The Dressing Factor and Crossing Equations

TL;DR

The paper investigates the BES dressing factor for the AdS$_5\times$S$^5$ world-sheet S-matrix, using the DHM integral representation and crossing equations to fix the principal branch on the rapidity torus. It then extends the construction to bound-state and mirror-theory scattering, showing that the fused bound-state dressing factor or the improved factor $\Sigma^{QQ'}$ depends only on bound-state kinematics and is universal across internal constituent configurations. The authors provide explicit analytic continuations via the $\Phi$- and $\Psi$-functions, verify crossing for fundamental particles and bound states, and derive a final bound-state factor for the mirror theory that is suitable for completing the TBA equations. Overall, the results establish a crossing-consistent branch of the BES phase and demonstrate the universality of the mirror bound-state S-matrix, enabling a consistent finite-size spectral analysis through the mirror TBA.

Abstract

We utilize the DHM integral representation for the BES dressing factor of the world-sheet S-matrix of the AdS_5xS^5 light-cone string theory, and the crossing equations to fix the principal branch of the dressing factor on the rapidity torus. The results obtained are further used, in conjunction with the fusion procedure, to determine the bound state dressing factor of the mirror theory. We convincingly demonstrate that the mirror bound state S-matrix found in this way does not depend on the internal structure of a bound state solution employed in the fusion procedure. This welcome feature is in perfect parallel to string theory, where the corresponding bound state S-matrix has no bearing on bound state constituent particles as well. The mirror bound state S-matrix we found provides the final missing piece in setting up the TBA equations for the AdS_5xS^5 mirror theory.

Figures (11)

• Figure 1: On the left figure the torus is divided by the curves $|x^+|=1$ and $|x^-|=1$ into four non-intersecting regions. The middle figure represents the torus divided by the curves ${\rm Im}(x^+)=0$ and ${\rm Im}(x^-)=0$, also in four regions. The right figure contains all the curves of interest.
• Figure 2: The curves x$_\pm^{(n)}=1/x(u\pm {2i\over g} n)$ with $-2\le u=w+{1\over w} \le 2$ for $g=3$ and $n=1,2,3,4$. The endpoints of the curves correspond to $w=\pm 1$. The curves closest to the circle correspond to $n=1$.
• Figure 3: A little dragging of the variable $x_1$ inside the integration contour results into an extra contribution given by the integral around $x_1$ with integration performed in the clock-wise direction.
• Figure 4: On the left figure analytic continuation paths are shown on the $z$-torus. On the right figure blue curves represent the lines $x^+(z)$ corresponding to the torus variable $z$ going upward from the real line to the line with Im$(z)={\omega_2}/i$ and they have $|$Re$(z)|\le {\omega_1\over 4}$. Black curves $x^+(z)$ correspond to $z$ going upward and have $|$Re$(z)|\ge {\omega_1\over 4}$. Any black or blue curve intersects the lowest curve inside the circle. Paths sufficiently close to the lines $|$Re$(z)|\le {\omega_1\over 4}$ do not intersect any cut except the lowest curve and, therefore, they are used for analytic continuation.
• Figure 5: Blue and black curves on the right figure represent the curves $x^-(z)$ corresponding to the curves $x^+(z)$ in Figure 4. No black curve intersects the cuts inside the circle, and the blue curves close enough to Re$(z)= {\omega_1\over 4}$ do not intersect the cuts either.