Deriving the isotropic QTM reference formulas¶

Usage, measures and limitations · Reference source

This page records the algebra implemented by the two pointwise references. Equation numbers refer to the published papers: JMP Theorem 4 is Eq. (282), whereas arXiv v3 calls it (284). JMP means Babenko et al., J. Math. Phys. 62, 041901 (2021); PRL means Babenko et al., Phys. Rev. Lett. 126, 210602 (2021).

Both return the symmetric Pauli-zz amplitude, with its integration measure exposed separately. The independently normalized real-rapidity vertex control uses the operator/Jacobian conversion in the usage guide. The computations below do not establish a global contour deformation or a general proof of higher-sector equivalence.

Shared ordinary special functions¶

Writing \(G\) for the Barnes function, JMP (275)–(276) give

\[ \widehat p(z)=\frac{\pi}{2} -i\log\frac{\cosh[\pi(z+i/2)/2]}{\cosh[\pi(z-i/2)/2]}, \]

\[ \Psi(z)= \Gamma\!\left(\frac12-\frac{iz}{2}\right) \Gamma\!\left(1-\frac{iz}{2}\right) \frac{G(1-iz/2)^2}{G(1/2-iz/2)^2}. \]

Define

\[ Q=\frac12\sum_{j=1}^{\ell}[\widehat p(v_j)-\widehat p(u_j)], \qquad S(z)=i\, \frac{\Gamma(1+iz/2)\Gamma(1/2-iz/2)} {\Gamma(1-iz/2)\Gamma(1/2+iz/2)}. \]

In products over \(U\ominus V\), a hole has sign \(+1\) and a particle sign \(-1\). A double product includes both signs, including diagonal self-pairs. Logarithms are used to form Gamma/Barnes products; the final amplitudes, rather than individual log branches, are compared.

JMP: direct quadrature and the distinguished-column trap¶

JMP (279)–(280) use

\[ g(x,y)=\frac{\pi}{\sinh[\pi(y-x)]},\qquad e(z)=\frac1z-\frac1{z-i}, \]

\[ \phi^\sigma(t)= \prod_{w\in U\ominus V} \Gamma\!\left(\frac12-\frac{\sigma i(t-w)}2\right) \Gamma\!\left(1+\frac{\sigma i(t-w)}2\right). \]

The two matrices are

\[ \Omega_{jk}=g(u_j,v_k) -\int_{\mathbb R}\frac{dt}{2\pi i} \frac{\phi^-(v_k)}{\phi^+(t)}g(u_j,t)e(t-v_k), \]

\[ \overline\Omega_{jk}=g(v_j,u_k) -\int_{\mathbb R}\frac{dt}{2\pi i} \frac{\phi^-(t)}{\phi^+(u_k)}g(v_j,t)e(u_k-t). \]

Construct all possible modified columns, not just the first:

\[ D_{jk}=\int_{\mathbb R}\frac{dt}{\pi} \frac{\phi^-(v_k)}{\phi^+(t)}\frac{g(u_j,t)}{v_k-t}, \qquad \overline D_{jk}=\int_{\mathbb R}\frac{dt}{\pi} \frac{\phi^-(t)}{\phi^+(u_k)}\frac{g(v_j,t)}{u_k-t}. \]

For matrices \(B,C\), let

\[ T(B,C)=\sum_{k=1}^{\ell} \det(B\text{ with column }k\text{ replaced by column }k\text{ of }C). \]

The symmetric amplitude used for the pointwise comparison is

\[ A_\ell^{\rm JMP} =(-1)^{\ell-1}\sin^2 Q \left[\prod_{x,y\in U\ominus V}\Psi(x-y)\right] T(\Omega,D)T(\overline\Omega,\overline D). \]

Why not use Eq. (282) literally? After Eq. (243), the authors use symmetry of the outer integral to single out the first column of each determinant. Eq. (282) therefore has measure \(1/[(\ell-1)!]^2\) and only those first-column replacements. Expressed with our common measure, that literal integrand is

\[ A_{\ell,\mathrm{first}}^{\rm JMP} =(-1)^{\ell-1}\ell^2\sin^2 Q \left[\prod_{x,y\in U\ominus V}\Psi(x-y)\right] \det\Omega^{[1]}\det\overline\Omega^{[1]}. \]

It is not generally equal pointwise to the symmetric PRL amplitude. Averaging independently over the \(\ell\) choices in each determinant restores \(A_\ell^{\rm JMP}\). This is an integration-symmetry convention, not a defect in the paper. The code returns both versions, and a regression test checks that they differ before symmetrization and agree afterwards.

Quadrature uses the real line, split at zero and all real rapidity parts. split=False and a larger maxdegree provide alternative convergence checks. Returned quadrature errors are mpmath estimates for the matrix-entry integrals, not certified errors on the final determinant product.

PRL: the isotropic limit cannot be taken entrywise alone¶

Set \(q=e^{-\gamma}\), rescale the paper's rapidities by \(\gamma\), and put \(b=q^4=e^{-4\gamma}\). A multiplicative parameter becomes

\[ q^c e^{2i\gamma z}=b^{c/4-iz/2}. \]

Thus the two basic hypergeometric functions in PRL (6) become ordinary \({}_{2\ell}F_{2\ell-1}(1)\). Their parameters are implemented directly in hyper_parameters. In both cases their excess is

\[ \sum_r b_r-\sum_r a_r=1. \]

However, \(\det M\) and \(\det\widehat M\) both vanish as \(\gamma\to0\). Just substituting \(q=1\) gives zero and loses the leading answer.

The first correction comes from the series tail¶

For either of these excess-one series, define

\[ \Phi={}_pF_{p-1}(\{a\};\{b\};1), \qquad K=\frac{\prod_r\Gamma(b_r)}{\prod_r\Gamma(a_r)}. \]

The ordinary summand has tail \(K/n^2\). At any fixed summation index, the first-order change of its basic-hypergeometric counterpart cancels, including the argument \(b=e^{-4\gamma}\). Differentiating a finite truncation therefore misses the actual leading correction.

With \(h=4\gamma\), the transition region \(n\sim1/h\) instead gives

\[ \Phi_q-\Phi =Kh\int_0^\infty \left[\frac{e^{-x}}{(1-e^{-x})^2}-\frac1{x^2}\right]dx+o(h) =-\frac{Kh}{2}+o(h). \]

The integrand is finite at zero and integrable at infinity. Its primitive is \(1/x-1/(e^x-1)\), with endpoint values \(1/2\) and zero. Consequently the analytic first coefficient used here is

\[ \left.\frac{d\Phi_q}{d\gamma}\right|_{0^+}=-2K. \]

This matched-tail derivation applies at generic parameters away from poles; it is not a uniform error bound near all degenerate rapidity configurations. The test suite checks it directly against the unexpanded basic series.

Singular determinants without inverses or finite differences¶

Let \(a(z)=-iz/2\), \(a_{ij}=a(v_i-v_j)\). The limit of PRL (7) is

\[ r_{ji}=\frac1{4a_{ij}(1+a_{ij})} \prod_{u\in U}\frac{a(v_i-u)}{a(v_i-u)+1/2} \prod_{r\ne i,j}\frac{a(v_i-v_r)+1/2}{a(v_i-v_r)}. \]

Its first derivative in \(\gamma\) vanishes. The scattering factor also has zero first derivative: the \(e^{i\gamma z}\) phase in the finite-\(q\) definition must be retained to cancel the first correction from q-Gamma. Thus \(M_0,M_1\) follow by using \(\Phi,-2K\), respectively, in PRL (8)–(9).

The bar in that paper means invert all \(P,H\), equivalently negate all rapidities after rescaling; it does not mean complex conjugation. The second matrix is obtained by replacing \((U,V)\) with \((-V,-U)\).

Set

\[ d_M=T(M_0,M_1),\qquad d_{\widehat M}=T(\widehat M_0,\widehat M_1). \]

These are the coefficients of \(\gamma\) in the respective determinants. No inverse of a singular matrix and no numerical differentiation is needed. At \(\ell=1\), Gauss summation provides a separate closed-form check:

\[ \Phi_1=\frac{\Gamma(1/2+a)}{\sqrt{\pi}\Gamma(1+a)},\qquad \Phi'_1=a\Phi_1,\qquad (M_1)_{11}=i(v-u)\overline\Phi_1,\quad a=-i(v-u)/2. \]

Limiting scalar prefactor¶

Define

\[ F(z)=\sinh^2(\pi z/2)\Psi(z)\Psi(-z), \qquad s=\sum_j(v_j-u_j). \]

The resulting symmetric isotropic amplitude is

\[ A_\ell^{\rm PRL} =d_Md_{\widehat M} \left[\frac{\pi\Psi(0)}{2}\right]^{2\ell} \frac{\pi^2\sin^2 Q}{\sinh^2(\pi s/2)} \frac{\prod_{i\lt j}F(u_i-u_j)F(v_i-v_j)} {\prod_{i,j}F(u_i-v_j)}. \]

For checking constants, modular transformation gives

\[ \vartheta_1(\gamma z\mid e^{-a\gamma}) \sim B_a\sinh(\pi z/a),\qquad B_a=2\sqrt{\frac{\pi}{a\gamma}}e^{-\pi^2/(4a\gamma)}. \]

The exponential factors cancel in PRL (10). Before the determinant product, the remaining scalar scales as \(\gamma^{-2\ell-2}\). The rescaled rapidity measure contributes \(\gamma^{2\ell}\), and the two determinant zeros contribute \(\gamma^2\). This accounts for the finite amplitude above. Here \(k\) is the paper's discrete branch label, not the physical momentum. The staggered \(k=1\) part vanishes in this isotropic limit; only \(k=0\) is evaluated here, consistently with JMP (277).

Removing the common contour offset analytically¶

The complex-partition audit makes the fixed-rapidity boundary step simpler: no offset extrapolation is needed for this purpose. This follows directly from the structure of the isotropic PRL formula, not from a fit to the vertex result.

Write the symmetric amplitude as

\[ A_H(\eta)=K(\boldsymbol\beta,\eta)B_H(\boldsymbol\beta). \]

All factors in the PRL expression other than its explicit momentum sine depend on rapidity differences: the hypergeometric parameters, scattering ratios, determinant coefficients, pair products, and the sum \(\sum(v-u)\). They are unchanged by a common translation \(i\eta\). This is visible already in the structure of PRL Eqs. (6)–(11). Here we use the explicit isotropic determinant coefficients derived in the preceding limiting algebra.

The elementary identity

\[ e^{i\widehat p(z-i)}=e^{-i\widehat p(z)} \]

follows from the cosh ratio in \(\widehat p\). Consequently the sine-squared factor is the same for every assignment:

\[ K(\boldsymbol\beta,\eta) =\sin^2\!\left[\frac12\sum_{j=1}^n \widehat p\left(\frac{\beta_j}{\pi}+i\eta\right)\right]. \]

Any \(2\pi\) log-branch changes disappear in the sine squared. At the real boundary this is simply

\[ K_0(\boldsymbol\beta) =\sin^2\!\left[\sum_{j=1}^n\arctan(e^{\beta_j})\right], \qquad A_H(0)=K_0B_H. \]

The audit computes \(B_H\) by omitting the explicit sine factor from the product, not by dividing \(A_H\) by a possibly small \(K\). This also avoids a numerical \(0/0\) at zero physical momentum, although other coincident-rapidity/removable-singularity limits are not thereby solved. It evaluates special functions at well-separated interior contour offsets, then inserts the exact physical \(K_0\).

This explains the offset dependence tested by the vertex audit. Near \(\eta=0\), \(K(-\eta)=K(\eta)^*\); when the summed reduced coefficient is real, its real correction starts at \(O(\eta^2)\) and its imaginary part at \(O(\eta)\). There is no residual offset dependence to fit in \(B_H\). For nonzero \(K\) and total, the cancellation condition is also independent of the common offset:

\[ \frac{\sum_H|A_H(\eta)|}{|\sum_H A_H(\eta)|} =\frac{\sum_H|B_H|}{|\sum_H B_H|}. \]

For the independent JMP route the corresponding coefficient is

\[ B_H^{\rm JMP} =(-1)^{\ell-1}\! \left[\prod_{x,y\in U\ominus V}\Psi(x-y)\right] T(\Omega,D)T(\bar\Omega,\bar D). \]

Here \(T\) is the sum of the distinguished-column determinants, not one column chosen without symmetrization. The fixed-real-line JMP quadrature and the PRL series are evaluated at identical \(U,V\) for each partition; their products \(B_H\) are compared before any sum over \(H\). Their individual determinant coefficients need not be equal.

The partition audit retains the full complex amplitudes, reduced coefficients, explicit coordinates and route-specific factors. It repeats the calculation at 30/42 decimal digits and offsets \(0.25/0.1\), changing the JMP quadrature splitting and resolution as well. It also reconstructs each \(\eta=10^{-10}\) complex amplitude from \(K(\eta)B_H\). These comparison inputs are generated afresh by default, or read from an explicitly supplied, source-verified vertex audit. The usage guide gives both commands.

This factorization is a fixed-rapidity algebraic statement within the implemented isotropic formula. It is not a proof that an outer integration contour may be shifted without residues, or that tails and collisions are uniformly controlled. Nor does an individual complex \(B_H\) represent a positive Lehmann weight.