Unitarity-Based Cutting Rules

• Unitarity-based cutting rules are systematic methods that express the discontinuous or absorptive parts of quantum amplitudes as sums over on-shell subdiagrams.
• They simplify loop calculations in quantum field, string, and cosmological theories by organizing perturbative expansions and enforcing probability conservation.
• These rules enable analytic reconstruction through dispersive representations and coproduct algebra, with applications ranging from dimensional regularization to finite-temperature and holographic frameworks.

Unitarity-based cutting rules are fundamental diagrammatic procedures that implement unitarity and causality in the perturbative expansion of quantum field theories, string theories, conformal field theories, and cosmological observables. They express the (anti-)hermitian or discontinuous part of a quantum amplitude, correlator, or wavefunction coefficient as a sum over products of on-shell sub-diagrams, encoding the constraints imposed by the conservation of probability. These rules underpin the optical theorem, organize perturbative calculations, and connect analytic properties of amplitudes to physical phase-space processes.

1. Foundations of Unitarity and the Optical Theorem

Unitarity requires that the time-evolution or scattering operator $U$ satisfies $U^\dagger U=1$, leading to constraints such as

$i(T - T^\dagger) = T^\dagger T,$

with $S = 1 + iT$ the $S$-matrix. For an amplitude $A = \langle f|T|i\rangle$, this implies

$$2\,\mathrm{Im}\,A = \sum_n \langle f|T|n\rangle\,\langle n|T|i\rangle^*,$$

which is the optical theorem. Diagrammatically, the anti-hermitian (absorptive) part of a Feynman diagram is constructed by placing internal lines on shell and sewing lower-order amplitudes—the essential idea behind cutting rules (Sen, 2020).

In cosmology, for the wavefunction of the universe $\Psi[\phi]\sim \exp[i\Gamma[\phi]]$, unitarity leads to an infinite set of relations among wavefunction coefficients $\psi_n$. At tree-level, these compose the "Cosmological Optical Theorem": $i\,\mathrm{Disc}[i \psi_n^{(0)}] = 0,$ and, for the four-point exchange diagram in the $s$-channel,

$$i\,\mathrm{Disc}_{p_s}[i\psi_4^{(s)}(k_1, k_2, k_3, k_4)] = \int_{q, q'} P_q\,\bigl[i\,\mathrm{Disc}_q[i\psi_3(k_1,k_2,q)]\bigr]\,\bigl[i\,\mathrm{Disc}_{q'}[i\psi_3(q',k_3,k_4)]\bigr]\,.$$

This structure parallels the absorptive part of S-matrix amplitudes but is formulated for cosmological boundary data (Melville et al., 2021).

2. Diagrammatic Cutting Rules Across Theories

The core of the cutting prescription is to represent a loop diagram's discontinuity as a sum over all possible ways to put internal lines on-shell, partitioning the diagram into on-shell subdiagrams corresponding to physical processes.

• Quantum Field Theory (QFT): The Cutkosky rules state that each propagator $i/(p^2-m^2+i\epsilon)$ in a Feynman diagram is replaced, when "cut," by $2\pi \delta(p^2-m^2)\theta(p^0)$, and the cut separates the diagram so that the left and right parts correspond to complex-conjugate amplitude contributions (Sen, 2020, Pius et al., 2018, Abreu et al., 2014).
• General $L$-loop Cosmological Cutting Rule: For a Feynman-Witten diagram $D$, the central rule is

$$i\,\underset{I}{\mathrm{Disc}}\left[i\psi^{(D)}\right] = \sum_{\varnothing\neq C\subseteq I} \left[\prod_{\ell\in C}\int_{p_\ell}P_{p_\ell}\right] \prod_a\left(-i\right)\underset{I_a\cup\{p_\ell:\ell\in C_a\}}{\mathrm{Disc}} \left[i\psi^{(D_C^{(a)})}\right].$$

This encodes the combinatorics of all possible (non-empty) cuts $C$ of internal lines, with the remaining diagram breaking into disconnected pieces whose discontinuities are recursively determined (Melville et al., 2021).

• Gauge Theories with Dimensional Regularization: The generalised unitarity method via the Four-Dimensional Formulation (FDF) represents $d$-dimensional cut conditions with 4D massive vectors and scalars carrying $(-2\epsilon)$-SR selection rules; for a $k$-fold cut, the replacement is

$$\frac{i}{D_i + i0} \rightarrow 2\pi \delta^+\big((\bar\ell + q_i)^2 - m_i^2\big),$$

with the cut integrand given by sewing $k$ tree-level amplitudes (Bobadilla et al., 2015).

3. Analytic Structure, Discontinuities, and Multiple Cuts

Unitarity-based cuts are intimately related to the analytic structure of amplitudes via their branch cuts and discontinuities:

• A single cut in a channel $s$ corresponds to the discontinuity,

$$\mathrm{Disc}_s\,I(s) = I(s + i\epsilon) - I(s - i\epsilon),$$

and for scalar integrals,

$\mathrm{Cut}_s\,I = -\mathrm{Disc}_s\,I.$

This relation extends to sequences of cuts (multiple channels) and is precisely encoded in the coproduct structure of generalized polylogarithmic Feynman integrals (Abreu et al., 2014).

• In cosmology, the discontinuity $\mathrm{Disc}$ extracts the imaginary part with respect to specific internal energies or partial energies, following a precise diagrammatic prescription (e.g., for cutting an internal line $p$ in a cosmological diagram, $\mathrm{Disc}_p$ applies to the relevant substructure) (Melville et al., 2021).
• In string theory, unitarity-based cuts can be performed directly in the worldsheet formalism. Application of Witten's $i\varepsilon$ prescription to worldsheet moduli allows the computation of the imaginary part of amplitudes by "circling" degenerate cycles, and the resulting structure is in exact correspondence with field-theoretic Cutkosky rules (Eberhardt et al., 2022). Sums over infinite towers of internal string states are implicit in polynomial coefficients of cut integrals.

4. Specialized and Generalized Cutting Rules

Unitarity-based cutting rules adapt to diverse settings, each with specific subtleties:

• Finite Temperature and Nonequilibrium: Vacuum diagrams are drawn on a thermal cylinder. Cut lines correspond to on-shell propagation, while windings around the thermal circle generate Bose-Einstein and Fermi-Dirac factors, producing the correct statistical weights in finite-temperature rates and kinetic equations (Blažek et al., 2021, Maták et al., 2022).
• Complex-Mass Scheme: For unstable-particle propagators, the cutting rule is modified: the usual delta function is replaced by a Breit–Wigner "smeared delta," reflecting the spread of the resonance,

$$i(\Delta - \Delta^\dagger) \rightarrow 2\pi \tilde\delta(p^2-M^2),$$

with

$$\tilde\delta(p^2-M^2) = \frac{M\Gamma/\pi}{(p^2-M^2)^2 + (M\Gamma)^2}.$$

Cuts through unstable lines only contribute near resonance and reorganize into stable-particle self-energy cuts (Denner et al., 2014, Bauer et al., 2012).

• Special Kinematics: In two dimensions, the cut-constructibility of S-matrices (i.e., the full amplitude being reconstructed solely from cuts) holds in supersymmetric and certain integrable models. In generic cases, non-cut (rational) terms correspond to coupling renormalization (Forini et al., 2014).
• AdS/CFT and CFT Correlators: Cutting rules generalize as "AdS Cutkosky rules," where the absorptive part of a Witten diagram is computed by integrating over the product of "transition amplitudes" built from on-shell bulk-to-boundary and bulk-to-bulk propagators, correlating precisely with the CFT optical theorem and the double commutator in Lorentzian inversion formulas (Fitzpatrick et al., 2011, Meltzer et al., 2020).

5. Cosmological Cutting Rules: Applications and Bounds

In cosmological perturbation theory (e.g., inflationary EFTs), cutting rules have multiple applications:

• The discontinuity of an $n$-loop cosmological diagram is recursively fixed by lower-loop (and tree-level) data. At one loop in the EFT of inflation, the only nonvanishing discontinuities arise from double cuts of cubic vertices: $$i\,\mathrm{Disc}[i\psi_{k,-k}^{1\mathrm{-loop}}] = \frac{H^2}{f_\pi^4}\,\frac{i\,k^3}{16\pi}\,\gamma(c_s,\tilde c_3),$$ with $\gamma$ a calculable function of the EFT coefficients (Melville et al., 2021).
• These rules enable efficient bootstrapping of higher-order corrections and derivation of perturbative unitarity bounds on Wilson coefficients, such as

$|\gamma(c_s,\tilde c_3)| < 32\pi\,f_\pi^4/H^4.$

• Cutting rules fix not only the imaginary but also real parts via causality and analyticity constraints, often involving the structure of cosmological polytopes. The combinatorics of "optical polytopes" underlie the equivalence of various cutting representations and explain the emergence of the flat-space optical theorem in appropriate limits (Albayrak et al., 2023).

6. Methodological Relations, Outlook, and Generalizations

Unitarity-based cutting rules underlie multiple synergistic techniques in modern field theory, cosmology, and string theory:

• Integrand-Level Unitarity: Direct extraction of loop-integrand coefficients via systematic multiple cuts ("generalized unitarity") is now a central computational tool, including in dimensionally regulated amplitudes (Bobadilla et al., 2015).
• Analytic Reconstruction: Knowledge of all possible cuts (including multiple and sequential cuts) allows full reconstructibility of Feynman integrals through dispersive representations, coproduct algebra, and symbol calculus. This is central both to practical multi-loop calculations and to analytic understanding (Abreu et al., 2014).
• Connection to Causality and Locality: Tree-loop duality and the Bogoliubov causality condition clarify the algebraic structure of cuts, avoiding ambiguities (e.g., contact terms) and making the methodology fully covariant and generalizable to non-local theories (Tomboulis, 2017, Pius et al., 2018).
• Cosmological Observables: The in-in (Schwinger–Keldysh) formalism supports a complete cutting rule for cosmological correlators, in which the discontinuity is a sum over products of lower-order correlators (including "barred" combinations to close the algebra), with the absence of $\Theta$-functions allowing conformal time integrals to factorize efficiently (Colipí-Marchant et al., 27 Dec 2025, Ema et al., 2024).
• Worldsheet and String Cutting: String amplitudes at higher genus admit direct computation of their imaginary parts via moduli-space contour deformations, with "tropical analysis" revealing the detailed pattern of physical thresholds and phase-space structure (Eberhardt et al., 2022).
• Nonperturbative Sectors and D-instantons: The extension of cutting rules to nonperturbative sectors (such as D-instantons) requires carefully restricting cuts to those that leave all boundaries of a D-instanton on one side. Reality of the closed-string effective action is essential for unitarity to remain valid (Sen, 2020).

7. Summary Table: Unitarity Cutting Rules Across Fields

Context	Key Diagrammatic Rule	Principal References
QFT (stable particles)	Replace cut $$i/(p^2-m^2+i\epsilon) \to 2\pi\delta(p^2-m^2)\theta(p^0)$$, sum all partitions	(Sen, 2020, Pius et al., 2018)
Cosmology (wavefunction, correl.)	$i\,\mathrm{Disc}[i\psi_n]$ recursively in terms of lower-loop/tree diagrams, via spatial power spectra $P_q$	(Melville et al., 2021, Albayrak et al., 2023)
Dimensionally regulated QFT	Cut propagators $i/(D_i+i0)\to 2\pi\delta^+(D_i)$, FDF rules for $\epsilon$-tracking	(Bobadilla et al., 2015)
Finite-temperature field theory	Cylinder method: cut lines sum over windings, generating $f_B(E), f_F(E)$ statistical weights	(Blažek et al., 2021, Maták et al., 2022)
Complex-mass (unstable particles)	Cut $i/(p^2-M^2+iM\Gamma)\to 2\pi \tilde\delta(p^2-M^2)$, only near resonance	(Denner et al., 2014, Bauer et al., 2012)
(A)dS/CFT and Holography	Causal double-commutators sum over cuts on bulk-to-bulk lines, matching OPE decomposition	(Fitzpatrick et al., 2011, Meltzer et al., 2020)
String theory (worldsheet)	$i\varepsilon$ on moduli; imaginary part as contour winding and worldsheet tropical analysis	(Eberhardt et al., 2022)
Cosmological in-in correlators	SK formalism cuts via Wightman functions, subdiagrams as fully retarded objects	(Ema et al., 2024, Colipí-Marchant et al., 27 Dec 2025)

The unitarity-based cutting rules provide a unifying infrastructure for enforcing probability conservation, organizing loop-level computations, uncovering analytic structure, and systematically bootstrapping amplitudes and correlators in flat, curved, and thermal quantum field theories, as well as in string theory and holographic settings. Their combinatorial and geometric structure encodes deep relations between causality, locality, and physical phase space across disciplines.