FR05-B · Which scattering functions survive?

S-matrix Bootstrap

How much does consistency tell us about a scattering amplitude? Crossing relates two descriptions of the same process, the pole spectrum fixes which stable particles may appear, and unitarity limits the elastic probability. Together they leave fewer candidate scattering functions.

Drag a proposed amplitude through a coefficient plane. Watch crossing, the pole spectrum and probability conservation remove different parts of the plane.

Largest elastic magnitude $0.786$

Crossing mismatch $0$

Pole-spectrum defects $0$

Selected elastic probability $0.304$

Model assumptions

This finite rational ansatz tests necessary conditions on an identical-particle elastic S-matrix element in one spatial dimension. The external mass is one and the prescribed bound-state mass is the square root of three. Analyticity includes the fixed physical-sheet pole spectrum and regular high-energy and threshold boundaries. Generic subunitary candidates lose elastic probability down to threshold. With a unique lightest external species and no other channel open at that threshold, a complete theory has a purely elastic interval; this ansatz does not enforce it or supply inelastic channels. The allowed interval is exact within this ansatz; it does not classify complete quantum field theories. Curves and exported values are calculated model data.

Finite ansatz · calculated curve

Following the amplitude around the complex plane

CandidateExact elastic CDD referenceSelected rapidity

\theta=1.00,\quad s/m^2=5.09

The violet reference has unit magnitude at every real rapidity. A candidate outside the unit circle assigns too much probability to the elastic channel.

Paulos et al. · 1607.06110v2 · Figure 5a

Two variables, one analytic scattering function

The reference uses the one-pole CDD solution at bound-state mass square root of three. The independent energy-domain formula below reproduces the analytic curves compared in Figure 5a; the paper’s numerical optimizer is outside this reproduction.

Real partImaginary partMagnitude

S(s)=\frac{\sqrt{s}\sqrt{4-s}+\sqrt3}{\sqrt{s}\sqrt{4-s}-\sqrt3}

Lines use the energy-domain formula; dots independently use the rapidity CDD factor. Above threshold the physical upper rim uses the negative imaginary square root. Both calculations, their grid and discrepancies are saved in the model snapshot.

\max|S_s-S_\theta|<10^{-12}

Source equations 9 and 14 Read the course

Start with the essentials

Focus question: How much does consistency tell us about a scattering amplitude?
One-sentence intuition: A function can conserve probability on the real energy axis and still have forbidden poles inside the physical sheet.

Core mathematical model

Putting the physical sheet inside a disk

\begin{aligned}s&=4m^2\cosh^2(\theta/2),\quad m=1\\z&=\tanh\!\left(\frac{\theta-i\pi/2}{2}\right)\\\theta\mapsto i\pi-\theta&\quad\Longrightarrow\quad z\mapsto-z\end{aligned}

Rapidity labels the collision energy. The physical sheet becomes the strip between zero and pi in imaginary rapidity; the change to z maps its interior into the unit disk. Crossing exchanges the two sides of that disk.

A fixed bound state and a finite candidate family

\begin{aligned}C_\alpha(\theta)&=\frac{\sinh\theta+i\sin\alpha}{\sinh\theta-i\sin\alpha},\quad\alpha=\frac\pi3\\S_{a,b,r}(\theta)&=C_\alpha(\theta)\frac{r+iaz}{1-bz^2}\\m_1&=2\cos(\alpha/2)=\sqrt3\end{aligned}

The CDD factor supplies the prescribed bound-state pole and its crossed partner. The rational multiplier changes the amplitude. Exact crossing sets a to zero. Keeping the prescribed pole spectrum and regular boundaries requires the magnitude of b to be smaller than one.

Testing probability conservation everywhere

\begin{aligned}t&=\operatorname{sech}\theta\in[0,1]\\F(t)&=r^2+a^2-(1-b)^2+2rat-4bt^2\\|S(\theta)|\le1\quad&\Longleftrightarrow\quad\max_{0\le t\le1}F(t)\le0\\a=0,\ |b|<1\quad&\Longrightarrow\quad |b|\le1-r\end{aligned}

For this ansatz unitarity reduces to the maximum of one quadratic polynomial. Checking both endpoints and its interior maximum tests the entire real rapidity axis, including the high-energy limit. The colored region uses this algebraic test; the plots sample the same function.

Common difficulties

An allowed amplitude is only one consistency test

Typical misconceptionEvery allowed point defines a complete quantum field theory.

Better mental modelThe test concerns one elastic matrix element and a finite ansatz. Other scattering channels, residue signs and the existence of a full theory require additional conditions.

A line really has zero width

Typical misconceptionThe bright line is a narrow tolerance band.

Better mental modelCrossing is an exact functional identity. The plot draws the zero-width set with a visible stroke; dragging near its center snaps the odd coefficient to zero.

Run the experiment

01
Remove one requirement
Predict what happens if you uncheck Crossing. Then drag the gold point sideways.
What to observe: A two-dimensional region opens up. Restoring crossing leaves only the vertical line because the odd coefficient must vanish exactly.
02
Look away from the real axis
Choose Hidden physical-sheet poles. Uncheck Fixed pole spectrum while keeping Unitarity enabled.
What to observe: The candidate passes boundary probability conservation. Restoring the pole requirement rejects it: two additional denominator zeros sit inside the disk.
03
Reach the elastic solution
Choose Elastic CDD solution. Lower Numerator scale and predict whether the allowed interval grows or shrinks.
What to observe: At unit scale the allowed interval contracts to a point, and the complex trajectory lies on the unit circle. A smaller scale opens an interval inside the disk. Completing such a function with physical inelastic channels needs further conditions.

S-matrix Bootstrap

Following the amplitude around the complex plane

Two variables, one analytic scattering function