Consumption Externalities in partial equilibrium analysis

I’ve been working on a standard partial equilibrium with externalities problem and have been having some difficulty with these (relatively) basic concepts. My issue is primarly with solving for Pareto optimality and comparing it to a competitive outcome.

$$u^A(y_A, x_A, x_B)=y_A+ln x_A+frac{x_B^2}{2}$$
$$u^B(y_B,x_A,x_B)=y_B+ln x_B-x_A $$

Solving for CE is fairly straight forward using the price normalization method of $(p_x,p_y)=(p,1)$
$$max_{x_A,y_A} mathcal{L}=y_A+ln x_A+frac{x_B^2}{2}+lambda_A(m-px_A-y_A)$$
$$max_{x_B,y_B} mathcal{L}=y_B+ln x_B-x_A+lambda_B(m-px_B-y_B)$$
The allocation of goods in CE amounts of $x_A, x_B$ in this context is:
$${x_A^*,x_B^*}=left{frac{1}{p},frac{1}{p}right}$$

Further for the pareto optimal allocation we solve:

$$max_{x_A,x_B,y_A,y_B} u^A(y_A, x_A, x_B)+u^B(y_B,x_A,x_B)$$
$$max_{x_A,x_B,y_A,y_B} {y_A+ln x_A+frac{x_B^2}{2}+y_B+ln x_B-x_A}$$
Solving for $x_A$ is fairly straight forward via the FOC, we have:
$$x_A^{**}=1$$
for dealing with $x_B$ there is difficulty as:
$$x_B+frac{1}{x_B}=0$$
where solving gives us
$$x_B=sqrt{-1}$$
which makes no sense.

Im definately doing something wrong but cant figure it out.