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For the all-kink sector, keep ${\alpha }_{ij}={\tanh }^2[({\theta }_i-{\theta }_j)/2]$ and $E_i=e^{X_i}$. Enumerating the even and odd binary configurations gives the Hirota tau functionsThe last minus sign is the product of three negative pair coefficients. As in the two-body limit, the three velocities are $v_i={\beta }_i/{\kappa }_i=\tanh {\theta }_i$.

To follow the first kink, keep $X_1$ bounded and take $t\to \pm \infty$. Let $J_{\pm }$ be the set of spectators whose exponential diverges in that limit:For no large spectators, $g/f\to E_1$. For one large spectator $j$, $g/f\to -1/({\alpha }_{1j}{E}_1)$. For two large spectators, the dominant terms give $g/f\to {\alpha }_{12}{\alpha }_{13}{E}_1$. On a continuous branch of a multivalued function, each case has local profile $4\arctan \exp (X_1+c_{\pm })$ plus the appropriate vacuum offset, whereConsequently the incoming and outgoing intercepts are $-({\gamma }_1+c_{-})/{\kappa }_1$ and $-({\gamma }_1+c_{+})/{\kappa }_1$. This provesAgain the time expression requires $v_1\ne 0$ and distinct velocities; the corresponding spatial-shift identity holds also when $v_1=0$. With mixed orientations, use the general pair shift established above and determine growing spectators by the sign of ${\kappa }_j(v_1-v_j)$; the same multiplication of pair coefficients proves additivity.

There is no independent three-body contribution to the asymptotic shift. The pairwise additivity of soliton shifts is a classical manifestation of factorized scattering in an integrable partial differential equation. The collision preserves the individual asymptotic rapidities and profiles, and the net shift is independent of the sequence of separated pair collisions. In the quantum theory, consistency of the corresponding species-changing S-matrices becomes the Yang-Baxter equation; the classical scalar shift identity is its physical precursor, rather than a derivation of all quantum matrix identities.

Keep the particle labels specified under 2. There are two connected tree-level Feynman diagrams: the quartic contact interaction vertex and an $s$-channel light antiparticle exchange. The initial Noether charge is $1-2=-1$, exactly the charge of $\bar{\varphi }$. At the first cubic Feynman vertex, the incoming $\varphi$ and $\Phi$ leave the second $\varphi$ leg internal; the conjugate Feynman vertex supplies both outgoing particles.

ThusHere $s=(p_1+p_2{)}^2$ for incoming $\varphi (p_1),\Phi (p_2)$. Both terms have the same sign in the chosen S-matrix convention.

The distinction from antiparticles matters. If the paper's heavy “$\Phi$ particle” is instead intended to mean the charge-$+2$ state $\bar{\Phi }$, so that its printed decay into $\varphi \varphi$ is allowed, the exchange diagram is crossed. For that alternative physical process,This is a consistent alternative naming convention only if it is also applied to the decay in iii; one cannot use the same heavy particle label for both charge assignments.

Rapidity is a dimensionless parameter for a signed collinear velocity $u$ with $|u|<c$, defined using the inverse hyperbolic functions byHere $c$ is the speed of light, and $\tanh$ is the hyperbolic tangent. Zero velocity has zero rapidity; negative velocity has negative rapidity. Every finite real rapidity gives $|u|<c$, while $u\to \pm c$ requires $\theta \to \pm \infty$. Thus a light ray has no finite rapidity in this parametrization. The header's formula uses natural units $c=1$: its $v$ is the numerical velocity $u/c$.

For the inertial frames in the velocity-addition formula, $S^{\prime }$ moves at signed velocity $V$ along the $x$-axis of $S$. Set $\varphi =\mathrm{artanh}(V/c)$. Re-expressing the same particle in the moving inertial frame givesThe subtraction is a passive change of reference frame; the inverse adds the particle's rapidity in $S^{\prime }$ to the frame's rapidity relative to $S$. For example, $V=0.6c$ and $u^{\prime }=0.6c$ give $\varphi ={\theta }^{\prime }=\ln 2$, so $\theta =\ln 4$ and $u=c\tanh (\ln 4)=15c/17$, exactly as in the velocity-addition formula.

For a massive particle of rest mass $m>0$ moving in one spatial dimension, its energy and signed momentum areThe hyperbolic cosine is the particle's Lorentz factor, and the hyperbolic sine is $\gamma u/c$. The hyperbolic Pythagorean identity immediately gives the relativistic energy-momentum relation $E^2-c^2{p}^2=m^2{c}^4$. In the older $c=1$ notation, writing the rest mass as $M$ recovers $v=\tanh \theta$, $E=M\cosh \theta$, and $p=M\sinh \theta$. These expressions use natural units in which the factors of $c$ are numerically one; the preceding formulas restore those factors explicitly. ^[ref].

The useful simplification follows from the hyperbolic-function identity $\tanh (a+b)=(\tanh a+\tanh b)/(1+\tanh a\tanh b)$: the nonlinear velocity-addition formula becomes ordinary addition of rapidities. See iterated collinear boosts with equal rapidity for a plotted example. A common collinear Lorentz boost shifts every rapidity by the same constant, so a rapidity difference is invariant. This makes rapidity useful for relativistic factorized scattering in one spatial dimension, where two-body S-matrices depend on the difference of the incoming rapidities, with particle species fixed.

A specified asymptotic configuration of the particles and their internal quantum states in quantum scattering. An incoming channel can scatter into several outgoing channels; S-matrix elements give their probability amplitudes with a stated probability current normalization. A channel can be energetically inaccessible even when other channels are open.

For a conserved charge $Q$ and invariant vacuum, its global symmetry commutes with the S-matrix. Matrix elements between charge eigenstates give the displayed selection rule. Constant electromagnetic phases cancel at every QED vertex, and nonzero physical amplitudes conserve total charge. Local gauge invariance strengthens this global rule to the Ward-Takahashi identity.

The S-matrix maps incoming asymptotic states to outgoing asymptotic states. Its connected transition matrix elements contain an overall energy-momentum Dirac delta function and an invariant scattering amplitude. In quantum field theory, the LSZ reduction formula obtains these elements from amputated field correlators.

Consider one chosen model at a fixed total energy, with two open scattering channels, no absorption and no omitted outgoing modes. Channel 1 leaves the target unchanged; channel 2 excites it. Let $v_1=v>0$ and $v_2=v/2$. For the same projectile mass, conservation of energy then requires a target excitation $\Delta =3m{v}^2/8$. These are hypothetical parameters, not a fit to a particular atom. Each incoming or outgoing mode is normalized to unit probability current; the outgoing modes will have the angular profiles specified below.

Choose the S-matrixThis phenomenological unitary matrix specifies the channel conversion; it is not derived here from an interaction potential energy. For incidence in channel 1, its first column gives the outgoing flux amplitudesTo recover ordinary wave coefficients relative to an incident coefficient one, undo the velocity normalization:Thus $|c_2{|}^2=32/25>1$ is allowed: channel 2 travels more slowly. Its outgoing probability current fraction is $16/25$, not $32/25$. Renormalizing the unweighted squares would give $9/41$ and $32/41$, the wrong probabilities. More generally the ordinary coefficient map is $A=V^{-1/2}S{V}^{1/2}$, with $V=\mathrm{diag}(v,v/2)$; it obeys $A^{†}VA=V$. The conserved quantity is the squared coefficient norm weighted by velocity, which represents probability current.

Now specify an original angular realization of these two outgoing modes, with scattering angle $\theta$ and no azimuth dependence:The first normalization follows from $2\pi {\int }_{-1}^1(1+u^2)du=16\pi /3$, where $u=\cos \theta$. These angular profiles define the modes of this model; they are not asserted to follow from Born's approximation or an atomic potential energy. The joint directional probability densities are

Suppose a detector accepts the cone $D:0\le \theta \le \pi /3$, with full azimuth and unit efficiency. Its angular fractions within the channels areMultiplying the angular fractions by the outgoing flux weights givesFor $1600$ independent incident events, the expected values are therefore $171$ and $256$ accepted detections, compared with $576$ and $1024$ over all directions. The remaining expected $1173$ events fall outside the cone. The chosen incident number makes the expectations integers; actual counts fluctuate. Conditional on a detection inside the cone, the channel probabilities are $171/427$ and $256/427$, rather than $9/25$ and $16/25$. Angular acceptance changes the observed channel proportions even with equal detector efficiencies.

The final atomic states are orthogonal, so ignoring the atomic readout adds the two channel intensities; it does not produce $|b_1{g}_1+b_2{g}_2{|}^2$. Coherent paths that reach the same final state can instead interfere, as the existing probability amplitude and quantum superposition examples demonstrate. The distinction is physical distinguishability of final states, not whether the detector operator chooses to display their labels. Flux normalization and angular integration make the scattering interpretation useful: they turn complex outgoing waves into conserved event weights and then into the statistics of a specified detector.

A contribution to the unitarity relation for the S-matrix obtained by placing a set of intermediate lines on shell and summing their intermediate states. In a physical-state formulation only physical states enter the sum. When covariant gauge boson index sums include auxiliary polarizations, matching Faddeev-Popov ghost field contributions are needed to reproduce the physical result in a Non-Abelian gauge theory.

For real $u$, this scalar S-matrix block satisfies $S_u(\theta )S_u(-\theta )=1$ and Hermitian analyticity of a two-particle S-matrix. It therefore has modulus one for real rapidity differences. For $0<u<\pi$ its pole at $\theta =iu$ has residue $2i\sin u$ and admits a direct-channel bound-state pole interpretation. Its crossed block is $\cosh [(\theta -iu)/2]/\cosh [(\theta +iu)/2]$. The endpoints $u=0$ and $u=\pi$ are removable degeneracies giving constant amplitudes, so a particle cannot be inferred there from the uncancelled denominator alone. This block specifies an amplitude; a complete theory must also obey the remaining crossing symmetry and bootstrap fusion constraints.