Mandelstam variables

Mandelstam variables are a set of three Lorentz-invariant quantities, denoted s, t, and u, that fully characterize the kinematics of two-particle to two-particle scattering processes in relativistic quantum field theory.^[1] For incoming particles with four-momenta p_1 and p_2, and outgoing particles with four-momenta p_3 and p_4, they are defined as s = (p_1 + p_2)^2, t = (p_1 - p_3)^2, and u = (p_1 - p_4)^2, where the squares denote the Minkowski inner product.^[1] These variables satisfy the relation s + t + u = m_1^2 + m_2^2 + m_3^2 + m_4^2, where m_i are the rest masses of the particles, ensuring conservation of four-momentum.^[1] Introduced by physicist Stanley Mandelstam in his 1958 work on analytic properties of transition amplitudes, they provide a frame-independent description essential for perturbative calculations and dispersion relations in quantum field theory.^[2] In particle physics, Mandelstam variables simplify the analysis of scattering amplitudes by parameterizing the center-of-mass energy squared (s), the momentum transfer squared (t), and the u-channel exchange (u). The variable s corresponds to the total energy available in the center-of-mass frame, often exceeding the threshold for particle production, while t and u relate to the exchanged particles in t- and u-channel diagrams, respectively.^[1] Their invariance under Lorentz transformations makes them indispensable for comparing theoretical predictions with experimental data across different frames, such as lab or collider setups. Beyond kinematics, Mandelstam variables underpin key theoretical developments, including the Mandelstam representation, which expresses scattering amplitudes as double spectral integrals over these variables, enabling the study of analyticity and unitarity in S-matrix theory.^[2] This framework influenced the bootstrap hypothesis and Regge theory in the 1960s, providing insights into hadron interactions without relying on field-theoretic Lagrangians.^[3] Today, they remain fundamental in high-energy physics computations, such as those in the Standard Model and beyond, for processes like deep inelastic scattering and jet production at accelerators like the LHC.^[1]

Definition and Notation

Two-to-two scattering process

In relativistic quantum field theory, the two-to-two scattering process describes the interaction of two incoming particles, labeled 1 and 2 with four-momenta p_1 and p_2, producing two outgoing particles, labeled 3 and 4 with four-momenta p_3 and p_4. Each particle is on-shell, satisfying the mass-shell condition p_i^2 = m_i^2 for i = 1, 2, 3, 4, where m_i denotes the rest mass of the i-th particle and the metric convention is the mostly minus signature \eta_{\mu\nu} = \mathrm{diag}(+1, -1, -1, -1). This setup forms the foundational kinematic configuration for analyzing scattering amplitudes in particle physics, ensuring that the particles' energies and momenta are physically realizable.^[4] Conservation of four-momentum is imposed by the relation p_1 + p_2 = p_3 + p_4, which follows from the translational invariance of the underlying theory. To characterize the kinematics in a frame-independent manner, suitable variables must be Lorentz scalars, invariant under Lorentz transformations of the four-momenta. Such invariants are constructed from bilinear combinations of the momenta, providing a complete description of the process without reference to a specific observer's frame. The Mandelstam variables represent one such set of these invariants derived from the four-momenta.^[4] The center-of-mass frame offers an intuitive reference for visualizing the scattering, where the total three-momentum vanishes: \vec{p}_1 + \vec{p}_2 = \vec{0} = \vec{p}_3 + \vec{p}_4. In this frame, the incoming particles approach from opposite directions with equal and opposite momenta, and the outgoing particles recede similarly, with the scattering parameterized by relative angles and energies determined by the total center-of-mass energy. This frame simplifies calculations of differential cross-sections and highlights the azimuthal symmetry around the beam axis. These variables were introduced by Stanley Mandelstam in 1958 within the framework of S-matrix theory, providing a basis for dispersion relations in pion-nucleon scattering.^[5]

Kinematic invariants

In the two-to-two scattering process involving incoming particles with four-momenta p_1 and p_2 and outgoing particles with four-momenta p_3 and p_4, the Mandelstam variables are defined as the Lorentz-invariant combinations

s = (p_1 + p_2)^2 = (p_3 + p_4)^2,

t = (p_1 - p_3)^2 = (p_2 - p_4)^2,

u = (p_1 - p_4)^2 = (p_2 - p_3)^2.

These variables were introduced by Stanley Mandelstam in his work on dispersion relations for pion-nucleon scattering. The variable s represents the square of the total four-momentum of the initial (or final) state, while t is the square of the four-momentum transfer between p_1 and p_3 in the direct channel, and u is the square of the four-momentum transfer in the crossed channel. In natural units where \hbar = c = 1, the Mandelstam variables carry dimensions of energy squared; in high-energy particle physics, they are conventionally expressed in GeV². A key kinematic relation among the variables is s + t + u = m_1^2 + m_2^2 + m_3^2 + m_4^2, where m_i are the rest masses of the particles; this sum rule sets constraints on the physical region, analogous to a triangle inequality in the space of invariants.

Physical Interpretation

S-channel and crossing

In the s-channel, the Mandelstam variable s corresponds to the square of the total center-of-mass energy available in the direct two-to-two scattering process, given exactly by s = (E_1 + E_2)^2 in the center-of-mass frame, where E_1 and E_2 are the energies of the incoming particles.^[6] This configuration describes the physical region where the incoming particles collide head-on, allowing s to determine the energy threshold for producing intermediate resonances or other excited states in the interaction.^[7] The s-channel thus captures the time-like momentum transfer, emphasizing the forward evolution of the scattering amplitude along the collision axis. Crossing symmetry arises from the analytic structure of scattering amplitudes in quantum field theory, enabling the continuation of the amplitude across different kinematic regions by interchanging incoming and outgoing particles.^[6] This symmetry relates the s-channel process to equivalent descriptions in the t- or u-channels through permutations of the Mandelstam variables, ensuring that the same underlying amplitude governs physically distinct but kinematically connected reactions.^[8] For instance, in electron-muon scattering (e^- \mu^- \to e^- \mu^-), which proceeds via space-like photon exchange in the t-channel, crossing to the s-channel yields the time-like process e^- e^+ \to \mu^- \mu^+, where a virtual photon mediates the annihilation and pair production at the center-of-mass energy squared s > 0.^[8] The Mandelstam representation formalizes this crossing symmetry by expressing the scattering amplitude as an integral over double spectral functions that encode the absorptive contributions from both s- and t-channel cuts, providing a unified framework for amplitudes analytic in the complex Mandelstam plane except for branch cuts corresponding to physical thresholds.^[6] Conceptually, these spectral functions capture the discontinuity across the cuts, allowing the amplitude to satisfy unitarity in one channel while incorporating crossing to others, thus bridging direct scattering with crossed processes without explicit perturbation theory.^[6]

T-channel and u-channel

In particle physics, the t-channel describes a kinematic configuration in two-to-two scattering where the Mandelstam variable t = (p_1 - p_3)^2 represents the square of the four-momentum transfer between the incoming particle 1 and outgoing particle 3, analogous to the center-of-mass energy squared in a crossed process involving particles 1 and \bar{3} (or antiparticles in the crossed diagram). For physical scattering in the s-channel, t < 0, corresponding to space-like momentum transfer, which facilitates the exchange of virtual particles that mediate the interaction without on-shell production. A representative example is nucleon-nucleon scattering, where t-channel meson exchange, such as one-pion exchange, contributes to the force between nucleons via virtual pion propagation with momentum squared t.^[7]^[9] The u-channel is similarly defined by u = (p_1 - p_4)^2, representing the momentum transfer squared in the alternative crossing where incoming particle 1 interacts with outgoing particle 4 (or \bar{4}), again with u < 0 in the physical s-channel region for space-like exchange. This channel arises from crossing the outgoing particles in the t-channel diagram and describes virtual particle exchange in the crossed kinematics, often symmetric to the t-channel when the scattering involves identical particles, leading to u \leftrightarrow t interchange invariance in the amplitude. In such cases, the contributions from t- and u-channel exchanges are indistinguishable for identical bosons or fermions, enhancing the role of symmetric potentials in the interaction.^[7] The physical regions for these channels are visualized in the Mandelstam plot, a representation in the complex s-t plane (with u = \Sigma m_i^2 - s - t, where \Sigma m_i^2 is the sum of squared masses), where the s-channel physical domain lies above the threshold curve s > (m_1 + m_2)^2, bounded laterally by the curves t_{\min} and u_{\min} (both negative), corresponding to forward (\cos\theta = 1) and backward (\cos\theta = -1) scattering angles in the center-of-mass frame. These boundaries delineate the allowed kinematics for real scattering events, with t and u confined to negative values to ensure positive scattering angles and energy conservation, while the t- and u-channel physical regions occupy analogous domains rotated by crossing symmetry. For instance, in Compton scattering (\gamma e^- \to \gamma e^-), the u-channel diagram involves electron exchange with momentum squared u, illustrating virtual fermion propagation in the backward scattering regime.^[10]^[11]

Mathematical Relations

Sum rule for invariants

In the context of two-to-two particle scattering, the Mandelstam variables s, t, and u satisfy the algebraic relation s + t + u = m_1^2 + m_2^2 + m_3^2 + m_4^2, where m_i are the rest masses of the incoming particles with four-momenta p_1 and p_2, and outgoing particles with p_3 and p_4.^[1] This sum rule holds generally for processes involving particles of unequal masses and follows directly from four-momentum conservation and the on-shell conditions p_i^2 = m_i^2.^[12] The derivation begins with the definitions s = (p_1 + p_2)^2, t = (p_1 - p_3)^2, and u = (p_1 - p_4)^2. Expanding the sum gives:

s + t + u = (p_1 + p_2)^2 + (p_1 - p_3)^2 + (p_1 - p_4)^2 = p_1^2 + p_2^2 + 2 p_1 \cdot p_2 + p_1^2 + p_3^2 - 2 p_1 \cdot p_3 + p_1^2 + p_4^2 - 2 p_1 \cdot p_4.

Simplifying the expression yields:

s + t + u = 3 p_1^2 + p_2^2 + p_3^2 + p_4^2 + 2 p_1 \cdot (p_2 - p_3 - p_4).

From four-momentum conservation, p_1 + p_2 = p_3 + p_4, so p_2 - p_3 - p_4 = -p_1. Substituting this in, the cross term becomes $2 p_1 \cdot (-p_1) = -2 p_1^2, and thus:

s + t + u = 3 p_1^2 + p_2^2 + p_3^2 + p_4^2 - 2 p_1^2 = p_1^2 + p_2^2 + p_3^2 + p_4^2 = m_1^2 + m_2^2 + m_3^2 + m_4^2.

This identity is Lorentz-invariant and applies to any frame.^[12]^[1] The sum rule has key implications for kinematic analysis: once two of the variables are specified, the third is determined, providing a consistency check in scattering calculations. For example, in elastic scattering of identical particles of mass m, it simplifies to s + t + u = 4m^2.^[1]

Relativistic approximations

In the high-energy limit, where the center-of-mass energy \sqrt{s} greatly exceeds the rest masses of the participating particles, the Mandelstam variables approximate the relation s + t + u \approx 0. This simplification stems from the exact sum rule s + t + u = \sum m_i^2, where the masses m_i become negligible compared to the kinematic scales set by s, t, and u. As a result, the third variable can be expressed in terms of the other two, such as u \approx -s - t, facilitating calculations in scattering processes.^[13] For processes involving massless particles, like photons in quantum electrodynamics (QED) or gluons in quantum chromodynamics (QCD), the relation holds exactly: s + t + u = 0. This massless case eliminates mass-dependent terms entirely, simplifying the kinematics and allowing direct focus on momentum transfers.^[14] In the center-of-mass frame for such high-energy, massless approximations, the Mandelstam variables t and u relate directly to the scattering angle \theta between incoming and outgoing particles:

t \approx -\frac{s}{2} (1 - \cos\theta),

u \approx -\frac{s}{2} (1 + \cos\theta).

These expressions link the invariants to the angular distribution of scattered particles, with t dominating forward scattering (\theta \approx 0) and u backward scattering (\theta \approx \pi).^[13] These relativistic approximations are valid for QED processes, such as electron-muon scattering, and QCD interactions, like quark-gluon scattering, at energies well above the electroweak scale (around 100 GeV), where lepton and hadron masses are insignificant relative to \sqrt{s}.^[14]

Applications in Particle Physics

Role in Feynman diagrams

In perturbative quantum field theory, Mandelstam variables play a crucial role in Feynman diagram calculations by providing Lorentz-invariant labels for the kinematic channels corresponding to different intermediate states in scattering processes. These variables parameterize the momentum transfers and center-of-mass energies, allowing amplitudes to be expressed in a frame-independent manner and facilitating the identification of singularities associated with on-shell intermediate particles.^[15] The correspondence between Mandelstam variables and diagram topologies is evident in the pole structure of propagators. In s-channel diagrams, where incoming particles annihilate to form an intermediate state that then produces the outgoing particles, the propagator for the intermediate particle of mass m contributes a factor of $1/(s - m^2 + i\epsilon), resulting in a pole at s = m^2 when the invariant mass squared equals the physical mass squared. Similarly, t-channel diagrams, involving the exchange of an intermediate particle between one incoming and one outgoing leg, feature propagators with $1/(t - m^2 + i\epsilon), yielding poles at t = m^2. The u-channel follows analogously, with poles at u = m^2. This structure highlights how the variables encode the thresholds and resonances in each channel.^[15] A concrete illustration occurs in tree-level 2-to-2 scattering diagrams for theories with Yukawa-like couplings, such as a scalar field interacting via g \phi \bar{\psi} \psi, where the amplitude includes contributions from s-, t-, and u-channel exchanges. The squared matrix element | \mathcal{M} |^2 is then expressed as g^4 \left[ \frac{1}{(s - m^2)^2} + \frac{1}{(t - m^2)^2} + \frac{1}{(u - m^2)^2} \right] (up to spinor traces and normalization factors for identical particles), demonstrating the direct dependence on the Mandelstam variables through the propagator denominators. In the specific case of \phi^4 theory at tree level, the contact diagram yields a momentum-independent amplitude \mathcal{M} = -\lambda, so | \mathcal{M} |^2 = \lambda^2, but the overall kinematics remain parameterized by s, t, u via the relation s + t + u = \sum m_i^2.^[16]^[17] Mandelstam variables also facilitate the partial wave expansion of scattering amplitudes, which decomposes the process into contributions from definite angular momentum projections. In the center-of-mass frame, the scattering amplitude depends on s (fixing the energy) and the angle \theta through t = - (s/2) (1 - \cos\theta) (for equal masses and massless exchange, approximately), allowing projection onto partial waves via integrals involving Legendre polynomials P_\ell(\cos\theta): f(s, \theta) = \sum_\ell (2\ell + 1) a_\ell(s) P_\ell(\cos\theta), where the coefficients a_\ell relate to the projections of the full amplitude in the Mandelstam representation. This expansion connects the invariant variables to the orbital angular momentum \ell, aiding in the analysis of resonances and unitarity.^[18]

Use in scattering amplitudes

In quantum field theory, the two-to-two scattering amplitude \mathcal{M}(s, t, u) is expressed as an analytic function of the Mandelstam variables, enabling the use of dispersion relations to decompose it into real and imaginary parts. For fixed momentum transfer t, the forward scattering amplitude (at t = 0) can be written via unsubtracted or subtracted dispersion relations in the energy variable s, where the real part is given by a principal value integral over the imaginary part along the physical cut, reflecting causality and unitarity. This fixed-t dispersion relation, \operatorname{Re} \mathcal{M}(s, t) = \frac{1}{\pi} \mathcal{P} \int ds' \frac{\operatorname{Im} \mathcal{M}(s', t)}{s' - s}, allows extraction of the amplitude from experimental data on the imaginary part, such as from total cross sections, and is foundational for non-perturbative analyses in strong interactions. Crossing symmetry further enriches the structure of \mathcal{M}(s, t, u), positing that the amplitude is a single analytic function in the complex Mandelstam plane, with physical processes in different channels related by analytic continuation. For scattering of identical particles, this manifests as \mathcal{M}(s, t, u) = \mathcal{M}(t, s, u) = \mathcal{M}(u, t, s), ensuring the same functional form describes s-channel scattering, t-channel exchange, and u-channel processes under particle interchange, without introducing new parameters. This symmetry, derived from the underlying Lorentz invariance and the S-matrix analyticity, constrains the amplitude's form and facilitates double dispersion relations integrating over both s and t cuts, as in the Mandelstam representation. In Regge theory, the high-energy behavior of scattering amplitudes at fixed t is dominated by the exchange of Regge poles, leading to \mathcal{M}(s, t) \sim \beta(t) s^{\alpha(t)} as s \to \infty, where \alpha(t) is the Regge trajectory interpolating the spin J of exchanged particles as a function of t, typically linear \alpha(t) = \alpha_0 + \alpha' t. This power-law growth captures the pomeron and reggeon contributions in hadron physics, explaining the near-constant total cross sections observed at accelerators, and extends perturbation theory to resum infinite Regge cuts for improved high-energy predictions. Unitarity imposes strong constraints on \mathcal{M}(s, t, u) through the optical theorem, which relates the imaginary part of the forward amplitude to the total cross section: \sigma_{\rm tot}(s) = \frac{1}{s} \Im \mathcal{M}(s, t=0), ensuring probability conservation via the S-matrix condition S^\dagger S = 1. At high energies, this links the amplitude's growth to \sigma_{\rm tot}, with perturbative expectations yielding \sigma_{\rm tot} \sim 1/s from constant \mathcal{M}, though non-perturbative effects like Regge exchanges allow slower growth up to \sigma_{\rm tot} \sim (\ln s)^2 per the Froissart bound, bounding the theory's ultraviolet behavior.^[19]