Also at ]University of Chinese Academy of Sciences

# Update of hadronic decays of $J/\psi$ and $\psi(2S)$ though virtual photons

Libo Liao<EMAIL_ADDRESS>Guangxi Key Laboratory of Machine Vision and
Intelligent Control, Wuzhou University, Wuzhou, Guangxi, 543002, China
Shudong Wang  [  Gang Li<EMAIL_ADDRESS>Institute of High Energy Physics,
Chinese Academy of Sciences, Shijingshan District, Beijing, 100049, China
Zhaoling Zhang  Weimin Song  College of Physics, Jilin University, Changchun,
Jilin, 130012, China

###### Abstract

The hadronic decay branching ratios of $J/\psi$ and $\psi(2S)$ through virtual
photons $B(J/\psi,\psi(2S)\rightarrow\gamma^{*}\rightarrow\text{hadrons})$ are
updated by using the latest published measurements of the $R$ value and the
branching ratios of $J/\psi,\psi(2S)\rightarrow l^{+}l^{-}$. The precision in
$B(J/\psi\rightarrow\gamma^{*}\rightarrow\text{hadrons})$ and
$B(\psi(2S)\rightarrow\gamma^{*}\rightarrow\text{hadrons})$ improve by factors
of 4 and 3, respectively.

The $J/\psi(1^{3}S_{1})$ and $\psi(2^{3}S_{1})$, $J^{PC}(1^{--})$ states of
charmonium have extremely small width. The reason is usually referred to by
the phenomenological Okubo-Zweig-Iizuka rule, which states that processes in
which the initial quark pairs cannot appear as part of the final state
particles are suppressed. Therefore, the $J/\psi$ only decays through virtual
photons (vacuum polarization), three gluons, two gluons plus a single photon,
or into $\eta_{c}$ via M1 transition and three photons. The determination of
the partial widths could play an important role in understanding the nature of
$J/\psi$ decays. The hadronic final state through virtual photons and gluons
is indistinguishable. Fortunately, the branching ratios
$B(J/\psi\rightarrow\gamma^{*}\rightarrow\text{hadrons})$ can be calculated
using the relation from Ref. [1]

$\begin{split}B(J/\psi\rightarrow\gamma^{*}\rightarrow\text{hadrons})&=\frac{\sigma_{h}}{\sigma_{l}}\times
B(J/\psi\rightarrow l^{+}l^{-})\\\ &\equiv R\times B(J/\psi\rightarrow
l^{+}l^{-})~{}~{}~{}~{}~{}(l=e,\mu),\end{split}$ (1)

where assumes that the ratios of the cross sections for virtual photons
decaying to hadrons and leptons, denoted as $\sigma_{h}$ and $\sigma_{l}$
respectively, are the same for both electron-positron and $c\bar{c}$
annihilations. This relationship also holds for the decay of $\psi(2S)$.

The best published determinations on
$B(J/\psi,\psi(2S)\rightarrow\gamma^{*}\rightarrow\text{hadrons})$ are from
2004 and were determined by Seth [2]:

$B(J/\psi\rightarrow\gamma^{*}\rightarrow\text{hadrons})=(13.5\pm 0.3)\%,$ (2)
$B(\psi(2S)\rightarrow\gamma^{*}\rightarrow\text{hadrons})=(1.66\pm 0.10)\%.$
(3)

The $R$ value used in this result was $2.28\pm 0.04$.

Recently, more precise results for the $R$ value and
$B(J/\psi,\psi(2S)\rightarrow l^{+}l^{-})$ became available. The KEDR
experiment measured the $R$ value between 1.84 and 3.05 GeV [3] and between
1.84 and 3.72 GeV [4, 5]. The BESII measured the $R$ value at energy points of
3.650, 3.6648 and 3.773 GeV [6], between 3.650 and 3.872 GeV [7], and at 2.60,
3.07 and 3.65 GeV [8]. The newest measurements by BESIII range from 2.2324 to
3.6710 GeV [9].

In the studies of $B(J/\psi,\psi(2S)\rightarrow l^{+}l^{-})$, the most recent
and the most precise results were published by CLEO $c$ and BESIII, which
reached relative precision of 0.6% and 2.1%, respectively. Reasons for these
improvements are better detectors, larger data samples, and carefully studied
systematic uncertainties. By combining all available results [10], the
precisions on $B(J/\psi\rightarrow l^{+}l^{-})$ and $B(\psi(2S)\rightarrow
l^{+}l^{-})$ improved by factors of 4 and 2.5, respectively, as shown in Table
1.

Table 1: Summary of the branching ratios $B(J/\psi,\psi(2S)\rightarrow l^{+}l^{-})$ and their averages in PDG 2002 [11] and 2022 [10]. In the 3rd and 5th columns, the branching ratios $B(J/\psi\rightarrow l^{+}l^{-})$ and $\psi(2S)\rightarrow l^{+}l^{-})$ were assuming lepton universality and neglecting the mass difference between electrons and muons. The relative uncertainty of the branching ratios is also provided. | PDG 2002 (%) | Average (%) | Relative uncertainty (%) | PDG 2022 (%) | Average (%) | Relative uncertainty (%)
---|---|---|---|---|---|---
$B(J/\psi\rightarrow e^{+}e^{-})$ | (5.93$\pm$0.10) | (5.90$\pm$0.09) | 1.6 | (5.971$\pm$0.032) | (5.967$\pm$0.023) | 0.4
$B(J/\psi\rightarrow\mu^{+}\mu^{-})$ | (5.88$\pm$0.10) | (5.961$\pm$0.033)
$B(\psi(2S)\rightarrow e^{+}e^{-})$ | (0.73$\pm$0.04) | (0.73$\pm$0.04) | 5.5 | (0.793$\pm$0.017) | (0.794$\pm$0.017) | 2.2
$B(\psi(2S)\rightarrow\mu^{+}\mu^{-})$ | (0.70$\pm$0.09) | (0.80$\pm$0.06)

The $R$ values between 2.0 and 3.73 GeV are used to approximate the ones at
$J/\psi$ and $\psi(2S)$ peaks, as shown in Fig. 1. The results of the $R$
scans used in this study are from the MARK-I, the DM 2, the BESII, the KEDR,
the BESIII, etc. It can be seen that the precision of $R$ values has improved
significantly in the last decades, benefiting from better detectors,
increasing data samples, and precise knowledge of initial state radiation
correction and systematic uncertainties.

Various combinations of the existing measurements of the $R$ value are tested
to determine the $R$. The results are summarized in Table. 2. It can be seen
clearly that the BESII, the KEDR, and the BESIII can achieve comparable
precision, and all three are better than the one used in Ref. [2]. It should
be noted that the BESII published new measurements of the $R$ value in 2006
[6, 7] and 2009 [8] after the publication of Ref [2]. The precision reaches
0.4% if all the results are combined, which is going to be used in the
following calculations.

Table 2: $R$ value scan results from several experiments in the center of mass energy range between 2.0 to 3.73 GeV. The combined fit result is used to approximate the $R$ values underneath the $J/\psi$ and $\psi(2S)$ peaks. Only experiments on more than five energy points are included. | $R$ | Relative uncertainty
---|---|---
MARK-I | $2.56\pm 0.08$ | 3.1%
DM 2 | $2.19\pm 0.07$ | 3.3%
BESII | $2.20\pm 0.02$ | 1.0%
KEDR | $2.22\pm 0.02$ | 0.7%
BESIII | $2.30\pm 0.01$ | 0.6%
Combined fit | $2.26\pm 0.01$ | 0.4%
Ref. [2] | $2.28\pm 0.04$ | 1.8%
Figure 1: Summary of $R$ measurements in the c.m. energy region from 2.0 to
3.73 GeV. Red dots are results from BESIII [9]. The $R$ values of the BESII
experiment [12, 13, 6, 7, 8] are indicated as green dots. Blue rectangles
represent the results of KEDR detector [4, 3, 5]. MARK-1 [14] and DM 2 [15]
results are shown as cyan stars and orange crosses, respectively. The values
from PLUTO [16] are denoted by brown diamonds. Magenta triangles stand for the
results of the Crystal Ball Collaboration [17].

The branching ratios
$B(J/\psi,\psi(2S)\rightarrow\gamma^{*}\rightarrow\text{hadrons})$ are
calculated using equation (1) and the combined fit result on $R$ from Table 2.
The results in

$B(J/\psi\rightarrow\gamma^{*}\rightarrow\text{hadrons})=(13.46\pm
0.07)\%~{},$

and

$B(\psi(2S)\rightarrow\gamma^{*}\rightarrow\text{hadrons})=(1.79\pm
0.04)\%~{},$

are compared to the previous results in Table 3. Benefiting from the improved
precision of the $R$ values and $B(J/\psi,\psi(2S)\rightarrow l^{+}l^{-})$,
the relative uncertainties are reduced significantly. The
$B(J/\psi\rightarrow\gamma^{*}\rightarrow\text{hadrons})$ and
$B(\psi(2S)\rightarrow\gamma^{*}\rightarrow\text{hadrons})$ are improved by
about 4 times and 3 times, respectively. But it should be noted that the
correlations among R values within a single experiment are neglected, as well
as the correlation between the branching ratios for $B(J/\psi\to e^{+}e^{-})$
and $B(J/\psi\to\mu^{+}\mu^{-})$.

Table 3: Comparison of $B(J/\psi,\psi(2S)\rightarrow\gamma^{*}\rightarrow\text{hadrons})$ in Ref [2] and those in this study.  Channel | Ref. [2] | This study | Relative uncertainty
---|---|---|---
$B(J/\psi\rightarrow\gamma^{*}\rightarrow h)$ | (13.5$\pm$0.3)% | (13.46$\pm$0.07)% | 2.2%$\rightarrow$0.5%
$B(\psi(2S)\rightarrow\gamma^{*}\rightarrow h)$ | (1.66$\pm$0.10)% | (1.79$\pm$0.04)% | 6.1%$\rightarrow$2.1%

The update on
$B(J/\psi,\psi(2S)\rightarrow\gamma^{*}\rightarrow\text{hadrons})$ will help
to constrain branching ratios $B(J/\psi,\psi(2S)\rightarrow\gamma gg,ggg)$ and
enhance our understanding of the two charmonia. Taking the $J/\psi$ as an
example, two important aspects arise. Firstly,
$B(J/\psi\rightarrow\gamma^{\star}\rightarrow\text{hadrons})$ can be utilized
to test the theoretical calculation presented in equation (8) of Ref. [1],
including its higher-order corrections. Secondly, it enables the calculation
of branching ratios such as $B(J/\psi\rightarrow\gamma gg,ggg)$ and the
relative fractions of $B(ggg)$, $B_{elm}(\text{had})$, $B(\gamma gg)$, and
$B(\gamma\eta_{c})$, in conjunction with $B(J/\psi\rightarrow\gamma\eta_{c})$,
$B(J/\psi\rightarrow l^{+}l^{-})$. Notably, the precision of the ratio
$\Gamma(\gamma gg)/\Gamma(ggg)$ introduces the dominant uncertainty,
emphasizing the need for future updates to improve its precision.

## References

  * L. Köpke and N. Wermes [1989] L. Köpke and N. Wermes, J/psi decays, Phys. Rep. 174, 67 (1989).
  * Seth [2004] K. K. Seth, Hadronic decays of $J/\psi(1^{3}S_{1})$ and $\psi^{\prime}(2^{3}S_{1})$ through virtual photons, Phys. Rev. D 69, 097503 (2004).
  * Anashin  _et al._ [2017] V. V. Anashin _et al._ , Measurement of $R$ between 1.84 and 3.05 GeV at the KEDR detector, Phys. Lett. B 770, 174 (2017).
  * Anashin  _et al._ [2016] V. V. Anashin _et al._ , Measurement of $R_{\text{uds}}$ and $R$ between 3.12 and 3.72 GeV at the KEDR detector, Phys. Lett. B 753, 533 (2016).
  * Anashin  _et al._ [2019] V. V. Anashin _et al._ (KEDR Collaboration), Precise measurement of $R_{\text{uds}}$ and $R$ between 1.84 and 3.72 GeV at the KEDR detector, Phys. Lett. B 788, 42 (2019).
  * Ablikim  _et al._ [2006a] M. Ablikim _et al._ (BES Collaboration), Measurements of the cross-sections for $e^{+}e^{-}\rightarrow$ hadrons at 3.650 GeV, 3.6648 GeV, 3.773 GeV and the branching fraction for $\psi(3770)\rightarrow\text{non-}D\bar{D}$, Phys. Lett. B 641, 145 (2006a).
  * Ablikim  _et al._ [2006b] M. Ablikim _et al._ , Measurements of the Continuum $R_{uds}$ and $R$ Values in $e^{+}e^{-}$ Annihilation in the Energy Region Between 3.650 and 3.872 GeV, Phys. Rev. Lett. 97, 262001 (2006b).
  * Ablikim  _et al._ [2009] M. Ablikim _et al._ (BES Collaboration), R value measurements for $e^{+}e^{-}$ annihilation at 2.60, 3.07 and 3.65 GeV, Phys. Lett. B 677, 239 (2009).
  * Ablikim  _et al._ [2022] M. Ablikim _et al._ (BESIII Collaboration), Measurement of the Cross Section for $e^{+}e^{-}\rightarrow$ Hadrons at Energies from 2.2324 to 3.6710 GeV, Phys. Rev. Lett. 128, 062004 (2022).
  * Workman  _et al._ [2022] R. L. Workman _et al._ (Particle Data Group), Review of particle physics, Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
  * Hagiwara  _et al._ [2002] K. Hagiwara _et al._ (Particle Data Group), Review of particle physics, Phys. Rev. D 66, 010001 (2002).
  * Bai  _et al._ [2000] J. Z. Bai _et al._ (BES Collaboration), Measurement of the Total Cross-Section for Hadronic Production by $e^{+}e^{-}$ Annihilation at Energies Between 2.6-5 GeV, Phys. Rev. Lett. 84, 594 (2000).
  * Bai  _et al._ [2002] J. Z. Bai _et al._ (BES Collaboration), Measurements of the Cross-Section for $e^{+}e^{-}\rightarrow$ Hadrons at Center-of-Mass Energies from 2 to 5 GeV, Phys. Rev. Lett. 88, 101802 (2002).
  * Siegrist  _et al._ [1982] J. Siegrist _et al._ , Hadron production by $e^{+}e^{-}$ annihilation at center-of-mass energies between 2.6 and 7.8 GeV. I. Total cross-section, multiplicities and inclusive momentum distributions, Phys. Rev. D 26, 969 (1982).
  * Bacci  _et al._ [1979] C. Bacci _et al._ , Total cross-section for hadronic production by $e^{+}e^{-}$ annihilation in the total cm energy range 1.42-3.09 GeV, Phys. Lett. B 86, 234 (1979).
  * Crigee and Knies [1982] L. Crigee and G. Knies, $e^{+}e^{-}$ physics with the PLUTO detector, Phys. Rep. 83, 151 (1982).
  * [17] A. Osterheld _et al._ , Measurements of total hadronic and inclusive $D^{*}$ cross-sections in $e^{+}e^{-}$ annihilations between 3.87 GeV and 4.5 GeV, SLAC Report No. SLAC-PUB-4160, 1986 .


$\displaystyle=2\sup_{x\in\frac{3}{4}B_{r\theta^{-(k-1)}}}\sum_{j=1}^{\infty}\int_{B_{r\theta^{-k+j+1}}\backslash
B_{r\theta^{-k+j}}}\left[\left(\frac{d\left(y,x_{0}\right)}{r\theta^{-k}}\right)^{\beta}-1\right]J(x,dy)$
$\displaystyle\leq
2\sum_{j=1}^{\infty}\sup_{x\in\frac{3}{4}B_{r\theta^{-(k-1)}}}\int_{B_{r\theta^{-k+j+1}}\backslash
B_{r\theta^{-k+j}}}\left(\theta^{(j+1)\beta}-1\right)J(x,dy)$
$\displaystyle\leq
2\sum_{j=1}^{\infty}\left(\theta^{(j+1)\beta}-1\right)\sup_{x\in\frac{3}{4}B_{r\theta^{-(k-1)}}}\int_{B_{r\theta^{-k+j}}^{c}}J(x,dy)$
$\displaystyle\leq
2\sum_{j=1}^{\infty}\left(\theta^{(j+1)\beta}-1\right)\sup_{x\in\frac{3}{4}B_{r\theta^{-(k-1)}}}\int_{B(x,r\theta^{-k+j}/4)^{c}}J(x,dy)$
$\displaystyle\leq
2C\sum_{j=1}^{\infty}\left(\theta^{(j+1)\beta}-1\right)\sup_{x\in\frac{3}{4}B_{r\theta^{-(k-1)}}}\frac{1}{w(x,r\theta^{-k+j}/4)},$
(6.13)

where $C>0$ is the same constant as in (1.13). Since

$\frac{w(x,r\theta^{-k+j}/4)}{w(x_{0},r\theta^{-k})}\geq\frac{w(x,r\theta^{-k+j})}{C_{2}4^{\beta_{2}}w(x_{0},r\theta^{-k})}\geq\frac{C_{1}\theta^{j\beta_{1}}}{C_{2}4^{\beta_{2}}}$

for any $x\in\frac{3}{4}B_{r\theta^{-(k-1)}}$ by using (1.5), it follows from
(6.13) that

$w\left(x_{0},r\theta^{-k}\right)T_{\frac{3}{4}B_{r\theta^{-(k-1)}},B_{r\theta^{-(k-1)}}}(h_{-})\leq\frac{2CC_{2}4^{\beta_{2}}}{C_{1}}\sum_{j=1}^{\infty}\frac{\theta^{(j+1)\beta}-1}{\theta^{j\beta_{1}}}.$
(6.14)

Therefore, substituting (6.12), (6.14) into (6.11), we obtain

$\displaystyle\operatorname*{einf}_{B_{r\theta^{-k}}}h$
$\displaystyle\geq\left(C_{H}2^{1/p}\right)^{-1}-w\left(x_{0},r\theta^{-k}\right)T_{\frac{3}{4}B_{r\theta^{-(k-1)}},B_{r\theta^{-(k-1)}}}(h_{-})$
$\displaystyle\geq\left(C_{H}2^{1/p}\right)^{-1}-\frac{2CC_{2}4^{\beta_{2}}}{C_{1}}\sum_{j=1}^{\infty}\frac{\theta^{(j+1)\beta}-1}{\theta^{j\beta_{1}}}.$
(6.15)

Since $\theta^{-1}\leq\delta$, we see that for any
$\beta\in\left(0,\beta_{1}/2\right)$

$\sum_{j=l+1}^{\infty}\theta^{-j\beta_{1}}\left(\theta^{(j+1)\beta}-1\right)\leq\sum_{j=l+1}^{\infty}\theta^{-j\beta_{1}}\theta^{(j+1)\beta_{1}/2}=\frac{\theta^{-l\beta_{1}/2}}{1-\theta^{-\beta_{1}/2}}\leq\frac{\delta^{l\beta_{1}/2}}{1-\delta^{\beta_{1}/2}}\leq\left(\frac{8CC_{2}4^{\beta_{2}}}{C_{1}}C_{H}2^{1/p}\right)^{-1},$

provided that the number $l$ is sufficiently large, which depends only on
$\delta$ but is independent of $\beta,\theta$. For such a number $l$, we now
choose $\beta\in\left(0,\beta_{1}/2\right)$ to be so small that

$\displaystyle\sum_{j=1}^{l}\theta^{-j\beta_{1}}\left(\theta^{(j+1)\beta}-1\right)$
$\displaystyle\leq$
$\displaystyle\theta^{-\beta_{1}}\sum_{j=1}^{l}\left(\theta^{(j+1)\beta}-1\right)\leq
l\theta^{-\beta_{1}}\left(\theta^{(l+1)\beta}-1\right)$ $\displaystyle\leq$
$\displaystyle
l\delta^{\beta_{1}}\left(\theta^{(l+1)\beta}-1\right)\leq\left(\frac{8CC_{2}4^{\beta_{2}}}{C_{1}}C_{H}2^{1/p}\right)^{-1}.$

It follows that

$\sum_{j=1}^{\infty}\frac{\theta^{(j+1)\beta}-1}{\theta^{j\beta_{1}}}=\sum_{j=1}^{l}\theta^{-j\beta_{1}}\left(\theta^{(j+1)\beta}-1\right)+\sum_{j=l+1}^{\infty}\theta^{-j\beta_{1}}\left(\theta^{(j+1)\beta}-1\right)\leq
2\left(\frac{8CC_{2}4^{\beta_{2}}}{C_{1}}C_{H}2^{1/p}\right)^{-1},$

from which, we see by (6.15) that

$\operatorname*{einf}_{B_{r\theta^{-k}}}h\geq\left(C_{H}2^{1/p}\right)^{-1}-\frac{2CC_{2}4^{\beta_{2}}}{C_{1}}\cdot
2\left(\frac{8CC_{2}4^{\beta_{2}}}{C_{1}}C_{H}2^{1/p}\right)^{-1}=\left(2C_{H}2^{1/p}\right)^{-1}=\lambda.$

Therefore, $v\leq 1-\lambda$ in $B_{r\theta^{-k}}$, thus showing (6.10) when
(6.8) is satisfied, as desired.

It remains to consider the case when (6.9) is satisfied. We need to show

$v\geq-1+\lambda\text{ \ in }B_{r\theta^{-k}}\text{.}$ (6.16)

Indeed, consider the function $h=1+v$. Similar to the argument above, setting
$M_{k}=M_{k-1}$ and $m_{k}=M_{k}-K\theta^{-k\beta}$, one can obtain (6.16).
The proof is complete. ∎

From the above, we immediately get the Hölder continuity of harmonic
functions.

######  Lemma 6.2.

Let $(\mathcal{E},\mathcal{F})$ be a regular Dirichlet form in $L^{2}$ without
killing part. If conditions $(\mathrm{wEH})$ and $(\mathrm{TJ})$ hold, then
there exist three constants $C>0,\theta\in(0,1]$ and $\varepsilon\in(0,1)$
such that, for any ball $B\left(x_{0},r\right)$ with $r<\sigma\overline{R}$
and for any globally bounded function $u$, which is harmonic in
$B\left(x_{0},r\right)$,

$|u(x)-u(y)|\leq C\left(\frac{d(x,y)}{r}\right)^{\theta}\|u\|_{L^{\infty}}.$
(6.17)

for almost every points $x,y\in B\left(x_{0},\varepsilon r\right)$.

###### Proof.

Let the function $u\in L^{\infty}$ be harmonic in $B\left(x_{0},r\right)$ with
$r<\sigma\overline{R}$. By Lemma 6.1,

$\operatorname*{eosc}_{B(x_{0},\rho)}u\leq
C||u||_{L^{\infty}}\left(\frac{\rho}{r}\right)^{\beta},\ \ 0<\rho\leq r.$
(6.18)

We show that (6.17) holds for $\theta=\beta,\varepsilon=1/4$.

Indeed, let $x$ be any point $x$ in $B(x_{0},r/4)$, the function $u$ is
harmonic in $B(x,\frac{3}{4}r)\subseteq B(x_{0},r)$. Let $y$ be a point in
$B(x_{0},r/4)$. Applying (6.18) with $x_{0}$ replaced by $x$, $r$ by
$\frac{3}{4}r$ and with $\rho=\frac{3}{2}d(x,y)$, we obtain

$|u(x)-u(y)|\leq\operatorname*{eosc}_{B(x,\frac{3}{2}d(x,y))}u\leq
C||u||_{L^{\infty}}\left(\frac{3d(x,y)/2}{3r/4}\right)^{\beta}=C2^{\beta}\left(\frac{d(x,y)}{r}\right)^{\beta}||u||_{L^{\infty}},$

thus showing (6.17). The proof is complete. ∎

Another consequence of the weak elliptic Harnack inequality is that it implies
a lower bound of the mean exit time on a ball, as we will see below.

Recall that the operator $\mathcal{L}^{\Omega}$ is the generator of the
Dirichlet form $(\mathcal{E},\mathcal{F}(\Omega))$ for any non-empty open
subset $\Omega$ of $M$. For a ball $B\subset M$, let the function $E^{B}$ be a
weak solution of the Poisson-type equation $-\mathcal{L}^{B}u=1$ in $B$, that
is

$\mathcal{E}(E^{B},\varphi)=(1,\varphi)\text{ \ for any
}0\leq\varphi\in\mathcal{F}(B).$ (6.19)

We say that _condition_ $(\mathrm{E}_{\geq})$ holds if there exist three
constants $C>0$ and $\sigma,\delta$ in $(0,1)$ such that, for all balls
$B\subset M$ with radius less than $\sigma\overline{R}$,

$\operatorname*{einf}_{x\in\delta B}E^{B}(x)\geq Cw(B).$ (6.20)

We say that _condition_ $(\mathrm{E}_{\leq})$ holds if there exist two
constants $C>0$ and $\sigma$ in $(0,1)$ such that, for all balls $B\subset M$
with radius less than $\sigma\overline{R}$,

$||E^{B}||_{L^{\infty}}\leq Cw(B).$ (6.21)

######  Lemma 6.3.

Let $(\mathcal{E},\mathcal{F})$ be a regular Dirichlet form in $L^{2}$. Then

$(\mathrm{VD})+(\mathrm{Cap}_{\leq})+(\mathrm{FK})+(\mathrm{wEH})\Rightarrow(\mathrm{E}_{\geq})+(\mathrm{E}_{\leq}).$
(6.22)

###### Proof.

Note that by [24, Theorem 9.4 p.1542]

$(\mathrm{FK})\Rightarrow(\mathrm{E}_{\leq}),$ (6.23)

(observing that we only use condition $(\mathrm{FK})$ at this stage).

It remains to show the implication

$(\mathrm{VD})+(\mathrm{Cap}_{\leq})+(\mathrm{FK})+(\mathrm{wEH})\Rightarrow(\mathrm{E}_{\geq}).$
(6.24)

Let $\delta$ be the same constant as in condition $(\mathrm{wEH})$. Without
loss of generality, assume that $\delta<\frac{2}{3}$. Let $B:=B(x_{0},R)$ be a
ball in $M$ with $R<\sigma\overline{R}$. Let $u$ be the unique weak solution
such that

$\mathcal{E}(u,\varphi)=(1_{\delta B},\varphi)\text{ \ for any
}0\leq\varphi\in\mathcal{F}(B).$ (6.25)

It is known that $u\in\mathcal{F}(B)$, $u\geq 0$ in $M$, and $u$ is
superharmonic in $B$, see for example [22, Lemma 5.1]. Applying (1.16) in
condition $(\mathrm{wEH})$ on the function $u$ and the ball $B$, and with
$f=0$, $r=\delta R$, and noting that $u_{-}=0$ in $M$, we obtain

$\left(\fint_{\delta B}u^{p}d\mu\right)^{1/p}\leq
C_{H}\operatorname*{einf}_{\delta B}u.$ (6.26)

On the other hand, we have by condition $(\mathrm{Cap}_{\leq})$

$\mathcal{E}(\phi,\phi)\leq C\frac{\mu(B)}{w(B)}$ (6.27)

for some $\phi\in\mathrm{cutoff}\left((2/3)B,B\right)$.

Taking $\varphi=\phi$ in (6.25) and using condition $(\mathrm{VD})$, we see
that

$\mathcal{E}(u,\phi)=(1_{\delta B},\phi)=\int_{\delta B}\phi d\mu=\mu(\delta
B)\geq C_{\mu}^{-1}\delta^{d_{2}}\mu(B).$ (6.28)

Taking $\varphi=u$ in (6.25) and using the Cauchy-Schwarz inequality and
(6.27), it follows that

$\mathcal{E}(u,\phi)\leq\sqrt{\mathcal{E}(u,u)\mathcal{E}(\phi,\phi)}=\sqrt{(1_{\delta
B},u)}\sqrt{\mathcal{E}(\phi,\phi)}\leq C\sqrt{\int_{\delta
B}ud\mu}\sqrt{\frac{\mu(B)}{w(B)}}.$ (6.29)

Therefore, combining (6.28) and (6.29), we obtain

$\int_{\delta B}ud\mu\geq C\mu(B)w(B).$ (6.30)

Since by (6.23)

$||u||_{L^{\infty}}\leq||E^{B}||_{L^{\infty}}\leq Cw(B),$

we conclude by (6.26) that

$\displaystyle\int_{\delta B}ud\mu$ $\displaystyle=\int_{\delta B}u^{p}\cdot
u^{1-p}d\mu\leq\left(Cw(B)\right)^{1-p}\int_{\delta
B}u^{p}d\mu=\left(Cw(B)\right)^{1-p}\mu(\delta B)\fint_{\delta B}u^{p}d\mu$
$\displaystyle\leq C^{\prime}w(B)^{1-p}\mu(B)(\operatorname*{einf}_{\delta
B}u)^{p}\leq C^{\prime\prime}w(B)^{1-p}\mu(B)(\operatorname*{einf}_{\delta
B}E^{B})^{p},$

thus showing (6.20) by (6.30). The proof is complete. ∎

Finally, we show that the weak elliptic Harnack inequality also implies a
Lemma of growth, termed _condition_ $(\mathrm{LG}_{0})$, for any _global non-
negative superharmonic_ function.

We say that _condition_ $(\mathrm{LG}_{0})$ holds if there exist four
constants $\sigma,\epsilon_{0},\eta,\delta\in(0,1)$ such that, for any ball
$B:=B(x_{0},R)$ with radius $R\in(0,\sigma\overline{R})$ and for any
$u\in\mathcal{F}^{\prime}\cap L^{\infty}$ that is superharmonic in $B$ and
non-negative globally in $M$, if

$\frac{\mu(\delta B\cap\\{u<a\\})}{\mu(\delta B)}\leq\epsilon_{0}$ (6.31)

for some $a>0$, then

$\operatorname*{einf}_{\delta B}u\geq\eta a.$ (6.32)

We remark that the superharmonic function $u$ in condition $(\mathrm{LG}_{0})$
is required to be non-negative _globally_ in $M$, instead of being non-
negative _locally_ in condition __ $(\mathrm{LG})$ given in Definition 3.2.

######  Lemma 6.4.

Let $(\mathcal{E},\mathcal{F})$ be a regular Dirichlet form in $L^{2}$. Then

$(\mathrm{wEH})\Rightarrow(\mathrm{LG}_{0}).$ (6.33)

###### Proof.

Let $u\in\mathcal{F}^{\prime}\cap L^{\infty}$ be superharmonic in $B$ and non-
negative globally in $M$. Assume that (6.31) holds, namely,

$\frac{\mu(\delta B\cap\\{u\geq a\\})}{\mu(\delta B)}=1-\frac{\mu(\delta
B\cap\\{u<a\\})}{\mu(\delta B)}\geq 1-\epsilon_{0}.$

Since $u_{-}=0$ in $M$, we see that

$T_{\frac{3}{4}B_{R},B_{R}}(u_{-})\equiv 0.$

Applying (1.16) with $r=\delta R$ and $f\equiv 0$, it follows that

$\operatorname*{einf}_{\delta B}u\geq C_{H}^{-1}\left(\fint_{\delta
B}u^{p}d\mu\right)^{1/p}\geq C_{H}^{-1}a\left(\frac{\mu(\delta B\cap\\{u\geq
a\\})}{\mu(\delta B)}\right)^{1/p}\geq C_{H}^{-1}(1-\epsilon_{0})^{1/p}a,$

thus showing that (6.32) is true with $\eta=C_{H}^{-1}(1-\epsilon_{0})^{1/p}$.
The proof is complete. ∎

Lemma 6.3 above gives a direct, simpler proof of obtaining a lower bound of
the mean exit time from the weak elliptic Harnack inequality. We remark that
this conclusion can also be obtained in a more indirect way, without recourse
to condition $(\mathrm{FK})$. Indeed, the implication

$(\mathrm{VD})+(\mathrm{Cap}_{\leq})+(\mathrm{LG}_{0})\Rightarrow(\mathrm{E}_{\geq})$

has been proved in a forthcoming paper [21] for any regular Dirichlet form in
$L^{2}$. Combining this with (6.33), we have

$(\mathrm{VD})+(\mathrm{Cap}_{\leq})+(\mathrm{wEH})\Rightarrow(\mathrm{VD})+(\mathrm{Cap}_{\leq})+(\mathrm{LG}_{0})\Rightarrow(\mathrm{E}_{\geq}),$

from which, we also obtain condition $(\mathrm{E}_{\geq})$ from the weak
elliptic Harnack inequality but without using $(\mathrm{FK})$. We do need
condition $(\mathrm{FK})$ in Lemma 6.3, not only in deriving condition
$(\mathrm{E}_{\leq})$ but also in deriving condition $(\mathrm{E}_{\geq})$.

##  7\. An example

In this section we give an example to illustrate Theorem 1.8. We show that the
assumptions $(\mathrm{VD})$, $(\mathrm{RVD})$, $(\mathrm{Gcap})$,
$(\mathrm{TJ})$, $(\mathrm{PI})$ are all satisfied so that the weak Harnack
inequality holds, but the jump kernel does not exist. This example is
essentially taken from [6, Section 15], see also [21].

######  Example 7.1 (Ultra-metric space).

Let $\beta,\alpha_{1},\alpha_{2}$ be three positive numbers. Let
$\left(M_{i},d_{i},\mu_{i}\right)$ for $i=1,2$ be two ultrametric spaces,
where $d_{i}$ is an ultra-metric:

$d_{i}(x,y)\leq\max\\{d_{i}(x,z),d_{i}(z,y)\\}\text{ \ for all }x,y,z\in
M_{i},$

and the measure $\mu_{i}$ is Ahlfors-regular:

$C^{-1}r^{\alpha_{i}}\leq\mu_{i}\left(B\left(x_{i},r\right)\right)\leq
Cr^{\alpha_{i}}\text{ for all }x_{i}\in M_{i}\text{ and all }r>0$ (7.1)

for some constant $C\geq 1$. Let $J_{i}$ be a function on $M_{i}\times M_{i}$
for $i=1,2$ such that for $\mu_{i}$-almost all $x_{i},y_{i}\in M_{i}$,

$J_{i}\left(x_{i},y_{i}\right)=d_{i}\left(x_{i},y_{i}\right)^{-\left(\alpha_{i}+\beta\right)}.$
(7.2)

Consider the product space $M:=M_{1}\times M_{2}$ equipped with product
measure $\mu:=\mu_{1}\times\mu_{2}$ and the metric

$d(x,y):=\max\left\\{d_{1}\left(x_{1},y_{1}\right),d_{2}\left(x_{2},y_{2}\right)\right\\}\text{
for }x=\left(x_{1},x_{2}\right),y=\left(y_{1},y_{2}\right)\text{ in }M.$

Clearly, $(M,d,\mu)$ is an ultrametric space and for any point
$x=\left(x_{1},x_{2}\right)$ in $M$, the metric ball $B(x,r)$ in $M$ can be
written as

$B(x,r)=B(x_{1},r)\times B(x_{2},r)\text{ for any }r>0.$ (7.3)

From this, we see that for any point $x=\left(x_{1},x_{2}\right)$ in $M$ and
any $r>0$,

$V(x,r)=\mu(B(x,r))=\mu_{1}\left(B\left(x_{1},r\right)\right)\mu_{2}\left(B\left(x_{2},r\right)\right)\asymp
r^{\alpha_{1}+\alpha_{2}}=r^{\alpha},$ (7.4)

where $\alpha:=\alpha_{1}+\alpha_{2}$. For simplicity, let the scaling
function $w(x,r)$ be defined by

$w(x,r)=a(x)r^{\beta}\ \ \text{for any point}\ x\in M\ \text{and any }\ r>0,$

where $a(x)$ is a measurable function on $M$ with $C^{-1}\leq a(x)\leq C$ for
all $x\in M$ ($C\geq 1$). Clearly, such a function $w$ satisfies (1.5) and

$C^{-1}r^{\beta}\leq w(x,r)\leq Cr^{\beta}.$ (7.5)

Define the measure $J$ on $\mathcal{B}(M\times M)$ by
$J(dx,dy)=J(x,dy)\mu(dx)$, where $J(x,dy)$ is a transition function on
$M\times\mathcal{B}(M)$ given by

$J(x,dy)=J_{1}(x_{1},y_{1})\mu_{1}(dy_{1})\delta_{x_{2}}(dy_{2})+J_{2}(x_{2},y_{2})\mu_{2}(dy_{2})\delta_{x_{1}}(dy_{1})$
(7.6)

for any points $x=\left(x_{1},x_{2}\right),y=\left(y_{1},y_{2}\right)$ in $M$,
where $\delta_{b}(dx)$ is the Dirac measure concentrated at point $b$. By
(7.4) and (7.6), we have for any $r>0$ and any point $x=(x_{1},x_{2})\in M$,

$\displaystyle\int_{B(x,r)^{c}}J(x,dy)$
$\displaystyle=\int_{B(x,r)^{c}}\left(J_{1}(x_{1},y_{1})\mu_{1}(dy_{1})\delta_{x_{2}}(dy_{2})+J_{2}(x_{2},y_{2})\mu_{2}(dy_{2})\delta_{x_{1}}(dy_{1})\right)$
$\displaystyle=\int_{B(x_{1},r)^{c}}J_{1}(x_{1},y_{1})\mu_{1}(dy_{1})+\int_{B(x_{2},r)^{c}}J_{2}(x_{2},y_{2})\mu_{2}(dy_{2})$
$\displaystyle\leq\frac{C}{r^{\beta}}+\frac{C}{r^{\beta}}=\frac{2C}{r^{\beta}}\leq\frac{C^{\prime}}{w(x,r)}\text{
\ \ \ \ (using (\ref{e1}), (\ref{e2}) and (\ref{ew}))},$ (7.7)

which is exactly condition $(\mathrm{TJ})$.

Let $\left(\mathcal{E},\mathcal{F}\right)$ be a Dirichlet form in
$L^{2}\left(M,\mu\right)$ defined by

$\mathcal{E}(u,v)=\iint_{M\times
M}\left(u\left(x\right)-u\left(y\right)\right)\left(v\left(x\right)-v\left(y\right)\right)J\left(x,dy\right)\mu(dx),\quad
u,v\in\mathcal{F},$

where the space $\mathcal{F}$ is the closure of the set

$\left\\{\sum_{i=0}^{n}c_{i}\mathbf{1}_{B_{i}}:n\in\mathbb{N},c_{i}\in\mathbb{R},B_{i}\text{
is a compact ball }\right\\}$

under the inner product

$\sqrt{\mathcal{E}(\cdot,\cdot)+(\cdot,\cdot)_{L^{2}\left(M,\mu\right)}}.$

By [6, Theorem 2.2], the form $\left(\mathcal{E},\mathcal{F}\right)$ is
regular and non-local. By (7.4), the measure $\mu$ satisfies conditions
$(\mathrm{VD})$ and $(\mathrm{RVD})$, whilst condition $(\mathrm{Gcap})$
automatically holds since it follows directly from condition $(\mathrm{TJ})$
and the ultrametric property. Hence, conditions $(\mathrm{VD})$,
$(\mathrm{RVD})$, $(\mathrm{Gcap})$, $(\mathrm{TJ})$ in Theorem 1.8 are
satisfied.

It remains to verify condition $(\mathrm{PI})$. Indeed, let
$B:=B(x^{\prime},r)$ be a metric ball in $M$. Writing up
$x^{\prime}=(x_{0},y_{0})$ with $x_{0}\in M_{1}$, $y_{0}\in M_{2}$, we see
$B=B(x_{0},r)\times B(y_{0},r)$ by using (7.3). By (7.6), (7.2)

$\displaystyle\int_{B}\int_{B}(u(x)-u(y))^{2}J(x,dy)\mu(dx)$ $\displaystyle=$
$\displaystyle\int_{B}\left\\{\int_{B(x_{0},r)}\frac{(u(x_{1},x_{2})-u(y_{1},x_{2}))^{2}}{d_{1}(x_{1},y_{1})^{\alpha_{1}+\beta}}\mu_{1}(dy_{1})\right.$
$\displaystyle+\left.\int_{B(y_{0},r)}\frac{(u(x_{1},x_{2})-u(x_{1},y_{2}))^{2}}{d_{2}(x_{2},y_{2})^{\alpha_{2}+\beta}}\mu_{2}(dy_{2})\right\\}\mu(dx).$

The first integral on the right-hand side is estimated as follows: for any
$(x_{1},x_{2})\in B(x_{0},r)\times B(y_{0},r)$,

$\displaystyle\int_{B(x_{0},r)}\frac{(u(x_{1},x_{2})-u(y_{1},x_{2}))^{2}}{d_{1}(x_{1},y_{1})^{\alpha_{1}+\beta}}\mu_{1}(dy_{1})$
$\displaystyle\geq$
$\displaystyle\int_{B(x_{0},r)}\frac{(u(x_{1},x_{2})-u(y_{1},x_{2}))^{2}}{r^{\alpha_{1}+\beta}}\mu_{1}(dy_{1})$
$\displaystyle\geq$ $\displaystyle
C^{-1}\int_{B(y_{0},r)}\int_{B(x_{0},r)}\frac{(u(x_{1},x_{2})-u(y_{1},x_{2}))^{2}}{r^{\alpha_{1}+\alpha_{2}+\beta}}\mu_{1}(dy_{1})\mu_{2}(dy_{2})\
\ \text{(using (\ref{e1}))}$ $\displaystyle=$ $\displaystyle
C^{-1}\int_{B}\frac{(u(x_{1},x_{2})-u(y_{1},x_{2}))^{2}}{r^{\alpha+\beta}}\mu(dy)$

by using the fact that $\alpha_{1}+\alpha_{2}=\alpha$, from which, we have

$\int_{B}\int_{B(x_{0},r)}\frac{(u(x_{1},x_{2})-u(y_{1},x_{2}))^{2}}{d_{1}(x_{1},y_{1})^{\alpha_{1}+\beta}}\mu_{1}(dy_{1})\mu(dx)\geq
C^{-1}\int_{B}\int_{B}\frac{(u(x_{1},x_{2})-u(y_{1},x_{2}))^{2}}{r^{\alpha+\beta}}\mu(dy)\mu(dx).$

Similarly, the second integral is estimated by

$\displaystyle\int_{B}\int_{B(y_{0},r)}\frac{(u(x_{1},x_{2})-u(x_{1},y_{2}))^{2}}{d_{2}(x_{2},y_{2})^{\alpha_{2}+\beta}}\mu_{2}(dy_{2})\mu(dx)$
$\displaystyle\geq$ $\displaystyle
C^{-1}\int_{B}\int_{B}\frac{(u(x_{1},x_{2})-u(x_{1},y_{2}))^{2}}{r^{\alpha+\beta}}\mu(dy)\mu(dx)$
$\displaystyle=$ $\displaystyle
C^{-1}\int_{B}\int_{B}\frac{(u(y_{1},y_{2})-u(y_{1},x_{2}))^{2}}{r^{\alpha+\beta}}\mu(dx)\mu(dy)\text{
\ (swapping }(x_{1},x_{2})\text{ with }(y_{1},y_{2})\text{).}$

Therefore, we conclude from above that, using the elementary inequality
$a^{2}+b^{2}\geq(a+b)^{2}/2$,

$\displaystyle\int_{B}\int_{B}(u(x)-u(y))^{2}J(x,dy)\mu(dx)$
$\displaystyle\geq$ $\displaystyle
C^{-1}\left\\{\int_{B}\int_{B}\frac{(u(x_{1},x_{2})-u(y_{1},x_{2}))^{2}}{r^{\alpha+\beta}}\mu(dy)\mu(dx)\right.$
$\displaystyle+\left.\int_{B}\int_{B}\frac{(u(y_{1},y_{2})-u(y_{1},x_{2}))^{2}}{r^{\alpha+\beta}}\mu(dx)\mu(dy)\right\\}$
$\displaystyle\geq$ $\displaystyle
C^{-1}\int_{B}\int_{B}\frac{(u(x_{1},x_{2})-u(y_{1},y_{2}))^{2}}{2r^{\alpha+\beta}}\mu(dx)\mu(dy)$
$\displaystyle\geq$
$\displaystyle\frac{C^{\prime}r^{-\beta}}{\mu(B)}\int_{B}\int_{B}(u(x)-u(y))^{2}\mu(dx)\mu(dy)\
\ \text{(using (\ref{e3}))}$ $\displaystyle=$ $\displaystyle
2C^{\prime}r^{-\beta}\int_{B}(u-{u_{B}})^{2}d\mu$ $\displaystyle\geq$
$\displaystyle\frac{C}{w(B)}\int_{B}(u-{u_{B}})^{2}d\mu\ \ \text{(using
(\ref{ew})),}$

thus showing that condition $\mathrm{(PI)}$ with $\kappa$ $=1$ is satisfied.

Therefore, all the hypotheses in Theorem 1.8 are satisfied, and the weak
elliptic Harnack inequality follows. We mention that the jump kernel does not
exist by (7.6) in this case.

##  8\. Appendix

In this appendix, we collect some known results that have been cited in this
paper. Recall the John-Nirenberg inequality for BMO functions on a doubling
space.

######  Definition 8.1 (BMO function).

For a locally integrable function $u$ on an open set $\Omega$, the seminorm
$||u||_{\mathrm{BMO}(\Omega)}$ is defined by

$||u||_{\mathrm{BMO}(\Omega)}:=\sup_{B\subset\Omega}\fint_{B}|u-u_{B}|d\mu,$

where the supremum is taken over all the balls contained in $\Omega$. The
space $\mathrm{BMO}(\Omega)$ consists of all locally integrable functions $u$
on $\Omega$ such that $||u||_{\mathrm{BMO}(\Omega)}<\infty$.

The following was addressed in [1, Theorem 5.2].

######  Lemma 8.2 (John-Nirenberg inequality).

Let $(M,d,\mu)$ be a metric measure space satisfying condition
$(\mathrm{VD})$. If $u\in\mathrm{BMO}(\Omega)$ for a non-empty open subset
$\Omega$ of $M$, then

$\mu(\\{x\in B:|u-u_{B}|>\lambda\\})\geq
c_{1}\mu(B)\exp\left(-\frac{c_{2}\lambda}{||u||_{\mathrm{BMO}(\Omega)}}\right)$

for any ball with $12B\subseteq\Omega$ and any $\lambda>0$, where constants
$c_{1},c_{2}$ are independent of $u,\lambda,\Omega$ and ball $B$.

The following is a folklore, see for example [7, Corollary 5.6].

######  Lemma 8.3.

Let $(M,d,\mu)$ be a metric measure space satisfying condition
$(\mathrm{VD})$. Let $B_{0}:=B(x_{0},R)$ be a ball in $M$. Then for any
$u\in\mathrm{BMO}(B_{0})$

$\left\\{\fint_{B}\exp\left(\frac{c_{2}}{2b}u\right)d\mu\right\\}\left\\{\fint_{B}\exp\left(-\frac{c_{2}}{2b}u\right)d\mu\right\\}\leq(1+c_{1})^{2}$
(8.1)

for any ball $B$ with $12B\subseteq B_{0}$ and any
$b\geq||u||_{\mathrm{BMO}(B_{0})}$, where the constants $c_{1},c_{2}$ are the
same as in Lemma 8.2.

The following has been proved in a forthcoming paper [21].

######  Proposition 8.4.

Let $(\mathcal{E},\mathcal{F})$ be a regular Dirichlet form in $L^{2}$ without
killing part. Assume that a function $F\in C^{2}(\mathbb{R})$ satisfies

$F^{\prime\prime}\geq
0,~{}\sup_{\mathbb{R}}|F^{\prime}|<\infty,~{}\sup_{\mathbb{R}}F^{\prime\prime}<\infty.$

Then for any $u,\varphi\in\mathcal{F}^{\prime}\cap L^{\infty}$, both functions
$F(u),F^{\prime}(u)\varphi$ belong to the space $\mathcal{F}^{\prime}\cap
L^{\infty}$. Moreover, if further $\varphi\geq 0$ in $M$, then

$\mathcal{E}(F(u),\varphi)\leq\mathcal{E}(u,F^{\prime}(u)\varphi).$ (8.2)

The following is taken from in [32, Lemma 2.12].

######  Lemma 8.5.

Let $(\mathcal{E},\mathcal{F})$ be a Dirichlet form in $L^{2}$. If each
$f_{n}\in\mathcal{F}$ and

$f_{n}\overset{L^{2}}{\rightarrow}f,\text{ \
}\sup_{n}\mathcal{E}(f_{n})<\infty,$

then $f\in\mathcal{F}$, and there exists a subsequence, still denoted by
$\\{f_{n}\\}$, such that $f_{n}\overset{\mathcal{E}}{\rightharpoonup}f$
weakly, that is,

$\mathcal{E}(f_{n},\varphi)\rightarrow\mathcal{E}(f,\varphi)$

as $n\rightarrow\infty$ for any $\varphi\in\mathcal{F}$. Moreover, we have

$\mathcal{E}(f)\leq\liminf_{n\rightarrow\infty}\mathcal{E}(f_{n}).$

## References

  * [1] D. Aalto, L. Berkovits, O. E. Kansanen, and H. Yue, _John-Nirenberg lemmas for a doubling measure_ , Studia Math., 204 (2011), pp. 21–37.
  * [2] M. T. Barlow, _Some remarks on the elliptic Harnack inequality_ , Bull. London Math. Soc., 37 (2005), pp. 200–208.
  * [3] M. T. Barlow, Z.-Q. Chen, and M. Murugan, _Stability of EHI and regularity of MMD spaces_ , Preprint, (2020).
  * [4] M. T. Barlow and M. Murugan, _Stability of elliptic Harnack inequality_ , Ann. of Math. (2), 187 (2018), pp. 777–823.
  * [5] R. F. Bass and Z.-Q. Chen, _Regularity of harmonic functions for a class of singular stable-like processes_ , Math. Z., 266 (2010), pp. 489–503.
  * [6] A. Bendikov, A. Grigor’yan, E. Hu, and J. Hu, _Heat kernels and non-local Dirichlet forms on ultrametric spaces_ , Ann. Sc. Norm. Super. Pisa Cl. Sci., XXII (2021), pp. 399–461.
  * [7] M. Biroli and U. Mosco, _A Saint-Venant type principle for Dirichlet forms on discontinuous media_ , Ann. Mat. Pura Appl. (4), 169 (1995), pp. 125–181.
  * [8] , _Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces_ , Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 6 (1995), pp. 37–44.
  * [9] A. Björn and J. Björn, _Nonlinear potential theory on metric spaces_ , vol. 17 of EMS Tracts in Mathematics, European Mathematical Society (EMS), Zürich, 2011.
  * [10] E. Bombieri and E. Giusti, _Harnack’s inequality for elliptic differential equations on minimal surfaces_ , Invent. Math., 15 (1972), pp. 24–46.
  * [11] S. Boutayeb, T. Coulhon, and A. Sikora, _A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces_ , Adv. Math., 270 (2015), pp. 302–374.
  * [12] X. Cabré, _Nondivergent elliptic equations on manifolds with nonnegative curvature_ , Comm. Pure Appl. Math., 50 (1997), pp. 623–665.
  * [13] Z.-Q. Chen, T. Kumagai, and J. Wang, _Elliptic Harnack inequalities for symmetric non-local Dirichlet forms_ , J. Math. Pures Appl. (9), 125 (2019), pp. 1–42.
  * [14] , _Stability of heat kernel estimates for symmetric non-local Dirichlet forms_ , Mem. Amer. Math. Soc., 271 (2021), pp. v+89.
  * [15] E. Di Benedetto and N. S. Trudinger, _Harnack inequalities for quasi-minima of variational integrals_ , Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 1 (1984), pp. 295–308.
  * [16] A. Di Castro, T. Kuusi, and G. Palatucci, _Nonlocal Harnack inequalities_ , J. Funct. Anal., 267 (2014), pp. 1807–1836.
  * [17] , _Local behavior of fractional $p$-minimizers_, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), pp. 1279–1299.
  * [18] B. Dyda and M. Kassmann, _Regularity estimates for elliptic nonlocal operators_ , Anal. PDE, 13 (2020), pp. 317–370.
  * [19] M. Fukushima, Y. Oshima, and M. Takeda, _Dirichlet forms and symmetric Markov processes_ , Walter de Gruyter & Co., Berlin, 2011.
  * [20] A. Grigor’yan, E. Hu, and J. Hu, _Two-sided estimates of heat kernels of jump type Dirichlet forms_ , Adv. Math., 330 (2018), pp. 433–515.
  * [21] , _Parabolic mean value inequality and on-diagonal upper bound of the heat kernel_ , 2022.  to appear.
  * [22] A. Grigor’yan and J. Hu, _Heat kernels and Green functions on metric measure spaces_ , Canad. J. Math., 66 (2014), pp. 641–699.
  * [23] , _Upper bounds of heat kernels on doubling spaces_ , Mosc. Math. J., 14 (2014), pp. 505–563.
  * [24] A. Grigor’yan, J. Hu, and K.-S. Lau, _Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric measure spaces_ , J. Math. Soc. Japan, 67 (2015), pp. 1485–1549.
  * [25] J. Heinonen, _Lectures on analysis on metric spaces_ , Universitext, Springer-Verlag, New York, 2001.
  * [26] J. Kinnunen and N. Shanmugalingam, _Regularity of quasi-minimizers on metric spaces_ , Manuscripta Mathematica, 105 (2001), pp. 401–423.
  * [27] N. V. Krylov and M. V. Safonov, _An estimate for the probability of a diffusion process hitting a set of positive measure_ , Dokl. Akad. Nauk SSSR, 245 (1979), pp. 18–20.
  * [28] , _A property of the solutions of parabolic equations with measurable coefficients_ , Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), pp. 161–175, 239.
  * [29] E. M. Landis, _Some questions in the qualitative theory of second-order elliptic equations (case of several independent variables)_ , Uspehi Mat. Nauk, 18 (1963), pp. 3–62.
  * [30] E. M. Landis, _Second order equations of elliptic and parabolic type_ , vol. 171 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1998\.  Translated from the 1971 Russian original by Tamara Rozhkovskaya, with a preface by Nina Ural’tseva.
  * [31] G. Liu and M. Murugan, _Parabolic Harnack inequality implies the existence of jump kernel_ , Potential Analysis, (2020).
  * [32] Z. M. Ma and M. Röckner, _Introduction to the theory of (nonsymmetric) Dirichlet forms_ , Universitext, Springer-Verlag, Berlin, 1992.
  * [33] D. Maldonado, _On the elliptic Harnack inequality_ , Proc. Amer. Math. Soc., 145 (2017), pp. 3981–3987.
  * [34] D. Maldonado and P. R. Stinga, _Harnack inequality for the fractional nonlocal linearized Monge-Ampère equation_ , Calc. Var. Partial Differential Equations, 56 (2017), pp. Paper No. 103, 45.
  * [35] J. Moser, _A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations_ , Comm. Pure Appl. Math., 13 (1960), pp. 457–468.
  * [36] J. Moser, _On Harnack’s theorem for elliptic differential equations_ , Comm. Pure Appl. Math., 14 (1961), pp. 577–591.
  * [37] M. V. Safonov, _Harnack’s inequality for elliptic equations and Hölder property of their solutions_ , Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 96 (1980), pp. 272–287, 312.  Boundary value problems of mathematical physics and related questions in the theory of functions, 12.
  * [38] L. Saloff-Coste, _A note on Poincaré, Sobolev, and Harnack inequalities_ , Internat. Math. Res. Notices, 2 (1992), pp. 27–38.
  * [39] , _Aspects of Sobolev-type inequalities_ , Cambridge University Press, Cambridge, 2002.
  * [40] L. Silvestre, _Hölder estimates for solutions of integro-differential equations like the fractional Laplace_ , Indiana Univ. Math. J., 55 (2006), pp. 1155–1174.
  * [41] K.-T. Sturm, _Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality_ , J. Math. Pures Appl. (9), 75 (1996), pp. 273–297.


# WAVE-FRONTS OF GRAVITATIONAL WAVES PARTIALLY TRAPPED IN ULTRA-COMPACT STARS
111Based on an unpublished paper by Horák, Klimovičová and Abramowicz [3].

Marek Abramowicz  Department of Physics, Göteborg University, SE-412-96
Göteborg, Sweden
Department of Physics, Silesian University in Opava, Czech Republic
N.Copernicus Astronomical Centre (Camk, PAN), Warsaw, Poland  Jiří Horák
Institute of Astronomy, Czech Academy of Sciences, Prague, Czech Republic
Kateřina Klimovičová  Department of Physics, Silesian University in Opava,
Czech Republic

###### Abstract

We dedicate this work to Dr Omer Blaes, professor of physics at UCSB, on the
occasion of his sixtieth birthday. We have been collaborating now and then
with Dr Blaes on problems involving oscillations, waves and stability [2].
Happy birthday, Omer. Enjoy the analytic treatment of damping of the
gravitational waves trapped inside ultra compact stars and its possible
connection to Quantum Gravity in the context of the LIGO-Virgo efforts in
accurately measuring ringdowns and echoes.

##  1 Quantum Gravity is coming to town.

There is a widespread hope that the recent advances in technology for
observing the black hole high energy phenomena — in particular the
gravitational wave detections by LIGO-Virgo (LV) and the close-up imaging of
black holes in M87* and SgrA* by the Event Horizon Telescope (EHT) — could
provide means of a direct check on some Quantum Gravity (QG) ideas. Cardoso
and Pani in an excellent recent review [4] elaborate on the consensus opinion
that some of the features anticipated in observations by LV (e.g. the echoes
and/or the non-standard ringdown shape) and by EHT (e.g. “the second ring”)
may distinguish between the Kerr black hole and some of its alternatives —
e.g. strange stars, gravastars, wormholes, firewalls — by proving that the
objects observed are horizonless.

However, there is a minority opposition to this, most eloquently expressed by
the late Freeman Dyson [5]${}^{,\,}$[6], who points out that QG is meaningless
if we have no apparatus able to detect a single graviton. Dyson formally
proved that LV, even when updated, cannot for a fundamental reason detect a
single graviton. In this paper we reply to this criticism by recalling an
analogy with electromagnetic radiation. Physicists long before being able to
detect single photons, had at hand proofs of the quantum nature of light, e.g.
by observing the Fraunhofer lines. Here we suggest, although we do not prove,
that in the observed ringdown phenomenon there could be detectable imprints of
the fact that the energy of the gravitational radiation is carried in finite
portions — the gravitons.

##  2 The Teukolsky wave equation for the internal Schwarzschild metric in the
optical geometry representation.

Let us rewrite the well-known Schwarzschild solution describing the vacuum
metric outside a spherically symmetric, static body with mass $M$ (and
gravitational radius $r_{\rm G}=GM/c^{2}$) in the “optical geometry” form:

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle\left(1-\frac{r_{\rm
G}}{r}\right)\left[dt^{2}-\left(1-\frac{r_{\rm
G}}{r}\right)^{-2}dr^{2}-\left(1-\frac{r_{\rm
G}}{r}\right)^{-1}r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})\right]$ (1)
$\displaystyle=$
$\displaystyle\mathrm{e}^{-2\Phi}\left[dt^{2}-d{r_{*}}^{2}-{\tilde{r}}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right]=\mathrm{e}^{-2\Phi}\left[dt^{2}-dh^{2}\right].$ (2)

The optical geometry appears in the square brackets — it is conformal to the
original Schwarzschild metric. The second line (2) represents optical geometry
for any static, space-time geometry in spherical coordinates. The 3-D “optical
space” has the metric $-dh^{2}=\gamma_{ik}dx^{i}dx^{k}$.

In the Minkowski metric, $\Phi=0$, and $r_{*}={\tilde{r}}=r$, but in general,
one has

$\Phi=-\frac{1}{2}\ln g_{tt},\leavevmode\nobreak\ \leavevmode\nobreak\
r_{*}={\int}{{\sqrt{-\frac{g_{rr}}{g_{tt}}}}}dr,\leavevmode\nobreak\
\leavevmode\nobreak\ {\tilde{r}}=\sqrt{-\frac{g_{\phi\phi}}{g_{tt}}}.$ (3)

Along a null geodesic, i.e. along a light or gravitational wave-front
trajectory $ds=0$. From the Fermat principle, $\delta\int dt=0$ along light
trajectories in static space-times. Thus, from (2) one concludes that
$\delta\int dh=0$ along light trajectories: they are geodesic lines also in
the 3-D optical space [7].

The optical geometry corresponding to the interior of a constant density
Schwarzschild star is spherical and isometric with spatial sections of the
static Einstein Universe with the curvature scalar ${\cal
R}=6/{\tilde{a}}^{2}=$ const. The 3-D optical space corresponding to the
internal Schwarzschild solution has the metric,

$dh^{2}=\left[dr_{\ast}^{2}+{\tilde{r}}^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right)\right],\leavevmode\nobreak\
\leavevmode\nobreak\ {\tilde{r}}={\tilde{a}}\sin(r_{\ast}/{\tilde{a}}).$ (4)

The spherical geometry of the 2-D equatorial plane $\theta=\pi/2,\,d\theta=0$
is shown as an embedding diagram in Figure 4. The interior (indicated by the
color) and the exterior Schwarzschild solution, join at the radius of the star
$r=R<(3/2)r_{\rm G}$. Note that the radius ${\tilde{a}}$ of the spherical
bulge equals (we use geometrical units with $c=1=G$),

${\tilde{a}}=\frac{R}{2}\left(\frac{R}{M}\right)^{1/2}\left(1-\frac{9}{4}\frac{M}{R}\right)^{-1/2}$
(5)

where $R$ is the radius of the star and $M$ is its mass. For compactness
$M/R=4/9$ the radius of the bulge tends to infinity. This is the maximal
compactness a non-singular Schwarzschild star could have — the Buchdahl-Bondi
limit [8].

Let the wave function be described in the Minkowski space-time by the standard
formula,

$\Psi(t,r,\theta,\phi)=\mathop{{\sum_{\ell=0}^{\infty}}}\mathop{{\sum_{m=-\ell}^{\ell}}}\left\\{W_{\ell
m}(r)P^{m}_{\ell}(\cos\theta)\,\rm{e}^{-\rm{i}\omega
t}\,\rm{e}^{-\rm{i}m\phi}\right\\}.$ (6)

$P^{m}_{\ell}(\cos\theta)$ are the Legendre polynomials. After the standard
separation of variables, the radial part of the wave equation in the axially
symmetric case $\partial_{\phi}=0,\,m=0$, takes the familiar form:

$\mbox{Minkowski:}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \setstackgap{S}{0pt}\setstackEOL{\\\ }\scalerel*{\kern
1.0pt\Shortstack{\rule{12.0pt}{0.6pt}\\\ \rule{0.6pt}{10.0pt}\kern
10.0pt\rule{1.4pt}{10.0pt}\\\ \rule{12.0pt}{1.4pt}}\kern 1.0pt}{\Delta}_{\rm
r}W=\left[\,\omega^{2}-\frac{{\partial^{2}}}{{\partial
r^{2}}}+V(r)\right]W=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{where}\leavevmode\nobreak\ \leavevmode\nobreak\
V(r)=\frac{\ell(1+\ell)}{r^{2}}.$ (7)

Using the optical geometry reasoning based on the spirit of equation (2), one
may immediately guess the correct form of the wave equation (“the Teukolsky
equation”) in a general static space-time, just by removing the Minkowski
degeneracy $r=r_{*}={\tilde{r}}$.

$\mbox{General:}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\setstackgap{S}{0pt}\setstackEOL{\\\ }\scalerel*{\kern
1.0pt\Shortstack{\rule{12.0pt}{0.6pt}\\\ \rule{0.6pt}{10.0pt}\kern
10.0pt\rule{1.4pt}{10.0pt}\\\ \rule{12.0pt}{1.4pt}}\kern 1.0pt}{\Delta}_{\rm
r}W=\left[\,\omega^{2}-\frac{{\partial^{2}}}{{\partial
r_{*}^{2}}}+V(r)\right]W=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{where}\leavevmode\nobreak\ \leavevmode\nobreak\
V(r)=\frac{\ell(1+\ell)}{{\tilde{r}}^{2}}+\delta V(r).$ (8)

The frequency of the wave is a complex quantity, with ${\rm
Re}(\omega)=\sigma$ being the frequency of the quasi-normal modes, and ${\rm
Im}(\omega)=1/t_{0}$ being the inverse of a characteristic damping time of
these modes, i.e. of the “ringdown” in the LV language.

##  3 Formal solution of the Teukolsky equation.

It would be convenient to introduce new variables (see Figure 4), $\chi\equiv
r_{\ast}/\tilde{a},\leavevmode\nobreak\ \kappa\equiv\tilde{a}\omega$, and
write the Teukolsky equation (8) in a dimensionless form,

$\frac{{d^{2}W}}{{d\chi^{2}}}+\left[\kappa^{2}-\tilde{a}^{2}\left(V_{\ell}+\delta
V\right)\right]W=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{where}\leavevmode\nobreak\ \leavevmode\nobreak\
V_{\ell}\equiv\frac{\ell(\ell+1)}{{\tilde{r}^{2}}}\gg\delta V$ (9)

We solve the Teukolsky equation separately for the exterior and interior
Schwarzschild solutions, adopting the regularity condition at the center and
no ingoing wave at infinity,

$W^{\prime}(0)=0,\quad\left[W^{\prime}-\mathrm{i}\kappa
W\right]_{\chi\rightarrow\infty}=0$ (10)

The exterior solution is given by the WKBJ approximation,

$W_{\mathrm{ext}}=\left(\frac{\Theta}{\gamma}\right)^{1/4}\left[\mathrm{Ai}(\Theta)-\mathrm{i}\mathrm{Bi}(\Theta)\right],$
(11)

with $\mathrm{Ai}(\Theta)$ and $\mathrm{Bi}(\Theta)$ being the Airy functions,
and

$\Theta\equiv\left[\frac{3}{2}{\int}_{r(\chi)}^{r_{\mathrm{t}}}\gamma(r)\left(\frac{d\chi}{dr}\right)dr\right]^{2/3},\leavevmode\nobreak\
\leavevmode\nobreak\
\gamma(r)\equiv\sqrt{\tilde{a}^{2}V_{\mathrm{ext}}(r)-\kappa^{2}}.$ (12)

The interior solution is given in terms of the associated Legendre functions:

$W_{\mathrm{int}}(\chi)=(\sin\chi)^{1/2}Q^{\mu}_{\nu}(\cos\chi),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mu=\ell+1/2,\leavevmode\nobreak\ \leavevmode\nobreak\ \nu=\kappa-1/2$ (13)

Matching the interior and exterior solutions at the surface of the star $r=R$
reveals the eigenvalue nature of the problem. The matching is possible only
for the eigenfrequencies $\kappa$ characterized by

$\mathrm{Re}(\kappa)=n,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{Im}\left(\kappa\right)=-A_{\ell}\frac{{n^{2\lambda}(\ell+n)!}}{{(n-\ell-1)!}}\times\exp\left(\frac{32}{27}\pi
nx^{1/2}\right)x^{\lambda+\ell+1/2}.$ (14)

Here $\lambda^{2}=\ell(\ell+1)$, $A_{2}=1.165\times 10^{-2}$ and $x=1-9M/4R$.

##  4 Optical geometry solution for ${\rm Re}(\kappa)$ and ${\rm Im}(\kappa)$.

Figure 4: The Schwarzschild optical space ($\theta=\pi/2$ cut). The interior
solution is indicated by the color.  $\chi$ is the rescaled proper distance
from the center of the star, and $\tilde{a}$ is the circumferential radius of
great circles at the spherical bulge of the stellar interior: the great
circles are geodesic lines in the metric (4), therefore they are light
trajectories in the optical space. It was noticed [9] that a standing wave
that has $n$ nodes along a great circle, obviously has frequency

$\kappa=n$ (a)

This agrees with the formally derived equation (14) for Re($\kappa$).  One may
estimate the mode decay time $t_{0}$ by considering the black body radiation
with a uniform energy density $\varepsilon$ enclosed in a container. The
radiation flux that leaks from a small hole in the container is
$f=4\varepsilon/c$. Therefore, $t_{0}={\varepsilon\cal{V}}/{f{{\cal A}}}$,
where ${\cal V}(R)$ is the volume of the container and ${\cal A}(R)$ is the
surface area of the hole. Using the explicit form of these two functions that
are known in the Schwarzschild internal solution, one arrives at (for
$\ell=2$):

$t_{0}={\rm const}\,x^{-3/2}$ (b)

##  5 Discussion

Figure 5: Solid lines: equations (14) and (a). Points: numerical [10].

Solid lines: (14). Dashed line bb: eq. (b). They do not coincide.  Our formal
calculation of $t_{0}$ is based on the classical, non-quantum, Teukolski
equation, while the bb line in the Figure is calculated from the quantum black
body EM Planck formula: radiation in the container consists of individual
photons and is radiated away photon by photon. If this makes any practical
difference, the ringdown shape, observed sufficiently accurately, should
therefore be affected by the existence of gravitons!

##  6 Conclusions and plans

Discrete orbital structures (i.e. ones described by natural numbers) are
typical of many aspects of classical celestial mechanics: well-known examples
are Newton’s orbital resonances or Einstein’s holonomy invariance of parallel
transport [11], [12], and the kilohertz quasi-periodic oscillations [13]-
[16]. In all of these situations the gravity has no discrete aspects, only the
orbits have them. All of these effects are observed.

In this paper we described the discrete features of the “ringdowns” i.e.
damped gravitational wave-fronts: (a) classically, by solving the Teukolsky
equation and (b) quantum-mechanically, by employing a supposed graviton-photon
analogy, based on the black body formulae. These two descriptions give
different results. We argue that in a more sophisticated theoretical analysis
of the ringdowns than presented here, the discrete nature will mature into a
kind of the “Planck formula for gravitational radiation”. Because the
ringdowns are observed by the LV interferometers, this will be a sure step
towards experimental quantum gravity.

We are now working on a few follow-ups: (i) A “topology catalogue” of
ringdowns, reflections and echoes for “topology different” ultra-compact
objects. (ii) A non-linear Teukolsky equation and possible GW gravitational
resonances. (iii) Application of the Rayleigh-Jeans law into wave turbulence:
different wave modes exchange their energy (like nonlinear harmonic
oscillators) to reach equipartition. One may apply the same to GWs and perhaps
repeat the RJ derivation with a Planck-like ansatz.

## Acknowledgments

This work was supported by the Czech Science Foundation grant EXPRO 21-06825X.

## References

## References

  * [1]
  * [2] M. Abramowicz, O. Blaes, J. Horák, W. Kluźniak, P. Rebusco, Class. Quantum Grav. 23, 1689 (2006)
  * [3] J. Horák, K. Klimovičová and M Abramowicz, Class. Quantum Grav., in press, (2021)
  * [4] V. Cardoso, P. Pani, Liv. Rev. Gen. Relat. 22, 4 (2019)
  * [5] F. Dyson, Poincaré Prize Lecture Aalborg, Denmark , (2012)
  * [6] F. Dyson, New York Rev. Books , (May 13, 2004)
  * [7] M. Abramowicz, B. Carter and J.-P. Lasota, Gen. Relat. Grav. 20, 1173 (1988)
  * [8] H.A. Buchdahl, Phys. Rev. 116, 1027 (1959)
  * [9] M. Abramowicz, N. Andersson, M. Bruni, P. Ghosh P, S, Sonego, Class. Quantum Grav. 14, L189 (1997)
  * [10] N. Andersson, Y. Kojima, K.D. Kokkotas, Ap.J. 462, 855 (1996)
  * [11] T. Rothman, G.F.R. Ellis, J. Murugan, Class. Quantum Grav. 18, 1217 (2001)
  * [12] M.A. Abramowicz, G.J.E. Almergren, W. Kluźniak, A.V. Thampan, F. Wallinder, Class. Quantum Grav. 19, L57 (2002)
  * [13] G. Török, M. Abramowicz, W. Kluźniak, Z. Stuchlík, Astron. Astroph. 436, 1 (2005)
  * [14] J. Horák, Astron. Astroph. 486, 1 (2008)
  * [15] O.M. Blaes, P. Arras, P.C. Fragile, Mon. Not. Roy. astr. Soc. 369, 1235 (2006)
  * [16] P.C. Fragile, O. Straub, O.M. Blaes, Mon. Not. Roy. astr. Soc. 461, 1356 (2016)
  * [17]