Cutkosky Cutting Rules in Feynman Diagrams

• Cutkosky cutting rules are relations for Feynman amplitudes that compute diagram discontinuities by replacing selected propagators with on-shell delta functions.
• They link analytic structures—such as Landau singularities, pinch geometry, and monodromy—to combinatorial operations like edge shrinking and on-shell cuts.
• The method underpins dispersion relations and perturbative unitarity, with extensions to cosmological, AdS/CFT, thermal, and strong-field quantum theories.

Searching arXiv for primary and related sources on Cutkosky cutting rules.

Searching "Cutkosky Rules from Outer Space"

Cutkosky cutting rules are relations for Feynman amplitudes that compute a discontinuity, variation, or anti-Hermitian part of a diagram by replacing selected internal propagators with on-shell delta functions and summing over admissible cuts. In the modern mathematical formulation, they are simultaneously a statement about the monodromy of Feynman amplitudes around Landau singularities and a combinatorial-geometric structure on graphs generated by two operations—shrinking internal edges and putting internal lines on the mass shell. This viewpoint organizes normal thresholds, anomalous thresholds, dispersion relations, and the optical theorem in a common framework (Kreimer, 2016, Bloch et al., 2015).

1. The theorem and its standard form

Let $\Gamma$ be a Feynman graph with edge set $E_\Gamma$, external momenta $p$, and propagators $\ell_e=(k_e^2-m_e^2+i\epsilon)$. Choose a spanning forest $F$ in $\Gamma$. Denote by $E'$ the set of edges that do not connect distinct components of $F$, and by $E''=E_\Gamma\setminus E'$ the cut edges, namely those that do connect different components. If the reduced graph $\Gamma''$ obtained by shrinking all edges in $E_\Gamma$0 has a physical Landau singularity at $E_\Gamma$1, then the variation of the unrenormalized Feynman amplitude $E_\Gamma$2 as $E_\Gamma$3 winds around that singularity is

$E_\Gamma$4

where $E_\Gamma$5 puts the line $E_\Gamma$6 on its physical mass shell, while uncut edges retain the Feynman denominator (Kreimer, 2016).

Equivalently, for a chosen channel variable $E_\Gamma$7, one often writes the discontinuity across the branch cut as

$E_\Gamma$8

In the formulation of connected graphs $E_\Gamma$9, loop number $p$0, and channel variables defined by $p$1-forests, the same statement appears as

$p$2

with $p$3 the number of edges in a chosen minimal separating cut (Bloch et al., 2015).

These formulas encode the operational rule familiar from perturbative unitarity: a cut line is no longer integrated with a Feynman propagator but is instead placed on shell by a positive-energy delta function. In perturbative quantum field theory this is the diagrammatic content of

$p$4

or, equivalently, the relation $p$5 following from $p$6 and $p$7 (Sen, 2020, Pius et al., 2016).

2. Landau singularities, pinch geometry, and monodromy

The analytic origin of the cutting rules lies in the geometry of pinch singularities. Singularities of a Feynman amplitude occur when propagator quadrics fail to intersect transversely. In the parametric representation this is expressed by the Landau equations: there exist nonzero real parameters $p$8 such that

$p$9

with the point $\ell_e=(k_e^2-m_e^2+i\epsilon)$0 lying in the interior of the simplex $\ell_e=(k_e^2-m_e^2+i\epsilon)$1; such a point is a physical Landau singularity (Bloch et al., 2015).

The local mechanism is contour pinching in complexified loop-momentum space. When a subset of propagator quadrics $\ell_e=(k_e^2-m_e^2+i\epsilon)$2 pinches the integration contour, analytic continuation around the singular point produces a nontrivial variation. In the Bloch–Kreimer formulation, Landau analysis identifies physical singularities by the simultaneous equations $\ell_e=(k_e^2-m_e^2+i\epsilon)$3 for $\ell_e=(k_e^2-m_e^2+i\epsilon)$4 together with derivative conditions, and Pham’s theory of vanishing cycles, within Picard–Lefschetz theory, identifies the resulting variation with a sum of residues of the form $\ell_e=(k_e^2-m_e^2+i\epsilon)$5 (Kreimer, 2016).

The local model near an ordinary double-point pinch is

$\ell_e=(k_e^2-m_e^2+i\epsilon)$6

For $\ell_e=(k_e^2-m_e^2+i\epsilon)$7 there is a real vanishing sphere, and the Picard–Lefschetz formula describes how monodromy adds the corresponding vanishing cycle to the original integration chain. In a physical pinch, the Hessian is definite of one sign, so the vanishing sphere lies entirely in the real locus of the propagator intersection, and integrating the residue over that real sphere is exactly the replacement of each on-shell propagator by $\ell_e=(k_e^2-m_e^2+i\epsilon)$8 (Bloch et al., 2015).

A central structural statement is that the admissible pinch configurations are precisely spanning-forest cuts. This excludes arbitrary on-shell subsets: the singular configurations relevant for the discontinuity are those whose removal disconnects the graph in the appropriate way. This is what ties the analytic structure of the amplitude to graph combinatorics (Kreimer, 2016).

3. Shrinking and cutting as operations on a cubical chain complex

A distinctive feature of the outer-space approach is that two operations usually treated separately in perturbation theory become boundary operations of a single cubical complex. For a bridge-free graph $\ell_e=(k_e^2-m_e^2+i\epsilon)$9 and a spanning tree $F$0 with $F$1 edges, one associates the cube $F$2. The assignment is that $F$3 means shrink edge $F$4 to length zero, and $F$5 means put edge $F$6 on the mass shell, i.e. mark it by a Cutkosky cut (Kreimer, 2016).

Shrinking an internal edge $F$7 corresponds graphically to the contraction $F$8, and geometrically to the boundary operator at the face $F$9. In momentum-space language it corresponds to removing the propagator in the high-energy limit, or in parametric language to the hyperface $\Gamma$0. Putting an internal line on shell replaces

$\Gamma$1

graphically removing the line from the integration and marking it as cut; geometrically this is the boundary operator at $\Gamma$2 (Kreimer, 2016).

These two operations anti-commute appropriately, so the graphs obtained from $\Gamma$3 by various shrinkings and cuttings form a cubical chain complex. In Culler–Vogtmann Outer Space, each top-dimensional cell corresponds to a marked metric graph of fixed rank, and faces correspond either to shrinking edges or to imposing on-shell conditions. The analytic structure of amplitudes—Landau thresholds, monodromies, and anomalous thresholds—is thereby encoded in the boundary structure of these cubes (Bloch et al., 2015).

The same organization appears algebraically in lower-triangular matrices $\Gamma$4 indexed by an ordering of a spanning tree. Their diagonal entries correspond to leading thresholds, while lower sub-diagonals are obtained recursively by dispersion integrals. This suggests a systematic bookkeeping device for iterated cuts and reconstructions of amplitudes from their discontinuities (Kreimer, 2016).

4. Thresholds, dispersion relations, and standard examples

The one-loop bubble is the canonical example. For a two-point graph with propagators $\Gamma$5,

$\Gamma$6

The only nontrivial cut is $\Gamma$7, and Cutkosky’s rule gives

$\Gamma$8

which is the standard phase-space integral for two on-shell intermediate states (Kreimer, 2016). In the corresponding parametric analysis, the second Symanzik polynomial has discriminant

$\Gamma$9

and the pinch occurs at $E'$0 (Bloch et al., 2015).

The one-loop triangle illustrates the distinction between normal and anomalous thresholds. With edges $E'$1 and external legs $E'$2 satisfying $E'$3, the parametric form is

$E'$4

with

$E'$5

and

$E'$6

For the cut $E'$7 in the $E'$8 channel, the reduced Landau equations for $E'$9 yield the normal threshold

$F$0

Writing $F$1 with $F$2, $F$3, and $F$4 as given in the parametric decomposition, the vanishing of the discriminant

$F$5

determines the anomalous threshold (Kreimer, 2016).

After cutting edges $F$6, one obtains

$F$7

so the normal and anomalous thresholds are read from the domain of the remaining $F$8-integration (Kreimer, 2016). A common simplification is to identify Cutkosky rules only with leading normal thresholds; the outer-space formulation instead treats normal-threshold and anomalous-threshold transitions within one boundary calculus.

Once $F$9 is known above threshold, one may reconstruct the amplitude by a once-subtracted dispersion relation,

$E''=E_\Gamma\setminus E'$0

where $E''=E_\Gamma\setminus E'$1 is the lowest threshold in the channel (Kreimer, 2016). In the forward limit, the optical theorem identifies $E''=E_\Gamma\setminus E'$2 and thus $E''=E_\Gamma\setminus E'$3, with the two-point version of Cutkosky’s rule furnishing the corresponding on-shell decomposition (Kreimer, 2016).

5. Perturbative unitarity beyond local relativistic field theory

In superstring field theory, the same unitarity statement persists, but the proof requires a contour analysis adapted to vertices that are entire in the external momenta, decay exponentially for large spacelike momenta, and grow exponentially for large timelike energies. Loop-energy contours are therefore taken with endpoints at $E''=E_\Gamma\setminus E'$4 and analytically continued from purely imaginary external energies. The discontinuity of an amplitude $E''=E_\Gamma\setminus E'$5 is then

$E''=E_\Gamma\setminus E'$6

and Pius–Sen prove this to all orders in perturbation theory by an induction on loop number using reduced diagrams, the classification into $E''=E_\Gamma\setminus E'$7VR and $E''=E_\Gamma\setminus E'$8VI graphs, and contour deformations through pinching poles (Pius et al., 2016).

Euclidean nonlocal quantum field theories provide another extension. There amplitudes are first defined with all loop energies integrated along the imaginary axis and external energies purely imaginary, then analytically continued to real energies by deforming contours when propagator poles cross. The resulting discontinuity obeys the same replacement rule

$E''=E_\Gamma\setminus E'$9

but only cut diagrams corresponding to normal thresholds contribute to the imaginary part. Because the nonlocal vertex factors are entire and pole-free, anomalous thresholds do not generate new physical discontinuities. In gauge and gravity theories, BRST or diffeomorphism Ward identities then cancel unphysical cuts, so that only physical polarizations propagate on shell (Briscese et al., 2018).

D-instanton amplitudes require a modified cutting rule. In perturbative amplitudes the anti-Hermitian part is written as a sum over cut diagrams, but for D-instanton contributions only those cuts are allowed for which all world-sheet boundaries ending on a given D-instanton lie on the same side of the cut. Open-string propagators are never cut, since open-string modes on the D-instanton are not asymptotic closed-string states. Sen proves that these rules follow from the closed-string effective action after integrating out open-string modes, provided the effective action is real in the Lorentzian continuation. The failure of this reality condition explains unitarity violation in the closed-string sector of two-dimensional string theory, whereas in critical superstring theory Picard–Lefschetz theory selects contours yielding a real effective action (Sen, 2020).

6. Cosmological, AdS, thermal, and strong-field generalizations

The basic logic of cutting extends well beyond flat-space scattering amplitudes. In cosmology, unitarity of in-in evolution leads to “Cosmological Cutting Rules” for wavefunction coefficients. For a diagram $\Gamma''$0 contributing to $\Gamma''$1, one introduces a discontinuity operation on a chosen set of internal lines and sums over all nonempty subsets of cuts. Each cut line is removed and replaced by two bulk-to-boundary legs together with a factor of the late-time power spectrum $\Gamma''$2, giving schematic relations of the form

$\Gamma''$3

These rules apply for arbitrary interactions of fields of any mass and any spin with a Bunch–Davies vacuum around a very general class of FLRW spacetimes, and they reduce loop discontinuities to lower-loop data (Melville et al., 2021).

A related de Sitter/EAdS perspective is that flat-space on-shell deltas uplift to discontinuity operators in exchanged-energy variables under dressing maps. In this representation,

$\Gamma''$4

and one has the schematic correspondence

$\Gamma''$5

Successive cuts together with momentum-space dispersion relations reconstruct tree-level de Sitter Witten diagrams from lower-point contact objects up to contact diagram ambiguities (Das et al., 5 Feb 2026, Ansari et al., 13 Jan 2026).

In holographic CFTs, the AdS Cutkosky rules compute the causal double-commutator entering the Lorentzian inversion formula by cutting Witten diagrams. A cut bulk-to-bulk propagator $\Gamma''$6 is replaced by the on-shell Wightman propagator $\Gamma''$7, and a split representation factorizes it into on-shell bulk-to-boundary lines divided by the boundary Wightman two-point function. The cut diagram therefore takes the factorized form

$\Gamma''$8

which is the AdS analogue of summing over physical intermediate states. In the flat-space limit, one recovers the ordinary $\Gamma''$9-matrix optical theorem (Meltzer et al., 2020).

At finite temperature in many-body perturbation theory, cutting rules are formulated in terms of retarded $E_\Gamma$00-point functions. Cut lines are represented by lesser/greater propagators carrying thermal occupation factors, and the cut expansion writes, for example,

$E_\Gamma$01

This factorized structure implies positivity: in the operator sense, $E_\Gamma$02 is positive semi-definite, and therefore the spectral function remains positive. The construction is stated to preserve positivity in the $E_\Gamma$03, second Born, and $E_\Gamma$04-matrix approximations at finite temperature (Hyrkäs et al., 2022).

In strong-field QED with a plane-wave background, Veltman’s largest-time equation can be reformulated in the Furry picture. The resulting cutting equation relates the difference between a diagram and its inversion to a sum over cuts of Furry-picture subdiagrams with on-shell intermediate states. Applied to two-loop elastic electron scattering, the central cut through a polarization insertion yields the direct and exchange tree-level trident diagrams. The formulation is stated to apply to any loop order and to admit extensions to nonperturbative regimes, although reconstruction of differential rates requires additional care even when total rates follow reliably from cuts (Selivanov et al., 12 May 2026).