= Fixed-target antiproton production threshold
{title2=$K_{\mathrm{thr}}=6m_pc^2$}
The <fixed-target production threshold> concerns the incident accelerator beam, whereas <antiproton mass from momentum and velocity> concerns an outgoing secondary beam. To see the distinction, consider a free <proton> hitting a stationary free <proton>, producing
$$
p+p\longrightarrow p+p+p+\bar p.
$$
The final <electric charge> is $+2e$ and final <baryon number> is $3-1=2$, as initially. Approximate all four final <rest masses> by the same <proton> <rest mass> $m$. With <metric signature> $(+---)$, define the squared <centre-of-mass energy> by $W^2=c^2(P_1+P_2)^2$. The incident <proton> has total <energy> $E=mc^2+K$, and the target <four-momentum> is $(mc,\mathbf0)$. <Four-momentum conservation> and the <mass shell> relation give
$$
W^2=2m^2c^4+2mc^2E=4m^2c^4+2mc^2K.
$$
At threshold the four final particles have zero relative <momentum> in their <centre-of-momentum frame>, so $W_{\mathrm{thr}}=4mc^2$. Hence
$$
16m^2c^4=4m^2c^4+2mc^2K_{\mathrm{thr}},\qquad K_{\mathrm{thr}}=6mc^2\simeq5.63\,\mathrm{GeV}.
$$
Only $2mc^2$ of new <rest energy> is created. The larger incident <kinetic energy> is required because the final system must also carry the initial laboratory <momentum>. The incident total <energy> at threshold is $7mc^2$; neither this nor $K_{\mathrm{thr}}$ is the selected secondary <momentum>.
The 1955 authors explicitly qualified the stationary-nucleon threshold: target <nucleon> motion inside an <atomic nucleus> can lower the required incident <energy>.https://escholarship.org/content/qt46p0z8w7/qt46p0z8w7.pdf[UCRL-3172, p. 3]{ref}
The kinematic reason is visible for two on-shell incoming <nucleons>:
$$
W^2=2m^2c^4+2\left(E_bE_t-c^2\mathbf p_b\cdot\mathbf p_t\right).
$$
A target <momentum> directed against the beam increases $W^2$. Applying this illustration to a bound <nucleon> also requires <binding energy> and residual-<nucleus> recoil; the free stationary-target $6mc^2$ result is therefore not a universal sharp threshold for a <copper> target. A low observed secondary <momentum> is entirely compatible with a much larger accelerator <kinetic energy>.
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