all degrees of the polynomials that define the integration contour, i.e.
and 2n+1 is the number of integration variables, i.e. (Λ, σ 1 , . . . , σ n , y 1 , . . . , y n ). Clearly (5.2) has a pole at infinity and it must be integrated when the global residue theorem is performed. Since the integrand in (5.2) is not well defined when Λ = 0, then this implies that the (5.2) integrand is given on the Λ = 0 chart . Thus, so as to explore the pole at infinity we consider the following transformation Under this transformation (5.2) becomes invariant, i.e.
10) where γ is the contour defined by the equations and H(σ , y ) is defined with the τ a:b 's forms Note that the minus sign dΛ/Λ → −dΛ /Λ is used to reorient the Λ contour. Finally, we can now integrate around the point Λ = 0 which is the pole at infinity, therefore performing the global residue theorem the (5.2) integral could be read as , (5.13) where the newΓ contour is now defined by the 2n equations Λ = 0, y 2 b − σ 2 b + Λ 2 = 0, for b = l,Ẽ a (σ, y) = 0, for a ∈ {1, 2, . . . , n}. (5.14)

Recovering The Curve
Note that in the result obtained in (5.13) the y 2 l − σ 2 l + Λ 2 = 0 constraint is lost, i.e. we are not anymore on the support of the curve y 2 l = σ 2 l − Λ 2 . Since our aim is to be on a sphere then this constraint must be recovered. In order to get back the y 2 l − σ 2 l + Λ 2 = 0 equation we perform the residue theorem but now using the y l variable.
Before applying the residue theorem it is useful to remember the following, first, in full view, the integration contour is defined by polynomials over the y i 's variables and secondly, the integrand , has just one singularity over the y i 's variables given by (y 2 l − σ 2 l + Λ 2 ). With this in mind, we are ready to use the residue theorem over y l . The integral by the y l variable is read as  Finally, the (5.2) integral is written as Fixing the (σ m , σ n , σ p , σ q ) puctures and the (E m , E n , E q ) scattering equations the above integral becomes Note that we have chosen the same labels for the punctures and scattering equa-tions, this will be useful when we will formulate the Λ−algorithm in section 6.2. The Λ−prescription, (5.18), recovered the support on the curves y 2 a = σ 2 a − Λ 2 , moreover, it must be computed around the cycles Λ → 0 and Λ → ∞, which are exactly the same, as one can see in In the next section we will learn to use this new prescription and we will propose a new algorithm (the Λ−algorithm).

Λ-Diagrams and A New Algorithm
Here we present a new algorithm, which is a consequence of the new prescription given in (5.18).
Before formulating the algorithm we introduce some notations. Let us remember the s a 1 ...an Mandelstam variables are defined as 3 s a 1 ...an := 1 2 (k a 1 + · · · + k an ) 2 . (6.1) Nevertheless, it will be useful for us to use the variables k a 1 ...an := n a i <a j k a i · k a j , (6.2) Clearly, when the particles are massless, i.e. k 2 i = 0, then s a 1 ...an = k a 1 ...an . In the next two section, 6.1 and 6.1.1, we give all tools to formulate our new algorithm in section 6.2. While we develope the sections 6.1 and 6.1.1, we apply all these tools on a simple and particular example and at the end we obtain the result for the (5.18) integral.

More Notations and a Simple Example
In the same way as in [17], any H(σ, y) integrand over M 0,n can be written as a linear combinations of integrands with no zeros , i.e. integrands with just 2n − τ a:b factors. We call this kind of integrands as 4 H D (σ). Each H D (σ) integrand has associated a 4-regular graph 5 (bijective map), which we denoted by G = (V G , E G ) [17,58,59]. The vertex set of G is given by the n-labels (punctures) Since τ a:b always appears into a chain, for instance, let us remember the smallest chain is given by then the graph is not a directed graph, as well as in [17]. This is useful to clarify that the G graph must be draw such that the number of intersection among the edges is as small as possible.
Note that the G graph does not have any information of the GL(2, C) symmetry and the Λ parameter or branch cut. In order to introduce this information on the graph we coloured the vertex set in the following way The G graph can now contain the whole information of the integrand, i.e, it now represents the total integrand I = |ijk|∆ F P (ijk, d)H(σ).
For example, using the P SL(2, C) symmetry to fix the (σ 1 , σ 2 , σ 3 ) punctures and the scale symmetry to fix the σ 4 puncture, the graph in where Fig.6.3(c) shows the whole possibles non-zero contributions or configurations, up to Z 2 symmetry y a → −y a , after performing the Λ integral around Λ = 0. It is explained in detail in the next section.

Configurations and Λ−Theorem
Although we previously have already used the word "configuration", in this section we give a formal definition. So, the first thing we do in this section is to define what is a configuration • Configuration: A configuration, which we denoted by C, is the integration over the (y 1 , . . . y n ) variables around one of the 2 n solutions of the equations y 2 a − σ 2 a + Λ 2 = 0, for a = 1 . . . n. (6.6) This definition means that a C configuration is the choosing of the 2 n possibilities given by (y 1 = ± σ 2 1 − Λ 2 , ..., y n = ± σ 2 n − Λ 2 ), i.e. a configuration fixed the punctures on the upper or lower sheet. Now, with this in mind we are ready to come back to our example and note that besides of two configurations given in Fig.6.3(c), there are more possibles configurations (up to Z 2 symmetry) such as , where the red line enclose the punctures on the same branch cut, i.e. the red line is the branch cut, which is controled by the Λ integration variable.
However, these five configurations vanish trivially because the P SL(2, C) symmetry is breaking on upper and lower sheet when Λ → 0. This computation is straightforward. So as to classify the different kind of configurations we introduce the following terminology • Allowable Configuration: Let C be a configuration. We say C is an allowable configuration if the number of fixed punctures on the upper and lower sheet is two. This implies that in the Λ → 0 limit the P SL(2, C) symmetry is well defined (gauged) on each sheet.
Clearly, for the diagram in Fig.6.3 there is one more allowable configuration given by , (Fig.6.4) Allowable configuration which vanishes.
but this one also vanishes. The vanishing of this last configuration is a consequence of the following theorem Λ−Theorem Let C be an allowable configuration, then the integrand I = |ijk|∆ F P (ijk, d)H D (σ) on the C configuration has the Λ−behavior around Λ = 0, where L is the number of edges which are intersected by the red line.
This theorem is proved in appendix A.
So far, we have defined what is a configuration, an allowable configuration and we have formulated the Λ−theorem. Now, with the intention to set down the Λ−algorithm it is useful to define a new kind of configuration • Singular Configuration: Let C be a configuration. We say C is an singular configuration if C is an allowable configuration and the integrand, I = |ijk|∆ F P (ijk, d)H(σ) ∼ Λ −s , s > 0 around Λ = 0.
Following with our example, we note that expanding the I = |123|∆ which is the right answer.
We call this method the Λ−algorithm. In the next section we explain carefully this algorithm.

The Λ−Algorithm
In this section we introduce formally the Λ−algorithm, which is given up to Z 2 symmetry,, y a → −y a . We describe step by step the method. Λ−Algorithm Steps • (1) To draw the graph to be computed. Let us remember that the graph must be drawn such that the intersection number of the edges is the minimum. The gauge fixing from the step (1) must be chosen such that there are not singular configurations. This fact becomes clearer in section 7.
If it is not possible to choose a gauge fixing such that it avoids singular configurations then the Λ−algorithm can not be applied directly. k upper 0 = k p + · · · + k q + · · · + k r + · · · , (6.7) and the momentum of the red massive puncture on the lower-sheet is the sum over all momenta of the particles inside of the red line, i.e.
k lower 0 = k m + · · · + k n + · · · . (6.8) The scattering equation associates to the puncture in the green triangle, Using the scattering equations (at Λ = 0) located on the same sheet as the green puncture, in figure(c) it is the lower sheet, i.e E r , . . ., it is straightforward to prove that factor becomes one of the two Faddeev-Popov determinants on the lower brach and the numerator, (σ q −σ 0 )(σ 0 −σ q ), cancels out with the |m, n, q| ∆ FP (mnq, p) Faddeev-Popov expansion given in appendix A. Therefore, one can say that the E p scattering amplitude becomes the propagator • (4) To come back to the step (1).
It is useful to remember that a 4-regular graph with 3 vertices is just 1 (Figure (d)) 3-point 4-regular graph

Building Blocks
Since that the Λ-algorithm is an iterative process then it is useful to construct fundamental graphs or irreducible graphs (building blocks). Our building blocks are given by the following diagrams of 4 and 5 vertices ( Fig.6.7) Building Blocks.

) Building Blocks (I) and (II).
In order to compute the the (II) and (III) building blocks, one can note that on the support of the E d scattering equation (before performing the residue theorem, section 5) Finally, the (V ) building block in Fig.6.7 can not be computed using the Λ−algorithm, because this graph has a singular configuration. So, we use the algorithm given in [17] (the general KLT algorithm).

General KLT algorithm and computation of the (V ) building block
In order to apply the general KLT algorithm [17] on the (V ) building block, one must first note that this building block has the following decomposition (in two 2-regular graphs) . (Fig.6.

10) Decomposition in two 2-regular graphs
The second step is to find a left and right (Parke-Taylor) base compatible 6 with the the I L and I R graphs. Choosing the left and right base as where the relative sign was explained in [2] .
So as to be consistent with the initial gauge fixing we must keep it 7 , i.e. the color of the vertices.
Although in [2] was given an algorithm to computed the diagrams found in Fig.6.12, we apply the Λ−algorithm since it works when one of the particles is off-shell.
Let us consider the second component of the first vector given in In Fig.6.13 we describe step by step the Λ−algorithm for a particular diagram in Fig.6.12: • (1) We draw the graph to be computed, including the gauge fixing (colored vertices).
• (2) We find the all non-zero allowable configurations, which is only one.
-(i) The scattering equation 1/E b becomes the propagator 1/k ab .
-(ii) The subdiagram obtained on the upper-sheet is a 4-regular graph at three point, which is trivial,, i.e. 1. On the other hand, the 4-regular subdigram obtains on the lower-sheet is a 4-point graph, which is the (II) building block given in Fig.6.7.
-(iii) The new massive particle in the graph on the lower-sheet has momen- • (4) we used the (II) building block in Fig.6.7 to obtain the final answer.
Following the same simple procedure one can compute all graphs in Fig.6.12, for example, the m Therefore, the I (V ) building block is (6.10)

Examples
Although in the previous section we have already applied the Λ−algorithm, the idea here is to give some non-trivial examples in order to show the power of this new algorithm. This section is divided as follows, the first example show us how to use the Λ algorithm, which will be applied over a six point highly non trivial diagram. The idea of the second one is to mix the Λ algorithm with the KLT general algorithm [17], where we will compute a six point diagram which cannot be performed just with the Λ algorithm. Finally, the last one is given in order to illustrate the using of all building blocks, with this in mind we choose a non trivial eight point diagram. The first one, I (A) , will be computed just using the Λ−algorithm. For the second one, I (B) , the Λ−algorithm is not enough. We will combine the Λ and the general KLT algorithm [17] to compute it.

I (A) -Computation
In order to avoid singular configurations we choose the following gauge fixing . (Fig.7.2) Gauge Fixing. This is straightforward to see that there are two kind of allowable configurations. The first one is given by the diagrams , (Fig.7.3) Allowable configurations of type 1. and the second one by . (Fig.7.4) Allowable configurations of type 2.
Since the elements of each type are totally analogues then we only compute one of each set.
Let us begin by computing the (I) configuration in Fig.7.3. Applying the Λ−algorithm, the E 5 scattering equation becomes the 1/k 356 propagator and the diagram breaks into two graphs (upper and lower sheet) in the original CHY approach . (Fig.7.5) Computing the (I) diagram.
Using the building blocks given in section 6.3 (see Fig.6.8 and Fig.6.9), we are able to find the final answer for the (I) configuration in Fig.7.3. Thus, following the same procedure for the (II) and (III) configurations one obtains It is simple to see that the non-zero allowable configurations in where it is useful to remember that k 0 = k 2 + k 3 , Fig.7.7. From the building blocks of the section 6.3, Fig.6.9, we obtain the final answer for the (a) configuration in Fig.7 Therefore, summing over all allowable configurations we obtain the total answer for the I (A) graph, which is given by the non trivial expression and the labels A, B, C, D, I and J mean a index set, for example k A := k a 1 ···am . The (7.1) result was checked numerically.

I (B) -Computation (General KLT and Λ Algorithms)
In section 6.3 we have combined the general KLT algorithm [17] and the Λ−algorithm, respectively, in order to compute the (V ) building block, however, in this section our idea is the opposite. First, we apply the Λ−algorithm as far as it is possible. From this method we will obtain subdiagrams with less vertices than the original one. Second, we perform the general KLT algorithm on these subdiagrams and finally we will be able to use the Λ−algorithm, again, to compute the diagrams into the vectors and matrix, such as it was done in section 6.3.1. Let us remember that in the general KLT algorithm [17] one must find a base, left (L) and right (R), such that all graphs have a Hamiltonian decomposition, i.e. the integrands are product of two Parke-Taylor factors. One of its main drawback is to compute the inverse of the Gram matrix given by the product among the left and right base, m L|R . For example, in six-point it is necessary to invert a 6 × 6 matrix.
However, since our idea is first to apply the Λ−algorithm then this drawback is softened.
Let us consider the I (B) example in Fig.7.1. In order to avoid singular allowable configurations we set the following gauge fixing . (Fig.7.13) Gauge Fixing .
Since these three configurations are the same up to relabel the (1,2,3) vertices then it is enough just to compute one of them, for example we choose the first one, (i) configuration. Following the techniques presented in the section 7.1.1 one obtains . (Fig.7.15) Computing the (i) configuration.
The 5 point graph on the right hand side can not be computed using the Λ−algorithm presented in section 6.2, therefore we use the general KLT-algorithm [17].
Following the same procedure used to compute the (V ) building block in such that the diagrams in the (LI L ) and (RI R ) vectors have a Hamiltonian decomposition [58,59]. Thus, we can write the 5-point diagram in Fig.7.16 as the matrix product (LI L )(m L|R ) −1 (RI R ), diagrammatically one has .
Using the Λ−algorithm we compute each diagram in Fig.7 , which was checked numerically.

Eight-Point
In this section we consider a non-trivial 8 point graph, which has the following left and right decomposition . In addition to continue testing the power of the algorithm, this kind of graph was chosen in order to use the (V ) building block (Fig.6.7).
First we fix a gauge such that there is no a singular configuration, for example we choose the gauge fixing . where we have fixed the "a"-puncture (by scale symmetry) in order to avoid singular configurations. This graph is very similar to one given in Fig.7.2 and its computation is totally analog. Using the Λ−algorithm the result for this graph is the function The first one, F 5 13 (k a , k b , k c , k d , k e ), is a subdiagram obtained from the (1) and (3) configurations and the second one, F 5 2 (k a , k b , k c , k d , k e ), is obtained from the (2) configuration. These two subdiagrams are very similar to one obtained in Fig.7.8 and their computations are very simple using the Λ−algorithm. The results for these two graphs are Note that the two answers are totally different, this is because k ac = k bde since there is a massive particle. We can now write the results for the Type (II) configurations, Finally, from the Type (III) configuration, {(i)}, one obtains the subdiagrams .
The first one, F V (k a , k b , k c , k d , k e ), is the (V ) building block computed in section 6.3.1 and it is given by The second one was also computed in section 6.3.1 using the Λ−algorithm and its result is very simple Thus, the (i) configuration can be read as (i) = F V (k 1 , k 2 , k 3 + k 4 + k 7 + k 8 , k 5 , k 6 ) F (i) (k 7 , k 8 , k 1 + k 2 + k 5 + k 6 , k 3 , k 4 ) k 3478 (7.14) The full answer is the sum over all configurations given in Fig.7.22, i.e.
This result was checked numerically.

The Baadsgaard, Bohr, Bourjaily and Damgaard Rules (BBBD)
Vs The Λ-Algorithm In [18], Baadsgaard et al, formulated some rules in order to compute the same kind of integrals or diagrams that we have studied so far. Nevertheless, although their rules are a certain sum over all possible factorization limits, similar to the Λ-algorithm, these two algorithms present important differences. For example, the Λ-algorithm depends of the gauge fixing, such as it has been explained and shown in section 6. This particular characteristic is in fact a powerful tool, for instance, using the BBBD rules, which are independent of the choice of gauge, it is not possible to compute directly integrands such as ones given by the diagrams in Fig. 7 At the same way as in [18], the Λ-algorithm can also be directly used to integrands with non trivial numerator. For example, let us consider the same diagram as in [18] , ( . Note that the lines and the antilines connecting the same two vertices cancel each other 9 . Moreover, due to the presence of a non trivial numerator, the denominator has more factors than otherwise, this is so as to retain the SL(2, C) invariance. Obviously, the N 6 diagram in Fig.8.1 is not a 4-regular graph, but the subtraction between the number of lines and antilines must always be four (on each vertex) in order to keep the SL(2, C) symmetry.
To compute the N 6 diagram we are obliged to extend the Λ-Theorem • Λ−Theorem (Extension) Let C be an allowable configuration, then the integrand I = |ijk|∆ F P (ijk, d) H(σ) on the C configuration has the Λ−behavior around Λ = 0, where L is the number of lines and A is the number of antilines which are intersected by the red line.
Using the Λ−Theorem (Extension) it is simple to see there are only two non zero allowable configurations . (Fig.8.2) Non zero configurations for the N 6 diagram.
These two configurations are easily calculated from the rules in section 6.2, so the first one configuration reads which is the same answer found in [18]. This show how powerful is the Λ−algorithm, which can be applied to solve highly non trivial integrands.

Discussions
In this paper we gave a new representation for the CHY integrals. We call this new representation as the Λ−prescription. The Λ−prescription is supported on an algebraic curve of degree two, which is embedded in CP 2 , i.e. this is a sphere. This curve can be thought as a Riemann surface with two sheets connected by a branch cut.
The new scattering equations (the Λ−scattering equations) must contain information about the branch where the particles (punctures) are localized. For example, the Λ scattering equations are given by the expression with a = 1, . . . n. When y a = σ 2 a − Λ 2 one says that the particle (puncture) is on the upper sheet and when y a = − σ 2 a − Λ 2 then one says that the particle is on the lower sheet. Note that the quadratic curves, y a , have an additional parameter, Λ, which controls the opening of the branch cut. When this parameter is promoted as a variable then a new symmetry arise (scale symmetry), which can be used to fix one more particle (puncture).
In section 5 we performed the global residue theorem over this new variable, Λ. After integrating Λ one obtains that the Λ prescription must be evaluated at the point Λ = 0 (Λ = ∞), i.e. at the limit when the branch cut collapses in a line. So, the initial integral is broken into two new smaller integrals, which are now written as in the original CHY approach. In addition, these two new integrals are multiplied by a propagator, which is associated to the collapsed branch cut, it is kind a factorization limit. This is an iterative process, i.e. it can be applied over each one of these two new integrals. All this procedure is encoded into what we call the Λ−algorithm.
The Λ−algorithm allow us to expand a given integral in terms of fundamental building blocks, given in Fig.6.7.
Unlike to the other algorithms, the Λ−algorithm depends totally of the gauge fixing. Although this does not look like to be a good thing, in fact it is. For example, diagrams such as ones given in Fig.7.1, which are very complicated using other type of algorithms, they are easily computed from the Λ algorithm, obviously, after choosing a good gauge.
The Λ algorithm is a powerful, simple and beautiful tool because it is a pictorial algorithm. Nevertheless, this mechanism has some limitations, i.e. there are some CHY integrals which can not be performed just using this algorithm. This is due we do not know the behavior of the singular allowable configurations, which is the reason why one must choose a good gauge. It will be very interesting to know how to extend the Λ algorithm to singular allowable configurations.
We know that the Λ algorithm can be used on a big spectrum of CHY integrals, the main idea is to choose a gauge such that the all allowable configurations will not be singular. In particular, we know diagrams on which this fact always happens. These diagrams are given by all possible combinations of the following two 2-regular graphs . (Fig.9.1) Two 2-regular graphs. I a is a Parker-Taylor graph. I b is a bubble with a regular polygon graph.
The I a graph is clearly a Parker-Taylor factor, therefore, the diagram given by the integrand H(σ) = I a I a is just the m(α|β) kernel, which is very simple to compute. The other two options given by the integrands H(σ) = I a I b and H(σ) = I b I b , which are non trivial diagrams, they can be easily computed using the Λ algorithm.
The Λ algorithm has two more advantages. As we saw, some massive particles arise in the process, so this algorithm supports off shell particles. The other one is that this algorithm could be used on integrands with non trivial numerators, such as one given in Fig.8.1. These two characteristics are very important in order to compute diagrams at loop level, for example, the diagram given by , ( Fig.9.2) 5-gon CHY diagram representation.
which appears at 1-loop computation of the 5-gon, it can easily be computed using the Λ−algorithm [60].
Finally, note that the integrand in the Λ prescription is basically obtained from the original CHY approach just changing the 1 z ab form by the τ a:b form. However, although 1 z ab is an antisymmetric form, i.e 1 z ab = − 1 z ba , the τ a:b form is not, τ a:b = −τ a:b . So, the antisymmetric matrix, Ψ αβ , which was defined in [1], it is not any more antisymmetric when (z ab ) −1 is replaced by τ ab . Therefore, the Pfaffian of Ψ αβ (τ a:b ) is not well defined. Naively, in order to give an interpretation for the Yang-Mills theory from the Λ prescription one can replace the Pfaffian of Ψ αβ (z a:b ) by det (Ψ αβ (τ a:b )), but we leave this for future research.

A Λ-Theorem
In this appendix we prove the Λ-theorem, which was given in section 6.1.1.

Λ−Theorem
Let C be an allowable configuration, then the integrand I = |ijk|∆ F P (ijk, d)H D (σ) on the C configuration has the Λ−behavior means that σ a and σ b are on the upper branch cut. For the others three more configurations, (a → upper, b → lower), (a → lower, b → upper) and (a → lower, b → lower), the Λ expansion is read as Now, let us remember that the H D (σ) integrand is given by the products of chains, i.e. the products of factors such as [a 1 , . . . a k ] = (τ a 1 :a 2 τ a 2 :a 3 · · · τ a k−1 :a k τ a k :a 1 ).
( where L is the number of edges which are intersected by the red line. Thus the Λ−theorem has been proved The proof for the Λ−theorem (Extension), given in section 8, is completely analogous.