Relativistic Scattering Kinematics

For a two-body reaction, on-shell masses and four-momentum conservation leave two independent Lorentz invariants. The Mandelstam variables $s,t,u$ package the center-of-mass energy, momentum transfer, and crossed channels; the Källén function determines thresholds and momentum magnitudes; and a bounded interval of $t$ is exactly the physical scattering-angle range.

Required background. S-Matrix and T-Matrix Normalization fixes which overall delta function has been removed from the invariant amplitude.

Helpful background. Lorentz Field Representations and Poincaré Particle Representations supplies the orbit and invariant-product language used here.

Two-to-two invariants

Consider

$$p_1+p_2\longrightarrow p_3+p_4,$$

with every displayed $p_i$ future-directed and on shell, $p_i^2=m_i^2$. Define

$$\begin{aligned} s&=(p_1+p_2)^2=(p_3+p_4)^2,\\ t&=(p_1-p_3)^2=(p_2-p_4)^2,\\ u&=(p_1-p_4)^2=(p_2-p_3)^2. \end{aligned}$$

Expanding and using momentum conservation gives

$$\boxed{s+t+u=m_1^2+m_2^2+m_3^2+m_4^2.}$$

Only two of the three variables are independent. In the physical $s$-channel, $\sqrt{s}$ is the total center-of-mass energy; $t$ is the squared transfer from particle 1 to particle 3. These definitions and the sum rule agree with the convention-explicit treatment in Schwartz 2014, § 7.4.1, pp. 98–99.

The Källén function and thresholds

Define

$$\lambda(x,y,z) =x^2+y^2+z^2-2xy-2xz-2yz =\big[x-(\sqrt y+\sqrt z)^2\big] \big[x-(\sqrt y-\sqrt z)^2\big].$$

In the center-of-mass frame, $P=p_1+p_2=(\sqrt{s},\mathbf0)$. Solving $P\cdot p_1=(s+m_1^2-m_2^2)/2$ yields

$$\begin{aligned} E_1&=\frac{s+m_1^2-m_2^2}{2\sqrt{s}}, &E_2&=\frac{s+m_2^2-m_1^2}{2\sqrt{s}},\\ |\mathbf p_i|&=\frac{\sqrt{\lambda(s,m_1^2,m_2^2)}}{2\sqrt{s}}. \end{aligned}$$

Similarly,

$$\begin{aligned} E_3&=\frac{s+m_3^2-m_4^2}{2\sqrt{s}}, &E_4&=\frac{s+m_4^2-m_3^2}{2\sqrt{s}},\\ |\mathbf p_f|&=\frac{\sqrt{\lambda(s,m_3^2,m_4^2)}}{2\sqrt{s}}. \end{aligned}$$

A physical reaction therefore requires

$$s\ge (m_1+m_2)^2, \qquad s\ge (m_3+m_4)^2.$$

The other zero, $s=(m_a-m_b)^2$, is a pseudothreshold of the algebraic square root; it is not the production threshold for two future-directed particles of masses $m_a,m_b$. At the final-state threshold, $|\mathbf p_f|\to0$.

These center-of-momentum energies, square roots, and physical threshold branches are worked through with the same covariant normalization in Srednicki 2007, § 11, pp. 93–101.

The scattering angle is an invariant interval

Choose the center-of-mass $z$ axis along $\mathbf p_1$ and let $\theta$ be the angle between $\mathbf p_1$ and $\mathbf p_3$. Then

$$t=m_1^2+m_3^2-2E_1E_3 +2|\mathbf p_i||\mathbf p_f|\cos\theta.$$

Substituting the invariant energies gives

$$\begin{aligned} t={}&m_1^2+m_3^2 -\frac{(s+m_1^2-m_2^2)(s+m_3^2-m_4^2)}{2s}\\ &+\frac{\sqrt{\lambda(s,m_1^2,m_2^2)} \sqrt{\lambda(s,m_3^2,m_4^2)}}{2s}\cos\theta. \end{aligned}$$

Thus the physical interval is $t_-\le t\le t_+$, where $t_-$ is obtained at $\cos\theta=-1$ and $t_+$ at $\cos\theta=+1$. The Jacobian is

$$\frac{\mathrm dt}{\mathrm d\cos\theta} =2|\mathbf p_i||\mathbf p_f| =\frac{\sqrt{\lambda_i\lambda_f}}{2s}.$$

Equivalently, proposed invariants $(s,t)$ describe a real two-body collision only if both Källén functions are nonnegative and

$$\left| \frac{2s(t-m_1^2-m_3^2) +(s+m_1^2-m_2^2)(s+m_3^2-m_4^2)} {\sqrt{\lambda(s,m_1^2,m_2^2)} \sqrt{\lambda(s,m_3^2,m_4^2)}} \right|\le1.$$

The quantity inside the absolute value is $\cos\theta$. This invariant inequality is often a cleaner physical-region check than constructing explicit four-vectors.

For equal masses $m_i=m$, elastic scattering has

$$t=-2|\mathbf p|^2(1-\cos\theta), \qquad u=-2|\mathbf p|^2(1+\cos\theta),$$

so the physical $s$-channel obeys $s\ge4m^2$ and $t,u\le0$.

An independent invariant treatment of two-body collision and decay kinematics appears in Weinberg 1995, § 3.4, pp. 134–141.

The figure locates this region and its two crossed counterparts. Inspect the threshold lines and note that the small exchange graphs label which invariant can flow through a propagator; they do not by themselves prove analytic crossing.

Equal-mass $2\to2$ kinematics with $s+t+u=4m^2$. The $s$-channel has $s\ge4m^2$ and $t,u\le0$; the $t$- and $u$-channel regions follow by permutation. Threshold locations are exact, the displayed wedges are cropped, and the contact and exchange sketches are schematic and not to scale. Dashed arrows denote crossed continuation as a kinematic relation, not a proof of an analytic crossing theorem.

The same information has a nonvisual form:

Physical interpretation for equal masses	Timelike channel invariant	Other two invariants
$1+2\to3+4$	$s\ge4m^2$	$t\le0$, $u\le0$
$1+\bar3\to\bar2+4$	$t\ge4m^2$	$s\le0$, $u\le0$
$1+\bar4\to3+\bar2$	$u\ge4m^2$	$s\le0$, $t\le0$

Beyond two-to-two scattering

For $2\to n$, useful invariants include subsystem masses

$$s_I=\left(\sum_{i\in I}p_i\right)^2.$$

They are constrained by on-shell conditions, total momentum conservation, and Gram determinants. For any selected vectors $v_1,\ldots,v_r$, define

$$G_{ij}=v_i\cdot v_j.$$

In four spacetime dimensions, any five four-vectors are linearly dependent, so $\det G=0$ for their $5\times5$ Gram matrix. The condition is necessary, not sufficient: the on-shell equations, energy signs, and physical-region inequalities must also hold. A list of pairwise invariants that violates any of these constraints describes no real momentum configuration.

For a three-body decay $P\to p_1+p_2+p_3$, two pair masses such as $s_{12}$ and $s_{23}$ provide coordinates on a Dalitz plot. Its boundary occurs when the remaining decay angle reaches $\cos\theta=\pm1$, equivalently when the relevant Gram determinant vanishes. Phase space, rather than kinematics alone, supplies the integration measure on that domain.

Common pitfalls

“$s,t,u$ are always independent.” On-shell four-point kinematics imposes one linear relation. Special masses and dimensions can impose further constraints.

“A zero of $\lambda$ is always a physical production threshold.” The factorization has both threshold and pseudothreshold zeros. Future-directed positive-energy kinematics selects the physical branch.

“Crossing is just renaming $s,t,u$.” Permuting external momenta maps kinematic formulas, but relating physical amplitudes in separated regions requires analytic continuation, statistics phases, and state conventions.

“Any chosen $s$ and $t$ define a real collision.” The Källén functions must be nonnegative and $t$ must lie in its angular interval. Multi-particle configurations obey additional Gram constraints.

Check your understanding

1. Derive $s+t+u=\sum_i m_i^2$ by expanding the definitions.

   Answer

   Expand $s= m_1^2+m_2^2+2p_1\cdot p_2$, $t=m_1^2+m_3^2-2p_1\cdot p_3$, and $u=m_1^2+m_4^2-2p_1\cdot p_4$. Momentum conservation gives $p_2-p_3-p_4=-p_1$, so the dot-product combination equals $-2m_1^2$. The remaining masses give the stated sum.

2. For massless $2\to2$ scattering, show that $t=-\tfrac{s}{2}(1-\cos\theta)$.

   Answer

   With every mass zero, $E_i=E_f=\sqrt{s}/2$ and $|\mathbf p_i|=|\mathbf p_f|=\sqrt{s}/2$. Substitution in the angle formula gives $t=-s/2+(s/2)\cos\theta$.

3. At fixed $s$ above both thresholds, show that the endpoints $t_\pm$ coalesce when either the incoming or outgoing pair reaches threshold.

   Answer

   Their separation is

   $$t_+-t_-=4|\mathbf p_i||\mathbf p_f| =\frac{\sqrt{\lambda(s,m_1^2,m_2^2)} \sqrt{\lambda(s,m_3^2,m_4^2)}}{s}.$$

   If either Källén function vanishes, the corresponding center-of-mass momentum is zero and no scattering angle remains to vary, so the allowed $t$ interval collapses to one point.

Continue

Lorentz-Invariant Phase Space turns the allowed momentum domain into an integration measure. Mandelstam Channels and Tree-Level Crossing develops channel permutations and analytic continuation. High-Energy and Regge Limits treats the special large-$s$ limits.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.