Cosmic string solution to general relativity

I’m having a difficulty in finalizing a resolution of the Einstein equation for a static cosmic string. I start with the following metric ansatz, for a static straight string oriented along the $z$ direction. Using cylindrical coordinates:
$$tag{1}
ds^2 = mathcal{P}^2(r) , dt^2 – dr^2 – mathcal{Q}^2(r), dvartheta^2 – dz^2.
$$
Plugging this metric into the Einstein equation gives the following equations, after a few pages of calculations (I’m using the $eta = (1, -1, -1, -1)$ convention and $G_{ab} = -, kappa T_{ab}$. Indices are associated to the "flat" local inertial frames):
begin{align}
G_{00} &= frac{mathcal{Q}^{prime prime}}{mathcal{Q}} = -, kappa T_{00} = -, kappa rho, tag{2} [1ex]
G_{11} &= -, frac{mathcal{P}^{prime} mathcal{Q}^{prime}}{mathcal{P} mathcal{Q}} = -, kappa T_{11} = 0, tag{3} [1ex]
G_{22} &= -, frac{mathcal{P}^{prime prime}}{mathcal{P}} = -, kappa T_{22} = 0, tag{4} [1ex]
G_{33} &= -, frac{mathcal{P}^{prime prime}}{mathcal{P}} – frac{mathcal{P}^{prime} mathcal{Q}^{prime}}{mathcal{P} mathcal{Q}} – frac{mathcal{Q}^{prime prime}}{mathcal{Q}} = -, kappa T_{33} = +, kappa tau. tag{5}
end{align}
Here, $rho > 0$ and $sigma = -, tau < 0$ are the string’s energy density and tension, respectively. Now, (2) implies $mathcal{Q}^{prime} ne 0$, so (3) and (4) give $mathcal{P}^{prime} = 0$ and $mathcal{P}^{prime prime} = 0$. Then (5) implies $tau = rho$, which is good for a relativistic string. Then, the trouble begins when I attempt to solve (2). I was expecting a thin string, so a Dirac’s delta for the density $rho$. But (2) suggest another solution and I wasn’t expecting this. Assuming a constant density $rho = rho_0$ for $r < R$ (the string’s radius) and $rho = 0$ for $r > R$, I get
$$tag{6}
mathcal{Q}^{prime prime} + kappa rho_0 , mathcal{Q} = 0.
$$
Then this is an harmonic equation which have the general solution (I write $lambda equiv sqrt{kappa rho_0}$ to simplify things)
$$tag{7}
mathcal{Q}(r) = alpha sin(lambda r) + beta cos(lambda r).
$$
For the exterior metric: $rho = 0$, (2) reduces to $mathcal{Q}^{prime prime} = 0$, which have solution $mathcal{Q}(r) = a r + b$. Matching the solution at $r = R$ gives
$$tag{8}
mathcal{Q}(R) = alpha sin(lambda R) + beta cos{lambda R} = a R + b.
$$
So my problem is to find the constants $alpha$, $beta$, $a$ and $b$ (four constants!), from regularity and junction conditions.

---

EDIT: From A.V.S and Michael answers below, I should apply the regularity at $r = 0$:
begin{align}
mathcal{Q}_{text{int}}(0) &= 0, tag{9} [1ex]
mathcal{Q}_{text{int}}^{prime}(0) &= 1. tag{10}
end{align}
This gives $alpha = lambda ^{-1}$ and $beta = 0$, so
$$tag{11}
mathcal{Q}_{text{int}}(r) = frac{1}{lambda} , sin(lambda r).
$$
This is fine. But then I imposes the junction at the string’s surface. Apparently, there’s a subtlety that I don’t understand here. According to some obscure papers I’ve found, the radial coordinate $r$ isn’t the same on the interior side and on the exterior of the string, so $R_{text{int}} ne R_{text{ext}}$:
begin{align}
mathcal{Q}_{text{int}}(R_{text{int}}) &= mathcal{Q}_{text{ext}}(R_{text{ext}}), tag{12} [1ex]
mathcal{Q}_{text{int}}^{prime}(R_{text{int}}) &= mathcal{Q}_{text{ext}}^{prime}(R_{text{ext}}). tag{13}
end{align}
This gives the following junction conditions:
begin{align}
frac{1}{lambda} , sin(lambda R_{text{int}}) &= a R_{text{ext}} + b, tag{13} [1ex]
cos(lambda R_{text{int}}) &= a. tag{14}
end{align}
I can’t solve this system of equations without an extra constraint (??). I get the relation of some paper if I impose $b = 0$ so
$$tag{15}
R_{text{ext}} = frac{1}{lambda} , tan(lambda R_{text{int}}).
$$
Defining the energy per unit length $mu = rho_0 A_{text{int}}$, I get $a = 1 – 4 G mu$, which relates to the angle deficit.

But how can I justify that $b = 0$ and that the radial coordinate isn’t the same on both side of the string’s surface? Why can’t I just use $R_{text{int}} = R_{text{ext}} = R$, and then find $a$ and $b ne 0$ ?

An "obscure" paper that shows some details (with pesky weird notation!), without explaining the different radial coordinate and why $b$ should be 0. See expressions (6), (7), (8), on page 2:

https://arxiv.org/abs/hep-th/0107026

See also pages 3 and 4 of this paper:

https://arxiv.org/abs/gr-qc/9508055