bounded and $\cap_{n}\Delta_{n}^{*}\setminus\Delta=\emptyset$ so
$\mu(\Delta_{n}^{*}\setminus\Delta)\to 0$ as $n\to\infty$. The third integral
is always non-negative. Hence, as $\tilde{\kappa}_{n}(x)\leq 0$, assumption
$(A)$ of Theorem 4.1 is clearly fulfilled. To verify assumption $(B)$, observe
that

$\log\tilde{v}_{n}(x)=2n\tilde{\kappa}_{n}(x)+\log
v_{\Delta_{n}^{*}}(l_{n}(x)),$

where $\tilde{v}_{n}$ is the Radon-Nikodym derivative of $\tilde{\mu}_{n}$
with respect to $\omega_{\Delta}$, see (1.7). We readily have that

$|\log v_{\Delta}(x)-\log
v_{\Delta_{n}^{*}}(l_{n}(x))|=|\log\mu^{\prime}(x)-\log\mu^{\prime}(l_{n}(x))-\log
a_{n}|.$

Since $a_{n}\to 1$, it follows from (3.2) that $\|\log v_{\Delta}-\log
v_{\Delta_{n}^{*}}(l_{n})\|_{L^{1}(\omega_{\Delta})}\to 0$ as $n\to\infty$.
Furthermore, we have that

$\int_{\Delta}|\tilde{\kappa}_{n}(x)|d\omega_{\Delta}(x)=\int_{\Delta_{n}^{*}}|\kappa_{n}(x)|d\omega_{\Delta_{n}^{*}}(x)\lesssim\int_{\Delta_{n}^{*}\setminus\Delta_{n}}|w_{\Delta_{n}}(x)|^{\varkappa}d\omega_{\Delta_{n}^{*}}(x),$

where we used assumption (1) of Theorem 4.2. Hence, it readily follows from
the definition of $\Delta_{n}^{*}$ that

(4.9)
$2n\int_{\Delta}|\tilde{\kappa}_{n}(x)|d\omega_{\Delta}(x)\lesssim_{\Delta}n\int_{0}^{\delta_{n}}x^{(\varkappa+1)/2}d\omega_{[0,\delta_{n}]}(x)\lesssim_{\Delta}n\delta_{n}^{(\varkappa+1)/2}=\xi_{n}.$

Since $\lim_{n\to\infty}\xi_{n}=0$ by our assumptions, we get
$\|2n\tilde{\kappa}_{n}\|_{L^{1}(\omega_{\Delta})}\to 0$ as $n\to\infty$,
which shows that condition $(B)$ of Theorem 4.1 is also satisfied. Altogether,
we have that the triples $(\tilde{\mu}_{n},\tilde{h}_{n},\tilde{\omega}_{n})$
satisfy all the conditions of Theorem 4.1 and therefore

$T_{n}\left(e^{2nV^{\tilde{\omega}_{n}}+\tilde{h}_{n}}\tilde{\mu}_{n}\right)(z)=\left(1+o_{\mathcal{K}}(1)\right)\exp\left(n\int\log(z-x)d\tilde{\omega}_{n}(x)\right)\frac{G(e^{\tilde{h}_{n}}\tilde{\mu}_{n},\infty)}{G(e^{\tilde{h}_{n}}\tilde{\mu}_{n},z)}$

holds locally uniformly in $D_{\Delta}$. Computations in Lemma 4.3 now show
that the above formula is equivalent to the statement of the lemma. ∎

The next lemma provides a simple uniform estimate on the boundary behavior for
the sequence of outer functions. Recall the definition of function $G$ given
in (1.8) and (1.4).

######  Lemma 4.7.

There exists a non-decreasing function $\epsilon_{\mu}(t)$ such that
$\lim_{t\downarrow 0}\epsilon_{\mu}(t)=0$ and

$\log
G^{-2}\big{(}e^{h_{n}}\mu_{|\Delta_{n}},\beta_{n}+t\big{)}\leq\frac{\epsilon_{\mu}(t)}{\sqrt{t}}$

for every $t\in[0,\beta(\mu)-\beta_{n}]$ and every $n$. Moreover, an analogous
estimate holds with $\beta_{n}+t$ replaced by $\alpha_{n}-t$ for
$t\in[0,\alpha_{n}-\alpha(\mu)]$.

###### Proof.

By (1.4) and (1.8) that

$\log
G^{-2}\big{(}e^{h_{n}}\mu_{|\Delta_{n}},\beta_{n}+t\big{)}\leq\frac{w_{\Delta_{n}}(\beta_{n}+t)}{2}\int_{\Delta_{n}}\frac{|h_{n}(x)|+|\log
v_{\Delta_{n}}(x)|}{\beta_{n}+t-x}d\omega_{\Delta_{n}}(x).$

It follows from the Cauchy integral formula that

$\frac{1}{w_{\Delta_{n}}(z)}=\frac{1}{\pi\mathrm{i}}\int_{\Delta_{n}}\frac{1}{x-z}\frac{dx}{w_{\Delta_{n}+}(x)}=\int_{\Delta_{n}}\frac{d\omega_{\Delta_{n}}(x)}{z-x},$

where we used the fact that $w_{\Delta_{n}+}(x)=\mathrm{i}|w_{\Delta_{n}}(x)|$
for $x\in\Delta_{n}$. Since $\mathcal{K}$ is a compact, the functions
$|h_{n}(x)|$ are uniformly bounded. Therefore, we get that

$w_{\Delta_{n}}(\beta_{n}+t)\int_{\Delta_{n}}\frac{|h_{n}(x)|}{\beta_{n}+t-x}d\omega_{\Delta_{n}}(x)\lesssim_{\mathcal{K}}1.$

Recall further that $v_{\Delta_{n}}(x)=\pi\mu^{\prime}(x)|w_{\Delta_{n}}(x)|$.
We have that

$\displaystyle
w_{\Delta_{n}}(\beta_{n}+t)\int_{\Delta_{n}}\frac{|\log|w_{\Delta_{n}}(x)||}{\beta_{n}+t-x}d\omega_{\Delta_{n}}(x)$
$\displaystyle\lesssim_{\Delta}\sqrt{t}+\sqrt{t}\int_{\beta_{n}-1}^{\beta_{n}}\frac{-\log(\beta_{n}-x)}{\beta_{n}+t-x}\frac{dx}{\sqrt{\beta_{n}-x}}$
$\displaystyle\lesssim_{\Delta}\sqrt{t}+|\log t|.$

Moreover,

$\displaystyle\int_{\Delta_{n}}\frac{|\log\mu^{\prime}(x)|}{\beta_{n}+t-x}d\omega_{\Delta_{n}}(x)\lesssim_{\Delta}\frac{1}{\sqrt{t}}\int_{\alpha_{n}}^{\beta_{n}-\sqrt{t}}\frac{|\log\mu^{\prime}(x)|}{\sqrt{\beta_{n}-x}}dx+\frac{1}{t}\int_{\beta_{n}-\sqrt{t}}^{\beta_{n}}\frac{|\log\mu^{\prime}(x)|}{\sqrt{\beta_{n}-x}}dx$
$\displaystyle\lesssim\frac{\|\log\mu^{\prime}\|_{L^{1}(\omega_{\Delta})}}{\sqrt{t}}+\frac{1}{t}\int_{\beta_{n}-\sqrt{t}}^{\beta_{n}}\frac{|\log\mu^{\prime}(x)|}{\sqrt{\beta_{n}-x}}dx,$

where we used (3.2) for the last estimate. Set

$\epsilon(t):=\sup_{n>N_{0}}\int_{\beta_{n}-\sqrt{t}}^{\beta_{n}}\frac{|\log\mu^{\prime}(x)|}{\sqrt{\beta_{n}-x}}dx,$

where $N_{0}$ is sufficiently large to make sure that integrals converge.
Clearly, this is a non-decreasing function of $t$. We claim that
$\lim_{t\downarrow 0}\epsilon(t)=0$. Indeed, assume to the contrary that there
exist $\epsilon_{0}>0$, a sequence $\\{t_{m}\\}$ decreasing to $0$, and a set
$\\{m_{n}\\}$ such that

(4.10)
$\int_{\beta_{n_{m}}-\sqrt{t}_{m}}^{\beta_{n_{m}}}\frac{|\log\mu^{\prime}(x)|}{\sqrt{\beta_{n_{m}}-x}}dx\geq\epsilon_{0}.$

If $\\{m_{n}\\}$ is bounded, it contains a constant sequence. However, (4.10)
cannot hold along this sequence as the integrals of a fixed integrable
function over sets of decreasing measure must vanish. On the other hand, if
$\\{n_{m}\\}$ contains a strictly increasing sequence, (4.10) contradicts
Proposition 3.3(iii) since $\mu\in\textrm{USz}(\Delta)$. Hence, altogether,

$w_{\Delta_{n}}(\beta_{n}+t)\int_{\Delta_{n}}\frac{|\log\mu^{\prime}(x)|}{\beta_{n}+t-x}d\omega_{\Delta_{n}}(x)\lesssim_{\mu,\Delta}1+\frac{\epsilon(t)}{\sqrt{t}}.$

Collecting all the previous estimates gives us

$\log
G^{-2}\big{(}e^{h_{n}}\mu_{|\Delta_{n}},\beta_{n}+t\big{)}\lesssim_{\Delta,\mu,\mathcal{K}}\frac{t+\sqrt{t}+\sqrt{t}|\log
t|+\epsilon(t)}{\sqrt{t}}.$

Since the numerator above can be easily estimated by a non-decreasing function
that has zero limit at zero, the claim of the lemma follows. ∎

Before stating the next lemma, we deduce the following estimate. If $f$ is a
function in the Hardy space $H^{2}(\mathbb{D})$, then the Cauchy integral
formula and Cauchy-Schwarz inequality give

$|f(z)|\leq\int_{\mathbb{T}}\frac{|f(\eta)|}{|z-\eta|}\frac{|d\eta|}{2\pi}\leq\frac{\|f\|_{L^{2}(\mathbb{T})}}{\sqrt{1-|z|}}.$

Thus, if $F$ is a function in the Hardy space
$H^{2}(\overline{\mathbb{C}}\setminus\Delta)$, then $F\circ\phi_{\Delta}$ is a
function in $H^{2}(\mathbb{D})$, see (1.3), and therefore

(4.11)
$|F(\beta+\delta)|\leq\frac{\|F\|_{L^{2}(\omega_{\Delta})}}{\sqrt{1-\phi_{\Delta}(\beta+\delta)}}\lesssim_{\Delta}\frac{\|F\|_{L^{2}(\omega_{\Delta})}}{\sqrt[4]{\delta}}.$

######  Lemma 4.8.

The sequence $\\{\xi_{n}\\}$ in the definition of $\delta_{n}$ in (4.7) can be
chosen so that

$\int_{\Delta(\mu)\setminus\Delta_{n}^{*}}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x)\to
0\quad\text{as}\quad n\to\infty.$

###### Proof.

Let $l_{n}(x)=a_{n}x+b_{n}$, $a_{n}>0$, be the linear function that maps of
$\Delta$ onto $\Delta_{n}^{*}$, as in the proof of Lemma 4.6. Due to the
compactness of $\mathcal{K}$, it holds that the sequence $h_{n}-h_{n}\circ
l_{n}$ converges to zero uniformly on $\Delta$. The uniform Szegő condition
implies

(4.12) $\displaystyle G\big{(}e^{h_{n}}\mu_{|\Delta_{n}^{*}},z\big{)}$
$\displaystyle=\big{(}1+o_{\mathcal{K}}(1)\big{)}G\big{(}e^{h_{n}}\mu_{|\Delta},z\big{)},$
$\displaystyle G\big{(}e^{h_{n}}\mu_{|\Delta_{n}},z\big{)}$
$\displaystyle=\big{(}1+o_{\mathcal{K}}(1)\big{)}G\big{(}e^{h_{n}}\mu_{|\Delta},z\big{)},$

locally uniformly in $D_{\Delta}$. Hence, the last asymptotic formula of
Theorem 4.1 pulled back to the intervals $\Delta_{n}^{*}$ as in Lemma 4.6
shows that

(4.13)
$G^{-2}(e^{h_{n}}\mu_{|\Delta_{n}},\infty)\int_{\Delta_{n}^{*}}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x)=2+o_{\mathcal{K}}(1).$

Using the fact that $\kappa_{n}(x)\equiv 0$ on $\Delta_{n}$, we get that

$|F_{n\pm}(x)|^{-2}=G^{-2}(e^{h_{n}}\mu_{|\Delta_{n}},\infty)e^{\theta_{n}(x)}v_{\Delta_{n}}(x)$

almost everywhere on $\Delta_{n}$ by (1.5), (1.7) and (1.8), where

$F_{n}(z)=\exp\left(n\int\log(z-x)d\omega_{n}(x)\right)\frac{G(e^{h_{n}}\mu_{|\Delta_{n}},\infty)}{G(e^{h_{n}}\mu_{|\Delta_{n}},z)}$

(here we again restricted $\mu$ to $\Delta_{n}$ and not $\Delta_{n}^{*}$ as in
Lemma 4.6). Therefore, it holds that

$2+o_{\mathcal{K}}(1)\geq\int_{\Delta_{n}}\big{|}P_{n}^{*}(x)/F_{n\pm}(x)\big{|}^{2}d\omega_{\Delta_{n}}(x),$

where we first reduce the interval of integration from $\Delta_{n}^{*}$ to
$\Delta_{n}$ in (4.13) and then drop the singular part of $\mu$. Since the
potentials $V^{\omega_{n}}$ are continuous in $\mathbb{C}$, each function
$P_{n}^{*}/F_{n}$ is a product of a Szegő function and a bounded analytic
function. As such, it belongs to $H^{2}(D_{\Delta_{n}})$. Hence, it follows
from (4.11) that

$\big{|}(P_{n}^{*}/F_{n})(x)\big{|}^{2}\lesssim_{\Delta,\mathcal{K}}1/\sqrt{x-\beta_{n}},\quad
x\in[\beta_{n},\beta(\mu)].$

The compactness of $\mathcal{K}$ as well as (4.12) imply that
$G(e^{h_{n}}\mu_{|\Delta_{n}},\infty)\lesssim_{\mu,\mathcal{K}}1$.

Now, we are ready to estimate the integrals in the statement of the lemma. We
only carry out the estimates on $[\beta_{n}^{*},\beta(\mu)]$ as the estimate
on $[\alpha(\mu),\alpha_{n}^{*}]$ can be done analogously. Assume that
$\beta_{n}^{*}<\beta(\mu)$ (otherwise we have nothing to prove). In this case
$\beta_{n}^{*}=\beta_{n}+\delta_{n}$. Using the bounds we just obtained, Lemma
4.7 and condition (1) in Theorem 4.2 give

$\int_{\beta_{n}^{*}}^{\beta(\mu)}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x)=\int_{\beta_{n}^{*}}^{\beta(\mu)}\left|\frac{P_{n}^{*}(x)}{F_{n}(x)}\right|^{2}\frac{G^{2}(e^{h_{n}}\mu_{|\Delta_{n}},\infty)}{G^{2}(e^{h_{n}}\mu_{|\Delta_{n}},x)}e^{2n\kappa_{n}(x)+h_{n}(x)}d\mu(x)\\\
\lesssim_{\Delta,\mu,\mathcal{K}}\int_{\beta_{n}^{*}}^{\beta(\mu)}\exp\left(-C_{\Delta}n(x-\beta_{n})^{\varkappa/2}+\frac{\epsilon_{\mu}(x-\beta_{n})}{\sqrt{x-\beta_{n}}}\right)\frac{d\mu(x)}{\sqrt{x-\beta_{n}}}.$

We further get that

$\displaystyle\int_{\beta_{n}^{*}}^{\beta(\mu)}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x)$
$\displaystyle\lesssim_{\Delta,\mu,\mathcal{K}}\int_{\delta_{n}}^{\beta(\mu)-\beta_{n}}\exp\left(C\left(-nt^{\varkappa/2}+\frac{\epsilon_{\mu}(t)}{\sqrt{t}}\right)\right)d\mu(t+\beta_{n})$
(4.14) $\displaystyle\lesssim_{\Delta,\mu,\mathcal{K}}\max_{\delta_{n}\leq
t\leq
T}\exp\left(C\left(-nt^{\varkappa/2}+\frac{\epsilon_{\mu}(t)}{\sqrt{t}}\right)\right),$

where $T:=\sup_{n}(\beta(\mu)-\beta_{n})$ and $T>0$. Let

(4.15) $T_{n}:=(2\epsilon_{\mu}(T)/n)^{2/(\varkappa+1)}.$

When

(4.16) $T_{n}\leq t\leq T$

we get $\epsilon_{\mu}(t)\leq\epsilon_{\mu}(T)$ since $\epsilon_{\mu}(t)$ is
non-decreasing. Also, for such $t$, we have
$\epsilon_{\mu}(T)\leq(n/2)t^{(\varkappa+1)/2}$ by the choice of $T_{n}$ and
combining these bounds gives

(4.17) $\epsilon_{\mu}(t)\leq(n/2)t^{(\varkappa+1)/2}\,.$

Respectively, we get for these $t$ that

$-nt^{\varkappa/2}+\epsilon_{\mu}(t)/\sqrt{t}\stackrel{{\scriptstyle\eqref{ser1}}}{{\leq}}-(n/2)t^{\varkappa/2}\stackrel{{\scriptstyle\eqref{ser2}}}{{\leq}}-(n/2)T_{n}^{\varkappa/2}\stackrel{{\scriptstyle\eqref{ser2a}}}{{=}}-\big{(}n\epsilon_{\mu}^{\varkappa}(T)/2\big{)}^{1/(\varkappa+1)},$

and the right-hand side converges to $-\infty$ as $n\to\infty$. Since
$\epsilon_{\mu}(T_{n})\to 0$, we can choose $\xi_{n}$ in (4.7) so that

(4.18) $2\epsilon_{\mu}(T_{n})\leq\xi_{n}.$

Respectively, for

(4.19) $\delta_{n}\leq t\leq T_{n}$

we have that $\epsilon_{\mu}(t)\leq\epsilon_{\mu}(T_{n})$ by monotonicity and
$\epsilon_{\mu}(T_{n})\leq(n/2)t^{(\varkappa+1)/2}$ by $t>\delta_{n}$, (4.18)
and formula (4.7). Thus, combining these bounds gives

(4.20) $\epsilon_{\mu}(t)\leq(n/2)t^{(\varkappa+1)/2}\,.$

Hence, we get for such $t$ that

$-nt^{\varkappa/2}+\epsilon_{\mu}(t)/\sqrt{t}\stackrel{{\scriptstyle\eqref{ser6}}}{{\leq}}-(n/2)t^{\varkappa/2}\stackrel{{\scriptstyle\eqref{ser7}}}{{\leq}}-(n/2)\delta_{n}^{\varkappa/2}\stackrel{{\scriptstyle\eqref{delta_n}}}{{=}}-(1/2)(n\xi_{n}^{\varkappa})^{1/(\varkappa+1)}.$

Now, we choose positive sequence $\\{\xi_{n}\\}$ so that

(4.21)
$\lim_{n\to\infty}\xi_{n}=0,\quad\lim_{n\to\infty}n\xi_{n}^{\varkappa}=+\infty,\quad\text{and}\quad
2\epsilon_{\mu}(T_{n})\stackrel{{\scriptstyle\eqref{ser3}}}{{\leq}}\xi_{n}.$

This finishes the proof of the lemma because the right-hand side of (4.14)
converges to zero when $n\to\infty$. ∎

######  Lemma 4.9.

Theorem 4.2 takes place.

###### Proof.

Let $F_{n}^{*}(z)$ be as in (4.8) and

$\Omega_{\kappa_{n}}(z):=\Omega_{\Delta_{n}^{*}}\big{(}e^{-n\kappa_{n}};z),$

which is an outer function in $H^{2}(D_{\Delta_{n}^{*}})$ whose traces satisfy
$|\Omega_{\kappa_{n}\pm}(x)|^{2}=e^{-2n\kappa_{n}(x)}$ almost everywhere on
$\Delta_{n}^{*}$, see (1.5). Thus, it holds that

$|(\Omega_{\kappa_{n}}F_{n}^{*})_{\pm}(x)|^{-2}=G_{n}^{-2}e^{\theta_{n}(x)}v_{\Delta_{n}^{*}}(x)$

almost everywhere on $\Delta_{n}^{*}$ due to (1.5), (1.7) and (1.8), where
$G_{n}:=G(e^{h_{n}}\mu_{|\Delta_{n}^{*}},\infty)$. For brevity, put
$P_{n}=T_{n}(e^{\theta_{n}}\mu)$. Then, it follows from the Cauchy integral
formula that

$\displaystyle\frac{P_{n}(z)-P_{n}^{*}(z)}{(\Omega_{\kappa_{n}}F_{n}^{*})(z)w_{\Delta_{n}^{*}}(z)}$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\oint_{\Gamma_{n}}\frac{P_{n}(s)-P_{n}^{*}(s)}{z-s}\frac{1}{(\Omega_{\kappa_{n}}F_{n}^{*})(s)}\frac{ds}{w_{\Delta_{n}^{*}}(s)}$
$\displaystyle=\int_{\Delta_{n}^{*}}\frac{P_{n}(x)-P_{n}^{*}(x)}{z-x}\mathrm{Re}\left(\frac{1}{(\Omega_{\kappa_{n}}F_{n}^{*})_{+}(x)}\right)d\omega_{\Delta_{n}^{*}}(x),$

where $\Gamma_{n}$ is any counter-clockwise oriented Jordan curve that
separates $\Delta_{n}^{*}$ and $z$ and we used the following facts

$F_{n-}^{*}(x)=\overline{F_{n+}^{*}(x)}\quad\text{and}\quad
w_{\Delta_{n}^{*}+}(x)=-w_{\Delta_{n}^{*}-}(x)=\mathrm{i}|w_{\Delta_{n}^{*}}(x)|,\quad
x\in\Delta_{n}^{*},$

as well as (1.2). Since the absolute value of the real part does not exceed
the absolute value of a complex number, we get from the Cauchy-Schwarz
inequality that

$\displaystyle\left|\frac{P_{n}(z)-P_{n}^{*}(z)}{(\Omega_{\kappa_{n}}F_{n}^{*})(z)}\right|^{2}$
$\displaystyle\leq
G_{n}^{-2}\frac{|w_{\Delta_{n}^{*}}(z)|^{2}}{\mathrm{dist}(z,\Delta_{n}^{*})^{2}}\int_{\Delta_{n}^{*}}\big{(}P_{n}(x)-P_{n}^{*}(x)\big{)}^{2}e^{\theta_{n}(x)}v_{\Delta_{n}^{*}}(x)d\omega_{\Delta_{n}^{*}}(x)$
(4.22) $\displaystyle\leq
G_{n}^{-2}\frac{|w_{\Delta_{n}^{*}}(z)|^{2}}{\mathrm{dist}(z,\Delta_{n}^{*})^{2}}\int_{\Delta_{n}^{*}}\big{(}P_{n}(x)-P_{n}^{*}(x)\big{)}^{2}e^{\theta_{n}(x)}d\mu(x).$

It follows from the compactness of $\mathcal{K}$ and (4.12) that the constants
$G_{n}^{-2}$ are uniformly bounded above. Because $P_{n}^{*}$ is the $n$-th
monic orthogonal polynomials with respect to
$e^{\theta_{n}}\mu_{|\Delta_{n}^{*}}$ and $P_{n}$ is a monic polynomials of
degree $n$, we have that

$\int_{\Delta_{n}^{*}}P_{n}(x)P_{n}^{*}(x)e^{\theta_{n}(x)}d\mu(x)=\int_{\Delta_{n}^{*}}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x),$

from which we easily deduce that

(4.23)
$\int_{\Delta_{n}^{*}}\big{(}P_{n}(x)-P_{n}^{*}(x)\big{)}^{2}e^{\theta_{n}(x)}d\mu(x)=\int_{\Delta_{n}^{*}}P_{n}(x)^{2}e^{\theta_{n}(x)}d\mu(x)-\int_{\Delta_{n}^{*}}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x).$

Recall that for the $n$-th monic orthogonal polynomial $P_{n}$ it holds that

(4.24)
$\int_{\Delta(\mu)}P_{n}^{2}(x)e^{\theta_{n}(x)}d\mu(x)\leq\int_{\Delta(\mu)}P^{2}(x)e^{\theta_{n}(x)}d\mu(x)$

for any monic polynomial $P$ of degree $n$, and $P_{n}^{*}$ in particular.
Therefore, we obtain from (4.23), simple majorization, and (4.24) that

$\displaystyle\int_{\Delta_{n}^{*}}\big{(}P_{n}(x)-P_{n}^{*}(x)\big{)}^{2}e^{\theta_{n}(x)}d\mu(x)$
$\displaystyle\leq\int_{\Delta(\mu)}P_{n}(x)^{2}e^{\theta_{n}(x)}d\mu(x)-\int_{\Delta_{n}^{*}}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x)$
$\displaystyle\leq\int_{\Delta(\mu)}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x)-\int_{\Delta_{n}^{*}}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x)$
$\displaystyle=\int_{\Delta(\mu)\setminus\Delta_{n}^{*}}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x)\to
0$

as $n\to\infty$, where the last conclusion was shown in Lemma 4.8. Since the
intervals $\Delta_{n}^{*}$ converge to $\Delta$, we get from (4.22), the above
estimates, and Lemma 4.6 that

$P_{n}(z)=P_{n}^{*}(z)+o_{\mathcal{K}}(1)(\Omega_{\kappa_{n}}F_{n}^{*})(z)=\big{(}1+o_{\mathcal{K}}(1)\big{)}F_{n}^{*}(z)$

locally uniformly in $D_{\Delta}$, where the last equality also used the fact
that $\Omega_{\kappa_{n}}(z)=1+o(1)$ locally uniformly in the complement of
$\Delta$ as is clear from (1.4), (4.7), and (4.9). The first claim of Theorem
4.2 now follows from (4.12). The second claim of the theorem is a consequence
of (4.12) for $\Delta_{n}$ and of (4.13) if we observe that

$\int_{\Delta_{n}^{*}}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x)\leq\int_{\Delta_{n}^{*}}P_{n}(x)^{2}e^{\theta_{n}(x)}d\mu(x)\leq\int_{\Delta(\mu)}P_{n}(x)^{2}e^{\theta_{n}(x)}d\mu(x)\\\
\leq\int_{\Delta(\mu)}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x)=\int_{\Delta_{n}^{*}}P_{n}^{*}(x)^{2}e^{\theta_{n}(x)}d\mu(x)+o(1),$

where we used Lemma 4.8 for the last equality. As we just mentioned, the last
claim of the theorem follows from the second line of (4.12). ∎

##  5\. Strong Asymptotics of OPs with Reciprocal Polynomial Weights

In this section, we describe strong asymptotics of polynomials orthogonal with
respect to a sequence of measures that are ratios of certain bounded
perturbations of a Szegő measure and polynomials of growing degrees. Below, we
mostly follow [38] and generalize some results obtained in the pioneering work
[10]. Theorem 5.8 further below was used in a crucial way in Lemma 4.4 on the
way to proving Theorem 4.1. Theorem 5.8 itself is a straightforward
consequence of Theorem 5.1 that we state in the next subsection.

###  5.1. Orthogonality on $\mathbb{T}$

Let $\\{(\sigma_{n},g_{n},W_{n})\\}$ be triples of a finite positive Borel
measure on $\mathbb{T}$, a continuous real-valued function on $\mathbb{T}$,
and a monic polynomial of degree $n$ with all its zeros inside the unit disk.
For each $n$ we then define an inner product on the unit circle by

$\langle f,k\rangle_{\mathbb{T},n}=\int
f(\xi)\overline{k(\xi)}\,\frac{e^{g_{n}(\xi)}d\sigma_{n}(\xi)}{|W_{n}(\xi)|^{2}}.$

We are interested in orthonormal polynomials $\phi_{n}(z)$, $\deg\phi_{n}=n$,
satisfying

(5.1)
$\begin{cases}\langle\phi_{n},\xi^{m}\rangle_{\mathbb{T},n}=0,&m\in\\{0,1,\ldots,n-1\\},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\langle\phi_{n},\phi_{n}\rangle_{\mathbb{T},n}=1,\end{cases}$

and normalized to have a positive leading coefficient, i.e.,
$\phi_{n}(z)=\alpha_{n}z^{n}+\ldots$, $\alpha_{n}>0$ (when $g_{n}=0$ and
$\sigma_{n}=\sigma$, this is exactly the setting considered in [38]).

In this section, we let the symbol $*$ denote the transformation

$p^{\ast}(z)=z^{n}\overline{p(1/{\bar{z}})}$

defined on the set of polynomials of degree at most $n$. In particular, if we
denote the zeros of $W_{n}(z)$ by $b_{n,j}$, $j\in\\{1,2,\ldots,n\\}$, which
all belong to the unit disk, then

$W_{n}(z)=\prod_{j=1}^{n}(z-b_{n,j})\quad\text{and}\quad
W_{n}^{*}(z)=\prod_{j=1}^{n}(1-\overline{b_{n,j}}z).$

Write $d\sigma_{n}=\upsilon_{n}dm+d\sigma_{n}^{s}$, where the measures
$d\sigma_{n}^{s}$ are singular to $dm(\xi)=(2\pi)^{-1}|d\xi|$, the normalized
Lebesgue measure on $\mathbb{T}$. In this section, we shall assume the
following: __

  * $(A_{\mathbb{T}})$

_there exists a finite measure_ $d\sigma=\upsilon dm+d\sigma^{s}$, where
$d\sigma^{s}$ is singular to $dm$, such that for any non-negative continuous
function $f$ on $\mathbb{T}$ it holds that

$\limsup_{n\to\infty}\int fd\sigma_{n}\leq\int fd\sigma;$

  * $(B_{\mathbb{T}})$

_the densities_ $\upsilon_{n}$ and $\upsilon$ have logarithms integrable with
respect to $dm$ and $\|\log\upsilon_{n}-\log\upsilon\|_{L^{1}(\mathbb{T})}\to
0$ as $n\to\infty$;

  * $(C_{\mathbb{T}})$

_the functions_ $g_{n}$ belong to $\mathcal{E}$, a fixed compact subset of
$C(\mathbb{T})$;

  * $(D_{\mathbb{T}})$

_the zeros_ $\\{b_{n,j}\\}$ satisfy $\sum_{j=1}^{n}(1-|b_{n,j}|)\to\infty$ as
$n\to\infty$.

It is known, see [17, Section II.2], that the condition $(D_{\mathbb{T}})$ is
equivalent to

(5.2) $\frac{W_{n}(z)}{W_{n}^{*}(z)}=o(1)$

locally uniformly in $\mathbb{D}$. To describe the results, let us introduce
the Szegő function of a measure $d\sigma=\upsilon dm+d\sigma^{s}$, which is
given by

$D(\sigma,z):=\exp\left(\frac{1}{2}\int\frac{\xi+z}{\xi-z}\log\upsilon(\xi)dm(\xi)\right)$

and is independent of the singular part $\sigma^{s}$. When $\log\upsilon$ is
integrable with respect to $dm$, the function $D(\sigma,z)$ is analytic in
$\overline{\mathbb{C}}\setminus\mathbb{T}$, in fact, it is an outer function
in both $\mathbb{D}$ and
$\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$, its values inside and
outside of the unit disk are related via the identity

$D^{-1}(\sigma,z)=\overline{D(\sigma,1/\bar{z})},\quad z\not\in\mathbb{T},$

and it has a non-tangential limit on $\mathbb{T}$ (taken within $\mathbb{D}$)
that satisfies $|D(\sigma,\xi)|^{2}=\upsilon(\xi)$ for almost every $\xi$ on
$\mathbb{T}$.

######  Theorem 5.1.

With the above definitions, assume that the orthogonality measures in (5.1)
satisfy conditions $(A_{\mathbb{T}})-(D_{\mathbb{T}})$. Then,

(5.3)
$\frac{\phi_{n}^{*}(z)}{W_{n}^{*}(z)}D_{n}(z)=1+o_{\mathcal{E}}(1)\quad\text{and}\quad\frac{\phi_{n}(z)}{\phi_{n}^{*}(z)}=o_{\mathcal{E}}(1)$

locally uniformly in $\mathbb{D}$ as $n\to\infty$, where
$D_{n}(z)=D(e^{g_{n}}\sigma_{n},z)$. In particular, we have that

(5.4) $\alpha_{n}D_{n}(0)=1+o_{\mathcal{E}}(1)\quad\text{as}\quad n\to\infty.$

###  5.2. Proof of Theorem 5.1

As in the previous sections, we shall prove Theorem 5.1 in a sequence of steps
that we organize as separate lemmas. For the first step we define the
Caratheodory function of a measure $\sigma$ by

$F(\sigma,z):=\int\frac{\xi+z}{\xi-z}d\sigma(\xi).$

Further, for each orthonormal polynomial $\phi_{n}(z)$ we define its companion
polynomial as a polynomial $\psi_{n}(z)$ of degree at most $n$ such that
$\psi_{n}^{*}(z)$ interpolates $\phi_{n}^{*}(z)F(e^{g_{n}}\sigma_{n},z)$ at
the zeros of $zW_{n}(z)$, see [38, Equation (3.6)].

######  Lemma 5.2.

Under the conditions of Theorem 5.1, we have

$F(e^{g_{n}}\sigma_{n},z)-\frac{\psi^{*}_{n}(z)}{\phi^{*}_{n}(z)}=o_{\mathcal{E}}(1)\quad\text{as}\quad
n\to\infty$

locally uniformly in $\mathbb{D}$. Moreover,

(5.5) $F(e^{g_{n}}\sigma_{n},0)=\frac{\psi^{*}_{n}(0)}{\phi^{*}_{n}(0)}\,.$

###### Proof.

Combining formulas (3.23), (3.24), and (3.26) from [38], we get

$\left|F(e^{g_{n}}\sigma_{n},z)-\frac{\psi^{*}_{n}(z)}{\phi^{*}_{n}(z)}\right|\leq
2\sqrt{2}\left|z\frac{W_{n}(z)}{W_{n}^{*}(z)}\right|\frac{|e^{g_{n}}\sigma_{n}|}{(1-|z|)^{3/2}}$

for $z\in\mathbb{D}$, where $|\mu|$ denotes the total mass of the measure
$\mu$. Taking $f=1$ in $(A_{\mathbb{T}})$, we get

(5.6)
$\limsup_{n\to\infty}|\sigma_{n}|\leq|\sigma|\quad\text{and}\quad\limsup_{n\to\infty}|e^{g_{n}}\sigma_{n}|\lesssim_{\mathcal{E}}|\sigma|,$

where the last bound follows from $(C_{\mathbb{T}})$. Applying (5.2) finishes
the proof of the lemma. ∎

Notice that Jensen’s inequality and $(C_{\mathbb{T}})$ yield that

(5.7)
$\liminf_{n\to\infty}|\sigma_{n}|\geq\liminf_{n\to\infty}\exp\left(\int\log\upsilon_{n}dm\right)=\exp\left(\int\log\upsilon
dm\right)>-\infty$

and $(B_{\mathbb{T}}$) gives a lower bound

(5.8)
$\liminf_{n\to\infty}|e^{g_{n}}\sigma_{n}|\gtrsim_{\mathcal{E}}\exp\left(\int\log\upsilon
dm\right).$

Put

(5.9)
$\lambda_{n}(z)=\mathrm{Re}\left(\frac{\psi^{*}_{n}(z)}{\phi^{*}_{n}(z)}\right),\quad|z|\leq
1,$

which is a harmonic function in some neighborhood of $\overline{\mathbb{D}}$.
It has been shown in [38, Equation (3.9)] that

(5.10)
$\frac{\psi^{*}_{n}(z)}{\phi^{*}_{n}(z)}=\int\frac{\xi+z}{\xi-z}\left|\frac{W_{n}(\xi)}{\phi_{n}(\xi)}\right|^{2}dm(\xi).$

Hence, $\lambda_{n}(z)$ is a real part of the Caratheodory function of an
absolutely continuous measure with strictly positive density and therefore is
a strictly positive harmonic function in $\overline{\mathbb{D}}$. It readily
follows from the second claim of Lemma 5.2 that

(5.11)
$|\lambda_{n}m|=\lambda_{n}(0)=F(e^{g_{n}}\sigma_{n},0)=|e^{g_{n}}\sigma_{n}|.$

######  Lemma 5.3.

Let $\mathcal{Z}$ be a compact set in $C(\mathbb{T})$. Under the conditions of
Theorem 5.1, it holds that for any $\epsilon>0$ there exists
$N_{\mathcal{E},\mathcal{Z}}(\epsilon)$ such that

$\left|\int he^{g_{n}}d\sigma_{n}-\int h\lambda_{n}dm\right|\leq\epsilon$

for all $n\geq N_{\mathcal{E},\mathcal{Z}}(\epsilon)$ and each
$h\in\mathcal{Z}$.

###### Proof.

We use a standard approximation argument. Comparing the Taylor coefficients of
functions $F(e^{g_{n}}\sigma_{n},z)$ and $\psi^{*}_{n}(z)/\phi^{*}_{n}(z)$ at
$z=0$ and applying Lemma 5.2 gives

$\int
h(\xi)\big{(}e^{g_{n}}d\sigma_{n}-\lambda_{n}dm\big{)}=o_{\mathcal{E},h}(1)$

as $n\to\infty$, where $h(\xi)=\xi^{j}$, $j\in\mathbb{Z}$. Hence, this claim
remains true for each trigonometric polynomial $h$. Now, given $\epsilon>0$,
we can use compactness and denseness of trigonometric polynomials in
$C(\mathbb{T})$ to find a finite collection of trigonometric polynomials
$\\{h_{1},h_{2},\ldots,h_{K(\epsilon)}\\}$ such that for each
$h\in\mathcal{Z}$ there is $k\in\\{1,2,\ldots,K(\epsilon)\\}$ for which

$\|h-h_{k}\|_{\infty}<\frac{\epsilon}{3M_{\mathcal{E}}},$

where $\|e^{g_{n}}\sigma_{n}\|\leq M_{\mathcal{E}}$. Then,

$\left|\int(h-h_{k})e^{g_{n}}d\sigma_{n}\right|\leq\frac{\epsilon}{3}\quad\text{and}\quad\left|\int(h-h_{k})\lambda_{n}dm\right|\leq\frac{\epsilon}{3}$

by (5.11). As just observed, for each $h_{k}$ one can find a natural number
$N_{\mathcal{E},h_{k}}(\epsilon)$ such that

$\left|\int
h_{k}\big{(}e^{g_{n}}d\sigma_{n}-\lambda_{n}dm\big{)}\right|\leq\frac{\epsilon}{3}$

for $n\geq N_{\mathcal{E},h_{k}}(\epsilon)$. Taking
$N_{\mathcal{E},\mathcal{Z}}(\epsilon)=\max_{1\leq k\leq
K(\epsilon)}N_{\mathcal{E},h_{k}}(\epsilon)$ yields the desired claim. ∎

For the next step, we shall need the mutual entropy of two measures. Let $\mu$
and $\nu$ be two measures on $\mathbb{T}$ such that $\mu$ is absolutely
continuous with respect to $\nu$. The entropy $S(\mu|\nu)$ is defined as

$S(\mu|\nu)=-\int\log\left(\frac{d\mu}{d\nu}\right)d\mu.$

It is known that $S(\mu|\nu)\leq\log|\nu|$ and the following representation
holds, see [36, Lemma 2.3.3, p.137]:

(5.12) $S(\mu|\nu)=\inf_{f}\left(\int fd\nu-\int(1+\log f)d\mu\right),$

where the infimum is taken over all positive continuous functions on
$\mathbb{T}$. Moreover, see [36, Example 2.3.2], if $\mu=m$ and
$d\nu=\nu^{\prime}dm+d\nu^{s}$, then

(5.13)
$S(m|\nu)=-\int\log\left(\frac{dm}{d\nu}\right)dm=\int\log\nu^{\prime}dm.$

######  Lemma 5.4.

Under the conditions of Theorem 5.1, it holds that for any $\epsilon>0$ there
exists $N_{\mathcal{E}}(\epsilon)$, that depends on $\mathcal{E}$ but not a
particular choice of $\\{g_{n}\\}\subset\mathcal{E}$, such that

$\int\log\lambda_{n}dm\leq\int\big{(}g_{n}+\log\upsilon_{n}\big{)}dm+\epsilon,\quad
n\geq N_{\mathcal{E}}(\epsilon).$

###### Proof.

It follows from (5.12) and (5.13), applied with $\mu=m$ and $\nu=\sigma$, as
well as the condition $(B_{\mathbb{T}})$ that there exists a positive
continuous function $f_{\epsilon}$ and a natural number $N_{1}(\epsilon)$ such
that

$\int f_{\epsilon}d\sigma-\int(1+\log f_{\epsilon})dm\leq\int\log\upsilon
dm+\frac{\epsilon}{4}\leq\int\log\upsilon_{n}dm+\frac{\epsilon}{2}$

for all $n\geq N_{1}(\epsilon)$. Hence, we get again from (5.12) and (5.13),
applied this time with $\mu=m$ and $d\nu=\lambda_{n}dm$, that

$\displaystyle\int\log\lambda_{n}dm$
$\displaystyle\leq\int(e^{-g_{n}}f_{\epsilon})\lambda_{n}dm-\int\big{(}1+\log(e^{-g_{n}}f_{\epsilon})\big{)}dm$
(5.14)
$\displaystyle\leq\int\big{(}g_{n}+\log\upsilon_{n}\big{)}dm+\int(e^{-g_{n}}f_{\epsilon})\lambda_{n}dm-\int
f_{\epsilon}d\sigma+\frac{\epsilon}{2}$

for all $n\geq N_{1}(\epsilon)$. Furthermore, it follows from the condition
$(A_{\mathbb{T}})$ that there exists a natural number $N_{2}(\epsilon)$ such
that

$\int f_{\epsilon}d\sigma_{n}\leq\int f_{\epsilon}d\sigma+\frac{\epsilon}{4}$

for all $n\geq N_{2}(\epsilon)$. Thus, we get from Lemma 5.3, applied with
$\mathcal{Z}=\\{e^{-g}f_{\epsilon}:g\in\mathcal{E}\\}$, that there exists
$N_{\mathcal{E}}(\epsilon)\geq\max\\{N_{1}(\epsilon),N_{2}(\epsilon)\\}$ such
that

(5.15)
$\int(e^{-g_{n}}f_{\epsilon})\lambda_{n}dm\leq\int(e^{-g_{n}}f_{\epsilon})e^{g_{n}}d\sigma_{n}+\frac{\epsilon}{4}\leq\int
f_{\epsilon}d\sigma+\frac{\epsilon}{2}$

for all $n\geq N_{\mathcal{E}}(\epsilon)$. Clearly, inequalities (5.14) and
(5.15) yield the desired claim. ∎

######  Lemma 5.5.

Under the conditions of Theorem 5.1, (5.4) takes place.

###### Proof.

It follows directly from (5.9) and (5.10), see also [38, Equation (3.8)], that

$\left|\frac{W_{n}^{*}(\xi)}{\phi_{n}^{*}(\xi)}\right|^{2}=\left|\frac{W_{n}(\xi)}{\phi_{n}(\xi)}\right|^{2}=\lambda_{n}(\xi),\quad|\xi|=1.$

As $W_{n}^{*}(z)/\phi_{n}^{*}(z)$ is an analytic and non-vanishing function in
the closed unit disk, the logarithm of its absolute value is harmonic there.
Thus, we get from the mean-value property that

$\log\alpha_{n}=-\log\left|\frac{W_{n}^{*}(0)}{\phi_{n}^{*}(0)}\right|=-\frac{1}{2}\int\log\lambda_{n}dm,$

where one also needs to recall that $\phi_{n}^{*}(0)=\alpha_{n}>0$ and
$W_{n}^{*}(0)=1$. Hence, it follows from Lemma 5.4 and the very definition of
$D_{n}(z)$ that

$\log\alpha_{n}\geq-\frac{1}{2}\int\big{(}g_{n}+\log\upsilon_{n}\big{)}dm+o_{\mathcal{E}}(1)=-\log
D_{n}(0)+o_{\mathcal{E}}(1).$

On the other hand, we get from (5.1) that

$0=\log\int\left|\frac{\phi_{n}}{W_{n}}\right|^{2}e^{g_{n}}d\sigma_{n}\geq\log\int\left|\frac{\phi_{n}}{W_{n}}\right|^{2}e^{g_{n}}\upsilon_{n}dm.$

Therefore, Jensen’s inequality gives

$0\geq\int\log\frac{e^{g_{n}}\upsilon_{n}}{\lambda_{n}}dm=2\log\big{(}\alpha_{n}D_{n}(0)\big{)},$

which finishes the proof of the lemma. ∎

######  Lemma 5.6.

Under the conditions of Theorem 5.1, it holds that

$\int\left|\frac{\phi_{n}^{*}}{W_{n}^{*}}D_{n}-1\right|^{2}dm=o_{\mathcal{E}}(1)$

as $n\to\infty$, where $D_{n}(\xi)$ denotes above the non-tangential boundary
values of $D_{n}(z)$ on $\mathbb{T}$ taken from within $\mathbb{D}$. In
particular, the first asymptotic formula in (5.3) takes place.

###### Proof.

Denote the integral in the statement of the lemma by $I$. Then, we have that

$I=\int\left|\frac{\phi_{n}^{*}}{W_{n}^{*}}D_{n}\right|^{2}dm+1-2\int\mathrm{Re}\left(\frac{\phi_{n}^{*}}{W_{n}^{*}}D_{n}\right)dm.$

Since $|D_{n}(\xi)|^{2}=e^{g_{n}(\xi)}\upsilon_{n}(\xi)$ for almost every
$|\xi|=1$, the mean-value property for harmonic functions yields that

$I=\int\left|\frac{\phi_{n}}{W_{n}}\right|^{2}e^{g_{n}}\upsilon_{n}dm+1-2\mathrm{Re}\left(\frac{\phi_{n}^{*}(0)}{W_{n}^{*}(0)}D_{n}(0)\right)\leq
2-2\alpha_{n}D_{n}(0).$

The first claim of the lemma now follows from (5.4). In particular, we have
shown that the functions $1-\phi_{n}^{*}D_{n}/W_{n}^{*}$ belong to the Hardy
space $H^{2}(\mathbb{D})$. Thus, the second claim of the lemma now follows
from the first and the Cauchy integral formula for functions in
$H^{2}(\mathbb{D})$. ∎

######  Lemma 5.7.

Under the conditions of Theorem 5.1, the second asymptotic formula in (5.3)
takes place.

###### Proof.

It follows from the first asymptotic formula in (5.3) that

$\frac{\phi_{n}(z)}{\phi_{n}^{*}(z)}=\frac{W_{n}^{*}(z)}{\phi_{n}^{*}(z)}\frac{\phi_{n}(z)}{W_{n}^{*}(z)}=(1+o_{\mathcal{E}})\frac{\phi_{n}(z)D_{n}(z)}{W_{n}^{*}(z)}.$

Thus, it is sufficient for us to study the behavior of
$\phi_{n}D_{n}/W_{n}^{*}$ in the unit disk. Since these functions have
integrable traces on $\mathbb{T}$ and $dm(\xi)=d\xi/(2\pi\mathrm{i}\xi)$, it
follows from the Cauchy integral formula that

$\frac{\phi_{n}(z)D_{n}(z)}{W_{n}^{*}(z)}=\int\frac{\phi_{n}(\xi)D_{n}(\xi)}{W_{n}^{*}(\xi)}\frac{dm(\xi)}{1-z\overline{\xi}}=\int
B_{n}(\xi)U_{n}(\xi)\frac{\phi_{n}(\xi)\overline{D_{n}(\xi)}}{W_{n}(\xi)}\frac{dm(\xi)}{1-z\overline{\xi}},$

where $B_{n}(z)=W_{n}(z)/W_{n}^{*}(z)$ and
$U_{n}(\xi)=D_{n}(\xi)/\overline{D_{n}(\xi)}$. Since
$|B_{n}(\xi)U_{n}(\xi)|=1$ for $|\xi|=1$, it follows from the Cauchy-Schwarz
inequality that

$\left|\int\left(\frac{\phi_{n}(\xi)\overline{D_{n}(\xi)}}{W_{n}(\xi)}-1\right)B_{n}(\xi)U_{n}(\xi)\frac{dm(\xi)}{1-z\overline{\xi}}\right|\leq\frac{\|\phi_{n}\overline{D_{n}}/W_{n}-1\|_{L^{2}(\mathbb{T})}}{\sqrt{1-|z|^{2}}}.$

Thus, we deduce from the first claim of Lemma 5.6 that

$\frac{\phi_{n}(z)D_{n}(z)}{W_{n}^{*}(z)}=o_{\mathcal{E}}(1)+\int
B_{n}(\xi)U_{n}(\xi)\frac{dm(\xi)}{1-z\overline{\xi}},$

where $o_{\mathcal{E}}(1)$ holds locally uniformly in the unit disk.111When
$U_{n}(\xi)=U(\xi)$, the last integral above can be understood as
$(\mathcal{T}_{U}B_{n})(z)$, where $\mathcal{T}_{U}$ is a Toeplitz operator
with symbol $U$. Since $\mathcal{T}_{U}^{*}=\mathcal{T}_{\overline{U}}$ and
the Blaschke products $B_{n}(z)$ converge weakly to zero in
$H^{2}(\mathbb{D})$ by (5.2), the functions $(\mathcal{T}_{U}B_{n})(z)$ also
converge weakly to zero in $H^{2}(\mathbb{D})$, which finishes the proof of
the lemma in this case. This observation somewhat simplifies the arguments
given in [38, pages 182-188]. Let $\\{b_{n,k}\\}$ be the Fourier coefficients
of $B_{n}(\xi)$ and $\\{u_{n,k}(z)\\}$ be the Fourier coefficients of
$U_{n}(\xi)/(1-z\overline{\xi})$. Then,

$\int
B_{n}(\xi)U_{n}(\xi)\frac{dm(\xi)}{1-z\overline{\xi}}=\sum_{k=0}^{\infty}b_{n,k}u_{n,-k}(z).$

Since both $B_{n}(\xi)$ and $U_{n}(\xi)$ are unimodular functions,

$\sum_{k\in\mathbb{Z}}|b_{n,k}|^{2}=1,\quad\sum_{k\in\mathbb{Z}}|u_{n,k}|^{2}=\int\frac{dm(\xi)}{|1-z\bar{\xi}|^{2}}=\frac{1}{1-|z|^{2}}\,.$

The Cauchy-Schwarz inequality gives

$\displaystyle\left|\int
B_{n}(\xi)U_{n}(\xi)\frac{dm(\xi)}{1-z\overline{\xi}}\right|$
$\displaystyle\displaystyle\leq\left|\sum_{k=0}^{K}b_{n,k}u_{n,-k}(z)\right|+\left|\sum_{k=K+1}^{\infty}b_{n,k}u_{n,-k}(z)\right|$
$\displaystyle\displaystyle\leq\sqrt{\sum_{k=0}^{K}\frac{|b_{n,k}|^{2}}{1-|z|^{2}}}+\sqrt{\sum_{k=K+1}^{\infty}|u_{n,-k}(z)|^{2}}$

for any natural number $K$. Asymptotic formula (5.2) yields that for any
$\epsilon>0$ and any $K$, there exists $N_{K}(\epsilon)$ such that the first
sum above is bounded by $\epsilon/(1-|z|^{2})^{1/2}$ for all $n\geq
N_{K}(\epsilon)$. Thus, to prove the lemma we are only left to argue that
there exists $K=K_{\mathcal{E},z}(\epsilon)$, which is locally uniformly
bounded with respect to $z$, such that the second sum is bounded by
$\epsilon$. To prove this claim, it is enough to show that the functions (as
functions in variable $\xi$)

$\frac{1}{1-z\overline{\xi}}\frac{D(e^{g},\xi)}{\overline{D(e^{g},\xi)}}\frac{D(\upsilon_{n},\xi)}{\overline{D(\upsilon_{n},\xi)}}$

form a precompact set in $L^{2}(\mathbb{T})$ for $g\in\mathcal{E}$,
$n\in\mathbb{N}$, and $z$ in any fixed compact subset of $\mathbb{D}$. Let
$\mathcal{H}$ be the Hilbert transform. The functions above can be written as

$\big{(}1-z\overline{\xi}\big{)}^{-1}e^{2\mathrm{i}(\mathcal{H}g)(\xi)}e^{2\mathrm{i}(\mathcal{H}\log\upsilon_{n})(\xi)}.$

Consider the following subsets of $L^{2}(\mathbb{T})$:

$\begin{cases}S_{1}&=\big{\\{}\big{(}1-z\overline{\xi}\big{)}^{-1}:~{}z\in\text{compact
in }\mathbb{D}\big{\\}},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
S_{2}&=\left\\{e^{2\mathrm{i}(\mathcal{H}g)(\xi)}:g\in\mathcal{E}\right\\},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
S_{3}&=\left\\{e^{2\mathrm{i}(\mathcal{H}\log\upsilon_{n})(\xi)}:n\in\mathbb{N}\right\\}.\end{cases}$

The compactness of $S_{1}$ in $L^{2}(\mathbb{T})$ is trivial. Since
$\mathcal{E}$ is compact in $C(\mathbb{T})$ it is also compact in
$L^{2}(\mathbb{T})$. The operator $\mathcal{H}$ is a bounded on
$L^{2}(\mathbb{T})$. Hence, $\mathcal{H}\mathcal{E}$ is a compact subset of
real-valued functions in $L^{2}(\mathbb{T})$. Given two real numbers $f$ and
$g$, we have

(5.16) $|e^{\mathrm{i}f}-e^{\mathrm{i}h}|^{2}=4\sin^{2}((f-h)/2)\leq
4|f-h|^{p},\quad p\in(0,2].$

Therefore, $S_{2}$ is compact in $L^{2}(\mathbb{T})$. Finally, since
$\log\upsilon_{n}\to\log\upsilon$ in $L^{1}(\mathbb{T})$ due to our condition
$(B_{\mathbb{T}})$, it holds that
$\mathcal{H}\log\upsilon_{n}\to\mathcal{H}\log\upsilon$ in $L^{p}(\mathbb{T})$
for any $p\in(0,1)$ by Kolmogorov’s theorem. Using (5.16) with any such $p$
yields that the functions $\exp(2\mathrm{i}\mathcal{H}\log\upsilon_{n})$
converge to $\exp(2\mathrm{i}\mathcal{H}\log\upsilon)$ in $L^{2}(\mathbb{T})$,
which shows that $S_{3}$ is precompact in $L^{2}(\mathbb{T})$. Since
$S_{1},S_{2}$, and $S_{3}$ are bounded in $L^{\infty}(\mathbb{T})$, the
product set $S_{1}S_{2}S_{3}$ is precompact in $L^{2}(\mathbb{T})$ as claimed.
That finishes the proof of the lemma. ∎

###  5.3. Orthogonality on $[-1,1]$

We will now translate the results of Theorem 5.1 to the case of polynomials
orthonormal on the interval $[-1,1]$. Here, we look at triples
$(\tilde{\mu}_{n},h_{n},\tau_{n})$, where $\tilde{\mu}_{n}$ is a finite
positive Borel measure, $h_{n}$ is a continuous function, and $\tau_{n}$ is a
polynomial of degree at most $2n$ with real coefficients that does not vanish
on $[-1,1]$ and is normalized to have value $1$ at $0$. For each $n$ we define
an inner product on $[-1,1]$ by

$\langle f,g\rangle_{[-1,1],n}=\int
f(x)\overline{g(x)}\,\frac{e^{h_{n}(x)}d\tilde{\mu}_{n}(x)}{\tau_{n}(x)}.$

We are interested in the orthonormal polynomials with the above varying
weights. That is, we study polynomials $p_{n}(z)$, $\deg p_{n}=n$, satisfying
orthogonality relations

(5.17) $\begin{cases}\langle
p_{n},x^{m}\rangle_{[-1,1],n}=0,&m\in\\{0,1,\ldots,n-1\\},\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ \langle p_{n},p_{n}\rangle_{[-1,1],n}=1,\end{cases}$

which are normalized to have positive leading coefficient that we denote by
$\gamma_{n}$, i.e., $p_{n}(z)=\gamma_{n}z^{n}+\text{lower degree terms}$,
$\gamma_{n}>0$.

The assumptions we made about the polynomials $\tau_{n}$ can be equivalently
stated in the following way. Let $\\{a_{n,1},a_{n,2},\ldots,a_{n,2n}\\}$ be a
conjugate-symmetric multi-set (points $a_{n,j}$ can coincide and be either
real or come in complex-conjugate pairs) such that $a_{n,j}\notin[-1,1]$
(these points can be equal to $\infty$). Then,

$\tau_{n}(x)=\prod_{j=1}^{2n}\left(1-\frac{x}{a_{n,j}}\right),$

where we understand $x/a_{n,j}$ as $0$ when $a_{n,j}=\infty$. Below, we assume
the following: __

  * $(A_{[-1,1]})$

_there exists a finite measure_ $\mu$ on $[-1,1]$ such that for any non-
negative continuous function $f$ on $[-1,1]$ it holds that

$\limsup_{n\to\infty}\int fd\tilde{\mu}_{n}\leq\int fd\mu;$

  * $(B_{[-1,1]})$

_the Radon-Nikodym derivatives of_ $\tilde{\mu}_{n}$ and $\mu$ with respect to
$\omega=\omega_{[-1,1]}$, say $\tilde{v}_{n}$ and $v$, see (1.2) and (1.7),
satisfy $\|\log\tilde{v}_{n}-\log v\|_{L^{1}(\omega)}\to 0$ as $n\to\infty$;

  * $(C_{[-1,1]})$

_the functions_ $h_{n}$ belong to $\mathcal{K}$, a fixed compact subset of
$C[-1,1]$;

  * $(D_{[-1,1]})$

_the zeros_ $\\{a_{n,j}\\}$ of the polynomials $\tau_{n}$ satisfy
$\sum_{j=1}^{2n}(1-|\phi(a_{n,j})|)\to\infty$ as $n\to\infty$, where
$\phi(z)=\phi_{[-1,1]}(z)$, see (1.3).

The next result is proven exactly as [38, Theorem 1]. Nevertheless, we provide
most of the details for the reader’s convenience.

######  Theorem 5.8.

With the above definitions, assume that the triples
$(\tilde{\mu}_{n},h_{n},\tau_{n})$ satisfy conditions
$(A_{[-1,1]})-(D_{[-1,1]})$. Let $p_{n}(z)$ be as in (5.17). Then,

(5.18)
$2\widetilde{G}_{n}^{2}(z)\,\frac{p_{n}^{2}(z)}{\tau_{n}(z)}\,\prod_{j=1}^{2n}\frac{\phi(z)-\phi(a_{n,j})}{1-\overline{\phi(a_{n,j})}\phi(z)}=1+o_{\mathcal{K}}(1)$

locally uniformly in $\overline{\mathbb{C}}\setminus[-1,1]$ as $n\to\infty$,
where $\widetilde{G}_{n}(z)=G(e^{h_{n}}\tilde{\mu}_{n},z)$ and the meaning of
$o_{\mathcal{K}}(1)$ is the same as in Theorem 4.1. In particular, we have
that

(5.19)
$\widetilde{G}_{n}^{2}(\infty)\frac{\gamma^{2}_{n}}{2^{2n-1}}\,\prod_{j:a_{n,j}\neq\infty}\big{(}2a_{n,j}\phi(a_{n,j})\big{)}=1+o_{\mathcal{K}}(1)\quad\text{as}\quad
n\to\infty.$

###### Proof.

The results of this theorem follow from Theorem 5.1 after we connect
orthogonality on $[-1,1]$ to the orthogonality on the circle by the Joukovski
map

$J(z)=\big{(}z+z^{-1}\big{)}/2.$

To translate conditions $(A_{[-1,1]})-(D_{[-1,1]})$ into conditions
$(A_{\mathbb{T}})-(D_{\mathbb{T}})$, set $b_{2n,j}=\phi(a_{n,j})$,
$j\in\\{1,2,\ldots,2n\\}$. That is, $J(b_{2n,j})=a_{n,j}$. Then

$\tau_{n}(z)=\prod_{a_{n,j}\neq\infty}\frac{(\zeta-b_{2n,j})(1-\zeta
b_{2n,j})}{2a_{n,j}b_{2n,j}\zeta}=\frac{W_{2n}(\zeta)W_{2n}^{*}(\zeta)}{c_{n}\zeta^{2n}},$

$z=J(\zeta)$, where $c_{n}=\prod_{a_{n,j}\neq\infty}2a_{n,j}b_{2n,j}$,
$W_{2n}(\zeta)=\prod_{j=1}^{2n}(\zeta-b_{2n,j})$, and we used the conjugate
symmetry of the multi-set $\\{b_{2n,1},b_{2n,2},\ldots,b_{2n,2n}\\}$. Clearly,
conditions $(D_{[-1,1]})$ and $(D_{\mathbb{T}})$ are equivalent to each other.
Define $\mathcal{E}=\\{h\circ J:h\in\mathcal{K}\\}$ and let $g_{2n}=h_{n}\circ
J$. It trivially holds that condition $(C_{[-1,1]})$ implies condition
$(C_{\mathbb{T}})$. Every measure $\mu$ defined on $[-1,1]$ can be mapped to a
measure $\sigma$ on $\mathbb{T}$ by the formula

$2\sigma(A)=\mu(J(A_{+}))+\mu(J(A_{-})),$

where $A$ is any Borel subset in $\mathbb{T}$,
$A_{+}=A\cap\\{e^{\mathrm{i}t}:t\in[0,\pi)\\}$, and $A_{-}=A\setminus A_{+}$.
For example, the mapping of the arcsine law $\omega$ results in the normalized
Lebesgue measure $m$ on the circle. More generally, if one has a function $v$
that is integrable with respect to $\omega$, the measure $vd\omega$ is mapped
to $\upsilon dm$, where $\upsilon=v\circ J$. We use this map to define
measures $\sigma_{2n}$ on $\mathbb{T}$ that correspond to the measures
$\tilde{\mu}_{n}$ on $[-1,1]$. This gives
$d\sigma_{2n}=\upsilon_{2n}dm+d\sigma_{2n}^{s}$ and $d\sigma=\upsilon
dm+d\sigma^{s}$, where $\upsilon_{2n}=\tilde{v}_{n}\circ J$ and
$\upsilon=v\circ J$. Notice that condition $(B_{[-1,1]})$ implies
$(B_{\mathbb{T}})$. Similarly, condition $(A_{[-1,1]})$ implies condition
$(A_{\mathbb{T}})$.

Let now $\phi_{2n}$ be the polynomials satisfying orthogonality relations
(5.1) with the above defined $(\sigma_{2n},g_{2n},W_{2n})$. It is well-known,
see [40, Theorem 11.5] or [38, Lemma 4.13], that

$p_{n}^{2}(z)=\frac{\phi_{2n}^{*}(\zeta)^{2}}{2c_{n}\zeta^{2n}}\frac{(1+\phi_{2n}(\zeta)/\phi_{2n}^{*}(\zeta))^{2}}{1+\phi_{2n}(0)/\phi_{2n}^{*}(0)}\quad\text{and}\quad\gamma_{n}^{2}=2^{2n-1}\frac{\alpha_{2n}^{2}}{c_{n}}\left(1+\frac{\phi_{2n}(0)}{\phi_{2n}^{*}(0)}\right),$

where $z=J(\zeta)$. It can be readily verified that
$\widetilde{G}_{n}(z)=D_{2n}(\zeta)$, see [38, Lemma 4.3]. Thus, asymptotic
formulae (5.18) and (5.19) follow from (5.3) and (5.4), where one needs to use
the identity

$\frac{\widetilde{G}_{n}^{2}(z)}{\tau_{n}(z)}\cdot\prod_{j=1}^{2n}\frac{\phi(z)-\phi(a_{n,j})}{1-\overline{\phi(a_{n,j})}\phi(z)}=c_{n}\left(\frac{\zeta^{n}D_{2n}(\zeta)}{W_{2n}^{*}(\zeta)}\right)^{2}.\qed$

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∎

e1e-mail<EMAIL_ADDRESS>
11institutetext: George Washington University, Washington, DC 20052, USA
22institutetext: Massachusetts Institute of Technology, Cambridge, MA 02139,
USA

# Target-Normal Single Spin Asymmetries Measured with Positrons

G. N. Grauvogelgw  T. Kutzgw,mit  A. Schmidte1,gw

(Received: date / Accepted: date)

###### Abstract

Two-photon exchange and the larger class of hadronic box diagrams are
difficult to calculate without a large degree of model-dependence. At the same
time, these processes are significant radiative corrections in parity-
violating electron scattering, in neutron decay, and may even be responsible
for the proton’s form factor ratio discrepancy. New kinds of experimental data
are needed to help constrain models and guide future box-diagram calculations.
The target-normal single spin asymmetry, $A_{n}$, formed with an unpolarized
beam scattering from a target that is polarized normal to the scattering
plane, is sensitive to the imaginary part of the two-photon exchange
amplitude, and can provide a valuable constraint. A measurement with both
electrons and positrons can reduce sources of experimental error, and
distinguish between the effects of two-photon exchange and those of time-
reversal symmetry violation. This article describes a proposed experiment in
Hall A, using the new Super Big-Bite Spectrometer that can cover a momentum
transfer range in the critical zone of uncertainty between where hadronic
calculations and those based on partonic degrees of freedom are expected to be
accurate.

###### Keywords:

two-photon exchange positrons single-spin asymmetries polarized target

††journal: Eur. Phys. J. A

##  1 Introduction

Hadronic box diagrams in elastic electron scattering are difficult to
calculate without significant model dependence. Unfortunately, they are also
produce significant radiative corrections in a number of measurements, for
example, $\gamma Z$-exchange in parity-violating electron scattering and
$\gamma W^{\pm}$-exchange in measurements of beta-decay widths. Two-photon
exchange (TPE) in elastic electron-proton scattering is hypothesized to be
responsible for the discrepancy between unpolarized and polarized extractions
of the proton’s form factor ratio Guichon:2003qm ; Blunden:2003sp . All of
these applications require a better understanding of box-diagram processes,
and new experimental constraints are needed to help improve theoretical
calculations. There are several experimental observables that are directly
sensitive to box-diagram contributions, and because they provide orthogonal
constraints, it is advantageous to pursue a variety.

One such observable is a target-normal single-spin asymmetry (SSA), denoted
here by $A_{n}$. This asymmetry is measured by scattering an unpolarized
electron (or positron) beam on a target polarized in a direction perpendicular
to the scattering plane, and comparing cross sections for “up” and “down”
target polarizations. In the limit of one-photon exchange, single-spin
asymmetries in elastic scattering are forbidden, so $A_{n}$ is a direct
measure of multi-photon exchange.

Following the formalism of Ref. Carlson:2007sp , $A_{n}$ for a proton target
can be related to the proton’s higher-order form factors,
$\delta\tilde{G}_{E}$, $\delta\tilde{G}_{M}$, and $\delta\tilde{F}_{3}$, by

$A_{n}=\frac{\sqrt{2\varepsilon(1+\varepsilon)}}{\sqrt{\tau}\left(G_{M}^{2}+\frac{\varepsilon}{\tau}G_{E}^{2}\right)}\times\Bigg{[}-G_{M}\textrm{Im}\left(\tilde{G}_{E}+\frac{\nu}{m_{p}^{2}}\tilde{F}_{3}\right)\\\
+G_{E}\textrm{Im}\left(\tilde{G}_{M}+\frac{2\varepsilon\nu}{m_{p}^{2}(1+\varepsilon)}\tilde{F}_{3}\right)\Bigg{]}+\mathcal{O}(\alpha^{4}),$
(1)

$G_{E}$ and $G_{M}$ are the proton’s standard electric and magnetic form
factors, $m_{p}$ is the mass of the proton, $\tau$ is the dimensionless
quantity $Q^{2}/4M^{2}$, where $Q^{2}$ is the magnitude of the squared
invariant 4-momentum transfer, $\varepsilon$ is the virtual photon
polarization parameter, defined by $\varepsilon^{-1}\equiv
1+2(1+\tau)\tan^{2}\frac{\theta}{2}$, where $\theta$ is the lepton scattering
angle in the rest frame of the target, and $\nu$ is the Lorentz-invariant
parameter formed by contracting the sum of the initial and final lepton
4-momenta with the sum of the initial and final proton 4-momenta, i.e.
$\nu\equiv(p_{e}+p_{e^{\prime}})_{\mu}(p_{p}+p_{p^{\prime}})^{\mu}$. Eq. 1
shows that $A_{n}$ is sensitive to the imaginary parts of the higher-order
form factors, meaning that it provides a completely independent constraint
from measurements of the unpolarized positron-proton/electron-proton cross
section ratio or of polarization transfer, which probe the real parts. As with
other observables of TPE, the sign of the effect would be reversed in positron
scattering.

In addition to two-photon exchange, any process that violates time-reversal
symmetry (i.e., T-violating) will also lead to a non-zero $A_{n}$
Carlson:2007sp . T-violating effects would be symmetric for both electrons and
positrons Christ:1966 . By measuring with both beam species, TPE and
T-violation can be distinguished. It is possible for a T-violating
electromagnetic current to be generated either by particles with spin 1 or
greater Glaser:1957  (e.g., the deuteron Schildknecht:1966zz ), or through
inelastic scattering Christ:1966 . In elastic electron-nucleon scattering,
T-violation could only arise from a non-electromagnetic scattering process
Bernstein:1965hj . The amount of T-violation within the Standard Model would
produce an insignificant $A_{n}$, so a detection would immediately indicate a
beyond-Standard Model effect.

There are several different theoretical approaches for calculating hard two-
photon exchange, but they mostly fall into two classes: hadronic (for example,
Tomalak:2014sva ; Blunden:2017nby ; Ahmed:2020uso  and others) and partonic
(for example, Chen:2004tw ; Afanasev:2005mp ; Kivel:2009eg  and others). The
hadronic calculations are expected to be most valid at lower momentum
transfer, i.e., $Q^{2}<3$ GeV${}^{2}/c^{2}$, while the partonic calculations
are applicable for very high momentum transfer, i.e., $Q^{2}>5$
GeV${}^{2}/c^{2}$. While validation of these calculations is needed at all
kinematics, measurements in the zone from $3\lessapprox Q^{2}\lessapprox 5$
GeV${}^{2}/c^{2}$, in between the regions of validity for hadronic and
partonic calculations, would be especially useful.

## Previous Measurements

$A_{n}$ has been measured previously in only a few different reactions, and no
measurements have been made in elastic electron-proton scattering with enough
precision to resolve a non-zero effect. SLAC published a measurement in 1970
on polarized protons in a butanol target at very forward scattering angles
($\approx 3^{\circ}$), but could not exclude a zero asymmetry Powell:1970qt .
Earlier, less precise results from Frascati, Orsay, and Stanford measured
recoil polarization from an unpolarized target, which is an equivalent
observable DiGiorgio:1965 ; Bizot:1968 ; Prepost:1968bow . A number of
experiments have looked for T-violation in inelastic scattering Chen:1968mm ;
Appel:1970ni ; Rock:1970sj ; Airapetian:2009ab ; Katich:2013atq . Two recent
measurements have been made at Jefferson Lab of quasi-elastic scattering from
polarized 3He Zhang:2015kna ; Long:2019iig . Precision measurements of elastic
scattering on protons have yet to be made with electrons or positrons. Ref.
Zhang:2015kna  measured an asymmetry of a few parts per thousand in 3He, which
corresponds to an asymmetry of a few percent from polarized neutrons. It would
be reasonable to expect an asymmetry of similar size from polarized protons.

Figure 1: Layout of the proposed measurement in Hall A. BigBite and SBS will
cover the same scattering angle allowing a simultaneous left/right asymmetry
measurement. Due to the bending of charged particles in the holding field,
they will sit at slightly different angles in the Hall coordinate system. The
holding field will also require that the incoming beam be steered on to the
target slightly off axis using a magnetic chicane.

The measurement of Ref. Zhang:2015kna  used the two high-resolution
spectrometers in Jefferson Lab’s Hall A to simultaneously measure at 17∘ both
left and right of the beam direction. The target was polarized in either the
up or the down direction. By using two spectrometers, the target asymmetry was
measured as a left/right asymmetry. The target polarization was additionally
flipped to reduce systematic uncertainty. This left-right approach has the
advantage of being a simultaneous measurement; both the left and right arms
experience the same time-varying beam conditions. As a result, the systematic
uncertainty on the raw asymmetry was impressively small, ranging from
0.014–0.026%. The measurement was statistics limited. For technical reasons,
this level of accuracy would be difficult to match with a polarized proton
target, but, nevertheless, the Hall A measurement illustrates how this
asymmetry is a systematically clean observable.

Table 1:  Proposed Measurement Plan |  |  |  |  | Big-Bite | SBS |  |
---|---|---|---|---|---|---|---|---
Beam [GeV] |  $Q^{2}$ [GeV2] | $\varepsilon$ |  $\theta_{e}$ [∘] |  $p_{e}$ [GeV$/c$] |  $\theta_{\text{lab}}$ [∘] |  $R$ [m] |  $\theta_{\text{lab}}$ [∘] |  $R$ [m] |  $\Omega$ [msr] | Days
6.6 | 4.0 | 0.8696 | 21.2 | 4.34 | 23.2 | 2.1 | 21.2 | 3.0 | 26 | 9
6.6 | 3.0 | 0.9207 | 17.3 | 5.00 | 17.9 | 2.8 | 16.7 | 3.6 | 18 | 3
4.4 | 3.0 | 0.8065 | 28.6 | 2.80 | 30.3 | 1.6 | 26.9 | 2.5 | 38 | 4
4.4 | 2.0 | 0.9004 | 21.3 | 3.33 | 22.3 | 2.2 | 20.3 | 3.0 | 26 | 1
2.2 | 2.0 | 0.5600 | 53.2 | 1.13 | 58.9 | 1.1 | 47.5 | 1.6 | 70 | 2
2.2 | 1.0 | 0.8419 | 30.3 | 1.67 | 32.3 | 1.5 | 28.3 | 2.4 | 40 | 0.5
2.2 | 0.5 | 0.9353 | 19.7 | 1.93 | 20.7 | 2.4 | 18.7 | 3.2 | 23 | 0.5
Total |  |  |  |  |  |  |  |  |  | 20

The measurement of Ref. Long:2019iig  also looked at quasi-elastic scattering
from 3He in Hall A. However, instead of using simultaneous left and right
electron arms, the experiment made a coincidence measurement of the
3He$(e,e^{\prime}n)$ reaction. While this is sensitive to TPE in principle,
the requirement of a detected neutron introduces new asymmetry-generating
effects, such as final state interactions, meson exchange currents. In fact,
Ref. Long:2019iig  drew conclusions about the magnitude of non-plane wave
reaction effects, rather than TPE or T-violation. It is relevant, however,
that Ref. Long:2019iig  achieved an approximately $9\%$ _relative_ systematic
uncertainty, i.e., equivalent to an absolute uncertainty of 0.09% on 1%
asymmetry. The uncertainty was dominated by the time-variance of the target
polarization. We can expect a similar level of systematic uncertainty for a
measurements made by a single spectrometer.

## Proposed Measurement

The addition of a positron source at Jefferson Lab’s CEBAF accelerator would
make possible a first ever measurement of the single-spin asymmetry $A_{n}$ in
positron scattering. We propose here a concept for a measurement of $A_{n}$ in
both electron and positron elastic scattering from polarized protons, which
could distinguish TPE and T-violation effects. This measurement would also
provide the first $A_{n}$ data on protons for $Q^{2}>1$ GeV${}^{2}/c^{2}$,
reaching up to 4 GeV${}^{2}/c^{2}$, in between the regions where hadronic
calculations and partonic calculations are expected to be most accurate. This
is a more ambitious proposal than that first suggested in Ref. Schmidt:2017gky
. This is made possible by employing Hall A’s new Super BigBite Spectrometer
(SBS), paired with the upgraded BigBite Spectrometer (BB), providing larger
acceptance than Hall A’s HRSs. Both SBS and BB are non-focusing magnetic
spectrometers, each with a single dipole magnet bending charged particles
upwards from horizontal plane. At their most favorable positions, SBS can have
an acceptance over 70 msr SBS:CDR  and BB can have an acceptance over 90 msr
transversity , compared to the approximately 6 msr acceptance of each HRS
Alcorn:2004sb . The use of an even larger acceptance spectrometer—Hall B’s
CLAS12—is unfeasible due to the need for a strong transverse magnetic field to
hold the target polarization. Such a holding field would not be possible
inside CLAS12’s longitudinal solenoid magnet Burkert:2020akg . A top-view
illustration of the experiment layout is shown in Fig. 1.

The unpolarized beam would be directed on a vertically polarized target.
Polarized proton targets present a number of technical hurdles, and these are
compounded when the polarization is transverse. Maintaining polarization
requires a holding field of several tesla in strength; this is much greater
than for polarized helium-3 targets, where 30 G = 3 mT is sufficient. Several
different polarized proton targets have been used at Jefferson Lab
Averett:1999nz ; Keith:2003ca ; Keith:2012ad ; Pierce:2013pua , with the
target from the $g_{2}^{p}$ and $G_{E}^{p}$ Experiments Pierce:2013pua
demonstrating the best performance under high-luminosity conditions. In this
target, protons are dynamically polarized in frozen beads of ammonia (NH3),
within a 2.5–5 T holding field, achieving approximately 70% average
polarization with a beam current of up to 100 nA. This corresponds to a
luminosity of roughly $10^{35}$ cm2 s-1, given a 3 cm long target, and a rough
estimate of a 60% packing fraction for the ammonia beads. For this proposal,
we assume that a similar target could be produced, which could achieve 66%
polarization at a luminosity of $10^{35}$ cm2 s-1.

The $g_{2}^{p}$/$G_{E}^{p}$ target was never used with vertical polarization,
though there are no technological reasons preventing the target magnet from
being oriented vertically. However, the vertical holding field has a major
impact on the steering of the beam, and on the trajectories of charged
particles emerging from the target. Assuming a total field integral of 1.5 Tm,
a 2.2 GeV beam would be deflected horizontally by $\approx 11.4^{\circ}$ when
passing through the target. To correct for this deflection, it would be
necessary to use a magnetic chicane to steer the beam onto the target at an
angle, so that the target holding field itself could bend the beam back toward
the beam dump. Fig. 1 shows how this chicane would be oriented in the proposed
experiment. One consequence of this design is that the beam will cross the
target at a slight angle relative to the hall coordinate system, and this,
along with the deflection of outgoing particles must be accounted for when
determining the placement of the spectrometers.

Figure 2: An overhead view of the target region showing the curvature of
charged particles caused by the holding field. In the proposed measurement,
the magnet will be oriented to direct the “sheet of flame” to the left side of
the beam line, covered by BB. The BB dipole magnet will need to deflect low-
momentum sheet-of-flame particles away from the detectors.

Transverse polarized targets also have the adverse property of producing a
large background of low-energy electrons that is spread in a so-called “sheet
of flame.” Beam particles that lose energy in the target, e.g., through
bremsstrahlung, are deflected strongly in the holding field. For the proposed
vertical holding magnet, the sheet of flame will be bent in the horizontal
plane, into the aperture of one of the spectrometers, illustrated in Fig. 2.
Fortunately, the sheet-of-flame particles will be much lower momentum than the
particles of interest, and the spectrometer dipoles will deflect the sheet out
of plane by large angles. Care must be taken to ensure that the sheet of flame
misses the active detectors; otherwise they would be swamped with ionization.

In this experiment, leptons would be scattered elastically from a polarized
proton target, and detected in either the SBS or BB spectrometers to the left
and right of the beam line, respectively (see Fig. 1). The two spectrometers
would cover identical scattering angles so that the measurement of $A_{n}$
could be made from the left-right asymmetry alone, reducing systematic effects
from time-varying running conditions. Because the target holding field will
deflect the beam and outgoing particles, the placement of the spectrometers in
the hall will be at two slightly different angles. We propose orienting the
chicane and holding field so that the beam and the sheet of flame are bent to
the left, i.e., toward BB and away from SBS, for both electron and positron
running modes; to switch between electrons and positrons, the chicane and
target holding field polarities will also switch. Conveniently, the target
polarization direction can be flipped independently of the holding field
polarity. Frequent target polarization flips can be used to further reduce
systematics coming from the target and from acceptance differences between the
two spectrometers.

Figure 3:  Kinematic map of the proposed measurement (diamonds) in
$\varepsilon$,$Q^{2}$ space. The kinematics of the SLAC measurements from 1970
(Powell et al., open circles) Powell:1970qt  are shown for comparison. The
Hall A results for the neutron using a polarized 3He target (Zhang et al.,
open triangles) are from Ref. Zhang:2015kna . The partonic calculation of Ref.
Afanasev:2005mp  is claimed to be valid when Mandelstam variables $|t|>2$
GeV${}^{2}/c^{2}$ and $|u|>m_{p}^{2}$, the region in the upper right area of
the map. This region will be probed by three of the proposed kinematic points.

SBS and BB would require detector packages optimized for electron detection.
Of the currently planned measurements in the Hall A SBS program, the GMn GMn ,
GEn GEn , GEn-RP GEnRP , and Transversity transversity  experiments plan to
use Big Bite as the electron arm. Though the specifics of their detector
packages vary slightly, they all employ GEM planes for tracking, a gas
Cherenkov detector for electron ID, a lead glass calorimeter, and a
scintillator hodoscope positioned between the pre-shower and shower layers of
the calorimeter. We envision using a similar detector configuration here.

None of the currently planned measurements use SBS for electron detection. To
minimize systematics, it would be preferable to make the SBS detector package
as similar as possible to that of BB. While suitable GEM tracking stations
have been built for SBS, a new gas Cherenkov detector would need to be
constructed, as well as a calorimeter. It is possible that the BigCal from the
GEp Experiment GEP5  could suffice as the calorimeter.

The proposed measurement plan is shown in Table 1, and consists of a total of
20 running days, with the assumption of 50% running efficiency built in.
Overhead for changing between electron and positron running modes, changing
the number of passes in CEBAF, and repositioning the spectrometers is not
considered. In this plan, the SBS and BB spectrometers will simultaneously
cover $Q^{2}$ values of 0.5, 1, 2, 3, and 4 GeV${}^{2}/c^{2}$. The use of
three different beam energies, standard first-pass, second-pass, and third-
pass CEBAF energies, will allow measurements of $Q^{2}=2$ and 3
GeV${}^{2}/c^{2}$ to be made at multiple values of $\varepsilon$. The exact
energy per pass is not critical, and this plan could be adjusted slightly
depending on how CEBAF is performing when this experiment takes place. The
number of days listed includes both electron and positron running, i.e., 4
days means 2 days with an electron beam and 2 days with a positron beam. To
minimize any time-varying systematics, it would be desirable to be able to
switch between electrons and positrons frequently, though this would likely
need to be balanced against the time needed to flip the polarity of the
accelerator. Similarly, the target polarization direction should be flipped
frequently. Since the target would likely need to be re-annealed every few
hours to restore polarization, this would be the sensible timescale for target
spin flips.

The solid angle coverages of SBS and BB depend greatly on how closely the
spectrometer magnets can be positioned to the target. At forward angles, BB
must be moved further away from the target to avoid hitting the downstream
beam pipe. While the SBS dipole magnet has a cut-out to allow closer placement
to the beam pipe, it must still be positioned further away to avoid hitting
BB. This limits of the angular acceptance of both spectrometers. Table 1
includes columns for our estimates of the nominal distance to the target for
the BB and SBS magnets and the solid angle, $\Omega$, common to both
spectrometers based on where they might be placed to avoid hitting each other
or the beam pipe. A rigorous optimization of these spectrometer positions
would need to be conducted based on the equipment and obstructions in Hall A
at the time the experiment is to be run.

As mentioned above, the positioning of the BB detectors must take the sheet of
flame into account. The BB dipole can produce a field integral of $\approx 1$
Tm, enough to bend the sheet of flame more than 25∘ out of plane at the most
unfavorable forward kinematic point (6.6 GeV beam, $17^{\circ}$). As long as
the sheet misses the BB GEMs, it will not pose a problem to the experiment.
This may require repositioning the GEMs, or reducing their area, in turn
decreasing the BB acceptance. However, since in the proposed experiment SBS
defines the common acceptance, a slight reduction in BB’s acceptance may not
have any impact on the measurement. In our rate estimates, we assume that BB
can safely match the SBS acceptance while avoiding the sheet-of-flame
background, but acknowledge that a detailed simulation will be vital to ensure
this background can be deflected, blocked, or avoided so that the experiment
can be successful.

A kinematic map of the proposed measurement plan in $\varepsilon,Q^{2}$ space
is shown in Fig. 3.

## Statistical Uncertainty

Figure 4:  Anticipated statistical precision of the proposed measurement for
the three different beam energies. The projected statistical uncertainties are
approximately 0.75% or better for all kinematic points. The Hall A results for
the neutron using a polarized 3He target are from Ref. Zhang:2015kna  (open
triangles). Slightly different beam energies were used, and we place the
points on the panel with the closest beam energy. The curves show the
predictions of the Gaussian GPD model of Afanasev et al. Afanasev:2005mp  for
both an electron (red) and positron (blue) beam. The calculation is claimed to
be valid for $-t>2$ GeV2 and $-u>m_{p}^{2}$. When those conditions are not
met, we show the calculation as a dashed line rather than a solid line.

The statistical precision for the proposed measurement was estimated as:

$\delta
A_{n}=\frac{1}{PD\sqrt{2\frac{d\sigma}{d\Omega}\Omega\mathcal{L}T\epsilon}},$
(2)

where $P$ is the target polarization (assumed 66.7%), $D$ is the target
dilution factor (assumed 50% for NH3 based on the experience of Ref.
Jones:2006kf ), $\frac{d\sigma}{d\Omega}$ is the elastic scattering cross
section, $\Omega$ is the single spectrometer acceptance, $\mathcal{L}$ is the
luminosity ($10^{35}$ cm-2s-1), $T$ is the run time per target polarization
setting, and $\epsilon$ is the running efficiency (assumed 50%). For these
estimates, SBS and BB were assumed to cover the common angular acceptance
listed in Table 1. The proposed measurement would have approximately 0.75%
statistical precision or smaller on $A_{n}$. The results of Ref. Zhang:2015kna
indicate that, at least for the neutron, $A_{n}$ is a percent-level asymmetry.
Fig. 4 shows the size of the statistical error bars for the three different
beam energies in comparison to theoretical predictions as well as the neutron
results from Ref. Zhang:2015kna . The GPD-based partonic calculation of Ref.
Afanasev:2005mp  predicts a sub-percent asymmetry in the proposed kinematics,
which would be difficult to discern for the highest $Q^{2}$ points. However,
we note that the validity of partonic calculations in the kinematics is an
open question, that $A_{n}$ for the neutron was measured to be several
percent, and that the proposed measurement would have higher precision and
much greater kinematic reach.

## Systematics

The proposed measurement would have several sources of systematic uncertainty
to overcome. The dominant source would be the time-variation of the target
polarization. The target polarization is one of the multiplicative factors
that is needed to extract $A_{n}$ from the measured count-rate asymmetry, and
uncertainty in this polarization goes directly into uncertainty in $A_{n}$.
For the target described in Ref. Pierce:2013pua , polarization was monitored
continuously through NMR, and a similar procedure would be vital for the
proposed measurement. The stability of the NMR system would be critical in
order to ascertain the charge-weighted target polarization for every
measurement setting. Ref. Pierce:2013pua  observed that in between annealings,
the target polarization degraded significantly with accumulated dose: though
peak polarizations of 90% were achieved, this would degrade steadily, and the
average polarization obtained was only 70%. The $g_{2}^{p}$-Experiment claimed
an uncertainty of 2–4.5% on the target polarization Zielinski:2017gwp . For
the proposed measurement, this would translate to a _relative_ uncertainty on
the asymmetry, i.e. a 5% target polarization uncertainty would produce an
uncertainty of 0.05% on a 1% asymmetry. This is comparable to what was
achieved with the polarized 3He target in Ref. Long:2019iig . Note that the
time-dependence would be somewhat mitigated, since we propose a _simultaneous_
left/right measurement.

Another systematic uncertainty would come from the knowledge of the dilution
factor of the NH3 target. Elastic scattering events from polarized protons in
the target would be accompanied by a quasi-elastic background from nucleons in
nitrogen nuclei. This background can be subtracted since the signal is
strongly peaked at $W^{2}=m_{p}^{2}$, where $W$ invariant mass of the final
state hadronic system, while the background is broad and smoothly varying due
to Fermi motion. However, there would be systematic uncertainty associated
with this subtraction, and it would be common to both positron and electron
measurements, i.e., it would have no effect on T-violation, but would bias the
measurement of TPE. The potential bias of this dilution effect can be reduced
by enriching the target material with 15N, which is only one proton-hole shy
of doubly-magic 16O. By contrast 14N has both an unpaired proton and an
unpaired neutron. A 15N-enriched target and a subtraction of the quasi-elastic
background was employed in a measurement in Hall C Jones:2006kf  that achieved
a 1% uncertainty on the dilution factor. While the success of this approach
depends on the momentum resolution of the spectrometer used, one could expect
similar uncertainty for the experiment proposed here.

Other systematics, including differences in spectrometer efficiencies,
spectrometer alignment, and beam current monitoring, would be mitigated by
making a simultaneous left/right asymmetry, and by flipping the target
polarization.

## Summary

The target-normal single spin asymmetry $A_{n}$ is sensitive to the imaginary
part of the two-photon exchange amplitude, and has never been measured on a
polarized proton target. By using both a positron and an electron beam, not
only can a measurement of $A_{n}$ provide new independent constraints on TPE,
but some experimental systematics can be reduced, and the effects of TPE can
be distinguished from those of T-violation. Using the new SBS in Hall A in
conjunction with the refurbished BB spectrometer, a 20-day measurement would
allow a comprehensive scan of $Q^{2}$ at modest $\varepsilon$, and push up to
$Q^{2}=4$ GeV${}^{2}/c^{2}$, in-between the region of validity for hadronic
calculations and those using GPDs or other partonic approaches. These data
would constrain models of TPE, and assist in the effort to understand and
model electroweak hadronic box processes.

###### Acknowledgements.

We thank J. D. Maxwell of the Jefferson Lab target group for helpful
discussions about working with transversely polarized targets.

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