How are $R$ and $P$ branches defined in rovibrational transition?

In rovibrational transitions we have following selection rules

$$\Delta v = \pm 1$$

$$\Delta J = \pm 1$$

where $v$ is the vibrational quantum number and $J$ is rotational quantum number.

• All $\Delta J = +1$ transitions give rise to R-branch lines.
• All $\Delta J = -1$ transitions give rise to P-branch lines.

Literature also notes that R-branch lines are higher in frequency (and thus energy) than P-branch lines.

This is how it's defined but notice how I emphasized transition above? What confuses me is this: it is not specified which transition is meant, is it absorption or emission (of a photon)? Because it seems to me that it makes a difference.

Let's take one example transition, say between levels (v=0, J=0) to the (v=1, J=1) then we have:

• in case of absorption J goes from 0 to 1 so: $\Delta J = +1$
• in case of emission J goes from 1 to 0 so: $\Delta J = -1$

So it's the same transition but different $\Delta J$ based on whether it's emission or absorption. So to which branch does this transition belongs then?

You're quite correct that the simple definition for the $R$ branch $\Delta J = +1$ is ambiguous unless you also specify that $\Delta v > 0$. The $R$ branch is the higher energy branch so it's $\Delta J = +1, \Delta v = +1$ in absorption and $\Delta J = -1, \Delta v = -1$ in emission.