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## Appendix D Formation Time and Brick Estimates 

## D.1 γ$_{c}$ for Bethe-Heitler to LPM Regime 

## D.1.1 Formation Physics 

By the Heisenberg Uncertainty Principle, the formation time, τ ′ $_{f}$, for a gluon (or photon) of energy k 0 ′ separating from its parent quark (or gluon)—without loss of generality taken as a quark from now on—in the rest frame of the quark, is 

<img src='content_image/309012.jpg'>

where the primed coordinates are in the rest frame of the quark and unprimed coordinates will be in the lab frame. To find τ$_{f}$ in the lab frame, one must first boost the gluon energy in the lab frame into the rest frame of the quark, then boost back to the lab frame. Take a quark with 4-momentum p = ( p $^{0}$, p $^{z}$, p ) and gluon of momentum k = ( k $^{0}$, k $^{z}$, k ). The velocity of the quark is β = p $^{z}$/p $^{0}$. 

