$\displaystyle-2M_{\pi}^{2}\tilde{C}^{r}_{35}-2(2M_{K}^{2}+M_{\pi}^{2})\tilde{C}^{r}_{36}-4(M_{K}^{2}-M_{\pi}^{2})\tilde{C}^{r}_{37}-2s\tilde{C}^{r}_{88}+2(s-2M_{K}^{2})\tilde{C}^{r}_{89}$
$\displaystyle-4M_{K}^{2}\tilde{C}^{r}_{106}+2(s-2M_{K}^{2})\tilde{C}^{r}_{107}+\frac{s-4M_{K}^{2}}{4F_{\pi}^{2}}J^{r}_{KK}(s)~{},$
$\displaystyle F^{\bar{s}s}_{T,\pi\pi}(s)$ $\displaystyle=$ $\displaystyle
0\quad(\text{by isospin and Bose symmetry}),$ $\displaystyle
F^{\bar{s}s}_{T,\eta\eta}(s)$ $\displaystyle=$ $\displaystyle 0\quad(\text{by
Bose symmetry}).$ (D.10)

###  D.8 Tensor Form Factors for the $I=\frac{1}{2}$ System

The tensor form factors for the $I=1/2$ systems up to NLO in ChPT are given by

$\displaystyle\sqrt{\frac{2}{3}}F^{\bar{u}s}_{T,K\pi}(s)$ $\displaystyle=$
$\displaystyle
1+\frac{-20M_{K}^{2}+16M_{\pi}^{2}+s}{4(M_{K}^{2}-M_{\pi}^{2})}\mu_{\pi}+\frac{-12M_{K}^{2}+8M_{\pi}^{2}+s}{2(M_{K}^{2}-M_{\pi}^{2})}\mu_{K}+\frac{44M_{K}^{2}-8M_{\pi}^{2}-9s}{12(M_{K}^{2}-M_{\pi}^{2})}\mu_{\eta}$
$\displaystyle-\frac{5M_{K}^{2}+M_{\pi}^{2}-s}{96\pi^{2}F_{\pi}^{2}}-\frac{4(M_{K}^{2}-M_{\pi}^{2})}{F_{\pi}^{2}}L_{5}^{r}-4M_{K}^{2}\tilde{C}^{r}_{34}-2M_{\pi}^{2}\tilde{C}^{r}_{35}-2(2M_{K}^{2}+M_{\pi}^{2})\tilde{C}^{r}_{36}$
$\displaystyle-2s\tilde{C}^{r}_{88}-2(M_{K}^{2}+M_{\pi}^{2}-s)\tilde{C}^{r}_{89}-2(M_{K}^{2}+M_{\pi}^{2})\tilde{C}^{r}_{106}-2(M_{K}^{2}+M_{\pi}^{2}-s)\tilde{C}^{r}_{107}$
$\displaystyle+\frac{M_{\eta}^{4}-2M_{\eta}^{2}(M_{K}^{2}+s)+(M_{K}^{2}-s)^{2}}{8F_{\pi}^{2}s}\bar{J}_{K\eta}(s)$
$\displaystyle+\frac{M_{K}^{4}-2M_{K}^{2}(M_{\pi}^{2}+s)+(M_{\pi}^{2}-s)^{2}}{8F_{\pi}^{2}s}\bar{J}_{\pi
K}(s)~{},$ $\displaystyle\sqrt{\frac{2}{3}}F^{\bar{u}s}_{T,K\eta}(s)$
$\displaystyle=$ $\displaystyle
1+\frac{-20M_{K}^{2}+16M_{\pi}^{2}+s}{4(M_{K}^{2}-M_{\pi}^{2})}\mu_{\pi}+\frac{-12M_{K}^{2}+8M_{\pi}^{2}+s}{2(M_{K}^{2}-M_{\pi}^{2})}\mu_{K}+\frac{44M_{K}^{2}-8M_{\pi}^{2}-9s}{12(M_{K}^{2}-M_{\pi}^{2})}\mu_{\eta}$
(D.11)
$\displaystyle-\frac{5M_{K}^{2}+M_{\pi}^{2}-s}{96\pi^{2}F_{\pi}^{2}}-\frac{28(M_{K}^{2}-M_{\pi}^{2})}{3F_{\pi}^{2}}L_{5}^{r}-4M_{K}^{2}\tilde{C}^{r}_{34}+\frac{2(M_{\pi}^{2}-4M_{K}^{2})}{3}\tilde{C}^{r}_{35}$
$\displaystyle-2(2M_{K}^{2}+M_{\pi}^{2})\tilde{C}^{r}_{36}-2s\tilde{C}^{r}_{88}+\frac{2(-7M_{K}^{2}+M_{\pi}^{2}+3s)}{3}\tilde{C}^{r}_{89}+\frac{2(M_{\pi}^{2}-7M_{K}^{2})}{3}\tilde{C}^{r}_{106}$
$\displaystyle+\frac{2(-7M_{K}^{2}+M_{\pi}^{2}+3s)}{3}\tilde{C}^{r}_{107}+\frac{M_{\eta}^{4}-2M_{\eta}^{2}(M_{K}^{2}+s)+(M_{K}^{2}-s)^{2}}{8F_{\pi}^{2}s}\bar{J}_{K\eta}(s)$
$\displaystyle+\frac{M_{K}^{4}-2M_{K}^{2}(M_{\pi}^{2}+s)+(M_{\pi}^{2}-s)^{2}}{8F_{\pi}^{2}s}\bar{J}_{\pi
K}(s)~{}.$

###  D.9 Tensor Form Factors for the $I=1$ System

The tensor form factors for the isovector systems up to NLO in ChPT are given
by

$\displaystyle-\frac{F^{\bar{u}d}_{T,\pi\pi}(s)}{\sqrt{2}}$ $\displaystyle=$
$\displaystyle
1-\frac{13}{3}\mu_{\pi}-\frac{8}{3}\mu_{K}-\frac{1}{3}\mu_{\eta}+\frac{-2M_{K}^{2}-4M_{\pi}^{2}+s}{96\pi^{2}F_{\pi}^{2}}-4M_{\pi}^{2}\tilde{C}^{r}_{34}-2M_{\pi}^{2}\tilde{C}^{r}_{35}$
$\displaystyle-2(2M_{K}^{2}+M_{\pi}^{2})\tilde{C}^{r}_{36}-2s\tilde{C}^{r}_{88}+2(s-2M_{\pi}^{2})\tilde{C}^{r}_{89}-4M_{\pi}^{2}\tilde{C}^{r}_{106}+2(s-2M_{\pi}^{2})\tilde{C}^{r}_{107}$
$\displaystyle+\frac{s-4M_{\pi}^{2}}{6F_{\pi}^{2}}J^{r}_{\pi\pi}(s)+\frac{s-4M_{K}^{2}}{12F_{\pi}^{2}}J^{r}_{KK}(s)~{},$
$\displaystyle-F^{\bar{u}d}_{T,K\bar{K}}(s)$ $\displaystyle=$ $\displaystyle
1-\frac{13}{3}\mu_{\pi}-\frac{8}{3}\mu_{K}-\frac{1}{3}\mu_{\eta}+\frac{-2M_{K}^{2}-4M_{\pi}^{2}+s}{96\pi^{2}F_{\pi}^{2}}-\frac{8(M_{K}^{2}-M_{\pi}^{2})}{F_{\pi}^{2}}L^{r}_{5}-4M_{\pi}^{2}\tilde{C}^{r}_{34}$
$\displaystyle+2(M_{\pi}^{2}-2M_{K}^{2})\tilde{C}^{r}_{35}-2(2M_{K}^{2}+M_{\pi}^{2})\tilde{C}^{r}_{36}-2s\tilde{C}^{r}_{88}+2(s-2M_{K}^{2})\tilde{C}^{r}_{89}-4M_{K}^{2}\tilde{C}^{r}_{106}$
$\displaystyle+2(s-2M_{K}^{2})\tilde{C}^{r}_{107}+\frac{s-4M_{\pi}^{2}}{6F_{\pi}^{2}}J^{r}_{\pi\pi}(s)+\frac{s-4M_{K}^{2}}{12F_{\pi}^{2}}J^{r}_{KK}(s)~{},$
$\displaystyle F^{\bar{u}d}_{T,\pi\eta}(s)$ $\displaystyle=$ $\displaystyle
0\quad(\text{by C parity}).$ (D.12)

##  Appendix E $B_{s}\to\pi^{+}\pi^{-}$ Form Factors

Normalized by the scalar form factor, the $S$-wave $\pi^{+}\pi^{-}$ LCDAs are
defined as Diehl:1998dk ; Polyakov:1998ze ; Kivel:1999sd ; Diehl:2003ny ;
Meissner:2013hya

$\displaystyle\langle{(\pi^{+}\pi^{-})_{S}}|\bar{s}(x)\gamma_{\mu}s(0)|0\rangle$
$\displaystyle=$ $\displaystyle
F_{S,\pi\pi}^{\bar{s}s}(m_{\pi\pi}^{2})p_{{\pi\pi},\mu}\int_{0}^{1}du\,e^{iup_{\pi\pi}\cdot
x}\phi_{\pi\pi}(u)\,,$
$\displaystyle\langle{(\pi^{+}\pi^{-})_{S}}|\bar{s}(x)s(0)|0\rangle$
$\displaystyle=$ $\displaystyle
F_{S,\pi\pi}^{\bar{s}s}(m_{\pi\pi}^{2})B_{0}\int_{0}^{1}du\,e^{iup_{\pi\pi}\cdot
x}\phi_{\pi\pi}^{s}(u)\,,$
$\displaystyle\langle{(\pi^{+}\pi^{-})_{S}}\bar{s}(x)\sigma_{\mu\nu}s(0)|0\rangle$
$\displaystyle=$ $\displaystyle-
F_{S,\pi\pi}^{\bar{s}s}(m_{\pi\pi}^{2})B_{0}\frac{1}{6}(p_{{\pi\pi}\mu}x_{\nu}-p_{{\pi\pi}\nu}x_{\mu})\int_{0}^{1}du\,e^{iup_{\pi\pi}\cdot
x}{\phi_{\pi\pi}^{\sigma}(u)}\,.$ (E.1)

$\phi_{\pi\pi}$ and $\phi_{\pi\pi}^{s},\phi_{\pi\pi}^{\sigma}$ are twist-2 and
twist-3 LCDAs, respectively. They are normalized as

$\displaystyle\int_{0}^{1}du\phi_{\pi\pi}^{s}(u)=\int_{0}^{1}du\phi_{\pi\pi}^{\sigma}(u)=1\,.$
(E.2)

According to the conformal symmetry in QCD Braun:2003rp , the twist-3 LCDAs
have the asymptotic form Diehl:1998dk ; Polyakov:1998ze ; Kivel:1999sd ;
Diehl:2003ny :

$\displaystyle\phi_{\pi\pi}^{s}(u)=1,\;\;\;\phi_{\pi\pi}^{\sigma}(u)=6u(1-u)\,,$
(E.3)

while the twist-2 LCDA can be expanded in terms of the Gegenbauer moments as

$\displaystyle\phi_{\pi\pi}(u)=6u(1-u)\sum_{n}a_{n}C_{n}^{3/2}(2u-1)\,.$ (E.4)

Since the contributions from higher Gegenbauer moments are suppressed, it is
enough to only consider the lowest moment $a_{1}$. The first Gegenbauer moment
for the $f_{0}(980)$ is Cheng:2005nb

$\displaystyle a_{1}=-1.35\,.$ (E.5)

The $B_{s}\to\pi^{+}\pi^{-}$ form factors in terms of the $\pi^{+}\pi^{-}$
LCDAs read as Meissner:2013hya ,

$\displaystyle{\cal F}^{B_{s}\to\pi\pi}_{1}(M_{\pi\pi}^{2},q^{2})$
$\displaystyle=N_{F}\bigg{\\{}\int_{u_{0}}^{1}\frac{du}{u}{\rm
exp}\left[-\frac{m_{b}^{2}+u\bar{u}M_{\pi\pi}^{2}-\bar{u}q^{2}}{uM^{2}}\right]\bigg{[}-\frac{m_{b}}{B_{0}}\Phi_{\pi\pi}(u)+u\Phi_{\pi\pi}^{s}(u)+\frac{1}{3}\Phi_{\pi\pi}^{\sigma}(u)$
$\displaystyle+\frac{m_{b}^{2}+q^{2}-u^{2}M_{\pi\pi}^{2}}{uM^{2}}\frac{\Phi_{\pi\pi}^{\sigma}(u)}{6}\bigg{]}+\exp{\left[-\frac{s_{0}}{M^{2}}\right]}\frac{\Phi_{\pi\pi}^{\sigma}(u_{0})}{6}\frac{m_{b}^{2}-u_{0}^{2}M_{\pi\pi}^{2}+q^{2}}{m_{b}^{2}+u_{0}^{2}M_{\pi\pi}^{2}-q^{2}}\bigg{\\}},$
(E.6) $\displaystyle{\cal F}^{B_{s}\to\pi\pi}_{-}(M_{\pi\pi}^{2},q^{2})$
$\displaystyle=N_{F}\left\\{\int_{u_{0}}^{1}\frac{du}{u}{\rm
exp}\left[-\frac{m_{b}^{2}+u\bar{u}M_{\pi\pi}^{2}-\bar{u}q^{2}}{uM^{2}}\right]\bigg{[}\frac{m_{b}}{B_{0}}\Phi_{\pi\pi}(u)+(2-u)\Phi_{\pi\pi}^{s}(u)\right.$
$\displaystyle\;\;\;\left.+\frac{1-u}{3u}\Phi_{\pi\pi}^{\sigma}(u)-\frac{u({m_{b}^{2}+q^{2}-u^{2}M_{\pi\pi}^{2}})+2(m_{b}^{2}-q^{2}+u^{2}M_{\pi\pi}^{2})}{u^{2}M^{2}}\frac{\Phi_{\pi\pi}^{\sigma}(u)}{6}\bigg{]}\right.$
$\displaystyle\;\;\;\left.-\frac{u_{0}({m_{b}^{2}+q^{2}-u_{0}^{2}M_{\pi\pi}^{2}})+2(m_{b}^{2}-q^{2}+u_{0}^{2}M_{\pi\pi}^{2})}{u_{0}(m_{b}^{2}+u_{0}^{2}M_{\pi\pi}^{2}-q^{2})}\exp{\left[-\frac{s_{0}}{M^{2}}\right]}\frac{\Phi_{\pi\pi}^{\sigma}(u_{0})}{6}\right\\},$
(E.7) $\displaystyle{\cal F}^{B_{s}\to\pi\pi}_{0}(M_{\pi\pi}^{2},q^{2})={\cal
F}^{B_{s}\to\pi\pi}_{1}(M_{\pi\pi}^{2},q^{2})+\frac{q^{2}}{M_{B_{s}}^{2}-M_{\pi\pi}^{2}}{\cal
F}^{B_{s}\to\pi\pi}_{-}(M_{\pi\pi}^{2},q^{2}),$ (E.8) $\displaystyle{\cal
F}^{B_{s}\to\pi\pi}_{T}(M_{\pi\pi}^{2},q^{2})$
$\displaystyle=2N_{F}(M_{B_{s}}+M_{\pi\pi})\bigg{\\{}\int_{u_{0}}^{1}\frac{du}{u}{\rm
exp}\left[-\frac{(m_{b}^{2}-\bar{u}q^{2}+u\bar{u}M_{\pi\pi}^{2})}{uM^{2}}\right]\left[-\frac{\Phi_{\pi\pi}(u)}{2B_{0}}+m_{b}\frac{\Phi_{\pi\pi}^{\sigma}(u)}{6uM^{2}}\right]$
$\displaystyle\;\;\;+m_{b}\frac{\Phi_{\pi\pi}^{\sigma}(u_{0})}{6}\frac{\exp[-s_{0}/M^{2}]}{m_{b}^{2}-q^{2}+u_{0}^{2}M_{\pi\pi}^{2}}\bigg{\\}},$
(E.9)

##  Appendix F Vector Meson Dominance and Tensor Form Factors at the Vector
Pole

The antisymmetric tensor interaction is not a part of the elementary SM
interaction structure, so it is quite difficult to compare theoretical results
of tensor form factor calculations with data. Therefore, for an independent
cross-check one should rely on future lattice QCD calculations. One could of
course calculate the two-meson tensor form factors directly on the lattice,
but since the results of different theory predictions of the tensor form
factors differ mainly around the $\rho$-peak (see Fig. 21), a more
straightforward cross-check is to calculate the tensor charge of the
$\rho$-meson. Combining its value and the vector meson dominance (VMD)
picture, one is able to compare the two-meson tensor form factors with lattice
predictions around the vector-meson mass pole. We outline the method below.

The tensor charge $f_{\mathcal{V}}^{T}$ of a vector meson $\mathcal{V}$ is
defined through

$\left\langle\mathcal{V}(p)|\bar{q}^{\prime}\sigma^{\mu\nu}q|0\right\rangle=-if^{T}_{\mathcal{V},\bar{q}^{\prime}q}(\varepsilon^{\mu*}p^{\nu}-\varepsilon^{\nu*}p^{\mu})\,,$
(F.1)

where $\varepsilon^{\mu}$ is the polarization vector of $\mathcal{V}$. Suppose
now we are interested in the tensor form factor
$F_{T,\phi\phi^{\prime}}^{\bar{q}^{\prime}q}(s)$ defined in Eq. (5.1) at
$s\approx M_{\mathcal{V}}^{2}$. If one is able to parameterize the
$\mathcal{V}\rightarrow\phi\phi^{\prime}$ amplitude as

$iM_{\mathcal{V}\rightarrow\phi\phi^{\prime}}=if_{\mathcal{V}\phi\phi^{\prime}}(p_{\phi}-p_{\phi^{\prime}})_{\mu}\varepsilon^{\mu}\,,$
(F.2)

then the VMD picture provides an approximate expression of
$F_{T,\phi\phi^{\prime}}^{\bar{q}^{\prime}q}(s)$ at $s\approx
M_{\mathcal{V}}^{2}$,

$\frac{\Lambda_{2}}{F_{\pi}^{2}}F_{T,\phi\phi^{\prime}}^{\bar{q}^{\prime}q}(s)\approx-\frac{2f_{\mathcal{V}\phi\phi^{\prime}}f_{\mathcal{V},\bar{q}^{\prime}q}^{T}}{s-M_{\mathcal{V}}^{2}+i\Theta(s-4M_{\pi}^{2})M_{\mathcal{V}}\Gamma_{\mathcal{V}}}~{},$
(F.3)

where $\Gamma_{\mathcal{V}}$ is the total decay width of $\mathcal{V}$. An
interesting consequence of this formula is that one expects
$\mathrm{Re}F_{T,\phi\phi^{\prime}}^{\bar{q}^{\prime}q}(s)$ to vanish and
$\mathrm{Im}F_{T,\phi\phi^{\prime}}^{\bar{q}^{\prime}q}(s)$ to peak at
$s=M_{\mathcal{V}}^{2}$. Furthermore, with future lattice inputs of
$f_{\mathcal{V},\bar{q}^{\prime}q}^{T}$, the equation above serves as a
consistency check of the theoretical result for two-meson tensor form factors
at the vector-meson pole.

## References

  * (1) R. Aaij et al. [LHCb Collaboration], Measurement of Form-Factor-Independent Observables in the Decay $B^{0}\to K^{*0}\mu^{+}\mu^{-}$, Phys. Rev. Lett. 111, 191801 (2013) [arXiv:1308.1707 [hep-ex]].
  * (2) R. Aaij et al. [LHCb Collaboration], Test of lepton universality with $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$ decays, JHEP 1708, 055 (2017) [arXiv:1705.05802 [hep-ex]].
  * (3) A. Abdesselam et al. [Belle Collaboration], Test of lepton flavor universality in ${B\to K^{\ast}\ell^{+}\ell^{-}}$ decays at Belle, arXiv:1904.02440 [hep-ex].
  * (4) K. M. Watson, The Effect of final state interactions on reaction cross-sections, Phys. Rev. 88, 1163 (1952).
  * (5) A. B. Migdal, Bremsstrahlung and pair production in condensed media at high-energies, Phys. Rev. 103, 1811 (1956).
  * (6) U.-G. Meißner and W. Wang, Generalized Heavy-to-Light Form Factors in Light-Cone Sum Rules, Phys. Lett. B 730, 336 (2014) [arXiv:1312.3087 [hep-ph]].
  * (7) M. Albaladejo, J. T. Daub, C. Hanhart, B. Kubis and B. Moussallam, How to employ ${\bar{B}}_{d}^{0}\to J/\psi\left(\pi\eta,\bar{K}K\right)$ decays to extract information on $\pi$$\eta$ scattering, JHEP 04, 010 (2017) [arXiv:1611.03502 [hep-ph]].
  * (8) J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250, 465 (1985).
  * (9) J. Gasser and U.-G. Meißner, Chiral expansion of pion form-factors beyond one loop, Nucl. Phys. B 357, 90 (1991).
  * (10) U.-G. Meißner and J. A. Oller, $J/\psi\to\phi\pi\pi$ ($K\bar{K}$) decays, chiral dynamics and OZI violation, Nucl. Phys. A 679, 671 (2001) [hep-ph/0005253].
  * (11) T. A. Lähde and U.-G. Meißner, Improved Analysis of $J/\psi$ Decays into a Vector Meson and Two Pseudoscalars, Phys. Rev. D 74, 034021 (2006) [hep-ph/0606133].
  * (12) J. F. Donoghue, J. Gasser and H. Leutwyler, The Decay of a Light Higgs Boson, Nucl. Phys. B 343, 341 (1990)
  * (13) B. Moussallam, $N_{f}$ dependence of the quark condensate from a chiral sum rule, Eur. Phys. J. C 14, 111 (2000) [hep-ph/9909292.
  * (14) S. Descotes-Genon, Zweig rule violation in the scalar sector and values of low-energy constants, JHEP 03, 002 (2001) [hep-ph/0012221.
  * (15) M. Hoferichter, C. Ditsche, B. Kubis and U.-G. Meißner, Dispersive analysis of the scalar form factor of the nucleon, JHEP 06, 063 (2012) [arXiv:1204.6251 [hep-ph]].
  * (16) J. T. Daub, C. Hanhart and B. Kubis, A model-independent analysis of final-state interactions in ${\bar{B}}_{d/s}^{0}\to J/\psi\pi\pi$, JHEP 02, 009 (2016) [arXiv:1508.06841 [hep-ph]].
  * (17) J. T. Daub, H. K. Dreiner, C. Hanhart, B. Kubis and U.-G. Meißner, Improving the Hadron Physics of Non-Standard-Model Decays: Example Bounds on R-parity Violation, JHEP 01, 179 (2013) [arXiv:1212.4408 [hep-ph]].
  * (18) A. Celis, V. Cirigliano and E. Passemar, Lepton flavor violation in the Higgs sector and the role of hadronic $\tau$-lepton decays, Phys. Rev. D 89, 013008 (2014) [arXiv:1309.3564 [hep-ph]].
  * (19) A. Monin, A. Boyarsky and O. Ruchayskiy, Hadronic decays of a light Higgs-like scalar, Phys. Rev. D 99, 015019 (2019) [arXiv:1806.07759 [hep-ph]].
  * (20) M. W. Winkler, Decay and detection of a light scalar boson mixing with the Higgs boson, Phys. Rev. D 99, 015018 (2019) [arXiv:1809.01876 [hep-ph]].
  * (21) S. Ropertz, C. Hanhart and B. Kubis, A new parametrization for the scalar pion form factors, Eur. Phys. J. C 78, 1000 (2018) [arXiv:1809.06867 [hep-ph]].
  * (22) M. Albaladejo and B. Moussallam, Form factors of the isovector scalar current and the $\eta\pi$ scattering phase shifts, Eur. Phys. J. C 75, 488 (2015) [arXiv:1507.04526 [hep-ph]].
  * (23) J. Lu and B. Moussallam, The $\pi\eta$ interaction and $a_{0}$ resonances in photon–photon scattering, Eur. Phys. J. C 80, 436 (2020) [arXiv:2002.04441 [hep-ph]].
  * (24) M. Jamin, J. A. Oller and A. Pich, Strangeness changing scalar form-factors, Nucl. Phys. B 622, 279 (2002) [hep-ph/0110193].
  * (25) M. Jamin, J. A. Oller and A. Pich, S wave $K\pi$ scattering in chiral perturbation theory with resonances, Nucl. Phys. B 587, 331 (2000) [hep-ph/0006045].
  * (26) M. Jamin, J. A. Oller and A. Pich, Scalar $K\pi$ form factor and light quark masses, Phys. Rev. D 74, 074009 (2006) [hep-ph/0605095].
  * (27) V. Bernard and E. Passemar, Chiral Extrapolation of the Strangeness Changing $K\pi$ Form Factor, JHEP 1004, 001 (2010) [arXiv:0912.3792 [hep-ph]].
  * (28) J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158, 142 (1984).
  * (29) D. R. Boito, R. Escribano and M. Jamin, The $K\pi$ vector form factor and constraints from $K_{l3}$ decays, arXiv:1101.2887 [hep-ph].
  * (30) D. R. Boito, R. Escribano and M. Jamin, Constraining the $K\pi$ vector form factor by $\tau\to K\pi\nu_{\tau}$ and $K_{\ell 3}$ decay data, Nucl. Phys. Proc. Suppl. 207-208, 148 (2010) [arXiv:1010.0100 [hep-ph]].
  * (31) D. R. Boito, R. Escribano and M. Jamin, Improving the $K\pi$ vector form factor through $K_{l3}$ constraints, AIP Conf. Proc. 1343, 271 (2011) [arXiv:1012.3493 [hep-ph]].
  * (32) F. Guerrero and A. Pich, Effective field theory description of the pion form-factor, Phys. Lett. B 412, 382 (1997) [hep-ph/9707347].
  * (33) J. F. De Trocóniz and F. J. Ynduráin, Precision determination of the pion form-factor and calculation of the muon $g-2$, Phys. Rev. D 65, 093001 (2002) [hep-ph/0106025.
  * (34) A. Pich and J. Portoles, The Vector form-factor of the pion from unitarity and analyticity: A Model independent approach, Phys. Rev. D 63, 093005 (2001) [hep-ph/0101194].
  * (35) J. F. de Trocóniz and F. J. Ynduráin, The Hadronic contributions to the anomalous magnetic moment of the muon, Phys. Rev. D 71, 073008 (2005) [hep-ph/0402285.
  * (36) F.-K. Guo, C. Hanhart, F. J. Llanes-Estrada and U.-G. Meißner, Quark mass dependence of the pion vector form factor, Phys. Lett. B 678, 90 (2009) [arXiv:0812.3270 [hep-ph]].
  * (37) C. Hanhart, A New Parameterization for the Pion Vector Form Factor, Phys. Lett. B 715, 170 (2012) [arXiv:1203.6839 [hep-ph]].
  * (38) D. Gómez Dumm and P. Roig, Dispersive representation of the pion vector form factor in $\tau\to\pi\pi\nu_{\tau}$ decays, Eur. Phys. J. C 73, 2528 (2013) [arXiv:1301.6973 [hep-ph]].
  * (39) G. Colangelo, M. Hoferichter and P. Stoffer, Two-pion contribution to hadronic vacuum polarization, JHEP 02, 006 (2019) [arXiv:1810.00007 [hep-ph]].
  * (40) S. Gonzàlez-Solís and P. Roig, A dispersive analysis of the pion vector form factor and $\tau^{-}\rightarrow K^{-}K_{S}\nu_{\tau}$ decay, Eur. Phys. J. C 79, 436 (2019) [arXiv:1902.02273 [hep-ph]].
  * (41) G. Colangelo, M. Hoferichter and P. Stoffer, Constraints on the two-pion contribution to hadronic vacuum polarization, arXiv:2010.07943 [hep-ph].
  * (42) J. Bijnens, G. Colangelo and P. Talavera, The Vector and scalar form-factors of the pion to two loops, JHEP 9805, 014 (1998) [hep-ph/9805389].
  * (43) J. Bijnens and P. Talavera, Pion and kaon electromagnetic form-factors, JHEP 0203, 046 (2002) [hep-ph/0203049].
  * (44) J. A. Oller, E. Oset and J. E. Palomar, Pion and kaon vector form-factors, Phys. Rev. D 63, 114009 (2001) [hep-ph/0011096].
  * (45) C. Bruch, A. Khodjamirian and J. H. Kühn, Modeling the pion and kaon form factors in the timelike region, Eur. Phys. J. C 39, 41 (2005) [hep-ph/0409080].
  * (46) A. Pich, I. Rosell and J. J. Sanz-Cillero, The vector form factor at the next-to-leading order in $1/N_{c}$: chiral couplings $L_{9}(\mu)$ and $C_{88}(\mu)-C_{90}(\mu)$, JHEP 1102, 109 (2011) [arXiv:1011.5771 [hep-ph]].
  * (47) C. Hanhart, B. Kubis and J. R. Peláez, Investigation of $a_{0}-f_{0}$ mixing, Phys. Rev. D 76, 074028 (2007) [arXiv:0707.0262 [hep-ph]].
  * (48) S. Descotes-Genon and B. Moussallam, Analyticity of $\eta\pi$ isospin-violating form factors and the $\tau\rightarrow\eta\pi\nu$ second-class decay, Eur. Phys. J. C 74, 2946 (2014) [arXiv:1404.0251 [hep-ph]].
  * (49) R. Escribano, S. Gonzàlez-Solís and P. Roig, Predictions on the second-class current decays $\tau^{-}\to\pi^{-}\eta^{(\prime)}\nu_{\tau}$, Phys. Rev. D 94, 034008 (2016) [arXiv:1601.03989 [hep-ph]].
  * (50) J. F. Donoghue and H. Leutwyler, Energy and momentum in chiral theories, Z. Phys. C 52, 343 (1991).
  * (51) B. Kubis and U.-G. Meißner, Virtual photons in the pion form-factors and the energy momentum tensor, Nucl. Phys. A 671, 332 (2000) [Erratum: Nucl. Phys. A 692, 647 (2001)] [hep-ph/9908261.
  * (52) O. Catà and V. Mateu, Chiral perturbation theory with tensor sources, JHEP 0709, 078 (2007) [arXiv:0705.2948 [hep-ph]].
  * (53) J. A. Miranda and P. Roig, Effective-field theory analysis of the $\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ decays, JHEP 1811, 038 (2018) [arXiv:1806.09547 [hep-ph]].
  * (54) V. Cirigliano, A. Crivellin and M. Hoferichter, No-go theorem for nonstandard explanations of the $\tau\to K_{S}\pi\nu_{\tau}$ CP asymmetry, Phys. Rev. Lett. 120, 141803 (2018) [arXiv:1712.06595 [hep-ph]].
  * (55) M. Hoferichter, B. Kubis, J. Ruiz de Elvira and P. Stoffer, Nucleon Matrix Elements of the Antisymmetric Quark Tensor, Phys. Rev. Lett. 122, 122001 (2019) [Erratum: Phys. Rev. Lett. 124, 199901 (2020)] [arXiv:1811.11181 [hep-ph]].
  * (56) A. Gomez Nicola and J. R. Peláez, Meson meson scattering within one loop chiral perturbation theory and its unitarization, Phys. Rev. D 65, 054009 (2002) [hep-ph/0109056].
  * (57) S. Weinberg, Phenomenological Lagrangians, Physica A 96, 327 (1979).
  * (58) S. R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177, 2239 (1969).
  * (59) J. Bijnens, G. Colangelo and J. Gasser, $K_{l4}$ decays beyond one loop, Nucl. Phys. B 427, 427 (1994) [hep-ph/9403390].
  * (60) J. Bijnens and G. Ecker, Mesonic low-energy constants, Ann. Rev. Nucl. Part. Sci. 64, 149 (2014) [arXiv:1405.6488 [hep-ph]].
  * (61) A. Manohar and H. Georgi, Chiral Quarks and the Nonrelativistic Quark Model, Nucl. Phys. B 234, 189 (1984).
  * (62) T. N. Truong, Chiral Perturbation Theory and Final State Theorem, Phys. Rev. Lett. 61, 2526 (1988).
  * (63) A. Dobado and J. R. Peláez, The Inverse amplitude method in chiral perturbation theory, Phys. Rev. D 56, 3057 (1997) [hep-ph/9604416].
  * (64) A. Gomez Nicola, J. R. Peláez and G. Rios, The Inverse Amplitude Method and Adler Zeros, Phys. Rev. D 77, 056006 (2008) [arXiv:0712.2763 [hep-ph]].
  * (65) M.-L. Du, D. Gülmez, F.-K. Guo, U.-G. Meißner and Q. Wang, Interactions between vector mesons and dynamically generated resonances, Eur. Phys. J. C 78, 988 (2018) [arXiv:1808.09664 [hep-ph]].
  * (66) D. Iagolnitzer, J. Zinn-Justin and J. B. Zuber, Yang-Mills fields and pseudoscalar meson scattering, Nucl. Phys. B 60, 233 (1973).
  * (67) A. M. Badalian, L. P. Kok, M. I. Polikarpov and Y. A. Simonov, Resonances in Coupled Channels in Nuclear and Particle Physics, Phys. Rept. 82, 31 (1982).
  * (68) Z. Xiao and H. q. Zheng, The Use of dispersion relations in the $\pi\pi$ and $K\bar{K}$ coupled channel system, Commun. Theor. Phys. 48, 685 (2007) [hep-ph/0103042].
  * (69) Z. Xiao and H. q. Zheng, The Use of analyticity in the $\pi\pi$ and $K\bar{K}$ coupled channel system, hep-ph/0107188.
  * (70) R. García-Martín, R. Kamiński, J. R. Peláez, J. Ruiz de Elvira and F. J. Ynduráin, The Pion-pion scattering amplitude. IV: Improved analysis with once subtracted Roy-like equations up to 1100 MeV, Phys. Rev. D 83, 074004 (2011) [arXiv:1102.2183 [hep-ph]].
  * (71) R. Mercer et al., $K\pi$ scattering phase shifts determined from the reactions $K^{+}p\to K^{+}\pi^{-}\Delta^{++}$ and $K^{+}p\to K^{0}\pi^{0}\Delta^{++}$, Nucl. Phys. B 32, 381 (1971).
  * (72) H. H. Bingham et al., A New analysis of $K\pi$ scattering as observed in $K^{+}p\to K^{+}\pi^{0}\Delta^{++}$ from 3-GeV/c to 13-GeV/c, Nucl. Phys. B 41, 1 (1972).
  * (73) D. Linglin et al., $K\pi$ elastic scattering cross-section measured in 14.3 Gev/c $Kp$ interactions, Nucl. Phys. B 57, 64 (1973).
  * (74) S. L. Baker et al., A Study of $K^{+}\pi^{-}$ Elastic Scattering in the Reaction $K^{+}n\to K^{+}\pi^{-}p$ Between 2.0-GeV/c and 3.0-GeV/c, Nucl. Phys. B 99, 211 (1975).
  * (75) P. Estabrooks, R. K. Carnegie, A. D. Martin, W. M. Dunwoodie, T. A. Lasinski and D. W. G. S. Leith, Study of $K\pi$ Scattering Using the Reactions $K^{\pm}p\to K^{\pm}\pi^{+}n$ and $K^{\pm}p\to K^{\pm}\pi^{-}\Delta^{++}$ at 13-GeV/c, Nucl. Phys. B 133, 490 (1978).
  * (76) D. Aston et al., A Study of $K^{-}\pi^{+}$ Scattering in the Reaction $K^{-}p\to K^{-}\pi^{+}n$ at 11-GeV/c, Nucl. Phys. B 296, 493 (1988).
  * (77) P. del Amo Sanchez et al. [BaBar Collaboration], Analysis of the $D^{+}\to K^{-}\pi^{+}e^{+}\nu_{e}$ decay channel, Phys. Rev. D 83, 072001 (2011) [arXiv:1012.1810 [hep-ex]].
  * (78) M. Ablikim et al. [BESIII Collaboration], Study of $D^{+}\to K^{-}\pi^{+}e^{+}\nu_{e}$, Phys. Rev. D 94, 032001 (2016) [arXiv:1512.08627 [hep-ex]].
  * (79) A. D. Martin and E. N. Ozmutlu, Analyses of $K\bar{K}$ Production and Scalar Mesons, Nucl. Phys. B 158, 520 (1979).
  * (80) D. H. Cohen, D. S. Ayres, R. Diebold, S. L. Kramer, A. J. Pawlicki and A. B. Wicklund, Amplitude Analysis of the $K^{-}K^{+}$ System Produced in the Reactions $\pi^{-}p\to K^{-}K^{+}n$ and $\pi^{+}n\to K^{-}K^{+}p$ at 6-GeV/c, Phys. Rev. D 22, 2595 (1980).
  * (81) R. Kamiński, L. Lesniak and K. Rybicki, Separation of S wave pseudoscalar and pseudovector amplitudes in $\pi^{-}p\text{(polarized)}\to\pi^{+}\pi^{-}n$ reaction on polarized target, Z. Phys. C 74, 79 (1997) [hep-ph/9606362].
  * (82) T. A. Armstrong et al. [WA76 and Athens-Bari-Birmingham-CERN-College de France Collaborations], Study of the $\eta\pi^{+}\pi^{-}$ system centrally produced in the reaction $pp\to p_{f}(\eta\pi^{+}\pi^{-})p_{s}$ at 300-GeV/c, Z. Phys. C 52, 389 (1991).
  * (83) F. James and M. Roos, Minuit: A System for Function Minimization and Analysis of the Parameter Errors and Correlations, Comput. Phys. Commun. 10, 343 (1975).
  * (84) iminuit team, iminuit – A Python interface to Minuit, https://github.com/scikit-hep/iminuit.
  * (85) F.-K. Guo, A Julia wrapper of iminuit, https://github.com/fkguo/IMinuit.jl.
  * (86) B. Ananthanarayan, G. Colangelo, J. Gasser and H. Leutwyler, Roy equation analysis of $\pi\pi$ scattering, Phys. Rept. 353, 207 (2001) [hep-ph/0005297].
  * (87) G. Colangelo, J. Gasser and H. Leutwyler, $\pi\pi$ scattering, Nucl. Phys. B 603, 125 (2001) [hep-ph/0103088].
  * (88) I. Caprini, G. Colangelo and H. Leutwyler, Regge analysis of the $\pi\pi$ scattering amplitude, Eur. Phys. J. C 72, 1860 (2012) [arXiv:1111.7160 [hep-ph]].
  * (89) R. Kamiński, J. R. Peláez and F. J. Ynduráin, The Pion-pion scattering amplitude. III. Improving the analysis with forward dispersion relations and Roy equations, Phys. Rev. D 77, 054015 (2008) [arXiv:0710.1150 [hep-ph]].
  * (90) P. Büttiker, S. Descotes-Genon and B. Moussallam, A new analysis of $\pi K$ scattering from Roy and Steiner type equations, Eur. Phys. J. C 33, 409 (2004) [hep-ph/0310283].
  * (91) J. R. Peláez and A. Rodas, $\pi\pi\rightarrow K{\bar{K}}$ scattering up to 1.47 GeV with hyperbolic dispersion relations, Eur. Phys. J. C 78, 897 (2018) [arXiv:1807.04543 [hep-ph]].
  * (92) J. R. Peláez and A. Rodas, Pion-kaon scattering amplitude constrained with forward dispersion relations up to 1.6 GeV, Phys. Rev. D 93, 074025 (2016) [arXiv:1602.08404 [hep-ph]].
  * (93) J. R. Peláez and A. Rodas, Dispersive $\pi K\to\pi K$ and $\pi\pi\to K\bar{K}$ amplitudes from scattering data, threshold parameters and the lightest strange resonance $\kappa$ or $K^{*}_{0}(700)$, arXiv:2010.11222 [hep-ph].
  * (94) J. A. Oller, E. Oset and J. R. Peláez, Meson meson interaction in a nonperturbative chiral approach, Phys. Rev. D 59, 074001 (1999) [Errata: Phys. Rev. D 60, 099906 (1999); Phys. Rev. D 75, 099903 (2007)] [hep-ph/9804209.
  * (95) J. A. Oller and E. Oset, $N/D$ description of two meson amplitudes and chiral symmetry, Phys. Rev. D 60, 074023 (1999) [hep-ph/9809337].
  * (96) T. Hannah, The Inverse amplitude method and chiral zeros, Phys. Rev. D 59, 057502 (1999) [hep-ph/9703403].
  * (97) S. J. Brodsky and G. R. Farrar, Scaling Laws at Large Transverse Momentum, Phys. Rev. Lett. 31, 1153 (1973).
  * (98) M. Döring, U.-G. Meißner and W. Wang, Chiral Dynamics and S-wave Contributions in Semileptonic B decays, JHEP 1310, 011 (2013) [arXiv:1307.0947 [hep-ph]].
  * (99) F. Noël, Towards a consistent description of $K\pi$ scattering, Master’s thesis, University of Bonn (2020).
  * (100) L. von Detten, A new parametrization for scalar $K\pi$ scattering and production, Master’s thesis, University of Bonn (2020).
  * (101) D. Müller, D. Robaschik, B. Geyer, F.-M. Dittes and J. Hořejši, Wave functions, evolution equations and evolution kernels from light ray operators of QCD, Fortsch. Phys. 42, 101 (1994) [hep-ph/9812448].
  * (102) M. Diehl, T. Gousset, B. Pire and O. Teryaev, Probing partonic structure in $\gamma^{*}\gamma\to\pi\pi$ near threshold, Phys. Rev. Lett. 81, 1782 (1998) [hep-ph/9805380].
  * (103) M. V. Polyakov, Hard exclusive electroproduction of two pions and their resonances, Nucl. Phys. B 555, 231 (1999) [hep-ph/9809483].
  * (104) N. Kivel, L. Mankiewicz and M. V. Polyakov, NLO corrections and contribution of a tensor gluon operator to the process $\gamma^{*}\gamma\to\pi\pi$, Phys. Lett. B 467, 263 (1999) [hep-ph/9908334].
  * (105) M. Diehl, Generalized parton distributions, Phys. Rept. 388, 41 (2003) [hep-ph/0307382].
  * (106) P. Hägler, B. Pire, L. Szymanowski and O. V. Teryaev, Hunting the QCD-Odderon in hard diffractive electroproduction of two pions, Phys. Lett. B 535, 117 (2002) [Erratum: Phys. Lett. B 540, 324 (2002)] [hep-ph/0202231].
  * (107) B. Pire, F. Schwennsen, L. Szymanowski and S. Wallon, Hard Pomeron-Odderon interference effects in the production of $\pi^{+}\pi^{-}$ pairs in high energy gamma-gamma collisions at the LHC, Phys. Rev. D 78, 094009 (2008) [arXiv:0810.3817 [hep-ph]].
  * (108) H. Y. Cheng, C. K. Chua and K. C. Yang, Charmless hadronic B decays involving scalar mesons: Implications to the nature of light scalar mesons, Phys. Rev. D 73, 014017 (2006) [hep-ph/0508104].
  * (109) W. Wang and R. L. Zhu, To understand the rare decay $B_{s}\to\pi^{+}\pi^{-}\ell^{+}\ell^{-}$, Phys. Lett. B 743, 467-471 (2015) [arXiv:1502.05104 [hep-ph]].
  * (110) Y. J. Shi and W. Wang, Chiral Dynamics and S-wave contributions in Semileptonic $D_{s}/B_{s}$ decays into $\pi^{+}\pi^{-}$, Phys. Rev. D 92, 074038 (2015) [arXiv:1507.07692 [hep-ph]].
  * (111) Y. J. Shi, W. Wang and S. Zhao, Chiral dynamics, S-wave contributions and angular analysis in $D\rightarrow\pi\pi\ell\bar{\nu}$, Eur. Phys. J. C 77, 452 (2017) [arXiv:1701.07571 [hep-ph]].
  * (112) C. H. Chen and H. n. Li, Three body nonleptonic B decays in perturbative QCD, Phys. Lett. B 561, 258 (2003) [hep-ph/0209043].
  * (113) W. F. Wang, H. n. Li, W. Wang and C. D. Lü, $S$-wave resonance contributions to the $B^{0}_{(s)}\to J/\psi\pi^{+}\pi^{-}$ and $B_{s}\to\pi^{+}\pi^{-}\mu^{+}\mu^{-}$ decays, Phys. Rev. D 91, 094024 (2015) [arXiv:1502.05483 [hep-ph]].
  * (114) A. J. Ma, Y. Li, W. F. Wang and Z. J. Xiao, $S$-wave resonance contributions to the $B^{0}_{(s)}\to\eta_{c}{(2S)}\pi^{+}\pi^{-}$ in the perturbative QCD factorization approach, Chin. Phys. C 41, 083105 (2017) [arXiv:1701.01844 [hep-ph]].
  * (115) D. Boito, J.-P. Dedonder, B. El-Bennich, R. Escribano, R. Kamiński, L. Lesniak and B. Loiseau, Parametrizations of three-body hadronic $B$\- and $D$-decay amplitudes in terms of analytic and unitary meson-meson form factors, Phys. Rev. D 96, 113003 (2017) [arXiv:1709.09739 [hep-ph]].
  * (116) Z. R. Liang, F. B. Duan and X. Q. Yu, Study of the quasi-two-body decays $B^{0s}\rightarrow\psi(3770)(\psi(3686))\pi^{+}\pi^{-}$ with perturbative QCD approach, Eur. Phys. J. C 79, 370 (2019) [arXiv:1810.12507 [hep-ph]].
  * (117) Y. Xing and Z. P. Xing, $S$-wave contributions in $\bar{B}_{s}^{0}\to(D^{0},\bar{D}^{0})\pi^{+}\pi^{-}$ within perturbative QCD approach, Chin. Phys. C 43, 073103 (2019) [arXiv:1903.04255 [hep-ph]].
  * (118) C. Hanhart, S. Holz, B. Kubis, A. Kupść, A. Wirzba and C.-W. Xiao, The branching ratio $\omega\rightarrow\pi^{+}\pi^{-}$ revisited, Eur. Phys. J. C 77, 98 (2017) [Erratum: Eur. Phys. J. C 78, 450 (2018)] [arXiv:1611.09359 [hep-ph]].
  * (119) B. Ananthanarayan, I. Caprini and D. Das, Electromagnetic charge radius of the pion at high precision, Phys. Rev. Lett. 119, 132002 (2017) [arXiv:1706.04020 [hep-ph]].
  * (120) U.-G. Meißner, Low-Energy Hadron Physics from Effective Chiral Lagrangians with Vector Mesons, Phys. Rept. 161, 213 (1988)
  * (121) F. Klingl, N. Kaiser and W. Weise, Effective Lagrangian approach to vector mesons, their structure and decays, Z. Phys. A 356, 193 (1996) [hep-ph/9607431].
  * (122) Y. S. Amhis et al. [HFLAV Collaboration], Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of 2018, arXiv:1909.12524 [hep-ex].
  * (123) V. Bernard, D. R. Boito and E. Passemar, Dispersive representation of the scalar and vector $K\pi$ form factors for $\tau\to K\pi\nu_{\tau}$ and $K_{\ell 3}$ decays, Nucl. Phys. B Proc. Suppl. 218, 140 (2011) [arXiv:1103.4855 [hep-ph]].
  * (124) S. Z. Jiang, Q. Wang and Y. Zhang, Computation of the $p^{6}$ order low-energy constants with tensor sources, Phys. Rev. D 87, 094014 (2013) [arXiv:1203.0712 [hep-ph]].
  * (125) S. Z. Jiang, Z. L. Wei, Q. S. Chen and Q. Wang, Computation of the $O(p^{6})$ order low-energy constants: An update, Phys. Rev. D 92, 025014 (2015) [arXiv:1502.05087 [hep-ph]].
  * (126) J. Rendón, P. Roig and G. Toledo Sánchez, Effective-field theory analysis of the $\tau^{-}\rightarrow(K\pi)^{-}\nu_{\tau}$ decays, Phys. Rev. D 99, 093005 (2019) [arXiv:1902.08143 [hep-ph]].
  * (127) F. Z. Chen, X. Q. Li, Y. D. Yang and X. Zhang, CP asymmetry in $\tau\to K_{S}\pi\nu_{\tau}$ decays within the Standard Model and beyond, Phys. Rev. D 100, 113006 (2019) [arXiv:1909.05543 [hep-ph]].
  * (128) E. A. Garcés, M. Hernández Villanueva, G. López Castro and P. Roig, Effective-field theory analysis of the $\tau^{-}\to\eta^{(\prime)}\pi^{-}\nu_{\tau}$ decays, JHEP 1712, 027 (2017) [arXiv:1708.07802 [hep-ph]].
  * (129) G. López Castro, Non-standard interactions in $\tau^{-}\to(\pi^{-}\eta,\pi^{-}\pi^{0})\nu_{\tau}$ decays, SciPost Phys. Proc. 1, 008 (2019) [arXiv:1812.05892 [hep-ph]].
  * (130) V. M. Braun, G. P. Korchemsky and D. Müller, The Uses of conformal symmetry in QCD, Prog. Part. Nucl. Phys. 51, 311 (2003) [hep-ph/0306057].


$\displaystyle\det\begin{bmatrix}\displaystyle\frac{\partial
I_{\mathcal{A}}}{\partial s}&\displaystyle\frac{\partial
I_{\mathcal{A}}}{\partial E}\\\\[9.0pt] \displaystyle\frac{\partial
I_{\mathcal{B}}}{\partial s}&\displaystyle\frac{\partial
I_{\mathcal{B}}}{\partial E}\end{bmatrix}=i\pi.$ (5.37)

Proof. The determinant gives

$\displaystyle\det\mathbb{J}=\frac{\partial I_{\mathcal{A}}}{\partial
s}\frac{\partial I_{\mathcal{B}}}{\partial E}-\frac{\partial
I_{\mathcal{B}}}{\partial s}\displaystyle\frac{\partial
I_{\mathcal{A}}}{\partial E}.$ (5.38)

Since $Q(z;{s},E)=z^{4}+{s}z^{2}+2z+E,$ the derivative w.r.t. $E$ gives the
holomorphic periods and the derivative in $s$ gives second-kind periods. Thus
we can compute the above expressions with the Riemann bilinear identity to
give

$\displaystyle\det\mathbb{J}$
$\displaystyle=\frac{1}{4}\int_{\mathcal{A}}\frac{z^{2}{\mathrm{d}}z}{\sqrt{Q(z;s,E)}}\int_{\mathcal{B}}\frac{{\mathrm{d}}z}{\sqrt{Q(z;s,E)}}-\frac{1}{4}\int_{\mathcal{A}}\frac{z^{2}{\mathrm{d}}z}{\sqrt{Q(z;s,E)}}\int_{\mathcal{B}}\frac{{\mathrm{d}}z}{\sqrt{Q(z;s,E)}}$
$\displaystyle=2i\pi\frac{1}{2}\mathop{\rm
res}_{z=\infty^{+}}\frac{z^{2}{\mathrm{d}}z}{\sqrt{Q(z;s,E)}}\int_{\tau_{1}}^{z}\frac{{\mathrm{d}}\xi}{\sqrt{Q(\xi;s,E)}}=i\pi$

where we have used that

$\displaystyle\frac{z^{2}{\mathrm{d}}z}{\sqrt{Q(z;s,E)}}=\left(1+\mathcal{O}(z^{-2})\right){\mathrm{d}}z\qquad\int_{\tau_{1}}^{z}\frac{{\mathrm{d}}\xi}{\sqrt{Q(\xi;s,E)}}=\int_{\tau_{1}}^{\infty}\frac{{\mathrm{d}}\xi}{\sqrt{Q(\xi;s,E)}}-\frac{1}{z}+\mathcal{O}(z^{-2}).$
(5.39)

and the contribution from the point $\infty^{-}$ is the same as the point
$\infty^{+}$. $\blacksquare$

We observe that

$\omega:=\frac{\partial I_{\mathcal{A}}}{\partial
E}=\int_{\tau_{2}}^{\tau_{0}}\frac{{\mathrm{d}}z}{\sqrt{Q(z_{+};s,E)}},\quad\omega^{\prime}:=\frac{\partial
I_{\mathcal{B}}}{\partial
E}=\int_{\tau_{1}}^{\tau_{0}}\frac{{\mathrm{d}}z}{\sqrt{Q(z_{+};s,E)}}$ (5.40)

are the half periods of the holomorphic differential
$\frac{{\mathrm{d}}z}{\sqrt{Q(z;s,E)}}$. The lemma is useful in that it allows
us to explore the geometry of the quantization conditions;

######  Proposition 5.6.

Let $(s_{0},E_{0})$ correspond to the first-order quantization conditions
(5.23) or (5.24) in the bulk, namely, $m_{j}/n\simeq c_{j}\neq 0$. Then the
neighbour points in the $s$–plane form a slowly modulated hexagonal lattice in
the sense that the six closest neighbours of $s_{0}$ are

$s_{0}+2\hbar\left(\omega\Delta m_{1}-\omega^{\prime}\Delta m_{2}\right)$
(5.41)

where $\omega$ and $\omega^{\prime}$ are the half periods of the holomorphic
differentials in (5.40) and

$\displaystyle\Delta m_{j}\in\\{-1,0,1\\},\qquad\ |\Delta m_{1}+\Delta
m_{2}|\leq 1,\qquad|\Delta m_{1}|+|\Delta m_{2}|\geq 1.$ (5.42)

Proof. Let $(m_{1},m_{2},m_{3})$ be a triple of quantization numbers for
either (5.23) or (5.24). The neighbour points correspond to adding/subtracting
$1$ from each, subject to the constraints

$\displaystyle\Delta m_{1}+\Delta m_{2}+\Delta m_{3}=0,\ \ \ \ \Delta
m_{j}\in\\{-1,0,1\\}.$ (5.43)

There are six elementary possibilities

$(\Delta m_{1},\Delta m_{2},\Delta
m_{3})\in\bigg{\\{}(1,-1,0),(1,0,-1),(0,1,-1),(-1,1,0),(-1,0,1),(0,-1,1)\bigg{\\}}.$

The values of the periods
$\int_{\tau_{\ell}}^{\tau_{0}}\sqrt{Q(z_{+};s,E)}{\mathrm{d}}z$, $\ell=1,2,3$,
change by $\hbar{\Delta m_{\ell}}$, where $\hbar=(n+1)^{-1}$ in the ST case
and $\hbar(n+1/2)^{-1}$ in the VY case.

Let $(s,E)=(s_{0},E_{0})+(\Delta s,\Delta E)$ be a neighbour point in the
lattice. We want to estimate $(\Delta s,\Delta E)$: we observe that

$\int_{\tau_{2}}^{\tau_{0}}\sqrt{Q(z_{+};s,E)}{\mathrm{d}}z=\frac{1}{2}I_{\mathcal{A}},\quad\int_{\tau_{1}}^{\tau_{0}}\sqrt{Q(z_{+};s,E)}{\mathrm{d}}z=\frac{1}{2}I_{\mathcal{B}},\quad\int_{\tau_{3}}^{\tau_{0}}\sqrt{Q(z_{+};s,E)}{\mathrm{d}}z=-\frac{1}{2}I_{\mathcal{A}}-\frac{1}{2}I_{\mathcal{B}}$

where $I_{\mathcal{A}}$ and $I_{\mathcal{B}}$ are defined in Lemma 5.5. If we
take for example the first two periods and expand (5.23) or (5.24) to linear
order, we obtain

$\displaystyle\begin{bmatrix}\Delta s\\\ \Delta E\end{bmatrix}\simeq
2\hbar\begin{bmatrix}\displaystyle\frac{\partial I_{\mathcal{A}}}{\partial
E}&\displaystyle\frac{-\partial I_{\mathcal{B}}}{\partial E}\\\\[9.0pt]
\displaystyle\frac{-\partial I_{\mathcal{A}}}{\partial
s}&\displaystyle\frac{\partial I_{\mathcal{B}}}{\partial
s}\end{bmatrix}\begin{bmatrix}\Delta m_{1}\\\ \Delta m_{2}\end{bmatrix}$
(5.44)

so that, from the definition (5.40) one recovers (5.41). This local lattice
generators $\omega$ and $\omega^{\prime}$ are slowly modulated across the
elliptic region. $\blacksquare$

##### Near the origin.

If $(s,E)=\mathcal{O}(\hbar)$ then the elliptic surface is
$w^{2}=z^{4}+\mathcal{O}(\hbar)z^{2}+2z+\mathcal{O}(\hbar)$ and then a direct
computation shows that $\boldsymbol{\tau}={\rm
e}^{\frac{2i\pi}{3}}+\mathcal{O}(\hbar)$. Note that the $\mathbb{Z}_{3}$
symmetry of the limiting elliptic curve gives

$\displaystyle{\rm
e}^{2i\pi/3}=\boldsymbol{\tau}=\frac{-1}{1+\boldsymbol{\tau}}=-1-\frac{1}{\boldsymbol{\tau}}.$
(5.45)

We are now going to show that two quantization conditions yield the same
lattices to order $\hbar^{2}$.

######  Theorem 5.7.

The rescaled lattices of the zeroes of the VY Polynomials, and the of ST
problem coincide to within order $\mathcal{O}(\hbar^{2})=\mathcal{O}(n^{-2})$
in a $\mathcal{O}(\hbar)$ neighbourhood of the origin in the $s$–plane. More
precisely the quantization conditions (5.23), (5.24) corresponding to the
triples $(m_{1},m_{2},m_{3}),\ \ m_{1}+m_{2}+m_{3}=n-1$ and
$(k_{1},k_{2},k_{3}),\ \ k_{1}+k_{2}+k_{3}=n-1$ with $m_{j}=k_{j}$ single out
values of $s,E$ that differ by a discrepancy of order
$\mathcal{O}(\hbar^{2})$, provided that $m_{j}-\frac{n-1}{3}$ remain bounded
as $n\to\infty$.

Proof. Let $(s,E)=\mathcal{O}(\hbar)$. Then the two quantization conditions
(5.23), (5.24) read, to order $\hbar$,

$\displaystyle\begin{split}&2(n+1)\int_{\tau_{j}}^{\tau_{0}}\sqrt{Q(z_{+};s,E)}{\mathrm{d}}z=-2i\pi\left(m_{j}+\frac{2}{3}\right)\\\
&(2n+1)\int_{\tau_{j}}^{\tau_{0}}\sqrt{Q(z_{+};s,E)}{\mathrm{d}}z=-2i\pi\left(k_{j}+\frac{1}{2}\right)\end{split}$
(5.46)

Let us now set $s=\hbar\delta s$, $E=\hbar\delta E$ for some fixed values
$\delta s,\delta E$. Then we can use the following linear approximation

$\displaystyle\int_{\tau_{j}}^{\tau_{0}}\sqrt{Q(z_{+};s,E)}{\mathrm{d}}z$
$\displaystyle\simeq\int_{\tau_{j}}^{\tau_{0}}\sqrt{Q(z_{+};0,0)}{\mathrm{d}}z+\hbar\delta
s\int_{\tau_{j}}^{\tau_{0}}\frac{z^{2}{\mathrm{d}}z}{2\sqrt{Q(z_{+};0,0)}}+\hbar\delta
E\int_{\tau_{j}}^{\tau_{0}}\frac{{\mathrm{d}}z}{2\sqrt{Q(z_{+};0,0)}}=$ (5.47)
$\displaystyle=-\frac{i\pi}{3}+\hbar\left(c_{2}{\rm
e}^{\frac{2i\pi}{3}(j-1)}\delta s+c_{0}{\rm e}^{-\frac{2i\pi}{3}(j-1)}\delta
E\right){\rm e}^{\frac{7i\pi}{6}}$ (5.48) $\displaystyle c_{\ell}$
$\displaystyle=\int_{-2^{\frac{1}{3}}}^{0}\frac{z^{\ell}{\mathrm{d}}z}{2\sqrt{|z^{4}+2z|}},\
\ \ c_{0}\simeq 1.9276,\ \ c_{2}\simeq 0.9409.$ (5.49)

where $\hbar$ means either $(n+1)^{-1}$ or $(n+1/2)^{-1}$ depending on the
case we are considering. Replacing the above expansions in (5.46) we obtain
(all to within $\mathcal{O}(\hbar)$);

$\displaystyle 2(n+1)\left(-\frac{i\pi}{3}+\frac{1}{n+1}\left(c_{2}{\rm
e}^{\frac{2i\pi}{3}(j-1)}\delta s+c_{0}{\rm e}^{-\frac{2i\pi}{3}(j-1)}\delta
E\right){\rm e}^{\frac{7i\pi}{6}}\right)=-2i\pi\left(m_{j}+\frac{2}{3}\right)$
(5.50)
$\displaystyle(2n+1)\left(-\frac{i\pi}{3}+\frac{1}{n+\frac{1}{2}}\left(c_{2}{\rm
e}^{\frac{2i\pi}{3}(j-1)}\delta s+c_{0}{\rm e}^{-\frac{2i\pi}{3}(j-1)}\delta
E\right){\rm e}^{\frac{7i\pi}{6}}\right)=-2i\pi\left(k_{j}+\frac{1}{2}\right)$
(5.51)

Simplifying we get the quantization rules for $\delta s,\delta E$ in identical
form provided we identify $m_{j}=k_{j}$:

$\displaystyle 2\left(c_{2}{\rm e}^{\frac{2i\pi}{3}(j-1)}\delta s+c_{0}{\rm
e}^{-\frac{2i\pi}{3}(j-1)}\delta E\right){\rm
e}^{\frac{7i\pi}{6}}=-2i\pi\left(m_{j}-\frac{n-1}{3}\right)+\mathcal{O}(\hbar).$
(5.52)

Since the uncertainty on $\delta s,\delta E$ is of order $\mathcal{O}(\hbar)$,
the uncertainty on $s=\hbar\delta s,E=\hbar\delta E$ is of order
$\mathcal{O}(\hbar^{2})$, namely, the two quantization conditions give two
approximate lattices that differ by $\mathcal{O}(\hbar^{2})$ as long as
$m_{j}-\frac{n-1}{3}$ remain bounded as $n\to\infty$. $\blacksquare$

######  Remark 5.8.

The differential equation (3.1) for $t=0=\Lambda$ can be solved in terms of
Whittaker ${\rm W}_{\mu,\nu},{\rm M}_{\mu,\nu}$ functions (i.e. confluent
hypergeometric functions) [DLMF 13.14] as follows:

$\displaystyle y^{\prime\prime}-(z^{4}+2Jz)y=0\ \ ,\ y_{1}=\frac{1}{z}{\rm
M}_{-\frac{J}{3},\frac{1}{6}}\left(\frac{2z^{3}}{3}\right),\ \
y_{2}=\frac{1}{z}{\rm
W}_{-\frac{J}{3},\frac{1}{6}}\left(\frac{2z^{3}}{3}\right).$ (5.53)

Writing $J=n+1$, $n=0,1,2,\dots$ we have that for $n\equiv 1\mod 3$ ($J\equiv
2\mod 3$) the solution $y_{1}$ is our quasi-polynomial solution. For example

$\displaystyle J=2$ $\displaystyle
y_{1}(z)=\left(\frac{2}{3}\right)^{\frac{2}{3}}z\,{\rm e}^{z^{3}/3}$
$\displaystyle J=5$ $\displaystyle
y_{1}(z)=\left(\frac{2}{3}\right)^{\frac{2}{3}}\frac{z}{2}(z^{3}+2)\,{\rm
e}^{z^{3}/3}$ $\displaystyle J=8$ $\displaystyle
y_{1}(z)=\left(\frac{2}{3}\right)^{\frac{2}{3}}\frac{z}{7}(z^{6}+7z^{3}+7)\,{\rm
e}^{z^{3}/3}$ $\displaystyle J=11$ $\displaystyle
y_{1}(z)=\left(\frac{2}{3}\right)^{\frac{2}{3}}\frac{z}{70}(2z^{9}+30z^{6}+105z^{3}+70)\,{\rm
e}^{z^{3}/3}$

et cetera. This corresponds to the quantization conditions
$m_{j}=\frac{n-1}{3}$, $j=1,2,3$.

##  Appendix A Contribution of the integrals near the turning points

In order to estimate integrals $\int\Psi^{2}{\mathrm{d}}z$ near turning points
we need, as a model, to estimate the corresponding integrals for the solutions
of the normal form of the ODE near a simple turning point. Namely, we need to
estimate integrals involving squares of solutions of the Airy equation. This
simple lemma will be expedient.

######  Lemma A.1.

Let $f=f(x),g=g(x)$ be two arbitrary solutions of
$y(x)^{\prime\prime}-xy(x)=0$. There are polynomials
$A_{n},B_{n},C_{n}\in\mathbb{C}[x]$ of degrees $n+1,n,n-1$, respectively, such
that

$\displaystyle
x^{n}f^{2}(x)=\frac{{\mathrm{d}}}{{\mathrm{d}}x}\left(A_{n}(x)f(x)^{2}-B_{n}(x)(f^{\prime}(x))^{2}+C_{n}(x)f(x)f^{\prime}(x)\right).$
(A.1) $\displaystyle
x^{n}f\,g=\frac{{\mathrm{d}}}{{\mathrm{d}}x}\left(A_{n}\,f\,g-B_{n}\,f^{\prime}g^{\prime}+C_{n}\frac{fg^{\prime}+gf^{\prime}}{2}\right).$
(A.2)

They are related to each other by the following equations:

$\displaystyle A_{n}(x)=xB_{n}(x)-\frac{1}{2}B_{n}^{\prime\prime}(x),\qquad
C_{n}(x)=B_{n}^{\prime}(x),\qquad
B_{n}(x)+2xB_{n}^{\prime}(x)-\frac{1}{2}B_{n}^{\prime\prime\prime}(x)=x^{n}.$
(A.3)

The first few are:

$\displaystyle\begin{array}[]{cc|c|c}n&A_{n}&B_{n}\\\ \hline\cr 0&x&1\\\
\hline\cr 1&\frac{x^{2}}{3}&\frac{x}{3}\\\ \hline\cr
2&\frac{x^{3}}{5}-\frac{1}{5}&\frac{x^{2}}{5}\\\ \hline\cr
3&\frac{x^{4}}{7}&\frac{x^{3}}{7}-\frac{3}{7}\\\ \hline\cr
4&\frac{x^{2}(x^{3}-2)}{9}&\frac{x(x^{3}+4)}{9}\end{array}.$ (A.10)

The leading coefficients of $A_{n},B_{n}$ are

$\displaystyle A_{n}(x)=\frac{x^{n+1}}{2n+1}+\mathcal{O}(x^{n}),\ \ \
B_{n}=\frac{x^{n}}{2n+1}+\mathcal{O}(x^{n-1}).$ (A.11)

Proof. It suffices to prove the first formula since the second is obtained by
polarization. Let $A,B,C$ be polynomials of degrees at most $n+1,n,n-1$
respectively and consider the Anzatz

$\displaystyle J=Af^{2}-B(f^{\prime})^{2}+Cff^{\prime}.$ (A.12)

Then, using the ODE, we get

$\displaystyle
J^{\prime}=\big{(}A^{\prime}+xC\big{)}f^{2}+\big{(}2A-2xB+C^{\prime}\big{)}ff^{\prime}+\big{(}C-B^{\prime}\big{)}(f^{\prime})^{2}.$
(A.13)

Thus we seek polynomials such that

$\displaystyle A^{\prime}+xC=x^{n},\ \ \ \ C=B^{\prime},\ \ \
2A-2xB+C^{\prime}=0$ (A.14) $\displaystyle A^{\prime}+xB^{\prime}=x^{n},\ \ \
\ C=B^{\prime},\ \ \ 2A-2xB+B^{\prime\prime}=0$ (A.15) $\displaystyle
A=xB-\frac{1}{2}B^{\prime\prime},\ \ C=B^{\prime},\ \ \
B+2xB^{\prime}-\frac{1}{2}B^{\prime\prime\prime}=x^{n}.$ (A.16)

The latter equation for $B$ has a unique polynomial solution obtained by
inverting the (finite-dimensional) linear operator on the space of polynomials
of degree $\leq n$. The relationship of the leading coefficients follows by
plugging into (A.1). $\blacksquare$

######  Proposition A.2.

Let $f(x)$ be an analytic function in a neighbourhood of the origin.

  1. [1]

Let $a,b>0$ . Then

$\displaystyle\int_{-a}^{b}{\mathrm{Ai}}^{2}(\hbar^{-\frac{2}{3}}x)f(x){\mathrm{d}}x=i\hbar^{\frac{1}{3}}\int_{-a}^{0}\frac{f(x){\mathrm{d}}x}{2\pi\sqrt{x}_{+}}+\mathcal{O}(\hbar^{\frac{4}{3}}),$
(A.17)

where $\sqrt{x}$ is analytic in $\mathbb{C}\backslash(-\infty,0]$ and positive
for $x>0$ and $\sqrt{x}_{+}$ denotes the boundary value from the upper half
plane.

  2. [2]

Let $a,b>0$. Then

$\displaystyle\int_{-a}^{{\rm
e}^{i\pi/3}b}{\mathrm{Ai}}\left(\hbar^{-\frac{2}{3}}\omega^{k}x\right){\mathrm{Ai}}\left(\hbar^{-\frac{2}{3}}\omega^{j}x\right)f(x){\mathrm{d}}x=q_{jk}\int_{-a}^{0}\frac{i\hbar^{\frac{1}{3}}f(x){\mathrm{d}}x}{2\pi\sqrt{x}_{+}}+r_{jk}\int_{0}^{{\rm
e}^{i\pi/3}b}\frac{i\hbar^{\frac{1}{3}}f(x){\mathrm{d}}x}{2\pi\sqrt{x}}+\mathcal{O}(\hbar^{\frac{4}{3}}).$
(A.18)

Here $\omega=e^{\frac{2\pi i}{3}}$ and $\sqrt{x}$ and $\sqrt{x}_{+}$ are
defined as above and

$\displaystyle q_{jk}=\left\\{\begin{array}[]{cc}1&j=k=0\\\\[4.0pt] \frac{{\rm
e}^{\frac{i\pi}{3}}}{2}&j=0,\ k=1\\\\[4.0pt] \frac{{\rm
e}^{-\frac{i\pi}{3}}}{2}&j=0,\ k=2\\\\[4.0pt] 0&j=1=k\\\\[4.0pt]
\frac{1}{2}&j=1,\ k=2\\\\[4.0pt] 0&j=k=2\end{array}\right.\ \ \ \ \
r_{jk}=\left\\{\begin{array}[]{cc}0&j=k=0\\\\[4.0pt] -\frac{{\rm
e}^{\frac{i\pi}{3}}}{2}&j=0,\ k=1\\\\[4.0pt] \frac{{\rm
e}^{-\frac{i\pi}{3}}}{2}&j=0,\ k=2\\\\[4.0pt] {\rm
e}^{-\frac{i\pi}{3}}&j=1=k\\\\[4.0pt] \frac{1}{2}&j=1,\ k=2\\\\[4.0pt]
0&j=k=2\end{array}\right..$ (A.31)

Proof. We will consider $f(x)=x^{n}$, with the proof being completed simply by
summing the series.
[1] Then we need to compute
$\int_{-a}^{b}{\mathrm{Ai}}^{2}(\hbar^{-\frac{2}{3}}x)x^{n}{\mathrm{d}}x.$ We
can use Lemma A.1 and setting
$\alpha=a\hbar^{-\frac{2}{3}},\beta=b\hbar^{-\frac{2}{3}}$ we obtain

$\displaystyle\int_{-a}^{b}{\mathrm{Ai}}^{2}(\hbar^{-\frac{2}{3}}x)x^{n}{\mathrm{d}}x=\hbar^{\frac{2(n+1)}{3}}\int_{-\alpha}^{\beta}{\mathrm{Ai}}^{2}(\xi)\xi^{n}{\mathrm{d}}\xi=$
$\displaystyle=\hbar^{\frac{2(n+1)}{3}}\bigg{(}A_{n}(\xi){\mathrm{Ai}}^{2}(\xi)-B_{n}(\xi)\,({\mathrm{Ai}}^{\prime}(\xi))^{2}\
+C_{n}(\xi){\mathrm{Ai}}(\xi){\mathrm{Ai}}(\xi)^{\prime}\bigg{)}\bigg{|}_{-\alpha}^{\beta}.$
(A.32)

Recall now the asymptotic expansion of the Airy function

$\displaystyle{\mathrm{Ai}}(s)$ $\displaystyle=\frac{{\rm
e}^{-\frac{2}{3}s^{\frac{3}{2}}}}{2\sqrt{\pi}s^{\frac{1}{4}}}\big{(}1+\mathcal{O}(s^{-\frac{3}{2}})\big{)},\
\ \ |s|\to+\infty,\ \ |\arg(s)|<\pi-\epsilon$ (A.33)
$\displaystyle{\mathrm{Ai}}(-s)$
$\displaystyle=\frac{\sin\left(\frac{2}{3}s^{\frac{3}{2}}+\frac{\pi}{4}\right)}{\sqrt{\pi}s^{\frac{1}{4}}}\big{(}1+\mathcal{O}(s^{-3})\big{)}-\frac{\cos\left(\frac{2}{3}s^{\frac{3}{2}}+\frac{\pi}{4}\right)}{\sqrt{\pi}s^{\frac{1}{4}}}\mathcal{O}(s^{-\frac{3}{2}}),\
\ \ |s|\to\infty,|\arg s|<\frac{2\pi}{3}.$ (A.34)

It is then clear that the evaluation at $\beta=\hbar^{-\frac{2}{3}}b$ yields
exponentially small terms thanks to (A.33).

For the evaluation at $-\alpha=-\hbar^{-\frac{2}{3}}a$ we use instead (A.34);
thanks to (A.11) we see that

$\displaystyle\hbar^{\frac{2(n+1)}{3}}\int_{-\alpha}^{\beta}{\mathrm{Ai}}^{2}(\xi)\xi^{n}{\mathrm{d}}\xi=\hbar^{\frac{2(n+1)}{3}}\frac{(-\alpha)^{n+1}}{2n+1}\frac{1}{\pi\sqrt{\alpha}}\left(1+\mathcal{O}(\alpha^{-\frac{3}{2}})\right)=$
$\displaystyle=\hbar^{\frac{1}{3}}\frac{(-a)^{n+1}}{2n+1}\frac{1}{\pi\sqrt{a}}\left(1+\mathcal{O}(\alpha^{-\frac{3}{2}})\right)=\hbar^{\frac{1}{3}}\int_{-a}^{0}\frac{x^{n}{\mathrm{d}}x}{2\pi\sqrt{-x}}+\mathcal{O}(\hbar^{\frac{4}{3}}).$
(A.35)

Finally, we observe that $\sqrt{-x}=-i\sqrt{x}_{+}$ for $x\in\mathbb{R}_{-}$.
[2] All these integrals involve oscillatory direction for each of the
arguments. The computation is of entirely similar nature, where it is
particularly important to pay great care at the relative phases.

Using Lemma A.1 we need to estimate integrals of the form
(${\mathrm{Ai}}_{j}(\xi):={\mathrm{Ai}}(\omega^{j}\xi)$)

$\displaystyle\int_{-a}^{{\rm
e}^{\frac{i\pi}{3}}b}x^{n}{\mathrm{Ai}}_{j}(x\hbar^{-\frac{2}{3}}){\mathrm{Ai}}_{k}(x\hbar^{-\frac{2}{3}}){\mathrm{d}}x=\hbar^{\frac{2(n+1)}{3}}\int_{-\alpha}^{{\rm
e}^{\frac{i\pi}{3}}\beta}\xi^{n}{\mathrm{Ai}}_{j}(\xi){\mathrm{Ai}}_{k}(\xi){\mathrm{d}}\xi=\hbar^{\frac{2(n+1)}{3}}H_{jk}(\xi)\Bigg{|}_{-\alpha}^{{\rm
e}^{\frac{i\pi}{3}}\beta}$ $\displaystyle
H_{n}^{(j,k)}:=\left[A_{n}{\mathrm{Ai}}_{j}{\mathrm{Ai}}_{k}-B_{n}{\mathrm{Ai}}_{j}^{\prime}{\mathrm{Ai}}_{k}^{\prime}-C_{n}\frac{{\mathrm{Ai}}_{j}^{\prime}{\mathrm{Ai}}_{k}+{\mathrm{Ai}}_{j}{\mathrm{Ai}}_{k}^{\prime}}{2}\right]$
(A.36)

At the point $\alpha=\hbar^{-\frac{2}{3}}a$ a direct computation using the
asymptotic properties (A.33), (A.34) yields:

$\displaystyle-\hbar^{\frac{2(n+1)}{3}}H_{n}^{(j,k)}(-\hbar^{-\frac{2}{3}}a)\simeq
i\int_{-a}^{0}\frac{\hbar^{\frac{1}{3}}x^{n}{\mathrm{d}}x}{2\pi\sqrt{x_{+}}}\times
q_{jk}+\mathcal{O}(\hbar^{\frac{4}{3}})$ (A.37)

Similarly one finds

$\displaystyle\hbar^{\frac{2(n+1)}{3}}H_{n}^{(j,k)}(\hbar^{-\frac{2}{3}}b{\rm
e}^{i\pi/3})\simeq i\int_{0}^{b{\rm
e}^{i\pi/3}}\frac{\hbar^{\frac{1}{3}}x^{n}{\mathrm{d}}x}{2\pi\sqrt{x}}\times
r_{jk}+\mathcal{O}(\hbar^{\frac{4}{3}})$ (A.38)

with $q_{jk},r_{jk}$ as in (A.31). Now the integral (A.18) with $f(x)=x^{n}$
becomes

$\displaystyle\int_{-a}^{{\rm
e}^{i\pi/3}b}{\mathrm{Ai}}(\hbar^{-\frac{2}{3}}\omega^{k}x){\mathrm{Ai}}(\hbar^{-\frac{2}{3}}\omega^{j}x)x^{n}{\mathrm{d}}x=\hbar^{\frac{2(n+1)}{3}}H_{n}^{(j,k)}(\xi)\bigg{|}_{-\hbar^{-\frac{2}{3}}a}^{\hbar^{-\frac{2}{3}}{\rm
e}^{\frac{i\pi}{3}}b}.$ (A.39)

Putting together the two terms we obtain the proof. $\blacksquare$

As a consequence of the Lemma A.1 we can estimate the contribution of
integrals involving $\Psi^{2}$ in a neighbourhood of a turning point.

######  Theorem A.3.

Let $\tau$ be a simple turning point for the potential $Q(x)$, $\gamma_{d}$ a
steepest descent $\ominus$ path and $\gamma_{o}$ the oscillatory path on the
opposite side of $\tau$: we choose two points
$\tau_{+}\in\gamma_{d},\tau_{-}\in\gamma_{o}$, in the neighbourhood of $\tau$
at finite positive distance. Let $\Psi$ be a solution of the ODE

$\displaystyle\hbar^{2}\Psi^{\prime\prime}(x)-Q(x)\Psi(x)=0.$ (A.40)

Suppose that $\Psi$ is asymptotic to the recessive formal solution
$\psi_{+}^{(\tau)}$. Then

$\displaystyle\int_{\tau_{-}}^{\tau_{+}}\Psi^{2}{\mathrm{d}}z=\int_{\tau_{-}}^{\tau}\frac{2i\hbar\,\differential
z}{\sqrt{Q(z_{+};s,E)}}+\mathcal{O}(\hbar^{2}).$ (A.41)

with the branch-cut of $\sqrt{Q}$ running along the oscillatory path and the
determination the one that has regative real part on the descent path
$\gamma_{d}$.

Figure 11: The integration for Theorem A.3.

Proof. In [Vör83] it is shown that there is a conformal mapping
$\xi(z;\hbar^{2})\in\mathcal{O}(z)\otimes\mathbb{C}[[\hbar^{2}]]$ that maps
$z=\tau(=0)$ to $\xi(\tau;\hbar^{2})\equiv 0$ (identically in $\hbar$) that
maps the equation to the Airy equation:

$\displaystyle\hbar^{2}\partial_{z}^{2}\Psi(z)-Q(z)\Psi(z)=0\ \ \ \mapsto\ \ \
\ \hbar^{2}\partial_{\xi}^{2}f(\xi)-\xi f(\xi)=0.$ (A.42)

The relationship between the two equations is as follows

$\displaystyle Q(z)=\left(\frac{{\mathrm{d}}\xi}{{\mathrm{d}}x}\right)^{2}\
\xi-\frac{\hbar^{2}}{2}\\{\xi,z\\},\ \ \ \
\Psi(z)=\frac{f(\xi)}{\sqrt{{\mathrm{d}}\xi/{\mathrm{d}}z}}$ (A.43)

where $\\{,\\}$ is the Schwarzian derivative

$\displaystyle\\{\xi,z\\}=\frac{\xi^{\prime\prime\prime}}{\xi^{\prime}}-\frac{3}{2}\left(\frac{\xi^{\prime\prime}}{\xi^{\prime}}\right)^{2}.$
(A.44)

The formal series $\xi(z;\hbar^{2})$ is Borel–resummable to an analytic
conformal mapping whose asymptotic expansion as $\hbar\to 0_{+}$ coincides
with the formal series [KT05].

The arc $[\tau,\tau_{+}]$ on the contour $\gamma_{d}$ is mapped to a segment
$[0,\xi_{+}]\subset\mathbb{R}_{+}$ in the $\xi$–plane, and similarly the arc
$[\tau_{-},\tau]\subset\gamma_{o}$ maps to $[\xi_{-},0]\subset\mathbb{R}_{-}$.

The function $\Psi$ is the recessive solution on $\gamma_{+}$ and hence it
must map to a multiple of the ${\mathrm{Ai}}$ function within a whole
neighbourhood of $\tau$. Thus the integral to be estimated translates to

$\displaystyle\int_{\tau_{-}}^{\tau_{+}}\Psi(z;\hbar)^{2}{\mathrm{d}}z=C(\hbar)\int_{\xi_{-}}^{\xi_{+}}\frac{{\mathrm{Ai}}\left(\hbar^{-\frac{2}{3}}\xi\right)^{2}}{\xi^{\prime}(z^{-1}(\xi);\hbar^{2})^{2}}{\mathrm{d}}\xi,$
(A.45)

where $C(\hbar)$ is an appropriate proportionality constant that depends on
the chosen normalization of $\Psi$. Our choice is to normalize $\Psi$ so that
it is asymptotic to the recessive formal solution $\psi_{+}^{(\tau)}$. To fix
$C(\hbar)$ we can consider the asymptotics of the Airy function at $\tau_{+}$
(A.33) and compare it to the WKB solution. Indeed we find that

$\displaystyle\frac{{\mathrm{Ai}}\left(\hbar^{-\frac{4}{3}}\xi_{+}\right)^{2}}{\xi^{\prime}(z^{-1}(\xi_{+});\hbar^{2})}\simeq\frac{\hbar^{\frac{1}{3}}{\rm
e}^{-\frac{4}{3\hbar}\xi_{+}^{\frac{3}{2}}}}{4\pi\xi^{\prime}(z^{-1}(\xi_{+};\hbar^{2})\sqrt{\xi_{+}}}\simeq\frac{\hbar^{\frac{1}{3}}{\rm
e}^{\frac{2}{\hbar}\int_{\tau}^{\tau_{+}}\sqrt{Q}{\mathrm{d}}z}}{4\pi\sqrt{Q(\tau_{+};s,E)}}\simeq\frac{1}{4\hbar^{\frac{2}{3}}\pi}\left(\psi_{-}^{(\tau)}\right)^{2},$
(A.46)

so that $C(\hbar)=4\hbar^{\frac{2}{3}}\pi(1+\mathcal{O}(\hbar^{2}))$. The
integral involving the Airy function in (A.45) is of the general form of
Proposition A.2 and hence, to within $\mathcal{O}(\hbar^{\frac{4}{3}})$ we
have

$\displaystyle\int_{\xi_{-}}^{\xi_{+}}\frac{{\mathrm{Ai}}\left(\hbar^{-\frac{2}{3}}\xi\right)^{2}}{\xi^{\prime}(z^{-1}(\xi);\hbar^{2})^{2}}{\mathrm{d}}\xi=\frac{i\hbar^{\frac{1}{3}}}{2\pi}\int_{\xi_{-}}^{0}\frac{{\mathrm{d}}\xi}{\xi^{\prime}(z^{-1}(\xi);\hbar^{2})^{2}\sqrt{\xi(z^{-1}(\xi)_{+};\hbar^{2})}}.$
(A.47)

Now from (A.43) we get

$\displaystyle\xi^{\prime}\sqrt{\xi}=\sqrt{Q(z)}+\mathcal{O}(\hbar^{2}),$
(A.48)

and hence, to within $\mathcal{O}(\hbar^{2})$, we have

$\displaystyle\int_{\xi_{-}}^{0}\frac{{\mathrm{d}}\xi}{\xi^{\prime}(z^{-1}(\xi);\hbar^{2})^{2}\sqrt{\xi(z^{-1}(\xi);\hbar^{2})^{2}}}=\int_{\xi_{-}}^{0}\frac{{\mathrm{d}}\xi}{\xi^{\prime}(z^{-1}(\xi);\hbar^{2})\sqrt{Q(z^{-1}(\xi))}}=\int_{\tau_{-}}^{\tau}\frac{{\mathrm{d}}z}{\sqrt{Q(z)}}.$
(A.49)

Using the estimate of $C(\hbar)$ and substituting into (A.45) we obtain the
statement of the theorem. $\blacksquare$

###  A.1 Proof of Theorem 5.4

We will consider in detail only the case $D$ because it contains all the
intricacies.

Since we want to impose that the integral $I_{13}$ vanishes, it suffices to
determine the integral of a function that is proportional to $y_{n}=p_{n}{\rm
e}^{2\theta}$. Keeping this in mind we then observe that in the region
$(A_{1})$ along the Stokes curve $(\tau_{1},\infty_{1})$ the function is
recessive (as $n\to\infty$ and as $x\to\infty_{1}$) and hence it must be
proportional to $\Psi^{(A_{1})}_{+}(z;\hbar)$ and thus asymptotic to
$\psi_{+}^{(\tau_{1})}$. We will then forget about $y_{n}$ and integrate
instead the function $\Psi^{(A_{1})}_{+}(z;\hbar)^{2}$ along $\gamma$: recall
that this function is entire since it is the solution of the ODE (4.6).

Let us refer to Fig. 8. We choose the contour $\gamma$ to follow the steepest
descent from $\infty_{1}$ to $\tau_{1}$, then the branch-cut to $\tau_{0}$,
then to $\tau_{2}$ and the descent path to $\infty_{3}$. The branch-cut will
be chosen to follow the anti–Stokes curve in a neighbourhood of each turning
point.

Figure 12: The splitting of the integration of (A.62) along the three contours
for each of the three integrands near $\tau_{0}$.

We then split our contour $\gamma=\bigcup_{j=1}^{7}\gamma_{j}$ in several
pieces

$\displaystyle\infty_{1}\mathop{\longrightarrow}^{\gamma_{1}}\tau_{1}^{+}\mathop{\longrightarrow}^{\gamma_{2}}\tau_{1}^{-}\mathop{\longrightarrow}^{\gamma_{3}}\tau_{0}^{+}\mathop{\longrightarrow}^{\gamma_{4}}\tau_{0}^{-}\mathop{\longrightarrow}^{\gamma_{5}}\tau_{2}^{+}\mathop{\longrightarrow}^{\gamma_{6}}\tau_{2}^{-}\mathop{\longrightarrow}^{\gamma_{7}}\infty_{3}.$
(A.50)

Observe next that the solution $\Psi:=\Psi^{(A_{1})}_{+}$ is recessive both on
the Stokes line $\infty_{1}\to\tau_{1}$ as well as on the Stokes line
$\tau_{2}\to\infty_{3}$ because we are already on a point of the ES spectrum,
where the Stokes matrices satisfy
$\mathbb{S}_{0}=\mathbb{S}_{2}=\mathbb{S}_{4}=\mathbf{1}$. More precisely it
follows that $\Psi_{+}^{(A_{1})}=\Psi_{+}^{(D_{3})}={\rm
e}^{v_{12}}\Psi_{+}^{(D_{1})}={\rm e}^{v_{12}}\Psi_{+}^{(A_{2})}$.

##### Contributions of $\gamma_{1},\gamma_{7}$.

On both these contours the integral is exponentially small as $\hbar\to 0_{+}$
since $\Psi=\Psi_{+}^{(A_{1})}$ is recessive on $\gamma_{1}$ and $\Psi={\rm
e}^{v_{1}2}\Psi_{+}^{(D_{1})}$ is recessive on $\gamma_{7}$.

##### Contributions of $\gamma_{2},\gamma_{6}$.

These are estimated by the use of Theorem A.3

$\displaystyle\int_{\tau_{1}^{+}}^{\tau_{1}^{-}}\Psi^{2}{\mathrm{d}}z$
$\displaystyle=\int_{\tau_{1}^{+}}^{\tau_{1}^{-}}\frac{2i\hbar\,\differential
z}{\sqrt{Q(z_{-};s,E)}}+\mathcal{O}(\hbar^{2}),$ (A.51)
$\displaystyle\int_{\tau_{2}^{+}}^{\tau_{2}^{-}}\Psi^{2}{\mathrm{d}}z$
$\displaystyle={\rm
e}^{2v_{12}}\int_{\tau_{2}^{+}}^{\tau_{2}^{-}}\left(\Psi_{+}^{(D_{1})}\right)^{2}{\mathrm{d}}z={\rm
e}^{2v_{12}}\left(\int_{\tau_{2}^{+}}^{\tau_{2}}\frac{2i\hbar\,\differential
z}{\sqrt{Q(z_{-};s,E)}}+\mathcal{O}(\hbar^{2})\right).$ (A.52)

The boundary value is the $-$ because the orientation of the branch-cut that
we have chosen is the one egressing from the turning point, while in Theorem
A.3 it was the one towards the turning point (and the boundary value was the
$+$ one).

##### Contributions of $\gamma_{3},\gamma_{5}$.

Along either paths we are away from the turning points and can substitute the
asymptotic expressions in terms of the WKB solutions. We can use the
Riemann–Hilbert problem 4.7 to see that the function $\Psi=\Psi_{+}^{(A_{1})}$
has the asymptotic behaviour

$\displaystyle\Psi_{+}^{(A_{1})}(z;\hbar)$
$\displaystyle\simeq\left\\{\begin{array}[]{ll}\psi_{+}^{(\tau_{1})}(z;\hbar)+i\psi_{-}^{(\tau_{1})}(z;\hbar)&z\in(B_{1}^{r})\\\\[4.0pt]
{\rm e}^{-v_{10}}\psi_{+}^{(\tau_{0})}(z;\hbar)+i{\rm
e}^{v_{10}}\psi_{-}^{(\tau_{0})}(z;\hbar)&z\in(B_{1}^{\ell})\\\\[4.0pt] {\rm
e}^{v_{12}}\left({\rm e}^{v_{20}}\psi_{+}^{(\tau_{0})}(z;\hbar)-i{\rm
e}^{-v_{20}}\psi_{-}^{(\tau_{0})}(z;\hbar)\right)&z\in(C_{1}^{r})\\\\[4.0pt]
{\rm
e}^{v_{12}}\left(\psi_{+}^{(\tau_{2})}(z;\hbar)-i\psi_{-}^{(\tau_{2})}(z;\hbar)\right)&z\in(C_{1}^{\ell})\end{array}\right.$
(A.57)

The different expressions in $(B_{1}^{\ell,r})$ are due simply to the
different choice of normalization point for the WKB formal solutions, but we
can use either of the two expressions also as asymptotic expansion in the
other. Ditto for the pair $(C_{1}^{\ell,r})$. Consequently we can estimate the
contribution of the whole $\gamma_{3}$ using the asymptotic expression for
$(B_{1}^{r})$ as follows:

$\displaystyle\int_{\tau_{1}^{-}}^{\tau_{0}^{+}}\Psi^{2}{\mathrm{d}}z$
$\displaystyle\simeq\int_{\tau_{1}^{-}}^{\tau_{0}^{+}}\left((\psi_{+}^{(\tau_{1})})^{2}-(\psi_{-}^{(\tau_{1})})^{2}\right){\mathrm{d}}z+2i\int_{\tau_{1}^{-}}^{\tau_{0}^{+}}\psi_{+}^{(\tau_{1})}\psi_{-}^{(\tau_{1})}\,{\mathrm{d}}z,$
(A.58)

with similar expression for the integral along $\gamma_{5}$. The contribution
of the first integral in the right side of (A.58) is of order
$\mathcal{O}(\hbar^{2})$ as we show a few lines below. The main contribution
comes instead from the last integral and it yields

$\displaystyle
2i\int_{\tau_{1}^{-}}^{\tau_{0}^{+}}\psi_{+}^{(\tau_{1})}\psi_{-}^{(\tau_{1})}\,{\mathrm{d}}z=2i\int_{\tau_{1}^{-}}^{\tau_{0}^{+}}\frac{\hbar{\mathrm{d}}z}{\sqrt{Q(z_{-},s,E)}}+\mathcal{O}(\hbar^{2}),$
(A.59)

where the $-$ boundary value in the last integral is due to the fact that the
regions $(B_{1}^{\ell,r})$ lie on the $-$ side of the branch-cut, with the
orientation that we have chosen. In particular this contribution together with
the contribution (A.51) yield a single integration from $\tau_{1}$ to
$\tau_{0}^{+}$. Similar considerations apply to the integral along
$\gamma_{5}$ which yield

$\displaystyle\int_{\tau_{0}^{-}}^{\tau_{2}^{+}}\Psi^{2}{\mathrm{d}}z\simeq\xi_{12}\left(\int_{\tau_{0}^{-}}^{\tau_{2}^{+}}\frac{2i\hbar{\mathrm{d}}z}{\sqrt{Q(z_{+};s,E)}}+\mathcal{O}(\hbar^{2})\right)$
(A.60)

To see that the first integral in (A.58) is sub-dominant, for example consider
the integration of $(\psi_{+}^{(\tau_{1})})^{2}$. We can deform the path of
integration (at fixed endpoints) into the region $(C_{3})$ where the formal
solution is recessive. Then a simple estimate shows that the only contribution
come from the neighbourhood of the endpoints of integration (at which points
the real part of the exponential is zero). Then we can use Laplace’s method to
easily estimate their contribution of order777Note that the function being
integrated has already an $\hbar$ in front, and the integral contributes
another order $\mathcal{O}(\hbar)$. $\mathcal{O}(\hbar^{2})$. A similar
reasoning applies to the integration of $(\psi_{-}^{(\tau_{1})})^{2}$ where
instead we deform the contour within the region $(B_{1})$.

##### Contribution of $\gamma_{4}$.

We are going to show that this integral is estimated as follows:

$\displaystyle-\int_{\tau_{0}^{+}}^{\tau_{0}}\frac{2i\hbar(1+\mathcal{O}(\hbar))\differential
z}{\sqrt{Q(z_{+};s.E)}}-\left(1+{\rm
e}^{2v_{12}}\right)\int_{\tau_{0}}^{\tau_{0}^{-}}\frac{2i\hbar(1+\mathcal{O}(\hbar))\differential
z}{\sqrt{Q(z_{+};s.E)}}$ (A.61)

Our function
$\Psi=\Psi^{(A_{1})}_{+}=\Psi_{+}^{(B_{1}^{l})}+i\Psi_{-}^{(B_{1}^{l})}$. Near
$\tau_{0}$ the function $\Psi$ is, along the contours of integration, an
oscillatory solution. We express it in terms of the normalized solutions in
$(B_{1}^{r})$:

$\displaystyle\Psi^{(A_{1})}_{+}={\rm e}^{v_{10}}\Psi_{+}^{(B_{1}^{r})}+i{\rm
e}^{-v_{10}}\Psi_{-}^{(B_{1}^{r})}\ \ \Rightarrow\ \ \ $
$\displaystyle(\Psi^{(A_{1})}_{+})^{2}={\rm
e}^{-2v_{10}}\underbrace{\left(\Psi_{+}^{(B_{1}^{r})}\right)^{2}}_{{\bf(a)}}-{\rm
e}^{2v_{10}}\underbrace{\left(\Psi_{-}^{(B_{1}^{r})}\right)^{2}}_{{\bf(b)}}+2i\underbrace{\Psi_{+}^{(B_{1}^{r})}\Psi_{-}^{(B_{1}^{r})}}_{\bf(c)}.$
(A.62)

Refer now to Fig. 12: the term marked ${\bf(a)}$ is recessive in the blue-
shaded region, the term (b) in the pink-shaded region while (c) is oscillatory
throughout. Consequently we will split the integration of $\Psi^{2}$ into
three paths from $\tau_{0}^{+}$ to $\tau_{0}^{-}$ and integrate (a) along the
blue path, (b) along the red path and (c) along the green path.

The whole contribution of ${\bf(b)}$ is then sub-leading and of order
$\mathcal{O}(\hbar^{2})$ where the main contributions come from the
neighourhoods of $\tau_{0}^{\pm}$.

The contribution of ${\bf(a)}$ near $\tau_{0}^{+}$ and $\xi_{0}$ similarly is
of order $\mathcal{O}(\hbar)^{2}$ (with the main contribution coming solely
from the neighbourhood of $\tau_{0}^{-}$). The integration from $\xi_{0}$ to
$\tau_{0}^{-}$ is achieved by Theorem A.3 and we have

$\displaystyle{\bf(a)}\mapsto{\rm
e}^{-2v_{10}}\int_{\xi_{0}}^{\tau_{0}^{-}}\left(\Psi_{+}^{(B_{1}^{r})}\right)^{2}{\mathrm{d}}z={\rm
e}^{-2v_{10}}\int_{\tau_{0}}^{\tau_{0}^{-}}\frac{2i\hbar(1+\mathcal{O}(\hbar)){\mathrm{d}}z}{\sqrt{Q(z_{+};s,E)}}.$
(A.63)

We then have the contribution of (c) along the green path; for this we need to
use a similar reasoning as in the proof of Theorem A.3. The function
$\Psi_{+}^{(B_{1}^{r})}$ is recessive in the blue-shaded region of Fig. 12;
thus in the coordinate $\xi(z;\hbar)$ along the direction $\arg\xi=0$ and is
therefore proportional to
${\mathrm{Ai}}(\hbar^{-\frac{2}{3}}\xi)/\sqrt{\xi^{\prime}}$ after the change
of coordinates. Similarly the function $\Psi_{-}^{(B_{1}^{r})}$ is recessive
in the pink-shaded region and hence proportional to
${\mathrm{Ai}}(\hbar^{-\frac{2}{3}}\omega^{2}\xi)/\sqrt{\xi^{\prime}}$ , with
$\omega={\rm e}^{2i\pi/3}$

$\displaystyle\int_{\tau_{0}^{+}}^{\tau_{0}^{-}}\Psi_{+}^{(B_{1}^{r})}\Psi_{-}^{(B_{1}^{r})}{\mathrm{d}}z=C(\hbar)\int_{\xi_{+}}^{\xi_{-}}\frac{{\mathrm{Ai}}(\hbar^{-\frac{2}{3}}\xi){\mathrm{Ai}}(\hbar^{-\frac{2}{3}}\omega^{2}\xi){\mathrm{d}}\xi}{(\xi^{\prime})^{2}}\simeq
C(\hbar)\left(\int_{\tau_{0}^{+}}^{\tau_{0}^{-}}\frac{i\hbar^{\frac{1}{3}}{\mathrm{d}}z}{4\pi\sqrt{Q(z_{+};s,E)}}+\mathcal{O}(\hbar^{\frac{4}{3}})\right)$
(A.64)

The latter integral was estimated as in the first part of this proof using
Lemma A.2 by using part [2] of Proposition A.2: it requires the case $j=0,k=2$
in (A.31). The proportionality constant $C(\hbar)$ is now estimated by
matching the asymptotic behaviour;

$\displaystyle\frac{\hbar}{\sqrt{Q(z;s,E)_{+}}}\simeq\Psi_{+}^{(B_{1}^{r})}\Psi_{-}^{(B_{1}^{r})}\simeq
C(\hbar)\frac{\hbar^{\frac{1}{3}}{\rm
e}^{\frac{i\pi}{3}}}{4\pi\xi^{\frac{1}{2}}\ \xi^{\prime}}$ (A.65)

So, the coefficient $(1+{\rm e}^{2v_{12}})$ in (A.61) comes from the
contribution of (c) and (a) while the first term in (A.61) is coming only from
the integration of (a) from (A.62). We have thus established (A.61). To
complete the proof we now recall that the parameters $\xi_{jk}={\rm
e}^{2v_{jk}}$ satisfy the relation (5.8) so that $1+\xi_{10}=-\xi_{20}^{-1}$
so that all the integrals combine nicely. Indeed putting together all
contributions we obtain

$\displaystyle\int_{\gamma}\Psi^{2}{\mathrm{d}}z=$
$\displaystyle\overset{(\gamma_{2})}{\int_{\tau_{1}}^{\tau_{1}^{-}}\frac{2i\hbar\,\differential
z}{\sqrt{Q(z_{-};s,E)}}}+\overset{(\gamma_{3})}{\int_{\tau_{1}^{-}}^{\tau_{0}^{+}}\frac{2i\hbar{\mathrm{d}}z}{\sqrt{Q(z_{-},s,E)}}}\overset{(\gamma_{4})}{-\int_{\tau_{0}^{+}}^{\tau_{0}}\frac{2i\hbar(1+\mathcal{O}(\hbar))\differential
z}{\sqrt{Q(z_{+};s.E)}}}+$ (A.66)
$\displaystyle-\overset{(\gamma_{4})^{\prime}}{\left(1+{\rm
e}^{-2v_{10}}\right)\int_{\tau_{0}}^{\tau_{0}^{-}}\frac{2i\hbar(1+\mathcal{O}(\hbar))\differential
z}{\sqrt{Q(z_{+};s.E)}}}+\overset{(\gamma_{5})}{\xi_{12}\int_{\tau_{0}^{-}}^{\tau_{2}^{+}}\frac{2i\hbar(1+\mathcal{O}(\hbar)){\mathrm{d}}z}{\sqrt{Q(z_{+};s,E)}}}+$
(A.67)
$\displaystyle+\overset{(\gamma_{6})}{\xi_{12}\int_{\tau_{2}^{+}}^{\tau_{2}}\frac{2i\hbar(1+\mathcal{O}(\hbar))\,\differential
z}{\sqrt{Q(z_{-};s,E)}}}$ (A.68)

where above each term we have recalled with contribution it comes from. We now
see that the contribution indicated with $(\gamma_{4})^{\prime}$ and
$(\gamma_{5})$ combine to give a single integral: this is due to the fact that
${\rm e}^{-2v_{10}}=\xi_{10}^{-1}$ and from (5.8) we see

$1+\xi_{10}^{-1}=-\xi_{30}.$

Then, using the identity

$\displaystyle\int_{\tau_{3}}^{\tau_{0}}S_{\rm{odd}}(z_{+};\hbar){\mathrm{d}}z=2i\pi(n+1)-\int_{\tau_{1}}^{\tau_{0}}S_{\rm{odd}}(z_{+};\hbar){\mathrm{d}}z-\int_{\tau_{1}}^{\tau_{0}}S_{\rm{odd}}(z_{+};\hbar){\mathrm{d}}z=2i\pi(n+1)+\int_{\tau_{1}}^{\tau_{2}}S_{\rm{odd}}(z_{+};\hbar){\mathrm{d}}z$
(A.69)

we deduce that $\xi_{30}=(\xi_{10}\xi_{20})^{-1}=\xi_{12}={\rm e}^{2v_{12}}$,
which allows us to write the two terms on line (A.67) as a single integral.
Adding all the contributions yields the final statement. $\blacksquare$

##  Appendix B Numerical verifications

We performed numerical verifications of the formulas in view of the adage,
valid in many walks of life, of “trust but verify”. For the JM case we proceed
as indicated below.

  1. 1.

Compute numerically in arbitrary precision the list of zeros,
$\mathscr{Z}_{n}$, of the VY polynomial $Y_{n}(t)$, i.e., the poles of the
rational solution $u_{n}$ (1.8).

For each of $a\in\mathscr{Z}_{n}$, compute the corresponding value of the
parameter $b$ appearing in (2.8). Thus we can define, for each
$a\in\mathscr{Z}_{n}$ a corresponding value of $\Lambda=\frac{7a^{2}}{36}+10b$
as in (2.17). Let us call $\mathscr{L}_{n}$ the list of the corresponding
values of $\Lambda$.

  2. 2.

Let $\mathscr{S}_{n}:=\mathscr{Z}_{n}/(n+\frac{1}{2})^{\frac{2}{3}}$ and
$\mathscr{E}_{n}:=\mathscr{L}_{n}/(n+\frac{1}{2})^{\frac{4}{3}}$ as per
scaling (4.2). For each pair $(s_{\ell},E_{\ell})$ in
$\mathscr{S}_{n},\mathscr{E}_{n}$ we construct the corresponding potential
$Q(z;s_{\ell},E_{\ell})=z^{4}+s_{\ell}z^{2}+2z+E_{\ell}$ and evaluate
numerically the Voros symbols including the subleading term, using Prop. 4.2,
along the relevant cycles of the Riemann surface of $\sqrt{Q}$. The numerical
verification consists in checking

$\displaystyle(2n+1)\oint\sqrt{Q(z;s_{\ell},E_{\ell})}{\mathrm{d}}z+\frac{1}{2n+1}\oint
S_{1}(z){\mathrm{d}}z\simeq i\pi+2i\pi m_{\gamma},$ (B.1)

i.e. an odd multiple of $i\pi$. We tested up to $n=26$ and the numerics indeed
supports the formula. It is interesting to observe that the leading order
computation yields (unsurprisingly) less accurate approximations of odd
multiples of $i\pi$, in particular in that it has a small but still non-
negligible real part.

For the ST case we proceed similarly as follows.

  1. 1.

Compute numerically in arbitrary precision the list of zeros,
$\mathscr{Z}_{n}$, of the ST discriminant $D_{n}$ (1.7).

For each of $t\in\mathscr{Z}_{n}$, compute the characteristic polynomial of
the matrix $M_{n}(t)$ (1.5) and find the double eigenvalue $\lambda$: due to
the numerical error in the evaluation of the zero of the discriminant, one has
to find the two roots that are very close to each other. Once we find
$\lambda$ we define $\Lambda=\lambda+\frac{t^{2}}{4}$ as per (3.5). We thus
construct corresponding lists $\mathscr{Z}_{n},\mathscr{L}_{n}$ as in the
previous case, for the values in the ES spectrum.

  2. 2.

We perform the appropriate scaling
$\mathscr{S}_{n}:=\mathscr{Z}_{n}/(n+1)^{\frac{2}{3}}$ and
$\mathscr{E}_{n}:=\mathscr{L}_{n}/(n+1)^{\frac{4}{3}}$ as per scaling (4.1).
For each pair $(s_{\ell},E_{\ell})$ in $\mathscr{S}_{n},\mathscr{E}_{n}$ we
construct the corresponding potential
$Q(z;s_{\ell},E_{\ell})=z^{4}+s_{\ell}z^{2}+2z+E_{\ell}$ and we compute the
Voros symbols $v_{jk}$ to subleading order as for the previous case. Let us
call $\xi_{jk}={\rm e}^{2v_{jk}}$ as in the main text. Then the numerical
verification consists in two checks:

    1. (a)

We verify that the three Fock–Goncharov parameters
$\xi_{10},\xi_{20},\xi_{30}$ satisfy the rational parametrization (5.8).

    2. (b)

We verify the equation (5.12), which returns numerically correct within less
than $1\%$ for $n=80$ with a higher accuracy of up to $0.01\%$,
unsurprisingly, when we check zeros that lie away from the boundary of the
triangular region (elliptic region) with boundaries described in Section
(5.4).

###  B.1 Overlapping patterns

Figure 13: The ES spectrum (black) and the VY zeros (red) for $n=40$. On the
left the zeros of the ST discriminant are scaled as $(n+1)^{-\frac{2}{3}}$,
which is the “natural” scaling. On the right they are scaled by
$n^{-\frac{2}{3}}$.

According to our analysis, the natural scaling for the points in the ES
spectrum is the one indicated in (4.1) for the ST case, and (4.2) for the VY
polynomial case. In particular for the ST case $\hbar=(n+1)^{-1}$ and for the
VY case $\hbar=(n+\frac{1}{2})^{-1}$.

If we plot the zeros with these exact scalings, we obtain the two lattices
shown in Fig. 13, on the left.

However, if we use the scaling $\hbar=n^{-\frac{2}{3}}$ for the ES spectrum,
we obtain an almost perfect match as shown on the right pane of Fig. 13.

We cannot find a mathematical justification of this coincidence, beyond what
we already have proved; namely we can justify the coincidence of the two
lattices in a $\mathcal{O}(\hbar)$ neighbourhood of the origin, as well as the
matching, slowly modulated, local geometry of either lattices (Prop. 5.6).
However we cannot quantify the reason why the “wrong” scaling seems to yield a
much better match.

We have, however, verified that the discrepancy between the two (scaled)
lattices decreases indeed like $\mathcal{O}(n^{-1})$ in the regions at finite
distance from the origin, and as $\mathcal{O}(n^{-2})$ in the
$\mathcal{O}(\hbar)$–region around the origin.

To effect this test, we choose a point, $s_{0}$ in the elliptic region and
find the closest points of either lattices to $s_{0}$. Let $\Delta_{n}(s_{0})$
denote the distance between these two points belonging to the different
lattices; we call it the “local discrepancy”. Then we plot
$\ln\Delta_{n}(s_{0})$ against $\ln n$. If we choose $s_{0}$ at finite
positive distance from the origin, the slope of this graph is $-1$ while for
$s_{0}\simeq 0$ the slope is $-2$. This plot of local discrepancy
$\Delta_{n}(s_{0})$ is reported in Fig. 14 in the two regimes of $s_{0}\simeq
0$ and $s_{0}\neq 0$, verifying that the decrease of the discrepancy is
$\mathcal{O}(n^{-2})$ in the first case and $\mathcal{O}(n^{-1})$ in the
second.

. .

Figure 14: The local discrepancies $\Delta_{n}(s_{0})$ against $n$ in a log-
log plot for two values of $s_{0}$. The red dots correspond to the discrepancy
with the “wrong” scaling $n^{-\frac{2}{3}}$. Although the “wrong” scaling is
better, the rate of convergence is the same in the local lattices. In the left
picture $s_{0}=0$ and the slope is $-2$ (the dotted line is plotted for
reference). In the right picture $s_{0}=1+i$ and the slope is $-1$ as
expected.

Acknowledgments This project has received funding from the European Union’s
H2020 research and innovation programme under the Marie Skłodowska–Curie grant
No. 778010 IPaDEGAN. TG and MB acknowledge the support of GNFM-INDAM group and
the research project Mathematical Methods in Non Linear Physics (MMNLP),
Gruppo 4-Fisica Teorica of INFN. The work of MB was supported in part by the
Natural Sciences and Engineering Research Council of Canada (NSERC) grant
RGPIN-2016-06660.

## References

  * [AKT96] T. Aoki, T. Kawai, Y. Takei,  “WKB analysis of Painlevé transcendents with a large parameter, II. Multiple-scale analysis of Painlevé transcendents”,  Structure of solutions of differential equations,  World Sci. Publ., River Edge, NJ, 1996.
  * [BB98] C. M. Bender, S. Boettcher,  “Quasi-exactly solvable quartic potential”,  J. Phys. A, 31 (1998), no. 14, 273–277.
  * [Ber11] M. Bertola.  Boutroux curves with external field: equilibrium measures without a variational problem.  Anal. Math. Phys., 1(2-3):167–211, 2011.
  * [BCG22] M. Bertola, E. Chavez–Heredia, T. Grava, “The Stieltjes-Fekete problem and degenerate orthogonal polynomials”. Preprint: https://arxiv.org/pdf/2206.06861.pdf
  * [BB15] M. Bertola, T. Bothner, “Zeros of Large Degree Vorob’ev-Yablonski Polynomials via a Hankel Determinant Identity”, IMRN, 2015, no. 19, 9330–9399.
  * [BK19] M. Bertola, D.Korotkin, “WKB expansion for a Yang–Yang generating function and the Bergman tau function”, Teoret. Mat. Fiz., 206, no 3. 295–338 (2021).
  * [BT16] M. Bertola, A. Tovbis, “On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight”,  SIGMA, 12(118):50, December 2016.
  * [BM12] R. Buckingham, P. Miller,  “The sine-Gordon equation in the semiclassical limit: critical behavior near a separatrix”,  J. Anal. Math. 118 (2012), no. 2, 397-492
  * [BM14] R. Buckingham, P. Miller, “Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour”, Nonlinearity 27 (2014), 2489-2577.
  * [BM15] R. Buckingham, P. Miller, “Large-degree asymptotics of rational Painlevé-II functions: critical behaviour”, Nonlinearity 28 (2015), 1539–1596.
  * [CM03] P. Clarkson, E. Mansfield, “The second Painlevé equation, its hierarchy and associated special polynomials”, Nonlinearity 16 (2003), R1-R26.
  * [Dei99] P. Deift,  “Orthogonal polynomials and random matrices: a Riemann-Hilbert approach”,  volume 3 of Courant Lecture Notes in Mathematics.  New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.
  * [EG11] A. Eremenko, A. Gabrielov,  “Quasi-exactly solvable quartic: elementary integrals and asymptotics”,  J. Phys. A Math. Theor., 44 (2011), no. 31, 312001-14.
  * [EG12] A. Eremenko, A. Gabrielov,  “Quasi-exactly solvable quartic: real algebraic spectral locus”,  J. Phys. A Math. Theor., 45 (2012), 175205.
  * [FIKN06] A. Fokas, A. Its, A. Kapaev, V. Novokshenov,  ”Painlevé Transcendents: The Riemann-Hilbert Approach”,  Mathematical Surveys Monographs, 128, American Mathematical Society, Providence, RI, 2006.
  * [GLS02] V. Gromak, I, Laine, S. Shimomura,  “Painlevé differential equations in the complex plane”,  volume 28 of De Gruyter Studies in Mathematics.  Walter de Gruyter & Co., Berlin, 2002.
  * [IN14] K. Iwaki, T. Nakanishi,  “Exact WKB analysis and cluster algebras”,  J. Phys. A, 47 (2014), no. 47, 474009, 98 pp.
  * [Inc44] E. L. Ince,  “Ordinary Differential Equations”,  Dover Publications, New York, 1944.
  * [IN86] A. Its, V. Novokshenov,  “The isomonodromic deformation method in the theory of Painlevé equations”,  Lecture Notes in Mathematics 1191.  Springer-Verlag, Berlin, 1986.
  * [JM81] M. Jimbo, T. Miwa,  “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II”.  Phys. D, 2(3) (1981), 407–448.
  * [KO96] K. Kajiwara, Y. Ohta,  “Determinant structure of the rational solutions for the Painlevé II equation”,  J. Math. Phys. 37(9), 4693–4704 (1996).
  * [KT05] Takahiro Kawai and Yoshitsugu Takei,  Algebraic analysis of singular perturbation theory, volume 227 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, 2005.  Translated from the 1998 Japanese original by Goro Kato, Iwanami Series in Modern Mathematics.
  * [Mas10a] D. Masoero,  “Poles of intégrale tritronquée and anharmonic oscillators. Asymptotic localization from WKB analysis”,  Nonlinearity 23 (2010), no. 10, 2501–2507.
  * [Mas10b] D. Masoero,  “Poles of intégrale tritronquée and anharmonic oscillators. A WKB approach”,  J. Phys. A, 43(9) (2010), 095201.
  * [Mas18] D. Masoero, P. Roffelsen,  “Poles of Painlevé IV rationals and their distribution”, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 002, 49 pp.
  * [MT13] E. Mukhin, V. Tarasov,  “On conjectures of A. Eremenko and A. Gabrielov for quasi-exactly solvable quartic”  Lett. Math. Phys. 103(6), 653–663 (2013)
  * [Olv74] F. Olver,  “Asymptotics and Special Functions”,  Elsevier, 1974.
  * [ST22] B. Shapiro, M. Tater, “On spectral asymptotic of quasi-exactly solvable quartic potential”,Anal. Math. Phys. 12 (2022), no. 1, Paper no. 2, 35 pp.
  * [Str84] K. Strebel, “Quadratic differentials”, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 5, Berlin 1984.
  * [Tan00] M. Taneda,  “Remarks on the Yablonskii-Vorob’ev polynomials”,  Nagoya Math. J., 159:87–111, 2000.
  * [Vör83] A. Vöros.  “The return of the quartic oscillator: the complex WKB method”,  Ann. Inst. H. Poincaré Sect. A (N.S.), 39(3):211–338, 1983\.
  * [Vor65] A. Vorob’ev, “On rational solutions of the second Painlevé equation”, Diff. Eqns 1 (1965), 58-9 (in Russian).
  * [Yab59] A. Yablonskii, “On rational solutions of the second Painlevé equation”, Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3 (1959), 30-5 (in Russian).
  * [Wa87] W. Wasow, “Asympotic expansions for ordinary differential equations”, Dover (1987).