Cross Sections and Decay Rates

For stable external particles, a convention-normalized differential cross section or decay width is obtained by multiplying the squared invariant amplitude by Lorentz-invariant phase space and dividing by the normalization appropriate to the initial state. A two-particle initial state contributes the invariant flux; a one-particle decay contributes $2M$ in its rest frame. Unobserved final quantum numbers are summed, unprepared initial degeneracies are averaged, and a factorial removes any overcounting of identical final configurations. Calling the result an observable additionally requires a finite, specified measurement—an important qualification in theories with massless quanta.

Required background. Lorentz-Invariant Phase Space fixes $\mathrm d\Phi_n$. LSZ Reduction fixes the residue-normalized invariant amplitude $\mathcal M$ and its stable-state domain.

Helpful background. Plane Waves, Spin Sums, and Bilinears supplies the spin projectors used in unpolarized rates.

The invariant flux for two incoming particles

For $p_1+p_2\to f$, define

$$\mathcal F =4\sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2} =2\sqrt{\lambda(s,m_1^2,m_2^2)}.$$

In a frame description,

$$\mathcal F=4E_1E_2v_{\mathrm M},$$

where

$$v_{\mathrm M} =\frac{\sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2}}{E_1E_2}$$

is the frame-dependent Møller flux velocity; the product $E_1E_2v_{\mathrm M}$ is invariant. For collinear counter-propagating beams, $v_{\mathrm M}=|\mathbf v_1-\mathbf v_2|$. It is a flux factor and can exceed one, so it should not be confused with the physical speed of either particle measured in the rest frame of the other. In the center-of-mass frame,

$$\mathcal F=4\sqrt{s}\,|\mathbf p_i|.$$

This factor divides out the incident particle densities and their encounter rate. It is not a symmetry factor and does not depend on the final state.

The master cross-section formula

With the chapter convention

$$\langle f|S|i\rangle_c =i(2\pi)^4\delta^{(4)}(P_f-P_i)\mathcal M_{fi},$$

the differential cross section is

$$\boxed{ \mathrm d\sigma_{2\to n} =\frac{1}{\mathcal F} \frac{1}{S_f}\, \overline{|\mathcal M_{fi}|^2}\, \mathrm d\Phi_n. }$$

$S_f=\prod_a n_a!$ if the chosen labeled integration domain counts all permutations of $n_a$ identical final particles of each species $a$. If the integration region already chooses one representative per permutation orbit, set $S_f=1$.

The bar means “sum and average according to the measurement,” not a universal algebraic operation. For an unpolarized process with initial degeneracies $g_1,g_2$,

$$\overline{|\mathcal M|^2} =\frac{1}{g_1g_2} \sum_{\text{initial labels}} \sum_{\text{unobserved final labels}} |\mathcal M|^2.$$

Do not average an initial polarization that was prepared, and do not average over final states that are being resolved. Schwartz derives the amplitude, flux, and phase-space combination with the same state normalization in Schwartz 2014, § 5.1, pp. 57–63.

More generally, a partially polarized or coherent initial ensemble is described by density matrices:

$$\overline{|\mathcal M|^2} =\sum_{r_1r_2r'_1r'_2,f} \rho^{(1)}_{r_1r'_1}\rho^{(2)}_{r_2r'_2} \mathcal M_{f;r_1r_2} \mathcal M^*_{f;r'_1r'_2}.$$

The familiar average is the special choice $\rho^{(a)}=1/g_a$. Off-diagonal entries retain interference between coherently prepared spin states, so replacing every preparation by an unpolarized average can discard physical angular information.

Two-body scattering

Using

$$\mathrm d\Phi_2 =\frac{1}{16\pi^2} \frac{|\mathbf p_f|}{\sqrt{s}}\,\mathrm d\Omega$$

and $\mathcal F=4\sqrt{s}|\mathbf p_i|$ gives

$$\boxed{ \frac{\mathrm d\sigma}{\mathrm d\Omega} =\frac{1}{64\pi^2s} \frac{|\mathbf p_f|}{|\mathbf p_i|} \frac{1}{S_f}\, \overline{|\mathcal M|^2}. }$$

For azimuthally symmetric scattering,

$$\frac{\mathrm d\sigma}{\mathrm dt} =\frac{1}{16\pi\lambda(s,m_1^2,m_2^2)} \frac{1}{S_f}\, \overline{|\mathcal M|^2},$$

provided $t$ is used over its physical interval. The two formulas agree because $\mathrm dt/\mathrm d\cos\theta=2|\mathbf p_i||\mathbf p_f|$ and $\mathrm d\Omega=2\pi\,\mathrm d\cos\theta$.

The reduction from the invariant master formula to these two-body expressions is also given in Srednicki 2007, § 11, pp. 93–101.

As a normalization check, take equal-mass real scalars with $\mathcal L_{\mathrm{int}}=-\lambda\phi^4/4!$. At tree level $\mathcal M=-\lambda$ and $|\mathbf p_f|=|\mathbf p_i|$. Over the full labeled solid angle,

$$\frac{\mathrm d\sigma}{\mathrm d\Omega} =\frac{\lambda^2}{128\pi^2s}$$

for two identical final scalars, where $S_f=2!$. Omitting the factorial would count the configurations $(p_3,p_4)$ and $(p_4,p_3)$ separately.

Decay widths

For a particle of physical mass $M$ decaying at rest,

$$\boxed{ \mathrm d\Gamma_{1\to n} =\frac{1}{2M} \frac{1}{S_f}\, \overline{|\mathcal M|^2}\, \mathrm d\Phi_n. }$$

Here an overline averages over the parent spin only when the parent is unpolarized; final labels are summed if unobserved. For a moving parent, the coordinate-time rate is smaller by $M/E$, expressing time dilation. The rest-frame width is the invariant quantity usually quoted.

For a two-body decay,

$$\frac{\mathrm d\Gamma}{\mathrm d\Omega} =\frac{|\mathbf p_*|}{32\pi^2M^2} \frac{1}{S_f}\, \overline{|\mathcal M|^2}.$$

If the amplitude is angle independent,

$$\Gamma =\frac{|\mathbf p_*|}{8\pi M^2} \frac{1}{S_f}\, \overline{|\mathcal M|^2}.$$

For $\mathcal L_{\mathrm{int}}=-\tfrac g2X\phi^2$, with $M>2m$ and two identical final $\phi$ particles,

$$\mathcal M=-g, \qquad \Gamma(X\to\phi\phi) =\frac{g^2}{32\pi M} \sqrt{1-\frac{4m^2}{M^2}}.$$

The threshold square root comes entirely from phase space. The amplitude is finite there at this order.

An unstable parent is not an exact in-state at $t=-\infty$. The formula is understood as the width extracted in perturbation theory over times long compared with microscopic interaction scales but short compared with the lifetime, or equivalently from the corresponding pole expansion. Weinberg states this time-window qualification in Weinberg 1995, § 3.4, pp. 136–141. Broad resonances and measured line shapes require the dedicated unstable-particle treatment.

Dimensional checks

In four dimensions,

$$[\mathrm d\Phi_n]=2n-4, \qquad [\mathcal M_{2\to n}]=2-n.$$

Therefore $|\mathcal M|^2\mathrm d\Phi_n$ is dimensionless, and division by $\mathcal F$ gives $[\sigma]=-2$. For a $1\to n$ amplitude, $[\mathcal M]=3-n$, so $|\mathcal M|^2\mathrm d\Phi_n/(2M)$ has dimension one, as a width must.

From partonic rate to measured observable

The master formulas are not yet detector predictions. Experimental acceptance, resolution, cuts, unstable-particle reconstruction, and hadronic initial states introduce measurement functions, factorization inputs, and often inclusive sums needed for infrared safety. A finite exclusive amplitude is not automatically an observable in a theory with massless quanta.

Continue to Measurement Functions and Inclusive Observables for the measurement map, Infrared and Collinear Safety for massless limits, and Unstable-Particle Observables and Controlled Resonance Approximations for line shapes.

Common pitfalls

“Sum over initial spins for an unpolarized beam.” Sum and then divide by the number of equally populated initial states. A prepared spin state is not averaged.

“Identical initial particles require a $1/2!$.” The cross section is defined per incident flux of the prepared beams. The factorial corrects overcounted identical final configurations in the integration domain.

“A decay width is an ordinary eternal S-matrix transition from an unstable ket.” An unstable particle is not an exact asymptotic state. The width is a controlled pole or finite-time quantity.

“The flux is $4E_1E_2|\mathbf v_1-\mathbf v_2|$ in every geometry.” The invariant statement uses $\sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2}$. The simple velocity difference applies to collinear beams.

Check your understanding

1. Derive the invariant flux in the center-of-mass frame.

   Answer

   Since $\lambda(s,m_1^2,m_2^2)=4s|\mathbf p_i|^2$, $\mathcal F=2\sqrt\lambda=4\sqrt{s}|\mathbf p_i|$.

2. Why is there no final-spin average in an inclusive unpolarized rate?

   Answer

   Distinct final spin states are distinct allowed outcomes, so their probabilities add. An average is used only over an initial statistical mixture whose total incident flux is fixed.

3. For massless distinguishable $2\to2$ scattering with an angle-independent amplitude $\mathcal M_0$, integrate the full solid angle.

   Answer

   Here $|\mathbf p_f|/|\mathbf p_i|=1$ and $S_f=1$, so

   $$\sigma=4\pi\frac{|\mathcal M_0|^2}{64\pi^2s} =\frac{|\mathcal M_0|^2}{16\pi s}.$$

   For two identical final particles integrated over the same labeled solid angle, divide this result by $2!$.

Continue

The Optical Theorem and Cut Interpretation relates inclusive rates to the forward amplitude. Phase-Space Integration and Monte Carlo Estimators develops validated numerical integration.

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.