# LFU violations in leptonic $\tau$ decays and $B$-physics anomalies

Lukas Allwicher<EMAIL_ADDRESS>Physik-Institut, Universität Zürich,
CH-8057 Zürich, Switzerland  Gino Isidori<EMAIL_ADDRESS>Physik-
Institut, Universität Zürich, CH-8057 Zürich, Switzerland  Nudžeim Selimović
<EMAIL_ADDRESS>Physik-Institut, Universität Zürich, CH-8057 Zürich,
Switzerland

###### Abstract

We present a complete analysis of Lepton Flavor Universality (LFU) violations
in leptonic $\tau$ decays in motivated models addressing the $B$-physics
anomalies, based on the $SU(4)\times SU(3)\times SU(2)\times U(1)$ gauge
group. We show that the inclusion of vector-like fermions, required by
$B$-physics data, leads to sizable modifications of the leading-log results
derived within an Effective Field Theory approach. In the motivated parameter-
space region relevant to the $B$-physics anomalies, the models predict a few
per-mil decrease of the effective $W$-boson coupling to $\tau$, within the
reach of future experiments.

††preprint: ZU-TH-42/21

##  I Introduction

The per-mil level tests of Lepton Flavor Universality in $\tau$ decays
Pich:2013lsa  are among the most stringent constraints on physics beyond the
Standard Model (SM) close to the electroweak scale. These tests are
particularly interesting and challenging in view of the hints of LFU
violations reported in semileptonic $B$ decays, the so-called $B$-physics
anomalies, whose evidence has been rising over the years LHCb:2021trn ;
deSimone:2020kwi . Already in the early attempts to address the $B$ anomalies,
these constraints provided serious limitations on the proposed new physics
(NP) explanations (see e.g. Ref. Greljo:2015mma ). In this context, a key
observation was made in Ref. Feruglio:2016gvd ; Feruglio:2017rjo : even if
$\tau$ decays are not affected at the tree level by NP models addressing the
$B$ anomalies, the latter necessarily affect $\tau$ decays at the one-loop
level. More precisely, NP models addressing $b\to c\tau\bar{\nu}$ anomalies
via a modification of the left-handed (semileptonic) $b$-decay amplitudes,
lead to sizable one-loop corrections in $\tau$ decays. The leading-log
contribution is model independent, and is determined by the RG evolution of
the semileptonic operators in SM Effective Field Theory (SMEFT)
Jenkins:2013wua . The size of the discrepancy between data and theory in $b\to
c\tau\bar{\nu}$ transitions naturally implies LFU violations in purely
leptonic $\tau$ decays at the few per-mil level.

So far, all analyses of these effects have been based on leading-log Effective
Field Theory (EFT) results. However, finite one-loop corrections arising from
matching conditions at the NP scale might be relevant, both given the large
values of the effective couplings in the most motivated NP models and the
small separation between electroweak and NP scales. This is particularly true
in ultraviolet (UV) complete models which predict a non-trivial spectrum for
the heavy states.

In this paper we analyse such finite corrections in the so-called 4321 models,
i.e. models based on the gauge group $SU(4)\times SU(3)\times SU(2)\times
U(1)$ DiLuzio:2017vat ; Bordone:2017bld ; Greljo:2018tuh ; DiLuzio:2018zxy ;
Fuentes-Martin:2020bnh ; Fuentes-Martin:2020hvc ; Fuentes-Martin:2020pww ,
where the color group, $SU(3)_{c}$, is the diagonal (unbroken) subgroup of
$SU(4)\times SU(3)$. The spontaneous symmetry breaking $4321\to{\rm SM}$ leads
to a massive vector leptoquark (LQ), $U_{1}$, which is a very effective tree-
level mediator for the $B$ anomalies Barbieri:2015yvd ; Buttazzo:2017ixm . We
focus in particular on flavor non-universal 4321 models Bordone:2017bld ;
Greljo:2018tuh ; Fuentes-Martin:2020bnh ; Fuentes-Martin:2020pww ; Fuentes-
Martin:2020hvc , where only third-generation fermions are charged under
$SU(4)$, providing a natural justification for the flavor structure of the
$U_{1}$ couplings Barbieri:2015yvd .

The one-loop structure of 4321 models, which naturally include also vector-
like fermions and scalar fields, has been investigated in Fuentes-
Martin:2019ign ; Fuentes-Martin:2020luw ; Fuentes-Martin:2020hvc . Recent
phenomenological analyses Cornella:2021sby  suggest a non-trivial hierarchy in
the spectrum of the different NP states, with heavy vectors and relatively
light vector-like fermions. As we shall see, the latter can play a relevant
role in the LFU breaking effects in $\tau$ decays.

##  II EFT expressions for the LFU ratios

The observables we are interested in are the purely leptonic LFU ratios

$\displaystyle\left|g^{(\tau)}_{e}/g^{(\mu)}_{e}\right|^{2}$
$\displaystyle\equiv$ $\displaystyle\frac{\Gamma(\tau\to
e\nu\bar{\nu})}{\Gamma(\mu\to e\nu\bar{\nu})}\left[\frac{\Gamma_{\rm
SM}(\tau\to e\nu\bar{\nu})}{\Gamma_{\rm SM}(\mu\to
e\nu\bar{\nu})}\right]^{-1},\quad$ (1)

with $|g^{(\tau)}_{\mu}/g^{(\mu)}_{e}|^{2}$ and
$|g^{(\tau)}_{\mu}/g^{(\tau)}_{e}|^{2}$ defined in complete analogy. By
construction, these ratios are expected to be equal to one within the SM.
Their current experimental world averages can be found in Ref. Pich:2013lsa .

We work under the assumption that the new degrees of freedom modifying $\tau$
(and $\mu$) decays occur above the electroweak scale. Under this assumption,
we can describe the relevant NP contributions via the so-called low-energy EFT
(LEFT) Lagrangian, obtained by integrating out new degrees of freedom and
heavy SM fields ($W$, $Z$, $t$, and $H$):

$\displaystyle\mathcal{L}_{\rm
LEFT}=-\frac{2}{v^{2}}\sum_{k}\,\mathcal{C}_{k}\mathcal{O}_{k}\,.$ (2)

Using the notation of Ref. Jenkins:2017jig , where the RG structure of
$\mathcal{L}^{\rm LEFT}$ can also be found, the operators contributing at the
tree level to pure leptonic decays are

$\displaystyle[\mathcal{O}_{\nu e}^{V,LL}]_{\alpha\beta\gamma\delta}$
$\displaystyle=(\bar{\nu}_{L}^{\alpha}\gamma_{\mu}\nu_{L}^{\beta})(\bar{e}_{L}^{\gamma}\gamma^{\mu}e_{R}^{\delta})\,,$
(3) $\displaystyle[\mathcal{O}_{\nu e}^{V,LR}]_{\alpha\beta\gamma\delta}$
$\displaystyle=(\bar{\nu}_{L}^{\alpha}\gamma_{\mu}\nu_{L}^{\beta})(\bar{e}_{R}^{\gamma}\gamma^{\mu}e_{R}^{\delta})\,.$
(4)

Since the SM decay amplitude is purely left-handed (LH) and we work under the
hypothesis of small NP corrections, quadratic NP effects and the contributions
of the right-handed (RH) operators can be safely neglected. To a very good
accuracy, we can write

$\displaystyle R_{\beta\alpha}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma(\ell_{\beta}\to\ell_{\alpha}\nu\bar{\nu})}{\Gamma_{\rm
SM}(\ell_{\beta}\to\ell_{\alpha}\nu\bar{\nu})}\equiv 1+\delta R_{\beta\alpha}$
(5) $\displaystyle\approx$ $\displaystyle 1+2\,{\rm Re}[\mathcal{C}_{\nu
e}^{V,LL}]^{\rm NP}_{\alpha\beta\beta\alpha}\,,$ (6)

where we have used $[\mathcal{C}_{\nu e}^{V,LL}]^{\rm
SM}_{\alpha\beta\beta\alpha}=1$, up to tiny scale-independent electroweak
corrections that we can safely neglect.

The evaluation of the leptonic LFU ratios thus reduces to the evaluation of
the NP contributions to $\mathcal{C}_{\nu e}^{V,LL}$, at the electroweak
scale. To achieve this goal we need to match the explicit NP model onto the
SMEFT Lagrangian at the heavy scale, which we normalise as

$\displaystyle\mathcal{L}_{\rm SMEFT}=-\frac{2}{v^{2}}\sum_{k}\,C_{k}O_{k}\,,$
(7)

run down to the electroweak scale, and finally match the SMEFT onto the LEFT.
Starting from the leading SMEFT semileptonic operators relevant to the
$B$-physics anomalies, namely

$\displaystyle[O_{\ell q}^{(1)}]_{\alpha\beta ij}$ $\displaystyle=$
$\displaystyle(\bar{\ell}_{L}^{\alpha}\gamma_{\mu}\ell_{L}^{\beta})(\bar{q}_{L}^{i}\gamma^{\mu}q_{L}^{j})\,,$
$\displaystyle{}[O_{\ell q}^{(3)}]_{\alpha\beta ij}$ $\displaystyle=$
$\displaystyle(\bar{\ell}_{L}^{\alpha}\sigma^{I}\gamma_{\mu}\ell_{L}^{\beta})(\bar{q}_{L}^{i}\sigma^{I}\gamma^{\mu}q_{L}^{j})\,,$
(8)

performing a tree-level matching, and considering the leading-log contribution
in the RG evolution of the SMEFT operators, leads to Feruglio:2016gvd ;
Feruglio:2017rjo

$[\mathcal{C}_{\nu e}^{V,LL}]^{\rm NP-
LL}_{\alpha\beta\beta\alpha}=-\frac{m_{t}^{2}N_{\rm
c}}{4\pi^{2}v^{2}}\log\frac{\mu^{2}}{m_{t}^{2}}\sum_{\gamma=\alpha,\beta}[C_{\ell
q}^{(3)}]_{\gamma\gamma 33}~{}.$ (9)

where $N_{\rm c}=3$ is the number of colors and $\mu$ denotes the UV matching
scale.

In this paper we go one step forward in precision, both using one-loop SMEFT-
LEFT matching conditions at the low scale, and taking into account the high-
scale one-loop matching of the 4321 model onto the SMEFT. This way we
systematically control not only the leading-log corrections but also all the
relevant finite terms (at the same order in the perturbative expansion in
terms of the LQ coupling $g_{U}$). Proceeding this way, Eq. (9) gets modified
as follows

$\displaystyle[\mathcal{C}_{\nu e}^{V,LL}]^{\rm NP-
full}_{\alpha\beta\beta\alpha}=-2\sum_{\gamma=\alpha,\beta}[C_{H\ell}^{(3)}]_{\gamma\gamma}(\mu)+$
$\displaystyle\quad+[C_{\ell\ell}]_{\alpha\beta\beta\alpha}+[C_{\ell\ell}]_{\beta\alpha\alpha\beta}+$
$\displaystyle\quad-\frac{m_{t}^{2}N_{\rm
c}}{8\pi^{2}v^{2}}\sum_{\gamma=\alpha,\beta}[C_{\ell q}^{(3)}]_{\gamma\gamma
33}\left(1+2\log\frac{\mu^{2}}{m_{t}^{2}}\right)~{}.$ (10)

Here $C_{H\ell}^{(3)}$, $C_{\ell\ell}$ are the coefficients of the operators

$\displaystyle[O_{H\ell}^{(3)}]_{\alpha\beta}$ $\displaystyle=$
$\displaystyle(\bar{\ell}^{\alpha}\gamma_{\mu}\sigma^{I}\ell^{\beta})(H^{\dagger}i\overleftrightarrow{D^{\mu}}\sigma^{I}H)\,,$
(11) $\displaystyle{}[O_{\ell\ell}]_{\alpha\beta\gamma\delta}$
$\displaystyle=$
$\displaystyle(\bar{\ell}_{L}^{\alpha}\gamma_{\mu}\ell_{L}^{\beta})(\bar{\ell}_{L}^{\gamma}\gamma^{\mu}\ell_{L}^{\delta})~{},$
(12)

obtained by the one-loop matching of the NP model onto the SMEFT. In section
IV we derive the explicit expressions of these coefficients in terms of masses
and couplings of the heavy fields in the 4321 model.

##  III The model

###  III.1 Simplified version: SM fermions only

It is convenient to consider first a simplified version of the model with
minimal fermion content. In this limit only three chiral fermions are charged
under $SU(4)$: they can be identified with the third generation of SM fermions
supplemented by a RH neutrino ($\nu_{R}^{3}$). The transformation properties
of these chiral fields under the complete 4321 gauge group is Fuentes-
Martin:2020hvc

$\displaystyle\psi_{L}$ $\displaystyle=(q_{L}^{3}\ \ell^{3}_{L})^{T}$
$\displaystyle\sim(\textbf{4, 1, 2})_{0}\,,$ (13) $\displaystyle\psi^{+}_{R}$
$\displaystyle=(t_{R}\ \nu_{R}^{3})^{T}$ $\displaystyle\sim(\textbf{4, 1,
1})_{1/2}\,,$ (14) $\displaystyle\psi^{-}_{R}$ $\displaystyle=(b_{R}\
\tau_{R})^{T}$ $\displaystyle\sim(\textbf{4, 1, 1})_{-1/2}\,,$ (15)

where $t_{R}$, $b_{R}$, and $\tau_{R}$ have been identified with the
corresponding mass-eigenstates, while $q_{L}^{3}$ and $\ell_{L}^{3}$ denote
the quark and lepton doublets. For the sake of concreteness, we assume
$q_{L}^{3}$ and $\ell_{L}^{3}$ are aligned to the down-quark and charged-
lepton mass basis, respectively (hence $\ell_{L}^{3}\equiv\ell_{L}^{\tau}$).
We comment on the impact of this assumption at the end of Section IV.4. These
quantum-number assignments give rise to the following interaction between SM
fermions and the vector LQ:

$\displaystyle\Delta\mathcal{L}_{U}$ $\displaystyle=$
$\displaystyle\frac{g_{U}}{\sqrt{2}}U_{1}^{\mu}J^{U}_{\mu}+\mathrm{h.c.}~{},\qquad$
$\displaystyle J^{U}_{\mu}$ $\displaystyle=$
$\displaystyle\bar{q}_{L}^{\,3}\gamma_{\mu}\ell^{3}_{L}+\bar{b}_{R}\gamma_{\mu}\tau_{R}+\bar{t}_{R}\gamma_{\mu}\nu^{3}_{R}\,.$
(16)

The tree-level exchange of the $U_{1}$ field leads to

$[C_{\ell q}^{(3)}]_{\tau\tau 33}=\frac{1}{2}C_{U}\,,\qquad
C_{U}=\frac{g_{U}^{2}v^{2}}{4m_{U}^{2}}~{}.$ (17)

In this simplified version of the model, the SM fermions of the first and
second generation, which are singlets under $SU(4)$, do not couple to the
$U_{1}$.

###  III.2 Inclusion of vector-like fermions

In order to generate a non-vanishing coupling of the $U_{1}$ to second
generation fermions, the field content is enlarged including an additional
$SU(4)$-charged left-handed fermion

$\displaystyle\chi_{L}$ $\displaystyle=(Q_{L}^{\prime}\ L_{L}^{\prime})^{T}$
$\displaystyle\sim(\textbf{4, 1, 2})_{0}\,,$ (18)

and a corresponding RH partner ($\chi_{R}$) with the same SM quantum
numbers.111For the purpose of this analysis, we do not need to distinguish the
case where $\chi_{R}$ transform as a $(\textbf{4, 1, 2})_{0}$, from the case
where $\chi_{R}$ indicates two separate fields ($Q_{R}$ and $L_{R}$)
transforming as $(\textbf{1, 3, 2})_{0}$ and $(\textbf{1, 1, 2})_{0}$,
respectively Fuentes-Martin:2020hvc .

After the $4321\to{\rm SM}$ symmetry breaking, the effective mass terms in the
Lagrangian lead to two vector-like (VL) states ($Q$ and $L$, with different
masses), whose LH components mix with the LH chiral fermions. The inclusion of
the new $SU(4)$-charged fields modifies the LH current in Eq. (16) into

$\bar{q}_{L}^{\,3}\gamma_{\mu}\ell^{3}_{L}\ \to\ \left(\bar{q}_{L}^{\,3}\
\bar{Q}^{\prime}_{L}\right)W\gamma_{\mu}\left(\begin{array}[]{c}\ell^{3}_{L}\\\
L^{\prime}_{L}\end{array}\right)\,,$ (19)

where $W$ is a $2\times 2$ unitary matrix with a potentially large mixing
angle controlling the mixing of the exotic fermions and third-generation
chiral fermions. The states $Q_{L}^{\prime}$ and $L_{L}^{\prime}$ are not mass
eigenstates due to the additional (small) mixing with second-generation chiral
fermions. Expressing them in terms of the the mass-eigenstates leads to

$\displaystyle Q_{L}^{\prime}$ $\displaystyle=$ $\displaystyle
c_{Q}Q_{L}-s_{Q}q_{L}^{2}\,,$ $\displaystyle L_{L}^{\prime}$ $\displaystyle=$
$\displaystyle c_{L}L_{L}-s_{L}\ell_{L}^{2}\,,$ (20)

with $s_{L,Q}\ll 1$ and $c_{L,Q}=\sqrt{1-s^{2}_{L,Q}}\approx 1$. The states
orthogonal to those in Eq. (20) are the would be second-generation chiral
fermions in absence of mixing that, by construction, do not interact with the
$U_{1}$ field (see Ref. Fuentes-Martin:2020hvc  for more details).

In principle, the model could be modified adding also heavy fermions which
could mix with the $SU(2)_{L}$-singlet chiral fermions $\psi^{\pm}_{R}$. This
addition, which implies a modification of the RH current in Eq. (16), has no
direct impact on the amplitudes we are interested in. However, it might have
an indirect impact changing the best-fit value of $C_{U}$ resulting from the
global fit of the $B$ anomalies Cornella:2021sby . After the inclusion of both
sets of heavy fermions, the LQ current in Eq. (16), expressed in terms of
mass-eigenstates, assumes the generic form

$J^{U}_{\mu}=\sum_{\\{\psi^{i}_{L}\\}}\beta_{L}^{\psi^{i}\psi^{j}}\bar{\psi}^{i}_{L}\gamma_{\mu}\psi^{j}_{L}+\sum_{\\{\psi^{i}_{R}\\}}\beta_{R}^{\psi^{i}\psi^{j}}\bar{\psi}^{i}_{R}\gamma_{\mu}\psi^{j}_{R}~{}.$
(21)

In addition to modifying the LQ current, the field $\chi_{L}$ couples to the
right-handed SM fermions and the SM Higgs field via a (4321 invariant) Yukawa
interaction

$\Delta\mathcal{L}_{Y}=Y^{\prime}_{-}\bar{\chi}_{L}\psi^{-}_{R}H+Y^{\prime}_{+}\bar{\chi}_{L}\psi^{+}_{R}\tilde{H}+{\rm
h.c.}\,,$ (22)

where $\tilde{H}=i\sigma_{2}H^{\dagger}$. Expressing the latter in term of
mass-eigenstates, leads to the following interactions between $u_{R}$,
$d_{R}$, and the heavy fermions

$\Delta\mathcal{L}_{Y}\supset
c_{Q}Y_{-}\bar{Q}_{L}d_{R}^{3}H+c_{Q}Y_{+}\bar{Q}_{L}u^{3}_{R}\tilde{H}+\rm
h.c.\,,$ (23)

where the difference between $Y^{\prime}_{\pm}$ and $Y_{\pm}$ takes into
account the possible mixing in the RH sector. Note that
$\Delta\mathcal{L}_{Y}$ induces also a contribution to the effective SM Yukawa
interaction:

$\Delta\mathcal{L}_{Y}\supset-
s_{Q}Y_{-}\bar{q}^{2}_{L}d_{R}^{3}H-s_{Q}Y_{+}\bar{q}^{2}_{L}u^{3}_{R}\tilde{H}+\rm
h.c.\,.$ (24)

This implies $|Y_{+}|\sim y_{t}|V_{cb}|/s_{Q}\gg|Y_{-}|$, where $y_{t}$ is the
top-quark Yukawa coupling and $V_{ij}$ denote the matrix elements of the
Cabibbo-Kobayashi-Maskawa matrix.

##  IV One-loop matching conditions

###  IV.1 $U_{1}$ \+ SM fields

We first derive the matching condition to $C_{H\ell}^{(3)}$ in the simplified
model with only SM fermions. To this purpose, we consider the off-shell
Green’s function

$\braket{\ell_{\beta}^{b}(0)\bar{\ell}_{\alpha}^{a}(0)H^{c}(q)H^{\dagger
d}(-q)},$ (25)

where $a,b,c,d$ are $SU(2)_{L}$ indices, and all momenta are taken incoming.
The one-loop diagrams in the UV theory contributing to this correlation
function are shown in Fig. 1. In this case $\psi_{A,B}$ is identified with
$q^{3}_{L}$ and, since we neglect the bottom Yukawa coupling, only the diagram
on the left contributes.

Figure 1: One-loop diagrams in the full theory contributing to the correlation
function $\braket{\ell\bar{\ell}HH^{\dagger}}$. $\psi_{A,B}$ generically
denote left-handed fermions charged under $SU(4)$.

Since we are interested only in the $SU(2)_{L}$-triplet component of the
correlation function, we concentrate on the part of the amplitude proportional
to the factor $(\sigma^{I})^{a}_{b}(\sigma^{I})_{c}^{d}$ (which is omitted in
the amplitudes reported below). Computing the amplitude in the full theory in
the limit $m^{2}_{t}\ll|q^{2}|\ll m^{2}_{U}$ leads to

$\displaystyle\left[\mathcal{A}_{\rm UV,0}\right]_{\tau\tau}$ $\displaystyle=$
$\displaystyle-\frac{4iN_{\rm c}}{16\pi^{2}v^{2}}\ |y_{t}|^{2}[C_{\ell
q}^{(3)}]_{\tau\tau 33}\times$ (26)
$\displaystyle\times\log\left(\frac{m_{U}^{2}}{-q^{2}}\right)\
\bar{v}(0)\not{q}P_{L}u(0)\,,$

where $P_{L}=(1-\gamma_{5})/2$, with $C_{\ell q}^{(3)}$ given in Eq. (17). As
can be seen, the amplitude exhibits an infrared singularity, which is
regularized by $q^{2}\not=0$.

Figure 2: Tree-level and one-loop diagrams in the SMEFT contributing to the
correlation function $\braket{\ell\bar{\ell}HH^{\dagger}}$.

In order to perform the matching we need to compute the same correlation
function in the SMEFT. At the one-loop level, the relevant diagrams are shown
in Fig. 2. The amplitude corresponding to the diagram in Fig. 2a is simply

$\left[\mathcal{A}_{\rm
EFT,a}\right]_{\tau\tau}=-\frac{4i}{v^{2}}[C_{H\ell}^{(3)}]_{\tau\tau}\bar{v}(0)\not{q}P_{L}u(0)\,,$
(27)

while the amplitude generated by the diagram in Fig. 2b, in the limit
$m^{2}_{t}\ll|q^{2}|$, reads

$\displaystyle\left[\mathcal{A}_{\rm EFT,b}\right]_{\tau\tau}$
$\displaystyle=-\frac{4iN_{\rm c}}{16\pi^{2}v^{2}}|y_{t}|^{2}[C_{\ell
q}^{(3)}]_{\tau\tau 33}\times$ (28)
$\displaystyle\times\left(1+\log\frac{\mu^{2}}{-q^{2}}\right)\
\bar{v}(0)\not{q}P_{L}u(0)\,.$

As expected, $\mathcal{A}_{\rm EFT,b}$ exhibits the same infrared structure of
$\mathcal{A}_{\rm UV}$. By imposing the relation

$\mathcal{A}_{\rm EFT,a}(\mu)=\mathcal{A}_{\rm UV}-\mathcal{A}_{\rm
EFT,b}(\mu)$ (29)

with $\mathcal{A}_{\rm UV}=\mathcal{A}_{\rm UV,0}$ we determine the matching
condition

$[C_{H\ell}^{(3)}]_{\tau\tau}=-\frac{1}{16\pi^{2}}N_{\rm c}|y_{t}|^{2}[C_{\ell
q}^{(3)}]_{\tau\tau 33}\left(1+\log\frac{\mu^{2}}{m_{U}^{2}}\right)\,.$ (30)

As a consistency check, from this result we deduce that the running of
$C_{H\ell}^{(3)}$ due to $C_{\ell q}$ is

$16\pi^{2}\mu\frac{\partial}{\partial\mu}[C_{H\ell}^{(3)}]_{\tau\tau}=-2N_{\rm
c}|y_{t}|^{2}[C_{\ell q}^{(3)}]_{\tau\tau 33}\,,$ (31)

which matches the known result in Jenkins:2013wua .

###  IV.2 UV amplitude in the complete model

We now proceed evaluating the contributions to the UV amplitude from the
additional heavy states present in the complete model. In this case the
contribution of the SM fermions proceeds as above, but the tree-level
expression for $C_{\ell q}^{(3)}$ changes because of the modified LQ current
in Eq. (19). In particular, one gets

$[C_{\ell q}^{(3)}]_{\tau\tau 33}=\frac{1}{2}|W_{11}|^{2}C_{U}\,,\quad[C_{\ell
q}^{(3)}]_{\mu\mu 33}=\frac{1}{2}|W_{12}|^{2}s_{L}^{2}C_{U}\,.$ (32)

The VL fermions lead to two additional terms. The diagrams in Fig. 1 where
both $\psi_{A}$ and $\psi_{B}$ are identified with VL fermions, and those
where only one of them is a VL fermion, the other being $q_{L}^{3}$. In the
first case both diagrams are non vanishing and yield222Here, we give the
amplitudes for third generation leptons in the external states. The analogous
result for the second generation leptons can be obtained by replacing
$W_{i1}\to-s_{L}W_{i2}$.

$\displaystyle\left[\mathcal{A}_{\rm VL,1}\right]_{\tau\tau}=\frac{2iN_{\rm
c}}{16\pi^{2}}\frac{g_{U}^{2}|W_{21}|^{2}c_{Q}^{4}}{4m_{U}^{2}}\times$
$\displaystyle\qquad\times(|Y_{+}|^{2}+|Y_{-}|^{2})B_{1}(x_{Q})\
\bar{v}(0)\not{q}P_{L}u(0)\,,$ (33)

where $x_{Q}=m_{Q}^{2}/m_{U}^{2}$, with $m_{Q}$ being the VL quark mass, and

$B_{1}(x_{Q})=\frac{1-x_{Q}+\log x_{Q}}{(1-x_{Q})^{2}}\,.$ (34)

In the second case, neglecting all the SM Yukawa couplings except for $y_{t}$,
the result is

$\displaystyle\left[\mathcal{A}_{\rm VL,2}\right]_{\tau\tau}$
$\displaystyle=\frac{2iN_{\rm
c}}{16\pi^{2}}\frac{g_{U}^{2}c_{Q}^{2}}{4m_{U}^{2}}2{\rm
Re}(W_{11}^{*}W_{21}Y_{+}^{*}y_{t})\times$ (35) $\displaystyle\times
B_{0}(x_{Q})\ \bar{v}(0)\not{q}P_{L}u(0)\,,$

with $B_{0}(x_{Q})=\log x_{Q}/(1-x_{Q})$.

The above results hold in the Feynman gauge. In this gauge we need to take
into account also the contributions from diagrams of the type in Fig. 1, with
the $U_{1}$ replaced by the corresponding Goldstone boson (GB). The GB
amplitudes with one or two SM fermions are vanishing, while the one with two
VL fermions yields

$\displaystyle\left[\mathcal{A}_{\rm GB}\right]_{\tau\tau}=-\frac{2iN_{\rm
c}}{16\pi^{2}}\frac{g_{U}^{2}|W_{21}|^{2}c_{Q}^{4}}{4m_{U}^{2}}\times$
$\displaystyle\qquad\times(|Y_{+}|^{2}+|Y_{-}|^{2})B_{2}(x_{Q})\
\bar{v}(0)\not{q}P_{L}u(0)\,,$ (36)

where

$B_{2}(x_{Q})=\frac{x_{Q}-x_{Q}^{2}+x_{Q}^{2}\log x_{Q}}{4(1-x_{Q})^{2}}\,.$
(37)

Finally, the contributions where the $U_{1}$ is replaced by the corresponding
radial excitation (Higgs mode, with mass $m_{h_{U}}$) should also taken into
account. In this case we find

$\displaystyle\left[\mathcal{A}_{\rm R}\right]_{\tau\tau}=-\frac{2iN_{\rm
c}}{16\pi^{2}}\frac{g_{U}^{2}|W_{21}|^{2}c_{Q}^{4}\tan^{2}\beta}{4m_{U}^{2}}\times$
$\displaystyle\qquad\times(|Y_{+}|^{2}+|Y_{-}|^{2})B_{2}(x_{Q}^{R})\
\bar{v}(0)\not{q}P_{L}u(0)\,,$ (38)

with $x_{Q}^{R}=m_{Q}^{2}/m_{h_{U}}^{2}$ and
$\tan\beta=\omega_{1}/\omega_{3}$, where $\omega_{1}$ and $\omega_{3}$ are the
vacuum expectation values of the scalar fields mediating the 4321 $\to$ SM
breaking Fuentes-Martin:2020hvc . We neglect model-dependent contributions
involving quartic scalar couplings of the radial modes.

###  IV.3 Complete matching for $C_{H\ell}^{(3)}$

We are now in the position to sum all the contributions and obtain the
matching conditions for both $[C_{H\ell}^{(3)}]_{\tau\tau}$ and
$[C_{H\ell}^{(3)}]_{\mu\mu}$ in the complete model. Proceeding as in Eq. (29)
with

$\mathcal{A}_{\rm UV}=\mathcal{A}_{\rm UV,0}+\mathcal{A}_{\rm
VL,1}+\mathcal{A}_{\rm VL,2}+\mathcal{A}_{\rm GB}+\mathcal{A}_{\rm R}\,,$ (39)

we obtain

$\displaystyle[C_{H\ell}^{(3)}]_{\tau\tau}(\mu)=-\frac{1}{16\pi^{2}}\frac{N_{\rm
c}C_{U}}{2}\Big{[}|W_{11}|^{2}|y_{t}|^{2}\left(1+\log\frac{\mu^{2}}{m_{U}^{2}}\right)$
$\displaystyle\qquad+c_{Q}^{2}2\text{Re}(W_{11}^{*}W_{21}Y_{+}^{*}y_{t})B_{0}(x_{Q})$
$\displaystyle\qquad+c_{Q}^{4}|W_{21}|^{2}(|Y_{+}|^{2}+|Y_{-}|^{2})F(x_{Q},x_{Q}^{R})\Big{]}\,,$
(40)

and

$\displaystyle[C_{H\ell}^{(3)}]_{\mu\mu}(\mu)=-\frac{1}{16\pi^{2}}\frac{N_{\rm
c}C_{U}}{2}s_{L}^{2}\Big{[}|W_{12}|^{2}|y_{t}|^{2}\left(1+\log\frac{\mu^{2}}{m_{U}^{2}}\right)$
$\displaystyle\qquad+c_{Q}^{2}2\text{Re}(W_{12}^{*}W_{22}Y_{+}^{*}y_{t})B_{0}(x_{Q})$
$\displaystyle\qquad+c_{Q}^{4}|W_{22}|^{2}(|Y_{+}|^{2}+|Y_{-}|^{2})F(x_{Q},x_{Q}^{R})\Big{]}\,,$
(41)

where $F(x_{Q},x_{Q}^{R})=B_{1}(x_{Q})-B_{2}(x_{Q})-\tan^{2}\beta
B_{2}(x_{Q}^{R})$. Having introduced a single VL fermion, the first generation
leptons do not couple to the new dynamics and $[C_{H\ell}^{(3)}]_{ee}=0$.

###  IV.4 Matching to $C_{\ell\ell}$

The one-loop (LQ-box) contributions to the SMEFT operator
$[O_{\ell\ell}]_{\alpha\beta\gamma\delta}$ have been calculated in Ref.
Fuentes-Martin:2020hvc . The coefficients relevant to our analysis are

$[C_{\ell\ell}]_{\tau\mu\mu\tau}=[C_{\ell\ell}]_{\mu\tau\tau\mu}=C_{U}\frac{g_{U}^{2}}{16\pi^{2}}s_{L}^{2}B_{\ell\ell}^{1212}\,,$
(42)

where the explicit expression for the functions $B_{\ell\ell}^{ijkl}$ can be
found in Ref. Fuentes-Martin:2020hvc . Also in this case, Wilson coefficients
involving first generation fermions have vanishing contributions.

So far, we assumed that the third-generation lepton doublet charged under
$SU(4)$, namely $\ell_{L}^{3}$, can be identified with the $\ell_{L}^{\tau}$
doublet, defined by the $\tau$ mass-eigenstate. In general, a small
misalignment is possible. If the RH current of the $U_{1}$ is close to its
expectation in the minimal setup (i.e. if $\beta_{R}^{b\tau}\approx 1$),
bounds from $\tau\to\mu\gamma$ and $B_{s}\to\mu^{+}\mu^{-}$ allow a mixing of
at most $O(10^{-2})$ between $\ell_{L}^{3}$ and the mass-eigenstate
$\ell_{L}^{\mu}$ Fuentes-Martin:2019mun . These bounds are less stringent if
$|\beta_{R}^{b\tau}|\ll 1$: in this case the $\ell_{L}^{3}$–$\ell_{L}^{\mu}$
mixing, that we parameterize via the angle $s_{\tau}$ defined as in Fuentes-
Martin:2019mun , could be as large as $O(10^{-1})$. A non-vanishing
$s_{\tau}$, up to $O(10^{-1})$, has a negligible impact in all the amplitudes
evaluated so far. However, it leads to an additional non-vanishing
contribution to $[C_{\ell\ell}]_{\tau\mu\mu\tau}$ via the tree-level
$Z^{\prime}$-exchange amplitude (which involves only $\ell_{L}^{3}$).
Neglecting the subleading terms of $\mathcal{O}(g_{\rm SM}^{2})$ Fuentes-
Martin:2020hvc , this contribution yields

$[C_{\ell\ell}]_{\tau\mu\mu\tau}=[C_{\ell\ell}]_{\mu\tau\tau\mu}=\frac{3g_{U}^{2}v^{2}}{16m_{Z^{\prime}}^{2}}s_{\tau}^{2}~{}.$
(43)

##  V Numerics

We have now all the ingredients to estimate the complete impact of 4321
dynamics on the leptonic LFU ratios. Putting all the pieces together, the
corrections to the leptonic decay widths defined in Eq. (5) assume the
following form

$\displaystyle\delta R_{\mu e}=\frac{N_{\rm
c}}{16\pi^{2}}C_{U}\Big{[}|\beta_{L}^{b\mu}|^{2}|y_{t}|^{2}\left(1+2L_{U}\right)$
$\displaystyle\qquad+4c_{Q}\text{Re}(\beta_{L}^{b\mu^{*}}\beta_{L}^{Q\mu\,}Y_{+}^{*}y_{t})B_{0}(x_{Q})$
$\displaystyle\qquad+2c_{Q}^{2}|\beta_{L}^{Q\mu}|^{2}(|Y_{+}|^{2}+|Y_{-}|^{2})F(x_{Q},x_{Q}^{R})\Big{]}\,,$
(44) $\displaystyle\delta R_{\tau e}=\frac{N_{\rm
c}}{16\pi^{2}}C_{U}\Big{[}|\beta_{L}^{b\tau}|^{2}|y_{t}|^{2}\left(1+2L_{U}\right)$
$\displaystyle\qquad+4c_{Q}\text{Re}(\beta_{L}^{b\tau^{*}}\beta_{L}^{Q\tau\,}Y_{+}^{*}y_{t})B_{0}(x_{Q})$
$\displaystyle\qquad+2c_{Q}^{2}|\beta_{L}^{Q\tau}|^{2}(|Y_{+}|^{2}+|Y_{-}|^{2})F(x_{Q},x_{Q}^{R})\Big{]}\,,$
(45) $\displaystyle\delta R_{\tau\mu}=\frac{N_{\rm
c}}{16\pi^{2}}C_{U}\Big{\\{}(|\beta_{L}^{b\tau}|^{2}+|\beta_{L}^{b\mu}|^{2})|y_{t}|^{2}\left(1+2L_{U}\right)$
$\displaystyle\qquad+4c_{Q}\text{Re}\left[\left(\beta_{L}^{b\tau^{*}}\beta_{L}^{Q\tau}+\beta_{L}^{b\mu^{*}}\beta_{L}^{Q\mu}\right)Y_{+}^{*}y_{t}\right]B_{0}(x_{Q})$
$\displaystyle\qquad+2c_{Q}^{2}(|\beta_{L}^{Q\tau}|^{2}+|\beta_{L}^{Q\mu}|^{2})(|Y_{+}|^{2}+|Y_{-}|^{2})F(x_{Q},x_{Q}^{R})\Big{\\}}$
$\displaystyle\qquad+C_{U}\frac{g_{U}^{2}}{4\pi^{2}}s_{L}^{2}B_{\ell\ell}^{1212}+\frac{3g_{U}^{2}v^{2}}{4x_{Z^{\prime}}m_{U}^{2}}s^{2}_{\tau}\,,$
(46)

where $L_{U}=\log(m_{t}^{2}/m_{U}^{2})$ and
$x_{Z^{\prime}}=m_{Z^{\prime}}^{2}/m_{U}^{2}$. In the above expressions we
have replaced the dependence from the $W_{ij}$ matrix elements via the
effective $U_{1}$ couplings defined as in Eq. (21):

$\displaystyle\begin{aligned}
\beta_{L}^{b\tau}&=W_{11}\,,&\beta_{L}^{b\mu}&=-s_{L}W_{12}\,,\\\
\beta_{L}^{Q\tau}&=c_{Q}W_{21}\,,&\beta_{L}^{Q\mu}&=-c_{Q}s_{L}W_{22}\,.\end{aligned}$
(47)

Figure 3:  Separate contributions to $\delta R_{\tau\mu}$ in the 4321 model,
using the benchmark values in Eqs. (48)–(51), as a function of $x_{Q}$: 1)
modified $W$-boson vertex from $U_{1}$+SM fermions (black line), 2) modified
$W$-boson vertex from box with one and two VL fermions, for $Y_{+}>0$ (red) or
$Y_{+}<0$ (green), 3) $Z^{\prime}$-exchange amplitude (grey band); 4) four-
lepton box amplitude (blue line).

To estimate the size of the different contributions to the LFU ratios we
choose the following benchmark values for the LQ couplings

$\displaystyle C_{U}=\frac{g_{U}^{2}v^{2}}{4m_{U}^{2}}$
$\displaystyle=0.01\,,\quad$ $\displaystyle m_{U}=4\ \rm TeV\,,\,$
$\displaystyle\beta_{L}^{b\tau}$ $\displaystyle=1,\quad$
$\displaystyle\beta_{L}^{b\mu}=-0.2\,.\quad$ (48)

These values are representative of the best-fit point obtained in Ref.
Cornella:2021sby  when fitting present data, in the limit of small RH
couplings ($|\beta_{R}^{b\tau}|\ll 1$), and are consistent with the model
expectation $W_{ij}=O(1)$ and $|s_{L}|\ll 1$. For the latter reason we further
set

$c_{Q}=1\,,\quad\beta_{L}^{Q\tau}=1,\quad\ \beta^{Q\mu}=-0.2\,.$ (49)

Given the discussion in Section III.2, we neglect $Y_{-}$ and vary $Y_{+}$ in
the interval

$0.2<|Y_{+}|<1.0~{},$ (50)

corresponding to $|V_{cb}|<|s_{Q}|<|V_{us}|$. In principle, the signs of
$c_{Q}$, $\beta_{L}^{Q\tau}$, and $\beta^{Q\mu}$ could be varied; however,
only the relative sign of these couplings and $Y_{+}$ is relevant in $\delta
R_{\alpha\beta}$. Therefore we effectively explore all relevant options
varying the sign of $Y_{+}$.

Concerning the $Z^{\prime}$ contribution, at fixed $C_{U}$ the result in Eq.
(43) depends only on the combination $s_{\tau}^{2}/x_{Z^{\prime}}$. For the
sake of simplicity, we set $x_{Z^{\prime}}=3/5$333Thos corresponds to a heavy
mass for coloron, $m^{2}_{G}=(9/5)m^{2}_{U}$, which better evades direct
constraints Cornella:2021sby . and vary $s_{\tau}$ in the interval

$0<s_{\tau}<0.1~{}.$ (51)

We finally consider the limit of heavy radial excitation, setting
$x_{Q}^{R}=x_{Q}/(4\pi)^{2}$.

In Fig. 3 we show all contributions separately in the case of $\delta
R_{\tau\mu}$, which is sensitive to all types of amplitudes. As expected, the
contribution from $U_{1}$+SM fermions, which includes the LL result, is
dominant. However, the contribution from VL fermions represents a significant
correction. On the other hand, the $Z^{\prime}$-exchange and the four-lepton
box amplitudes are clearly subleading and safely negligible in most of the
parameter space.444The $Z^{\prime}$-exchange amplitude is below $10\%$, in
size, of the leading contribution from $U_{1}$+SM fermions for
$|s_{\tau}|\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}0.07$.

Figure 4: Comparison of the full calculation for $\delta R_{\tau\mu}$ (green
band) in the case $Y_{+}<0$, with the LL result obtained using $m_{U}=4$ TeV
as UV matching scale (black line). The vertical scale on the right indicates
the effective UV matching scale necessary for the LL result to produce a value
of $\delta R_{\tau\mu}$ as indicated by the vertical scale on the left.
Figure 5: Predictions for $\delta R_{\tau e}$ and $\delta R_{\mu e}$ in the
4321 model, using the benchmark values in Eqs. (48)–(50), as a function of
$x_{Q}$. The bands correspond to the full model results. The right plot is a
zoom on the upper part of the left plot in order to illustrate the different
contributions to $\delta R_{\mu e}$.

In Fig. 4 we compare the LL result for $\delta R_{\tau\mu}$ with the full
calculation in the case of $Y_{+}<0$ (and $s_{\tau}=0$), where VL fermions
decrease the effect induced by SM fermions only. As expected, in this case the
effect is equivalent to that of decreasing the UV matching scale of the the LL
result, from its natural value (namely $m_{U}$). The correction is sizable,
corresponding to an effective decrease of the matching scale from 4 TeV to
about 2 TeV or less. This effect is very relevant in decreasing the present
tension with data when fitting the $B$ anomalies Cornella:2021sby .

In Fig. 5 we show the results for both $\delta R_{\tau e}$ and $\delta R_{\mu
e}$. As expected, the result for $\delta R_{\tau e}$ is almost identical to
that of $\delta R_{\tau\mu}$, whereas the breaking of universality in $\delta
R_{\mu e}$ is one order of magnitude smaller, reaching $\mathcal{O}(10^{-4})$
at most. Note that in both cases, the unambiguous predictions following from
$B$ anomalies is a reduction of the LFU ratios from one.

##  VI The effective $W$\- and $Z$-boson couplings.

Figure 6:  Comparison between data and model predictions for the effective
$W$-boson couplings to leptons ($g^{W}_{\ell}$). The hatched green areas
indicate the predictions of 4321 models (full model) in the region relevant to
the $B$-physics anomalies. The corresponding LL results are highlighted in
gray. The red bands are the experimental results obtained using purely
leptonic decays Pich:2013lsa . The blue band is the value of
$|g^{W}_{\tau}/g^{W}_{\mu}|^{2}$ determined in Ref. Arroyo-Urena:2021nil  from
$\Gamma(\tau\to\pi\nu)/\Gamma(\pi\to\mu\bar{\nu})$.

The smallness of the $Z^{\prime}$-exchange and the four-lepton box amplitudes
allow us to describe the breaking of universality in leptonic $\tau$ decays
occurring in 4321 models as modifications of the effective $W$-boson couplings
to leptons ($g^{W}_{\ell}$). Defining the latter as

${\cal{L}}^{(\ell,W)}_{\rm
eff}=-{g_{\ell}^{W}\over\sqrt{2}}~{}\overline{\nu}_{\ell}\gamma^{\mu}P_{L}\ell~{}W^{+}_{\mu}+{\rm
h.c.},$ (52)

the ratios introduced in Eq. (1) can be expressed as

$\left|\frac{g^{(\tau)}_{e}}{g^{(\mu)}_{e}}\right|^{2}\approx\left|\frac{g_{\tau}^{W}}{g_{\mu}^{W}}\right|^{2}~{}.$
(53)

The smallness of NP effects for one-particle irreducible amplitudes involving
the first generation of quarks, implies the same effective $W$-boson couplings
can also be extracted from $\Gamma(\tau\to\pi\nu)$ and
$\Gamma(\pi\to\mu\bar{\nu})$.

In Fig. 6 we compare our results with the extraction of
$|g^{W}_{\tau}/{g^{W}_{\mu}}|$ using both leptonic and pion decays:

$\displaystyle\left|g_{\tau}^{W}/g_{\mu}^{W}\right|^{2}_{\tau-\rm decays}$
$\displaystyle=$ $\displaystyle 1.0022\pm
0.0030~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Pich:2013lsa}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
(54) $\displaystyle\left|g_{\tau}^{W}/g_{\mu}^{W}\right|^{2}_{\pi-\rm decays}$
$\displaystyle=$ $\displaystyle 0.9928\pm
0.0076~{}\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Arroyo-
Urena:2021nil}{\@@citephrase{(}}{\@@citephrase{)}}}}\,.$ (55)

We also compare the model prediction for $|g^{W}_{\tau}/{g^{W}_{e}}|$ with

$\displaystyle\left|g_{\tau}^{W}/g_{e}^{W}\right|^{2}_{\tau-\rm decays}$
$\displaystyle=$ $\displaystyle 1.0060\pm
0.0030~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Pich:2013lsa}{\@@citephrase{(}}{\@@citephrase{)}}}}\,.$
(56)

In order to obtain robust estimates, we vary $C_{U}$ in the interval
$0.005<C_{U}<0.01$ (with fixed $m_{U}=4$ TeV) and consider both $Y_{+}>0$ and
$Y_{+}<0$ (with $s_{\tau}=0$). As can be seen, in the
$|g^{W}_{\tau}/{g^{W}_{\mu}}|$ case present data are not precise enough to
distinguish the SM from the 4321 model (in the region relevant to the
$B$-physics anomalies). In the $|g^{W}_{\tau}/{g^{W}_{e}}|$ case, the
inclusion of the contributions from VL fermions decreases the tension with
present data, which is reduced to about $2\sigma$ for $Y_{+}<0$. In both
cases, a reduction of the present error by a factor 2-3 on the $\tau$ decay
widths, which might be accessible at Belle-II, could allow to perform very
stringent test of 4321 models in the motivated parameter-space region.

For completeness, we note that in this framework also the left-handed
couplings of the $Z$ boson to charged leptons and neutrinos are modified. The
calculation proceeds very similarly to the one presented in Section IV for the
$W$-boson couplings, the only relevant difference being the presence of the
singlet operator $O_{H\ell}^{(1)}$. The corresponding matching conditions
reads

$\displaystyle[C_{H\ell}^{(1)}]_{\tau\tau}(\mu)=\frac{1}{16\pi^{2}}\frac{N_{\rm
c}C_{U}}{2}\Big{[}|W_{11}|^{2}|y_{t}|^{2}\left(1+\log\frac{\mu^{2}}{m_{U}^{2}}\right)$
$\displaystyle\qquad+c_{Q}^{2}2\text{Re}(W_{11}^{*}W_{21}Y_{+}^{*}y_{t})B_{0}(x_{Q})$
$\displaystyle\qquad+c_{Q}^{4}|W_{21}|^{2}(|Y_{+}|^{2}-|Y_{-}|^{2})F(x_{Q},x_{Q}^{R})\Big{]}\,,$
(57)
$\displaystyle[C_{H\ell}^{(1)}]_{\mu\mu}(\mu)=\frac{1}{16\pi^{2}}\frac{N_{\rm
c}C_{U}}{2}s_{L}^{2}\Big{[}|W_{12}|^{2}|y_{t}|^{2}\left(1+\log\frac{\mu^{2}}{m_{U}^{2}}\right)$
$\displaystyle\qquad+c_{Q}^{2}2\text{Re}(W_{12}^{*}W_{22}Y_{+}^{*}y_{t})B_{0}(x_{Q})$
$\displaystyle\qquad+c_{Q}^{4}|W_{22}|^{2}(|Y_{+}|^{2}-|Y_{-}|^{2})F(x_{Q},x_{Q}^{R})\Big{]}\,,$
(58)

while $[C_{H\ell}^{(1)}]_{ee}\approx 0$. Defining the effective left-handed
$Z$-boson couplings as

$\mathcal{L}_{\text{eff}}^{(\ell,Z)}=-\frac{g_{2}}{c_{W}}\left[g_{\ell_{L}}^{Z}(\bar{\ell}\gamma^{\mu}P_{L}\ell)+g_{\nu_{\ell}}^{Z}(\bar{\nu}_{\ell}\gamma^{\mu}P_{L}\nu_{\ell})\right]Z_{\mu}\,,$
(59)

where $g_{2}$ is the $SU(2)_{L}$ gauge coupling, $c_{W}$ denotes the cosine of
the weak angle, and
$g^{Z,\text{SM}}_{\nu_{\ell}}=-g^{Z,\text{SM}}_{\ell_{L}}=1/2$, the modified
couplings ($g^{Z}_{i}=g^{Z,\text{SM}}_{i}+\delta g^{Z}_{i}$) are

$\displaystyle\delta g^{Z}_{\nu_{\ell}}(\mu)$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{2}\left\\{[C_{H\ell}^{(1)}]_{\ell\ell}(\mu)-[C_{H\ell}^{(3)}]_{\ell\ell}(\mu)\right\\}\,,$
(60) $\displaystyle\delta g^{Z}_{\ell_{L}}(\mu)$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{2}\left\\{[C_{H\ell}^{(1)}]_{\ell\ell}(\mu)+[C_{H\ell}^{(3)}]_{\ell\ell}(\mu)\right\\}\,.$
(61)

Since the leading contributions controlled by $y_{t}$ and $Y_{+}$ are equal
and opposite in $C_{H\ell}^{(1)}$ and $C_{H\ell}^{(3)}$, Eqs. (60)–(61) imply
a sizable modification of $g^{Z}_{\nu_{\tau}}$ and negligible corrections to
all the $g^{Z}_{\ell_{L}}$. Neglecting the subleading contribution
proportional to $|Y_{-}|^{2}$ we get

$\left.\delta g^{Z}_{\ell_{L}}\right|_{Y_{-}=0}=0\,,\quad\left.\frac{\delta
g^{Z}_{\nu_{\ell}}}{g^{Z,{\rm
SM}}_{\nu_{\ell}}}\right|_{Y_{-}=0}=\left.\frac{\delta g^{W}_{\ell}}{g^{W,{\rm
SM}}_{\ell}}\right|_{Y_{-}=0}.$ (62)

According to this result, the most significant constraint on the model from
$Z$-pole observables arises by the invisible decay width of the $Z$-boson, or
the effective number of LH neutrinos ($N_{\nu}^{\rm eff}$) determined by this
observable ALEPH:2005ab . Assuming that only $\delta g^{Z}_{\nu_{\tau}}$
receives a sizable correction (as expected in our model), we find

$\left|\frac{g^{Z}_{\nu_{\tau}}}{g^{Z,\rm
SM}_{\nu_{\tau}}}\right|^{2}_{N_{\nu}^{\rm eff}}=N_{\nu}^{\rm eff}-2=0.9840\pm
0.0082~{},$ (63)

which is slightly less stringent than the constraints from the effective $W$
couplings in Eqs. (54)–(56).

##  VII Conclusion

The recent $B$-physics anomalies have strengthened the importance of precise
tests of LFU in all accessible processes involving charged leptons. In this
paper we have presented the first complete analysis of LFU violations in
leptonic $\tau$ decays, within the motivated class of 4321 models addressing
the $B$-physics anomalies DiLuzio:2017vat ; Bordone:2017bld ; Greljo:2018tuh ;
DiLuzio:2018zxy ; Fuentes-Martin:2020bnh ; Fuentes-Martin:2020hvc ; Fuentes-
Martin:2020pww . As originally pointed out in Ref. Feruglio:2016gvd ;
Feruglio:2017rjo  via a general EFT approach, the $b\to c\tau\nu$ anomaly
implies a decrease of the effective $W$-boson coupling to $\tau$ leptons in
the few per-mil range. While confirming this general conclusion, we have shown
that the inclusion of vector-like fermions, which is motivated by $B$-physics
data in this context, can lead to sizable modifications of the EFT results. In
particular, the inclusion of vector-like fermions can partially decrease the
present tension of 4321 models with data on leptonic decays. Most importantly,
the results presented in this work could lead to very stringent tests for this
class of models, in the region favored by $B$-physics data, with the help of
future precision measurements of leptonic $\tau$ decay widths.

### Acknowledgments

We thank Claudia Cornella and Javier Fuentes-Martín for useful discussions.
This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
under grant agreement 833280 (FLAY), and by the Swiss National Science
Foundation (SNF) under contract 200021-175940.

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# Blow-up problem for nonlinear nonlocal parabolic equation with absorption
under nonlinear nonlocal boundary condition

Alexander Gladkov  Alexander Gladkov
Department of Mechanics and Mathematics
Belarusian State University
4 Nezavisimosti Avenue
220030 Minsk, Belarus and Peoples’ Friendship University of Russia (RUDN
University), 6 Miklukho-Maklaya street, Moscow 117198, Russian Federation
<EMAIL_ADDRESS>
###### Abstract.

In this paper we consider initial boundary value problem for nonlinear
nonlocal parabolic equation with absorption under nonlinear nonlocal boundary
condition and nonnegative initial datum. We prove comparison principle, global
existence and blow-up of solutions.

###### Key words and phrases:

Nonlocal parabolic equation; nonlocal boundary condition; blow-up; global
existence

###### 2020 Mathematics Subject Classification:

35K20, 35K58, 35K61

##  1\. Introduction

In this paper we consider the initial boundary value problem for nonlinear
nonlocal parabolic equation

$u_{t}=\Delta u+au^{p}\int_{\Omega}u^{q}(y,t)\,dy-bu^{m},\;x\in\Omega,\;t>0,$
(1.1)

with nonlinear nonlocal boundary condition

$\frac{\partial
u(x,t)}{\partial\nu}=\int_{\Omega}{k(x,y,t)u^{l}(y,t)}\,dy,\;x\in\partial\Omega,\;t>0,$
(1.2)

and initial datum

$u(x,0)=u_{0}(x),\;x\in\Omega,$ (1.3)

where $a,\,b,\,p,\,q,\,m,\,l$ are positive numbers, $\Omega$ is a bounded
domain in $\mathbb{R}^{N}$ for $N\geq 1$ with smooth boundary
$\partial\Omega$, $\nu$ is unit outward normal on $\partial\Omega.$

Throughout this paper we suppose that the functions $k(x,y,t)$ and $u_{0}(x)$
satisfy the following conditions:

$k(x,y,t)\in
C(\partial\Omega\times\overline{\Omega}\times[0,+\infty)),\;k(x,y,t)\geq 0;$
$u_{0}(x)\in C^{1}(\overline{\Omega}),\;u_{0}(x)\geq 0\textrm{ in
}\Omega,\;\frac{\partial
u_{0}(x)}{\partial\nu}=\int_{\Omega}{k(x,y,0)u_{0}^{l}(y)}\,dy\textrm{ on
}\partial\Omega.$

Various phenomena in the natural sciences and engineering lead to the
nonclassical mathematical models subject to nonlocal boundary conditions. For
global existence and blow-up of solutions for parabolic equations and systems
with nonlocal boundary conditions we refer to [1, 5, 8, 16, 17, 18, 21, 22,
23, 30, 31, 32] and the references therein. In particular, the blow-up problem
for parabolic equations with nonlocal boundary condition

$u(x,t)=\int_{\Omega}k(x,y,t)u^{l}(y,t)\,dy,\;x\in\partial\Omega,\;t>0,$

was considered in [4, 6, 9, 12, 28]. Initial boundary value problems for
parabolic equations with nonlocal boundary condition (1.2) were studied in
[13, 14, 26, 27, 29]. So, the problem (1.1)–(1.3) with $a=0$ was investigated
in [10, 11]. Initial-boundary value problems for nonlocal parabolic equations
with nonlocal boundary conditions were addressed in many papers also (see, for
example, [3, 7, 15, 25, 34, 35, 36]). In particular, some global existence and
blow-up results for (1.1)–(1.3) were obtained in [33].

The aim of this paper is to investigate global existence and blow-up of
solutions of (1.1)–(1.3).

This paper is organized as follows. In the next section we prove comparison
principle. The global existence of solutions for any initial data is proved in
Section 3. In Section 4 we present finite time blow-up results for solutions
with large initial data.

##  2\. Comparison principle

In this section a comparison principle for (1.1)–(1.3) will be proved. We
begin with definitions of a supersolution, a subsolution and a maximal
solution of (1.1)–(1.3). Let
$Q_{T}=\Omega\times(0,T),\;S_{T}=\partial\Omega\times(0,T)$,
$\Gamma_{T}=S_{T}\cup\overline{\Omega}\times\\{0\\}$, $T>0$.

######  Definition 2.1.

We say that a nonnegative function $u(x,t)\in C^{2,1}(Q_{T})\cap
C^{1,0}(Q_{T}\cup\Gamma_{T})$ is a supersolution of (1.1)–(1.3) in $Q_{T}$ if

$u_{t}\geq\Delta u+au^{p}\int_{\Omega}u^{q}(y,t)\,dy-bu^{m},\;(x,t)\in Q_{T},$
(2.1) $\frac{\partial
u(x,t)}{\partial\nu}\geq\int_{\Omega}{k(x,y,t)u^{l}(y,t)}\,dy,\;x\in\partial\Omega,\;0<t<T,$
(2.2) $u(x,0)\geq u_{0}(x),\;x\in\Omega,$ (2.3)

and $u(x,t)\in C^{2,1}(Q_{T})\cap C^{1,0}(Q_{T}\cup\Gamma_{T})$ is a
subsolution of (1.1)–(1.3) in $Q_{T}$ if $u\geq 0$ and it satisfies
(2.1)–(2.3) in the reverse order. We say that $u(x,t)$ is a solution of
problem (1.1)–(1.3) in $Q_{T}$ if $u(x,t)$ is both a subsolution and a
supersolution of (1.1)–(1.3) in $Q_{T}$.

######  Definition 2.2.

We say that a solution $u(x,t)$ of (1.1)–(1.3) in $Q_{T}$ is a maximal
solution if for any other solution $v(x,t)$ of (1.1)–(1.3) in $Q_{T}$ the
inequality $v(x,t)\leq u(x,t)$ is satisfied for $(x,t)\in
Q_{T}\cup\Gamma_{T}$.

######  Theorem 2.3.

Let $\overline{u}$ and $\underline{u}$ be a supersolution and a subsolution of
problem (1.1)–(1.3) in $Q_{T},$ respectively. Suppose that
$\underline{u}(x,t)>0$ or $\overline{u}(x,t)>0$ in ${Q}_{T}\cup\Gamma_{T}$ if
$\min(p,q,l)<1.$ Then $\overline{u}(x,t)\geq\underline{u}(x,t)$ in
${Q}_{T}\cup\Gamma_{T}.$

###### Proof.

Suppose that $\min\\{p,q,l\\}\geq 1.$ Let $T_{0}\in(0,T)$ and
$\\{\varepsilon_{n}\\}$ be decreasing to $0$ a sequence such that
$0<\varepsilon_{n}<1.$ For $\varepsilon=\varepsilon_{n}$ let
$u_{0\varepsilon}(x)$ be the functions with the following properties:
$u_{0\varepsilon}(x)\in C^{1}(\overline{\Omega}),\,$
$u_{0\varepsilon}(x)\geq\varepsilon,\,$ $u_{0\varepsilon_{i}}(x)\geq
u_{0\varepsilon_{j}}(x)$ for $\varepsilon_{i}>\varepsilon_{j},\,$
$u_{0\varepsilon}(x)\to u_{0}(x)$ as $\varepsilon\to 0$ and

$\frac{\partial
u_{0\varepsilon}(x)}{\partial\nu}=\int_{\Omega}k(x,y,0)u^{l}_{0\varepsilon}(y)\,dy$

for $x\in\partial\Omega.$ Let $u_{\varepsilon}$ be the solution of the
following auxiliary problem:

$\displaystyle\left\\{\begin{array}[]{ll}u_{t}=\Delta
u+au^{p}\int_{\Omega}u^{q}(y,t)\,dy-
bu^{m}+b\varepsilon^{m}&\textrm{for}\,\,\,x\in\Omega,\,\,\,\,\,t>0,\\\
\frac{\partial
u(x,t)}{\partial\nu}=\int_{\Omega}k(x,y,t)u^{l}(y,t)dy&\textrm{for}\,\,\,x\in\partial\Omega,\,\,t>0,\\\
u(x,0)=u_{0\varepsilon}(x)&\textrm{for}\,\,\,x\in\Omega,\end{array}\right.$
(2.7)

where $\varepsilon=\varepsilon_{n}.$ It was proved in [33] (see also [11] for
similar problem) that $u_{m}(x,t)=\lim_{\varepsilon\to 0}u_{\varepsilon}(x,t)$
is a maximal solution of (1.1)–(1.3). To establish theorem we will show that

$\underline{u}(x,t)\leq
u_{m}(x,t)\leq\overline{u}(x,t)\,\,\,\textrm{in}\,\,\,\overline{Q}_{T_{0}}\,\,\,\textrm{for
any}\,\,\,T_{0}\in(0,T).$ (2.8)

We prove the second inequality in (2.8) only, since the proof of the first one
is similar. Let $\varphi(x,t)\in C^{2,1}(\overline{Q}_{T_{0}})$ be a
nonnegative function such that

$\frac{\partial\varphi(x,t)}{\partial\nu}=0,\;(x,t)\in S_{T_{0}}.$

If we multiply the first equation in (2.7) by $\varphi(x,t)$ and then
integrate over $Q_{t}$ for $0<t<T_{0},$ we get

$\displaystyle\int_{\Omega}u_{\varepsilon}(x,t)\varphi(x,t)\,dx$
$\displaystyle\leq$
$\displaystyle\int_{\Omega}u_{\varepsilon}(x,0)\varphi(x,0)\,dx+\varepsilon^{m}b\int_{0}^{t}\int_{\Omega}\varphi(x,\tau)\,dxd\tau$
(2.9) $\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\Omega}\left[u_{\varepsilon}(x,\tau)\varphi_{\tau}(x,\tau)+u_{\varepsilon}(x,\tau)\Delta\varphi(x,\tau)\right.$
$\displaystyle+$
$\displaystyle\left.au_{\varepsilon}^{p}(x,\tau)\int_{\Omega}u_{\varepsilon}^{q}(y,\tau)\,dy-
bu_{\varepsilon}^{m}\varphi(x,\tau)\right]\,dxd\tau$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\partial\Omega}\varphi(x,\tau)\int_{\Omega}k(x,y,\tau)u_{\varepsilon}^{l}(y,\tau)\,dy\,dS_{x}d\tau.$

On the other hand, $\overline{u}$ satisfies (2.9) with reversed inequality and
with $\varepsilon=0.$ Set $w(x,t)=u_{\varepsilon}(x,t)-\overline{u}(x,t).$
Then $w(x,t)$ satisfies

$\displaystyle\int_{\Omega}w(x,t)\varphi(x,t)\,dx$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}w(x,0)\varphi(x,0)\,dx+\varepsilon^{m}b\int_{0}^{t}\int_{\Omega}\varphi\,dxd\tau$
(2.10) $\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\Omega}(\varphi_{\tau}+\Delta\varphi-
bm\theta_{1}^{m-1}(x,\tau)\varphi)w\,dxd\tau$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\Omega}ap\theta_{2}^{p-1}(x,\tau)w(x,\tau)\int_{\Omega}u_{\varepsilon}^{q}(y,\tau)\,dydxd\tau$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\Omega}a\overline{u}^{p}(x,\tau)\int_{\Omega}q\theta_{3}^{q-1}(y,\tau)w(y,\tau)\,dydxd\tau$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\partial\Omega}\varphi(x,\tau)\int_{\Omega}k(x,y,\tau)l\theta_{4}^{l-1}w(y,\tau)\,dydS_{x}d\tau,$

where $\theta_{i},\,i=\overline{1,4}$ are some continuous functions between
$u_{\varepsilon}$ and $\overline{u}.$ Note here that by hypotheses for
$k(x,y,t)$, $u_{\varepsilon}(x,t)$ and $\overline{u}(x,t)$, we have

$\displaystyle 0\leq\overline{u}(x,t)\leq M,\,\,\varepsilon\leq
u_{\varepsilon}(x,t)\leq M\,\,\textrm{in}\,\,\overline{Q}_{T_{0}}$
$\displaystyle\textrm{and}\,\,\,0\leq k(x,y,t)\leq
M\,\,\,\textrm{in}\,\,\partial\Omega\times\overline{Q}_{T_{0}},$ (2.11)

where $M$ is some positive constant. Then it is easy to see from (2) that
$\theta_{1}^{m-1},\,$ $\theta_{2}^{p-1},\,$ $\theta_{3}^{q-1}$ and
$\theta_{4}^{l-1}$ are positive and bounded functions in
$\overline{Q}_{T_{0}}$ and, moreover, $\theta_{2}^{p-1}\leq M^{p-1},\,$
$\theta_{3}^{q-1}\leq M^{q-1},\,$ $\theta_{4}^{l-1}\leq M^{l-1}.$ Define a
sequence $\\{a_{k}\\}$ in the following way: $\,a_{k}\in
C^{\infty}(\overline{Q}_{T_{0}}),\,$ $a_{k}\geq 0\,$ and $\,a_{k}\to
bm\theta_{1}^{m-1}\,$ as $\,k\to\infty\,$ in $L^{1}({Q}_{T_{0}}).$ Now
consider a backward problem given by

$\displaystyle\left\\{\begin{array}[]{ll}\varphi_{\tau}+\Delta\varphi-
a_{k}\varphi=0&\textrm{for}\,\,\,x\in\Omega,\,\,\,0<\tau<t,\\\
\frac{\partial\varphi(x,\tau)}{\partial\nu}=0&\textrm{for}\,\,\,x\in\partial\Omega,\,\,\,0<\tau<t,\\\
\varphi(x,t)=\psi(x)&\textrm{for}\,\,\,x\in\Omega,\end{array}\right.$ (2.15)

where $\psi(x)\in C_{0}^{\infty}(\Omega)$ and $0\leq\psi(x)\leq 1.$ Denote a
solution of (2.15) as $\varphi_{k}(x,\tau).$ Then by the standard theory for
linear parabolic equations (see [24], for example), we find that
$\varphi_{k}\in C^{2,1}(\overline{Q}_{t}),$ $0\leq\varphi_{k}(x,\tau)\leq 1$
in $\overline{Q}_{t}.$ Putting $\varphi=\varphi_{k}$ in (2.10) and passing
then to the limit as $k\to\infty,$ we infer

$\displaystyle\int_{\Omega}w(x,t)\psi(x)\,dx$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}w(x,0)_{+}\,dx+\varepsilon^{m}bT_{0}|\Omega|$
(2.16) $\displaystyle+$
$\displaystyle[a(p+q)|\Omega|M^{p+q-1}+l|\partial\Omega|M^{l}]\int_{0}^{t}\int_{\Omega}w(y,\tau)_{+}\,dyd\tau,$

where $w_{+}=\max\\{w,0\\},$ $|\partial\Omega|$ and $|\Omega|$ are the
Lebesgue measures of $\partial\Omega$ in $\mathbb{R}^{N-1}$ and $\Omega$ in
$\mathbb{R}^{N},$ respectively. Since (2.16) holds for every $\psi(x),$ we can
choose a sequence $\\{\psi_{k}\\}$ converging in $L^{1}(\Omega)$ to
$\psi(x)=1$ if $w(x,t)>0$ and $\psi(x)=0$ otherwise. Hence, from (2.16) we get

$\displaystyle\int_{\Omega}w(x,t)_{+}\,dx$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}w(x,0)_{+}\,dx+\varepsilon^{m}bT_{0}|\Omega|$
$\displaystyle+$
$\displaystyle[a(p+q)|\Omega|M^{p+q-1}+l|\partial\Omega|M^{l}]\int_{0}^{t}\int_{\Omega}w(y,\tau)_{+}\,dyd\tau.$

Applying now Gronwall’s inequality and passing to the limit $\varepsilon\to
0,$ the conclusion of the theorem follows for $\min(p,q,l)\geq 1.$ For the
case $\min(p,q,l)<1$ we can consider
$w(x,t)=\underline{u}(x,t)-\overline{u}(x,t)$ and prove the theorem in a
similar way using the positiveness of a subsolution or a supersolution. ∎

######  Remark 2.4.

Theorem 2.3 holds for $q=0.$

######  Remark 2.5.

We improve a comparison principle in [33]. The authors of [33] supposed that
$\underline{u}(x,t)>0$ or $\overline{u}(x,t)>0$ in ${Q}_{T}\cup\Gamma_{T}$ if
$\min(m,p,q,l)<1.$

Next lemma shows the positiveness for $t>0$ of solutions of (1.1)–(1.3) with
$m\geq 1$ and $u_{0}\not\equiv 0.$

######  Lemma 2.6.

Let $u_{0}\not\equiv 0$ in $\Omega,$ $m\geq 1.$ Suppose $u$ is a solution of
(1.1)–(1.3) in $Q_{T}.$ Then $u>0$ in ${Q}_{T}\cup S_{T}.$

###### Proof.

We take any $\tau\in(0,T).$ As $u(x,t)$ is continuous in $\overline{Q}_{\tau}$
function, then we have

$u(x,t)\leq M,\,\,(x,t)\in\overline{Q}_{\tau}$ (2.17)

with some positive constant $M.$ Now we put $v=u\exp(\lambda t),$ where
$\lambda\geq bM^{m-1}.$ It is easy to verify that $v_{t}-\Delta v\geq 0.$
Since $v(x,0)=u_{0}(x)\not\equiv 0$ in $\Omega$ and $v(x,t)\geq 0$ in
$Q_{\tau},$ by the strong maximum principle $v(x,t)>0$ in $Q_{\tau}.$ Let
$v(x_{0},t_{0})=0$ in some point $(x_{0},t_{0})\in S_{T}.$ Then according to
Theorem 3.6 of [19] it yields $\partial v(x_{0},t_{0})/\partial\nu<0,$ which
contradicts (1.2). ∎

##  3\. Global existence

######  Theorem 3.1.

Let $\max(p+q,l)\leq 1$ or $1<\max(p+q,l)<m.$ Then every solution of
(1.1)–(1.3) is global.

###### Proof.

In order to prove global existence of solutions we construct a suitable
explicit supersolution of (1.1)–(1.3) in $Q_{T}$ for any positive $T.$ Suppose
at first that $\max(p+q,l)\leq 1.$ Since $k(x,y,t)$ is a continuous function,
there exists a constant $K>0$ such that

$k(x,y,t)\leq K$ (3.1)

in $\partial\Omega\times Q_{T}.$ Let $\lambda_{1}$ be the first eigenvalue of
the following problem

$\begin{cases}\Delta\varphi+\lambda\varphi=0,\;x\in\Omega,\\\
\varphi(x)=0,\;x\in\partial\Omega,\end{cases}$ (3.2)

and $\varphi(x)$ be the corresponding eigenfunction with
$\sup\limits_{\Omega}\varphi(x)=1$. It is well known, $\varphi(x)>0$ in
$\Omega$ and $\max\limits_{\partial\Omega}\partial\varphi(x)/\partial\nu<0.$

We now construct a supersolution of (1.1)–(1.3) in $Q_{T}$ as follows

$\overline{u}(x,t)=\frac{C\exp(\mu t)}{c\varphi(x)+1},$

where constants $C,\mu$ and $c$ are chosen to satisfy the inequalities:

$c\geq\max\left\\{K\int_{\Omega}\frac{dy}{(\varphi(y)+1)^{l}}\max_{\partial\Omega}\left(-\frac{\partial\varphi}{\partial\nu}\right)^{-1},1\right\\},$
$C\geq\max\\{\sup_{\Omega}(c\varphi(x)+1)u_{0}(x),1\\},\quad\mu\geq\lambda_{1}+2c^{2}\sup_{\Omega}\frac{|\nabla\varphi|^{2}}{(c\varphi(x)+1)^{2}}+a|\Omega|.$

It is easy to obtain

$\displaystyle\overline{u}_{t}-\Delta\overline{u}$
$\displaystyle-a\overline{u}^{p}\int_{\Omega}\overline{u}^{q}(y,t)\,dy+b\overline{u}^{m}$
$\displaystyle\geq\left(\mu-\frac{c\lambda_{1}\varphi}{c\varphi(x)+1}-2c^{2}\sup_{\Omega}\frac{|\nabla\varphi|^{2}}{(c\varphi(x)+1)^{2}}-a|\Omega|\right)\overline{u}\geq
0$ (3.3)

for $(x,t)\in Q_{T},$

$\displaystyle\frac{\partial\overline{u}}{\partial\nu}=$ $\displaystyle
cC\exp(\mu t)\left(-\frac{\partial\varphi}{\partial\nu}\right)\geq
KC^{l}\exp(l\mu t)\int_{\Omega}\frac{dy}{(\varphi(y)+1)^{l}}$
$\displaystyle\geq$
$\displaystyle\int_{\Omega}k(x,y,t)\overline{u}^{l}(y,t)\,dy$ (3.4)

for $(x,t)\in S_{T}$ and

$\overline{u}(x,0)\geq u_{0}(x)$ (3.5)

for $x\in\Omega.$ By virtue of (3.3)–(3.5) and Theorem 2.3, the solution of
(1.1)–(1.3) exists globally.

Suppose now that $1<\max(p+q,l)<m.$ Let $v_{0}(x)$ be the function with the
following properties:

$v_{0}(x)\in C^{1}(\overline{\Omega}),\;v_{0}(x)>u_{0}(x)\textrm{ in
}\overline{\Omega},\;\frac{\partial
v_{0}(x)}{\partial\nu}=\int_{\Omega}{Kv_{0}^{l}(y)}\,dy\textrm{ on
}\partial\Omega,$

where $K$ was defined in (3.1). Let $v(x,t)$ be the solution of (1.1) in
$Q_{T}$ with boundary and initial data

$\frac{\partial
u(x,t)}{\partial\nu}=\int_{\Omega}{Ku^{l}(y,t)}\,dy,\;x\in\partial\Omega,\;0<t<T,$
(3.6) $u(x,0)=v_{0}(x),\;x\in\Omega.$ (3.7)

We prove that $v(x,t)$ exists in $Q_{T}$ for any $T>0.$ By Lemma 2.6 and
Theorem 2.3

$u(x,t)\leq v(x,t)\textrm{ in }Q_{T}.$

Integrating (1.1) for $v(x,t)$ over $Q_{t}$ and using Green’s identity and
H$\ddot{o}$lder’s inequality, we have

$\displaystyle\int_{\Omega}v(x,t)\,dx=\int_{\Omega}v_{0}(x)\,dx+\int_{0}^{t}\int_{\partial\Omega}\int_{\Omega}Kv^{l}(y,\tau)\,dy\,dS_{x}\,d\tau$
$\displaystyle+a\int_{0}^{t}\int_{\Omega}v^{p}(y,\tau)\,dy\int_{\Omega}v^{q}(y,\tau)\,dy\,d\tau-b\int_{0}^{t}\int_{\Omega}v^{m}(y,\tau)\,dy\,d\tau$
$\displaystyle\leq\int_{\Omega}v_{0}(x)\,dx+K|\partial\Omega|(t|\Omega|)^{\frac{m-l}{m}}\left(\int_{0}^{t}\int_{\Omega}v^{m}(y,\tau)\,dy\,d\tau\right)^{\frac{l}{m}}$
$\displaystyle+at^{\frac{m-p-q}{m}}|\Omega|^{\frac{2m-p-q}{m}}\left(\int_{0}^{t}\int_{\Omega}v^{m}(y,\tau)\,dy\,d\tau\right)^{\frac{p+q}{m}}-b\int_{0}^{t}\int_{\Omega}v^{m}(y,\tau)\,dy\,d\tau.$
(3.8)

Since $v(x,t)>0,$ we obtain from (3.8) that

$\int_{0}^{t}\int_{\Omega}v^{m}(y,\tau)\,dy\,d\tau\leq C(T),\;t\leq T.$ (3.9)

Now we prove that $v(x,t)$ can not blow up in $\Omega.$

Let $G_{N}(x,y;t-\tau)$ be the Green function of the heat equation with
homogeneous Neumann boundary condition. Then we have the representation
formula:

$\displaystyle v(x,t)$
$\displaystyle=\int_{\Omega}{G_{N}(x,y;t)v_{0}(y)}dy+K\int_{0}^{t}{\int_{\partial\Omega}G_{N}(x,\xi;t-\tau)\int_{\Omega}v^{l}(y,\tau)}dydS_{\xi}d\tau$
$\displaystyle+a\int_{0}^{t}{\int_{\Omega}G_{N}(x,y;t-\tau)v^{p}(y,\tau)\int_{\Omega}v^{q}(z,\tau)}\,dz\,dyd\tau$
$\displaystyle-b\int_{0}^{t}{\int_{\Omega}}G_{N}(x,y;t-\tau)v^{m}(y,\tau)\,dyd\tau$
(3.10)

for $(x,t)\in Q_{T}.$ We now take a sequence of sets $\\{\Omega_{k}\\}$ such
that $\Omega_{k}\subset\subset\Omega,\,$ $\Omega_{k}\subset\Omega_{k+1},\,$
$\Omega=\cup_{k=1}^{\infty}\Omega_{k},\,$ $\partial\Omega_{k}\in C^{2}.$ It is
well known (see, for example, [19],[20]) that

$G_{N}(x,y;t-\tau)\geq 0,\;x,y\in\Omega,\;0\leq\tau<t<T,$ (3.11)
$\int_{\Omega}{G_{N}(x,y;t-\tau)}\,dy=1,\;x\in\Omega,\;0\leq\tau<t<T,$ (3.12)
$0\leq G_{N}(x,y;t-\tau)\leq
c_{0},\;x\in\Omega_{k},\;y\in\partial\Omega,\;0<\tau<t<T,$ (3.13) $c_{1}\leq
G_{N}(x,y;t-\tau)\leq
c_{2},\;x\in\overline{\Omega},\;y\in\overline{\Omega},\;t-\tau\geq\varepsilon_{0},$
(3.14)
$\frac{c_{3}}{\sqrt{t-\tau}}\leq\int_{\partial\Omega}G_{N}(x,\xi;t-\tau)dS_{\xi}\leq\frac{c_{4}}{\sqrt{t-\tau}},\;x\in\partial\Omega,\;0<t-\tau\leq\varepsilon_{0},$
(3.15)

where $\varepsilon_{0}>0,\,$ $c_{0}$ is a positive constant depending on $k.$
Here and below $c_{i},\,i\in\mathbb{N}$ are positive constants. We note that

$v^{p}(y,t)\int_{\Omega}v^{q}(z,t)\,dz\leq
v^{p+q}(y,t)+\left(\int_{\Omega}v^{q}(z,t)\,dz\right)^{\frac{p+q}{q}}\,\textrm{for}\,\,\,(y,t)\in
Q_{T}.$ (3.16)

Indeed, if

$\int_{\Omega}v^{q}(z,t)\,dz\leq v^{q}(y,t),$

then

$v^{p}(y,t)\int_{\Omega}v^{q}(z,t)\,dz\leq v^{p+q}(y,t).$

Otherwise, we have

$v^{p}(y,t)\leq\left(\int_{\Omega}v^{q}(z,t)\,dz\right)^{\frac{p}{q}}$

and

$v^{p}(y,t)\int_{\Omega}v^{q}(z,t)\,dz\leq\left(\int_{\Omega}v^{q}(z,t)\,dz\right)^{\frac{p+q}{q}}.$

Now using (3.12), (3.16) and H$\ddot{o}$lder’s inequality, we estimate the
third integral in the right hand side of (3.10):

$\displaystyle
a\int_{0}^{t}{\int_{\Omega}G_{N}(x,y;t-\tau)v^{p}(y,\tau)\int_{\Omega}v^{q}(z,\tau)}\,dz\,dyd\tau$
$\displaystyle\leq
a\int_{0}^{t}{\int_{\Omega}}G_{N}(x,y;t-\tau)\left[v^{p+q}(y,\tau)+\left(\int_{\Omega}v^{q}(z,\tau)\,dz\right)^{\frac{p+q}{q}}\right]\,dyd\tau$
$\displaystyle\leq\int_{0}^{t}{\int_{\Omega}}G_{N}(x,y;t-\tau)\left[bv^{m}(y,\tau)+a\left(\frac{a}{b}\right)^{\frac{p+q}{m-(p+q)}}\right.$
$\displaystyle+\left.a|\Omega|^{\frac{(m-q)(p+q)}{mq}}\left(\int_{\Omega}v^{m}(z,\tau)\,dz\right)^{\frac{p+q}{m}}\right]\,dyd\tau$
$\displaystyle\leq
b\int_{0}^{t}{\int_{\Omega}}G_{N}(x,y;t-\tau)v^{m}(y,\tau)\,dyd\tau+a\left(\frac{a}{b}\right)^{\frac{p+q}{m-(p+q)}}t$
$\displaystyle+a|\Omega|^{\frac{(m-q)(p+q)}{mq}}\int_{0}^{t}\left(\int_{\Omega}v^{m}(z,\tau)\,dz\right)^{\frac{p+q}{m}}\,d\tau.$
(3.17)

By (3.9) – (3.13), (3.17) and H$\ddot{o}$lder’s inequality we have

$\displaystyle\sup_{\Omega_{k}}v(x,t)\leq$
$\displaystyle\sup_{\Omega}v_{0}(x)+c_{0}|\partial\Omega|K\int_{0}^{t}\int_{\Omega}v^{l}(y,\tau)\,dyd\tau$
$\displaystyle+$ $\displaystyle
c_{5}T+c_{6}T^{\frac{m-p-q}{m}}\left[\int_{0}^{t}\int_{\Omega}v^{m}(y,\tau)\,dyd\tau\right]^{\frac{p+q}{m}}\leq
c_{7}(T).$ (3.18)

Suppose that $v(x,t)$ blows up at $t=T$ and

$\int_{\Omega}v^{q}(y,T)\,dy=+\infty.$ (3.19)

Then by H$\ddot{o}$lder’s inequality we derive that

$\int_{\Omega}v^{m}(y,T)\,dy=+\infty.$ (3.20)

Since $v(x,t)$ blows up at $t=T$ from (3.10), (3.14), (3.15), (3.17) we obtain

$\int_{\Omega}v^{l}(y,T)\,dy=+\infty$ (3.21)

and, moreover,

$v(x,t)\,\textrm{blows up at }\,t=T\,\textrm{at every point of
}\,\partial\Omega.$ (3.22)

Integrating (1.1) for $v(x,t)$ over $\Omega$ and using H$\ddot{o}$lder’s
inequality, we have

$\displaystyle\int_{\Omega}v_{t}(x,t)\,dx$
$\displaystyle=K\int_{\partial\Omega}\int_{\Omega}v^{l}(y,t)\,dy\,dS_{x}+a\int_{\Omega}v^{p}(y,t)\,dy\int_{\Omega}v^{q}(y,t)\,dy$
$\displaystyle-b\int_{\Omega}v^{m}(y,t)\,dy\leq
K|\partial\Omega||\Omega|^{\frac{m-l}{m}}\left(\int_{\Omega}v^{m}(y,t)\,dy\right)^{\frac{l}{m}}$
$\displaystyle+a|\Omega|^{\frac{2m-p-q}{m}}\left(\int_{\Omega}v^{m}(y,t)\,dy\right)^{\frac{p+q}{m}}-b\int_{\Omega}v^{m}(y,t)\,dy.$

Thus by (3.20)

$\lim_{t\to T}\int_{\Omega}v_{t}(x,t)\,dx=-\infty.$ (3.23)

On the other hand, integrating (1.1) for $v(x,t)$ over $\Omega_{k}$, we infer

$\int_{\Omega_{k}}v_{t}(x,t)\,dx=\int_{\partial\Omega_{k}}\frac{\partial
v(x,t)}{\partial\nu}\,dS_{x}+a\int_{\Omega_{k}}v^{p}(y,t)\,dy\int_{\Omega}v^{q}(y,t)\,dy-b\int_{\Omega_{k}}v^{m}(y,t)\,dy$

and using (3.6), (3.18), (3.19), (3.21), (3.22) we obtain the contradiction
with (3.23).

Hence,

$\int_{\Omega}v^{q}(y,t)\,dy\leq c_{8}(T),\;t\leq T.$ (3.24)

Now we consider the following equation

$Lu\equiv u_{t}-\Delta u-ac_{8}(T)u^{p}+bu^{m}=0,\;(x,t)\in Q_{T}.$ (3.25)

It is easy to see that $v(x,t)$ is the subsolution of the problem (3.25),
(3.6), (3.7) in $Q_{T}.$ To construct a supersolution we use the change of
variables in a neighborhood of $\partial\Omega$ as in [2]. Let
$\overline{x}\in\partial\Omega$ and $\widehat{n}(\overline{x})$ be the inner
unit normal to $\partial\Omega$ at the point $\overline{x}.$ Since
$\partial\Omega$ is smooth it is well known that there exists $\delta>0$ such
that the mapping $\psi:\partial\Omega\times[0,\delta]\to\mathbb{R}^{n}$ given
by $\psi(\overline{x},s)=\overline{x}+s\widehat{n}(\overline{x})$ defines new
coordinates ($\overline{x},s)$ in a neighborhood of $\partial\Omega$ in
$\overline{\Omega}.$ A straightforward computation shows that, in these
coordinates, $\Delta$ applied to a function $g(\overline{x},s)=g(s),$ which is
independent of the variable $\overline{x},$ evaluated at a point
$(\overline{x},s)$ is given by

$\Delta g(\overline{x},s)=\frac{\partial^{2}g}{\partial
s^{2}}(\overline{x},s)-\sum_{j=1}^{n-1}\frac{H_{j}(\overline{x})}{1-sH_{j}(\overline{x})}\frac{\partial
g}{\partial s}(\overline{x},s),$ (3.26)

where $H_{j}(\overline{x})$ for $j=1,...,n-1,$ denote the principal curvatures
of $\partial\Omega$ at $\overline{x}.$ For $0\leq s\leq\delta$ and small
$\delta$ we have

$\left|\sum_{j=1}^{n-1}\frac{H_{j}(\overline{x})}{1-sH_{j}(\overline{x})}\right|\leq\overline{c}.$
(3.27)

Let $0<\varepsilon<\omega<\min(\delta,1),$ $\max(1/l,2/(m-1))<\beta<2/(l-1),$
$0<\gamma<\beta/2,$ $A\geq\sup_{\Omega}v_{0}(x).$ As in [15] for points in
$Q_{\delta,T}=\partial\Omega\times[0,\delta]\times[0,T]$ of coordinates
$(\overline{x},s,t)$ define

$\overline{v}(x,t)=\overline{v}((\overline{x},s),t)=\left[(s+\varepsilon)^{-\gamma}-\omega^{-\gamma}\right]_{+}^{\frac{\beta}{\gamma}}+A,$
(3.28)

where $s_{+}=\max(s,0).$ For points in $\overline{Q_{T}}\setminus
Q_{\delta,T}$ we set $\overline{v}(x,t)=A.$ We prove that $\overline{v}(x,t)$
is the supersolution of (3.25), (3.6), (3.7) in $Q_{T}.$ It is not difficult
to check that

$\left|\frac{\partial\overline{v}}{\partial
s}\right|\leq\beta\min\left(\left[D(s)\right]^{\frac{\gamma+1}{\gamma}}\left[(s+\varepsilon)^{-\gamma}-\omega^{-\gamma}\right]_{+}^{\frac{\beta+1}{\gamma}},\,(s+\varepsilon)^{-(\beta+1)}\right),$
(3.29) $\left|\frac{\partial^{2}\overline{v}}{\partial
s^{2}}\right|\leq\beta(\beta+1)\min\left(\left[D(s)\right]^{\frac{2(\gamma+1)}{\gamma}}\left[(s+\varepsilon)^{-\gamma}-\omega^{-\gamma}\right]_{+}^{\frac{\beta+2}{\gamma}},\,(s+\varepsilon)^{-(\beta+2)}\right),$
(3.30)

where

$D(s)=\frac{(s+\varepsilon)^{-\gamma}}{(s+\varepsilon)^{-\gamma}-\omega^{-\gamma}}.$

Then $D^{\prime}(s)>0$ and for any $\overline{\varepsilon}>0$

$1\leq D(s)\leq 1+\overline{\varepsilon},\;0<s\leq{\overline{s}},$ (3.31)

where
${\overline{s}}=[\overline{\varepsilon}/(1+\overline{\varepsilon})]^{1/\gamma}\omega-\varepsilon,$
$\varepsilon<[\overline{\varepsilon}/(1+\overline{\varepsilon})]^{1/\gamma}\omega.$
By (3.25)–(3.31) we can choose $\overline{\varepsilon}$ small and $A$ large so
that in $Q_{{\overline{s}},T}$

$\displaystyle L\overline{v}\geq
b\left(\left[(s+\varepsilon)^{-\gamma}-\omega^{-\gamma}\right]_{+}^{\frac{\beta}{\gamma}}+A\right)^{m}-\beta(\beta+1)\left[D(s)\right]^{\frac{2(\gamma+1)}{\gamma}}\left[(s+\varepsilon)^{-\gamma}-\omega^{-\gamma}\right]_{+}^{\frac{\beta+2}{\gamma}}$
$\displaystyle-\beta\overline{c}\left[D(s)\right]^{\frac{\gamma+1}{\gamma}}\left[(s+\varepsilon)^{-\gamma}-\omega^{-\gamma}\right]_{+}^{\frac{\beta+1}{\gamma}}-ac_{8}(T)\left(\left[(s+\varepsilon)^{-\gamma}-\omega^{-\gamma}\right]_{+}^{\frac{\beta}{\gamma}}+A\right)^{p}\geq
0.$

Let $s\in[{\overline{s}},\delta].$ From (3.26)–(3.30) we have

$|\Delta\overline{v}|\leq\beta(\beta+1)\omega^{-(\beta+2)}\left(\frac{1+\overline{\varepsilon}}{\overline{\varepsilon}}\right)^{\frac{\beta+2}{\gamma}}+\beta\overline{c}\omega^{-(\beta+1)}\left(\frac{1+\overline{\varepsilon}}{\overline{\varepsilon}}\right)^{\frac{\beta+1}{\gamma}}$

and $L\overline{v}\geq 0$ for large values of $A.$ Obviously, in
$\overline{Q_{T}}\setminus Q_{\delta,T}$

$L\overline{v}=-ac_{8}(T)A^{p}+bA^{m}\geq 0$

for large values of $A.$

Now we prove the following inequality

$\frac{\partial\overline{v}}{\partial\nu}(\overline{x},0,t)\geq\int_{\Omega}K\overline{v}^{l}(\overline{x},s,t)\,dy,\quad(x,t)\in
S_{T}$ (3.32)

for a suitable choice of $\varepsilon.$ To estimate the integral $I$ in the
right hand side of (3.32) we use the change of variables in a neighborhood of
$\partial\Omega$ as above. Let

$\overline{J}=\sup_{0<s<\delta}\int_{\partial\Omega}|J(\overline{y},s)|\,d\overline{y},$

where $J(\overline{y},s)$ is Jacobian of the change of variables. Then we have

$\displaystyle I\leq$ $\displaystyle\theta
K\int_{\Omega}\left[(s+\varepsilon)^{-\gamma}-\omega^{-\gamma}\right]_{+}^{\frac{\beta
l}{\gamma}}\,dy+\theta KA^{l}|\Omega|$ $\displaystyle\leq$
$\displaystyle\theta
K\overline{J}\int_{0}^{\omega-\varepsilon}\left[(s+\varepsilon)^{-\gamma}-\omega^{-\gamma}\right]^{\frac{\beta
l}{\gamma}}\,ds+\theta KA^{l}|\Omega|$ $\displaystyle\leq$
$\displaystyle\frac{\theta K\overline{J}}{\beta l-1}\left[\varepsilon^{-(\beta
l-1)}-\omega^{-(\beta l-1)}\right]+\theta KA^{l}|\Omega|,$

where $\theta=\max(2^{l-1},1).$ On the other hand, since

$\frac{\partial\overline{v}}{\partial\nu}(\overline{x},0,t)=-\frac{\partial\overline{v}}{\partial
s}(\overline{x},0,t)=\beta\varepsilon^{-\gamma-1}\left[\varepsilon^{-\gamma}-\omega^{-\gamma}\right]_{+}^{\frac{\beta-\gamma}{\gamma}},$

the inequality (3.32) holds if $\varepsilon$ is small enough. At last,

$v(x,0)\leq\overline{v}(x,0)\,\,\,\textrm{in}\,\,\,\Omega.$

Hence, by comparison principle for (3.25), (3.6), (3.7) (see Remark 2.4) we
get

$v(x,t)\leq\overline{v}(x,t)\,\,\,\textrm{in}\,\,\,\overline{Q}_{T}.$

∎

######  Remark 3.2.

The authors of [33] have proved global existence of solutions of (1.1)–(1.3)
for $p+q<m,\,l\leq 1.$

##  4\. Blow-up in finite time

To formulate finite time blow-up result we introduce

$\underline{k}(t)=\inf_{\Omega}\int_{\partial\Omega}k(x,y,t)\,dS_{x}$

and suppose that

$\underline{k}(0)>0.$ (4.1)

######  Theorem 4.1.

Let either $r+p>\max(m,1)$ or $l>\max(m,1)$ and (4.1) hold. Then solutions of
(1.1)–(1.3) blow up in finite time if initial data are large enough.

###### Proof.

Suppose at first that $r+p>\max(m,1).$ To prove the existence finite time
blow-up solutions we construct a suitable subsolution. Let $f(t)$ be the
solution of the following problem

$\begin{cases}f^{\prime}(t)=a|\Omega|f^{p+q}-bf^{m},\;t>0,\\\
f(0)=f_{0}.\end{cases}$

If

$f_{0}>\left(\frac{b}{a|\Omega|}\right)^{\frac{1}{p+q-m}},$

then $f(t)\geq f_{0}$ and there exists $t_{0}<+\infty$ such that

$\lim_{t\to t_{0}}f(t)=+\infty.$

It is easy to check that $f(t)$ is a subsolution of (1.1)–(1.3) if
$u_{0}(x)\geq f_{0}.$ By Theorem 2.3 $u(x,t)$ blows up in finite time.

Now suppose $l>\max(m,1)$ and (4.1) holds. Then there exists $T_{0}>0$ such
that $\underline{k}(t)>0$ for $t\in[0,T_{0}].$ Denote

$k_{0}=\min_{[0,T_{0}]}\underline{k}(t),\quad V(t)=\int_{\Omega}u(x,t)\,dx.$

Integrating (1.1) over $\Omega$ and using Green’s identity and
H$\ddot{o}$lder’s inequality, we obtain for $t\leq T_{0}$

$\begin{split}&\int_{\Omega}u_{t}(x,t)\,dx=\int_{\partial\Omega}\int_{\Omega}k(x,y,t)u^{l}(y,t)\,dy\,dS_{x}+a\int_{\Omega}u^{p}(y,t)\,dy\int_{\Omega}u^{q}(y,t)\,dy\\\
&-b\int_{\Omega}u^{m}(y,t)\,dy\geq
k_{0}\int_{\Omega}u^{l}(y,t)\,dy-b\int_{\Omega}u^{m}(y,t)\,dy\\\
&\geq\left(\int_{\Omega}u^{l}(y,t)\,dy\right)^{\frac{m}{l}}\left[k_{0}\left(\int_{\Omega}u^{l}(y,t)\,dy\right)^{\frac{l-m}{l}}-b|\Omega|^{\frac{l-m}{l}}\right]\\\
&\geq\left(\int_{\Omega}u(y,t)\,dy\right)^{m}|\Omega|^{-\frac{m(l-1)}{l}}\left[k_{0}\left(\int_{\Omega}u(y,t)\,dy\right)^{l-m}|\Omega|^{-\frac{(l-m)(l-1)}{l}}-b|\Omega|^{\frac{l-m}{l}}\right]\end{split}$
(4.2)

if

$J(t)\equiv
k_{0}V^{l-m}(t)|\Omega|^{-\frac{(l-m)(l-1)}{l}}-b|\Omega|^{\frac{l-m}{l}}>1.$
(4.3)

Then from (4.2), (4.3) we have

$V^{\prime}(t)\geq|\Omega|^{-\frac{m(l-1)}{l}}V^{m}(t)$ (4.4)

for $t\leq T_{0}.$ Let $m>1.$ Obviously, $V(t)$ blows up at $t\leq T_{0}$
subject to

$V(0)\geq\left\\{(m-1)|\Omega|^{-\frac{m(l-1)}{l}}T_{0}\right\\}^{-\frac{1}{m-1}}.$
(4.5)

To provide (4.3) we assume that

$V(0)>k_{0}^{-\frac{1}{l-m}}\left(b|\Omega|^{\frac{l-m}{l}}+1\right)^{\frac{1}{l-m}}|\Omega|^{\frac{l-1}{l}}.$
(4.6)

Easily to check (4.6) is equivalent to the inequality

$k_{0}V^{l-m}(0)|\Omega|^{-\frac{(l-m)(l-1)}{l}}-b|\Omega|^{\frac{l-m}{l}}>1.$

Hence,

$J(0)>1.$

Suppose there exists $t_{1}\in(0,T_{0}]$ such that $J(t_{1})=1$ and $J(t)>1$
for $t\in[0,t_{1}).$ Since for $t\in[0,t_{1}]$ (4.4) holds, we have $V(t)\geq
V(0)$ and

$J(t)\geq J(0)>1.$

Therefore, if we take initial data to satisfy (4.5), (4.6) then any solution
of (1.1)–(1.3) with $m>1$ blows up at $t\leq T_{0}.$

Let $m\leq 1.$ From (4.2) and H$\ddot{o}$lder’s inequality we deduce that

$V^{\prime}(t)\geq
k_{0}\int_{\Omega}u^{l}(y,t)\,dy-b\int_{\Omega}u^{m}(y,t)\,dy\geq
k_{0}|\Omega|^{1-l}V^{l}(t)-b|\Omega|^{1-m}V^{m}(t).$ (4.7)

Assume

$V(0)\geq\max\left\\{1,k_{0}^{-\frac{1}{l-m}}\left(b|\Omega|^{\frac{l-m}{l}}+1\right)^{\frac{1}{l-m}}|\Omega|^{\frac{l-1}{l}}\right\\}.$
(4.8)

Then by (4.4)

$V(t)\geq 1\,\,\,\textrm{for}\,\,\,t\geq 0$ (4.9)

and from (4.7), (4.9) we obtain

$V^{\prime}(t)\geq k_{0}|\Omega|^{1-l}V^{l}(t)-b|\Omega|^{1-m}(t)V(t).$

Solving this inequality we find that $V(t)$ blows up at $t\leq T_{0}$ if (4.8)
holds and

$V(0)\geq\left\\{\frac{k_{0}|\Omega|^{1-l}}{b|\Omega|^{1-m}}[1-\exp(-b(l-1)|\Omega|^{1-m}T_{0})]\right\\}^{-\frac{1}{l-1}}.$

∎

### Acknowledgements

This work is supported by the RUDN University Strategic Academic Leadership
Program and the state program of fundamental research of Belarus (grant
1.2.03.1).

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framework can obtain architectures that generalize well beyond the dataset
they were optimized on.

Table 7: Transfer learning result of our network for different width ($m$) which training in FashionMNIST (domain dataset) for $20$ epochs and then training in MNIST (target dataset) for $2$ or $5$ epochs. (the accuracy in the table are displayed in percentages) Epochs | $m=64$ | $m=128$ | $m=256$ | $m=512$ | $m=1024$
---|---|---|---|---|---
$20+2$ | $94.13\pm 0.64$ | $95.18\pm 0.25$ | $94.73\pm 0.22$ | $94.40\pm 0.80$ | $\bm{95.41}\pm 0.03$
$20+5$ | $95.73\pm 0.28$ | $96.12\pm 0.32$ | $96.73\pm 0.29$ | $96.73\pm 0.11$ | $\bm{96.96}\pm 0.22$

###  F.7 DARTS experiment on CNN

Our theory relies on fully-connected matrices and we have indeed verified
experimentally the validity of our theoretical findings. To scrutinize our
method even further, we attempt to extend our results to the popular
convolutional neural networks. We believe this will provide some further
insights on future extensions of our theory. In particular, we use DARTS
(similarly with the experiment in sec. 5.1) with convolutional layers. The
standard dataset of CIFAR-10 is selected; the details of the dataset are
shared in sec. F.1. The search space and search strategy follow sec. 5.1 with
one differentiating point: we use convolutional layers instead of fully
connected layers in Eq. (2).

We select DARTS on a Convolutional Neural Network with $L=10$ and $m=1024$,
while we repeat the experiment for 5 times. After training, the probability of
these activation functions and skip connections in each layer are reported in
Figure 7(a) and 7(b), respectively. Compared with the Figure 1, the activation
function search exhibits similar characteristics with the results of the fully
connected network. Namely: (1) ReLU and LeakyReLU have the highest probability
to be selected, (2) the difference of probability between different activation
functions in the first layer is the largest. But for skip layer search, CNN
exhibits the opposite results with fully connected network, that is, almost
all of the skip connections have a probability of being selected less than
$50\%$.

Based on the above results, our theory can still explain some of the phenomena
observed in CNNs, e.g., activation functions search. Nevertheless, our theory
on skip connections search on CNNs mismatches with experimental demonstration
in practice to some extent, which motivates us to conduct a refined analysis
on CNNs for NAS.

(a) activation functions $\bm{\sigma}$

(b) skip connections $\bm{\alpha}$

Figure 7: Architecture search results on activation functions indicated by the
probability of $\bm{\sigma}$ in (a) and skip connections indicated by
$\bm{\alpha}$ in (b). We notice that for each layer, ReLU and LeakyReLU are
selected with the higher probability.

###  F.8 $\beta$-DARTS experiment on MLP

In this section, we use an improved DARTS-based algorithm, $\beta$-DARTS [Ye
et al., 2022], for doing the activation function search. Our experiments are
performed on a 5-layers MLP and the experimental results are presented in
Figure 8. Compared with the results of DARTS in Figure 1, the experimental
results of $\beta$-DARTS indicate that the probability difference between
different activation functions is smaller, which may verify that DART is more
easily to overfit . This is also the advantage mentioned in the $\beta$-DARTS
paper.

Figure 8: Architecture search results using $\beta$-DARTS on activation
functions indicated by the probability of $\bm{\sigma}$. We notice that for
each layer, ReLU and LeakyReLU are selected with the higher probability.

##  Appendix G Societal impact

This is a theoretical work that derived generalization bounds for the
architectures obtained by NAS. As such, we do not expect our work to have
negative societal bias, as we do not focus on obtaining state-of-the-art
results in a particular task. On the contrary, our work can have various
benefits for the community:

  * •

We provide the first generalization bounds for the class of NAS architectures,
which is expected to have a positive impact on the understanding and the
application of such architectures.

  * •

As we illustrate in sec. 5, we can use the minimum eigenvalue as a promising
metric to guide NAS. This can lead to further investigation on techniques for
efficient evaluation of NAS by avoiding solving the intensive bi-level
optimization of NAS explicitly.

Nevertheless, we encourage researchers to further investigate the impact of
different architectures and their inductive biases on the society.