$T$-duality in effective gauge theories of a $D(p+1)$-brane

I am considering a $D(p+1)$-brane in a space $mathbb R^{1,p}times S_R^1$ where $S_R^1$ is the circle of radius $R$. I am assuming low energies $ERll1$, so that only the massless spectrum of the open string is relevant. I know that from the spectrum, which is a set of vector fields ${A_mu}_{mu=0}^{p+1}$ and a set of scalar fields ${X^i}_{i=p+2}^{25}$ (the dimension of the space is D=26), I can build an effective gauge theory given by the action

$$S_{D(p+1)} = -2pi R T_{p+1} int d^{p+1}sigma left[frac{1}{4}(alpha’ F_{alphabeta})(alpha’ F_{alphabeta}) + frac{1}{2}(alpha’ partial_{alpha}A_z)(alpha’ partial^{alpha}A_z) +frac{1}{2}partial_alpha X^i partial^alpha X^j delta_{ij}right]$$

where $F_{alphabeta} = partial_alpha A_beta – partial_beta A_alpha$, $alpha.beta=0,…,p$ and $z$ is the compact direction. Obviously, all the fields in the action as 0 momentum in the compact direction, so $n=0$ (the momentum is discretised in that direction). The factor $2pi R$ comes from the integration of the compact direction, for this reason the integral is only over $p+1$ variables, the directions along the brane that are no compact.

I also know that after T-duality the $D(p+1)$-brane, which is wrapped around the compact direction, will become a $Dp$-brane which is not wrapped around the compact direction, since the Neumann and Dirichlet conditions are interchanged. Hence, after $T$-duality I expect to find an effective gauge theory with one less vector field and an extra scalar field, so it will have the action

$$S_{Dp} = -T_{p} int d^{p+1}sigma left[frac{1}{4}(alpha’ F_{alphabeta})(alpha’ F_{alphabeta}) +frac{1}{2}partial_alpha X^i partial^alpha X^j delta_{ij}right]$$

where in this action the $i= p+1,…,25$ because there is an extra scalar field. This action has not a factor $2pi R$ since the brane is not wrapped around the compact direction. Consequently, that direction is not along the brane and the fields does not depend on it, so that it is not integrated as in the previous action.

From the imposition that my theory is invariant under T-duality I am finding that

$$A_z rightarrow X^{p+1} = alpha’ A_z$$
$$T_{p+1} rightarrow T_p = 2pi R T_{p+1}$$

So I find a transformation of $A_z$ and $T_{p+1}$ under T-duality. I am quite sure of the transformation of $T_{p+1}$ since I found similar results in some papers, but not about $A_z$. I was told that the $alpha’$ were added to take into account the dimensions so that the kinetic term is dimensionless. However, I do not see why I should not be able to reabsorb the $alpha’$ in the field, but still I have doubt about this transformation.

On the other hand, I was also told that from dimensional analysis the constants $T_{p+1}$ should have the form

$$T_{p+1} = frac{tilde{T}}{(alpha’)^{(p+2)/2}}$$

where $tilde{T}$ is dimensionless. I understand the expression since the dimensions of $T_{p+1}$ must compensate the dimensions of $R$ and $d^{p+1}sigma$ to give a dimensionless action. Nonetheless, if the expression is valid for any $p$ and $tilde{T}$ do not change (as I was told) then form the relation between $T_p$ I get

$$alpha’^{1/2} = 2pi R$$

and that is what I do not understand. Why is the $alpha’$ related to the radius? $alpha’$ is related to the tension of the string and the radius is the compactification, they seem independent. My first guess was that for that value the dual space was the same as the "non-dual" and if our theory has to be invariant under T-duality, only that value is possible. However, the value for the equality of the two spaces is obviously $alpha’^{1/2} = R$ because then the dual radius is just $R’ = R = alpha’^{1/2}$. So now I am not sure about the equation with $tilde{T}$ and if it is correct, I am not sure about the interpretation of $alpha’^{1/2} = 2pi R$.