$\|\phi(f_{1},e_{1})-\phi(f_{2},e_{2})\|_{L^{\infty}(\mathcal{S})}\,\leq\,C\epsilon^{3/2}\big{[}\,\|f_{1}-f_{2}\|_{*}+\|e_{1}-e_{2}\|_{**}\,\big{]}.$

Hence,

$\|\Lambda_{1*}(f_{1},e_{1})-\Lambda_{1*}(f_{2},e_{2})\|_{L^{2}(0,1)}\,\leq\,C^{3+1/2}\big{[}\,\|f_{1}-f_{2}\|_{*}+\|e_{1}-e_{2}\|_{**}\,\big{]}.$
(6.33)

Another one comes from second derivative of $\phi$ in $z$

$\Lambda_{1**}\,=\,2\,\epsilon\,k\alpha\int_{\mathbb{R}}\Big{(}\frac{x}{\beta}+f+h\Big{)}\phi_{zz}\,w_{x}\,{\rm
d}x+\epsilon^{2}\,a_{1}(\epsilon s,\epsilon
z)\alpha\int_{\mathbb{R}}\Big{(}\frac{x}{\beta}+f+h\Big{)}^{2}\,\phi_{zz}\,w_{x}\,{\rm
d}x.$

Then

$\|\Lambda_{1**}(f_{1},e_{1})-\Lambda_{1**}(f_{2},e_{2})\|_{L^{2}(0,1)}\,\leq\,C\,\epsilon^{3}\,\Big{[}\|f_{1}-f_{2}\|_{*}+\|e_{1}-e_{2}\|_{**}\Big{]}.$
(6.34)

For fixed $\epsilon$, the remainder $\Lambda_{1}-\Lambda_{1*}-\Lambda_{1**}$
actually defines a compact operator of the pair $(f,e)$ from $H^{2}(0,1)$ into
$L^{2}(0,1)$. This is a consequence of the fact that weak convergence in
$H^{2}(\mathcal{S})$ implies local strong convergence in $H^{1}(\mathcal{S})$,
and the same is the case for $H^{2}(0,1)$ and $C^{1}[0,1]$. If $f_{j}$ and
$e_{j}$ are bounded sequences in $H^{2}(0,1)$, then clearly the functions
$\phi(f_{j},e_{j})$ constitute a bounded sequence in $H^{2}(\mathcal{S})$. In
the above remainder we can integrate by parts once in $x$ if necessary.
Averaging against $w_{x}$ which decays exponentially, localizes the situation
and the desired result follows.

We also observe that

$\Lambda_{2}(\epsilon
z)\,=\,\int_{\mathbb{R}}\eta^{\epsilon}_{\delta}(s)\,{\mathcal{N}}(\phi)w_{x}\,{\rm
d}x,$

can be estimated similarly. Using the definition of ${\mathcal{N}}(\phi)$ and
the exponential decay of $w_{x}$ we get

$\|\Lambda_{2}\|_{L^{2}(0,1)}\,\leq\,C\epsilon^{1/2}\|\phi\|_{H^{2}(\mathcal{S})}\,\leq\,C\epsilon^{3}.$

Finally, let us consider

$\Lambda_{3}(\epsilon
z)\,=\,\int_{\mathbb{R}}p\big{(}\beta^{-2}\chi(\epsilon|x|){\mathbf{W}}^{p-1}-w^{p-1}\big{)}\phi\,w_{x}\,{\rm
d}x.$

Since

$v_{5}\,=\,w\,+\,\phi_{1}\,+\,\phi_{2}\,+\,\epsilon
eZ\,+\,\phi_{3}\,+\,\phi_{4}\quad\mbox{ and
}\quad\phi_{1}\,+\,\phi_{2}\,+\,\epsilon eZ+\,\phi_{3}\,+\,\phi_{4}$

can be estimated as

$\,|\phi_{1}|\,+\,|\phi_{2}|\,+\epsilon|e\,Z|\,+\,|\phi_{3}|\,+\,|\phi_{4}|\,\leq\,C\epsilon(1+|x|^{2})e^{-|x|},$

we obtain that for some $\rho>0$ the following uniform bound holds

$\big{|}\big{(}\beta^{-2}\chi(\epsilon|x|){\mathbf{W}}^{p-1}-w^{p-1}\big{)}w_{x}\big{|}\,\leq\,C\,\epsilon\,e^{-\rho|x|}.$

From here we find that

$\|\Lambda_{3}\|_{L^{2}(0,1)}\,\leq\,C\,\epsilon^{3/2}\|\phi\|_{H^{2}(\mathcal{S})}\,\leq\,C\epsilon^{3}.$

These two terms $\Lambda_{2}$ and $\Lambda_{3}$ also define compact operators
similarly as before.

We observe that exactly the same estimates can be carried out in the terms
obtained from integration against $Z$.

###  6.3 Projection of errors on the boundary

In this section we compute the projection of errors on the boundary. Without
loss of generality, only the projections of the error components on
$\partial_{0}{\mathcal{S}}$ will be given.

The main errors on the boundary integrated against $w_{x}$ and $Z$ in the
variable $x$ can be computed as the following:

$\displaystyle\int_{\mathbb{R}}$ $\displaystyle g_{0}(x)w_{x}\,{\rm d}x\,$
$\displaystyle=$
$\displaystyle\,-\epsilon\beta\big{(}k_{1}f+f^{\prime}\big{)}\int_{\mathbb{R}}w_{x}^{2}\,{\rm
d}x\,-\,\epsilon^{2}b_{5}\,a_{11}\int_{\mathbb{R}}xw_{1,x}w_{x}\,{\rm d}x$
$\displaystyle-\epsilon^{2}\beta\big{(}k_{1}f+f^{\prime}\big{)}\int_{\mathbb{R}}\Big{\\{}a_{12}(f+h)w_{2,x}+eZ_{x}+\beta^{-1}\big{[}A(0)Z_{x}+\phi_{22}(x,0)\big{]}\Big{\\}}w_{x}\,{\rm
d}x$
$\displaystyle-\epsilon^{2}\,\int_{\mathbb{R}}\Big{\\{}b_{1}\,\Big{(}\beta^{-2}x^{2}+h^{2}+2\,f\,h\Big{)}\beta-k\,\big{[}\beta^{-2}\,\beta^{\prime}\,x^{2}-(f\,h^{\prime}+h\,f^{\prime}+h\,h^{\prime})\beta\big{]}\Big{\\}}\,w_{x}^{2}\,{\rm
d}x$
$\displaystyle+\epsilon^{2}(\alpha^{\prime}\alpha^{-1}a_{11}+a_{11}^{\prime})\int_{\mathbb{R}}w_{1}w_{x}\,{\rm
d}x\,+\,\epsilon^{2}k\alpha^{\prime}\,\alpha^{-1}\beta^{-1}\int_{\mathbb{R}}xww_{x}\,{\rm
d}x$ $\displaystyle+\epsilon^{2}k\beta^{-2}\int_{\mathbb{R}}\Big{[}\epsilon
A^{\prime}(0)\beta(0)Z(x)+\phi_{22,z}(x,0)\Big{]}xw_{x}\,{\rm
d}x+O(\epsilon^{3}).$

Using the following formulas

$\int_{\mathbb{R}}Z^{2}(x)\,{\rm
d}x\,=\,-2\int_{\mathbb{R}}x\,Z_{x}(x)\,Z(x)\,{\rm d}x\,=\,1,$

we get the following two estimates

$\begin{split}\int_{\mathbb{R}}g_{0}(x)Z(x)\,{\rm
d}x\,=\,&\frac{\epsilon^{2}}{2}b_{5}\,e(0)+\epsilon^{2}\big{[}\alpha^{\prime}(0)\alpha^{-1}(0)e(0)+e^{\prime}(0)\big{]}+O(\epsilon^{3}).\end{split}$

Higher order errors can be proceeded as follows:

$\displaystyle\int_{\mathbb{R}}$ $\displaystyle
D_{3}^{0}(\phi(x,0))w_{x}\,{\rm d}x\,$ $\displaystyle=$
$\displaystyle\,\epsilon\int_{\mathbb{R}}\big{[}b_{5}\,x+\beta(k_{1}f+f^{\prime})\big{]}\phi_{x}(x,0)w_{x}\,{\rm
d}x$
$\displaystyle-\epsilon\alpha^{\prime}\alpha^{-1}\int_{\mathbb{R}}\phi(x,0)w_{x}\,{\rm
d}x\,-\,\epsilon
k\int_{\mathbb{R}}\Big{(}\frac{x}{\beta}+f+h\Big{)}\phi_{z}(x,0)w_{x}\,{\rm
d}x$
$\displaystyle+\epsilon^{2}\int_{\mathbb{R}}\Big{[}b_{1}\Big{(}\frac{x}{\beta}+f+h\Big{)}^{2}\beta-k\Big{(}\frac{x}{\beta}+f+h\Big{)}\Big{(}\frac{\beta^{\prime}}{\beta}x-\beta
f^{\prime}-\beta h^{\prime}\Big{)}\Big{]}\phi_{x}(x,0)w_{x}\,{\rm d}x$
$\displaystyle-\epsilon^{2}k\frac{\alpha^{\prime}}{\alpha}\int_{\mathbb{R}}\Big{(}\frac{x}{\beta}+f+h\Big{)}\phi(x,0)w_{x}\,{\rm
d}x\,+\,\epsilon^{2}b_{2}\int_{\mathbb{R}}\Big{(}\frac{x}{\beta}+f+h\Big{)}^{2}\phi_{z}(x,0)w_{x}\,{\rm
d}x$ $\displaystyle=$ $\displaystyle\,O(\epsilon^{\frac{5}{2}}),$

and also

$\begin{split}\int_{\mathbb{R}}&D_{3}^{0}(\phi(x,0))Z\,{\rm d}x\,\\\
=&\,\epsilon\int_{\mathbb{R}}\big{[}b_{5}\,x+\beta(k_{1}f+f^{\prime})\big{]}\phi_{x}(x,0)Z\,{\rm
d}x\\\ &-\epsilon\alpha^{\prime}\alpha^{-1}\int_{\mathbb{R}}\phi(x,0)Z\,{\rm
d}x\,-\,\epsilon
k\int_{\mathbb{R}}\Big{(}\frac{x}{\beta}+f+h\Big{)}\phi_{z}(x,0)Z\,{\rm d}x\\\
&+\epsilon^{2}\int_{\mathbb{R}}\Big{[}b_{1}\Big{(}\frac{x}{\beta}+f+h\Big{)}^{2}\beta-k\Big{(}\frac{x}{\beta}+f+h\Big{)}\Big{(}\frac{\beta^{\prime}}{\beta}x-\beta
f^{\prime}-\beta h^{\prime}\Big{)}\Big{]}\phi_{x}(x,0)Z\,{\rm d}x\\\
&-\epsilon^{2}k\frac{\alpha^{\prime}}{\alpha}\int_{\mathbb{R}}\Big{(}\frac{x}{\beta}+f+h\Big{)}\phi(x,0)Z\,{\rm
d}x\,+\,\epsilon^{2}b_{2}\int_{\mathbb{R}}\Big{(}\frac{x}{\beta}+f+h\Big{)}^{2}\phi_{z}(x,0)Z\,{\rm
d}x\\\ =&\,O(\epsilon^{\frac{5}{2}}).\end{split}$

The term $D_{0}^{0}(\phi)$ on the boundary integrated against $w_{x}$ and $Z$
in the variable $x$ are of size of order $O(\epsilon^{3})$.

##  7 The system for $(f,e):$ proof of the theorem

Using the estimates in previous section and $\theta=\epsilon z$, and then
defining the operators

$\displaystyle{\mathbb{L}}_{1}(f)$
$\displaystyle\,\equiv\,f{{}^{\prime\prime}}\,+\,\big{[}\hbar_{1}(\theta)\,+\,\alpha_{1}(\theta/\epsilon)\big{]}f^{\prime}\,+\,\big{[}\hbar_{2}(\theta)\,+\,\alpha_{2}(\theta/\epsilon)\big{]}f,$
(7.1) $\displaystyle{\mathbb{L}}_{2}(e)$
$\displaystyle\,\equiv\,\epsilon^{2}\,(\beta^{-2}\,e{{}^{\prime\prime}}\,+\,\hbar_{5}(\theta)\,e^{\prime})\,+\,\lambda_{0}e,$
(7.2)

where $\hbar_{1}(\theta)$, $\hbar_{2}(\theta)$, $\alpha_{1}(z)$ and
$\alpha_{2}(z)$ are smooth functions defined in (4.67), (4.68), (4.59) and
(4.60), we find the following nonlinear system of differential equations for
the parameters $(f,e)$

$\displaystyle{\mathbb{L}}_{1}(f)$
$\displaystyle\,=\,\hbar_{3}(\theta)\,e\,+\,\epsilon^{2}\,\hbar_{4}(\theta)\,e{{}^{\prime\prime}}\,+\,\epsilon\,{\mathcal{M}}_{1,\epsilon},$
(7.3) $\displaystyle{\mathbb{L}}_{2}(e)$
$\displaystyle\,=\,\epsilon^{3}\,f\,\hbar_{6}(\theta)\,e{{}^{\prime\prime}}\,+{\epsilon}\,\varrho_{2}\,f^{\prime}\,h^{\prime}+{\epsilon}\,\varrho_{2}\,\varpi\,f^{\prime}\,h+\,\epsilon^{2}{\mathcal{M}}_{2,\epsilon},$
(7.4)

with the boundary conditions

$f^{\prime}(1)\,+\,k_{2}\,f(1)\,+\,{\mathcal{M}}_{1}^{1}(f,e)\,=\,0,$ (7.5)
$f^{\prime}(0)\,+\,k_{1}\,f(0)\,+\,{\mathcal{M}}_{1}^{2}(f,e)\,=\,0,$ (7.6)
$e^{\prime}(1)\,+\,\tilde{b}_{6}\,e(1)\,+\,{\mathcal{M}}_{2}^{1}(f,e)\,=\,0,$
(7.7)
$e^{\prime}(0)\,+\,\tilde{b}_{5}\,e(0)\,+\,{\mathcal{M}}_{2}^{2}(f,e)\,=\,0.$
(7.8)

The constants ${\tilde{b}}_{6}$ and ${\tilde{b}}_{5}$ are given by

$\tilde{b}_{6}=\frac{b_{6}}{2}+\frac{\alpha^{\prime}(1)}{\alpha(1)},\qquad\tilde{b}_{5}=\frac{b_{5}}{2}+\frac{\alpha^{\prime}(0)}{\alpha(0)},$

where $b_{5}$ and $b_{6}$ are constants defined in (3.32) and (3.36).
${\mathcal{M}}_{j}^{i}$’s are some terms of order $O(\epsilon^{1/2})$. The
operators ${\mathcal{M}}_{1,\epsilon}$ and ${\mathcal{M}}_{2,\epsilon}$ can be
decomposed in the following forms

${\mathcal{M}}_{l,\epsilon}(f,e)\,=\,\mathbf{A}_{l,\epsilon}(f,e)\,+\,\mathbf{K}_{l,\epsilon}(f,e),\
\ \ \ l=1,2$

where $\mathbf{K}_{l,\epsilon}$’s are uniformly bounded in $L^{2}(0,1)$ for
$(f,e)$ in $\mathcal{F}$ and are also compact. The operators
$\mathbf{A}_{l,\epsilon}$, $i=1,2$, are Lipschitz in this region, see
(6.33)-(6.34)

$\|\mathbf{A}_{l,\epsilon}(f_{1},e_{1})-\mathbf{A}_{l,\epsilon}(f_{2},e_{2})\|_{L^{2}(0,1)}\,\leq\,C\,\big{[}\,\|f_{1}-f_{2}\|_{*}\,+\,\|e_{1}-e_{2}\|_{**}\,\big{]}.$
(7.9)

Before solving (7.3)-(7.8), some basic facts about the invertibility of
corresponding linear operators will be derived. Firstly, we consider the
following problem

$\begin{split}f{{}^{\prime\prime}}(\theta)\,+\,\big{(}\hbar_{1}(\theta)\,+\,\alpha_{1}&(\theta/\epsilon)\big{)}f^{\prime}(\theta)\,+\,\big{(}\hbar_{2}(\theta)\,+\,\alpha_{2}(\theta/\epsilon)\big{)}f(\theta)\,=\,\mathbf{g}(\theta),\\\
f^{\prime}(0)+k_{1}\,f(0)&=0,\qquad\qquad\quad
f^{\prime}(1)+k_{2}\,f(1)=0.\end{split}$ (7.10)

(2). Moreover, if

${\mathbf{g}}(\theta)=G_{1}(\theta/\epsilon)+G_{2}(\theta/\epsilon)+G_{3}(\theta/\epsilon),$

where $G_{1}$, $G_{2}$ and $G_{3}$ are given in (4.61), (4.62) and (4.64),
then

$\|f\|_{*}\leq C\epsilon^{\frac{1}{2}}.$

###### Proof.

(1). Under the non-degeneracy condition (2.11), the existence part comes from
the a priori estimate and the continuity method. Hence, we focus on the
derivations of the estimates.

First of all, by the relation

$\vartheta\,=\,\frac{\mathfrak{a}(\theta)}{\ell},$

where $\ell$ and $a(\theta)$ are given in (4.23) and (4.27), we make the
following transformation:

$\eta(\vartheta)\,=\,f(\theta),\qquad\hat{\beta}(\vartheta)\,=\,\beta(\theta),\qquad\hat{{\mathbf{g}}}(\vartheta)={\mathbf{g}}(\theta),$
$\hat{\hbar}_{1}(\vartheta)\,=\,\hbar_{1}(\theta),\qquad\hat{\hbar}_{2}(\vartheta)\,=\,\hbar_{2}(\theta),\qquad\hat{\varpi}(\vartheta)\,=\,\varpi(\theta).$

Then (7.10) gets transformed into

$\begin{split}\eta_{\vartheta\vartheta}\,+\,\frac{\hat{\beta}^{\prime}(\vartheta)}{\hat{\beta}(\vartheta)}\,\eta_{\vartheta}\,+\,\frac{\hat{\hbar}_{1}(\vartheta)\ell}{\hat{\beta}(\vartheta)}\,\eta_{\vartheta}&\,+\,\frac{\hat{\hbar}_{2}(\vartheta)\,\ell^{2}}{\hat{\beta}^{2}(\vartheta)}\,\eta\,+\,\frac{\hat{\alpha}_{1}(\vartheta)\ell}{\hat{\beta}(\vartheta)}\,\eta_{\vartheta}\,+\,\frac{\hat{\alpha}_{2}(\vartheta)\ell^{2}}{\hat{\beta}^{2}(\vartheta)}\,\eta\,=\,\frac{\ell^{2}}{\hat{\beta}^{2}(\vartheta)}\,\hat{{\mathbf{g}}}(\vartheta),\\\
\eta^{\prime}(1)\,+\,k_{2}\,\eta(1)&\,=\,0,\qquad\eta^{\prime}(0)\,+\,k_{1}\,\eta(0)\,=\,0.\end{split}$
(7.11)

Here the terms $\hat{\alpha}_{1}$ and $\hat{\alpha}_{2}$ are define as

$\begin{split}\hat{\alpha}_{1}(\vartheta)&\,=\,\hat{\bar{\alpha}}_{1}(\vartheta)+\hat{\tilde{\alpha}}_{1}(\vartheta),\qquad\hat{\alpha}_{2}(\vartheta)\,=\,\hat{\bar{\alpha}}_{2}(\vartheta)+\hat{\tilde{\alpha}}_{2}(\vartheta),\end{split}$
(7.12)

where we have denoted

$\hat{\bar{\alpha}}_{2}(\vartheta)=\hat{\varpi}(\vartheta)\,\hat{\bar{\alpha}}_{1}(\vartheta),\qquad\hat{\tilde{\alpha}}_{2}(\vartheta)=\hat{\varpi}(\vartheta)\,\hat{\tilde{\alpha}}_{1}(\vartheta),$
$\hat{\bar{\alpha}}_{1}(\vartheta)\,=2\varrho_{1}^{-1}\,\epsilon\,\hat{\chi}(\vartheta)\,A^{\prime}({\ell}\vartheta)\,\hat{\beta}(\vartheta)\,\int_{\mathbb{R}}\,Z_{x}\,w_{x}\,{\rm
d}x,$
$\hat{\tilde{\alpha}}_{1}(\vartheta)\,=2\varrho_{1}^{-1}\,\hat{\chi}(\vartheta)\,\displaystyle{\int_{\mathbb{R}}}\phi_{22,xz}\Bigg{(}x,\frac{\mathfrak{a}^{-1}(\ell\vartheta)}{\epsilon}\Bigg{)}\,w_{x}\,{\rm
d}x,$

and

$\hat{\chi}(\vartheta)\,\equiv\,\frac{\,\hat{\chi}_{0}(\vartheta)\,}{\beta(0)}\,+\,\frac{\,1-\hat{\chi}_{0}(\vartheta)\,}{\,\beta(1)}.$

In the above, $\mathfrak{a}^{-1}$ is the inverse of the map $\mathfrak{a}$
given in (4.27).

Define the operator ${\mathbb{L}}_{0}$ in the form

${\mathbb{L}}_{0}(\eta):\,=\,\eta^{\prime\prime}(\vartheta)\,+\,\mathfrak{q}_{1}(\vartheta)\,\eta^{\prime}(\vartheta)\,+\,\mathfrak{q}_{2}(\vartheta)\,\eta,$

where

$\mathfrak{q}_{1}(\vartheta)\,=\,\frac{\hat{\beta}^{\prime}(\vartheta)}{\hat{\beta}(\vartheta)}\,+\,\frac{\hat{\hbar}_{1}(\vartheta)\,{\ell}}{\hat{\beta}(\vartheta)},\qquad\mathfrak{q}_{2}(\vartheta)\,=\,\frac{\hat{\hbar}_{2}(\vartheta)\,{\ell}^{2}}{\hat{\beta}^{2}(\vartheta)}.$

There exists an orthonormal basis of $L^{2}(0,1)$ constituted by
eigenfunctions $\\{y_{j}\\}$, associated to the eigenvalues
$\\{\lambda_{j}\\}$, of the following eigenvalue problem

$\begin{split}-\,{\mathbb{L}}_{0}\big{(}y(\vartheta)\big{)}&\,=\,\lambda\,y(\vartheta),\
\ \ \ \ 0<\vartheta<1,\\\ y^{\prime}(1)\,+\,k_{2}\,y(1)&\,=\,0,\ \ \ \ \ \ \
y^{\prime}(0)\,+\,k_{1}\,y(0)\,=\,0.\end{split}$ (7.13)

The result in [39] shows that, as $j\rightarrow\infty$,

$\sqrt{\lambda_{j}}=j\pi+\frac{k_{2}-k_{1}}{j\pi}+O\Big{(}\frac{1}{j^{3}}\Big{)}.$

It is easy to see that there exists a positive constant $C$ such that
$|y^{\prime}_{j}(\vartheta)|\leq Cj$ for all $j\in\mathbb{N}$.

We then expand

$\displaystyle\frac{{\ell}^{2}}{\hat{\beta}^{2}(\vartheta)}\,\hat{{\mathbf{g}}}(\vartheta)$
$\displaystyle\,=\,\sum_{j=0}^{\infty}\,{\mathbf{g}}_{j}\,y_{j}(\vartheta),\qquad\qquad\eta(\vartheta)\,=\,\sum_{j=0}^{\infty}\,a_{j}\,y_{j}(\vartheta),$
$\displaystyle\frac{\hat{\bar{\alpha}}_{1}{\ell}}{\hat{\beta}(\vartheta)}\,\eta^{\prime}(\vartheta)$
$\displaystyle\,=\,\sum_{j=0}^{\infty}\,\bar{d}_{j}\,y_{j}(\vartheta),\qquad\frac{\hat{\bar{\alpha}}_{2}{\ell}^{2}}{\hat{\beta}^{2}(\vartheta)}\,\eta(\vartheta)\,=\,\sum_{j=0}^{\infty}\,\bar{c}_{j}\,y_{j}(\vartheta),$
$\displaystyle\frac{\hat{\tilde{\alpha}}_{1}{\ell}}{\hat{\beta}(\vartheta)}\,\eta^{\prime}(\vartheta)$
$\displaystyle\,=\,\sum_{j=0}^{\infty}\tilde{d}_{j}\,y_{j}(\vartheta),\
\qquad\frac{\hat{\tilde{\alpha}}_{2}{\ell}^{2}}{\hat{\beta}^{2}(\vartheta)}\,\eta(\vartheta)\,=\,\sum_{j=0}^{\infty}\tilde{c}_{j}\,y_{j}(\vartheta),$
$\displaystyle\frac{\hat{\alpha}_{1}{\ell}}{\hat{\beta}(\vartheta)}\,\eta^{\prime}(\vartheta)$
$\displaystyle\,=\,\sum_{j=0}^{\infty}\,d_{j}\,y_{j}(\vartheta),\qquad\frac{\hat{\alpha}_{2}{\ell}^{2}}{\hat{\beta}^{2}(\vartheta)}\,\eta(\vartheta)\,=\,\sum_{j=0}^{\infty}\,c_{j}\,y_{j}(\vartheta).$

Now, we will estimate the Fourier coefficients above by using the asymptotic
behaviors of the following two terms

$\Phi_{j,1}(\vartheta)=\int_{0}^{\vartheta}\sin\Big{(}\frac{\sqrt{\lambda}_{0}}{\epsilon}{\ell}s\Big{)}\,y_{j}(s)\,{\mathrm{d}}s,\qquad\Phi_{j,2}(\vartheta)=\int_{0}^{\vartheta}\cos\Big{(}\frac{\sqrt{\lambda}_{0}}{\epsilon}{\ell}s\Big{)}\,y_{j}(s)\,{\mathrm{d}}s.$

In the following formula, by using the equation for $y_{j}$ and integrating by
parts two times

$\displaystyle\Phi_{j,1}(\vartheta)$
$\displaystyle=-\frac{\epsilon^{2}}{\lambda_{0}{\ell}^{2}}\int_{0}^{\vartheta}\Big{[}\sin\Big{(}\frac{\sqrt{\lambda}_{0}}{\epsilon}\,{\ell}\,s\Big{)}\Big{]}^{\prime\prime}y_{j}(s)\,{\rm
d}s$
$\displaystyle=-\frac{\epsilon}{\sqrt{\lambda_{0}}\,{\ell}}\Big{[}\cos\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)}\,y_{j}(\vartheta)\,-\,y_{j}(0)\Big{]}\,+\,\frac{\epsilon^{2}}{\lambda_{0}{\ell}^{2}}\,\sin\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)}\,y^{\prime}_{j}(\vartheta)$
$\displaystyle\qquad-\frac{\epsilon^{2}}{\lambda_{0}{\ell}^{2}}\int_{0}^{\vartheta}\sin\Big{(}\frac{\sqrt{\lambda}_{0}}{\epsilon}{\ell}s\Big{)}\,y^{\prime\prime}_{j}(s)\,{\rm
d}s$
$\displaystyle=-\frac{\epsilon}{\sqrt{\lambda_{0}}{\ell}}\Big{[}\cos\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)}\,y_{j}(\vartheta)\,-\,y_{j}(0)\Big{]}\,+\,\frac{\epsilon^{2}}{\lambda_{0}{\ell}^{2}}\,\sin\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)}\,y^{\prime}_{j}(\vartheta)$
$\displaystyle\qquad-\frac{\epsilon^{2}}{\lambda_{0}{\ell}^{2}}\int_{0}^{\vartheta}\sin\Big{(}\frac{\sqrt{\lambda}_{0}}{\epsilon}\,{\ell}\,s\Big{)}\big{[}y^{\prime\prime}_{j}(s)\,+\,\mathfrak{q}_{1}(s)\,y^{\prime}_{j}(s)+\mathfrak{q}_{2}(s)\,y_{j}(s)\big{]}\,{\rm
d}s$
$\displaystyle\qquad\quad+\frac{\epsilon^{2}}{\lambda_{0}{\ell}^{2}}\int_{0}^{\vartheta}\sin\Big{(}\frac{\sqrt{\lambda}_{0}}{\epsilon}\,{\ell}\,s\Big{)}\big{[}\mathfrak{q}_{1}(s)\,y^{\prime}_{j}(s)\,+\,\mathfrak{q}_{2}(s)\,y_{j}(s)\big{]}\,{\rm
d}s$
$\displaystyle=\frac{\epsilon^{2}}{\lambda_{0}{\ell}^{2}}\int_{0}^{\vartheta}\sin\Big{(}\frac{\sqrt{\lambda}_{0}}{\epsilon}{\ell}s\Big{)}\lambda_{j}y_{j}(s)\,{\mathrm{d}}s\,-\,\frac{\epsilon}{\sqrt{\lambda_{0}}{\ell}}\Big{[}\cos\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)}\,y_{j}(\vartheta)-y_{j}(0)\Big{]}$
$\displaystyle\quad\,+\,\frac{\epsilon^{2}}{\lambda_{0}{\ell}^{2}}\,\sin\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)}\,y^{\prime}_{j}(\vartheta)\,+\,\frac{\epsilon^{2}}{\lambda_{0}{\ell}^{2}}\int_{0}^{\vartheta}\sin\Big{(}\frac{\sqrt{\lambda}_{0}}{\epsilon}{\ell}s\Big{)}\big{[}\mathfrak{q}_{1}(s)\,y^{\prime}_{j}(s)+\mathfrak{q}_{2}(s)\,y_{j}(s)\big{]}\,{\rm
d}s,$

we can get

$\begin{split}\Big{(}1-\frac{\epsilon^{2}\lambda_{j}}{\lambda_{0}{\ell}^{2}}\Big{)}\,\Phi_{j,1}(\vartheta)=-&\frac{\epsilon}{\sqrt{\lambda_{0}}{\ell}}\,\Big{[}\cos\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)}\,y_{j}(\vartheta)\,-\,y_{j}(0)\Big{]}\,+\,\frac{\epsilon^{2}}{\lambda_{0}{\ell}^{2}}\,\sin\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)}\,y^{\prime}_{j}(\vartheta)\\\
&\,+\,\frac{\epsilon^{2}}{\lambda_{0}{\ell}^{2}}\int_{0}^{\vartheta}\sin\Big{(}\frac{\sqrt{\lambda}_{0}}{\epsilon}\,{\ell}\,s\Big{)}\big{[}\mathfrak{q}_{1}(s)\,y^{\prime}_{j}(s)+\mathfrak{q}_{2}(s)\,y_{j}(s)\big{]}\,{\rm
d}s.\end{split}$

This implies that

$|\Phi_{j,1}(\vartheta)|\leq\frac{C\epsilon(\epsilon
j+1)}{|\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j}|}.$ (7.14)

Similarly, it can be derived

$|\Phi_{j,2}(\vartheta)|\leq\frac{C\epsilon(\epsilon
j+1)}{|\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j}|}.$ (7.15)

By using the gap condition (1.14), we can write $\hat{\bar{\alpha}}_{2}$ in
the form

$\hat{\bar{\alpha}}_{2}=\bar{K}_{1}\,\hat{\varpi}(\vartheta)\hat{\chi}(\vartheta)\,\hat{\beta}(\vartheta)\,\sin\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)}\,+\,\bar{K}_{2}\,\hat{\varpi}(\vartheta)\hat{\chi}(\vartheta)\,\hat{\beta}(\vartheta)\cos\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)},$

for parameters $\bar{K}_{1},\bar{K}_{2}$ depending on $\epsilon$, which are
also bounded by a universal constant independent of $\epsilon$. Hence,

$\begin{split}\bar{c}_{j}&=\int_{0}^{1}\Big{[}\bar{K}_{1}\sin\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)}\,+\,\bar{K}_{2}\,\cos\Big{(}\frac{\sqrt{\lambda_{0}}}{\epsilon}\,{\ell}\,\vartheta\Big{)}\Big{]}\,\hat{\varpi}(\vartheta)\,\hat{\chi}(\vartheta)\,\frac{{\ell}^{2}}{\hat{\beta}(\vartheta)}\,\eta(\vartheta)\,y_{j}(\vartheta)\,{\rm
d}\vartheta\\\
&=\bar{K}_{1}\,{\ell}^{2}\,\int_{0}^{1}\,\hat{\varpi}(\vartheta)\hat{\chi}(\vartheta)\,\hat{\beta}^{-1}(\vartheta)\,\Phi^{\prime}_{j,1}(\vartheta)\,\eta(\vartheta)\,{\rm
d}\vartheta\,+\,\bar{K}_{2}\,{\ell}^{2}\,\int_{0}^{1}\,\hat{\varpi}(\vartheta)\hat{\chi}(\vartheta)\,\hat{\beta}^{-1}(\vartheta)\,\Phi^{\prime}_{j,2}(\vartheta)\,\eta(\vartheta)\,{\rm
d}\vartheta.\end{split}$

Integrating once by parts and using (7.14)-(7.15), we obtain

$|\bar{c}_{j}|\leq\frac{C\epsilon(\epsilon
j+1)}{|\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j}|}\Big{\\{}\|\eta^{\prime}\|_{L^{2}}+\|\eta\|_{L^{2}}\Big{\\}}.$
(7.16)

A similar approach will imply

$|\bar{d}_{j}|\leq\frac{C\epsilon(\epsilon
j+1)}{|\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j}|}\Big{\\{}\|\eta^{\prime}\|_{L^{2}}+\|\eta\|_{L^{2}}\Big{\\}}.$

The estimate of $\tilde{d}_{j}$ can be showed as

$\displaystyle|\tilde{d}_{j}|$
$\displaystyle=\Bigg{|}\int_{0}^{1}\frac{\hat{\tilde{\alpha}}_{1}{\ell}}{\hat{\beta}(\vartheta)}\,\eta^{\prime}(\vartheta)\,y_{j}(\vartheta)\,{\rm
d}\vartheta\Bigg{|}$
$\displaystyle=2\varrho_{1}^{-1}{\ell}\,\Bigg{|}\displaystyle{\int_{0}^{1}\int_{\mathbb{R}}}\hat{\chi}(\vartheta)\,\hat{\beta}^{-1}(\vartheta)\,\eta^{\prime}(\vartheta)\,\phi_{22,xz}\Bigg{(}x,\frac{\mathfrak{a}^{-1}(\ell\vartheta)}{\epsilon}\Bigg{)}\,w_{x}\,y_{j}(\vartheta)\,{\rm
d}x\,{\rm d}\vartheta\Bigg{|}$ (7.17) $\displaystyle\leq
C\,\Bigg{[}\int_{0}^{1}\int_{\mathbb{R}}\big{|}\hat{\chi}(\vartheta)\,\hat{\beta}^{-1}(\vartheta)\,\eta^{\prime}(\vartheta)\,y_{j}(\vartheta)\,w_{x}\big{|}^{2}\,{\rm
d}x\,{\rm
d}\vartheta\Bigg{]}^{\frac{1}{2}}\Bigg{[}\int_{0}^{1}\int_{\mathbb{R}}\Big{|}\phi_{22,xz}\Bigg{(}x,\frac{\mathfrak{a}^{-1}(\ell\vartheta)}{\epsilon}\Bigg{)}\Big{|}^{2}\,{\rm
d}x\,{\rm d}\vartheta\Bigg{]}^{\frac{1}{2}}$ $\displaystyle\leq
C\,\epsilon^{\frac{1}{2}}\,\|\eta^{\prime}\|_{L^{2}},$

where $\mathfrak{a}^{-1}$ is the inverse of the map $\mathfrak{a}$ given in
(4.27). Similar estimates hold for $\tilde{c}_{j}$

$|\tilde{c}_{j}|\leq C\,\epsilon^{\frac{1}{2}}\,\|\eta\|_{L^{2}}.$

Whence, from the equations

$\begin{split}-\lambda_{j}a_{j}\,+\,c_{j}\,+\,d_{j}\,=\,{\mathbf{g}}_{j},\\\
\bar{c}_{j}\,+\,\tilde{c}_{j}\,=\,c_{j},\quad\bar{d}_{j}\,+\,\tilde{d}_{j}\,=\,d_{j},\end{split}$

and estimates of the Fourier coefficients $\bar{c}_{j}$, $\bar{d}_{j}$,
$\tilde{c}_{j}$ and $\tilde{d}_{j}$, we get

$|a_{j}|\,\leq\,\Big{|}\frac{{\mathbf{g}}_{j}}{\lambda_{j}}\Big{|}+\frac{C\epsilon(\epsilon
j+1)}{|\lambda_{j}(\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j})|}\times\\{\|\eta^{\prime}\|_{L^{2}}+\|\eta\|_{L^{2}}\\}\,+\,\frac{C\,\epsilon^{\frac{1}{2}}}{\lambda_{j}}\times\\{\|\eta^{\prime}\|_{L^{2}}+\|\eta\|_{L^{2}}\\}.$
(7.18)

Moreover, from the asymptotic expression of $\lambda_{j}$ in (7.13), it can be
shown

$\|\eta^{\prime}\|_{L^{2}}^{2}+\|\eta\|_{L^{2}}^{2}\,\leq\,C\,\|{\mathbf{g}}\|_{L^{2}}^{2}+C_{*}\,\sum_{j}\frac{j^{2}\epsilon^{2}(\epsilon
j+1)^{2}}{|\lambda_{j}|^{2}(\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j})^{2}}\times\\{\|\eta^{\prime}\|_{L^{2}}^{2}+\|\eta\|_{L^{2}}^{2}\\},$

where the positive constant $C_{*}$ does not depend on $\epsilon$. From the
asymptotic expression of $\lambda_{j}$, there exists a positive constant
$\epsilon_{0}$ such that, for all positive $\epsilon<\epsilon_{0}$ satisfying
(1.14), elementary analysis will imply the following estimates

$\begin{split}\sum\limits_{2\epsilon^{2}\lambda_{j}\geq
3\lambda_{0}{\ell}^{2}}\frac{j^{2}\epsilon^{2}(\epsilon
j+1)^{2}}{|\lambda_{j}|^{2}(\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j})^{2}}&\,\leq\,C\,\epsilon^{2},\\\
\sum\limits_{\lambda_{0}{\ell}^{2}<2\epsilon^{2}\lambda_{j}<3\lambda_{0}{\ell}^{2}}\frac{j^{2}\epsilon^{2}(\epsilon
j+1)^{2}}{|\lambda_{j}|^{2}(\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j})^{2}}&\,\leq\,C\,\epsilon,\\\
\sum\limits_{2\epsilon^{2}\lambda_{j}\leq\lambda_{0}{\ell}^{2}}\frac{j^{2}\epsilon^{2}(\epsilon
j+1)^{2}}{|\lambda_{j}|^{2}(\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j})^{2}}&\,\leq\,C\,\epsilon^{2}.\end{split}$

Hence, we can prove that

$\|\eta^{\prime}\|_{L^{2}}^{2}+\|\eta\|_{L^{2}}^{2}\,\leq\,C\,\|{\mathbf{g}}\|_{L^{2}}^{2}.$

Since

$\|\eta^{\prime\prime}\|_{L^{2}}\,\leq\,C\,\big{(}\|{\mathbf{g}}\|_{L^{2}}+\|\eta^{\prime}\|_{L^{2}}+\|\eta\|_{L^{2}}^{2}\big{)},$

the final result then follows easily.

(2). If
${\mathbf{g}}(\theta)=G_{1}(\theta/\epsilon)+G_{2}(\theta/\epsilon)+G_{3}(\theta/\epsilon)$,
using the same argument as the proof of formulas (7.16) and (7.17), we can
easily carry out the estimate of ${\mathbf{g}}_{j}$ in the form

$|{\mathbf{g}}_{j}|\leq\frac{C\epsilon(\epsilon
j+1)}{|\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j}|}+C\epsilon^{\frac{1}{2}}.$

For all positive $\epsilon<\epsilon_{0}$ satisfying (1.14), a similar analysis
will imply the following estimates

$\begin{split}\sum\limits_{2\epsilon^{2}\lambda_{j}\geq
3\lambda_{0}{\ell}^{2}}\frac{\epsilon(\epsilon
j+1)}{|\lambda_{j}(\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j})|}&\,\leq\,C\epsilon^{\frac{1}{2}},\\\
\sum\limits_{\lambda_{0}{\ell}^{2}<2\epsilon^{2}\lambda_{j}<3\lambda_{0}{\ell}^{2}}\frac{\epsilon(\epsilon
j+1)}{|\lambda_{j}(\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j})|}&\,\leq\,C\epsilon^{\frac{1}{2}},\\\
\sum\limits_{2\epsilon^{2}\lambda_{j}\leq\lambda_{0}{\ell}^{2}}\frac{\epsilon(\epsilon
j+1)}{|\lambda_{j}(\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j})|}&\,\leq\,C\,\epsilon^{\frac{1}{2}}.\end{split}$

Therefore, from (7.18) and the estimate of ${\mathbf{g}}_{j}$, we get

$\|\eta^{\prime}\|_{L^{2}}^{2}+\|\eta\|_{L^{2}}^{2}\,\leq\,C\,{\epsilon}+C_{*}\,\sum_{j}\frac{j^{2}\epsilon^{2}(\epsilon
j+1)^{2}}{|\lambda_{j}|^{2}(\lambda_{0}{\ell}^{2}-\epsilon^{2}\lambda_{j})^{2}}\times\\{\|\eta^{\prime}\|_{L^{2}}^{2}+\|\eta\|_{L^{2}}^{2}\\},$

So, we can prove that

$\|\eta^{\prime}\|_{L^{2}}+\|\eta\|_{L^{2}}\,\leq\,C\,\epsilon^{\frac{1}{2}},$

the final result then follows easily. ∎

Secondly, we consider the following problem

$\begin{split}\,\epsilon^{2}\,\big{[}\beta^{-2}\,e{{}^{\prime\prime}}\,+\,\hbar_{5}(\theta)\,e^{\prime}\big{]}\,+\,\lambda_{0}\,e\,=\,\tilde{{\mathbf{g}}},\qquad\forall\,0<\theta<1,\\\
e^{\prime}(0)\,+\,\tilde{b}_{5}\,e(0)\,=\,0,\qquad\quad
e^{\prime}(1)\,+\,\tilde{b}_{6}\,e(1)\,=\,0.\end{split}$ (7.19)

######  Lemma 7.2.

If $\tilde{{\mathbf{g}}}\in L^{2}(0,1)$, then for all small $\epsilon$
satisfying (1.14) there is a unique solution $e\in H^{2}(0,1)$ to problem
(7.19), which satisfies

$\|e\|_{b}\leq C\,\epsilon^{-1}\,\|\tilde{{\mathbf{g}}}\|_{L^{2}(0,1)}.$

Moreover, if $\tilde{{\mathbf{g}}}\in H^{2}(0,1)$, then

$\epsilon^{2}\,\|e^{\prime\prime}\|_{L^{2}(0,1)}\,+\,\epsilon\,\|e^{\prime}\|_{L^{2}(0,1)}\,+\,\|e\|_{L^{\infty}(0,1)}\,\leq\,C\,\|\tilde{{\mathbf{g}}}\|_{H^{2}(0,1)}.$
(7.20)

###### Proof.

See Lemma 8.1 in [23]. ∎

Thirdly, we consider the following system

$\begin{split}{\mathbb{L}}(f,e)\,\equiv\,({\mathbb{L}}_{1}(f),\,{\mathbb{L}}_{2}(e))\,=\,({\mathbf{g}}(\theta),\tilde{{\mathbf{g}}}(\theta)),\qquad
0<\theta<1,\\\ f^{\prime}(1)\,+\,k_{2}\,f(1)\,=\,\Gamma_{1}^{1},\qquad
f^{\prime}(0)\,+\,k_{1}\,f(0)\,=\,\Gamma_{0}^{1},\\\
e^{\prime}(1)\,+\,\tilde{b}_{6}\,e(1)\,=\,\Gamma_{1}^{0},\qquad
e^{\prime}(0)\,+\,\tilde{b}_{5}\,e(0)\,=\,\Gamma_{0}^{0},\end{split}$ (7.21)

where $\Gamma_{j}^{i},i,\,j=0,\,1$ are some constants.

######  Lemma 7.3.

Under the non-degenerate condition (2.7), if
${\mathbf{g}},\,\tilde{{\mathbf{g}}}\in L^{2}(0,1)$, then there exists
$\epsilon_{0}$ such that for all $0<\epsilon<\epsilon_{0}$ satisfying (1.14)
there is a unique solution $(f,e)$ in $H^{2}(0,1)$ to problem (7.21) which
satisfies

$\|f\|_{*}\,+\,\|e\|_{**}\leq\Big{[}\|{\mathbf{g}}\|_{L^{2}(0,1)}\,+\,\epsilon^{-1}\|\tilde{{\mathbf{g}}}\|_{L^{2}(0,1)}\,+\,\sum_{i,j=0}^{1}|\Gamma_{j}^{i}|\Big{]}.$

###### Proof.

See Lemma 7.3 in [61]. ∎

Proof of Theorem 1.1. Let us observe now that the linear operator

${\mathbb{L}}^{*}(f,e)=\big{(}{\mathbb{L}}_{1}(f)-\hbar_{3}(\epsilon
z)e-\epsilon^{2}\hbar_{4}(\epsilon
z)e{{}^{\prime\prime}},\,{\mathbb{L}}_{2}(e)\big{)},$

is invertible with bounds for
${\mathbb{L}}^{*}(f,e)=({\mathbf{g}},\tilde{{\mathbf{g}}})$ given by

$\|f\|_{*}\,+\,\|e\|_{**}\leq
C\|{\mathbf{g}}\|_{L^{2}(0,1)}\,+\,\epsilon^{-1}\|\tilde{{\mathbf{g}}}\|_{L^{2}(0,1)}.$

From (7.9), $\epsilon\,\mathbf{A}_{1,\epsilon}$ and
$\epsilon^{2}\,\mathbf{A}_{2,\epsilon}$ are contraction mappings of their
arguments in $\mathcal{F}$. By Banach Contraction Mapping Theorem and Lemma
7.3, we can solve the nonlinear problem

$\big{[}{\mathbb{L}}^{*}-(\epsilon\,\mathbf{A}_{1,\epsilon},\epsilon^{2}\,\mathbf{A}_{2,\epsilon})\big{]}(f,e)=({\mathbf{g}},\tilde{{\mathbf{g}}})$

with the boundary conditions defined in (7.5)-(7.8) is uniquely solvable for
$(f,e)\in\mathcal{F}$ provided that

$\|{\mathbf{g}}\|_{L^{2}(0,1)}\,<\,\epsilon^{1/2+\rho}\quad\mbox{and}\quad\|\tilde{{\mathbf{g}}}\|_{L^{2}(0,1)}\,<\,\epsilon^{3/2+\rho},$

with some small positive constants $\rho$. The desired result for full problem
(7.3)-(7.8) then follows directly from Schauder’s fixed-point Theorem. ∎

Acknowledgements:  J. Yang is supported by NSFC(No.11371254). Part of this
work was done when the authors visited Chern Institute of Mathematics, Nankai
University in summer of 2014: we are very grateful to the institution for the
kind hospitality. ∎

##  Appendix A Proofs of Lemmas 2.1 and 2.2

Proof of Lemma 2.1:  In fact, the curves can be expressed in the following
forms

$\displaystyle{\mathcal{C}}_{1}:$
$\displaystyle~{}{\mathbb{H}}\big{(}{\tilde{t}},{\tilde{\varphi}}_{1}({\tilde{t}})\big{)}\,=\,\gamma\big{(}{\tilde{\varphi}}_{1}({\tilde{t}})\big{)}+{\tilde{t}}\,n\big{(}{\tilde{\varphi}}_{1}({\tilde{t}})\big{)},$
(A.1) $\displaystyle{\mathcal{C}}_{2}:$
$\displaystyle~{}{\mathbb{H}}\big{(}{\tilde{t}},{\tilde{\varphi}}_{2}({\tilde{t}})\big{)}\,=\,\gamma\big{(}{\tilde{\varphi}}_{2}({\tilde{t}})\big{)}+{\tilde{t}}\,n\big{(}{\tilde{\varphi}}_{2}({\tilde{t}})\big{)},$
(A.2) $\displaystyle\Gamma:$
$\displaystyle~{}{\mathbb{H}}(0,\tilde{\theta})\,=\,\gamma(\tilde{\theta}).$
(A.3)

It follows that the tangent vectors of ${\mathcal{C}}_{1}$ at $P_{1}$ can be
written as

$\frac{{\mathrm{d}}{\mathcal{C}}_{1}}{{\mathrm{d}}{\tilde{t}}}\Big{|}_{{\tilde{t}}=0}\,=\,\frac{\partial\gamma}{\partial\tilde{\theta}}\Big{|}_{{\tilde{\theta}}\,=\,{{\tilde{\varphi}}_{1}(0)}}\cdot\frac{{\mathrm{d}}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}}\Big{|}_{{\tilde{t}}=0}\,+\,n\big{(}{\tilde{\varphi}}_{1}(0)\big{)},$

and the tangent vector of $\Gamma$ at $P_{1}$ is

$\frac{\partial\gamma}{\partial\tilde{\theta}}\Big{|}_{\tilde{\theta}\,=\,{\tilde{\varphi}}_{1}(0)}.$

According to the condition: $\Gamma\bot\partial\Omega$, we have that

$\left<\frac{{\mathrm{d}}{\mathcal{C}}_{1}}{{\mathrm{d}}{\tilde{t}}}\Big{|}_{{\tilde{t}}=0},\,\,\frac{\partial\gamma}{\partial\tilde{\theta}}\Big{|}_{\tilde{\theta}={\tilde{\varphi}}_{1}(0)}\right>=0.$

Combining with

$\left<n({\tilde{\varphi}}_{1}(0)),\,\,\frac{\partial\gamma}{\partial\tilde{\theta}}\Big{|}_{\tilde{\theta}={\tilde{\varphi}}_{1}(0)}\right>=0,$

we have

$\tilde{\varphi}_{1}^{\prime}(0)=0.$

Similarly, we can show ${\tilde{\varphi}}_{2}^{\prime}(0)=0$.

In the coordinate system $(y_{1},y_{2})$, $\gamma(\tilde{\theta})$ and
$n(\tilde{\theta})$ can be expressed as follows

$\gamma(\tilde{\theta})=\big{(}\gamma_{1}(\tilde{\theta}),\gamma_{2}(\tilde{\theta})\big{)},\qquad
n(\tilde{\theta})=\big{(}n_{1}(\tilde{\theta}),n_{2}(\tilde{\theta})\big{)}.$

The relations

$\displaystyle|\gamma_{1}^{\prime}(\tilde{\theta})|^{2}\,+\,|\gamma_{2}^{\prime}(\tilde{\theta})|^{2}=1,\qquad|n_{1}(\tilde{\theta})|^{2}+|n_{2}(\tilde{\theta})|^{2}=1,\qquad\gamma_{1}^{\prime}(\tilde{\theta})n_{1}(\tilde{\theta})+\gamma_{2}^{\prime}(\tilde{\theta})n_{2}(\tilde{\theta})=0,$

will give that

$\displaystyle\gamma_{1}^{\prime}(\tilde{\theta})n_{2}(\tilde{\theta})-\gamma_{2}^{\prime}(\tilde{\theta})n_{1}(\tilde{\theta})\,=\,1,$

provided that the sign is taken as ’+’ by suitable choice of the natural
parameter of $\Gamma$. Then, the curve ${\mathcal{C}}_{1}$ can be expressed in
the following form

$\begin{split}{\mathcal{C}}_{1}:~{}{\mathbb{H}}\big{(}{\tilde{t}},{\tilde{\varphi}}_{1}({\tilde{t}})\big{)}\,&=\Big{(}\gamma_{1}\big{(}{\tilde{\varphi}}_{1}({\tilde{t}})\big{)}+\tilde{t}n_{1}\big{(}{\tilde{\varphi}}_{1}({\tilde{t}})\big{)},\,\,\gamma_{2}\big{(}{\tilde{\varphi}}_{1}({\tilde{t}})\big{)}+\tilde{t}n_{2}\big{(}{\tilde{\varphi}}_{1}({\tilde{t}})\big{)}\Big{)}\\\
&\equiv\,\big{(}y_{1}(\tilde{t})\,,\,y_{2}(\tilde{t})\big{)}.\end{split}$

The calculations

$\displaystyle y_{1}^{\prime}({\tilde{t}})=$
$\displaystyle\,\gamma_{1}^{\prime}({\tilde{\varphi}}_{1})\cdot\frac{{\mathrm{d}}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}}+n_{1}({\tilde{\varphi}}_{1})+\tilde{t}\cdot
n_{1}^{\prime}({\tilde{\varphi}}_{1})\cdot\frac{{\mathrm{d}}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}},$
$\displaystyle y_{2}^{\prime}({\tilde{t}})=$
$\displaystyle\,\gamma_{2}^{\prime}({\tilde{\varphi}}_{1})\cdot\frac{{\mathrm{d}}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}}+n_{2}({\tilde{\varphi}}_{1})+\tilde{t}\cdot
n_{2}^{\prime}({\tilde{\varphi}}_{1})\cdot\frac{{\mathrm{d}}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}},$
$\displaystyle y_{1}^{\prime\prime}({\tilde{t}})=$
$\displaystyle\,\gamma_{1}^{\prime\prime}({\tilde{\varphi}}_{1})\cdot\Big{(}\frac{{\mathrm{d}}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}}\Big{)}^{2}+\gamma_{1}^{\prime}({\tilde{\varphi}}_{1})\cdot\frac{{\mathrm{d}}^{2}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}^{2}}+2\,n_{1}^{\prime}({\tilde{\varphi}}_{1})\cdot\frac{{\mathrm{d}}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}}$
$\displaystyle+\tilde{t}\,n_{1}^{\prime\prime}({\tilde{\varphi}}_{1})\cdot\Big{(}\frac{{\mathrm{d}}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}}\Big{)}^{2}+\tilde{t}\,n_{1}^{\prime}({\tilde{\varphi}}_{1})\cdot\frac{{\mathrm{d}}^{2}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}^{2}},$
$\displaystyle y_{2}^{\prime\prime}({\tilde{t}})=$
$\displaystyle\,\gamma_{2}^{\prime\prime}({\tilde{\varphi}}_{1})\cdot\Big{(}\frac{{\mathrm{d}}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}}\Big{)}^{2}+\gamma_{2}^{\prime}({\tilde{\varphi}}_{1})\cdot\frac{{\mathrm{d}}^{2}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}^{2}}+2\,n_{2}^{\prime}({\tilde{\varphi}}_{1})\cdot\frac{{\mathrm{d}}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}}$
$\displaystyle+\tilde{t}\,n_{2}^{\prime\prime}({\tilde{\varphi}}_{1})\cdot\Big{(}\frac{{\mathrm{d}}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}}\Big{)}^{2}+\tilde{t}\,n_{2}^{\prime}({\tilde{\varphi}}_{1})\cdot\frac{{\mathrm{d}}^{2}{{\tilde{\varphi}}_{1}}}{{\mathrm{d}}{\tilde{t}}^{2}},$

imply that

$\displaystyle(y_{1}^{\prime}(t))^{2}+(y_{2}^{\prime}(t))^{2}\Big{|}_{\tilde{t}=0}\,=\,|n(0)|^{2}\,=\,1,$

and

$\displaystyle
y_{1}^{\prime}(\tilde{t})\,y_{2}^{\prime\prime}(\tilde{t})-y_{1}^{\prime\prime}(\tilde{t})\,y_{2}^{\prime}(\tilde{t})\Big{|}_{\tilde{t}=0}\,=\,\big{(}n_{1}(0)\,\gamma_{2}^{\prime}(0)\,-\,n_{2}(0)\,\gamma_{1}^{\prime}(0)\big{)}\,{\tilde{\varphi}}_{1}^{\prime\prime}(0)\,=\,{\tilde{\varphi}}_{1}^{\prime\prime}(0).$

Therefore, the signed curvature of the curve $\mathcal{C}_{1}$ at the point
$P_{1}$ is

$k_{1}=\frac{\,y_{1}^{\prime}(0)y_{2}^{\prime\prime}(0)-y_{1}^{\prime\prime}(0)y_{2}^{\prime}(0)\,}{\big{(}(y_{1}^{\prime}(0))^{2}+(y_{2}^{\prime}(0))^{2}\big{)}^{\frac{3}{2}}}=\tilde{\varphi}_{1}^{\prime\prime}(0).$

Similarly, we can show $k_{2}={\tilde{\varphi}}_{2}^{\prime\prime}(0)$. ∎

Proof of Lemma 2.2: Indeed, first comes the derivative of first order

$\displaystyle\begin{aligned}
\frac{{\partial}F}{{\partial}t}(0,\theta)&\,=\,\Big{[}\gamma{\,{}^{\prime}}(\Theta)\cdot\Theta_{t}\,+\,n(\Theta)\,+\,tn^{\prime}(\Theta)\cdot{\Theta}_{t}\Big{]}\Big{|}_{(0,\theta)}\\\
&\,=\,\gamma{\,{}^{\prime}}(\Theta)\cdot 0\,+\,n(\Theta(0,\theta))\\\
&\,=\,n(\theta).\end{aligned}$

So as $t$ is small enough, $\frac{{\partial}F}{{\partial}t}\neq 0$. Consider
the derivative of second order

$\begin{split}q_{1}(\theta)&\,\equiv\,\frac{{\partial}^{2}F}{{\partial}t^{2}}(0,\theta)\\\
&\,=\,\left[\gamma^{\prime\prime}(\Theta)\cdot(\Theta_{t})^{2}\,+\,\gamma^{\prime}(\Theta)\cdot\Theta_{tt}\,+\,2n^{\prime}(\Theta)\cdot\Theta_{t}\,+\,tn^{\prime\prime}(\Theta)\cdot(\Theta_{t})^{2}\,+\,tn^{\prime}(\Theta)\cdot\Theta_{tt}\right]\Big{|}_{(0,\theta)}\\\
&\,=\,\gamma^{\prime}(\theta)\cdot\Theta_{tt}(0,\theta)\;\;{\bot
n(\gamma(\theta))}.\end{split}$

Moreover, there hold

$\displaystyle\begin{aligned}
q_{1}^{\prime}(\theta)\,=\,&\Big{[}\gamma^{\prime\prime}(\Theta)\cdot\Theta_{\theta}\cdot\Theta_{tt}\,+\,\gamma^{\prime}(\Theta)\cdot\Theta_{tt\theta}|_{(0,\theta)}\Big{]}|_{(0,\theta)}\\\
\,=\,&\gamma^{\prime\prime}(\Theta)\cdot\Theta_{tt}|_{(0,\theta)}\,+\,\gamma^{\prime}(\Theta)\Theta_{tt\theta}|_{(0,\theta)},\end{aligned}$

and specially,

$\displaystyle
q_{1}^{\prime}(0)\,=\,\gamma{{}^{\prime\prime}}(0){k}_{1}\,+\,\gamma{\,{}^{\prime}}(0)(k_{2}-k_{1}),\qquad
q_{1}^{\prime}(1)\,=\,\gamma{{}^{\prime\prime}}(1){k}_{2}\,+\,\gamma{\,{}^{\prime}}(1)(k_{2}-k_{1}).$

The derivative of third order is

$\begin{split}q_{2}(\theta)&\,\equiv\,\frac{{\partial}^{3}F}{{\partial}t^{3}}(0,\theta)\\\
&\,=\,\Big{[}\gamma{{}^{\prime\prime\prime}}(\Theta)\cdot(\Theta_{t})^{3}+3\gamma{{}^{\prime\prime}}(\Theta)\cdot\Theta_{t}\cdot\Theta_{tt}+\gamma{\,{}^{\prime}}(\Theta)\cdot\Theta_{ttt}+3n{{}^{\prime\prime}}(\Theta)\cdot(\Theta_{t})^{2}\\\
&\qquad+3n{{}^{\prime}}(\Theta)\cdot\Theta_{tt}\Big{]}\Big{|}_{(0,\theta)}\\\
&\,=\,\gamma{\,{}^{\prime}}(\theta)\cdot\Theta_{ttt}(0,\theta)+3n{{}^{\prime}}(\theta)\cdot\Theta_{tt}(0,\theta)\qquad\bot
n(\theta).\end{split}$

This finishes the proof of the lemma. ∎

As a conclusion, as $t$ is small enough, there hold the asymptotic behavior

$\displaystyle
F(t,\theta)\,=\,\gamma(\theta)+tn(\theta)+\frac{t^{2}}{2}q_{1}(\theta)+\frac{t^{3}}{6}q_{2}(\theta)+O(t^{4}),~{}~{}\forall\,\theta\in[0,1],~{}t\in(-\delta_{0},\delta_{0}),$
(A.4)

where $\delta_{0}>0$ is a small constant. This gives us that

$\frac{{\partial}F}{{\partial}t}(t,\theta)\,=\,n(\theta)+tq_{1}(\theta)+\frac{t^{2}}{2}q_{2}(\theta)+O(t^{3}),$
$\frac{{\partial}F}{{\partial}\theta}(t,\theta)\,=\,\gamma^{\prime}(\theta)-k(\theta)t\gamma{\,{}^{\prime}}(\theta)+\frac{t^{2}}{2}q_{1}^{\prime}(\theta)+\frac{t^{3}}{6}q_{2}^{\prime}(\theta)+O(t^{4}).$

Specially, there hold

$\frac{{\partial}F}{{\partial}t}(t,0)\,=\,n(0)+tq_{1}(0)+\frac{t^{2}}{2}q_{2}(0)+O(t^{3}),$
$\frac{{\partial}F}{{\partial}t}(t,1)\,=\,n(1)+tq_{1}(1)+\frac{t^{2}}{2}q_{2}(1)+O(t^{3}),$
$\frac{{\partial}F}{{\partial}\theta}(t,0)\,=\,\gamma^{\prime}(0)-k(0)t\gamma{\,{}^{\prime}}(0)+\frac{t^{2}}{2}q_{1}^{\prime}(0)+\frac{t^{3}}{6}q_{2}^{\prime}(0)+O(t^{4}),$
$\frac{{\partial}F}{{\partial}\theta}(t,1)\,=\,\gamma^{\prime}(1)-k(1)t\gamma{\,{}^{\prime}}(1)+\frac{t^{2}}{2}q_{1}^{\prime}(1)+\frac{t^{3}}{6}q_{2}^{\prime}(1)+O(t^{4}).$

These formulas will play an important role in the derivation of the local form
of (1.1), which will be given in Appendix B.

##  Appendix B Local forms of the differential operators in (1.1)

In this section, we are devoted to presenting the expressions of the
differential operators $\Delta$ and $\partial/\partial\nu$ in (1.1). The
formulas will be provided in the local forms in the modified Fermi coordinates
given in Section 2.

We first derive the metric matrix. Note that

$\gamma^{\prime}(\theta)\perp n(\theta),\quad n^{\prime}(\theta)\bot
n(\theta),\quad q_{1}(\theta)\bot n(\theta),\quad q_{2}(\theta)\bot
n(\theta).$

As the first step, here are the computation of the metric matrix:

$\begin{split}g_{11}&\,=\,\left<\frac{{\partial}F}{{\partial}t},\,\frac{{\partial}F}{{\partial}t}\right>\\\
&\,=\,\left<n+tq_{1}+\frac{t^{2}}{2}q_{2}+O(t^{3}),\,\,n+tq_{1}+\frac{t^{2}}{2}q_{2}+O(t^{3})\right>\\\
&\,=\,1+t^{2}|q_{1}|^{2}+O(t^{3}),\end{split}$

and

$\begin{split}g_{12}&\,=\,\left<\frac{{\partial}F}{{\partial}t},\,\,\frac{{\partial}F}{{\partial}\theta}\right>\\\
&\,=\,\left<n+tq_{1}+\frac{t^{2}}{2}q_{2}+O(t^{3}),\,\,\gamma^{\prime}-kt\gamma{{}^{\prime}}+\frac{t^{2}}{2}q_{1}^{\prime}+O(t^{3})\right>\\\
&\,=\,\frac{t^{2}}{2}<n,q_{1}^{\prime}>+t<q_{1},\gamma^{\prime}>-t^{2}k<q_{1},\gamma^{\prime}>+\frac{t^{2}}{2}<q_{2},\gamma^{\prime}>+O(t^{3})\\\
&\,=\,t<q_{1},\gamma^{\prime}>+\frac{t^{2}}{2}\big{(}<q_{2},\gamma^{\prime}>-k<q_{1},\gamma^{\prime}>\big{)}+O(t^{3}),\end{split}$

where we have used the fact

$\displaystyle<q_{1}^{\prime},n>-k<q_{1},\gamma^{\prime}>\,=\,<q_{1}^{\prime},n>+<q_{1},n^{\prime}>\,=\,\frac{\partial}{\partial\theta}<q_{1},n>\,=\,0.$

The last element is

$\begin{split}g_{22}&\,=\,\left<\frac{{\partial}F}{{\partial}\theta},\,\,\frac{{\partial}F}{{\partial}\theta}\right>\\\
&\,=\,\left<\gamma^{\prime}-kt\gamma{{}^{\prime}}+\frac{t^{2}}{2}q_{1}^{\prime}+O(t^{3}),\,\,\gamma^{\prime}-kt\gamma{{}^{\prime}}+\frac{t^{2}}{2}q_{1}^{\prime}+O(t^{3})\right>\\\
&\,=\,1-2kt+t^{2}\left(<q_{1}^{\prime},\gamma^{\prime}>+k^{2}\right)+O(t^{3}).\end{split}$

So the determinant of the metric matrix is

$\begin{split}g&\,=\,\mbox{det}(g_{ij})\\\
&\,=\,1-2kt+t^{2}\left(<q_{1}^{\prime},\gamma^{\prime}>+k^{2}\right)+t^{2}|q_{1}|^{2}-t^{2}<q_{1},\gamma^{\prime}>^{2}+O(t^{3})\\\
&\,=\,1-2kt+t^{2}\left(<q_{1}^{\prime},\gamma^{\prime}>+k^{2}\right)+O(t^{3}),\end{split}$

where we have used $|q_{1}|^{2}-<q_{1},\gamma^{\prime}>^{2}=0$ due to the
expression of $q_{1}$ in Lemma 2.2.

We turn to the computation of the inverse of the metric matrix. By

$\big{[}1+at+bt^{2}+O(t^{3})\big{]}^{-1}\,=\,1-at+(a^{2}-b)t^{2}+O(t^{3}),$

we have

$\displaystyle g^{-1}\,\,=\,$
$\displaystyle\,1+2kt+t^{2}\left(3k^{2}-<q_{1}^{\prime},\gamma^{\prime}>\right)+O(t^{3}).$

Whence

$\begin{split}g^{11}&\,=\,g_{22}\cdot g^{-1}\\\
&\,=\,1-2kt+t^{2}\left(<q_{1}^{\prime},\gamma^{\prime}>+k^{2}\right)+2kt-4k^{2}t^{2}+t^{2}\left(3k^{2}-<q_{1}^{\prime},\gamma^{\prime}>\right)+O(t^{3})\\\
&\,=\,1+O(t^{3}),\\\ \\\ \end{split}$ $\begin{split}-g^{12}&\,=\,g_{12}\cdot
g^{-1}\\\
&\,=\,t<q_{1},\gamma^{\prime}>+2kt^{2}<q_{1},\gamma^{\prime}>+\frac{t^{2}}{2}\left(<q_{2}^{\prime},\gamma{{}^{\prime}}>-k<q_{1},\gamma^{\prime}>\right)+O(t^{3})\qquad\qquad\qquad\\\
&\,=\,t<q_{1},\gamma^{\prime}>+\frac{t^{2}}{2}\big{(}3k<q_{1},\gamma^{\prime}>+<q_{2},\gamma^{\prime}>\big{)}+O(t^{3}),\\\
\\\ \end{split}$

$\begin{split}g^{22}&\,=\,g_{11}\cdot
g^{-1}\,=\,1+2kt+t^{2}\left(3k^{2}-<q_{1}^{\prime},\gamma^{\prime}>+|q_{1}|^{2}\right)+O(t^{3}).\qquad\qquad\qquad\qquad\end{split}$

We can get the following formulas

$\begin{split}\sqrt{g}&\,=\,1-kt+\frac{1}{2}t^{2}<q_{1}^{\prime},\gamma^{\prime}>+O(t^{3}),\qquad\qquad\qquad\qquad\qquad\end{split}$

$\begin{split}(\sqrt{g})^{-1}&\,=\,1+kt+\frac{1}{2}t^{2}\left(2k^{2}-<q_{1}^{\prime},\gamma^{\prime}>\right)+O(t^{3}),\qquad\qquad\qquad\qquad\end{split}$

$\begin{split}\sqrt{g^{22}}&\,=\,1+kt+\frac{1}{2}t^{2}\Big{(}3k^{2}-<q_{1}^{\prime},\gamma^{\prime}>+|q_{1}|^{2}-\frac{1}{4}\cdot
4k^{2}\Big{)}+O(t^{3})\\\
&\,=\,1+kt+\frac{1}{2}\big{(}2k^{2}-<q_{1}^{\prime},\gamma^{\prime}>+|q_{1}|^{2}\big{)}t^{2}+O(t^{3}),\end{split}$

$\begin{split}\frac{1}{\sqrt{g^{22}}}&\,=\,1-kt+\Big{(}k^{2}-k^{2}+\frac{1}{2}<q_{1}^{\prime},\gamma^{\prime}>-\frac{1}{2}|q_{1}|^{2}\Big{)}t^{2}+O(t^{3})\qquad\\\
&\,=\,1-kt+\frac{1}{2}\big{(}<q_{1}^{\prime},\gamma^{\prime}>-|q_{1}|^{2}\big{)}t^{2}+O(t^{3}).\end{split}$

Now, recall the definition of Laplace-Beltrami operator in the form

$\begin{split}\triangle_{t,\theta}u\,=\,&\,g^{11}\frac{{\partial}^{2}u}{{\partial}t^{2}}+2g^{12}\frac{{\partial}^{2}u}{{\partial}t{\partial}\theta}+g^{22}\frac{{\partial}^{2}u}{{\partial}\theta^{2}}+(\sqrt{g})^{-1}\left[\frac{{\partial}}{{\partial}t}(\sqrt{g}g^{11})+\frac{{\partial}}{{\partial}\theta}(\sqrt{g}g^{21})\right]\frac{{\partial}u}{{\partial}t}\\\
&+(\sqrt{g})^{-1}\left[\frac{{\partial}}{{\partial}t}(\sqrt{g}g^{12})+\frac{{\partial}}{{\partial}\theta}(\sqrt{g}g^{22})\right]\frac{{\partial}u}{{\partial}\theta}.\end{split}$
(B.1)

The last two coefficients in (B.1) can be computed as follows. Since

$\begin{split}\sqrt{g}\cdot
g^{11}&\,=\,1-kt+\frac{t^{2}}{2}<q_{1}^{\prime},\gamma^{\prime}>+O(t^{3}),\end{split}$

we can get

$\begin{split}\frac{{\partial}}{{\partial}t}(\sqrt{g}g^{11})&\,=\,-k+t<q_{1}^{\prime},\gamma^{\prime}>+O(t^{2}).\end{split}$

Since

$\begin{split}\sqrt{g}\cdot
g^{12}&\,=\,-t<q_{1},\gamma^{\prime}>-\frac{t^{2}}{2}(3k<q_{1},\gamma^{\prime}>+<\gamma^{\prime},q_{2}>)+kt^{2}<q_{1},\gamma^{\prime}>+O(t^{3})\\\
&\,=\,-t<q_{1},\gamma^{\prime}>-\frac{t^{2}}{2}\left(k<q_{1},\gamma^{\prime}>+<\gamma^{\prime},q_{2}>\right)+O(t^{3}),\end{split}$

we can get

$\begin{split}\frac{{\partial}}{{\partial}\theta}(\sqrt{g}\cdot
g^{12})&\,=\,-t<q_{1},\gamma^{\prime}>^{\prime}-\frac{t^{2}}{2}\left(k<q_{1},\gamma^{\prime}>+<\gamma^{\prime},q_{2}>\right)^{\prime}+O(t^{3}),\\\
\frac{{\partial}}{{\partial}t}(\sqrt{g}\cdot
g^{12})&\,=\,-<q_{1},\gamma^{\prime}>-t\left(k<q_{1},\gamma^{\prime}>+<\gamma^{\prime},q_{2}>\right)+O(t^{2}).\end{split}$

So

$\begin{split}\frac{{\partial}}{{\partial}t}(\sqrt{g}\cdot
g^{11})+\frac{{\partial}}{{\partial}\theta}(\sqrt{g}\cdot
g^{12})&\,=\,-k+t\left(<q_{1}^{\prime},\gamma^{\prime}>-<q_{1},\gamma^{\prime}>^{\prime}\right)+O(t^{2})\\\
&\,=\,-k+O(t^{2}),\end{split}$

where we have used the relations

$<q_{1}^{\prime},\gamma^{\prime}>-<q_{1},\gamma^{\prime}>^{\prime}\,=\,-<q_{1},\gamma^{\prime\prime}>\,=\,0.$

We obtain the coefficient of the first bracket in (B.1) is

$\begin{split}&(\sqrt{g})^{-1}\left(\frac{{\partial}}{{\partial}t}(\sqrt{g}\cdot
g^{11})+\frac{{\partial}}{{\partial}\theta}(\sqrt{g}\cdot
g^{12})\right)\,=\,-k-k^{2}t+O(t^{2}).\end{split}$

On the other hand, by

$\begin{split}\sqrt{g}\cdot
g^{22}&\,=\,1-kt+\frac{t^{2}}{2}<q_{1},\gamma^{\prime}>+2kt-2k^{2}t^{2}+t^{2}\big{(}3k^{2}-<q_{1},\gamma^{\prime}>+|q_{1}|^{2}\big{)}+O(t^{3})\\\
&\,=\,1+kt+\frac{t^{2}}{2}\big{(}2k^{2}-<q_{1},\gamma^{\prime}>+2|q_{1}|^{2}\big{)}+O(t^{3}),\end{split}$

we can get

$\begin{split}\frac{{\partial}}{{\partial}\theta}(\sqrt{g}\cdot
g^{22})&\,=\,k^{\prime}t+\frac{t^{2}}{2}\big{(}2k^{2}-<q_{1},\gamma^{\prime}>+2|q_{1}|^{2}\big{)}^{\prime}+O(t^{3}).\end{split}$

This implies that

$\begin{split}&\frac{{\partial}}{{\partial}t}(\sqrt{g}\cdot
g^{12})+\frac{{\partial}}{{\partial}\theta}(\sqrt{g}\cdot
g^{22})\,=\,-<q_{1},\gamma^{\prime}>+t(k^{\prime}-k<q_{1},\gamma^{\prime}>-<q_{2},\gamma{\,{}^{\prime}}>)+O(t^{2}),\end{split}$

which gives the coefficient of the second bracket in (B.1) is

$\begin{split}&(\sqrt{g})^{-1}\left(\frac{{\partial}}{{\partial}t}(\sqrt{g}\cdot
g^{12})+\frac{{\partial}}{{\partial}\theta}(\sqrt{g}\cdot g^{22})\right)\\\
&\,=\,-<q_{1},\gamma^{\prime}>+t\big{(}k^{\prime}-2k<q_{1},\gamma^{\prime}>-<q_{2},\gamma^{\prime}>\big{)}+O(t^{2}).\end{split}$

Denote

$\displaystyle\varpi(\theta)\,\equiv\,<q_{1}(\theta),\gamma^{\prime}(\theta)>\,=\,\Theta_{tt}(0,\theta),$
(B.2)

where we have used the expression of $q_{1}$ in Lemma 2.2. Hence, the term
$\triangle_{y}u$ in (1.1) has the following form in the modified Fermi
coordinate system

$\begin{split}\triangle_{y}u&\,=\,\left[1+O(t^{3})\right]u_{tt}-2\left[\,\varpi\,t+\frac{t^{2}}{2}(3k\varpi+<q_{2},\gamma^{\prime}>)+O(t^{3})\right]u_{t\theta}\\\
&\quad+\left[1+2kt+t^{2}\big{(}3k^{2}-<q_{1}^{\prime},\gamma^{\prime}>+|q_{1}|^{2}\big{)}+O(t^{3})\right]u_{\theta\theta}\\\
&\quad+\left[-k-k^{2}t+O(t^{2})\right]u_{t}+\left[-\varpi+tk^{\prime}-t\big{(}2ka_{0}+<q_{2},\gamma^{\prime}>\big{)}+O(t^{2})\right]u_{\theta}\\\
&\,\equiv\,\left(1+a_{2}t^{3}\right)u_{tt}\,+\,\big{(}-2\,\varpi\,t+a_{3}t^{2}\big{)}u_{t\theta}\,+\,\big{(}1+2kt+a_{1}t^{2}\big{)}u_{\theta\theta}\\\
&\quad\,+\,\big{(}-k-k^{2}t+a_{4}t^{2}\big{)}u_{t}\,+\,\big{(}-\varpi+a_{5}t\big{)}u_{\theta}.\end{split}$
(B.3)

The terms in $\triangle_{y}u$ will be rearranged in the form

$\displaystyle\triangle_{y}u\,\,=\,\,u_{tt}+u_{\theta\theta}+\bar{B}_{1}(u)+\bar{B}_{0}(u),$
(B.4)

where

$\begin{split}\bar{B}_{1}(u)&\,=\,-(k+k^{2}t)u_{t}-2\,\varpi\,tu_{t\theta}-\varpi
u_{\theta},\end{split}$ (B.5)

and

$\begin{split}\bar{B}_{0}(u)&\,=\,\,2ktu_{\theta\theta}+a_{1}t^{2}u_{\theta\theta}+a_{2}t^{3}u_{tt}+a_{3}t^{2}u_{t\theta}+a_{4}t^{2}u_{t}+a_{5}tu_{\theta}.\end{split}$
(B.6)

We finally show the expression of $\nu$, the outward unit normal of
$\partial\Omega$ near $P_{1},P_{2}$, i.e., when $\theta=0,1$ in the modified
Fermi coordinates. This will provide the local expression of $\partial
u/\partial\nu$ in (1.1). Suppose

$\nu\,=\,\sigma_{1}\frac{{\partial}F}{{\partial}t}+\sigma_{2}\frac{{\partial}F}{{\partial}\theta}.$

Since ${{\partial}F}/{{\partial}t}\in T({\partial}\Omega)$, we have
$<{{\partial}F}/{{\partial}t},\nu>=0$. Hence

$\sigma_{2}\neq 0,\quad\frac{{\partial}F}{{\partial}\theta}\neq 0,$

and

$\sigma_{1}g_{11}+\sigma_{2}g_{12}\,=\,0.$

On the other hand, $<\nu,\nu>=1$, that is

$\left<\sigma_{1}\frac{{\partial}F}{{\partial}t}+\sigma_{2}\frac{{\partial}F}{{\partial}\theta},\,\sigma_{1}\frac{{\partial}F}{{\partial}t}+\sigma_{2}\frac{{\partial}F}{{\partial}\theta}\right>\,=\,1,$

which implies that

$\sigma_{1}^{2}g_{11}+\sigma_{2}^{2}g_{22}+2\sigma_{1}\sigma_{2}g_{12}\,=\,1.$

Combining above two equations, one can get

$\sigma_{1}\,=\,\pm\frac{g^{12}}{\sqrt{g^{22}}},\qquad\sigma_{2}\,=\,\pm\sqrt{g^{22}}.$

Recall that

$q_{1}(\theta)\,=\,\gamma{\,{}^{\prime}}(\theta)\big{[}(k_{2}-k_{1})\theta+k_{1}\big{]};$
$q_{1}^{\prime}(\theta)\,=\,\gamma{{}^{\prime\prime}}(\theta)\big{[}(k_{2}-k_{1})\theta+k_{1}\big{]}\,+\,\gamma{\,{}^{\prime}}(\theta)(k_{2}-k_{1});$
$q_{2}(\theta)\,=\,\gamma{\,{}^{\prime}}(\theta)\Big{[}\big{(}\tilde{\varphi}_{2}^{\prime\prime\prime}(0)-\tilde{\varphi}_{1}^{\prime\prime\prime}(0)\big{)}\theta+\tilde{\varphi}_{1}^{\prime\prime\prime}(0)\Big{]}\,-\,3k\gamma{\,{}^{\prime}}(\theta)\big{[}(k_{2}-k_{1})\theta+k_{1}\big{]}.$

By choosing the $+$, it is easy to check that

$\begin{split}\sigma_{2}\,=\,&1+kt+\frac{1}{2}\Big{(}2k^{2}-<q_{1}^{\prime},\gamma{\,{}^{\prime}}>+|q_{1}|^{2}\Big{)}t^{2}+O(t^{3}),\\\
\,=\,&\,1+kt+k^{2}t^{2}-\frac{1}{2}(k_{2}-k_{1})t^{2}+\frac{1}{2}\big{[}(k_{2}-k_{1})\theta+k_{1}\big{]}^{2}t^{2}+O(t^{3}).\end{split}$

Similarly, we get

$\begin{split}\sigma_{1}\,=\,&\,-\sigma_{2}\cdot g_{12}\cdot(g_{11})^{-1}\\\
\,=\,&-\Big{\\{}1+kt+k^{2}t^{2}-\frac{1}{2}(k_{2}-k_{1})t^{2}+\frac{1}{2}\big{[}(k_{2}-k_{1})\theta+k_{1}\big{]}^{2}t^{2}\Big{\\}}\\\
&\quad\times\Big{[}\,\varpi\,t+\frac{1}{2}(<q_{2},\gamma^{\prime}>-ka_{0})\Big{]}+O(t^{3})\\\
\,=\,&\,-\,\big{[}(k_{2}-k_{1})\theta+k_{1}\big{]}t\,+\,k\big{[}(k_{2}-k_{1})\theta+k_{1}\big{]}t^{2}-\frac{1}{2}\left[(\tilde{\varphi}_{2}^{\prime\prime\prime}(0)-\tilde{\varphi}_{1}^{\prime\prime\prime}(0))\theta+\tilde{\varphi}_{1}^{\prime\prime\prime}(0)\right]t^{2}\,+\,O(t^{3}).\end{split}$

In the modified Fermi coordinates $(t,\theta)$ in (2.4), the normal derivative
$\partial u/\partial\nu$ has a local form as follows

$\sigma_{1}\frac{\partial u}{\partial t}+\sigma_{2}\frac{\partial
u}{\partial\theta}.$

More precisely, for $\theta\,=\,0$, it is

$\displaystyle\begin{aligned}
&k_{1}tu_{t}\,+\,b_{1}t^{2}u_{t}\,-\,u_{\theta}\,-\,k(0)tu_{\theta}\,+\,b_{2}t^{2}u_{\theta}\,+\,{\bar{D}}_{0}^{0}(u),\end{aligned}$
(B.7)

where

$\displaystyle{\bar{D}}_{0}^{0}(u)\,=\,\sigma_{3}(t)\,u_{t}\,+\,\sigma_{4}(t)u_{\theta},$
(B.8)

and the constants $b_{1}$ and $b_{2}$ are given by

$\displaystyle
b_{1}\,=\,\frac{1}{2}\tilde{\varphi}_{1}^{\prime\prime\prime}(0)-k(0)k_{1},\qquad
b_{2}\,=\,\frac{1}{2}(k_{2}-k_{1})-k^{2}(0)-\frac{1}{2}k_{1}^{2}.$ (B.9)

On the other hand, for $\theta=1$, it has the form

$\displaystyle\begin{aligned}
&k_{2}tu_{t}\,+\,b_{3}t^{2}u_{t}\,-\,u_{\theta}\,-\,k(1)tu_{\theta}\,+\,b_{4}t^{2}u_{\theta}\,+\,{\bar{D}}_{0}^{1}(u),\end{aligned}$
(B.10)

with the notation

$\displaystyle{\bar{D}}_{0}^{1}(u)\,=\,\sigma_{5}(t)\,u_{t}\,+\,\sigma_{6}(t)u_{\theta},$
(B.11) $\displaystyle
b_{3}\,=\,\frac{1}{2}\tilde{\varphi}_{2}^{\prime\prime\prime}(0)-k(1)k_{2},\qquad
b_{4}\,=\,\frac{1}{2}(k_{2}-k_{1})-k^{2}(1)-\frac{1}{2}k_{2}^{2}.$ (B.12)

In the above, the functions $\sigma_{3},\cdots,\sigma_{6}$ are smooth
functions of $t$ with the properties

$|\sigma_{i}(t)|\leq C|t|^{3},\quad i=3,4,5,6.$

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# Muon $g-2$ anomaly and non-locality

A. Capolupo<EMAIL_ADDRESS>Dipartimento di Fisica “E.R. Caianiello”
Università di Salerno, and INFN — Gruppo Collegato di Salerno, Via Giovanni
Paolo II, 132, 84084 Fisciano (SA), Italy  G. Lambiase<EMAIL_ADDRESS>Dipartimento di Fisica “E.R. Caianiello” Università di Salerno, and INFN —
Gruppo Collegato di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA),
Italy  A. Quaranta<EMAIL_ADDRESS>Dipartimento di Fisica “E.R.
Caianiello” Università di Salerno, and INFN — Gruppo Collegato di Salerno, Via
Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy

###### Abstract

We show that the discrepancy between the observed value of the muon anomalous
moment and the standard model prediction can be explained in the framework of
nonlocal theories. We compute the leading order and next to leading order
nonlocal correction to the anomalous magnetic moment $\alpha_{NL}$ and we find
that it depends on the nonlocality scale $M_{f}$ and the fermion mass $m_{f}$
as $\alpha_{NL}\propto\frac{m_{f}^{2}}{M_{f}^{2}}$. Such a dependence of the
anomalous magnetic momentum allows to explain, in a flavor-blind nonlocality
scale, why the observed anomalous magnetic moment of the electron is much
closer to the standard model prediction, and permits to predict a large
anomaly that should exist for the $\tau$ particle. We also determine the lower
bounds on the nonlocality scale, for both flavor-blind and flavor-dependent
scenarios.

##  I Introduction

In recent years many phenomena, ranging from the strong CP problem, particle
mixing and oscillations, matter-antimatter asymmetry, to the nature of dark
matter and dark energy, as well as the lack of a quantum theory of gravity,
have lead to the development of theories beyond the Standard Model (SM) of
particles [1, 2, 3, 4, 5, 6]. An example of a phenomenon which cannot be
described within the SM may be represented by the muon magnetic moment
anomaly.

The successful prediction of the magnetic moment of the charged leptons is one
of the most celebrated accomplishments in Quantum Field Theory. The Dirac
theory predicts a $g$-factor $g_{Dirac}=2$, whereas the radiative corrections
shift the actual value slightly above $2$. This allows one to define the
anomalous magnetic moment $\alpha$ as the difference between the quantum field
theoretic prediction and the Dirac value $\alpha=\frac{g-2}{2}$. For the
electron, the anomalous moment $\alpha_{e}=\frac{(g-2)_{e}}{2}$ computed from
Quantum Electrodynamics matches with the experimental value up to a striking
precision of $10^{-12}$.

However recently, new experimental results from the Muon g-2 collaboration [7]
have confirmed a discrepancy between the observed value of the muon anomalous
moment $\alpha_{\mu}$ and the standard model prediction [8, 9]. The combined
data from the Brookhaven [10] and the Fermilab Muon g-2 [7] experiments lead
to a $4.2\sigma$ discrepancy

$\Delta\alpha_{\mu}=\alpha_{\mu,EXP}-\alpha_{\mu,SM}=(251\pm 59)\times
10^{-11}.$ (1)

The discrepancy has fueled the discussion upon the need for new physics,
since, if confirmed, it would be a clear indication of physics beyond the SM.
Many speculations about the origin of this discrepancy have been made in the
last years, including the contributions from axion-like particles [11] and
dark photons ([12] and refences therein), which, up to now, have not been
experimentally detected.

On the other hand, nonlocal theories in which the field quantities are not
restricted to be always point functions in the ordinary space, have
proliferated since the seminal work of Yukawa [13]. Nonlocal theories of
gravity have been developed as a extension of general relativity. Here
nonlocal (i.e. non-polynomial) form factors in the gravitational action can
help to solve the problem of ghosts as well as to improve the ultraviolet
behavior of the quantized theory [14, 15, 16, 17, 18, 19, 20, 21, 22]. In the
particle physics context, nonlocal theories are closely tied to string
theories [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], as well as
noncommutative field theories [37, 38]. Recently, a nonlocal extension of the
standard model has been proposed [39].

In this paper we speculate about the possibility that the $g-2$ discrepancy
for the muon be induced by a nonlocal theory of the kind discussed in [39]. We
compute the leading order and next to leading order nonlocal correction to the
anomalous magnetic moment $\alpha_{NL}$, and we find that it depends in a
simple fashion on the nonlocality scale $M_{f}$ and the fermion mass $m_{f}$
as $\alpha_{NL}\propto\frac{m_{f}^{2}}{M_{f}^{2}}$. We then compare the known
experimental anomaly $\Delta\alpha_{\mu}$ with the nonlocal prediction
$\alpha_{\mu,NL}$ in order to find a lower bound on the nonlocality scale. Our
results indicate that the nonlocal theory might explain the discrepancy
observed in the muon moment, and then $\alpha_{\mu,NL}$ might represent an
important contribution to $\Delta\alpha_{\mu}$. Moreover the nonlocal
contribution provides a very high discrepancy in the tau moment.

The paper is structured as follows. In sec.II, we analyze the muon magnetic
moment in a non-local theory, by computing the tree level correction
(subsection A) and the next to leading order correction (sebsection B) in non-
local electrodynamics, and then we provide lower bounds on the non-locality
scale. In sec. III, we report conclusions and discussions.

##  II Muon magnetic moment in a non-local theory

Figure 1: Reference Feynman diagram for the calculation of the magnetic dipole
moment. The vertex function $\Gamma^{\mu}(p^{\prime},p)$ has to include all
the relevant higher order diagrams (symbolized by the dots).

In deriving the magnetic moment of the charged leptons we follow the procedure
and notation used in [40], appropriately modified for the non-local theory
described in [39]. We briefly outline how the magnetic moment is computed
starting from Quantum Field Theory. The charged lepton field $\psi(x)$ is
assumed to be coupled to a classical background electromagnetic field
$A^{\mu}_{CL}(x)$ via the interaction Hamiltonian $H_{I}=e\int
d^{3}xA^{\mu}_{CL}(x)j_{\mu}(x)$, where $e$ is the electron charge and
$j_{\mu}=\bar{\psi}(x)\gamma_{\mu}\psi(x)$ is the charged lepton current. The
background field used to compute the magnetic moment is a static 3-vector
field $A^{\mu}_{CL}(x)\equiv(0,\pmb{A}_{CL}(\pmb{x}))$ producing the magnetic
field $\pmb{B}=\pmb{\nabla}\times\pmb{A}_{CL}$, with boldface letters denoting
$3d$ spatial vectors. The amplitude for the scattering off the classical field
$\pmb{A}_{CL}(\pmb{x})$ can be neatly expressed as

$i\mathcal{M}=ie\left[\bar{u}(p^{\prime})\pmb{\Gamma}(p^{\prime},p)u(p)\right]\pmb{\cdot}\pmb{A}_{CL}(\pmb{q})\
.$ (2)

In the above equation $u(p)$ is the Dirac spinor corresponding to momentum
$p$, $\pmb{q}=\pmb{p^{\prime}}-\pmb{p}$ denotes the 3-momentum exchange,
$\pmb{A}_{CL}(\pmb{q})$ is the Fourier transformed vector potential and
$\pmb{\Gamma}(p^{\prime},p)$ denotes the spatial part of the charged lepton
vertex function. The latter is computed perturbatively to any order in the
fine structure constant $\alpha$ by including all the relevant diagrams (the
reference diagram is shown in figure (1)) and can always be decomposed as

$\Gamma^{\mu}(p^{\prime},p)=F_{1}(q^{2})\gamma^{\mu}+F_{2}(q^{2})\frac{i\sigma^{\mu\nu}q_{\nu}}{2m}\
,$ (3)

where $q=p^{\prime}-p$ is the $4$-momentum exchange, $m$ is the charged lepton
mass and $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$ denotes the
commutator of Dirac gamma matrices. It is really the form factors
($F_{1},F_{2}$) that one wishes to compute from the underlying theory, and it
can be shown, by a straightforward argument, that they determine the
$g$-factor ($\pmb{\mu}=g\left(\frac{e}{2m}\right)\pmb{S}$) via the simple
relation

$g=2\left(F_{1}(q^{2}\rightarrow 0)+F_{2}(q^{2}\rightarrow 0)\right)\ .$ (4)

The zeroth order g-factor in $\alpha$ coincides with the Dirac result
$g_{0}=2$. Higher orders in perturbation theory, as well as the contributions
due to electroweak and strong interactions bring the actual value of $g$
slightly above $2$ yielding the complete standard model prediction $g_{SM}$.

###  II.1 Tree level correction in non-local electrodynamics

Figure 2: Lowest order non-local Feynman diagram for the computation of the
magnetic moment.

If standard electrodynamics is to be replaced with a non-local theory, the
corrections to the magnetic moment are already present at tree level. To be
precise, here we adopt the non-local theory proposed in [39]. The theory
introduces the new mass scales $M_{f}$, the non-locality scale for the
fermions, which may be either flavor-blind $M_{e}=M_{\mu}=M_{\tau}=M$ or
flavor dependent $M_{e}\neq M_{\mu}\neq M_{\tau}$ , and $M_{g}$, the non-
locality scale for the photons. The largest correction to the diagram of
figure (1) due to non-locality comes from the lowest order ($\alpha^{0}$) non-
local diagram of figure (2). Here one simply replaces the zeroth order local
vertex function $\Gamma^{\mu}=\gamma^{\mu}$ with the zeroth order (in
$\alpha$) non-local vertex function $\Gamma^{\mu}_{NL}$. Notice that at this
order the vertex function is insensible to $M_{g}$, because no internal photon
propagators appear. The non-local vertex function is found (apart from a $-ie$
factor) in [39], and reads

$\Gamma^{\mu}_{NL}(p^{\prime},p)=\frac{1}{2}\left[(p^{\mu}\not{p}^{\prime}+p^{{}^{\prime}\mu}\not{p})\left(\frac{e^{\frac{p^{{}^{\prime}2}}{M_{f}^{2}}}-e^{\frac{p^{2}}{M_{f}^{2}}}}{p^{{}^{\prime}2}-p^{2}}\right)+\left(e^{\frac{p^{{}^{\prime}2}}{M_{f}^{2}}}+e^{\frac{p^{2}}{M_{f}^{2}}}\right)\gamma^{\mu}\right]\
\ .$ (5)

The Feynman slash notation $\not{a}=\gamma^{\mu}a_{\mu}$ is used. In order to
deduce the form factors, we must bring the equation (5) into the form of
equation (3). It clearly suffices to transform only the first term of equation
(5). By sandwiching it between $\bar{u}(p^{\prime})$ and $u(p)$ we find

$\displaystyle\bar{u}(p^{\prime})\left[(p^{\mu}\not{p}^{\prime}+p^{{}^{\prime}\mu}\not{p})\left(\frac{e^{\frac{p^{{}^{\prime}2}}{M_{f}^{2}}}-e^{\frac{p^{2}}{M_{f}^{2}}}}{p^{{}^{\prime}2}-p^{2}}\right)\right]u(p)=m\left(\frac{e^{\frac{p^{{}^{\prime}2}}{M_{f}^{2}}}-e^{\frac{p^{2}}{M_{f}^{2}}}}{p^{{}^{\prime}2}-p^{2}}\right)\bar{u}(p^{\prime})(p^{{}^{\prime}\mu}+p^{\mu})u(p)$
$\displaystyle=m\left(\frac{e^{\frac{p^{{}^{\prime}2}}{M_{f}^{2}}}-e^{\frac{p^{2}}{M_{f}^{2}}}}{p^{{}^{\prime}2}-p^{2}}\right)\bar{u}(p^{\prime})\left[2m\gamma^{\mu}-i\sigma^{\mu\nu}q_{\nu}\right]u(p)\
.$ (6)

Here the first equality comes from the Dirac equation $\not{p}u(p)=mu(p)$ and
the second from the Gordon equality [41]. We remark that the spinors $u(p)$
satisfy the same free Dirac equation as in the local theory. This is because
the non-local action for the free ($A^{\mu}=0$) Dirac field [39] yields the
Dirac equation unmodified. Indeed the fermion action for $A^{\mu}=0$ is [39]

$\displaystyle S_{f}=\int
d^{4}x\bar{\psi}_{i}e^{-\frac{\square}{M_{f}^{2}}}(i\not{\partial}-m_{i})\psi_{i}$
(7)

and it is straightforward to see that upon variation it yields the ordinary
Dirac equation for $\psi_{i}$. Recognizing the first term of equation (5) from
equation (II.1), we immediately deduce the form factors

$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(e^{\frac{p^{{}^{\prime}2}}{M_{f}^{2}}}+e^{\frac{p^{2}}{M_{f}^{2}}}\right)+2m^{2}\left(\frac{e^{\frac{p^{{}^{\prime}2}}{M_{f}^{2}}}-e^{\frac{p^{2}}{M_{f}^{2}}}}{p^{{}^{\prime}2}-p^{2}}\right)\right]$
$\displaystyle F_{2}$ $\displaystyle=$
$\displaystyle-m^{2}\left(\frac{e^{\frac{p^{{}^{\prime}2}}{M_{f}^{2}}}-e^{\frac{p^{2}}{M_{f}^{2}}}}{p^{{}^{\prime}2}-p^{2}}\right)$
(8)

or
$F_{1}+F_{2}=\frac{1}{2}\left(e^{\frac{p^{{}^{\prime}2}}{M_{f}^{2}}}+e^{\frac{p^{2}}{M_{f}^{2}}}\right)$.
It is clear that for external on-shell momenta $p^{2}=m^{2}=p^{{}^{\prime}2}$,
the form factors, and in particular $F_{1}$, depend explicitly on the fermion
mass $m$. The nonlocal correction may be considered as giving rise to an
effective flavor-dependent electric charge (compare eq. (5) with the local
tree level vertex $\Gamma^{\mu}_{L}=e\gamma^{\mu}$). This can be interpreted
either as a violation of the lepton charge universality (implying, for
instance, different muon and electron charges) or a violation of the
electromagnetic gauge invariance and thus a violation of charge conservation.
In the spirit of the nonlocal theory proposed in [39], preserving the
electromagnetic gauge invariance, a violation of the lepton charge
universality is required. A process like the muon decay
$\mu^{-}\longrightarrow e^{-}+\bar{\nu}_{e}+\nu_{\mu}$ is consistent with
electric charge conservation only if neutrinos are allowed an electric charge.
This is possible in view of an eventual neutrino millicharge [42, 43, 44].
Therefore the possibility of a neutrino electric millicharge may justify a
flavor-dependent electric charge.

Since the background field is static, no energy exchange occurs
$q^{0}=0=p^{{}^{\prime}0}-p^{0}$ and the $q^{2}\rightarrow 0$ limit implies
$p^{{}^{\prime}\mu}\rightarrow p^{\mu}$. On the other hand, the external
fermion momenta are necessarily on-shell $p^{2}=p^{\prime 2}=m^{2}$, so that
from equation (4) we find the $g$-factor

$\displaystyle g$ $\displaystyle=$ $\displaystyle
2\left(F_{1}(q^{2}\rightarrow 0)+F_{2}(q^{2}\rightarrow
0)\right)=2e^{\frac{m^{2}}{M_{f}^{2}}}$ (9) $\displaystyle\simeq$
$\displaystyle 2\left(1+\frac{m^{2}}{M_{f}^{2}}\right)$

The last equivalence holds for $m\ll M_{f}$, as it is always the case.

Figure 3: Reference Feynman diagram for the order $\alpha$ contribution to the
vertex function $\Gamma^{\mu}$.

A couple of comments are now in order. First, the tree-level non-local
correction to the $g$-factor
$(g-2)_{NL}=2\left(e^{\frac{m^{2}}{M_{f}^{2}}}-1\right)\simeq\frac{2m^{2}}{M_{f}^{2}}$
represents an additional contribution to the full (local) standard model
prediction $(g-2)_{SM}$. Secondly, the dependence of the $g$-factor on the
fermion mass $\propto m^{2}$ prefigures an interesting quantitative
prediction. If $M_{f}$ is flavor-blind, we can immediately read off Eq. (9)
that the muon correction $(g-2)_{\mu,NL}$ is about $40.000$ times larger than
the correction for the electron, because
$(g-2)_{\mu,NL}\simeq\frac{m_{\mu}^{2}}{m_{e}^{2}}(g-2)_{e,NL}$. For the same
reason, the $\tau$ correction is about $\frac{m_{\tau}^{2}}{m_{\mu}^{2}}\simeq
282$ times larger than the muon correction. Therefore our computations allow
to describe the observed anomalies for electron and muon magnetic momentum and
to predict greater deviations from standard model prediction in the case of
the tau magnetic momentum.

###  II.2 Next to leading order non-local correction

The vertex function $\Gamma^{\mu}_{NL}(p^{\prime},p)$ has, roughly speaking,
two expansion parameters, the non-locality scale $M_{f},M_{g}$ and the fine
structure constant $\alpha$. Denoting by $(m,n)$ the expansion to order $m$ in
$\frac{1}{M_{f,g}^{2}}$ and of order $n$ in $\alpha$, the tree level
correction of the previous subsection corresponds to $(1,0)$. Here we wish to
compute the next order in $\alpha$, namely the expansion to order $(1,1)$. We
can immediately estimate that such a contribution shall be smaller by a factor
$\alpha\simeq 1/137$ with respect to the leading order. In particular we
consider the non-local version of the order $\alpha$ diagram of figure (3).

We closely follow the computation of the local correction shown in [40]. In
the non-local counterpart of diagram (3) each of the inner elements (the three
propagators and the three vertices) are replaced by their non-local form,
namely

$\Pi_{\gamma,NL}^{\mu\nu}(p^{2})=\frac{-i\eta^{\mu\nu}}{p^{2}+i\epsilon}e^{-\frac{p^{2}}{M_{g}^{2}}}\
;\ \ \ \
\Pi_{f,NL}(p^{\mu})=\frac{i(\not{p}+m)}{p^{2}-m^{2}+i\epsilon}e^{-\frac{p^{2}}{M_{f}^{2}}}\
,$ (10)

for the propagators and equation (5) for the vertices. The correction we wish
to compute is then, referring to figure (3)

$\delta\Gamma^{\mu}(p^{\prime},p)=\int\frac{d^{4}k}{(2\pi)^{4}}\Pi^{\gamma,NL}_{\nu\rho}((k-p)^{2})\bar{u}(p^{\prime})(-ie\Gamma^{\nu}_{NL}(p^{\prime},k^{\prime}))\Pi_{f,NL}(k^{\prime})\Gamma^{\mu}_{NL}(k^{\prime},k)\Pi_{f,NL}(k)(-ie\Gamma^{\rho}_{NL}(k,p))u(p)\
.$ (11)

Employing the Feynman parameters $x,y,z$ we can bring the denominator of the
integrand to the form $\frac{2}{D^{3}}$, with $D=l^{2}-\Delta+i\epsilon$,
$\Delta=-xyq^{2}+(1-z)^{2}m^{2}$ and $l=k+yq-zp$. The integration momentum
variable is switched to $l$ and a further integration on the Feynman
parameters $\int_{0}^{1}\ dx\ dy\ dz\ \delta(1-x-y-z)$ is introduced. Each of
the non-local elements brings an exponential factor of the form $e^{\frac{\pm
s^{2}}{M^{2}}}$ into the numerator of equation (11), where $s$ is some linear
combination of the internal momentum variable $l$ and the external momentum
variables $p,p^{\prime},q$ and $M$ stands for either $M_{g}$ or $M_{f}$.
Expanding the exponentials we get

$e^{\pm
s^{2}/M^{2}}=1\pm\frac{l^{2}+l\cdot(...)+(p_{ex})^{2}}{M^{2}}+\frac{1}{2M^{4}}\left(l^{4}+(...)+(p_{ex})^{4}\right)\pm...$
(12)

Where $p_{ex}$ denotes a combination of external momenta. The odd powers of
$l$ do not contribute to the integral, since the denominator is even
$D=D(l^{2})$. The even powers of $l$, on the other hand, all lead to divergent
expressions (logarithm and power law). When all the non-local terms are
included, these divergences go well beyond the local logarithmic divergence
arising from the local diagram [40] and therefore require a more sophisticated
renormalization scheme to be reabsorbed, with respect to local
electrodynamics. The full renormalization of the non-local theory is beyond
the scope of this paper, and is demanded to future studies. Here we focus on
the finite parts, arising from the terms of the form
$\frac{(p_{ex})^{2n}}{M^{2n}}$. Since each of the external momenta is either
on-shell ($p^{2}=p^{{}^{\prime}2}=m^{2}$) or shall be taken to vanish at the
end of the calculation $q^{2}\rightarrow 0$, it is sufficient to consider only
the lowest orders $\frac{p_{ex}^{2}}{M^{2}}$. The next finite terms are indeed
suppressed by higher powers of $m^{2}/M^{2}$ (they are
$O(\frac{m^{4}}{M^{4}})$).

Figure 4: Non-local vertex diagrams.

For a similar reason, to order $O(\frac{m^{2}}{M^{2}})$ it is sufficient to
consider the diagrams in which only one of the inner elements is replaced by
its non-local counterparts, as shown in the figure (4). The order $O(1)$
contribution for the diagrams of figure (4) is of course the same for all, and
corresponds to the local order $\alpha$ correction to the vertex function. For
instance, the diagram $(1)$ of figure (4) corresponds to

$\displaystyle\delta\Gamma_{1}^{\mu}(p^{\prime},p)$ $\displaystyle=$
$\displaystyle 2ie^{2}\int\ dx\ dy\ dz\
\delta(1-x-y-z)\int\frac{d^{4}l}{(2\pi)^{4}}\frac{2}{D^{3}}\left[1-\frac{(k-p)^{2}}{M_{g}^{2}}\right]\bar{u}(p^{\prime})$
$\displaystyle\Bigg{[}\gamma^{\mu}\left(\frac{-l^{2}}{2}+(1-x)(1-y)q^{2}+(1-4z+z^{2})m^{2}\right)+i\frac{\sigma^{\mu\nu}q_{\nu}}{2m}\left(2m^{2}z(1-z)\right)\Bigg{]}u(p)$

where it is understood that $(k-p)$ must be expressed in terms of $l$ and any
term containing $l$ has to be discarded (either for symmetry or because it
must be reabsorbed by the renormalization prescription). We distinguish two
kinds of terms. The “standard” contributions have the general form

$\delta\Gamma^{\mu,S}(p^{\prime},p)=\int d\Omega
f(p_{ex}^{2}))\bar{u}(p^{\prime})\Bigg{[}\gamma^{\mu}\left(\frac{-l^{2}}{2}+(1-x)(1-y)q^{2}+(1-4z+z^{2})m^{2}\right)+i\frac{\sigma^{\mu\nu}q_{\nu}}{2m}\left(2m^{2}z(1-z)\right)\Bigg{]}u(p)$
(13)

where $d\Omega=\frac{d^{4}l}{(2\pi)^{4}D^{3}}\ dx\ dy\ dz\ \delta(1-x-y-z)$.
These terms are essentially the same as the local integral, except for a
function depending only on the external momenta
$f(p_{ex}^{2})\propto\frac{m^{2}}{M^{2}}$. After discarding the higher powers
of $l$ and taking the $q^{2}\rightarrow 0$ limit, the computation of these
integrals is straightforward. All the propagator terms (diagrams $(1,3,5)$ of
figure 4) as well as the bare $\gamma^{\mu}$ part (see equation (5)) of the
vertex diagrams ($(2,4,6)$ of figure 4) yield standard contributions. Let us
work out the integral corresponding to the first diagram of figure 4.
Discarding the local term and taking the $q^{2}\rightarrow 0$ limit, the
integral becomes

$\displaystyle\delta\Gamma_{1}^{\mu}(p^{\prime},p)$ $\displaystyle=$
$\displaystyle-\frac{4ie^{2}}{M_{g}^{2}}\int_{0}^{1}dx\ dy\ dz\
\delta(1-x-y-z)\int\frac{d^{4}l}{(2\pi)^{4}}\frac{1}{(l^{2}-\Delta)^{3}}\left(m^{2}(1-z)^{2}\right)$
(14) $\displaystyle\times$
$\displaystyle\bar{u}(p^{\prime})\left[\gamma^{\mu}m^{2}(1-4z+z^{2})+\frac{i\sigma^{\mu\nu}q_{\nu}}{2m}\left(2m^{2}z(1-z)\right)\right]u(p)\
.$

Where it is understood that $\Delta=\Delta(q^{2}\rightarrow 0)=m^{2}(1-z)^{2}$
and we have omitted the divergent terms, which can be removed by means of an
appropriate regularization. The $l$ integration is most easily carried out by
means of a Wick rotation, yielding

$\displaystyle\delta\Gamma_{1}^{\mu}(p^{\prime},p)$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{\pi}\frac{m^{2}}{M_{g}^{2}}\int_{0}^{1}\ dx\ dy\
dz\
\delta(1-x-y-z)\bar{u}(p^{\prime})\left[\gamma^{\mu}(1-4z+z^{2})+\frac{i\sigma^{\mu\nu}q_{\nu}}{2m}\left(2z(1-z)\right)\right]u(p)\
.$

We can read off equation (II.2) the corrections to the form factors, which
after performing the integration over the Feynman parameters read $\delta
F_{1}=\frac{5\alpha}{3\pi}\frac{m^{2}}{M_{g}^{2}}$ and $\delta F_{2}=-\delta
F_{1}$. According to equation (4), the first diagram gives no net contribution
to the anomalous magnetic moment $\delta g=2(\delta F_{1}+\delta F_{2})=0$.
The other standard contributions are computed similarly and are collected in
the table (I).

The vertex diagrams (namely $2,4,6$ of figure 4) also contain a part
corresponding to the first term of equation (5). These terms cannot be
immediately reduced to the form of equation (13). For instance the “non
standard” part of diagram $2$ is

$\delta\Gamma_{2}^{\mu,NS}=\frac{ie^{2}}{M_{f}^{2}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\eta_{\nu\rho}}{(k-p)^{2}+i\epsilon}\bar{u}(p^{\prime})\left(k^{{}^{\prime}\nu}\not{p}^{\prime}+p^{{}^{\prime}\nu}\not{k}^{\prime}\right)\frac{i\left(\not{k}^{\prime}+m\right)}{k^{{}^{\prime}2}-m^{2}+i\epsilon}\gamma^{\mu}\frac{i\left(\not{k}+m\right)}{k^{2}-m^{2}+i\epsilon}\gamma^{\rho}u(p)\
.$ (15)

Evaluating the correction to the form factors from integrals of the kind (15)
is a lengthy but simple process, and requires a massive use of the properties
of the gamma matrices, along with the introduction of Feynman parameters. The
results are collected in the table (I). By summing all the corrections to the
form factors, we finally find

$\delta g=2\left(\delta F_{1}+\delta
F_{2}\right)=\frac{\alpha}{\pi}\frac{m^{2}}{M_{f}^{2}}\left(\frac{325}{12}-42\ln
2\right)\ .$ (16)

As expected this correction is about a factor $\alpha$ smaller than the tree
level correction.

Diagram | $\delta F_{1}$ | $\delta F_{2}$
---|---|---
1 | $\frac{5\alpha}{3\pi}\frac{m^{2}}{M_{g}^{2}}$ | $-\delta F_{1}$
2 (Standard part) | $\frac{\alpha}{4\pi}\frac{m^{2}}{M_{f}^{2}}\left(\frac{31}{6}-16\ln 2\right)$ | $\frac{\alpha}{2\pi}\frac{m^{2}}{M_{f}^{2}}\left(-\frac{19}{6}+4\ln 2\right)$
3 | $\frac{\alpha}{\pi}\frac{m^{2}}{M_{f}^{2}}\left(\frac{23}{6}-12\ln 2\right)$ | $\frac{\alpha}{\pi}\frac{m^{2}}{M_{f}^{2}}\left(2-4\ln 2\right)$
4 (Standard part) | $-\frac{\alpha}{\pi}\frac{m^{2}}{M_{f}^{2}}$ | $\frac{\alpha}{2\pi}\frac{m^{2}}{M_{f}^{2}}\left(-\frac{7}{6}+2\ln 2\right)$
5 | $\frac{\alpha}{\pi}\frac{m^{2}}{M_{f}^{2}}\left(\frac{23}{6}-12\ln 2\right)$ | $\frac{\alpha}{\pi}\frac{m^{2}}{M_{f}^{2}}\left(2-4\ln 2\right)$
6 (Standard part) | $\frac{\alpha}{4\pi}\frac{m^{2}}{M_{f}^{2}}\left(\frac{31}{6}-16\ln 2\right)$ | $\frac{\alpha}{2\pi}\frac{m^{2}}{M_{f}^{2}}\left(-\frac{19}{6}+4\ln 2\right)$
2 (Non standard) | $\frac{\alpha}{\pi}\frac{m^{2}}{M_{f}^{2}}\left(\frac{47}{12}-10\ln 2\right)$ | $\frac{\alpha}{\pi}\frac{m^{2}}{M_{f}^{2}}\left(-\frac{3}{2}+23\ln 2\right)$
4 (Non standard) | $\delta F_{1}$ | $-\delta F_{1}$
6 (Non standard) | $\frac{\alpha}{4\pi}\frac{m^{2}}{M_{f}^{2}}\left(\frac{13}{2}-12\ln 2\right)$ | 0
Table 1: List of the corrections from the diagrams of figure 4.

###  II.3 Lower bounds on the non-locality scale

The non-local correction might explain, at least partially, the so-called muon
$(g-2)$ anomaly, that is, the difference between the observed
$\alpha_{\mu,EXP}=\frac{(g-2)_{\mu,EXP}}{2}$ and the standard model prediction
$\alpha_{\mu,SM}=\frac{(g-2)_{\mu,SM}}{2}$. The observed anomalous magnetic
moment of the electron $\alpha_{e,EXP}$ is instead much closer to the standard
model prediction $\alpha_{e,SM}$. This fact is in line with the generic
prediction of the non-local theory, which for a flavor-blind scale
$M=M_{e}=M_{\mu}=M_{\tau}$ entails a larger anomaly for the muon (and for the
tau). Attributing the difference in the observed values of $\alpha$ and the
standard model computation to the non-local correction, we now determine the
lower bounds on $M_{f}$, for both flavor-blind and flavor-dependent scenarios.

In the flavor-blind case the relevant inequality is

$\displaystyle\frac{m_{\mu}^{2}}{M^{2}}\leq\alpha_{\mu,EXP}-\alpha_{\mu,SM}=0.00000000257$
(17)

which yields the lower bound $M\geq 4.384\ \mathrm{TeV}$. Here we have used
the latest available average values for the Standard Model prediction [8] and
the experimental result [7] .The corresponding non-local correction to the
electron anomaly is $\alpha_{e,NL}\leq 6\times 10^{-14}$ which is well within
the current experimental uncertainty $\sim 26\times 10^{-14}$ [45]. On the
other hand, the mass dependence implies that for a flavor-blind scale, the
corresponding maximum correction to the $\tau$ anomaly would be quite large
$\alpha_{\tau,NL}\leq 7.267\times 10^{-7}$.

In the flavor dependent case, the estimate for the muon scale is unaltered
$M_{\mu}\geq 4.384\mathrm{TeV}$. The electron scale should in any case be
large enough that the non-local correction be smaller than the current
experimental uncertainty on $\alpha_{e}$. This entails a slightly looser bound
on $M_{e}$, namely $M_{e}\geq 1.004\ \mathrm{TeV}$. In the end , the
experimental uncertainty on the $\tau$ magnetic moment is still too high to
deduce any plausible bound on $M_{\tau}$.

A comment is in order. The non-local scale, whose effect is to smear the
point-like vertices of the Standard Model [46], has been recently studied in
[47]. Here the authors show that a modification induced by non-locality to the
heavy-boson cross section may enhance the production of boosted Higgs boson,
providing a signal of non-local quantum field theory at the LHC. As pointed
out in [47], the data of LHC 2 run will be sensitive to to non-local scale
$\sim\text{few}\mathrm{TeV}$, which is of the same order of magnitude inferred
for muon $g-2$ anomaly, as discussed in the present paper.

##  III Discussion and Conclusions

In the framework of nonlocal theories, we have computed at tree level and at
next to leading order the lepton magnetic moments and shown that the
discrepancy between the observed value of the muon anomalous moment and the
standard model prediction can be completely explained by simply replacing the
zeroth order local vertex function $\gamma^{\mu}$ with the zeroth order (in
$\alpha$) non-local vertex function $\Gamma^{\mu}_{NL}$.

The correction derived for the magnetic momentum depends on the nonlocality
mass scale $M_{f}$ introduced by the theory and on the fermion mass $m_{f}$,
i.e.: $\alpha_{NL}\propto\frac{m_{f}^{2}}{M_{f}^{2}}$. The behavior of such
correction allows in principle to explain the muon magnetic momentum anomaly
if the nonlocality scale, according to equation (17), is about $M\simeq
4.384\mathrm{TeV}$. For larger $M$ the non-local theory is not sufficient to
account for the observed discrepancy, and other mechanisms must come into
play.

In addition the form $\alpha_{NL}\propto\frac{m_{f}^{2}}{M_{f}^{2}}$ is
compatible with the very small deviation of the electron magnetic moment from
the standard model prediction, and allows to predict a larger magnetic
momentum anomaly for the tau. Finally, we have determined the lower bounds on
the nonlocality scale, for both flavor-blind and flavor-dependent scenarios.

Our results are compatible with the ones derived in the local context, by
including the contributions from axion-like particles and dark photons. The
mechanism here proposed could then contribute, together with other mechanisms
beyond the Standard Model, to the explanation of the experimentally revealed
anomalous magnetic moments of the charged leptons.

## Acknowledgements

A.C., G.L. and A.Q. acknowledge partial financial support from MIUR and INFN,
A.C. and G.L. also acknowledge the COST Action CA1511 Cosmology and
Astrophysics Network for Theoretical Advances and Training Actions (CANTATA).

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