SplitHappens: Tree-Level Amplitude Splitting

• SplitHappens is a universal tree-level factorization pattern for particle and string scattering amplitudes, characterized by splitting an amplitude into two off-shell currents with an extra leg count.
• It leverages 2-split kinematics by imposing specific vanishing Mandelstam invariants and polarization constraints, leading to a nontraditional, multiplicative splitting away from physical poles.
• The approach unifies diverse theories—from CHY formulations to string theory—offering insights into soft theorems, integrand factorization, and novel worldsheet dynamics.

Searching arXiv for the relevant SplitHappens scattering-amplitude papers and closely related work. SplitHappens is the name given to a universal tree-level factorization pattern for particle and string scattering amplitudes in which, on a special kinematic subspace, an $n$-point amplitude does not factorize on a physical pole in the ordinary sense but instead splits directly into the product of two lower-point off-shell currents (Cao et al., 2024). The defining feature is that the two daughter objects carry $n+3$ external legs in total, rather than $n+2$ as in standard pole factorization. The phenomenon is formulated in terms of a distinguished bipartition of external data, special vanishing conditions on Mandelstam invariants, and, for spinning states, additional polarization orthogonality constraints. In this sense, SplitHappens organizes a broad class of tree amplitudes in open and closed string theory, CHY formulations, and multiple field-theory limits into a single worldsheet- and scattering-equation-based splitting structure (Cao et al., 2024).

1. Definition and kinematic setting

The basic setup chooses three distinguished labels $i,j,k$, partitions the remaining external legs into two nonempty sets $A$ and $B$, and imposes

$$A\cup B=\{1,\dots,n\}\setminus\{i,j,k\},\qquad A,B\neq\emptyset,$$

together with

$s_{ab}=0,\qquad \forall\, a\in A,\; b\in B.$

This locus is called 2-split kinematics (Cao et al., 2024). In a canonical choice one may take $k=n$, $i<j-1$, and

$n+3$0

(Cao et al., 2024).

On this locus, the amplitude splits into a left current and a right current with one off-shell leg each. The corresponding off-shell momenta are

$n+3$1

and the schematic factorization reads

$n+3$2

The two currents have $n+3$3 and $n+3$4 legs respectively, so the total number of external legs is $n+3$5 (Cao et al., 2024).

This is not ordinary pole factorization. A standard tree-level factorization channel produces two lower-point amplitudes sharing one internal on-shell state. SplitHappens instead gives a kinematic decomposition into two off-shell currents. The distinction is central: the split occurs away from a physical pole, even though the result is still multiplicative (Cao et al., 2024).

2. Worldsheet and CHY origin

The conceptual core of SplitHappens is that the splitting already occurs at the level of universal worldsheet building blocks. For strings, the scattering potential

$n+3$6

decomposes on the 2-split locus as

$n+3$7

Equivalently, in the formulation of (Cao et al., 2024),

$n+3$8

with

$n+3$9

This expresses the decomposition of the worldsheet dynamics into two sectors, each associated with a current carrying an off-shell leg (Cao et al., 2024, Cao et al., 2024).

On the CHY side, the same kinematics causes the scattering equations and CHY measure to split: $n+2$0 or, in open- and closed-string notation,

$n+2$1

(Cao et al., 2024, Cao et al., 2024).

A plausible implication is that SplitHappens is best viewed not as an isolated amplitude identity but as a universal localization phenomenon of the underlying worldsheet or CHY integral. This is reinforced by the all-order geometric reinterpretation of split factorizations in terms of joining smaller surfaces in the curve-integral formulation for $n+2$2 theory, where the split arises naturally from binary geometry and surface gluing (Arkani-Hamed et al., 2024).

3. Factorization of integrands and correlators

A systematic feature of the subject is that the principal integrands entering tree-level string and CHY formulas split individually on 2-split kinematics (Cao et al., 2024). The paper analyzes, in particular, the following objects:

Ingredient	Role in amplitudes	Split status
$n+2$3	Parke–Taylor factor	Splits universally
$n+2$4	Superstring correlator	Split follows from $n+2$5
$n+2$6	Bosonic string correlator	Splits under polarization constraints
$n+2$7	CHY half-integrands	Split systematically analyzed

For compatible orderings, the Parke–Taylor factor factorizes as

$n+2$8

and this factorization is kinematics-independent (Cao et al., 2024, Cao et al., 2024). Because open-string disk integrals $n+2$9 and closed-string integrals $i,j,k$0 are built directly from the Parke–Taylor structure, they themselves split into two stringy currents: $i,j,k$1 with an analogous formula for the $i,j,k$2-integrals (Cao et al., 2024).

For scalar and Goldstone-type theories, the matrix $i,j,k$3 becomes block diagonal in the relevant minors. A representative formula is

$i,j,k$4

which leads to a corresponding split for $i,j,k$5 (Cao et al., 2024). Parallel statements hold for $i,j,k$6, while $i,j,k$7 requires a classification of compatible perfect matchings into three cases before yielding mixed-current factorization (Cao et al., 2024).

For spin-1 and spin-2 theories, the CHY matrix

$i,j,k$8

splits under the appropriate polarization conditions, and the reduced Pfaffian gives the corresponding gluon and graviton splitting formulae. The superstring correlator is generated by differential operators acting on $i,j,k$9, so its splitting follows from the splitting of $A$0. The bosonic string correlator $A$1, built from $A$2 and $A$3, also splits once the cross-polarization terms vanish (Cao et al., 2024).

4. Theories covered and current structure

From the ingredient-level factorizations, SplitHappens yields amplitude-level splitting for a wide web of theories. The explicitly listed examples include bi-adjoint $A$4, $A$5 and $A$6 string integrals, the non-linear sigma model, Dirac–Born–Infeld, the special Galileon, Yang–Mills-scalar theory, Einstein–Maxwell-scalar theory, Yang–Mills, Einstein gravity, and bosonic and superstring extensions (Cao et al., 2024, Cao et al., 2024).

For scalar and pion-like theories, the left current often contains a mixed state in which the special legs $A$7 become bi-adjoint scalar legs (Cao et al., 2024). A representative formula is

$A$8

For Yang–Mills,

$A$9

while for gravity two variants are given: $B$0 (Cao et al., 2024).

The earlier paper emphasizes the same pattern in terms of stringy currents. In the low-energy limit of the deformed unified stringy amplitudes $B$1, the split interpolates between NLSM amplitudes for generic $B$2 and YMS amplitudes for $B$3, so that the same 2-split can produce an NLSM current times a mixed current with three $B$4's, or a YMS current times another mixed current, depending on the parity structure (Cao et al., 2024).

This suggests that SplitHappens is not tied to one Lagrangian realization but to a common factorization architecture shared by CHY- and worldsheet-representable tree theories.

5. Spinning amplitudes and polarization constraints

For amplitudes with spin, the Mandelstam constraints alone are insufficient. One must also impose vanishing conditions on Lorentz products involving polarizations. For gluons, the required conditions are

$B$5

with $B$6, $B$7, and $B$8 in the notation of (Cao et al., 2024), or $B$9 in the canonical presentation of (Cao et al., 2024). For gravitons, analogous conditions are imposed separately on $$A\cup B=\{1,\dots,n\}\setminus\{i,j,k\},\qquad A,B\neq\emptyset,$$0 and $$A\cup B=\{1,\dots,n\}\setminus\{i,j,k\},\qquad A,B\neq\emptyset,$$1 (Cao et al., 2024, Cao et al., 2024).

Under these conditions, gluon amplitudes in bosonic string and superstring theory split into a mixed current and a pure gluon current: $$A\cup B=\{1,\dots,n\}\setminus\{i,j,k\},\qquad A,B\neq\emptyset,$$2 (Cao et al., 2024). For closed strings and gravitons, imposing the conditions separately on left- and right-moving polarizations yields either a pure graviton current times a pure graviton current or an Einstein–Yang–Mills-type split into two mixed currents (Cao et al., 2024).

A common misconception is to regard these polarization conditions as ancillary technical restrictions. Within the formalism, they are structural: they decouple the polarization data of the two sectors and are precisely what allows the $$A\cup B=\{1,\dots,n\}\setminus\{i,j,k\},\qquad A,B\neq\emptyset,$$3, superstring correlator, and bosonic string correlator to split cleanly (Cao et al., 2024, Cao et al., 2024).

6. Relation to smooth splitting, hidden zeros, and soft theorems

SplitHappens is presented as a unifying mechanism for several previously observed factorization phenomena. The earlier “smooth splitting” or 3-split of scalar amplitudes arises by iterating the 2-split on one of the daughter currents: $$A\cup B=\{1,\dots,n\}\setminus\{i,j,k\},\qquad A,B\neq\emptyset,$$4 so the 3-split is a refinement of the more basic 2-split structure (Cao et al., 2024). This connects directly to the earlier analysis of “smooth splitting” and “semi-locality,” where scalar amplitudes on special split kinematics become products of exactly three amputated Berends–Giele currents, with pairwise overlap in one external leg and exactly $$A\cup B=\{1,\dots,n\}\setminus\{i,j,k\},\qquad A,B\neq\emptyset,$$5 nontrivial 3-splits (Cachazo et al., 2021).

Likewise, the “factorization near zeros” of Arkani-Hamed et al. is recovered by imposing extra vanishing conditions so that one of the currents collapses to a 4-point object. In the language of SplitHappens, the special “skinny” case $$A\cup B=\{1,\dots,n\}\setminus\{i,j,k\},\qquad A,B\neq\emptyset,$$6 is the seed for these zeros (Cao et al., 2024, Cao et al., 2024).

The skinny case also yields soft theorems. When $$A\cup B=\{1,\dots,n\}\setminus\{i,j,k\},\qquad A,B\neq\emptyset,$$7, one current becomes a universal 4-point object whose soft behavior reproduces standard and enhanced soft limits. The paper states that this gives the Weinberg soft gluon theorem and soft graviton theorem for Yang–Mills and gravity and their string extensions, as well as Adler-zero behavior for NLSM, DBI, and sGal (Cao et al., 2024). In the field-theory limit, the mixed 4-point gluon–scalar current becomes the gluon soft factor, while the Goldstone 4-point current scales as

$$A\cup B=\{1,\dots,n\}\setminus\{i,j,k\},\qquad A,B\neq\emptyset,$$8

for NLSM, DBI, and sGal respectively, reproducing the enhanced Adler zeros (Cao et al., 2024). The earlier account gives explicit 4-point factors and writes the gluon and graviton soft factors in standard amplitude form (Cao et al., 2024).

A plausible implication is that the subject reframes soft limits and hidden-zero factorizations as codimension-enhanced instances of one underlying split mechanism rather than separate phenomena.

7. Extensions, interpretation, and scope

The 2024 follow-up develops a broader geometric interpretation of split factorizations from the binary geometry of the curve-integral formulation for $$A\cup B=\{1,\dots,n\}\setminus\{i,j,k\},\qquad A,B\neq\emptyset,$$9 theory and extends the logic to all orders in the topological expansion (Arkani-Hamed et al., 2024). In that framework, splits arise from joining smaller surfaces into larger ones without introducing new internal edges, and the induced split kinematics makes higher amplitudes factor into lower ones. The same construction yields all-order loop-integrand splits and loop-integrated multi-soft limits, including pion and gluon cases related by a $s_{ab}=0,\qquad \forall\, a\in A,\; b\in B.$0-shift (Arkani-Hamed et al., 2024). This suggests that the tree-level SplitHappens phenomenon is the lowest-order manifestation of a more general geometric factorization principle.

Within its original domain, however, SplitHappens remains specifically a tree-level universal splitting behavior for particle and string amplitudes on 2-split kinematics (Cao et al., 2024). It concerns off-shell currents rather than ordinary on-shell factorization channels, and it is formulated most naturally through the splitting of the scattering potential, CHY measure, and standard CHY or string integrands. Its significance lies in unifying amplitude zeros, smooth splittings, and soft behavior under a single worldsheet/CHY mechanism (Cao et al., 2024, Cao et al., 2024).