††institutetext: Faculty of Fundamental Sciences, PHENIKAA University, Hanoi
12116, Vietnam

# Enhancing the doubly-longitudinal polarization in $WZ$ production at the LHC

Thi Nhung Dao  Duc Ninh Le<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present new results for the theoretical prediction of doubly-polarized
cross sections of $WZ$ events at the LHC using leptonic decays. Compared to
the previous studies, two new kinematic cuts are considered. These cuts are
designed to enhance the doubly-longitudinal (LL) polarization and, at the same
time, study the Radiation Amplitude Zero effect. We found a new cut on the
rapidity separation between the $Z$ boson and the electron from the $W$ decay
which makes the LL fraction largest, namely $|\Delta y_{Z,e}|<0.5$. This
result is obtained at the next-to-leading order in the strong and electroweak
couplings.

##  1 Introduction

With the new results from ATLAS ATLAS:2022oge  where doubly-polarized cross
sections of the diboson $W^{\pm}Z$ production at the Large Hadron Collider
(LHC) are measured for the first time using the Run-2 data set and that the
LHC Run 3 already began in July 2022, there is a foundation to expect that
more precise measurements of diboson joint-polarization cross sections from
ATLAS and CMS will come soon.

Measuring the doubly-polarized cross sections in diboson production processes
allows for testing the Standard Model (SM) at a deeper level as well as
finding possible new physics effects via polarization observables. Recent
theoretical works to define the signal part of the doubly-polarized cross
sections using the double-pole approximation (DPA) were able to provide
results at the next-to-leading-order (NLO) in QCD for $W^{+}W^{-}$
Denner:2020bcz , $WZ$ Denner:2020eck ; Le:2022lrp ; Le:2022ppa ;
Denner:2022riz , $ZZ$ Denner:2021csi  and at NLO in the electroweak (EW)
interactions for $ZZ$ Denner:2021csi  and $WZ$ Le:2022lrp ; Le:2022ppa . For
the case of $W^{+}W^{-}$, the next-to-next-to-leading-order QCD results are
available Poncelet:2021jmj .

In our previous works Le:2022lrp ; Le:2022ppa  the momenta of the final-state
leptons are selected according to the ATLAS fiducial phase-space cut (named
Cut 1 in this paper) as defined in Refs. Aaboud:2016yus ; ATLAS:2019bsc ;
ATLAS:2022oge . After discussions with experimental colleagues, we realized
that other phase-space cuts (Cut 2 and Cut 3 in this paper) should be explored
as well. These new kinematic setups are designed to enhance the doubly
longitudinal (LL) polarization Franceschini:2017xkh  and observe the Radiation
Amplitude Zero (RAZ) effect Baur:1994ia . The purpose of this work is to
explore these setups in the hope of finding optimal features for the study of
the LL polarization in $WZ$ events.

The paper is organized as follows. We first define the polarized cross
sections in Section 2, before presenting the numerical results in Section 3.
Conclusions are provided in Section 4.

##  2 Definition of polarized cross sections

In order to set up our notations, we briefly review here the definition of the
polarized cross sections. We use the same conventions and calculation setup as
in Le:2022ppa . The process of interest, which is measured at the LHC, reads

$\displaystyle
p(k_{1})+p(k_{2})\to\ell_{1}(k_{3})+\ell_{2}(k_{4})+\ell_{3}(k_{5})+\ell_{4}(k_{6})+X,$
(1)

where the final-state leptons can be either $e^{+}\nu_{e}\mu^{+}\mu^{-}$ or
$e^{-}\bar{\nu}_{e}\mu^{+}\mu^{-}$. Representative Feynman diagrams at leading
order (LO) are depicted in Fig. 1.

Figure 1: Double and single resonant diagrams at leading order. Group a)
includes both double ($WZ$) and single ($W\gamma$) resonant diagrams, while
group b) is only single resonant.

From Fig. 1 we see that the doubly-polarized $WZ$ events, which occur via the
double-resonant $WZ$ diagrams in the group a), are mixed with the single
resonant events. Theoretically, we cannot just select those $WZ$ double
resonant diagrams because they are linked with the single resonant diagrams by
gauge invariance. To separate the $WZ$ events we need to use a special
technique called the double-pole approximation, which selects only the gauge-
invariance part of the $WZ$ double resonant cross section. We note that the
DPA has been widely used in diboson production processes, see Ref.
Denner:2000bj  and references therein.

To be more concrete, the $WZ$ double resonant processes are written as

$\displaystyle p(k_{1})+p(k_{2})\to
V_{1}(q_{1})+V_{2}(q_{2})\to\ell_{1}(k_{3})+\ell_{2}(k_{4})+\ell_{3}(k_{5})+\ell_{4}(k_{6})+X,$
(2)

where the intermediate gauge bosons are $V_{1}=W^{\pm}$, $V_{2}=Z$. Hence the
double-pole unpolarized amplitude at leading order (LO) can be expressed as

$\displaystyle\mathcal{A}_{\text{LO,DPA}}^{\bar{q}q^{\prime}\to V_{1}V_{2}\to
4l}=\frac{1}{Q_{1}Q_{2}}\sum_{\lambda_{1},\lambda_{2}=1}^{3}\mathcal{A}_{\text{LO}}^{\bar{q}q^{\prime}\to
V_{1}V_{2}}(\hat{k}_{i})\mathcal{A}_{\text{LO}}^{V_{1}\to\ell_{1}\ell_{2}}(\hat{k}_{i})\mathcal{A}_{\text{LO}}^{V_{2}\to\ell_{3}\ell_{4}}(\hat{k}_{i}),$
(3)

with

$\displaystyle
Q_{j}=q_{j}^{2}-M_{V_{j}}^{2}+iM_{V_{j}}\Gamma_{V_{j}}\,(j=1,2),$ (4)

where $q_{1}=k_{3}+k_{4}$, $q_{2}=k_{5}+k_{6}$, $M_{V}$ and $\Gamma_{V}$ are
the physical mass and width of the gauge boson $V$, and $\lambda_{j}$ are the
polarization indices of the gauge bosons. Note that the helicity indices of
the initial quarks and final leptons are implicit. It is crucial that all
helicity amplitudes $\mathcal{A}$ in the r.h.s. are calculated using on-shell
(OS) momenta $\hat{k}_{i}$ for the final-state leptons as well as OS momenta
$\hat{q}_{j}$ for the intermediate gauge bosons, derived from the off-shell
(full process) momenta $k_{i}$ and $q_{j}$, in order to ensure that gauge
invariance in the amplitudes is preserved. An OS mapping is used to obtain the
OS momenta $\hat{k}_{i}$ from the off-shell momenta $k_{i}$. This OS mapping
is not unique, however the shift induced by different mappings is of order
$\alpha\Gamma_{V}/(\pi M_{V})$ Denner:2000bj . The OS mapping used in this
paper is the same as in Ref. Le:2022ppa .

Eq. (3) serves as the master equation to define the doubly polarized cross
sections. Since a massive gauge boson has three physical polarization states:
two transverse states $\lambda=1$ and $\lambda=3$ (left and right) and one
longitudinal state $\lambda=2$, the $WZ$ system has in total $9$ polarization
states. The unpolarized amplitude defined in Eq. (3) is the sum of these $9$
polarized amplitudes. The unpolarized cross section is then divided into the
following five terms:

  * •

$W_{L}Z_{L}$: The longitudinal-longitudinal (LL) contribution, obtained with
selecting $\lambda_{1}=\lambda_{2}=2$ in the sum of Eq. (3);

  * •

$W_{L}Z_{T}$: The longitudinal-transverse (LT) contribution, obtained with
selecting $\lambda_{1}=2$, $\lambda_{2}=1,3$. The LT cross section includes
the interference term between the $(21)$ and $(23)$ amplitudes.

  * •

$W_{T}Z_{L}$: The transverse-longitudinal (TL) contribution, obtained with
selecting $\lambda_{1}=1,3$, $\lambda_{2}=2$. The interference between the
$(12)$ and $(32)$ amplitudes is here included.

  * •

$W_{T}Z_{T}$: The transverse-transverse (TT) contribution, obtained with
selecting $\lambda_{1}=1,3$, $\lambda_{2}=1,3$. The interference terms between
the $(11)$, $(13)$, $(31)$, $(33)$ amplitudes are here included.

  * •

Interference: This includes the interference terms between the above LL, LT,
TL, TT amplitudes.

Our doubly-polarized cross-section results include not only the leading order
but also the NLO QCD and EW corrections. The expressions for double-pole
unpolarized amplitudes need to be extended to include also the virtual
corrections, the gluon/photon induced and radiation processes as done in Ref.
Le:2022ppa . In this short writing the above LO definition of the polarized
cross sections is enough for the reader to understand the numerical results
discussed in the next section.

##  3 Numerical results

Our numerical results are obtained for proton-proton collisions at a center-
of-mass energy of $13$ TeV. Fixed factorization and renormalization scales are
used, namely $\mu_{F}=\mu_{R}=\mu_{0}=(M_{W}+M_{Z})/2$, where
$M_{W}=80.385\,{\,\text{GeV}}$ and $M_{Z}=91.1876\,{\,\text{GeV}}$. For the
parton distribution functions (PDF) and value of the strong coupling constant,
the Hessian set LUXqed17_plus_PDF4LHC15_nnlo_30 Watt:2012tq ; Gao:2013bia ;
Harland-Lang:2014zoa ; Ball:2014uwa ; Butterworth:2015oua ; Dulat:2015mca ;
deFlorian:2015ujt ; Carrazza:2015aoa ; Manohar:2016nzj ; Manohar:2017eqh  via
the library LHAPDF6 Buckley:2014ana  is employed. More details about other
input parameters are provided in Ref. Le:2022ppa .

For NLO EW corrections, an additional photon can be emitted. Hence, the
lepton-photon recombination to define a dressed lepton is done before applying
the analysis cuts. A dressed lepton has the momentum of
$p^{\prime}_{\ell}=p_{\ell}+p_{\gamma}$ if the angular distance $\Delta
R(\ell,\gamma)\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}<0.1$, i.e. when
the photon is close enough to the bare lepton. Here the letter $\ell$ denotes
$e$ or $\mu$ and all momenta are calculated in the Lab frame. All leptons and
quarks except for the top quark are approximated as massless.

The doubly polarized cross sections and distributions depend on the reference
frame. We choose the $WZ$ center-of-mass frame, the same as in the ATLAS
measurement ATLAS:2022oge . We now specify the three cut setups used in this
paper. They read as follows.

#### Cut 1:

The baseline setup, called Cut 1, is the ATLAS fiducial set of cuts used in
Refs. Aaboud:2016yus ; ATLAS:2019bsc ; ATLAS:2022oge , which reads

$\displaystyle p_{T,e}>20{\,\text{GeV}},\quad
p_{T,\mu^{\pm}}>15{\,\text{GeV}},\quad|\eta_{\ell}|<2.5,$ $\displaystyle\Delta
R\left(\mu^{+},\mu^{-}\right)>0.2,\quad\Delta R\left(e,\mu^{\pm}\right)>0.3,$
(5) $\displaystyle
m_{T,W}>30{\,\text{GeV}},\quad\left|m_{\mu^{+}\mu^{-}}-M_{Z}\right|<10{\,\text{GeV}}\,,$

where $m_{T,W}=\sqrt{2p_{T,\nu}p_{T,e}[1-\cos\Delta\phi(e,\nu)]}$ with
$\Delta\phi(e,\nu)$ being the angle between the electron and the neutrino in
the transverse plane. This Cut 1 was used in our previous studies Le:2022lrp ;
Le:2022ppa .

#### Cut 2:

In addition to the cuts in Cut 1, we further require that the transverse
momentum of the $WZ$ system satisfies Franceschini:2017xkh

$\displaystyle p_{T,WZ}<70{\,\text{GeV}}.$ (6)

At NLO, this additional cut affects only the real-emission contributions with
an extra particle in the final state. The LO term and virtual corrections are
unaffected. The purpose of this cut is to observe the RAZ in the TT component,
which is smeared out by QCD radiation Baur:1994ia ; Franceschini:2017xkh .
This cut reduces higher-order QCD corrections. Dominant backgrounds, in
particular $t\bar{t}$, $t\bar{t}V$, $VVV$ are expected to decrease
significantly by this cut as well.

#### Cut 3:

In addition to the cuts in Cut 2, we further require Franceschini:2017xkh

$\displaystyle p_{T,Z}>200{\,\text{GeV}}.$ (7)

This additional cut reduces drastically the LO contribution as well as all NLO
corrections. The purpose of this cut is to focus more on the high energy
regime where new physics effects are expected to be present. As will be seen,
this cut will increase the LL fraction significantly.

###  3.1 Integrated polarized cross sections

  * | $\sigma_{\text{LO}}\,\text{[fb]}$ | $f_{\text{LO}}\,\text{[\%]}$ | $\sigma^{\text{EW}}_{\text{NLO}}\,\text{[fb]}$ | $f^{\text{EW}}_{\text{NLO}}\,\text{[\%]}$ | $\sigma^{\text{QCD}}_{\text{NLO}}\,\text{[fb]}$ | $f^{\text{QCD}}_{\text{NLO}}\,\text{[\%]}$ | $\sigma^{\text{QCDEW}}_{\text{NLO}}\,\text{[fb]}$ | $f^{\text{QCDEW}}_{\text{NLO}}\,\text{[\%]}$ | $\bar{\delta}_{\text{EW}}\,\text{[\%]}$
---|---|---|---|---|---|---|---|---|---
Unpol., Cut 1 | $18.934(1)^{+4.8\%}_{-5.9\%}$ | $100$ | $18.138(1)^{+4.9\%}_{-6.0\%}$ | $100$ | $34.071(2)^{+5.3\%}_{-4.2\%}$ | $100$ | $33.275(2)^{+5.4\%}_{-4.3\%}$ | $100$ | $-2.3$
Cut 2 | $18.934(1)^{+4.8\%}_{-5.9\%}$ | $100$ | $17.897(1)^{+4.9\%}_{-6.0\%}$ | $100$ | $25.860(3)^{+3.2\%}_{-2.5\%}$ | $100$ | $24.823(3)^{+3.4\%}_{-2.6\%}$ | $100$ | $-4.0$
Cut 3 | $0.392^{+1.4\%}_{-1.8\%}$ | $100$ | $0.343_{-1.5\%}^{+1.1\%}$ | $100$ | $0.445_{-1.5\%}^{+2.2\%}$ | $100$ | $0.396_{-1.2\%}^{+2.0\%}$ | $100$ | $-11.0$
$W^{+}_{L}Z_{L}$, Cut 1 | $1.492^{+5.1\%}_{-6.3\%}$ | $7.9$ | $1.428^{+5.2\%}_{-6.4\%}$ | $7.9$ | $1.938^{+2.7\%}_{-2.2\%}$ | $5.7$ | $1.874^{+2.8\%}_{-2.3\%}$ | $5.6$ | $-3.3$
Cut 2 | $1.492^{+5.1\%}_{-6.3\%}$ | $7.9$ | $1.420^{+5.3\%}_{-6.4\%}$ | $7.9$ | $1.786^{+1.9\%}_{-2.3\%}$ | $6.9$ | $1.714^{+2.0\%}_{-2.2\%}$ | $6.9$ | $-4.0$
Cut 3 | $0.105_{-0.7\%}^{+0.0\%}$ | $26.7$ | $0.092_{-0.5\%}^{+0.0\%}$ | $26.9$ | $0.100_{-0.6\%}^{+0.8\%}$ | $22.6$ | $0.088_{-1.0\%}^{+1.2\%}$ | $22.2$ | $-13.0$
$W^{+}_{L}Z_{T}$, Cut 1 | $2.018^{+5.8\%}_{-7.0\%}$ | $10.7$ | $1.951^{+5.8\%}_{-7.0\%}$ | $10.8$ | $5.273^{+7.3\%}_{-5.9\%}$ | $15.5$ | $5.207^{+7.4\%}_{-6.0\%}$ | $15.6$ | $-1.3$
Cut 2 | $2.018^{+5.8\%}_{-7.0\%}$ | $10.7$ | $1.928^{+5.8\%}_{-7.0\%}$ | $10.8$ | $3.419^{+4.9\%}_{-3.8\%}$ | $13.2$ | $3.329^{+5.1\%}_{-3.9\%}$ | $13.4$ | $-2.6$
Cut 3 | $0.017_{-0.4\%}^{+0.0\%}$ | $4.4$ | $0.016_{-0.5\%}^{+0.0\%}$ | $4.8$ | $0.023_{-2.9\%}^{+3.9\%}$ | $5.1$ | $0.022_{-2.9\%}^{+3.9\%}$ | $5.5$ | $-4.3$
$W^{+}_{T}Z_{L}$, Cut 1 | $1.903^{+5.7\%}_{-6.9\%}$ | $10.1$ | $1.893^{+5.7\%}_{-6.9\%}$ | $10.4$ | $5.024^{+7.4\%}_{-5.9\%}$ | $14.7$ | $5.013^{+7.4\%}_{-5.9\%}$ | $15.1$ | $-0.2$
Cut 2 | $1.903^{+5.7\%}_{-6.9\%}$ | $10.1$ | $1.826^{+5.8\%}_{-7.0\%}$ | $10.2$ | $3.281^{+5.0\%}_{-4.0\%}$ | $12.7$ | $3.204^{+5.1\%}_{-4.1\%}$ | $12.9$ | $-2.3$
Cut 3 | $0.017_{-0.4\%}^{+0.0\%}$ | $4.3$ | $0.017_{-0.5\%}^{+0.0\%}$ | $4.9$ | $0.021_{-2.5\%}^{+3.1\%}$ | $4.6$ | $0.020_{-2.3\%}^{+3.0\%}$ | $5.2$ | $0.0$
$W^{+}_{T}Z_{T}$, Cut 1 | $13.376^{+4.5\%}_{-5.6\%}$ | $70.6$ | $12.728(1)^{+4.6\%}_{-5.7\%}$ | $70.2$ | $21.626(2)^{+4.5\%}_{-3.6\%}$ | $63.5$ | $20.977(2)^{+4.7\%}_{-3.8\%}$ | $63.0$ | $-3.0$
Cut 2 | $13.376^{+4.5\%}_{-5.6\%}$ | $70.6$ | $12.587(1)^{+4.5\%}_{-5.7\%}$ | $70.3$ | $17.132(2)^{+2.6\%}_{-2.1\%}$ | $66.2$ | $16.342(2)^{+2.7\%}_{-2.2\%}$ | $65.8$ | $-4.6$
Cut 3 | $0.247_{-2.5\%}^{+2.2\%}$ | $63.1$ | $0.212_{-2.1\%}^{+1.9\%}$ | $61.9$ | $0.297_{-2.1\%}^{+3.1\%}$ | $66.7$ | $0.262_{-1.7\%}^{+3.0\%}$ | $66.1$ | $-11.8$
Inter., Cut 1 | $0.144(1)$ | $0.8$ | $0.138(1)$ | $0.8$ | $0.210(3)$ | $0.6$ | $0.204(3)$ | $0.6$ | $-2.9$
Cut 2 | $0.144(1)$ | $0.8$ | $0.137(1)$ | $0.8$ | $0.242(3)$ | $0.9$ | $0.235(3)$ | $0.9$ | $-2.9$
Cut 3 | $0.006$ | $1.5$ | $0.005$ | $1.6$ | $0.005$ | $1.0$ | $0.004$ | $1.0$ | $-20.0$

Table 1: Unpolarized (Unpol.) and doubly polarized cross sections in fb
together with polarization fractions calculated at LO, NLO EW, NLO QCD, and
NLO QCD+EW, all in the DPA, in the $WZ$ center-of-mass system for the $W^{+}Z$
process. The interference (Inter.) between the polarized amplitudes is
provided in the bottom row. The statistical uncertainties (in parenthesis) are
given on the last digits of the central prediction when significant. Seven-
point scale uncertainty is also provided for the cross sections as sub- and
superscripts in percent. In the last column the EW correction relative to the
NLO QCD prediction is given.

  * | $\sigma_{\text{LO}}\,\text{[fb]}$ | $f_{\text{LO}}\,\text{[\%]}$ | $\sigma^{\text{EW}}_{\text{NLO}}\,\text{[fb]}$ | $f^{\text{EW}}_{\text{NLO}}\,\text{[\%]}$ | $\sigma^{\text{QCD}}_{\text{NLO}}\,\text{[fb]}$ | $f^{\text{QCD}}_{\text{NLO}}\,\text{[\%]}$ | $\sigma^{\text{QCDEW}}_{\text{NLO}}\,\text{[fb]}$ | $f^{\text{QCDEW}}_{\text{NLO}}\,\text{[\%]}$ | $\bar{\delta}_{\text{EW}}\,\text{[\%]}$
---|---|---|---|---|---|---|---|---|---
Unpol., Cut 1 | $12.745^{+4.9\%}_{-6.2\%}$ | $100$ | $12.224^{+5.1\%}_{-6.3\%}$ | $100$ | $23.705(1)^{+5.5\%}_{-4.4\%}$ | $100$ | $23.184(1)^{+5.6\%}_{-4.5\%}$ | $100$ | $-2.2$
Cut 2 | $12.745^{+4.9\%}_{-6.2\%}$ | $100$ | $12.060(1)_{-6.3\%}^{+5.1\%}$ | $100$ | $17.905(2)_{-2.7\%}^{+3.4\%}$ | $100$ | $17.221(2)_{-2.8\%}^{+3.5\%}$ | $100$ | $-3.8$
Cut 3 | $0.209_{-1.8\%}^{+1.3\%}$ | $100$ | $0.184_{-1.5\%}^{+1.0\%}$ | $100$ | $0.259_{-2.7\%}^{+3.3\%}$ | $100$ | $0.234_{-2.5\%}^{+3.2\%}$ | $100$ | $-9.7$
$W^{-}_{L}Z_{L}$, Cut 1 | $1.094^{+5.2\%}_{-6.5\%}$ | $8.6$ | $1.048^{+5.3\%}_{-6.6\%}$ | $8.6$ | $1.407^{+2.6\%}_{-2.1\%}$ | $5.9$ | $1.361^{+2.7\%}_{-2.2\%}$ | $5.9$ | $-3.3$
Cut 2 | $1.094^{+5.2\%}_{-6.5\%}$ | $8.6$ | $1.043_{-6.6\%}^{+5.3\%}$ | $8.6$ | $1.308_{-2.3\%}^{+1.9\%}$ | $7.3$ | $1.257_{-2.3\%}^{+2.0\%}$ | $7.3$ | $-3.9$
Cut 3 | $0.059_{-0.8\%}^{+0.1\%}$ | $28.4$ | $0.052_{-0.6\%}^{+0.0\%}$ | $28.5$ | $0.057_{-0.4\%}^{+0.7\%}$ | $22.2$ | $0.051_{-0.7\%}^{+1.1\%}$ | $21.6$ | $-12.3$
$W^{-}_{L}Z_{T}$, Cut 1 | $1.508^{+5.8\%}_{-7.0\%}$ | $11.8$ | $1.456^{+5.9\%}_{-7.1\%}$ | $11.9$ | $3.921^{+7.3\%}_{-5.9\%}$ | $16.5$ | $3.869^{+7.4\%}_{-6.0\%}$ | $16.7$ | $-1.3$
Cut 2 | $1.508^{+5.8\%}_{-7.0\%}$ | $11.8$ | $1.440_{-7.1\%}^{+5.8\%}$ | $11.9$ | $2.605_{-4.0\%}^{+5.0\%}$ | $14.5$ | $2.536_{-4.1\%}^{+5.2\%}$ | $14.7$ | $-2.6$
Cut 3 | $0.010_{-0.5\%}^{+0.0}$ | $4.8$ | $0.010_{-0.4\%}^{+0.0\%}$ | $5.2$ | $0.015_{-4.2\%}^{+5.3\%}$ | $5.8$ | $0.014_{-4.2\%}^{+5.4\%}$ | $6.2$ | $0.0$
$W^{-}_{T}Z_{L}$, Cut 1 | $1.356^{+5.8\%}_{-7.0\%}$ | $10.6$ | $1.347^{+5.8\%}_{-7.0\%}$ | $11.0$ | $3.606^{+7.4\%}_{-6.0\%}$ | $15.2$ | $3.597^{+7.4\%}_{-6.0\%}$ | $15.5$ | $-0.2$
Cut 2 | $1.356^{+5.8\%}_{-7.0\%}$ | $10.6$ | $1.302_{-7.1\%}^{+5.9\%}$ | $10.8$ | $2.375_{-4.1\%}^{+5.1\%}$ | $13.3$ | $2.322_{-4.2\%}^{+5.2\%}$ | $13.5$ | $-2.3$
Cut 3 | $0.010_{-0.5\%}^{+0.0}$ | $4.7$ | $0.010_{-0.4\%}^{+0.0\%}$ | $5.2$ | $0.012_{-2.5\%}^{+3.9\%}$ | $4.7$ | $0.012_{-2.3\%}^{+3.7\%}$ | $5.1$ | $0.0$
$W^{-}_{T}Z_{T}$, Cut 1 | $8.833^{+4.6\%}_{-5.8\%}$ | $69.3$ | $8.416^{+4.8\%}_{-5.9\%}$ | $68.8$ | $14.664(1)^{+4.7\%}_{-3.8\%}$ | $61.9$ | $14.247(1)^{+4.9\%}_{-3.9\%}$ | $61.5$ | $-2.8$
Cut 2 | $8.833^{+4.6\%}_{-5.8\%}$ | $69.3$ | $8.321_{-6.0\%}^{+4.8\%}$ | $69.0$ | $11.549(1)_{-2.2\%}^{+2.8\%}$ | $64.5$ | $11.037(1)_{-2.3\%}^{+2.9\%}$ | $64.1$ | $-4.4$
Cut 3 | $0.129_{-2.5\%}^{2.2\%}$ | $61.8$ | $0.111_{-2.2\%}^{+1.7\%}$ | $60.7$ | $0.174_{-3.6\%}^{+4.8\%}$ | $67.1$ | $0.156_{-3.5\%}^{+4.8\%}$ | $66.8$ | $-10.3$
Inter., Cut 1 | $-0.046(1)$ | $-0.4$ | $-0.043(1)$ | $-0.4$ | $0.107(2)$ | $0.5$ | $0.110(2)$ | $0.5$ | $+2.8$
Cut 2 | $-0.046(1)$ | $-0.4$ | $-0.045(1)$ | $-0.4$ | $0.068(2)$ | $0.4$ | $0.069(2)$ | $0.4$ | $+1.5$
Cut 3 | $0.001$ | $0.4$ | $0.001$ | $0.4$ | $0.001$ | $0.3$ | $0.001$ | $0.3$ | $0.0$

Table 2: Same as Table 1 but for the $W^{-}Z$ process.

We first present results for the integrated cross sections at LO, NLO QCD, NLO
EW, and NLO QCD+EW for the unpolarized case, LL, LT, TL, TT polarizations and
the interference in Table 1 for the case of $W^{+}Z$ and in Table 2 for
$W^{-}Z$ for all three cut setups. The results for Cut 1 have already been
published in Refs. Le:2022lrp ; Le:2022ppa . They are re-provided here for the
sake of comparison.

Included in the two tables are also the polarization fractions, $f$,
calculated as ratios of the polarized cross sections over the unpolarized
cross section at each level of accuracy. The total EW correction relative to
the NLO QCD prediction is defined as

$\bar{\delta}_{\text{EW}}=(\sigma_{\text{NLO}}^{\text{QCDEW}}-\sigma_{\text{NLO}}^{\text{QCD}})/\sigma_{\text{NLO}}^{\text{QCD}}.$
(8)

This information is shown in the last column. Statistical errors are very
small and shown in a few places where they are significant. Scale
uncertainties are much bigger and are provided for the cross sections as sub-
and superscripts in percent. These uncertainties are calculated using the
seven-point method where the two scales $\mu_{F}$ and $\mu_{R}$ are varied as
$n\mu_{0}/2$ with $n=1,2,4$ and $\mu_{0}=(M_{W}+M_{Z})/2$ being the central
scale. Additional constraint $1/2\leq\mu_{R}/\mu_{F}\leq 2$ is used to limit
the number of scale choices to seven at NLO QCD. The cases
$\mu_{R}/\mu_{F}=1/4$ or $4$ are excluded, being considered too extreme.

From the tables, we see that the veto cut of $p_{T,WZ}<70$ GeV reduces the NLO
QCD+EW unpolarized cross section by around $25.5\%$ for both processes, which
is almost entirely due to the reduction of the QCD correction. This reduction
is however not equally distributed among different polarizations. For the
$W^{+}Z$ channel, they are $-8.5\%$, $-36.1$, $-36.1\%$, $-22.1\%$ for the
$W_{L}Z_{L}$, $W_{L}Z_{T}$, $W_{T}Z_{L}$, $W_{T}Z_{T}$, respectively. The
corresponding numbers for the $W^{-}Z$ case are $-7.6\%$, $-34.5\%$,
$-35.4\%$, $-22.5\%$. One notices that, for both processes, the $LL$ component
is least reduced while the mixed polarizations $LT$ and $TL$ are most affected
by the veto cut. This is reflected in the polarization fractions. The $f_{LL}$
is increased from $5.6\%$ ($5.9\%$) to $6.9\%$ ($7.3\%$) for the $W^{+}Z$
($W^{-}Z$) at NLO QCD+EW. Both $f_{LT},f_{TL}$ decrease two percent, but the
doubly-tranverse polarization fraction increases from $63.0\%$ ($61.5\%$) to
$65.8\%$ ($64.1\%$).

Moving to Cut 3, we see that the integrated cross sections are drastically
reduced, by around $99\%$ compared to Cut 1, for the unpolarized case and for
both processes. With $139\text{fb}^{-1}$ data, the numbers of signal events
for Cut 1 are $1190$, $1900$, $3100$, $10900$ for the LL, LT, TL, TT
polarizations, summing over the two processes, as shown in Table 1 (left) of
ATLAS:2022oge . Observing that the reduction is not uniform for different
polarizations, the corresponding results for Cut 3 are obtained as $51$, $8$,
$11$, $129$. These numbers will increased greatly when Run-3 data is added to
the analysis in the near future. The purpose of Cut 3 is to enhance the LL
fraction and this can be seen clearly in the tables. The $f_{LL}$ now reads
$22.2\%$ ($21.6\%$) at NLO QCD+EW for $W^{+}Z$ ($W^{-}Z$) process, being
ranked second after the TT fraction.

Concerning the EW corrections, they are all negative (except for the
interference) and their absolute values are all smaller than $5\%$ for Cut 1
and Cut 2. For Cut 3, they are greater than $10\%$ for the LL and TT cases,
signifying the importance of the EW corrections for the future measurements of
the LL fraction.

###  3.2 Kinematic distributions

In Fig. 2 ($W^{+}Z$) and Fig. 3 ($W^{-}Z$) we present new results for Cut 2
and Cut 3 on the distributions of rapidity separation between the positron (or
electron) and the $Z$ boson directions. The plots for Cut 1, already shown in
Fig. 5 of Ref. Le:2022ppa , are displayed here for the sake of comparison.

Comparing Cut 1 and Cut 2, we see that the above reduction in the cross
section comes from the phase space region of $|\Delta y_{Z,e}|\approx 0$,
where the TT, TL, and LT are most affected, while the LL changes slightly. The
TL and LT cross sections are still larger than the LL one, but the difference
is small.

Moving to Cut 3, we find, very surprisingly, that the LL cross section is
largest when the rapidity separation is smaller than $0.3$ for the $W^{+}Z$
case. The value is $0.1$ for the $W^{-}Z$ process. Since the LL cross section
is maximal at zero separation while the TT one is maximal at around $1.2$, we
can further suppress the TT events by imposing an additional cut on the
rapidity separation, namely $|\Delta y_{Z,e}|<\Delta y_{\text{cut}}$ with
$\Delta y_{\text{cut}}$ in the range $[0.5,1]$. Results for various values of
$\Delta y_{\text{cut}}$ are shown in Table 3. There, in the parentheses, we
also provide the acceptance $A$, defined as the ratio of the cross section
after applying the $\Delta y_{Z,e}$ cut with respect to the one before
applying this cut. From the table, we observe that the LL cross section is
largest when $\Delta y_{\text{cut}}$ smaller than $0.6$ for the $W^{+}Z$
process, and $0.4$ for the $W^{-}Z$ case. For the combined set of events,
choosing $\Delta y_{\text{cut}}=0.5$ guarantees that the LL fraction is
dominant with an acceptance of $53\%$. We are confident that this can be done
for the Run-3 data set.

---
Figure 2: Distributions in the rapidity separation (in absolute value) between
the positron and the $Z$ boson for Cut 1 (top left), Cut 2 (top right), and
Cut 3 (bottom) of the $W^{+}Z$ process. The big panel shows the absolute
values of the cross sections at NLO QCD+EW. The middle-up panel displays the
ratio of the NLO QCD cross sections to the corresponding LO ones. The middle-
down panel shows $\bar{\delta}_{\text{EW}}$, the EW corrections relative to
the NLO QCD cross sections, in percent. In the bottom panel, the normalized
shapes of the distributions are plotted to highlight differences in shape.

---
Figure 3: Same as Fig. 2 but for the $W^{-}Z$ process.

  * | $W^{+}Z$ | $W^{-}Z$
---|---|---
$\Delta y_{\text{cut}}$ | $\sigma_{\text{TT}}\,\text{[fb](A[\%])}$ | $\sigma_{\text{LL}}\,\text{[fb](A[\%])}$ | $\sigma_{\text{LT}}\,\text{[fb](A[\%])}$ | $\sigma_{\text{TL}}\,\text{[fb](A[\%])}$ | $\sigma_{\text{TT}}\,\text{[fb](A[\%])}$ | $\sigma_{\text{LL}}\,\text{[fb](A[\%])}$ | $\sigma_{\text{LT}}\,\text{[fb](A[\%])}$ | $\sigma_{\text{TL}}\,\text{[fb](A[\%])}$
$0.1$ | $0.008(3.0)$ | $0.010(11.4)$ | $0.002(7.2)$ | $0.001(6.0)$ | $0.005(3.4)$ | $0.006(11.4)$ | $0.001(7.2)$ | $0.001(6.0)$
$0.2$ | $0.016(6.0)$ | $0.020(22.6)$ | $0.003(14.4)$ | $0.002(12.1)$ | $0.011(6.9)$ | $0.011(22.6)$ | $0.002(14.5)$ | $0.001(12.0)$
$0.3$ | $0.024(9.1)$ | $0.029(33.3)$ | $0.005(21.6)$ | $0.004(18.4)$ | $0.016(10.4)$ | $0.017(33.3)$ | $0.003(21.7)$ | $0.002(18.2)$
$0.4$ | $0.033(12.4)$ | $0.038(43.3)$ | $0.006(28.7)$ | $0.005(24.7)$ | $0.022(14.1)$ | $0.022(43.3)$ | $0.004(28.8)$ | $0.003(24.6)$
$0.5$ | $0.042(16.0)$ | $0.046(52.5)$ | $0.008(35.7)$ | $0.006(31.3)$ | $0.028(18.0)$ | $0.027(52.5)$ | $0.005(35.8)$ | $0.004(31.1)$
$0.6$ | $0.052(19.7)$ | $0.053(60.7)$ | $0.009(42.6)$ | $0.008(37.9)$ | $0.034(22.1)$ | $0.031(60.8)$ | $0.006(42.7)$ | $0.005(37.7)$
$0.7$ | $0.062(23.7)$ | $0.060(68.0)$ | $0.011(49.3)$ | $0.009(44.5)$ | $0.041(26.3)$ | $0.034(68.0)$ | $0.007(49.3)$ | $0.005(44.3)$
$0.8$ | $0.073(28.0)$ | $0.065(74.2)$ | $0.012(55.6)$ | $0.010(51.0)$ | $0.048(30.8)$ | $0.038(74.3)$ | $0.008(55.6)$ | $0.006(50.9)$
$0.9$ | $0.085(32.4)$ | $0.070(79.5)$ | $0.013(61.6)$ | $0.012(57.3)$ | $0.055(35.3)$ | $0.040(79.7)$ | $0.009(61.5)$ | $0.007(57.2)$
$1.0$ | $0.097(37.0)$ | $0.074(84.0)$ | $0.015(67.1)$ | $0.013(63.2)$ | $0.063(40.1)$ | $0.043(84.1)$ | $0.010(67.0)$ | $0.008(63.1)$

Table 3: NLO QCD+EW cross sections using the combination of Cut 3 and $|\Delta
y_{Z,e}|<\Delta y_{\text{cut}}$ for various values of $\Delta y_{\text{cut}}$,
separately for the $W^{+}Z$ (left) and $W^{-}Z$ (right) processes. The
acceptance $A$, the percentage of the cross section after applying the $\Delta
y_{Z,e}$ cut, is provided in the parentheses.

##  4 Conclusions

We have presented new results for doubly-polarized cross sections of the $WZ$
production with fully leptonic decays at the LHC. Compared to the previous
studies, two new kinematic cut setups are considered. These cuts are designed
to observe the Radiation Amplitude Zero effect and to enhance the LL
polarization. Our results show that the new cuts suppress the mixed
polarizations drastically. For Cut 3, where $p_{T,WZ}<70$ GeV and
$p_{T,Z}>200$ GeV, the dominant polarization is the TT with $f_{TT}\approx
66\%$ and the second one being the LL with $f_{LL}\approx 22\%$. We found that
one can suppress the TT to make the LL dominant by imposing a new cut of
$|\Delta y_{Z,e}|<0.5$. The nice feature of this cut is that the cross
sections of all polarizations are large in the selected phase-space region,
therefore we still have enough number of events for the analysis.

###### Acknowledgements.

We thank our experimental colleagues, in particular Joany Manjarrés and Junjie
Zhu, for useful discussions. This research is funded by the Vietnam National
Foundation for Science and Technology Development (NAFOSTED) under grant
number 103.01-2020.17.

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# Distributions of consecutive level spacings of GUE and their ratio: an
analytic treatment

Shinsuke M. Nishigaki<EMAIL_ADDRESS>Graduate School of
Natural Science and Technology, Shimane University, Matsue 690-8504, Japan

###### Abstract

In recent studies of many-body localization in nonintegrable quantum systems,
the distribution of the ratio of two consecutive energy level spacings,
$r_{n}=(E_{n+1}-E_{n})/(E_{n}-E_{n-1})$ or
$\tilde{r}_{n}=\min(r_{n},r_{n}^{-1})$, has been used as a measure to quantify
the chaoticity, alternative to the more conventional distribution of the level
spacings, $s_{n}=\bar{\rho}(E_{n})(E_{n+1}-E_{n})$, as the former
unnecessitates the unfolding required for the latter. Based on our previous
work on the Tracy-Widom approach to the Jánossy densities, we present analytic
expressions for the joint probability distribution of two consecutive
eigenvalue spacings and the distribution of their ratio for the Gaussian
unitary ensemble (GUE) of random Hermitian $N\times N$ matrices at
$N\to\infty$, in terms of a system of differential equations. As a showcase of
efficacy of our results for characterizing an approach to quantum chaoticity,
we contrast them to arguably the most ideal of all quantum-chaotic spectra:
the zeroes of the Riemann $\zeta$ function on the critical line at increasing
heights.

A10,A13,A32,B83,B86

##  1 Introduction

Many-body localization that prohibits thermal equilibration of the wave
functions has been a core agenda of research in the field of quantum many-body
systems, including disordered and interacting fermions on a chain
Oganesyan:2007 , spin chains with transverse field, periodical kicks, or
disorder Atas:2013 ; Kim:2013 ; DAlessio:2014 ; Luitz:2015 , and the Sachdev-
Ye-Kitaev model and its deformed, coupled, or sparsed variant You:2017 ;
Garcia:2018 ; Garcia:2019 ; Garcia:2021 , to name but the few. In all of the
above studies, the ‘gap ratio distribution’, i.e. the distribution
$P_{\mathrm{r}}(r)$ of ratio of consecutive energy level spacings
$r_{n}=(E_{n+1}-E_{n})/(E_{n}-E_{n-1})$ or
$\tilde{r}_{n}=\min(r_{n},r_{n}^{-1})$, initiated by Ref. Oganesyan:2007 , has
been utilized as a criteria of (non)ergodicity. This trend is obviously due to
its computational advantage that unnecessitates the unfolding by the smoothed
density of states $\bar{\rho}(E)$, over the conventional distribution $P(s)$
of level spacings $s_{n}=\bar{\rho}(E_{n})(E_{n+1}-E_{n})$. More recently, the
use of the gap ratio distribution for characterizing spectral transitions
extended its reach beyond energy level statistics of many-body or chaotic
Hamiltonians, namely towards quantum entanglement spectra of reduced density
matrices in neural network states Deng:2017  and in quantum circuits
Zhang:2020 ; Jian:2022 .

The random matrix theory of the gap ratio distribution was introduced in Ref.
Atas:2013  and has since been quoted in practically every work in these fields
including Refs. Kim:2013 ; DAlessio:2014 ; Luitz:2015 ; You:2017 ; Garcia:2018
; Garcia:2019 ; Garcia:2021 ; Deng:2017 ; Zhang:2020 ; Jian:2022 . There, two
types of approximate expressions for $P_{\mathrm{r}}(r)$ were presented: one
from the Wigner-like surmise (substituting the large-$N$ limit of $N\times N$
matrices by $3\times 3$) for all three Dyson symmetry classes,
$P_{\mathrm{r}}(r)\approx C_{\beta}(r+r^{2})^{\beta}/(1+r+r^{2})^{1+3\beta/2}\
(\beta=1,2,4)$, and another from the quadrature discretization of the
resolvent operator
$\boldsymbol{\mathit{K}}_{I}(\mathbb{I}-\boldsymbol{\mathit{K}}_{I})^{-1}$
where $\boldsymbol{\mathit{K}}_{I}$ denotes the integral operator of
convolution with the sine kernel over an interval $I$ for the unitary class
$\beta=2$. Although the latter approximation is known to converge to the exact
value quickly as the quadrature order is increased Bornemann:2010a , the
analytical expression for $P_{\mathrm{r}}(r)$ is still missing. Moreover, the
Author cannot help feeling frustrated to find that every article that quotes
Ref. Atas:2013  almost always refers only to the Wigner-surmised form, and
calls that crude, uncontrolled approximation the ‘outcome from the random
matrix theory’. In view of these, this article aims to determine analytically
the joint distribution of consecutive eigenvalue spacings
$P_{\mathrm{c}}(a,b)$ and the distribution of their ratio $P_{\mathrm{r}}(r)$
for the unitary class, based on our recent work Nishigaki:2021  which provided
a generic prescription for determining the Jánossy density for any integrable
kernel as a solution to the Tracy-Widom system of PDEs Tracy:1994c .

This article is composed of the following parts: In §2 we list several facts
on the Jánossy density $J_{1}(0;I)$ for the sine kernel, i.e. the conditional
probability that an interval $I$ in the spectral bulk of the GUE contains no
eigenvalue except for the one at a designated locus. In §3 we follow the
prescription of Tracy and Widom and show that the Jánossy density and
associated distributions of consecutive eigenvalue spacings and their ratio
are analytically determined as a solution to a system of ODEs. As a showcase
of efficacy of our results for characterizing an approach to quantum
chaoticity without using $\bar{\rho}(E)$, in §4 we contrast them to the
distribution of zeroes of the Riemann $\zeta$ function on the critical line at
increasing heights. Numerical data of the Jánossy density and the program for
generating them are attached as Supplementary materials.

##  2 Jánossy density for the sine kernel

The sine kernel

$\displaystyle K(x,y)$ $\displaystyle=\frac{\sin(x-y)}{\pi(x-y)}=\frac{\sin
x\cos y-\cos x\sin y}{\pi(x-y)}$ (1)

governs the determinantal point process of unfolded*** We adopt a
normalization such that the mean eigenvalue spacing is $\pi$. eigenvalues or
eigenphases $\\{x_{i}\\}$ of random Hermitian or unitary matrices of infinite
rank $N\to\infty$, in the spectral bulk Mehta ; Forrester . From the very
defining property of the determinantal point process that the $p$-point
correlation function
${R}_{p}(x_{1},\ldots,x_{p})=\mathbb{E}[\prod_{i=1}^{p}\delta(x-x_{i})]$ is
given by $\det[K(x_{i},x_{j})]_{i,j=1}^{p}$, it follows that the conditional
$p$-point correlation function

$\displaystyle\tilde{R}_{p|1}(x_{1},\ldots,x_{p}|t)=\frac{\mathbb{E}[\delta(x-t)\prod_{i=1}^{p}\delta(x-x_{i})]}{\mathbb{E}[\delta(x-t)]}$
(2)

in which one of the eigenvalues is preconditioned at $x=t$ also takes a
determinantal form $\det[\tilde{K}(x_{i},x_{j})]_{i,j=1}^{p}$ governed by
another kernel Daley ,

$\displaystyle\tilde{K}(x,y)=K(x,y)-K(x,t)K(t,t)^{-1}K(t,y).$ (3)

In the case of the sine kernel (1), its translational invariance allows for
setting $t=0$ without loss of generality, so that Nagao:1993

$\displaystyle\tilde{K}(x,y)$
$\displaystyle=\frac{1}{\pi}\left(\frac{\sin(x-y)}{x-y}-\frac{\sin
x}{x}\frac{\sin
y}{y}\right)=\frac{\varphi(x)\psi(y)-\psi(x)\varphi(y)}{x-y}~{},$ (4)
$\displaystyle\varphi(x)$ $\displaystyle=\frac{1}{\sqrt{\pi}}\sin
x,~{}~{}\psi(x)=\frac{1}{\sqrt{\pi}}\left(\cos x-\frac{\sin x}{x}\right).$

The transformation of kernels (3) from $K$ to $\tilde{K}$ is associated with a
meromorphic $\mathrm{SL}(2,\mathbb{R})$ gauge transformation Nishigaki:2021

$\displaystyle\left[\begin{array}[]{c}\varphi(x)\\\
\psi(x)\end{array}\right]=\left[\begin{array}[]{cc}1&0\\\ -x^{-1}&1\\\
\end{array}\right]\left[\begin{array}[]{c}\pi^{-1/2}\sin x\\\ \pi^{-1/2}\cos
x\end{array}\right]$ (11)

on the two-component functions that consist respective kernels in the right
hand sides of (1) and (4). Accordingly, as stated in Theorem in Ref.
Nishigaki:2021 , the gauge-transformed section $[\varphi(x),\psi(x)]^{T}$
inherits from the original section $[\pi^{-1/2}\sin(x),\pi^{-1/2}\cos(x)]^{T}$
the covariant-constancy condition for a meromorphic
$\mathfrak{sl}(2,\mathbb{R})$ connection, i.e. a pair of linear differential
equations,

$\displaystyle m(x)\frac{d}{dx}\left[\\!\begin{array}[]{c}\varphi(x)\\\
\psi(x)\end{array}\\!\right]=\left[\\!\begin{array}[]{rr}A(x)\\!&\\!\\!B(x)\\\
-C(x)\\!&\\!\\!-A(x)\end{array}\\!\right]\left[\\!\begin{array}[]{c}\varphi(x)\\\
\psi(x)\end{array}\\!\right]\,\mbox{with polynomials}~{}m(x),A(x),B(x),C(x),$
(18)

which guarantees applicability of the Tracy-Widom method. In the present case
of spherical Bessel functions (4), polynomials consisting the connection (Lax
operator) $\frac{1}{m}\left[{~{}A~{}~{}B\atop-C\,-A}\right]$ are:

$\displaystyle A(x)=1,~{}~{}B(x)=C(x)=m(x)=x.$ (19)

Subsequently we shall use their nonzero coefficients,

$\displaystyle\alpha_{0}=\beta_{1}=\gamma_{1}=\mu_{1}=1.$ (20)

We take an interval $I=[a_{1},a_{2}]$ with $a_{1}<0<a_{2}$ so that the ordered
triple $(a_{1},0,a_{2})$ will serve as three consecutive eigenvalues, and
denote by $\tilde{\boldsymbol{\mathit{K}}}_{I}$ the integration operator
acting on the Hilbert space of square-integrable functions $L^{2}(I)$ with
convolution kernel $\tilde{K}(x,y)$,

$\displaystyle(\tilde{\boldsymbol{\mathit{K}}}_{I}f)(x):=\int_{I}dy\,\tilde{K}(x,y)f(y).$
(21)

Then by Gaudin and Mehta’s theorem Mehta , the Jánossy density
$J_{1}(0;[a_{1},a_{2}])$ Borodin:2003 , i.e. the conditional probability that
the interval $I=[a_{1},a_{2}]$ contains no eigenvalue except for the one
preconditioned at $x=0$, is expressed as the Fredholm deterninant of
$\tilde{\boldsymbol{\mathit{K}}}_{I}$:

$\displaystyle J_{1}(0;[a_{1},a_{2}])$
$\displaystyle=\mathrm{Det}\,(\mathbb{I}-\tilde{\boldsymbol{\mathit{K}}}_{I})=\exp\Bigl{(}-\sum_{n\geq
1}\frac{1}{n}\mathrm{Tr}\,\tilde{\boldsymbol{\mathit{K}}}_{I}^{n}\Bigr{)}$
$\displaystyle=\exp\left(-\int_{I}dx\,\tilde{K}(x,x)-\frac{1}{2}\int\\!\\!\\!\\!\int_{I}dxdy\,\tilde{K}(x,y)\tilde{K}(y,x)-\cdots\right).$
(22)

Note that the Jánossy density for a symmetric interval $I=[-t,t]$ was
previously expressed in terms of a Painlevé V transcendent, i.e. a special
solution to an ODE in $t$ Forrester:1996 . Our task in this article is to
extend their result to a generic interval and express $J_{1}(0;[a_{1},a_{2}])$
and an associated distribution $P_{\mathrm{r}}({r})$ in terms of a system of
PDEs in $a_{1}$ and $a_{2}$.

##  3 Tracy-Widom system

Tracy and Widom (TW) Tracy:1994c  established a systematic method of computing
the Fredholm determinant of an integrable integral kernel whose component
functions satisfy the condition (18). The quantities that appear in the TW
system [$j,k=1$ or $2$],

$\displaystyle R_{jk}$
$\displaystyle=\tilde{K}(a_{j},a_{k})+\int_{I}dx\,\tilde{K}(a_{j},x)\tilde{K}(x,a_{k})+\int\\!\\!\\!\\!\int_{I}dxdy\,\tilde{K}(a_{j},x)\tilde{K}(x,y)\tilde{K}(y,a_{k})+\cdots$
$\displaystyle=((\mathbb{I}-\tilde{\boldsymbol{\mathit{K}}}_{I})^{-1}\tilde{{K}})(a_{j},a_{k})=R_{kj},$
$\displaystyle q_{j}$
$\displaystyle=\varphi(a_{j})+\int_{I}dx\,\tilde{K}(a_{j},x)\varphi(x)+\int\\!\\!\\!\\!\int_{I}dxdy\,\tilde{K}(a_{j},x)\tilde{K}(x,y)\varphi(y)+\cdots$
$\displaystyle=((\mathbb{I}-\tilde{\boldsymbol{\mathit{K}}}_{I})^{-1}\varphi)(a_{j}),$
$\displaystyle p_{j}$
$\displaystyle=((\mathbb{I}-\tilde{\boldsymbol{\mathit{K}}}_{I})^{-1}\psi)(a_{j}),$
(23) $\displaystyle u$
$\displaystyle=\int_{I}dx\,\varphi(x)^{2}+\int\\!\\!\\!\\!\int_{I}dxdy\,\varphi(x)\tilde{K}(x,y)\varphi(y)+\int\\!\\!\\!\\!\int\\!\\!\\!\\!\int_{I}dxdydz\,\varphi(x)\tilde{K}(x,y)\tilde{K}(y,z)\varphi(z)+\cdots$
$\displaystyle=\int_{I}dx\,\varphi(x)((\mathbb{I}-\tilde{\boldsymbol{\mathit{K}}}_{I})^{-1}\varphi)(x),$
$\displaystyle v$
$\displaystyle=\int_{I}dx\,\psi(x)((\mathbb{I}-\tilde{\boldsymbol{\mathit{K}}}_{I})^{-1}\varphi)(x),~{}~{}w=\int_{I}dx\,\psi(x)((\mathbb{I}-\tilde{\boldsymbol{\mathit{K}}}_{I})^{-1}\psi)(x),$

are all treated as functions of the left and the right endpoints
$(a_{1},a_{2})$ of the interval $I$. Expanding the definitions (22), (23) in
$a_{1}$ and $a_{2}\ll 1$, the boundary conditions for these quantities read:

$\displaystyle\ln$ $\displaystyle
J_{1}(0;[a_{1},a_{2}])=\frac{a_{1}^{3}-a_{2}^{3}}{9\pi}-\frac{2(a_{1}^{5}-a_{2}^{5})}{225\pi}+\cdots,$
$\displaystyle q_{j}$
$\displaystyle=\frac{a_{j}}{\sqrt{\pi}}-\frac{a_{j}^{3}}{6\sqrt{\pi}}-\frac{(a_{1}^{3}-a_{2}^{3})a_{j}}{9\pi^{3/2}}+\frac{a_{j}^{5}}{120\sqrt{\pi}}+\cdots,$
(24) $\displaystyle p_{j}$
$\displaystyle=-\frac{a_{j}^{2}}{3\sqrt{\pi}}+\frac{a_{j}^{4}}{30\sqrt{\pi}}+\frac{(a_{1}^{4}-a_{2}^{4})a_{j}}{36\pi^{3/2}}+\cdots,$
$\displaystyle u$
$\displaystyle=-\frac{a_{1}^{3}-a_{2}^{3}}{3\pi}+\frac{a_{1}^{5}-a_{2}^{5}}{15\pi}+\cdots,~{}~{}v=\frac{a_{1}^{4}-a_{2}^{4}}{12\pi}+\cdots,~{}~{}w=-\frac{a_{1}^{5}-a_{2}^{5}}{45\pi}+\cdots,$

up to terms of $O(a_{1},a_{2})^{6}$. Substituting the coefficients (20) into
the TW system of PDEs (Eqs.(2.25), (2.26), (2.31), (2.32), (2.12)-(2.18),
(1.7a) of Ref.Tracy:1994c ), it takes the following form [below the pair of
indices $(j,k)$ assumes either $(1,2)$ or $(2,1)$]:

$\displaystyle R_{jk}$
$\displaystyle=\frac{q_{j}p_{k}-p_{j}q_{k}}{a_{j}-a_{k}}~{},$ $\displaystyle
a_{j}\frac{\partial q_{j}}{\partial a_{j}}$
$\displaystyle=Uq_{j}+(V+a_{j})p_{j}-(-1)^{k}a_{k}R_{jk}q_{k}~{},$
$\displaystyle a_{j}\frac{\partial p_{j}}{\partial a_{j}}$
$\displaystyle=-Up_{j}+(V-a_{j})q_{j}-(-1)^{k}a_{k}R_{jk}p_{k}~{},$ (25)
$\displaystyle\frac{\partial q_{j}}{\partial a_{k}}$
$\displaystyle=(-1)^{k}R_{jk}q_{k}~{},~{}~{}\frac{\partial p_{j}}{\partial
a_{k}}=(-1)^{k}R_{jk}p_{k}~{},$ $\displaystyle\frac{\partial u}{\partial
a_{j}}$ $\displaystyle=(-1)^{j}q_{j}^{2}~{},~{}~{}\frac{\partial v}{\partial
a_{j}}=(-1)^{j}q_{j}p_{j}~{},~{}~{}\frac{\partial w}{\partial
a_{j}}=(-1)^{j}p_{j}^{2}~{},$

where $U=1+u-w$ and $V=2v$. The ‘stiffness’ of the second and the third
equations of (25) at $a_{j}=0$ is only superficial, because $R_{jk}$, $q_{j}$,
and $p_{j}$ are of $O(a_{j})$ or higher orders in the limit $a_{j}\to 0$. We
remark that the above PDEs reduce to the ODEs (Eqs. (14), (15), (17)) of Ref.
Forrester:1996  and $V=0$ in the symmetric case, $|a_{1}|=a_{2}=t$. The
Fredholm determinant (22) is expressed by $R_{jj}$, which is composed of
$q_{j}$ and $p_{j}$ ((1.3), (1.7b) of Ref. Tracy:1994c ),

$\displaystyle(-1)^{j-1}\frac{\partial}{\partial a_{j}}\ln
J_{1}(0;[a_{1},a_{2}])=R_{jj}=p_{j}\frac{\partial q_{j}}{\partial
a_{j}}-q_{j}\frac{\partial p_{j}}{\partial a_{j}}~{}~{}~{}(j=1,2).$ (26)

For numerical evaluation of $J_{1}(0;[a,b])$, it is practically convenient to
start from the initial condition (24) at $(\epsilon a,\epsilon b)$ with a
sufficiently small $\epsilon>0$ and to integrate the TW system (25), (26) in
the radial direction $\left(a_{1}(s),a_{2}(s))=(sa,sb\right)$, $\epsilon\leq
s\leq 1$. The resultant system of ODEs in $s$, combined from the PDEs through
$s\frac{d}{ds}=a_{1}(s)\frac{\partial}{\partial
a_{1}}+a_{2}(s)\frac{\partial}{\partial a_{2}}$, reads:

$\displaystyle s\frac{d}{ds}\left[\begin{array}[]{l}q_{j}\\\
p_{j}\end{array}\right]=\left[\begin{array}[]{cc}U&V+a_{j}\\\
V-a_{j}&-U\end{array}\right]\left[\begin{array}[]{l}q_{j}\\\
p_{j}\end{array}\right]~{},$ (33) $\displaystyle
s\frac{dU}{ds}=-a_{1}(q_{1}^{2}-p_{1}^{2})+a_{2}(q_{2}^{2}-p_{2}^{2})~{},~{}s\frac{dV}{ds}=-2a_{1}q_{1}p_{1}+2a_{2}q_{2}p_{2}~{},$
$\displaystyle s\frac{d\ln
J_{1}}{ds}=a_{1}(q_{1}^{2}+p_{1}^{2})-a_{2}(q_{2}^{2}+p_{2}^{2})-(q_{1}p_{2}-p_{1}q_{2})^{2}+2U(q_{1}p_{1}-q_{2}p_{2})-V(q_{1}^{2}-p_{1}^{2}-q_{2}^{2}+p_{2}^{2}).$

Note that the form of the first line of Eq. (33) is directly inherited from
Eqs. (18), (19). It is quite plausible that the ODEs (33) can be expressible
as a Hamiltonian system, and $J_{1}(0;[a_{1},a_{2}])$ be regarded as a
$\tau$-function of an integrable hierarchy Forrester:2001 . However, these are
not immediately evident to the Author, and will be discussed in a separate
publication. As a cross-check of our formulation, we confirmed that the values
of $J_{1}(0;[a_{1},a_{2}])$ obtained by the above prescription [Fig. 1, left]
are identical, within the accuracy of numerical evaluation, to those from the
Nyström-type quadrature approximation of the Fredholm determinant
Bornemann:2010a

$\displaystyle\mathrm{Det}\,(\mathbb{I}-\tilde{\boldsymbol{\mathit{K}}}_{I})\simeq\det\left[\delta_{ij}-\tilde{K}(x_{i},x_{j})\sqrt{w_{i}w_{j}}\right]_{i,j=1}^{m}~{},$
(34)

where $\\{x_{i},w_{i}\\}_{i=1}^{m}$ is the $m$-th order quadrature of the
interval $I$ such that $\sum_{i=1}^{m}w_{i}f(x_{i})\stackrel{{\scriptstyle
m\to\infty}}{{\longrightarrow}}\int_{I}dx\,f(x)$, with a sufficiently large
$m$. Specifically, relative deviations of $J_{1}(0;[a_{1},a_{2}])$ computed by
the TW system (33) starting from the initial value $\epsilon=10^{-10}$ using
Mathematica’s NDSolve package with WorkingPrecision $\to$ 5 MachinePrecision,
and that evaluated by the Nyström-type approximation (34) with the Gauss-
Legendre quadrature of order $m=200$, do not exceed $10^{-27}$ for the whole
range of variables $|a_{j}|\leq 10$, and $<10^{-19}$ for $|a_{j}|\leq 20$.
Interested readers are invited to verify this statement by running the
Notebook Janossy_TW_N.nb included in Supplementary materials.

Figure 1:  The Jánossy density $J_{1}(0;[a_{1},a_{2}])$ for the sine kernel
[left], and the joint distribution $P_{\mathrm{c}}(a_{1},a_{2})$ of two
consecutive eigenvalue spacings [right]. For visual clarity, each coordinate
$a_{j}$ is rescaled from the text by $1/\pi$, so that the mean eigenvalue
spacing is unity.

Various probability density functions follow from the Jánossy density: The
level spacing distribution $P(s)$ Mehta , the distribution for the nearest
neighbor level spacing $P_{\mathrm{nn}}(t)$ Forrester:1996 , and the joint
distribution for the two consecutive level spacings
$P_{\mathrm{c}}(a_{1},a_{2})$ Atas:2013  are given by

$\displaystyle P(s)=-\frac{dJ_{1}(0;[0,s])}{ds},\ \ P_{\mathrm{nn}}(t)$
$\displaystyle=-\frac{dJ_{1}(0;[-t,t])}{dt},\ \
P_{\mathrm{c}}(a_{1},a_{2})=-\frac{\partial^{2}J_{1}(0;[a_{1},a_{2}])}{\partial
a_{1}\partial a_{2}},$ (35)

respectively [Fig. 1, right]. Finally, the distribution $P_{\mathrm{r}}(r)$
for the ratio $r=|a_{1}|/a_{2}$ of the two consecutive level spacings is given
by Atas:2013 ,

$\displaystyle P_{\mathrm{r}}(r)$
$\displaystyle=\int_{0}^{\infty}da_{2}\int_{-\infty}^{0}da_{1}P_{\mathrm{c}}(a_{1},a_{2})\,\delta(r-|a_{1}|/a_{2})=\int_{0}^{\infty}da\,a\,P_{\mathrm{c}}(-ra,a)~{}.$
(36)

If we switch the variable from $r$ to
$\tilde{r}:=\min(|a_{1}|,a_{2})/\max(|a_{1}|,a_{2})=\min(r,r^{-1})\in[0,1]$,
its distribution is twice the above $P_{\mathrm{r}}(r)$ [Fig. 2], yielding the
expectation values for its moments,

$\displaystyle\mathbb{E}[\tilde{r}^{k}]$
$\displaystyle=2\int_{0}^{\infty}da_{2}\int_{-a_{2}}^{0}da_{1}P_{\mathrm{c}}(a_{1},a_{2})\left(|a_{1}|/a_{2}\right)^{k}$
(37)
$\displaystyle=0.5997504209(1),\,0.4132049292(1),\,0.3100223500(1),\,0.2460560527(1)~{}~{}(k=1,2,3,4).$

Figure 2:  The distribution of the $\tilde{r}$-ratio,
$\tilde{r}=\min(|a_{1}|,a_{2})/\max(|a_{1}|,a_{2})$.

##  4 Zeroes of the Riemann $\zeta$ function

As a showcase of application of our analytic results to the judgement of
quantum chaoticity, we compare them to (arguably) the most ideal of all
quantum-chaotic spectra: the sequence of zeroes of the Riemann $\zeta$
function on the critical line, $\\{\frac{1}{2}+i\gamma_{n}\\}$. Their
imaginary parts are supposed to be the eigenvalues of the hypothetical
Hilbert-Pólya operator Wiki:Hilbert-Polya  that should be self-adjoint,
ergodic, and without any anti-unitary convolutive symmetry. After unfolding by
the Riemann-von Mangoldt formula for the asymptotic density of zeroes,
$\bar{\rho}(\gamma)=\frac{1}{2\pi}\log\frac{\gamma}{2\pi}$, the histogram
$P_{\mathrm{c}}(\delta_{-},\delta_{+})$ of two unfolded consecutive spacings
$\delta_{\pm}=\bar{\rho}(\gamma_{n})(\gamma_{n\pm 1}-\gamma_{n})$ of $10^{8}$
zeroes ending at $n=103800788359$ (the largest zero available at the LMFDB
LMFDB ) perfectly agree with the GUE result (35), as visualized in Fig. 3,
left. Moreover, had we not known the classical formula for
$\bar{\rho}(\gamma)$ a priori, the perfect match to the GUE could still be
deduced from the distributions of the ratios of consecutive spacings of zeroes
$r_{n}=(\gamma_{n+1}-\gamma_{n})/(\gamma_{n}-\gamma_{n-1})$ and
$\tilde{r}_{n}=\min(r_{n},r_{n}^{-1})$ [Fig. 3, right].

Figure 3:  The joint distribution $P_{\mathrm{c}}(\delta_{-},\delta_{+})$ of
two unfolded consecutive spacings
$\delta_{\pm}=\bar{\rho}(\gamma_{n})(\gamma_{n\pm 1}-\gamma_{n})$ of $10^{8}$
zeroes of the Riemann $\zeta$ function $\\{\frac{1}{2}+i\gamma_{n}\\}$ around
$n\simeq 1.037\cdot 10^{11}$ [left], and the distributions of the ratios
$r_{n}=(\gamma_{n+1}-\gamma_{n})/(\gamma_{n}-\gamma_{n-1})$ [right] and
$\tilde{r}_{n}=\min(r_{n},r_{n}^{-1})$ [right, inset]. Histograms from the
Riemann zeroes (black tiles and dots) are plotted versus the analytical
results (Figs. 1 and 2) for the GUE (orange surface and curves).

Indeed, this observation has been reported in Fig. 4 of the original article
Atas:2013  that inspired this work. Equipped with the high precision of the
moments of $\tilde{r}$ attained by our analytic derivation of
$P_{\mathrm{c}}(a_{1},a_{2})$, we can now revisit the rather crude observation
of Ref. Atas:2013  and improve it to the level of quantifying the systematic
convergence of the distribution of the $\tilde{r}$-ratios of the Riemann
zeroes to the GUE result (36). Mean values of the moments of $\tilde{r}_{n}$
in four windows of zeroes $[\gamma_{N},\gamma_{1.001N+1}]$ for
$N=10^{8},10^{9},10^{10}$, and $103700788358$ are summarized in Table 1.

Table 1: Moments of $\tilde{r}_{n}$ for the Riemann zeroes $N$ | $\gamma_{N}$ | $\langle\tilde{r}_{n}\rangle$ | $\langle\tilde{r}^{2}_{n}\rangle$ | $\langle\tilde{r}^{3}_{n}\rangle$ | $\langle\tilde{r}^{4}_{n}\rangle$
---|---|---|---|---|---
$10^{8}$ | $4.265354\cdot 10^{7}$ | 0.6032357 | 0.4168926 | 0.3133507 | 0.2489623
$10^{9}$ | $3.718702\cdot 10^{8}$ | 0.6021928 | 0.4158748 | 0.3125019 | 0.2482868
$10^{10}$ | $3.293531\cdot 10^{9}$ | 0.6014386 | 0.4149925 | 0.3116161 | 0.2474310
103700788358 | $3.058187\cdot 10^{10}$ | 0.6010277 | 0.4145862 | 0.3112812 | 0.2471641

As the height $\gamma_{N}$ increases, relative deviations from (37),
$\langle\tilde{r}^{k}_{n}\rangle/\mathbb{E}[\tilde{r}^{k}]-1$, are indeed
observed to vanish systematically from above, heuristically as proportional to
$\bar{\rho}(\gamma_{N})^{-3}$, indicating an approach to the complete quantum
chaoticity in the ‘thermodynamic limit’ $\gamma_{N}\to\infty$ [Fig. 4].

Figure 4: Relative deviations of the moments of the $\tilde{r}$-ratio
$\langle\tilde{r}^{k}_{n}\rangle\ (k=1,\ldots,4)$ in four windows of the
Riemann zeroes $\\{\frac{1}{2}+i\gamma_{n}\\}$, $n\in[N,1.001N+1]$ for
$N=10^{8},10^{9},10^{10}$, and $\simeq 10^{11}$, from the GUE results
$\mathbb{E}[\tilde{r}^{k}]$, plotted versus $\bar{\rho}(\gamma_{N})^{-3}$.
Error bars represent statistical fluctuations due to an arbitrary choice of
windows, estimated by the Jackknife method of splitting each data set in 10
bins. Dotted lines are linear fits with the least $\chi^{2}$ for each set of
four points in the same color.

We remark that the systematic deviations of the statistics of the Riemann
zeroes from the GUE (at infinite $N$) were previously studied in Refs.
Bogomolny:2006 ; Forrester:2015 ; Bornemann:2017 . There, the ‘finite-size
corrections’ in various statistical distributions for the Riemann zeroes were
reported to agree well with those for the circular unitary ensemble (CUE) at
finite $N_{\mathrm{eff}}\approx 1.446124\cdot\bar{\rho}(\gamma_{N})$. The
relationship between these observations and our finding above will need to be
clarified.

## Supplementary materials

Janossy_TW_N.nb: Mathematica Notebook for generating $J_{1}(0;[a_{1},a_{2}])$
analytically by the Tracy-Widom system (33), (24) or numerically by the
Nyström-type approximation (34).
Janossy_Sin.dat: numerics of $J_{1}(0;[a_{1},a_{2}])$ in the range
$(a_{1},a_{2})\in[-20,0]\times[0,20]$, provided upon request by email.

## Acknowledgments

I thank the LMFDB Collaboration LMFDB  for making the data of the Riemann
$\zeta$ function zeroes publicly downloadable, and Peter Forrester for an
informative comment on the manuscript. This work is supported by JSPS Grants-
in-Aids for Scientific Research (C) No.7K05416.

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