# Constraining $\lambda_{B_{s}}$ by $B_{s}\to\gamma^{*}$ and $B_{s}\to\phi$
form factors

Mikhail A. Ivanova, Dmitri Melikhova,b,c, and Silvano Simulad aJoint Institute
for Nuclear Research, 141980 Dubna, Russia
bD. V. Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov Moscow State
University, 119991, Moscow, Russia
cFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna,
Austria
dINFN, Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy

###### Abstract

We calculate the form factors $F_{V}(q^{2},q^{\prime 2})$ and
$F_{TV}(q^{2},q^{\prime 2})$ describing the $B_{s}\to\gamma^{*}$ transition
induced by the vector and tensor weak currents in a broad range of values of
$q^{2}$ and $q^{\prime 2}$ far below the quark thresholds in both $q^{2}$ and
$q^{\prime 2}$ channels. These form factors are calculated via the
distribution amplitudes of the $B_{s}$-meson. We then interpolate the obtained
results by a formula that contains pole at $q^{\prime 2}=M_{\phi}^{2}$ and
extract the residue which gives the $B_{s}\to\phi$ transition form factors
$V(q^{2})$ and $T_{1}(q^{2})$. In this way we obtain theoretical predictions
for these form factors without invoking quark-hadron duality and QCD sum
rules. Furthermore, we calculate the relationship between $V(0)$ and
$T_{1}(0)$ and the parameter $\lambda_{B_{s}}(\mu)$, the inverse moment of the
$B_{s}$-meson distribution amplitude. Using the available predictions for
$V(0)$ and $T_{1}(0)$ coming from approaches not referring to the
$B_{s}$-meson distribution amplitudes, we obtain the estimate
$\lambda_{B_{s}}(\mu\simeq m_{b})=(0.62\pm 0.10)$ GeV.

##  1 Introduction

The inverse momentum of the distribution amplitude (2DA) $\phi_{+}$ of
$B_{(s)}$-meson, $\lambda_{B_{(s)}}$, is an interesting parameter for the
phenomenology of weak $B_{(s)}$ decays as it enters a number of observables
related to semileptonic and multileptonic decays of $B_{(s)}$-mesons (see e.g.
korchemsky2000 ; braun2017 ; beneke2020 ; bbm2023 ; bbm2024 ; wang2023 ). In
spite of its importance, this parameter is presently not known with high
accuracy and theoretical predictions for this parameter vary in a rather broad
range, namely $\lambda_{B}(1\mbox{ GeV})=0.3\div 0.7$ GeV kou ; braun2004 ;
beneke2011 ; wang2016 ; zwicky2021 ; im2022 , whereas the ratio
$\lambda_{B_{s}}/\lambda_{B}$ is known with a better accuracy,
$\lambda_{B_{s}}/\lambda_{B}=1.19\pm 0.14$ khodjamirian2020 .

In this paper, we discuss a new way of constraining $\lambda_{B_{s}}$.

The first step is the calculation of the form factors $F_{V}(q^{2},q^{\prime
2})$ and $F_{TV}(q^{2},q^{\prime 2})$ mk2003 ; mnk2018  describing the
$B_{s}\to\gamma^{*}$ transition (see Fig. 1) in a broad range of values of
$q^{2}$ and $q^{\prime 2}$ far below quark thresholds via the 2DAs of the
$B_{s}$-meson. For the relevant 2DAs we make use of the local duality (LD)
model of braun2017 .

Then, we interpolate these results by an analytic formula which contains a
number of poles corresponding to vector meson resonances with quantum numbers
appropriate to $q^{2}$ and $q^{\prime 2}$ channels. This formula is based on a
dispersion representation for $F_{V,TV}(q^{2},q^{\prime 2})$ in $q^{\prime 2}$
with one subtraction mnp2004  and the spectral density of this representation
is saturated by two resonances - the known lowest resonances $\phi$ and an
effective vector meson pole with an unknown mass. The numerical parameters in
this analytic formula are obtained by fitting the results of our calculations
in a broad range of values of $q^{2}$ and $q^{\prime 2}$ far below the quark
thresholds in $q^{2}$ and $q^{\prime 2}$ channels. The analytic formula
contains a pole at $q^{\prime 2}=M_{\phi}^{2}$; the residue of this pole
provides the $B_{s}\to\phi$ form factor $V(q^{2})$ in the case of $F_{V}$ and
$T_{1}(q^{2})$ in the case of $F_{TV}$. Noteworthy, our estimate does not
employ quark-hadron duality and QCD sum rules and is therefore free of the
intrinsic uncertainties of the method of QCD sum rules. Our approach thus
provides an interesting complementary method to the widely used method of QCD
sum rules.

For comparison, we present in Appendix A the application of the standard QCD
sum rule machinery braun1994  in order to calculate $T_{1}(0)$ using the
$B_{s}$-meson 2DAs offen2007 . We recall that QCD sum rules calculate the
residue from the very same Green function which determines the form factor
$F_{TV}(q^{2},q^{\prime 2})$. However, the QCD sum-rule extraction contains
auxiliary parameters (Borel parameter and effective threshold $s_{\rm eff}$)
which are not fixed by any rigorous criteria, leading to uncertainties that
are difficult to control. Still we observe that using some commonly adopted
ways of fixing these auxiliary parameters leads to a good agreement between
the estimates for $T_{1}(0)$ obtained by a direct calculation of the
correlation function and extrapolation to the pole and by a direct evaluation
of the residue by a quark-hadron duality Ansatz.

The approach formulated above is based on using the representations for the
form factors $F_{V,TV}(q^{2},q^{\prime 2})$ up to leading-order in the strong
coupling constant $\alpha_{s}$. Making use of the known radiative corrections
for the form factor $F_{V}(q^{2},0)$ beneke2011 , we give arguments that for
the form factors $F_{V,TV}(q^{2},q^{\prime 2})$ the radiative corrections
evaluated at the scale $\mu=m_{b}$ are at the level of less than 0.5% in the
broad range of values of $q^{2}$ and $q^{\prime 2}$ used in our analysis. This
property means that the leading-order expression for $F_{V,TV}(q^{2},q^{\prime
2})$ expressed via the 2DAs at the scale $\mu=m_{b}$ should give a good
estimate for the physical form factors $F_{V}(q^{2},q^{\prime 2})$ and
$F_{TV}(q^{2},q^{\prime 2}|\mu=m_{b})$.

We then obtain the dependence of the $B_{s}\to\phi$ form factors $V(0)$ and
$T_{1}(0)$ on $\lambda_{B_{s}}(\mu=m_{b})$. Combining our results with the
predictions for $V(0)$ and $T_{1}(0)$ obtained by approaches not referring to
2DAs of the $B_{s}$-meson, we extract

$\displaystyle\lambda_{B_{s}}(\mu\simeq m_{b})=(0.62\pm 0.10)\mbox{ GeV}.$
(1.1)

##  2 The form factors $F_{V,TV}(q^{2},q^{\prime 2})$ and the $B_{s}\to\phi$
form factors $V(q^{2})$ and $T_{1}(q^{2})$

This Section provides details of the analytic calculation of
$F_{V,TV}(q^{2},q^{\prime 2})$ and the corresponding numerical results.

###  2.1 Analytic results for $F_{V,TV}(q^{2},q^{\prime 2})$ at leading order
in $\alpha_{s}$

Fig. 1: (a) Diagram corresponding to the form factor
$F^{(s)}_{TV}(q^{2},q^{\prime 2})$ containing $\phi$-meson pole in the
$q^{\prime 2}$-channel. (b) Diagram corresponding to parametrically suppressed
form factor $F^{(b)}_{TV}(q^{2},q^{\prime 2})$. The dashed blob stands for the
tensor current $\bar{s}\sigma_{\mu\nu}b$ and the solid blob stands for the
electromagnetic current, $j_{\alpha}^{\rm
e.m.}=Q_{s}\bar{s}\gamma_{\alpha}s+Q_{b}\bar{b}\gamma_{\alpha}b$.

Let us consider the transition amplitudes involving electromagnetic and weak
currents mk2003 ; mnk2018  (Fig. 1)111 Our notations and conventions are:
$\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$,
$\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$,
$\epsilon^{0123}=-1$,
$\epsilon_{abcd}\equiv\epsilon_{\alpha\beta\mu\nu}a^{\alpha}b^{\beta}c^{\mu}d^{\nu}$,
$e=\sqrt{4\pi\alpha_{\rm em}}$.:

$\displaystyle i\int dxe^{iq^{\prime}x}\langle 0|\left\\{Tj_{\alpha}^{\rm
e.m.}(x),\bar{s}\gamma_{\mu}q^{\nu}b(0)\right\\}|\bar{B}_{s}(p)\rangle$
$\displaystyle=$ $\displaystyle ie\,\epsilon_{\mu\alpha
qq^{\prime}}\frac{F_{V}(q^{2},q^{\prime 2})}{M_{B_{s}}},$ $\displaystyle i\int
dxe^{iq^{\prime}x}\langle 0|\left\\{Tj_{\alpha}^{\rm
e.m.}(x),\bar{s}\sigma_{\mu\nu}q^{\nu}b(0)\right\\}|\bar{B}_{s}(p)\rangle$
$\displaystyle=$ $\displaystyle ie\,\epsilon_{\mu\alpha
qq^{\prime}}F_{TV}(q^{2},q^{\prime 2}).$ (2.2)

The form factors $F_{i}(q^{2},q^{\prime 2})$, [$i=V,TV$] in Eq. (2.2) contains
two contributions, $F_{i}=F^{(s)}_{i}+F^{(b)}_{i}$ corresponding to the
diagrams in Fig. 1. We are interested in $F^{(s)}_{i}$ which contains a pole
at $q^{\prime 2}=M^{2}_{\phi}$ and neglect the form factor $F^{(b)}_{i}$
parametrically suppressed by $1/m_{b}$ compared to $F^{(s)}_{i}$. So,
hereafter $F_{i}$ denotes $F^{(s)}_{i}$.

The form factors $F_{i}$ may be calculated via the $B$-meson 2DAs using HQET
formula (see e.g. offen2007 )

$\displaystyle\langle 0|\bar{s}(x)\Gamma
b(0)|\bar{B}_{s}(p)\rangle=-\frac{if_{B_{s}}M_{B_{s}}}{4}\int d\xi e^{-i\xi
px}\,{\rm
Tr}\bigg{\\{}\gamma_{5}\Gamma(1+\not{v})\bigg{[}\phi_{+}(\xi)-\frac{\phi_{+}(\xi)-\phi_{-}(\xi)}{2vx}\not{x}\bigg{]}\bigg{\\}},\quad
p_{\mu}=M_{B_{s}}v_{\mu}.$ (2.3)

Omitting all contributions at order $O(m_{s}/M_{B_{s}})$ in the numerator, one
gets

$\displaystyle F_{V}(q^{2},q^{\prime 2})$ $\displaystyle=$ $\displaystyle-
Q_{s}f_{B_{s}}M_{B_{s}}\int
d\xi\bigg{[}\frac{\phi_{+}(\xi)}{m_{s}^{2}+\xi(1-\xi)M_{B_{s}}^{2}-q^{2}\xi-q^{\prime
2}(1-\xi)}+\frac{2m_{s}M_{B_{s}}\bar{\Phi}(\xi)}{(m_{s}^{2}+\xi(1-\xi)M_{B_{s}}^{2}-q^{2}\xi-q^{\prime
2}(1-\xi))^{2}}\bigg{]},$ $\displaystyle F_{TV}(q^{2},q^{\prime 2})$
$\displaystyle=$ $\displaystyle-Q_{s}f_{B_{s}}M_{B_{s}}\int
d\xi\frac{\phi_{+}(\xi)(1-\xi)+\bar{\Phi}(\xi)}{m_{s}^{2}+\xi(1-\xi)M_{B_{s}}^{2}-q^{2}\xi-q^{\prime
2}(1-\xi)}$ (2.4)

with

$\displaystyle\bar{\Phi}(\xi)=\int\limits_{0}^{\xi}d\xi^{\prime}\big{[}\phi_{+}(\xi^{\prime})-\phi_{-}(\xi^{\prime})\big{]}.$
(2.5)

The kinematical singularity at $vx\to 0$ in Eq. (2.3) cannot be a singularity
of the physical amplitude, so that the primitive $\bar{\Phi}(\xi)$ should
vanishe at the boundaries of the 2DA support region. Then the integration by
parts in $\xi$, necessary to handle the $1/vx$ term, does not contain any
nonzero surface term.

For the 2DAs we use a set referred to as Model IIB in braun2017  (our $\xi$
and $\omega_{0}$ are dimensionless):

$\displaystyle\phi_{+}(\xi)$ $\displaystyle=$
$\displaystyle\frac{5}{8\omega_{0}^{5}}\xi(2\omega_{0}-\xi)^{3}\theta(2\omega_{0}-\xi),$
(2.6) $\displaystyle\phi_{-}(\xi)$ $\displaystyle=$
$\displaystyle\frac{5}{192\omega_{0}^{5}}(2\omega_{0}-\xi)^{2}\bigg{[}6(2\omega_{0}-\xi)^{2}-\frac{7(\lambda_{E}^{2}-\lambda_{H}^{2})}{M_{B_{s}}^{2}\omega_{0}^{2}}(15\xi^{2}-20\omega_{0}\xi+4\omega_{0}^{2})\bigg{]}\theta(2\omega_{0}-\xi),$
(2.7)

where

$\displaystyle\omega_{0}=\frac{5}{2}\frac{\lambda_{B_{s}}}{M_{B_{s}}}.$ (2.8)

For numerical estimates we use $M_{B_{s}}$=5.367 GeV and $f_{B_{s}}$=0.230
GeV. The parameters $\lambda^{2}_{E,H}\simeq(0.1-0.2)$ GeV2 are related to
certain matrix elements of antiquark-quark-gluon operators braun2017 . The
contribution of the term proportional to $(\lambda_{E}^{2}-\lambda_{H}^{2})$
in Eq. (2.7) for the form factors $F_{V,TV}$ turns out to be numerically
negligible and may be safely omitted. For a detailed analysis of the analytic
properties of the form factor (2.4) for the case of the polynomial 2DAs we
refer to ims2020 .

###  2.2 Numerical results for $F_{V}(q^{2},q^{\prime 2})$ and
$F_{TV}(q^{2},q^{\prime 2})$

We now obtain the form factors $F_{i}(q^{2},q^{\prime 2})$, [$i=V,TV$] in a
broad range of values of $q^{2}$ and $q^{\prime 2}$ by the following
procedure:

(i) $F_{i}(q^{2},q^{\prime 2})$ is calculated using Eq. (2.4) with 2DAs
$\phi_{\pm}$ given in Eqs. (2.6) and (2.7) for $\lambda_{B_{s}}=0.6$ GeV.

(ii) We interpolate the numerical results in the range $0<q^{2}({\rm
GeV^{2}})<6$ and $-10<q^{\prime 2}({\rm GeV^{2}})<-0.6$ (i.e., far below the
quark thresholds located at $q^{2}=(m_{b}+m_{s})^{2}$ and $q^{\prime
2}=4m_{s}^{2}$) using a simple analytic fit formula

$\displaystyle F(y_{1},y_{2})$ $\displaystyle=$ $\displaystyle
f_{0}\left(1+a_{0}\frac{y_{1}}{1-y_{1}}\right)+R_{\phi}F_{0}\left(1+a_{1}\frac{y_{1}}{1-y_{1}}\right)\frac{y_{2}}{1-y_{2}}+R_{\phi}F_{1}\left(1+a_{2}\frac{y_{1}}{1-y_{1}}\right)\frac{y_{2}}{1-y_{2}/r^{2}},$
(2.9) $\displaystyle y_{1}\equiv q^{2}/M_{B^{*}_{s}}^{2},\;y_{2}\equiv
q^{\prime 2}/M^{2}_{\phi},$

where $R_{\phi}=-\frac{2f^{\rm e.m.}_{\phi}}{M_{\phi}},f^{\rm
e.m.}_{\phi}=Q_{s}f_{\phi},Q_{s}=-1/3,f_{\phi}=0.224\mbox{
GeV},M_{\phi}=1.020\mbox{ GeV},M_{B_{s}^{*}}=5.415\mbox{ GeV}$. The analytic
formula (2.9) reflects the correct location of the lowest physical poles at
$q^{2}=M_{B^{*}_{s}}^{2}$ and $q^{\prime 2}=M^{2}_{\phi}$.

The $B_{s}\to\phi$ transition form factors at $q^{2}=0$ are related to the
residues in the pole at $q^{\prime 2}=M_{\phi}^{2}$ such that
$V(0)=\frac{M_{B_{s}}+M_{\phi}}{M_{B_{s}}}F_{0,V}$ and $T_{1}(0)=F_{0,TV}$.

The formula (2.9) is suggested by the following theoretical considerations:

$\bullet$ We consider a dispersion representation in $q^{\prime 2}$ with one
subtraction and saturate its spectral density by the sum of two poles, a
physical pole at $q^{\prime 2}=M_{\phi}^{2}$ and an effective heavier pole at
$q^{\prime 2}=M_{\phi}^{2}/c$, $c<1$.

$\bullet$ The coefficients in this $q^{\prime 2}$-dispersion representation
are functions of $q^{2}$. For each of these functions we again write down a
spectral representation in $q^{2}$ with one subtraction and saturate the
spectral density by a lowest vector meson $B_{s}^{*}$. One may include also
the effective heavier vector pole in the $q^{2}$-channel, but this makes no
difference as already the $B_{s}^{*}$ lies far above the $q^{2}$-region of our
interest.

Table 1 gives the fit parameters obtained by this interpolation procedure.

Table 1: Parameters of the fit formula Eq. (2.9) obtained by the interpolation of our numerical results for $F_{V,TV}$ calculated for $\lambda_{B_{s}}=0.6$ GeV. The corresponding $B_{s}\to\phi$ form factors are $V(0)=0.35$ and $T_{1}(0)=0.33$. | $f_{0}$ | $a_{0}$ | $F_{0}$ | $a_{1}$ | $F_{1}$ | $a_{2}$ | $r$
---|---|---|---|---|---|---|---
$F_{V}$ | 0.13 | $1.36$ | $0.29$ | $4.57$ | $0.126$ | $-0.25$ | $2.17$
$F_{TV}$ | 0.11 | $1.23$ | $0.33$ | $2.62$ | $0.085$ | $0.26$ | $2.20$

The fit formula (2.9) interpolates the results of calculating
$F_{V,TV}(q^{2},q^{\prime 2})$ with an accuracy better than 1% in the full
adopted range of values of $q^{2}$ and $q^{\prime 2}$. Fig. 2 illustrates the
quality of reproducing the calculation results by the fit formula.

|
---|---
(a) | (b)
Fig. 2:  Solid (green) line is the results of direct calculations of the form
factor $F_{TV}(q^{2},q^{\prime 2})$ for $\lambda_{B_{s}}=0.6$ GeV. Dashed
(red) line is the results of the double fit to $F_{TV}(q^{2},q^{\prime 2})$ in
the range $-10<q^{\prime 2}({\rm GeV}^{2})<-0.6$ and $0<q^{2}({\rm
GeV}^{2})<6$.

One can reach a much better accuracy (at a per mille level) if one proceeds in
a bit different way: namely, performs a fit to the calculated
$F_{TV}(q^{2},q^{\prime 2})$ in the single variable $q^{\prime 2}$ in the
range $-10<q^{\prime 2}\;({\rm GeV}^{2})<-0.6$ for any fixed value of $q^{2}$
far below the quark threshold located at $(m_{b}+m_{s})^{2}$. We follow this
route when calculating the dependence of $T_{1}(0)$ and $V(0)$ on
$\lambda_{B_{s}}$ (see below).

As suggested by Eq. (2.9), in the region $0<q^{2}({\rm GeV}^{2})<6$ the
results of our calculation may be well approximated by a simple monopole
formula $F(q^{2})=F_{0}\left(1+a_{1}y_{1}/(1-y_{1})\right)$ with
$y_{1}=q^{2}/M^{2}_{B_{s}^{*}}$.

###  2.3 Radiative corrections and constraints on $\lambda_{B_{s}}(\mu)$

It is time to recall that the analysis of the previous Section does not
include the radiative corrections, whereas for the form factors of the tensor
current, $F_{TV}(q^{2},q^{\prime 2})$ and $T_{1}(q^{2})$ which depend on the
scale $\mu$, the radiative corrections are crucial for controlling this scale-
dependence. Also for the form factors $F_{V}(q^{2},q^{\prime 2})$ and
$V(q^{2})$, the radiative corrections may change the dependence of these form
factors on $\lambda_{B_{s}}(\mu)$.

Two remarks are in order.

$\bullet$ The radiative corrections are known for the case
$F_{V}(q^{2},q^{\prime 2}=0)$ as the function of
$E_{\gamma}=(M_{B_{s}}^{2}-q^{2})/(2M_{B_{s}})$, see beneke2011  and Appendix
B. The corresponding factor $R(E_{\gamma},\mu)$ is shown in Fig. 3 for
$\mu=m_{b}$.

Fig. 3:  The factor $R(E_{\gamma},\mu=m_{b})$ describing radiative corrections
to the form factor $F_{V}(E_{\gamma})$.

As seen from this plot, the correction evaluated at $\mu=m_{b}$ is negligible
in the range $2<E_{\gamma}({\rm GeV}^{2}<7$. Recall, however, that we have a
different situation: namely, $q^{2}\neq 0$ and $q^{\prime 2}<0$. We now give
arguments that the results for $F_{V}(q^{2},q^{\prime 2}=0)$ may be used to
estimate the radiative corrections for $F_{V}(q^{2},q^{\prime 2})$ at
$q^{\prime 2}<0$.

Let us introduce the “equivalent” energy $\bar{E}_{\gamma}(q^{2},q^{\prime
2})$ by the following expression222Note that $\bar{E}_{\gamma}$ is different
from the virtual photon energy for the process
$B_{s}\to\gamma(q)\gamma(q^{\prime})$ which is $E_{\gamma}(q^{2},q^{\prime
2})=(M_{B_{s}}^{2}+q^{\prime 2}-q^{2})/(2M_{B_{s}})$.:

$\displaystyle
m_{s}^{2}+M_{B_{s}}^{2}\bar{\xi}(1-\bar{\xi})-q^{2}\bar{\xi}-q^{\prime
2}(1-\bar{\xi})\equiv 2M_{B_{s}}\bar{\xi}\bar{E}_{\gamma}(q^{2},q^{\prime
2}),$ (2.10)

such that the denominator in Eq. (2.1) at $\xi=\bar{\xi}$ gives
$2M_{B_{s}}\bar{E}_{\gamma}\bar{\xi}$. Obviously, $\bar{E}_{\gamma}\to
E_{\gamma}$ for $m_{s}\to 0$, $q^{\prime 2}\to 0$, and $\bar{\xi}\to 0$.

The behaviour of the form factor and of the radiative corrections are related
to the behaviour of the denominator of Eq. (2.1). So, we expect that the
radiative correction in $F_{V}(q^{2},q^{\prime 2})$ may be estimated using the
function $R(\bar{E}_{\gamma},\mu)$ evaluated for $\bar{\xi}$, the value at
which $\phi_{+}(\xi)$ reaches its maximum. For the 2DAs considered here one
obtains $\bar{\xi}\simeq 0.2$, and it is easy to check that for the range used
in our analysis, namely $-10<q^{\prime 2}({\rm GeV}^{2})<0$ and $0<q^{2}({\rm
GeV}^{2})<6$, $\bar{E}_{\gamma}$ varies in the range $2<\bar{E}_{\gamma}({\rm
GeV}^{2})<7$. Thus, for the analysis of the previous Section, the radiative
corrections in the form factor $F_{V}$ are at the level of less than 1% and
may be neglected, if one expresses the leading-order contribution to the form
factor $F_{V}(q^{2},q^{\prime 2})$, Eq. (2.1), via
$\lambda_{B_{s}}(\mu=m_{b})$.

$\bullet$ The radiative corrections for the form factor
$F_{TV}(q^{2},q^{\prime 2})$ have not yet been calculated for any values of
$q^{2}$ and $q^{\prime 2}$. Again, since the behaviours of the form factors
and of the radiative correction are determined by the same denominator
appearing in the analytic formulas for $F_{TV}(q^{2},q^{\prime 2})$ and
$F_{V}(q^{2},q^{\prime 2})$, we conjecture that the radiative corrections at
$\mu=m_{b}$ in the considered range of values of $q^{2}$ and $q^{\prime 2}$
may be neglected too.

Moreover, the form factors of the tensor current depend on the scale, in
particular, one has $F_{TV}(0,0|\mu)$ and $T_{1}(0|\mu)$. Our arguments for a
negligible size of the radiative corrections at $\mu=m_{b}$ imply that the
analysis of the previous Section yields $F_{TV}(0,0|\mu=m_{b})$ and
$T_{1}(0|\mu=m_{b})$ as soon as the leading-order contributions in
$\alpha_{s}$, Eq. (2.1), are expressed via $\lambda_{B_{s}}(\mu=m_{b})$.

We would like to emphasize that for our extraction of $\lambda_{B_{S}}(\mu)$
it is not crucial that the radiative corrections vanish precisely at
$\mu=m_{b}$; it is only important that these corrections vanish at $\mu\sim
m_{b}$. We shall see later (Fig. 5) that the dependence of
$\lambda_{B_{S}}(\mu)$ in the region $\mu\sim m_{b}$ is relatively flat and,
therefore, the impact of the precise value of the scale at which the radiative
corrections vanish, has only a moderate impact on, e.g., the extracted value
of $\lambda_{B_{s}}(\mu=m_{b})$.

Fig. 4 presents the results based on Eqs. (2.1) showing strong correlations
between the value of $\lambda_{B_{s}}\equiv\lambda_{B_{s}}(m_{b})$ and the
form factor values $F_{V}(0,0)$, $V(0)$, $F_{TV}(0,0|\mu=m_{b})$ and
$T_{1}(0|\mu=m_{b})$.

|
---|---
(a) | (b)
|
(c) | (d)
Fig. 4:  (a) Form factor $V(0)$ calculated via the leading order diagram of
Fig. 1 (a) vs $\lambda_{B_{s}}=\lambda_{B_{s}}(m_{b})$. The band covers the
results for $V(0)$ in Table 2. (b) Form factor $T_{1}(0)$ at the scale
$\mu=m_{b}$ calculated via the leading order diagram of Fig. 1 (a) vs
$\lambda_{B_{s}}=\lambda_{B_{s}}(m_{b})$. The band covers the results for
$T_{1}(0)$ in Table 2. (c) Form factor $F_{V}(0,0)$ vs $\lambda_{B_{s}}$. The
band of values of $F_{V}(0,0)$ corresponds to the allowed band of
$\lambda_{B_{s}}(m_{b})$ from Plot. (a) (d) Form factor $F_{TV}(0,0)$ at the
scale $\mu=m_{b}$ vs $\lambda_{B_{s}}$. The band of values of $F_{TV}(0,0)$
corresponds to the allowed band of $\lambda_{B_{s}}(m_{b})$ from Plot. (b).

In order to disentangle the appropriate value of $\lambda_{B_{S}}(m_{b})$ we
use the existing estimates of the $B_{s}\to\phi$ form factors $V(0)$ and
$T_{1}(0|m_{b})$, obtained using approaches not directly referring to the 2DAs
of $B_{s}$-meson.

Table 2: Theoretical results for the $B_{s}\to\phi$ form factors $V(0)$ and $T_{1}(0|\mu=m_{b})$ and the corresponding $\lambda_{B_{s}}(m_{b})$ obtained from Fig. 4. | ms2000  | ballbraun1998  | ballzwicky2005  | lattice2014  | ivanov2016
---|---|---|---|---|---
$V(0)$ | $0.44\pm 0.05$ | $0.43\pm 0.06$ | $0.433\pm 0.035$ | $0.24\pm 0.07$ | $0.31\pm 0.03$
$\lambda_{B_{s}}(\mu=m_{b})[{\rm GeV}]$ | 0.54 | 0.55 | 0.55 | 0.72 | 0.65
$T_{1}(0|\,\mu=m_{b})$ | $0.38\pm 0.04$ | $0.35\pm 0.05$ | $0.349\pm 0.033$ | $0.31\pm 0.02$ | $0.27\pm 0.03$
$\lambda_{B_{s}}(\mu=m_{b})[{\rm GeV}]$ | 0.54 | 0.58 | 0.58 | 0.62 | 0.7

Table 2 shows a selection of such results obtained from quark models ms2000 ;
ivanov2016 333The results obtained with quark models do not include explicitly
the radiative corrections and therefore do not control explicitly the scale-
dependence of the form factors of the tensor current. There are however
arguments ms2000  that the tensor form factors obtained in quark models
correspond to the scale $\mu=m_{b}$., light-cone QCD sum rules based on the
$\phi$-meson DAs ballbraun1998 ; ballzwicky2005 , and lattice QCD lattice2014
.

Conservatively, the existing results provide the allowed range

$\displaystyle V(0)=0.34\pm 0.10,\qquad T_{1}(0|\mu=m_{b})=0.32\pm 0.06.$
(2.11)

Fig. 4(a) then yields the corresponding allowed range
$\lambda_{B_{S}}(m_{b})=(0.62\pm 0.10)$ GeV and from Fig. 4(b) one obtains the
estimate for the $B_{s}\to\gamma$ form factors $F^{(s)}_{V}(0,0)=0.13\pm 0.01$
and $F^{(s)}_{TV}(0,0|m_{b})=0.105\pm 0.015$. These values are in agreement
with the estimate obtained in mnk2018 , $F^{(s)}_{V}(0,0)=0.1$ and
$F^{(s)}_{TV}(0,0|m_{b})=0.11$, and consistent also with the recent lattice
QCD calculations of lattice2024 .

Making use of the results for $R(q^{2}=0,\mu)$ from Appendix A (Fig. 5(a)) and
the property that $R(q^{2}=0,\mu)/\lambda_{B_{s}}(\mu)$ is scale-independent,
we obtain the $\mu$-dependence of $\lambda_{B_{s}}(\mu)$ shown in Fig. 5.
Evolving the allowed range of $\lambda_{B_{s}}(\mu)$ from $\mu=m_{b}$ to
$\mu\simeq 1$ GeV, we obtain the estimate

$\displaystyle\lambda_{B_{s}}(1\mbox{ GeV})=0.49\pm 0.07\mbox{ GeV}.$ (2.12)

|
---|---
(a) | (b)
Fig. 5:  (a) The dependence of the factor $R(0,\mu)$ on $\mu$. (b) The
dependence of $\lambda_{B_{s}}(\mu)$ on $\mu$. Solid line describes the
evolution of the central value of the allowed interval of
$\lambda_{B_{s}}(m_{b})$ whereas the dash-dotted lines correspond to the lower
and the upper boundaries of the allowed interval of $\lambda_{B_{s}}(m_{b})$.

##  3 Discussion and conclusions

A. We have formulated an efficient algorithm of obtaining the form factors
$F_{V,TV}(q^{2},q^{\prime 2})$ in a broad range of momentum transfers
including the region of meson resonances where the form factors cannot be
calculated using finite-order QCD diagrams. Our algorithm is based on the
following steps:

i. We calculate the form factors $F_{V,TV}(q^{2},q^{\prime 2})$ at the lowest
order in $\alpha_{s}$ via the 2DAs of the $B_{s}$ meson in a broad range of
$q^{2}$ and $q^{\prime 2}$ far below quark thresholds where the calculation
based on finite order Feynman diagrams may be trusted. In this way we obtain,
in particular, the dependence of $F_{V,TV}(q^{2},q^{\prime 2})$ on the
parameter $\lambda_{B_{s}}$ making use of the lowest order in $\alpha_{s}$.

ii. We interpolate the obtained results by an analytic formula based on
dispersion representation for $F_{V,TV}(q^{2},q^{\prime 2})$ in $q^{\prime 2}$
with one subtraction. The considered analytic formula takes into account the
presence of the $\phi$-meson pole in the physical form factor and takes into
account contribution of the excited hadron states via an effective pole whose
mass and coupling are fitting parameters.

iii. We obtain predictions for the form factors $F_{V}(q^{2},q^{\prime 2})$
and $F_{TV}(q^{2},q^{\prime 2})$ in the physical region by using the fitting
formula in a broad range of momentum transfers including the resonance region.

B. We applied this algorithm and obtained the $B_{s}\to\phi$ form factors
$V(q^{2})$ ($T_{1}(q^{2})$) as a residue of $F_{V}(q^{2},q^{\prime 2})$
($F_{TV}(q^{2},q^{\prime 2})$) of the pole located at $q^{\prime
2}=M_{\phi}^{2}$ without using QCD sum rules. In this way we avoid the
systematic uncertainties related to the method of QCD sum rules which are not
possible to control in a rigorous way lms2007a ; lms2007b ; lms2009a ;
lms2009b ; m2009 .444 We are not saying that our method avoids systematic
uncertainties: e.g., the residue in the pole has sensitivity to the ranges of
values of $q^{2}$ and $q^{\prime 2}$ where the interpolation is performed.
However, taking into account the precise knowledge of the pole location (i.e.
the physical masses of the lightest vector mesons) reduces this uncertainty
considerably.

C. Our analysis is based on the correlation function calculated at leading
order in $\alpha_{s}$. Making use of the fact that the radiative corrections
for the form factor $F_{V}(q^{2},q^{\prime 2}=0)$ evaluated at the scale
$\mu=m_{b}$ turn out to be small in a broad range of photon energies
$E_{\gamma}$, we have given arguments that the same property is valid for the
form factors $F_{V}(q^{2},q^{\prime 2})$ in a broad range of values of $q^{2}$
and $q^{\prime 2}$ far below the quark thresholds, where our fitting procedure
is carried out. In the case of the form factor $F_{TV}(q^{2},q^{\prime 2})$
the radiative corrections are not known at all, but also in this case we may
conjecture that the same property is valid.

We emphasize that it is not necessary that the radiative corrections vanish
precisely at the scale $\mu=m_{b}$; it is sufficient that they vanish at some
scale not far from $m_{b}$. Then, the obtained estimate for
$\lambda_{B_{s}}(\mu=m_{b})$ would change only marginally as the dependence of
$\lambda_{B_{s}}(\mu)$ in the region $\mu\sim m_{b}$ is relatively flat. But,
obviously, for putting our analysis on a solid theoretical basis, a direct
evaluation of the radiative corrections for the form factors of the tensor
currents would be crucial.

Given the property that the radiative corrections are small at $\mu=m_{b}$,
our leading-order expressions for the form factors evaluated via the 2DAs at
the scale $\mu=m_{b}$, containing $\lambda_{B_{s}}(m_{b})$, should provide a
good approximation for the form factors $F_{V}(q^{2},q^{\prime 2})$ and
$V(q^{2})$ as well as for $F_{TV}(q^{2},q^{\prime 2}|\mu=m_{b})$ and
$T_{1}(q^{2}|\mu=m_{b})$.

Then, having at hand the dependence of $T_{1}(0)$ and $V(0)$ on
$\lambda_{B_{s}}(m_{b})$ and making use of the theoretical results for
$B_{s}\to\phi$ form factors $V(0)$ and $T_{1}(0|\mu=m_{b})$, obtained by
approaches which do not use the 2DAs of $B_{s}$-meson, we extract the
following allowed range s

$\displaystyle\lambda_{B_{s}}(\mu\simeq m_{b})=0.62\pm 0.10\mbox{ GeV}~{},~{}$
(3.13) $\displaystyle\lambda_{B_{s}}(1\mbox{ GeV})=0.49\pm 0.07\mbox{ GeV}.$
(3.14)

The latter result agrees nicely with thea recent estimate
$\lambda_{B_{s}}(1\mbox{ GeV})=0.48\pm 0.09$ GeV obtained using QCD sum rules
in lambdabs2024 .

Closing this paper, we believe that the proposed method for extracting the
meson-to-meson transition form factor as a residue of a dominant pole, like
$T_{1}(q^{2})$ in the form factor $F_{TV}(q^{2},q^{\prime 2})$, may be
promising for calculating other meson transition form factors and may be used
as a complementary method to QCD sum rules.

###### Acknowledgements.

We are grateful to Matthias Neubert, Sergio Leal-Gomez, and Giuseppe Gagliardi
for interesting discussions and valuable comments.

##  Appendix A Form factor $T_{1}(q^{2})$ for $B_{s}\to\phi$ from QCD sum rule

Here we would like to recall briefly a sum-rule calculation of the form factor
$T_{1}(q^{2})$. The starting point is the same correlation function (2.2) and
the form factor $F_{TV}(q^{2},q^{\prime 2})$. Eq. (2.4) based on the
calculation of QCD diagrams, may be identically rewritten in the form of a
dispersion representation in $q^{\prime 2}$:

$\displaystyle F_{TV}(q^{2},q^{\prime
2})=\int\limits_{0}^{2\omega_{0}}d\xi\frac{\phi_{B_{s}}(\xi)}{1-\xi}\int
ds^{\prime}\frac{1}{s^{\prime}-q^{\prime
2}}\delta\left(s^{\prime}-\frac{m_{s}^{2}+\xi(1-\xi)M_{B_{s}}^{2}-q^{2}\xi}{1-\xi}\right),$
(A.15)

where we introduce a short-hand notation

$\displaystyle\phi_{B_{s}}(\xi)\equiv-
Q_{s}M_{B_{s}}f_{B_{s}}\left[\phi_{+}(\xi)(1-\xi)+\bar{\Phi}(\xi)\right].$
(A.16)

We can now make a subtraction in this spectral representation and write:

$\displaystyle F_{TV}(q^{2},q^{\prime 2})=F_{TV}(q^{2},0)+q^{\prime
2}\int\limits_{0}^{2\omega_{0}}d\xi\frac{\phi_{B_{s}}(\xi)}{1-\xi}\int
ds^{\prime}\frac{1}{s^{\prime}(s^{\prime}-q^{\prime
2})}\delta\left(s^{\prime}-\frac{m_{s}^{2}+\xi(1-\xi)M_{B_{s}}^{2}-q^{2}\xi}{1-\xi}\right).$
(A.17)

Obviously, the representation (A.15) may not be used in the region of
$q^{\prime 2}$ close to the quark threshold at $4m_{s}^{2}$. On the other
hand, from the general property of the correlation function,
$F_{TV}(q^{2},q^{\prime 2})$ contains a pole at $q^{\prime 2}=M_{\phi}^{2}$.

---
Fig. 6:  The dual form factor $T_{1}(0|\tau,\Delta)$, a QCD sum rule estimate
for $T_{1}(0)$; $\Delta=0.5$ GeV.

We can now equate to each other two different representations for
$F_{TV}(q^{2},q^{\prime 2})$ – the phenomenological side of a QCD sum rule (in
the language of hadron states) and the theoretical side (based on QCD
diagrams):

$\displaystyle F_{TV}(q^{2},q^{\prime 2})$ $\displaystyle=$ $\displaystyle
R_{\phi}T_{1}(q^{2})\frac{1}{M_{\phi}^{2}-q^{\prime 2}}+\ldots$ (A.18)
$\displaystyle=$ $\displaystyle F_{TV}(q^{2},0)+q^{\prime
2}\int\limits_{0}^{1}d\xi\frac{\phi_{B_{s}}(\xi)}{1-\xi}\int
ds^{\prime}\frac{1}{s^{\prime}(s^{\prime}-q^{\prime
2})}\delta\left(s^{\prime}-\frac{m_{s}^{2}+\xi(1-\xi)M_{B_{s}}^{2}-q^{2}\xi}{1-\xi}\right),$

where $\dots$ denote the contribution of hadron continuum and the excited
resonances. To isolate the pole at $q^{\prime 2}=M_{\phi}^{2}$ we perform two
standard steps adopted in QCD sum rules:

  * (i)

Introduce the effective threshold $s_{\rm eff}$ and apply the duality cut
$s^{\prime}<s_{\rm eff}$. We shall make use of the standard hypothesis (“Braun
prescription”) based on the properties of hadron spectrum:

$\displaystyle s_{\rm eff}=(M_{\phi}+\Delta)^{2},\quad\Delta\simeq 500\;{\rm
MeV}.$ (A.19)

  * (ii)

Perform Borel transform $q^{\prime 2}\to\tau$ $(\equiv 1/M_{\rm Borel}^{2})$:

$\displaystyle 1/(s^{\prime}-q^{\prime 2})\to\exp(-s^{\prime}\tau),\quad
1/(M_{\phi}^{2}-q^{\prime 2})\to\exp(-M_{\phi}^{2}\tau).$ (A.20)

This Borel transform suppresses the contributions of higher-mass resonances on
the phenomenological side of the QCD sum rule for $F_{TV}(q^{2},q^{\prime 2})$
and kills all polynomial terms in $q^{\prime 2}$ on the OPE side of the QCD
sum rule, including possible subtraction terms.

Using the $\delta$-function to take the $s^{\prime}-$integral, we obtain the
dual form factor $T_{1}(q^{2}|\tau,\Delta)$ from which one should obtain a
sum-rule estimate for the physical form factor $T_{1}(0)$:

$\displaystyle
T_{1}(q^{2}|\tau,\Delta)=\frac{1}{R_{\phi}}\int\limits_{0}d\xi\frac{\phi_{B_{s}}(\xi)}{1-\xi}\exp\left[\tau\left(M_{\phi}^{2}-\frac{m_{s}^{2}+\xi(1-\xi)M_{B_{s}}^{2}-q^{2}\xi}{1-\xi}\right)\right]\theta\left(\frac{m_{s}^{2}+\xi(1-\xi)M_{B_{s}}^{2}-q^{2}\xi}{1-\xi}\leq
s_{\rm eff}\right).$ (A.21)

Note that the actual upper limit of the $\xi$-integration in Eq. (A.21) is
determined by the $\theta$-function (and not by the parameter $2\omega_{0}$
expected to be of the order of $\sim 0.5$) leaving the $\xi$-integration
region in the dual form factor closer to zero, i.e. $\xi\leq s_{\rm
eff}/M_{B_{s}}^{2}\sim 0.1$.

The dual form factor contains two auxiliary parameters, the effective
threshold $\Delta$ and the Borel parameter $\tau$. We shall not try to play
the game with stability windows but just fix $\Delta$ as mentioned above and
take the minimal value in the Borel window as the SR estimate for the form
factor $T_{1}(0)$. This gives $T_{1}(0)\simeq 0.32$ (see Fig. 6).

##  Appendix B Radiative corrections to the form factor $F_{V}$

The radiative corrections for the tensor current form factors $F_{TV,TA}$ are
not known. We therefore refer to the known radiative corrections for the form
factors $F_{A,V}$ beneke2011 . The necessary formulas from beneke2011  are
reproduced here for the convenience of the reader.

To the leading order in $1/m_{b}$, the form factors $F_{A,V}$ have the
following behaviour:

$\displaystyle F_{A,V}(E_{\gamma})$ $\displaystyle=$
$\displaystyle\frac{Q_{u}M_{B_{s}}f_{B_{s}}}{2E_{\gamma}\lambda_{B_{s}}(\mu)}\,R(E_{\gamma},\mu)+\dots$
(B.22)

The factor $R(E_{\gamma},\mu)$ describing the radiative corrections can be
written as a product of several factors,

$R(E_{\gamma},\mu)=C(E_{\gamma},\mu_{h1})K^{-1}(\mu_{h2})\times
U(E_{\gamma},\mu_{h1},\mu_{h2},\mu)\times J(E_{\gamma},\mu),$ (B.23)

which come from the different scales contributing to the process, namely:

$\bullet$ the factor relating the decay constant of the $B$-meson in QCD and
HQET

$K(\mu)=1+\frac{\alpha_{s}C_{F}}{4\pi}\left(\frac{3}{2}\ln\frac{m_{b}^{2}}{\mu^{2}}-2\right).$
(B.24)

$\bullet$ the factor related to the matching between the weak currents in QCD
and SCET

$C(E_{\gamma},\mu)=1+\frac{\alpha_{s}C_{F}}{4\pi}\left(-2\ln^{2}\frac{2E_{\gamma}}{\mu}+5\ln\frac{2E_{\gamma}}{\mu}-\frac{3-2x}{1-x}\ln
x-2{\rm Li}_{2}(1-x)-6-\frac{\pi^{2}}{12}\right),\quad x=2E_{\gamma}/m_{b}.$
(B.25)

$\bullet$ the hard-collinear radiative correction

$\displaystyle
J(E_{\gamma},\mu)=1+\frac{\alpha_{s}C_{F}}{4\pi}\left(\ln^{2}\frac{2E_{\gamma}\mu_{0}}{\mu^{2}}-2\sigma_{1}(\mu,\mu_{0})\ln\frac{2E_{\gamma}\mu_{0}}{\mu^{2}}-1-\frac{\pi^{2}}{6}+\sigma_{2}(\mu,\mu_{0})\right)$
(B.26)

with the inverse-logarithmic moments defined as

$\displaystyle\frac{1}{\lambda_{B_{s}}(\mu)}=\int_{0}^{\infty}\frac{d\omega}{\omega}\,\phi_{+}(\omega,\mu),\qquad\sigma_{n}(\mu,\mu_{0})=\lambda_{B_{s}}(\mu)\int_{0}^{\infty}\frac{d\omega}{\omega}\,\ln^{n}\frac{\mu_{0}}{\omega}\,\phi_{+}(\omega,\mu)$
(B.27)

where $\mu_{0}=1\mbox{ GeV}$ is a fixed reference scale which is part of the
definition of the inverse-logarithmic moments adopted in beneke2011 .
Parameter $\omega$ in (B.26) is related to dimensionless parameter $\xi$ in
(2.6) as $\omega=M_{B_{s}}\xi$.

In beneke2011  the authors notice the absence of a common scale $\mu$ that
avoids parametrically large logarithms of order $\ln m_{b}/\mu_{0}$ in (B.24),
(B.25) and (B.26) and introduce the evolution factor
$U(E_{\gamma},\mu_{h1},\mu_{h2},\mu)$ into (B.23). Its explicit expression is
given in the Appendix of beneke2011 .

Our observation based on the above formulas and on the explicit form of the
2DAs is however a bit different from the conclusion of beneke2011 . Indeed,
for the 2DA of the LD model (2.6), the dependence of $J(E_{\gamma},\mu)$ in
(B.26) on $\mu_{0}$ drops out. We have checked that this property holds as
well also for a different 2DA model referred to as Model IIA in braun2017 .
Thus, we expect that, at least at order ${\cal{O}}(\alpha_{s})$, the factor
$J(E_{\gamma},\mu)$ of $\mu_{0}$ is independent of $\mu_{0}$.

Because of this property, one can set $\mu_{0}$ equal to the hard scale $\mu$
thus avoiding large logs in all three factors (B.24), (B.25) and (B.26). In
this case, the expressions (2.13) and (2.14) from beneke2011  and the
expressions (42) and (43) from braun2004  (correspond to setting
$\mu_{0}\to\mu$) lead to the same results. If so, one can choose the common
hard scale in all factors (B.24), (B.25) and (B.26), and the evolution factor
$U(E_{\gamma},\mu_{h1},\mu_{h2},\mu)$ reduces to unity.

Moreover, at the scale $\mu=m_{b}$, the factor $R(E_{\gamma},\mu=m_{b})$ is
very close to unity in a broad range of values of $q^{2}$. Therefore, for
$\mu=m_{b}$ the radiative corrections may be neglected and the leading-order
analysis is expected to provide reliable results.

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$d^{E}_{w}(X_{1},X_{2})=\sup E(X_{2},w)\ominus\sup E(X_{1},w)$ if
$\emph{ec}(X_{1},w)=\emph{ec}(X_{2},w)$, $d^{E}_{w}(X_{1},X_{2})=1$ if
$\emph{ec}(X_{1},w)>\emph{ec}(X_{2},w)$ and $d^{E}_{w}(X_{1},X_{2})=0$ if
$\emph{ec}(X_{1},w)<\emph{ec}(X_{2},w)$. Then it can be shown that for
$X_{1},X_{2}\subseteq X$:

$\nu\mathrm{beh}(X_{1},X_{2})=\sup\nolimits_{w\in
A^{*}}d_{w}^{E}(X_{1},X_{2})$

As a concrete example we take the label set $A=\\{a,b\\}$ and the coalgebra
$c$ given below, which is inspired by a similar example in [bp:checking-nfa-
equiv]:

$x_{0}$$x_{1}$$x_{2}$$\dots$$y_{0}$$y_{1}$$y_{2}$$\dots$$z_{0}$$z_{1}$$z_{2}$$\dots$$x_{n}$$\nicefrac{{1}}{{4}}$$y_{n}$$\nicefrac{{1}}{{3}}$$z_{n}$$\nicefrac{{1}}{{2}}$$a$$a,b$$a,b$$a,b$$b$$a,b$$a,b$$a,b$$a,b$$a,b$$a,b$$a,b$$a,b$$a,b$$a,b$

Here, the outputs of the final states are $c(x_{n})=\nicefrac{{1}}{{4}}$,
$c(y_{n})=\nicefrac{{1}}{{3}}$ and $c(z_{n})=\nicefrac{{1}}{{2}}$, as
indicated. It holds that
$\nu\mathrm{beh}(\\{x_{0},y_{0}\\},\\{z_{0}\\})=\nicefrac{{1}}{{4}}$.
Intuitively this is true, since the largest distance is achieved if we follow
a path from $x_{0}$ with $a$ a $n$-last letter, yielding exception value
$\nicefrac{{1}}{{4}}$, while the same path results in the value
$\nicefrac{{1}}{{2}}$ from $z_{0}$, hence we obtain distance
$\nicefrac{{1}}{{2}}\ominus\nicefrac{{1}}{{4}}=\nicefrac{{1}}{{4}}$.

Note that the determinization of the transition system above is of exponential
size and the same holds for a representation of a post-fixpoint for witnessing
an upper bound for behavioural distance. We construct a $\mathcal{V}$-valued
relation $d$ of linear size witnessing that
$\nu\mathrm{beh}(\\{x_{0},y_{0}\\},\\{z_{0}\\})\leq\nicefrac{{1}}{{4}}$ via
up-to techniques. To this end let $d$ be defined by

$d(\\{x_{0},y_{0}\\},\\{z_{0}\\})=\nicefrac{{1}}{{4}}\qquad
d(\\{x_{i}\\},\\{z_{i}\\})=\nicefrac{{1}}{{4}}\qquad
d(\\{y_{i}\\},\\{z_{i}\\})=\nicefrac{{1}}{{6}}$

and distance $1$ for all other arguments.

It suffices to show that $d$ is a pre-fixed point of $\mathrm{beh}$ up-to $u$
(wrt. $\leq$). We can use the property that $u(d)(X_{1}\cup X_{2},Y_{1}\cup
Y_{2})\leq\max\\{d(X_{1},Y_{1}),d(X_{2},Y_{2})\\}$ (cf. Corollary 6.3 in the
appendix). Now the claim follows from unfolding the fixpoint equation by
considering $a$\- and $b$-transitions:

$\displaystyle\mathrm{beh}(u(d))(\\{x_{0},y_{0}\\},\\{z_{0}\\})$
$\displaystyle=\max\\{u(d)(\\{x_{0},x_{1},y_{0}\\},\\{z_{0},z_{1}\\}),u(d)(\\{x_{0},y_{0},y_{1}\\},\\{z_{0},z_{1}\\})\\}$
$\displaystyle\leq\max\\{d(\\{x_{0},y_{0}\\},\\{z_{0}\\}),d(\\{x_{1}\\},\\{z_{1}\\}),d(\\{y_{1}\\},\\{z_{1}\\})\\}=\nicefrac{{1}}{{4}}$

The arguments for $d(\\{x_{i}\\},\\{z_{i}\\})$, $d(\\{y_{i}\\},\\{z_{i}\\})$
are similar, concluding the proof.

##  7 Conclusion

#### Related work.

While the notion of Kantorovich distance on probability distributions is much
older, Kantorovich liftings in a categorical framework have first been
introduced in [bbkk:coalgebraic-behavioral-metrics] and have since been
generalized, for instance to codensity liftings [kwrk:composing-codensity-
bisimulations] or to lifting fuzzy lax extensions [ws:metrics-fuzzy-lax].

A coalgebraic theory of up-to techniques was presented in [bppr:general-
coinduction-up-to] and has been instantiated to the setting of coalgebraic
behavioural metrics in [bkp:up-to-behavioural-metrics-fibrations-journal]. The
latter paper concentrated on Wasserstein liftings, which leads to a
significantly different underlying theory. Furthermore, Wasserstein liftings
are somewhat restricted, since they rely only on a single evaluation map and
on couplings (which sometimes do not exist, making it difficult to define more
fine-grained metric). We are not aware of a way to handle the examples of
Section 6 directly in the Wasserstein approach.

Setting up the fibred adjunction (Section 3) and the definition of the
Kantorovich lifting (Section 4.1) has some overlap to [bgkmfsw:logics-
coalgebra-adjunction] and the very recent [kwrk:composing-codensity-
bisimulations]. Our focus is primarily on showing fibredness (naturality) via
a quantalic version of the extension theorem.

The aim of [kwrk:composing-codensity-bisimulations] is on combining
behavioural conformance via algebraic operations, which is different than our
notion of compositionality via functor liftings. There is some similarity in
the aims of both papers, namely the lifting of distributive laws and the
motivation to study up-to techniques. From our point of view, the obtained
results are largely orthogonal: while the focus of [kwrk:composing-codensity-
bisimulations] is on providing $n$-ary operations for composing conformances
and games and it provides a more general high-level account, we focus
concretely on compositionality of functor liftings (studying counterexamples
and treating the special case of polynomial functors), giving concrete recipes
for lifting distributive laws and applying the results to proving upper bounds
via up-to techniques.

#### Future work.

Our aim is to better understand the metric congruence results employed in
Section 6, comparing them with the similar proof rules in [mpp:quantitative-
algebraic-reasoning]. Compositionality of functor liftings fails in important
cases (cf. Example 4.3), which could be fixed by using different sets of
evaluation maps as in the Moss liftings in [ws:metrics-fuzzy-lax]. We also
plan to study case studies involving the convex powerset functor [bss:power-
convex-algebras]. Finally, we want to develop witness generation methods, by
constructing suitable post-fixpoint up-to on-the-fly, similar to [bblm:on-the-
fly-exact-journal].

#### Acknowledgments

We would like to thank Avi Craimer for helpful discussion and the organizers
of the ACT Adjoint School for enabling us to meet and write this paper
together!

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# Coverability is Undecidable in One-dimensional Pushdown Vector Addition
Systems with Resets

Sylvain Schmitz1,2 and Georg Zetzsche3 1 LSV, ENS Paris-Saclay & CNRS
Université Paris-Saclay
France  2 IUF
France  3 Max Planck Institute for Software Systems (MPI-SWS)
Germany

###### Abstract.

We consider the model of pushdown vector addition systems with resets. These
consist of vector addition systems that have access to a pushdown stack and
have instructions to reset counters. For this model, we study the coverability
problem. In the absence of resets, this problem is known to be decidable for
one-dimensional pushdown vector addition systems, but decidability is open for
general pushdown vector addition systems. Moreover, coverability is known to
be decidable for reset vector addition systems without a pushdown stack. We
show in this note that the problem is undecidable for one-dimensional pushdown
vector addition systems with resets.

Keywords. Pushdown vector addition systems; decidability

##  1\. Introduction

Vector addition systems with states (VASS) play a central role for modelling
systems that manipulate discrete resources, and as such provide an algorithmic
toolbox applicable in many different fields. Adding a pushdown stack to vector
addition systems yields so-called _pushdown VASS_ (PVASS), which are even more
versatile: one can model for instance recursive programs with integer
variables [2] or distributed systems with a recursive server and multiple
finite-state clients, and PVASS can be related to decidability issues in
logics on data trees [8]. However, this greater expressivity comes with a
price: the _coverability problem_ for PVASS is only known to be decidable in
dimension one [12]. This problem captures most of the decision problems of
interest and in particular safety properties, and is the stumbling block in a
classification for a large family of models combining pushdown stacks and
counters [16].

Another viewpoint on one-dimensional PVASS [12] is to see those systems as
extensions of two-dimensional VASS, where one of the two counters is replaced
by a pushdown stack. In this context, a complete classification with respect
to decidability of coverability, and of the more difficult _reachability
problem_ , was provided by Finkel and Sutre [6],

Table 1. Decidability status of the coverability and reachability problems in
extensions of two-dimensional VASS; our contribution is indicated in bold.

(a) Coverability problem. $\mathbb{N}$ |  $\mathbb{N}_{\mathsf{r}}$ |  $\mathbb{N}_{\mathsf{t}}$ |  $\mathbb{N}_{\mathsf{z}}$ |  $\mathsf{PD}$ |
---|---|---|---|---|---
D [7] |  D [1] |  D [4] |  D [15] |  D [12] |  $\mathbb{N}$
|  D [1] |  D [4] |  D [6] |  U |  $\mathbb{N}_{\mathsf{r}}$
|  |  D [4] |  U [6] |  U [6] |  $\mathbb{N}_{\mathsf{t}}$
|  |  |  U [14] |  U [14] |  $\mathbb{N}_{\mathsf{z}}$
|  |  |  |  U |  $\mathsf{PD}$

(b) Reachability problem. $\mathbb{N}$ |  $\mathbb{N}_{\mathsf{r}}$ |  $\mathbb{N}_{\mathsf{t}}$ |  $\mathbb{N}_{\mathsf{z}}$ |  $\mathsf{PD}$ |
---|---|---|---|---|---
D [9] |  D [15] |  D [15] |  D [15] |  ?? |  $\mathbb{N}$
|  D [6] |  D [6] |  D [6] |  U |  $\mathbb{N}_{\mathsf{r}}$
|  |  D [6] |  U [6] |  U [6] |  $\mathbb{N}_{\mathsf{t}}$
|  |  |  U [14] |  U [14] |  $\mathbb{N}_{\mathsf{z}}$
|  |  |  |  U |  $\mathsf{PD}$

whether one uses plain counters ($\mathbb{N}$), counters with resets
($\mathbb{N}_{\mathsf{r}}$), counters whose contents can be transferred to the
other counter ($\mathbb{N}_{\mathsf{t}}$), or counters with zero tests
($\mathbb{N}_{\mathsf{z}}$); see Table 1. In particular, two-dimensional VASS
with one counter extended to allow resets and one extended to allow zero tests
have a decidable reachability problem [6]: put differently, the coverability
problem for one-dimensional PVASS _with resets_ ($1$-PRVASS) is decidable if
the stack alphabet is of the form $\\{a,\bot\\}$ where $\bot$ is a
distinguished bottom-of-stack symbol.

#### Contributions.

In this note, we show that Finkel and Sutre’s decidability result does not
generalise to one-dimensional pushdown VASS with resets over an arbitrary
finite stack alphabet.

######  Theorem 1.

The coverability problem for $1$-PRVASS is undecidable.

As far as the coverability problem is concerned, this fully determines the
decidability status in extensions of two-dimensional VASS where one may also
replace counters by pushdown stacks ($\mathsf{PD}$); see Table 1(a).

Technically, the proof of Theorem 1 presented in Section 3 reduces from the
reachability problem in two-counter Minsky machines. The reduction relies on
the ability to _weakly implement_ [13] basic operations—like multiplication by
a constant—and their inverses—like division by a constant. This in itself
would not bring much; for instance, plain two-dimensional VASS can already
weakly implement multiplication and division by constants. The crucial point
here is that, in a $1$-PRVASS, we can also weakly implement the inverse of a
_sequence_ of basic operations performed by the system, by using the pushdown
stack to record a sequence of basic operations and later replaying it in
reverse, and relying on resets to “clean-up” between consecutive operations.
Note that without resets, while PVASS are known to be able to weakly implement
Ackermannian functions already in dimension one [11], they cannot weakly
compute sublinear functions [10]—like iterated division by two, i.e.,
logarithms.

##  2\. Pushdown Vector Addition Systems with Resets

A _( $1$-dimensional) pushdown vector addition system with resets
($1$-PRVASS)_ is a tuple $\mathcal{V}=(Q,\Gamma,A)$, where $Q$ is a finite set
of _states_ , $\Gamma$ is a finite set of _stack symbols_ , and $A\subseteq
Q\times I^{*}\times Q$ is a finite set of _actions_. Here, transitions are
labelled by finite sequences of _instructions_ from
$I\mathrel{\raisebox{-0.43057pt}{$\stackrel{{\scriptstyle\raisebox{-0.75346pt}{\scalebox{0.45}{{def}}}}}{{=}}$}}\Gamma\cup\bar{\Gamma}\cup\\{\mathsf{\mathord{+}},\mathsf{\mathord{-}},\mathsf{r}\\}$
where
$\bar{\Gamma}\mathrel{\raisebox{-0.43057pt}{$\stackrel{{\scriptstyle\raisebox{-0.75346pt}{\scalebox{0.45}{{def}}}}}{{=}}$}}\\{\bar{z}\mid
z\in\Gamma\\}$ is a disjoint copy of $\Gamma$.

A $1$-PRVASS defines a (generally infinite) transition system acting over
_configurations_ $(q,w,n)\in Q\times\Gamma^{*}\times\mathbb{N}$. For an
instruction $x\in I$, $w,w^{\prime}\in\Gamma^{\ast}$, and
$n,n^{\prime}\in\mathbb{N}$, we write
$(w,n)\xrightarrow{y}(w^{\prime},n^{\prime})$ in the following cases:

push:

if $x=z$ for $z\in\Gamma$, then $w^{\prime}=wz$ and $n^{\prime}=n$,

pop:

if $x=\bar{z}$ for $z\in\Gamma$, then $w=w^{\prime}z$ and $n^{\prime}=n$,

increment:

if $x=\mathsf{\mathord{+}}$, then $w^{\prime}=w$ and $n^{\prime}=n+1$.

decrement:

if $x=\mathsf{\mathord{-}}$, then $w^{\prime}=w$ and $n^{\prime}=n-1$, and

reset:

if $x=\mathsf{r}$, then $w^{\prime}=w$ and $n^{\prime}=0$.

Moreover, for a sequence of instructions $u=x_{1}\cdots x_{k}$ with
$x_{1},\ldots,x_{k}\in I$, we have $(w_{0},n_{0})\xrightarrow{u}(w_{k},n_{k})$
if for some
$(w_{1},n_{1}),\ldots,(w_{k-1},n_{k-1})\in\Gamma^{*}\times\mathbb{N}$, we have
$(w_{i},n_{i})\xrightarrow{x_{i}}(w_{i+1},n_{i+1})$ for all $0\leq i<k$.
Finally, for two configurations $(q,w,n),(q^{\prime},w^{\prime},n^{\prime})\in
Q\times\Gamma^{*}\times\mathbb{N}$, we write
$(q,w,n)\rightarrow_{\mathcal{V}}(q^{\prime},w^{\prime},n^{\prime})$ if there
is an action $(q,u,q^{\prime})\in A$ such that
$(w,n)\xrightarrow{u}(w^{\prime},n^{\prime})$.

The _coverability problem_ for 1-PRVASS is the following decision problem.

given:

a 1-PRVASS $\mathcal{V}=(Q,\Gamma,A)$, states $s,t\in Q$.

question:

are there $w\in\Gamma^{*}$ and $n\in\mathbb{N}$ with
$(s,\varepsilon,0)\rightarrow^{*}_{\mathcal{V}}(t,w,n)$?

##  3\. Reduction from Minsky Machines

We present in this section a reduction from reachability in two-counter Minsky
machines to coverability in 1-PRVASS.

###  3.1. Preliminaries

Recall that a _two-counter (Minsky) machine_ is a tuple $\mathcal{M}=(Q,A)$,
where $Q$ is a finite set of states and $A\subseteq
Q\times\\{0,1\\}\times\\{\mathsf{\mathord{+}},\mathsf{\mathord{-}},\mathsf{z}\\}\times
Q$ a set of actions. A _configuration_ is a now triple $(q,n_{0},n_{1})$ with
$q\in Q$ and $n_{0},n_{1}\in\mathbb{N}$. We write
$(q,n_{0},n_{1})\rightarrow_{\mathcal{M}}(q^{\prime},n^{\prime}_{0},n^{\prime}_{1})$
if there is an action $(q,c,x,q^{\prime})\in A$ such that
$n^{\prime}_{1-c}=n_{1-c}$ and

increment:

if $x=\mathsf{\mathord{+}}$, then $n^{\prime}_{c}=n_{c}+1$,

decrement:

if $x=\mathsf{\mathord{-}}$, then $n^{\prime}_{c}=n_{c}-1$, and

zero test:

if $x=\mathsf{z}$, then $n^{\prime}_{c}=n_{c}=0$.

The _reachability problem_ for two-counter machines is the following
undecidable decision problem [14].

given:

a two-counter machine $\mathcal{M}=(Q,A)$, and states $s,t\in Q$.

question:

does $(s,0,0)\rightarrow^{*}_{\mathcal{M}}(t,0,0)$ hold?

#### Gödel Encoding.

The first ingredient of the reduction is to use the well-known encoding of
counter values $(n_{0},n_{1})\in\mathbb{N}\times\mathbb{N}$ as a single number
$2^{n_{0}}3^{n_{1}}$; for instance, the pair
$(0,0)\in\mathbb{N}\times\mathbb{N}$ is encoded by $2^{0}3^{0}=1$. In this
encoding, incrementing the first counter means multiplying by $2$,
decrementing the second counter means dividing by $3$, and testing the second
counter for zero means verifying that the encoding is not divisible by $3$,
etc. Note that, in each case, we encode the instruction as a partial function
$g\colon\mathbb{N}\nrightarrow\mathbb{N}$; let us we define its _graph_ as the
binary relation
$R\mathrel{\raisebox{-0.43057pt}{$\stackrel{{\scriptstyle\raisebox{-0.75346pt}{\scalebox{0.45}{{def}}}}}{{=}}$}}\\{(m,n)\in\mathbb{N}\times\mathbb{N}\mid\text{$g$
is defined on $m$ and $g(m)=n$}\\}$. Thus the encoded instructions are the
partial functions with the following graphs:

$\displaystyle{\mathbin{R_{m_{f}}}}$
$\displaystyle\mathrel{\raisebox{-0.43057pt}{$\stackrel{{\scriptstyle\raisebox{-0.75346pt}{\scalebox{0.45}{{def}}}}}{{=}}$}}\\{(n,f\cdot
n)\mid n\in\mathbb{N}\\}$ for multiplication, $\displaystyle R_{d_{f}}$
$\displaystyle\mathrel{\raisebox{-0.43057pt}{$\stackrel{{\scriptstyle\raisebox{-0.75346pt}{\scalebox{0.45}{{def}}}}}{{=}}$}}\\{(f\cdot
n,n)\mid n\in\mathbb{N}\\}$ for division, and $\displaystyle R_{t_{f}}$
$\displaystyle\mathrel{\raisebox{-0.43057pt}{$\stackrel{{\scriptstyle\raisebox{-0.75346pt}{\scalebox{0.45}{{def}}}}}{{=}}$}}\\{(n,n)\mid
n\not\equiv 0\bmod{f}\\}$ for the divisibility test,

for a factor $f\in\\{2,3\\}$. This means that we can equivalently see

  * •

a two-counter machine with distinguished source and target states $s$ and $t$
as a regular language $M\subseteq\Delta^{*}$ over the alphabet
$\Delta\mathrel{\raisebox{-0.43057pt}{$\stackrel{{\scriptstyle\raisebox{-0.75346pt}{\scalebox{0.45}{{def}}}}}{{=}}$}}\\{m_{f},d_{f},t_{f}\mid
f\in\\{2,3\\}\\}$, and

  * •

reachability as the existence of a word $u=x_{1}\cdots x_{\ell}$ in the
language $M$, with $x_{1},\dots,x_{\ell}\in\Delta$, such that the pair $(1,1)$
belongs to the composition $R_{x_{1}}R_{x_{2}}\cdots R_{x_{\ell}}$.

#### Weak Relations.

Here, the problem is that it does not seem possible to implement these
operations (multiplication, division, divisibility test) directly in a
$1$-PRVASS. Therefore, a key idea of our reduction is to perform the
instructions of $u$ _weakly_ —meaning that the resulting value may be smaller
than the correct result—but _twice_ : once forward and once backward. More
precisely, for any relation $R\subseteq\mathbb{N}\times\mathbb{N}$, we define
the _weak forward_ and _backward_ relations $\overrightarrow{R}$ and
$\overleftarrow{R}$ by

$\displaystyle\overrightarrow{R}$
$\displaystyle\mathrel{\raisebox{-0.43057pt}{$\stackrel{{\scriptstyle\raisebox{-0.75346pt}{\scalebox{0.45}{{def}}}}}{{=}}$}}\\{(m,n)\in\mathbb{N}\times\mathbb{N}\mid\exists\tilde{n}\geq
n\colon(m,\tilde{n})\in R\\}$ $\displaystyle\overleftarrow{R}$
$\displaystyle\mathrel{\raisebox{-0.43057pt}{$\stackrel{{\scriptstyle\raisebox{-0.75346pt}{\scalebox{0.45}{{def}}}}}{{=}}$}}\\{(m,n)\in\mathbb{N}\times\mathbb{N}\mid\exists\tilde{m}\geq
m\colon(\tilde{m},n)\in R\\}.$

Let us call a relation $R\subseteq\mathbb{N}\times\mathbb{N}$ _strictly
monotone_ if for $(m,n)\in R$ and $(m^{\prime},n^{\prime})\in R$, we have
$m<m^{\prime}$ if and only if $n<n^{\prime}$. We shall rely on the following
Section, which is proven in Appendix A. propositiontwoapprox If
$R_{1},\ldots,R_{\ell}\subseteq\mathbb{N}\times\mathbb{N}$ are strictly
monotone relations, then $R_{1}R_{2}\cdots
R_{\ell}=\overrightarrow{R_{1}}\overrightarrow{R_{2}}\cdots\overrightarrow{R_{\ell}}\cap\overleftarrow{R_{1}}\overleftarrow{R_{2}}\cdots\overleftarrow{R_{\ell}}$.

We shall thus construct in Section 3.2 a $1$-PRVASS $\mathcal{V}$ in which a
particular state is reachable if and only if there exists a word $u\in M$ with
$u=x_{1}\cdots x_{\ell}$ and $x_{1},\ldots,x_{\ell}\in\Delta$, such that
$(1,1)\in\overrightarrow{R_{x_{1}}}\cdots\overrightarrow{R_{x_{\ell}}}$ and
$(1,1)\in\overleftarrow{R_{x_{1}}}\cdots\overleftarrow{R_{x_{\ell}}}$. Since
the relations $R_{m_{f}}$, $R_{d_{f}}$, and $R_{t_{f}}$ for $f\in\\{2,3\\}$
are strictly monotone, Section 3.1 guarantees that this is equivalent to
$(1,1)\in R_{x_{1}}\cdots R_{x_{\ell}}$. Intuitively, if we make a mistake in
the forward phase
$\overrightarrow{R_{x_{1}}}\cdots\overrightarrow{R_{x_{\ell}}}$, then at some
point, we produce a number $n$ that is smaller than the correct result
$\tilde{n}>n$. Then, the backward phase cannot compensate for that, because it
can only make the results even smaller, and cannot reproduce the initial
value.

###  3.2. Construction

We now describe the construction of our $1$-PRVASS $\mathcal{V}$. Its stack
alphabet
$\Gamma\mathrel{\raisebox{-0.43057pt}{$\stackrel{{\scriptstyle\raisebox{-0.75346pt}{\scalebox{0.45}{{def}}}}}{{=}}$}}\Delta\cup\\{\bot,\\#,a\\}$.
In $\mathcal{V}$, each configuration will be of the form $(q,\bot
w\\#a^{n},k)$, where $w\in\Delta^{*}$, and $n,k\in\mathbb{N}$. In the forward
phase, we simulate the run of the two-counter machine so that $n$ is the Gödel
encoding of the two counters. In order to perform the backward phase, the word
$w$ records the instruction sequence of the forward phase. The resettable
counter is used as an auxiliary counter in each weak computation step.

#### Gadgets.

$q_{1}$$q_{2}$$q_{3}$$\bar{a}\mathsf{\mathord{+}}^{f}$$\bar{\\#}m_{f}\\#$$\mathsf{\mathord{-}}a$$\mathsf{r}$
(a) $\mathcal{M}_{f}$: Multiplication by $f$.

$q_{1}$$q_{2}$$q_{3}$$\bar{a}^{f}\mathsf{\mathord{+}}$$\bar{\\#}\bar{m}_{f}\\#$$\mathsf{\mathord{-}}a$$\mathsf{r}$
(b) $\bar{\mathcal{M}}_{f}$: Backward multiplication by $f$.

$q_{1}$$q_{2}$$q_{3}$$\bar{a}^{f}\mathsf{\mathord{+}}$$\bar{\\#}d_{f}\\#$$\mathsf{\mathord{-}}a$$\mathsf{r}$
(c) $\mathcal{D}_{f}$: Division by $f$.

$q_{1}$$q_{2}$$q_{3}$$\bar{a}\mathsf{\mathord{+}}^{f}$$\bar{\\#}\bar{d}_{f}\\#$$\mathsf{\mathord{-}}a$$\mathsf{r}$
(d) $\bar{\mathcal{D}}_{f}$: Backward division by $f$.

$q_{1}$$q_{2}$$q_{3}$$\bar{a}^{f}\mathsf{\mathord{+}}^{f}$$\bar{a}^{g}\mathsf{\mathord{+}}^{g}\bar{\\#}t_{f}\\#$$\mathsf{\mathord{-}}a$$\mathsf{r}$
(e) $\mathcal{T}_{f}$: Divisibility test by $f$. The action from $q_{1}$ to
$q_{2}$ exists for all $1\leq g<f$.

$q_{1}$$q_{2}$$q_{3}$$\bar{a}^{f}\mathsf{\mathord{+}}^{f}$$\bar{a}^{g}\mathsf{\mathord{+}}^{g}\bar{\\#}\bar{t}_{f}\\#$$\mathsf{\mathord{-}}a$$\mathsf{r}$
(f) $\bar{\mathcal{T}}_{f}$: Backward divisibility test by $f$. The action
from $q_{1}$ to $q_{2}$ exists for all $1\leq g<f$.

Figure 1. Gadgets used in the reduction.

For each weak computation step, we use one of the gadgets from Fig. 1; note
that, for instance, “$+^{f}$” denotes the sequence of instructions $+\cdots+$
of length $f$. Observe that we have:

$\displaystyle(q_{1},\bot u\\#a^{m},0)$
$\displaystyle\rightarrow^{*}_{\mathcal{M}_{f}}(q_{3},\bot v\\#a^{n},0)$ iff
$\displaystyle v=um_{f}\text{ and }(m,n)\in\overrightarrow{R_{m_{f}}}$ (1)
$\displaystyle(q_{1},\bot u\\#a^{m},0)$
$\displaystyle\rightarrow^{*}_{\bar{\mathcal{M}}_{f}}(q_{3},\bot v\\#a^{n},0)$
iff $\displaystyle u=vm_{f}\text{ and }(n,m)\in\overleftarrow{R_{m_{f}}}$ (2)

and analogous facts hold for $\mathcal{D}_{f}$ and $\bar{\mathcal{D}}_{f}$
(with $d_{f}$ instead of $m_{f}$) and also for $\mathcal{T}_{f}$ and
$\bar{\mathcal{T}}_{f}$ (with $t_{f}$ instead of $m_{f}$). Let us explain this
in the case $\mathcal{M}_{f}$. In the loop at $q_{1}$, $\mathcal{M}_{f}$
removes $a$ from the stack and adds $f$ to the auxiliary counter. When $\\#$
is on top of the stack the automaton moves to $q_{2}$ and changes the stack
from $\bot u\\#$ to $\bot um_{f}\\#$. Therefore, once $\mathcal{M}_{f}$ is in
$q_{2}$, it has set the counter to $f\cdot m$. In the loop at $q_{2}$, it
decrements the counter and pushes $a$ onto the stack before it resets the
counter and moves to $q_{3}$. Thus, in state $q_{3}$, we have $0\leq n\leq
f\cdot m$.

#### Main Control.

Let $M\subseteq\Delta^{*}$ be accepted by the finite automaton
$\mathcal{A}=(\Delta,Q,A,s,t)$. Schematically, our $1$-PRVASS $\mathcal{V}$ is
structured as in the following diagram:

$s^{\prime}$$s$$t$$b$$t^{\prime}$$\bar{\mathcal{M}}_{f}$$\bar{\mathcal{D}}_{f}$$\bar{\mathcal{T}}_{f}$$\bot\\#a$$\bar{a}\bar{\\#}\\#a$$\bar{a}\bar{\\#}\bar{\bot}$

The part in the dashed rectangle is obtained from $\mathcal{A}$ as follows.
Whenever there is an action $(q,m_{f},q^{\prime})$ in $\mathcal{A}$, we glue
in a fresh copy of $\mathcal{M}_{f}$ between $q$ and $q^{\prime}$, including
$\varepsilon$-actions from $q$ to $q_{1}$ and from $q_{3}$ to $q^{\prime}$.
The original action $(q,m_{f},q^{\prime})$ is removed. We proceed analogously
for actions $(q,d_{f},q^{\prime})$ and $(q,t_{f},q^{\prime})$, where we glue
in fresh copies of $\mathcal{D}_{f}$ and $\mathcal{T}_{f}$, respectively.
Clearly, the part in the dashed rectangle realizes the forward phase as
described above.

Once it reaches $t$, $\mathcal{V}$ can check if the current number stored on
the stack equals $1$ and if so, move to state $b$. In state $b$, the backward
phase is implemented. The $1$-PRVASS $\mathcal{V}$ contains a copy of
$\bar{\mathcal{M}}_{f}$, $\bar{\mathcal{D}}_{f}$, and $\bar{\mathcal{T}}_{f}$
for each $f\in\\{2,3\\}$. Each of these copies can be entered from $b$ and
goes back to $b$ when exited.

Finally, the stack is emptied by an action from $b$ to $t^{\prime}$, which can
be taken if and only if the stack content is $\bot\\#a$. We can check that
from $(s^{\prime},\varepsilon,0)$, one can reach a configuration
$(t^{\prime},w,m)$ with $w\in\Gamma^{*}$ and $m\in\mathbb{N}$, if and only if
there exists $u\in M$, $u=x_{1}\cdots x_{\ell}$, and
$x_{1},\ldots,x_{\ell}\in\Delta$, with
$(1,1)\in\overrightarrow{R_{x_{1}}}\cdots\overrightarrow{R_{x_{\ell}}}\cap\overleftarrow{R_{x_{1}}}\cdots\overleftarrow{R_{x_{\ell}}}$.
According to Section 3.1, the latter is equivalent to $(1,1)\in
R_{x_{1}}\cdots R_{x_{\ell}}$.

##  4\. Concluding Remarks

In this note, we have proven the undecidability of coverability in one-
dimensional pushdown VASS with resets (c.f. Theorem 1). The only remaining
open question in Table 1 regarding extensions of two-dimensional VASS is a
long-standing one, namely the reachability problem for one-dimensional PVASS.
Another fruitful research avenue is to pinpoint the exact complexity in the
decidable cases of Table 1. Here, not much is known except regarding
coverability and reachability in two-dimensional VASS: these problems are
-complete if counter updates are encoded in binary [3] and -complete if
updates are encoded in unary [5].

##  Appendix A Proof of Section 3.1

It remains to prove Section 3.1. We will use the following lemma.

######  Lemma 2.

Let $R_{1},\ldots,R_{\ell}\subseteq\mathbb{N}\times\mathbb{N}$ be strictly
monotone relations and
$(m,n)\in\overrightarrow{R_{1}}\cdots\overrightarrow{R_{\ell}}$ and
$(m^{\prime},n^{\prime})\in\overleftarrow{R_{1}}\cdots\overleftarrow{R_{\ell}}$.
If $n^{\prime}\leq n$, then $m^{\prime}\leq m$. Moreover, if $n^{\prime}<n$,
then $m^{\prime}<m$.

###### Proof.

It suffices to prove the Lemma in the case $\ell=1$: then, the general version
follows by induction. Let $(m,n)\in\overrightarrow{R_{1}}$ and
$(m^{\prime},n^{\prime})\in\overleftarrow{R_{1}}$. Then there are
$\tilde{n}\geq n$ with $(m,\tilde{n})\in R_{1}$ and $\tilde{m}\geq m^{\prime}$
with $(\tilde{m},n^{\prime})\in R_{1}$. If $n^{\prime}<n$, then we have the
following relationships:

$\begin{matrix}m&R_{1}&\tilde{n}\\\ &&\rotatebox[origin={c}]{-90.0}{$\geq$}\\\
&&n\\\ &&\rotatebox[origin={c}]{-90.0}{$>$}\\\ \tilde{m}&R_{1}&n^{\prime}\\\
\rotatebox[origin={c}]{-90.0}{$\geq$}&&\\\ m^{\prime}&&\end{matrix}$

Since $R_{1}$ is strictly monotone, this implies $\tilde{m}<m$ and thus
$m^{\prime}<m$. The case $n^{\prime}\leq n$ follows by the same argument.∎

We are now ready to prove Section 3.1.

*

###### Proof.

Of course, for any relation $R\subseteq\mathbb{N}\times\mathbb{N}$, one has
$R\subseteq\overrightarrow{R}$ and $R\subseteq\overleftarrow{R}$. In
particular, $R_{1}R_{2}\cdots R_{\ell}$ is included in both
$\overrightarrow{R_{1}}\overrightarrow{R_{2}}\cdots\overrightarrow{R_{\ell}}$
and
$\overleftarrow{R_{1}}\overleftarrow{R_{2}}\cdots\overleftarrow{R_{\ell}}$.

For the converse inclusion, suppose
$(m,n)\in\overrightarrow{R_{1}}\overrightarrow{R_{2}}\cdots\overrightarrow{R_{\ell}}\cap\overleftarrow{R_{1}}\overleftarrow{R_{2}}\cdots\overleftarrow{R_{\ell}}$.
Then there are $p_{0},\ldots,p_{\ell}\in\mathbb{N}$ with $p_{0}=m$,
$p_{\ell}=n$, and $(p_{i-1},p_{i})\in\overrightarrow{R_{i}}$ for
$0<i\leq\ell$. There are also $q_{0},\ldots,q_{\ell}\in\mathbb{N}$ with
$q_{0}=m$, $q_{\ell}=n$, and $(q_{i-1},q_{i})\in\overleftarrow{R_{i}}$ for
$0<i\leq\ell$.

Towards a contradiction, suppose that $(p_{i-1},p_{i})\notin R_{i}$ for some
$0<i\leq\ell$. Then there is a $\tilde{p}_{i}>p_{i}$ with
$(p_{i-1},\tilde{p}_{i})\in R_{i}$. With this, we have

$\begin{matrix}m=p_{0}&\overrightarrow{R_{1}}\cdots\overrightarrow{R_{i}}&\tilde{p}_{i}&&\\\
&&\rotatebox[origin={c}]{-90.0}{$>$}&&\\\
&&p_{i}&\overrightarrow{R_{i+1}}\cdots\overrightarrow{R_{\ell}}&p_{\ell}\\\
&&&&\rotatebox[origin={c}]{-90.0}{$=$}\\\
m=q_{0}&\overleftarrow{R_{1}}\cdots\overleftarrow{R_{i}}&q_{i}&\overleftarrow{R_{i+1}}\cdots\overleftarrow{R_{\ell}}&q_{\ell}\end{matrix}$

Since $p_{\ell}=q_{\ell}$, Lemma 2 applied to $R_{i+1},\dots,R_{\ell}$ implies
$q_{i}\leq p_{i}$ and thus $q_{i}<\tilde{p}_{i}$. Applying Lemma 2 to
$R_{1},\dots,R_{i}$ then yields $q_{0}<p_{0}$, a contradiction. Therefore, we
have $(p_{i-1},p_{i})\in R_{i}$ for every $0<i\leq\ell$ and thus $(m,n)\in
R_{1}\cdots R_{\ell}$.∎

## Acknowledgements

Work partially funded by ANR-17-CE40-0028 BraVAS.

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