Shrink a family of S matrices and reproduce a published curve

Map the strip to a disk

Keep $m=1$ and $\alpha=\pi/3$ . The hyperbolic tangent maps the physical strip to the unit disk. Crossing sends $z$ to $-z$ ; real rapidity follows the lower semicircle. This disk coordinate shares a letter with the earlier BCFW parameter but serves a different purpose.

Multiply the fixed CDD factor by a finite rational function. Real coefficients $r,a,b$ control its shape, with $0.1\leq r\leq1$ . Freeze this family first: every allowed region refers to these functions under the stated necessary conditions.

z(\theta)=\tanh\frac{\theta-i\pi/2}{2},\qquad z(i\pi-\theta)=-z(\theta)

The prescribed CDD poles have been factored out explicitly. The extra factor still needs a check for new poles.

S(\theta)=C_{\pi/3}(\theta)\frac{r+iaz(\theta)}{1-bz(\theta)^2}

This is our teaching ansatz. It differs from the paper’s dispersive spline parameterization and defines a different finite numerical problem.

Source: M. F. Paulos · Patrick Dorey

See what each condition removes

Crossing requires the added factor to be invariant under $z\mapsto-z$ . Its denominator is already even, so the numerator’s linear term must vanish: $a=0$ . In coefficient space this collapses a two-dimensional region onto a line.

A fixed pole spectrum forbids extra denominator roots inside the disk. Their equation is $z^2=1/b$ . For $|b|>1$ they enter the disk. At $|b|=1$ they reach threshold or high-energy boundaries, which this model also requires to be regular. Positive $r$ prevents a linear numerator from canceling both roots. With crossing switched off, the numerator can also cancel a prescribed CDD pole on isolated coefficient lines. The fixed-spectrum check excludes those lines too.

Finally impose the modulus bound on the whole physical edge. The CDD factor already has unit modulus there. Expanding squared moduli reduces infinitely many real-rapidity inequalities to the maximum of a quadratic on one closed interval.

a=0,\qquad |b|<1

The first is exact crossing within this family. The second includes the explicitly stated regularity of its boundary limits.

t=-\operatorname{Im}z\in[0,1],\quad F(t)=r^2+a^2-(1-b)^2+2rat-4bt^2\leq0

This is numerator squared modulus minus denominator squared modulus. Endpoints include the corresponding boundary limits.

Worked example: solve the allowed region

Set $a=0$ by crossing. For $b\geq0$ the quadratic has its maximum at $t=0$ ; using $|b|<1$ gives $b\leq1-r$ . For $b<0$ it bends upward and its maximum is at $t=1$ , giving $-b\leq1-r$ . The cases join into one exact interval.

Choose $r=0.6$ . The allowed interval is $-0.4\leq b\leq0.4$ . At $b=0.3$ , the smallest denominator modulus is $0.7$ , so the largest boundary modulus is $0.6/0.7=6/7$ . At $b=0.5$ it becomes $1.2$ and violates the unitarity necessary condition.

Consider instead $r=0.6,a=0,b=2$ . The largest boundary modulus is only $0.6$ , but extra poles occur at $z=\pm1/\sqrt2$ inside the disk. A boundary-modulus test keeps this candidate; the fixed-spectrum condition excludes it.

\boxed{a=0,\qquad |b|\leq1-r\qquad(0.1\leq r\leq1)}

This is an analytic result for the stated conditions within this family. A complete-theory interpretation requires additional channels and threshold conditions.

Reproduce the specified scope of Figure 5a

Set $r=1,a=b=0$ to recover the exact CDD reference. Figure 5a of Paulos et al., arXiv:1607.06110v2, uses exchanged mass $m_1=\sqrt3$ . Equation (9) writes this function in energy squared and equation (14) in rapidity. Independent evaluations of real part, imaginary part and modulus expose branch and sign errors.

On the physical upper edge $s+i0$ , choose $Q(s)=-i\sqrt{s}\sqrt{s-4}$ . Thus $Q=-2i\sinh\theta$ , matching our positive CDD convention. At $s=6$ , $\sinh\theta=\sqrt3$ and $S=3/5+4i/5$ . This hand-calculable complex number supplies an independent reference without fitting.

The page and exported table recalculate function values. They do not digitize paper artwork or numerical optimization points. The reproduced scope is the specified analytic curve and residue. The paper’s dispersive spline optimization, full mass-dependent maximal-coupling curve and multiparticle spectra are outside this browser calculation.

S_{\rm ref}(s)=\frac{Q(s)+\sqrt3}{Q(s)-\sqrt3},\qquad Q(s)=-i\sqrt{s}\sqrt{s-4}\quad(s>4)

This specializes equation (9) to external mass one and exchanged mass square root of three on the chosen physical edge.

S_{\rm ref}(6)=\frac35+\frac45i,\qquad |S_{\rm ref}(6)|=1,\qquad\operatorname*{Res}_{s=3}S_{\rm ref}=-6

A complex value, unit boundary modulus and an interior-pole residue provide three distinct checks.

Source: M. F. Paulos

A grid finds counterexamples; a maximum controls the interval

For general parameters, test the quadratic at both endpoints. If it bends downward and its stationary point is inside the interval, test that point too. These finitely many candidates determine the whole boundary-modulus condition exactly in this family. An independent dense rapidity grid checks the implementation but does not replace the argument.

More general families make the difference harder to control. Passing a finite grid establishes only sampled inequalities. Increasing basis order can leave more candidate functions; adding test points can exclude some of them. A rigorous bound needs control of the continuous boundary, truncation remainders and numerical error.

A surviving function still needs a physical completion. This family supplies no other channels or inelastic thresholds. Many subunitary candidates lose elastic probability arbitrarily close to threshold; for the assumed lightest-external-particle spectrum with no other channel opening at or below its elastic threshold, a completion needs an elastic interval before its first inelastic threshold, plus the remaining spectrum and residue conditions. The interval answers a finite necessary-condition problem. The exact CDD reference has separate support from a known theory.

Source: M. F. Paulos

Try the reasoning in an experiment

Predict an interval before testing it

Set the scale to zero point six and calculate endpoints at plus and minus zero point four. Turn crossing, fixed spectrum and unitarity on separately and compare the eliminated regions.
Choose the preset whose boundary modulus passes but whose interior has extra poles. Explain why the boundary trajectory alone would mislead.
Return to the exact CDD reference and export its curve. Recalculate it from the energy-squared expression and compare real part, imaginary part, modulus and residue. Use the saved parameters in an external script to compare a general candidate on coarse and dense grids with the analytic maximum.

Exercises and explanations

Why does $r=1$ reduce the allowed interval to one point? Does this leave only one theory in all of theory space?
Read the explanation
Within this family $|b|\leq1-r$ forces $b=0$ , while crossing forces $a=0$ . The uniqueness concerns this ansatz, prescribed CDD spectrum and listed conditions. Other function families, spectra and channels remain outside it.
For $b>0$ , where is the quadratic’s stationary point and when must it be tested?
Read the explanation
Differentiation gives $t_*=ra/(4b)$ . It is an interior candidate maximum only if $0<t_*<1$ . Otherwise endpoints suffice. The linear case $b=0$ requires separate handling.
If a dense grid contradicts the analytic result, what should be checked first?
Read the explanation
Save parameters, precision and the largest violation. Then check the variable map, boundary branch, quadratic expansion and stationary-point handling. The figure must follow the same model; changing a display threshold to erase the contradiction would hide an error.

Sources and reproduced scope

Worked examples and schematic figures are authored for this course. Numerical values are calculated model references. The published comparison recalculates a specified analytic CDD curve; it does not copy experimental data, source artwork or the paper’s numerical optimization points. Source locations and reproduced scope are listed below.

M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, The S-matrix Bootstrap II: Two Dimensional Amplitudes, arXiv:1607.06110v2; JHEP 11 (2017) 143
Section 2 equations (1)–(3) fix crossing, elastic unitarity and coupling normalization. Equations (9), (13)–(14), Figures 5a and 6 give the sine-Gordon reference, CDD factor and physical strip. We recalculate the analytic Figure 5a reference at external mass one and exchanged mass square root of three, its pole residue and the complex value at energy squared six. We do not reproduce the dispersive optimization of section 2.1/Appendix A. The rational three-parameter family is our extension.
Patrick Dorey, Exact S-matrices, arXiv:hep-th/9810026v1 (1998)
Section 3, printed pages 12–16, derives the rapidity map, physical strip, real analyticity, matrix unitarity and crossing. Section 4, pages 16–21, relates imaginary rapidity to bound-state masses and discusses sine-Gordon poles, including the limits of identifying every pole with a new particle. The course uses the scalar identical-particle specialization; its finite rational ansatz is separately defined.