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On Multi-step BCFW Recursion Relations

In this paper, we extensively investigate the new algorithm known as the multi-step BCFW recursion relations. Many interesting mathematical properties are found and understanding these aspects, one can find a systematic way to complete the calculation of amplitude after finite, definite steps and get the correct answer, without recourse to any specific knowledge from field theories, besides mass dimension and helicities. This process consists of the pole concentration and inconsistency elimination. Terms that survive inconsistency elimination cannot be determined by the new algorithm. They include polynomials and their generalizations, which turn out to be useful objects to be explored. Afterwards, we apply it to the Standard Model plus gravity to illustrate its power and limitation. Ensuring its workability, we also tentatively discuss how to improve its efficiency by reducing the steps.


Introduction
In the past decade, the BCFW recursion relation [1,2] had been a very efficient on-shell method to calculate tree-level scattering amplitudes. Pedagogical reviews on this topic can be found in [3,4]. Still, it encounters certain difficulties when there exists no 'good' deformation [5,6], i.e., the real amplitude does not vanish under the large z limit, where z is the deformation parameter. The recursion relation then fails to capture a residual part called the boundary term, which corresponds to the residue at infinity of the deformed amplitude. Many related studies have been achieved including: introducing auxiliary fields to eliminate boundary terms [7,8], analyzing Feynman diagrams to isolate boundary terms [9,10,11], expressing boundary terms in terms of roots of amplitudes [12,13,14], collecting factorization limits to interpolate boundary terms [15] and using other types of deformations [16].
Recently, a new algorithm named as the multi-step BCFW recursion relations [17] was established to tackle this problem universally and systematically. It considerably widens the category of quantum field theories of solvable tree amplitudes [18]. However, some common puzzles encountered in practice still lacks a formal study. One core question is: How to reach the correct answer within finite, definite steps, if an amplitude is solvable by the algorithm?
In this paper, we will explore multi-step BCFW recursion relations extensively, by investigating the algebra of BCFW deformation generators and commutativity of constant extractions. Next, we will seek for a universal approach to reach the answer and ensure that it is correct. This safety promise relies on very little knowledge of a particular QFT, except mass dimension and helicities, hence the algorithm is expected to be able to solve all massless tree amplitudes, with certain limitation addressed below.
It is well known that on-shell methods heavily rely on factorization properties of amplitudes, and the latter is a reflection of locality and unitarity. These properties are mathematically implemented on poles of amplitudes and their residues. For amplitudes that admit polynomials, no on-shell methods so far can fix this ambiguity. One can list all possible forms as basis, but to determine the coefficients will unfavorably call for more traditional means such as Feynman rules. In this work, we will clarify the applicable range of multi-step BCFW recursion relations, and explore all possible forms of polynomials and their generalized cousins called pseudo polynomials and saturated fractions. These two generalized objects can be fixed by other types of deformations, and they have many interesting mathematical aspects to be explored.
The paper is organized as follows. In section 2, we review the multi-step BCFW recursion relations and explore the commutativity of constant extractions. In section 3, we propose the systematic process to calculate amplitudes after finite, definite steps, and clarify its applicable range and limitation. In section 4, we apply this process to Standard Model plus gravity to demonstrate its workability.

Multi-step BCFW Recursion Relations
In this section, we briefly review the multi-step BCFW recursion relations, in the novel language of extraction operators. After that, the commutativity of constant extractions is investigated.

Extraction operators
For a general BCFW deformation a i |b i ], namely let's define two operations on an amplitude-like rational function R(λ i ,λ i ) via 1 where P i and C i are the pole and constant 'extraction operators', which capture residues at finite locations except zero and infinity respectively. For a physical amplitude A, P i can capture only all or a part of its physical poles. But a general R such as P i A or C i A may also contain spurious poles, which is well known. Therefore the detectable poles at finite locations can be either physical or spurious.
By definition P i + C i = I, where I is the identity operator. When we calculate an amplitude, starting by the 0th step, the amplitude is unknown, so is the C 0 operation. However, the P 0 operation represents exactly the BCFW recursion relation, hence we actually reconstruct this part by employing factorization properties, rather than manipulating the unknown amplitude. Conventionally, C 0 is called the boundary term with respect to P 0 , which will be dissected into many parts to be determined. The dissection means, by expanding I for (n + 1) times repeatedly, we have I = P n + C n P n−1 + . . . + C n C n−1 · · · C 2 P 1 + C n C n−1 · · · C 2 C 1 P 0 + C n C n−1 · · · C 2 C 1 C 0 , (2.3) note that I acts on A implicitly. If the final boundary term C n C n−1 · · · C 2 C 1 C 0 vanishes, we have I = P n + C n P n−1 + . . . + C n C n−1 · · · C 2 P 1 + C n C n−1 · · · C 2 C 1 P 0 . (2.4) This identity formally represents the 'multi-step BCFW recursion relations'. Importantly, the workability of this multi-step approach relies on the existence of a series of deformations numbered by 0, 1, . . . , n for which C n C n−1 · · · C 2 C 1 C 0 = 0, and the latter is the key condition we will mainly focus on.
The operators above have a general algebraic property, namely the projectivity: To prove this, we first explicitly expand the deformed R as 2 with d k ≥ 1. In the expansion, when a 2k vanishes, b 1k must also vanish, otherwise a linear recombination of the numerator can further lower d k by one 3 . Now observe that performing the same deformation twice is equivalent to replacing z i by (z i + z i ), as Besides projectivity, a more intricate property is the commutativity: which demands certain condition, as will be investigated shortly. If it holds, again with P i = I − C i one can find that When all C's are chosen to be commutative in expansion (2.4), each term is 'orthogonal' to the others. This orthogonality has a nice meaning: Each term contains non-overlapping pole terms, consequently one can capture all pole terms step by step without checking back and forth. While commutativity may simplify the calculation considerably, it is obviously not necessary for (2.4) to work.
One last digression is when we do practical calculations, it is convenient to use P i + C i = I to switch between P i and C i , depending on which operation is easier. To check the equivalence between two visually different expressions, in appendix A we introduce a simple trick to solve all constraints and get a set of independent kinematic variables. This trick can uniquely fix the form of an expression, no matter by which means it is obtained (it is better to proceed it on computer programs).

Deformation generator algebra
Now we begin to explore the commutativity of C's, which can be decomposed into the commutativity at integrand level and at integral level. The former is encoded in two successive deformations, and the latter is encoded in two successive contour integrals, which will use Laurent expansion in w = 1/z. Before all of these, we need to first study the BCFW deformation generators and their algebra.
Let's define the BCFW deformation generator with respect to i|j] as then the familiar BCFW deformation becomes Although by default the spinorial partial derivatives treat all spinors as independent, we must also impose the momentum conservation constraint on physical amplitudes. Without doubt, this constraint will affect the independence of spinorial partial derivatives, but it will not affect the commutator algebra of D i|j] . Below we will provide a simple argument.
Note that any D i|j] automatically annihilates the sum of all external momenta, i.e., so we claim that momentum conservation is a trivial constraint. To get some intuition, one can consider a spherical surface, for which any rotation generator, say L xy , annihilates the constraint To parameterize one of the spherical symmetries explicitly, we can define an angle θ xy via then while x, y are no longer independent on the sphere, θ xy can be arbitrary, as this degree of freedom moves a point around on a subset of the spherical surface. From this viewpoint, the commutator algebra of L xy , L yz , L zx is obviously unchanged. More profoundly, it is these rotation generators that fully generate the spherical surface. Given a particular point in R 3 , rotation generators move it around to sweep over the entire surface of a fixed distance from the origin.
This picture can be exactly generalized to the case of BCFW deformation generators. We can define an 'angle' for each deformation in a complex spinorial sense, via then θ i|j] parameterizes one of the symmetries that preserve momentum conservation, and hence momentum conservation will not change the commutator algebra of D i|j] at all. Instead, this constraint is fully generated by 2C 2 n = n(n − 1) BCFW deformation generators. Given a particular point in C 4n , namely the complex spinorial space (λ i ,λ i ), BCFW deformation generators move it around to sweep over the entire codimension-4 surface of a fixed sum of external momenta. While physically, this sum is zero.
Since the commutator algebra is unchanged, we are free to treat all spinors as independent to derive the commutation relations. Imagine the 0th step of deformation is i|j], then the 1st step can be one of the four types as named below: k|l] = independent, i|l]/ k|j] = straight descendent, l|i]/ j|k] = skew descendent, j|i] = cross descendent. (2.18) The generators of first two types commute with that of i|j], i.e., For the last two types, 20) where h i is the helicity operator with respect to the i-th particle, for a function covariant under the little group (an amplitude does have this scaling property).
For the skew descendent case, use the Baker-Campbell-Hausdorff formula and due to commutativity of the straight descend, we have Hence skew descendent deformations i|j] and j|k] commute if D i|k] annihilates the amplitude, however, this is a too stringent condition which often trivializes the analysis. Therefore in general, skew descendent deformations do not commute.

Commutativity at integrand and integral levels
By applying the BCFW deformation generators, let's perform a constant extraction on rational function R(λ i ,λ i ), given by where we have changed variable w ≡ 1/z, so the infinity for z is the zero for w. However, the residue at this zero is not a naive one. Recall (2.6) in terms of w, it reads a naive substitution of w = 0 will cause divergence in the third term above. On the other hand, it is clear that after the expansion, the third term actually has no simple pole at w = 0, since there is already one w in the denominator of the integrand. To remove this divergent term, before the contour integration we must Laurent expand R(1/w) around w = 0, i.e., we need to first factor out a divergent factor 1/w d with d ≥ 1, leaving a finite fraction at w = 0, then Taylor expand it around w = 0. A simple example is then we can Taylor expand the function after 1/w around w = 0, and the contour integral will only pick up the constant part in this expression. Similarly, performing two successive constant extractions gives For independent and straight descendent cases, [D 1 , D 0 ] = 0, so the order of deformations is irrelevant, and hence commutativity of these two types holds at integrand level 4 . However, before performing the integral, double Laurent expansion of a fraction is a bit tricky. First, to properly factor out the overall factor in terms of w 0 and w 1 , we need to ensure that in both P (w 0 , w 1 ) and Q i (w 0 , w 1 ) are irreducible polynomials, i.e., there is no common factor w a 0 0 w a 1 1 of all monomials in each of them. Then, we find the expansions of a rational function (namely a fraction) in opposite orders may be different, which arises when one of the factors in the denominator contains no constant term. For example, take g(w 1 , w 0 ) = 1/(w 1 + w 0 ) and expand it around w 0 = 0 (it's impossible to further expand around w 1 = 0), we have and for the reverse order, hence they are clearly different. In general, if one of the Q i 's happens to satisfy Q i (0, 0) = 0, the double expansion depends on the order. Conversely, if all Q i 's obey Q i (0, 0) = 0, the order of double expansion is irrelevant. Since the contour integral simply picks up the constant part in the expansion, commutativity of the denominator expansion is equivalent to commutativity at integral level.
In practice, since it is clear that a detectable pole of merely one of the two successive deformations always contains a constant term after factoring out a proper factor, we should only focus on the overlap of two sets of detectable poles. But it's impractical to trace each term at each step, for figuring out the existence of a constant term.
Combining all these above, we may draw the conclusion: commutativity of C operators conditionally holds for the independent and straight descendent cases. Then these two types are considered as 'good', as they enjoy the orthogonal property (2.11). However, merely the good types are not sufficient to capture all physical pole terms, as will be explained in the end of appendix B. Hence we will not proceed further into commutativity of C operators, but return to seek for the condition of C n C n−1 · · · C 2 C 1 C 0 = 0 such that the (n + 1) steps can fully capture the amplitude.
Nevertheless, this investigation gives a crucial hint for the subsequent analysis: In the expansion, the coefficient of z after a deformation will be become a new pole of the corresponding boundary term, with power one or higher. To be concrete, consider the example below where b is the only source of poles after expansion.

Systematic Algorithm of Finite, Definite Steps
In this section, we propose the systematic process to capture the full amplitude after finite, definite steps of BCFW constant extractions. The condition for its calculation to be correctly completed is simply C n C n−1 · · · C 2 C 1 C 0 = 0. To achieve this, a form of all poles in the final boundary term is given, after a sequence of constant extractions, which is called the 'pole concentration'. This sequence is designed for covering all situations and how to optimize it case by case is set aside temporally. Having the final form of all poles, merely using the information of mass dimension and helicities is sufficient to judge whether the final boundary vanishes.

Pole concentration
Now we use pole concentration to capture all poles regardless of they are physical or spurious by applying the logic of (2.31). Explicitly, each time we perform a BCFW constant extraction on an amplitude, one or more of its physical poles will be filtered out, and consequently each corresponding boundary term will contain one or more pole(s) mutated from the original physical pole(s). For example, consider denominator 12 23 (the numerator is neglected for our purpose), under constant extraction 1|3], 1 12 23 where pole 12 has been replaced by 32 . And crucially, under a next constant extraction, pole 23 2 is either unchanged or replaced by another pole as a whole. This means once two poles are stacked, they are stacked forever. The same logic also works for anti-holomorphic poles. For a multi-particle pole, we first need to turn it into a product of holomorphic and anti-holomorphic poles, with a proper choice of deformation. As an example, under constant extraction 1|4], , (3.4) in either way we are again left with two-particle poles.
In general, one can first turn the multi-particle pole P 2 into i 1 |P |j 1 ], where P includes either i 1 or j 1 . Note that p i 1 or p j 1 in P is already filtered out by i 1 | or |j 1 ]. Next, one can continue to filter out more momenta in P by using i 1 |j 2 ] or i 2 |j 1 ], where P includes j 2 or i 2 , until this pole is split into a product of two-particle poles. Alternatively one can split i 1 |P |j 1 ] directly by using k|j 2 ] or i 2 |l], where P includes j 2 or i 2 but not k or l. Either way turns the pole into a product of one holomorphic and one anti-holomorphic poles in the end. Then for two-particle poles, once they are stacked, they must mutate as a whole afterwards. After finite steps, all poles can be encapsulated in only one holomorphic and one anti-holomorphic poles, with powers larger than one in general.
In appendix B, two example sequences of BCFW constant extractions are given to turn all poles of the final boundary term into a common denominator, given by 5 where i 1 , i 2 , i 3 , i 4 are four different arbitrary particle labels. The remaining factor is a rational function which is dimensionless and helicity-neutral, see (3.9) for example. Note that we have not reduced the denominator against the numerator. At the first glance, the reason to reach this final denominator is that one can use one more deformation, say i 1 |i 4 ], to get the maximal large z suppress, since all poles after concentration are vulnerable to it. But in fact, there is a less obvious argument for eliminating the final boundary term without introducing one more step, as will be shown later.
Here, m/m gets contribution from physical holomorphic/anti-holomorphic poles, and both of them get contributions from physical multi-particle poles. In general, m and m need not to be equal, since not all possible poles are physical for a particular amplitude. To see the range of m, m, we will analyze all possible physical poles for various n's. When n = 4, only a half of all two-particle poles can appear in the amplitude, since they are doubly duplicated by momentum conservation. When n = 5, there are only two-particle poles, as three-particle poles are equivalent to them by momentum conservation. When n ≥ 6, multi-particle poles arise. Their particle numbers range from 3 to (n − 3), to avoid duplications of two-particle poles by momentum conservation. To further avoid duplications of themselves by momentum conservation, one can fix the pole momentum by demanding it to always include one pivot particle, and then the number of multi-particle poles is reduced by one half.
According to this counting, the maxima of m, m are which nicely covers the special cases of n = 4 and n = 5.
However, there is one little subtlety in (3.5): For a given amplitude, while m, m can be easily read off by analyzing all of its non-vanishing factorization limits, its final boundary term, in general, contains not only poles where the term in parentheses is the remaining factor in (3.5). The advantage of packing up many pole terms into a dimensionless helicity-neutral factor is that, if we can show this representative factor cannot exist, all terms behind it including those with higher-power poles, must also be forbidden.
One digressive comment is that, so far we have found BCFW deformations to be the only type which admits a feasible pole concentration. A counterexample is, there is no straightforward pole concentration for Risager deformations [19]. We will not further explain the claim here, but it is not hard to confirm it. This is another specialty of BCFW deformations, in addition to that BCFW deformations automatically preserve (or generate) the momentum conservation constraint.

Kinematic mass dimension
To prepare for the later analysis, we need to understand some general information of amplitudes: mass dimension and helicities, with which the applicable range of multi-step BCFW recursion relations can be clarified.
First, for QFTs in 4-dmension, the mass dimension of an n-particle amplitude is (4 − n). We can use the LSZ reduction formula to prove this. Schematically, an n-particle amplitude A is defined via where Φ 1 . . . Φ n is the n-point function, ε and ∆ are the wave-function and kinematic operator for each field Φ. For a bosonic field, the mass dimensions of ε, ∆ and Φ are 0, 2 and 1 respectively, for a fermionic field, the mass dimensions of ε, ∆ and Φ are 1/2, 1 and 3/2 respectively. Hence the mass dimension of is −1. There are n such pieces, plus the momentum conservation delta function, the mass dimension of A is clearly (4 − n).
One special bosonic field is the graviton. By the perturbative definition g µν = η µν + h µν , it should be dimensionless. To treat it as ordinary bosonic fields, we need to redefine it via such that h µν carries mass dimension 1 and κ carries −1. Choosing κ to be √ 8πG, the free field part of Einstein-Hilbert Lagrangian will have an analogous form as that of ordinary bosonic fields. Consequently, κ becomes the coupling constant of gravity.
One can also rediscover κ via the on-shell method. For gravity, three-particle amplitudes including at least one graviton are where h = 0, 1/2, 1, 2, for all 'realistic' theories. No matter which value h takes, κ always carries mass dimension −1, since the mass dimension of three-particle amplitudes is 1.
In general, we can reverse this logic and define the 'kinematic mass dimension' of an n-particle amplitude as where (D c ) i is the mass dimension of the coupling constant for each vertex. From now on, we will focus on the kinematic part of an amplitude, which is constructed recursively.
For Standard Model 6 , all coupling constants are dimensionless so D = 4 − n, which is a non-positive number for various n's. For gravitational interactions D c = −1 and we will show that D = 2.
When D < 0, there is at least one irreducible denominator of the amplitude, which means a pole to be detected by BCFW deformations. Conversely, when D ≥ 0, the amplitude may admit some invulnerable terms to BCFW deformations which include polynomials and 'pseudo polynomials'. The classification of these objects can be found in appendix C.

The master formula
By using mass dimension and helicities, let's derive the master formula for subsequent discussions. After pole concentration, the final boundary term schematically reads 7 where we have temporally taken i 1,2,3,4 = 1, 2, 3, 4. The reason to use un-contracted spinors is that, this is more compact to capture the helicity information, and it can save the Schouten identity manipulations, as one can freely recombine them to get the desired spinorial products. Of course, the cost is that one needs to rule out all those illegitimate combinations. This treatment is similar to the method used in [20]. Now the helicity configuration enforces that where m, m are known for a particular amplitude. Note that there are 2n variables, with only n helicity constraints. We will fully exploit the n remaining degrees of freedom to derive the master formula.
The kinematic mass dimension of (3.15) is where obviously, α and β must be both even to form spinorial products. Also, we have m + m ≥ 1 with m, m ≥ 0, and α, β ≥ 0. For a legitimate final boundary term, D equals to D defined in (3.14). 6 The φ 3 interaction is not included in Standard Model. But since its coupling constant is of mass dimension 1, it is in fact more benign for the subsequent analysis. 7 In general, the final boundary term's numerator is a polynomial but we only focus on one term, as the identical analysis applies to all terms. The remaining factor in (3.5) is dropped from now on, since it is dimensionless and helicity-neutral.
When D = D under all circumstances, correct dimension and helicities cannot be satisfied simultaneously, then the final boundary term is eliminated. One direct way to achieve this inconsistency is to show D min is larger than D. First we need to figure out this minimum by eliminating one degree of freedom for each particle, and there are two variables α and β to be chosen. For i = 5, . . . , n, To manifest the non-negativity of D , our choice is Extending this logic for all particles, yields (3.20) which is the master formula and explicitly, which separates the sum into two parts according to the helicities. The final boundary term (3.15) now reads (p i = |i [i| is a helicity-neutral momentum with additional mass dimension 1) 22) where / means one of two candidate expressions is chosen to manifest the non-negativity of D , as this choice also manifests the 'extra neutral momenta', in addition to the 'net spinors' that carry the helicity information. While the latter content is mandatory, the former is optional since it is brought in to fill the extra capacity of mass dimension. There is no unique choice of picking these extra α's and β's as long as the total dimension is correct.
to maximize T 1234 , one must take h 1 , h 2 to be two minimal helicities and h 3 , h 4 to be two maximal ones 8 . On the other hand, even if one chooses h 1 , h 2 , h 3 , h 4 arbitrarily among all h i 's, D min is no less than zero. To see this, neglecting other non-negative parts in D , let's focus on the quantity by its definition T 1234 has a simple geometrical meaning: It is the sum of four distances from h 1 , h 2 to line h = m/2, and from h 3 , h 4 to h = −m/2. It's easy to find that its minimum is (m + m), when four points are on one horizontal line and the line is within the region between h = m/2 and h = −m/2. When this uniform line moves outside the region, T 1234 increases by 4 × (distance above or below). When this line is not uniform, one can always rearrange the four points to render T 1234 increase. Therefore T 1234 is always non-negative, so is D min . Also note that which all take integer values. According to (3.23), if |h| is fractional, D must be fractional.

Inconsistency elimination
We now use (3.20) to show that, all massless tree amplitudes except those admit (pseudo) polynomials of given mass dimension and helicities, shall be fully determined by multi-step BCFW recursion relations. In the following analysis, we pretend not to know any knowledge of QFT in the Lagrangian paradigm, except mass dimension and helicities. These are the only data needed to construct three-particle amplitudes, and recursions extend them to all higher-point cases.
First note that, after pole concentration, there is no need to find any further deformation to kill the final boundary term, because another choice will not change its pole form so one can always rearrange the entire series of pole concentration to reach the desired relabeling. Hence it must be the last step if chosen properly, and the direct way is to use the 'inconsistency elimination'.
Within its own framework, if inconsistency elimination can exclude the final boundary term, the new algorithm will be a powerful and completely independent approach to calculate corresponding amplitudes. Otherwise, terms that survive it need to be identified, similarly as (pseudo) polynomials. In fact, we do discover a new type of object called the 'saturated fraction' by this way.
From the master formula (3.20), it is already known that D min ≥ 0. Therefore if D < 0, D < D min always holds. This tells us the nontrivial cases are of D ≥ 0. When D ≥ D min , we have the inconsistency criteria below to eliminate the final boundary term: (1) Fractional Dimension (FD): If D = fractional. This arises when |h| = fractional, but Lorentz invariance demands the dimension of (3.15) to be an integer, which implies that fermions must appear in pairs to be consistent.
(2) Pair Mismatch (PM): If α = odd and β = odd, assume that FD is excluded already. In this case, (3.15) cannot be written as a fraction in terms of Lorentz invariant spinorial products, even though the dimension of (3.15) is an integer.
(3) Spinor Excess (SE): If there exists an i, such that α i > j =i α j or β i > j =i β j . In this case, spinorial contraction will force (3.15) to vanish, even though α = even and β = even.
Altogether, there are four layers of inconsistency criteria: (0) D < D min .
(2) If D ≥ D min , and FD is excluded, consider PM.
(3) If D ≥ D min , and both FD and PM are excluded, consider SE.
It's obvious that these inconsistency criteria only mention general properties of field theories. Hence inconsistency elimination is theory independent in general, while in practice, knowing some theory dependent properties would help simplify discussions case by case. If the final boundary term can survive all four criteria, it must admit 'saturated fractions' (SF). Note that we have already set aside polynomials and pseudo polynomials, because they can be found without using the master formula (3.20). Altogether, there are three types of objects invulnerable to BCFW deformations: (1) Polynomials, such as 12 .
(2) Pseudo polynomials of n = 4, such as [34]/ 12 . When 12 → 0, we must also have [34] → 0 since it is a BCFW deformation that is being used. Then the ratio [34]/ 12 is finite like a polynomial. Among these three objects, a polynomial is completely inert to BCFW constant extractions, in fact it is invulnerable to any type of deformation in on-shell methods. A pseudo polynomial is also completely inert to BCFW constant extractions, while this requires momentum conservation. A saturated fraction is form-inert to BCFW constant extractions 9 , but with its particle labels rearranged. The last two objects are vulnerable to other types of deformations, such as Risager deformations [19]. Detailed exploration of all these three types is presented in appendix C.
So far, we have witnessed how the systematic process of multi-step BCFW recursion relations can be arranged for solving a particular amplitude. In summary, there are four universal steps: (1) Analyze all non-vanishing factorization limits to determine the amplitude's common denominator, which is a product of all physical poles. This stage can be done almost purely diagrammatically.
(2) Figure out the amplitude's kinematic mass dimension, then combine this with its helicity configuration to determine all possible (pseudo) polynomials. If none of them arises, we assume the amplitude can be fully determined and proceed to the next step.
(3) Choose four particles of two maximal and two minimal helicities to determine the denominator's form of the final boundary term, and arrange a sequence of BCFW constant extractions to proceed pole concentration. The sequence must be able to capture all physical poles, such that each of them contributes to the final denominator via powers m, m. With this ensured, the sequence should be as short as possible. This optimization is an important future problem.
(4) Use all four inconsistency criteria layer by layer to eliminate the final boundary term. If it fails, identify all possible saturated fractions. Then discuss whether these saturated fractions are legitimate, if not, clarify the argument to rule them out as mentioned in the end of appendix C. This delicate treatment to remove all dependence on spurious poles is another valuable future problem.

Applications in Standard Model plus Gravity
Having a general guide of multi-step BCFW recursion relations, naturally we would like to see how it applies to specific theories. As familiar examples, let's first consider the realistic theories, i.e., Standard Model plus gravity 10 .
For reader's convenience, we rewrite the master formula below recall that one should take h 1 , h 2 to be two minimal helicities and h 3 , h 4 to be two maximal ones.

Two separated sectors
Let's first consider Standard Model and gravity separately. From the previous section, it is known that for Standard Model, D = 4 − n. On the other hand, since D min is no less than zero, when D ≤ −1, the final 9 Here we mean a pure saturated fraction. A mixed saturated fraction is a pure saturated fraction times a polynomial. Its transform under a constant extraction will be demonstrated in appendix C. 10 All other theories can be analyzed analogously. However, as the amplitude's kinematic mass dimension goes up, more types of polynomials, pseudo polynomials and saturated fractions may arise, and one needs to identify them carefully.
boundary term must be eliminated. This directly tells that all Standard Model amplitudes of n ≥ 5 are solvable, leaving amplitudes of n = 4 corresponding to D = 0 to be further analyzed. As will be shown later, the n = 4 case admits (pseudo) polynomials 1 and ( 34 / [12]) ±1 . For pure gravity, assume one of the Feynman diagrams of an n-particle amplitude contains v m m-leg vertices and p internal propagators, it is clear that and each m-leg vertex brings in (m − 2) κ's, hence from (3.14) we have where x, y are unspecified. And the all-but-one-minus case gives D min = 4 > 2 already. However, from three-particle amplitudes (3.13) (with h = 2) and non-vanishing factorization limits, it is not hard to show that any amplitude's helicity configuration of pure gravity must be between MHV and anti-MHV. For the MHV configuration, (4.1) gives D min = 8 > 2. Therefore pure gravity is in fact completely solvable.
In general, for Standard Model plus gravity we have D ≤ 2. Note that an amplitude which contains gravitational vertices only always obeys D = 2, regardless of how many or what kinds of external legs it owns. This specialty implies that one can arbitrarily attach more particles to a known amplitude without changing its mass dimension, via gravitational interaction.

Simplified diagrammatic rules
To simplify the general discussion of Standard Model plus gravity, we introduce the diagrammatic rules called 'stretch and shrink'. The first example is the gauge interaction, as shown in Figure 1.
Fixing four external gauge bosons, this 4-point vertex can be stretched into two connected 3-point vertices, without changing the vertex's mass dimension. This tricky equivalence holds at the level of mass dimension and helicities, which are the only information required for inconsistency elimination. In other words, we have chosen a representative sub-diagram to encode the same information of mass dimension and helicities, and reduce the types of equivalent sub-diagrams in the analysis. Following this logic, all higher-point vertices in Standard Model plus gravity can be stretched into a number of connected 3-point vertices, except the special φ 4 vertex. This simplified rule is notably advantageous in gravitational interaction.
As shown in Figure 2, gravitons can be shifted from any place to any place in a sub-diagram, without changing its mass dimension. Physically, this is because gravity is universal, gravitons can emit from any part of a system. Mathematically, this is because an m-point gravitational vertex carries coupling constant κ m−2 , where κ carries mass dimension −1. This vertex can contain gravitons only, or it can be attached by Standard Model lines. Therefore, the m-point vertex can be stretched into (m − 2) connected 3-point vertices, with the exception of φ 4 vertex.
For convenience we define the 'gravitational component', as shown in Figure 3. All vertices within this component are gravitational, while its external legs can be either Standard Model particles or gravitons, or both. There is one special graviton which will attach to another component. A trivial case is that there is no vertex at all, so this special graviton becomes the only component.
Gravitational components also obey the simplified rules, and for convenience they are usually shrunk into one component, as shown in Figure 4. This pack-up can reduce many sub-diagrams of gravitational components to one sub-diagram. It's free to attach or detach a gravitational component, since it will not change the mass dimension of the other component.
Summarizing the simplified diagrammatic rules, we are now left with the representative vertices only, as shown in Figure 5. In Standard Model plus gravity, the nontrivial cases are D = 0, 1, 2. By using the representative vertices in Figure 5, the following discussion is considerably shorten. For D min = 0, polynomials and pseudo polynomials arise when all helicities are the same. Then the last three diagrams in Figure 6 are excluded, since a three-point Standard Model vertex can never have three same helicities. The second diagram is also excluded since a single graviton has helicity ±2.
Therefore, (pseudo) polynomials come from the first and third diagrams, as listed in Figure 7. The first three diagrams in Figure  For the right diagram in Figure 8, when vertices of type (a), (b) and (c) are attached by a gravitational component, to consider the most conservative case, this component only contains external scalars, since higher-spin particles must appear in pairs of opposite helicities, which will not decrease D min . The helicities are (±1, +1, −1, 0, 0, . . .), (±1, +1/2, −1/2, 0, 0, . . .) and (±1, 0, 0, 0, 0, . . .) respectively, and . . . denotes more scalars besides the minimal five. Applying (4.1), corresponding D min 's are 3, 2 and 1, which excludes first two cases. However, the third case is also excluded even if its D min is allowed. The argument is that no spinorial product can be formed by only |1 2 or |1] 2 , which is known as the Spinor Excess of inconsistency elimination. The only polynomial comes from the vertex of type (d), as given in Figure 9. This polynomial is 12 or [12]. D = 2 case: The last case is D = 2. The possible amplitude is given in Figure 10, and corresponding polynomials are listed in Figure 11. The first one is P 2 xy , where x, y are two unspecified scalars. The second one is 1x [x2], with one pair of fermions of opposite helicities. The third one is 12 [34], with two pairs of fermions of opposite helicities. One may consider a fourth one, with one pair of gauge bosons of opposite helicities, which is allowed since its D min is 2. But this case is also excluded, as no spinorial product can be formed by only |1 2 |2] 2 or |1] 2 |2 2 .
Note that these D = 2 polynomials can be of either n = 4 or n ≥ 5 (for which all unspecified particles are scalars). For n = 4, there are dimensionless pseudo polynomials of the form ([34]/ 12 ) x , which can be an additional factor of the polynomials above. This factor will lead to a global shift of all four helicities. But incidentally, there is no extra legitimate pseudo polynomial after adding it. 11 Here ±1 and ±2 are independent, they do not necessarily have the same sign. Last but not the least, we need to check the analysis above has covered all possible diagrams, by using a compact formula given by where D is the kinematic mass dimension of a Standard Model 'skeleton', as shown in Figure 12. This The proof of this formula is simple. For S connected components, there are (S − 1) internal gravitons. Each internal graviton has two attached points, which bring in two κ's. Hence from (3.14), we have which is identical to (4.6) after a trivial rearrangement. Having this skeleton, to build a general amplitude, more gravitational components (either nontrivial components or single gravitons) can be attached to it. By applying the simplified diagrammatic rules, they can be packed into one single gravitational component. Now let's consider D = 0, we have S = 1, s 1 = 4 and S = 2, s 1 = s 2 = 3. These two cases correspond to the first and fourth diagrams in Figure 6. They can be attached by gravitational components, which gives the rest four diagrams. For D = 1, we only have S = 1, s 1 = 3. This diagram cannot stand alone, since there is no on-shell massless 3-particle amplitude. Hence it must be accompanied by gravitational components, which gives the two diagrams in Figure 8. For D = 2, the amplitude contains only gravitational vertices, which corresponds to the diagram in Figure 10. Therefore all possible diagrams have been covered.

Discussions
In this work, we are mainly concerned with the workability of the new algorithm known as multi-step BCFW recursion relations. The key techniques of this approach are pole concentration and inconsistency elimination. Its applicable range is also clarified and we find three types of objects invulnerable to BCFW deformations: polynomials, pseudo polynomials and saturated fractions. While the last two objects can be determined by other types of deformations, how to deal with polynomials is probably beyond usual on-shell methods, and it may lead to important generalizations of the present approach. Moreover, when saturated fractions arise, we need to discuss whether they are legitimate, if not, how to find an argument to rule them out is another valuable topic. Again, we would to emphasize this systematic algorithm relies on general properties of field theories only, such as Lorentz invariance, locality and unitarity. The major information we have used are mass dimension and helicities.
Ensuring its workability, we try to further improve the efficiency of multi-step BCFW recursion relations by taking two sophisticated aspects into account, as listed below: (a) Knowing the (final or intermediate) boundary term's schematic form (in terms of un-contracted spinors), it is also very natural to seek for a deformation which renders the boundary term vanish under the large z limit, other than employing inconsistency elimination. In practice, one can consider both ways at each step to shorten the sequence of pole concentration. More profoundly, inconsistency elimination is only an argument afterwards, as any boundary term that vanishes must be killed by a good deformation with respect to that step.
(b) In practice, it is often evident that there is no need to reach the final denominator (3.5). Merely a particular intermediate form is sufficient to complete the calculation correctly. This is due to the fact that pole concentration is for eliminating as many neutral momenta as possible in the denominator, and hence increasing the spinors in the numerator, in order to keep the helicities fixed.
Let's illustrate these two aspects through three simple examples. The first one is the MHV amplitude A(1 − , 2 + , 3 − , 4 + , . . . , n + ), where 1 − and 3 − are non-adjacent. Assuming all lower-point MHV amplitudes are known, non-vanishing factorization limits give the common denominator 1 12 23 . . . n − 1, n n1 , on the other hand, the kinematic dimension of this amplitude is (4 − n), as the gauge coupling constant is dimensionless. Then correct helicities require the numerator to be |1 4 |3 4 schematically, which uniquely fixes the amplitude as 13 4 12 23 . . . n − 1, n n1 , (5.2) from this example, we see the schematic form is a simple but powerful tool.
The second example is amplitude A(1 −1 , 2 +1 , 3 −1 , 4 +1 , 5 −2 ) in Einstein-Maxwell theory. Non-vanishing factorization limits give all physical poles as [12][32] [14][34] [15] are good deformations because each of them induces z 3 in the denominator, while the numerator can at most contain z 2 to avoid Spinor Excess, i.e., 4 |• in the numerator can at most contain two identical spinors to form non-vanishing spinorial products, which restricts 4 [•| in the same way. For this case, to find a deformation with maximal large z suppress is clearly more straightforward than using inconsistency elimination after a number of deformations.
The third example is even more interesting. Consider amplitude A(1 −1 , 2 +1 , 3 −1 , 4 +1 , 5 −1 , 6 +1 ) again in Einstein-Maxwell theory, factorization limits, mass dimension and helicities together fix its schematic form to be (5.8) then there is no need to proceed further, because in the numerator Spinor Excess already arises, as 18 |• can never saturate 1| 20 to form non-vanishing spinorial products. By this way, two steps can already get the correct answer, while a blind pole concentration in general requires 4(6 − 3) = 12 steps. Therefore, it is not always necessary to reach the final denominator, when eliminating part of neutral momenta in the denominator enforces the numerator to contain sufficient identical spinors for triggering Spinor Excess.
But this is not the end of the story. When proceeding the calculation while it is just shown that C 5|1] C 3|1] = 0, incidentally we also find C 5|1] P 3|1] = 0. This means 5|1] is a good deformation and hence one step is enough. Since particles 1 −1 , 3 −1 and 5 −1 are symmetric in the helicity configuration, 3|1] is also a good deformation. In general, there is a last good deformation corollary: After the n-th step, when I = (known terms) n + C n · · · C 0 is reached, we can further expand it by an (n + 1)-th step as I = P n+1 + C n+1 (known terms) n + C n+1 C n · · · C 0 , (5.10) assume the (n + 1)-th step is the last step for which C n+1 C n · · · C 0 = 0. And if C n+1 (known terms) n = 0, (5.11) then the (n + 1)-th step is a good deformation. This corollary is powerful in practical calculations, since unnecessary steps can be saved if we incidentally encounter the condition above. Back to the mainline, these two aspects (a) and (b) will be demonstrated more systematically, with more examples in our future work, with a possible joint use of the last good deformation corollary. The major goal is to improve the efficiency provided the workability is ensured. The exit of this maze is now located, and how to shorten the correct route is a complicated yet fascinating problem. Finally, we would like to highlight the power of simple analysis by mass dimension and helicities. These cheap information possibly lie in the core of the future study of efficiency.
We can pick |1 , |2 as two reference spinors, and for i, j = 1, 2 all independent Schouten identities can be solved via There are C 2 n ij 's and C 2 n−2 Schouten identities, so there are C 2 n − C 2 n−2 = 2n − 3 independent ij 's. We can do the same thing for anti-holomorphic spinorial products [ij]'s, and get (4n − 6) independent kinematic variables as below: 12 [

B. Examples of Pole Concentration
We have claimed that all poles of the final boundary term can be turned into a common denominator Here, two systematic sequences are presented to achieve it. where III and IV and can be copied from I and II, by reducing n to (n − 2) and swapping the holomorphic and anti-holomorphic deformed spinors. This sequence has 4(n − 3) steps in total. To see how this fully works, we assume that all physical poles occur (restrictions such as color order, must be disregarded). For convenience, let's define sets H = 1, . . . , n and A = [1, . . . , n] to denote all holomorphic and anti-holomorphic poles of n particles respectively.

B.1 First systematic sequence
Also, the set of all multi-particle poles is denoted by M . To fit the analysis of pole concentration, we will classify all multi-particle poles according to which particle is absent, in a default order. Concretely, they are categorized as (p 2 + X 23 ) 2 (P 23 + X 24 ) 2 (P 24 + X 25 ) 2 . . . (P 2,n−4 + X 2,n−3 ) 2 (P 2,n−3 + X 2,n−2 ) 2 (P 2,n−2 + X 2,n−1 ) 2 (B.2) where P ij = p i + . . . + p j , X ij is a sum of external momenta without those from p i , . . . , p j and i, . . . , j is the default order. As the pole momentum includes at least three particles, X 23 must at least include two particles, X 24 must at least include one and X 25 can be empty. Similarly, X 2,n−3 must at most include two particles, X 2,n−2 must at most include one and X 2,n−1 must be empty. Analogous restriction works for all X 2i 's in between. In this list, p 2 is the pivot momentum which is always included, and p 3 , . . . , p n−1 becomes the absent momentum one by one. Now for 2|3] in series I, the affected two-particle poles are here when X 23 includes two particles, and one is 4 which is filtered out by |4 , we are left with one particle x 24 , where x ij denotes one external momentum except that from p i , . . . , p j . A nice fact is that the split poles [2x 24 ] and x 24 4 are not the two-particle poles already filtered out by 2|3] and 3|4], in other words, the next steps will take care of them, so there is no need to look back. again the split two-particle poles will be taken care of by the next steps. Note the descendent poles from (P 24 + X 25 ) 2 , . . . , (P 2,n−3 + X 2,n−2 ) 2 , P 2 2,n−2 will no longer produce split poles. After step n − 2|n − 1], all descendent poles from M except split poles are After series II, it is easy to check that Note that H is in fact completely inert to series III and IV as only A is manipulated. Therefore we manage to turn all poles, regardless of two or more particles, physical of spurious, into a common denominator where X ni is a sum of external momenta without those from p n , p 1 , . . . , p i . Note that when i = n − 2, the spurious pole [n − 2|X n,n−2 |n is split into [n − 2, n − 1] n − 1, n . After series II, we are left with In general, not all physical poles can be detected by independent and straight descendent deformations alone. To calculate an n-particle amplitude by using only these two types of deformations, one can assign a labels for i| and (n − a) labels for |j] in a deformation series i|j]. Then two-particle poles [i 1 i 2 ] with i 1 , i 2 ∈ I a and j 1 j 2 with j 1 , j 2 ∈ I n−a cannot be detected, where I a and I n−a denote the sets of i| and |j] respectively.

C. (Pseudo) Polynomials and Saturated Fractions
This part gives the classification of all three types of objects that cannot be determined by multi-step BCFW recursion relations. They include (pseudo) polynomials and saturated fractions.

C.1 Polynomials and pseudo polynomials
Now, we list all (pseudo) polynomials that satisfy certain dimension and helicities, up to D = 2. This list can be similarly extended for D ≥ 3. One also needs to separate the cases for which one or two pair(s) of particle labels in the spinorial products are identical. Hence the helicity configuration can be where x = ±1, ±2, ±3, ±4, as the spin is restricted within 2.
Since the dimensionless pseudo polynomial behaves like a numerical factor, polynomials of n ≥ 5 can be multiplied by these factors to generate all possible pseudo polynomials. Hence, for n = 4 we have the following helicity configurations, corresponding to all (pseudo) polynomials: D = 0 : (y, y, y, y) − 2 ≤ y ≤ 2 D = 1 : where y = 0, x/2. Since (pseudo) polynomials of n = 4 are dimensionally equal, up to a global shift of all four helicities, for simplicity one can first select a set of representatives to analyze. As one example, the representative helicity configuration can be then y must be shifted around to cover all possible cases in (C.7).

C.2 Saturated fractions
The SF is an irreducible fraction, but impossible to be killed by BCFW constant extractions. A minimally z-power inducing constant extraction will only change its particle labels, but not its form. For example, note that the first term above is also a mixed SF again with relabeling. In general, under a non-minimally z-power inducing constant extraction, a mixed SF will transform into more than one term. These terms include a special one, which is related to the original SF by relabeling.
To investigate SF's more systematically, let's analyze one special category: When 0 ≤ h 1 , h 2 < m/2 and 0 ≤ h 3 , h 4 , the dependence on m, m dissolves in (3.20), then (C. 14) and the corresponding final boundary term schematically reads 1st type: Let's consider a first type of amplitudes 12 , of helicity configuration (h, h, h, h, h, . . .) with 0 ≤ h ≤ 2. We will explain why nontrivial SF's only exist for n ≥ 5 in the end. At this point, (C.14) becomes D = (n − 4)h + β 1 + β 2 + n i=3 α i , (C. 16) and the final boundary term schematically reads (C. 19) 12 All amplitudes in this part are not necessarily physical, as the investigation is purely of mathematical interest. For this type, we find that: When D = 0, there is no consistent choice. When D = 1, n = 5, h = 1/2 is SF, h = 0 is FD. When D = 1, n = 6, 7, 8, . . . , h = 0 is FD. When D = 2, n = 5, h = 3/2 is SF, h = 1 is FD, h = 1/2 is PM, h = 0 is FD. When D = 2, n = 6, h = 3/2, 1, 1/2, 0 is FD. When D = 2, n = 7, h = 1/2 is SF, h = 0 is FD. When D = 2, n = 8, 9, 10, . . . , h = 0 is FD. SF's above are  We will not continue to present a fifth type, as these toy examples have provided enough intuition for the generation of SF's. Naturally SF's are highly un-physical, but their existence in some artificial (effective) theories cannot be excluded. Sometimes even if an SF is admitted, we can use other arguments to exclude it, such as its denominator contains a spurious pole. Consider an SF whose denominator is 12 2 , or 12 but the amplitude has vanishing factorization limit under 12 → 0, then one needs to delicately remove all dependence on the spurious pole 12 2 or 12 of the known terms in (2.4), if it appears. For n = 4, an SF is equivalent to a pseudo polynomial. Consider under 1|2], the LHS is a polynomial while the RHS is an SF. Plus momentum conservation, for the RHS both its form and particle labels are preserved. For this reason, SF's of n = 4 are trivially inert to BCFW constant extractions, therefore they are in fact pseudo polynomials.