= Solution
Specify the sign convention for the <Lévy characteristic exponent> by
$$
\mathbb E e^{iuX_a}=e^{-a\Psi(u)}.
$$
Conditioning on $T_a$ and using the <Brownian first-passage Laplace transform> gives
$$
\mathbb E e^{iuW_{T_a}}=\mathbb E e^{-u^2T_a/2}
=e^{-a\sqrt{u^2}}=e^{-a|u|}.
$$
Therefore
$$
\boxed{\Psi(u)=|u|.}
$$
The absolute value is essential for negative $u$. The process is the standard symmetric <Cauchy process>; for $a>0$, $X_a$ has the <Cauchy distribution> with location zero and scale $a$. If the exponent convention instead uses $\mathbb E e^{iuX_a}=e^{a\psi(u)}$, the answer is $\psi(u)=-|u|$.
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