Breakdown of collinear factorisation in the photoproduction of a $\pi^0 \gamma$ pair with large invariant mass

We identify a $ 2 \to 3 $ exclusive process, where collinear factorisation is broken, namely the exclusive photoproduction of a $ \pi ^{0}\gamma $ pair with large invariant mass. This occurs because the process suffers from gluon exchanges trapped in the Glauber region. Using an explicit example, we show that the Glauber gluon, which is exchanged between a collinear spectator parton from the nucleon sector and a soft spectator parton from the outgoing pion, has both of its lightcone plus and minus components pinched. Therefore, it cannot be deformed to collinear/soft regions, as is often the case for processes that do factorise. We further confirm the leading power behaviour of the identified Glauber region, highlighting that this is the case although it relies on extracting a soft parton from the outgoing pion. We stress that the Glauber pinch for this process is of the leading power, due to the possibility of having two-gluon exchanges between the collinear nucleon sector and hard partonic scattering sub-process. In fact, the Glauber gluon that we identify is one of these two active gluons, and therefore, its effects are observed already at leading order. A direct consequence of our work is that collinear factorisation breaks in the same way for other $ 2 \to 3 $ exclusive processes, where two-gluon exchanges in the $ t $-channel are possible, like in the exclusive production of a photon pair from $ \pi ^{0} N $ collisions. However, we highlight that in cases where such two-gluon exchanges do not exist, like in the exclusive $ \pi ^{\pm}\gamma $ photoproduction, the Glauber exchanges that we discuss here do not occur, and hence they do not suffer from factorisation breaking effects.