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- W1966255950 startingPage "2011" @default.
- W1966255950 abstract "We calculate nonperturbatively and for small $Q$ the cross section $ensuremath{sigma}(Q)$ for a scattering process to occur with loss of four-momentum $Q$ to unobserved photons, a quantity which may be measured by observing the net recoil to all other particles. The calculation proceeds by means of a threshold theorem which asserts that for small $Q$, $ensuremath{sigma}(Q)ensuremath{sim}{ensuremath{sigma}}_{0}P(Q)$, where ${ensuremath{sigma}}_{0}$ is independent of $Q$, and $P(Q)$ is the spectral of the coherent state $ensuremath{Psi}$ of bremsstrahlung photons defined by ${a}^{ensuremath{mu}}(k)ensuremath{Psi}=i{(2ensuremath{pi})}^{ensuremath{-}frac{3}{2}}ensuremath{Sigma}{a}^{}{u}_{a}^{ensuremath{mu}}{({u}_{a}ifmmodecdotelsetextperiodcenteredfi{}k)}^{ensuremath{-}1}ensuremath{Psi}$, where ${a}^{ensuremath{mu}}(k)$ is the annihilation operator for a photon of four-momentum $k$, and ${e}_{a}$ and ${u}_{a}$ are the charges and four-velocities of the scattered charged particles. Although $ensuremath{Psi}$ is not in the Fock space, the evaluation of $P(Q)=〈ensuremath{Psi},{ensuremath{delta}}^{4}(Qensuremath{-}{P}_{mathrm{op}})ensuremath{Psi}〉$, where ${P}_{mathrm{op}}$ is the operator of total electromagnetic four-momentum, is straightforward. The resulting function $P(Q)$ simplifies if $Q$ is near the light cone, where the bulk of the probability is in fact located, $P(Q)ensuremath{sim}ensuremath{theta}({Q}^{0})ensuremath{theta}({Q}^{2}){[ensuremath{Gamma}(1+B)]}^{ensuremath{-}1}B{(frac{{Q}^{2}}{2})}^{Bensuremath{-}1}{I}_{0}(Q)mathrm{exp}[F(Q)]$, where ${I}_{0}(Q)=ensuremath{-}{(2ensuremath{pi})}^{ensuremath{-}3}{[ensuremath{Sigma}{a}^{}{e}_{a}{u}_{a}{({u}_{a}ifmmodecdotelsetextperiodcenteredfi{}Q)}^{ensuremath{-}1}]}^{2}>0$, $B=ensuremath{int}dstackrel{^}{k} {I}_{0}({Q}^{0}=1, stackrel{ensuremath{rightarrow}}{Q}=stackrel{^}{k})$, and $F(Q)$ is given explicitly in the text, satisfies $F(ensuremath{lambda}Q)=F(Q)ensuremath{-}B mathrm{ln}ensuremath{lambda}$, and is a smooth function as the light cone is approached. The spectral function exhibits two scaling laws, one governing the approach to the origin along a ray ${mathrm{lim}}_{ensuremath{lambda}ensuremath{rightarrow}0}{ensuremath{lambda}}^{4ensuremath{-}B}ensuremath{sigma}(ensuremath{lambda}Q)={ensuremath{sigma}}_{0}P(Q)$, the other governing the approach to the light cone at fixed energy ${Q}^{0}=E$ and angle $stackrel{^}{k}{mathrm{lim}}_{|Q|ensuremath{rightarrow}E}[{(Eensuremath{-}|stackrel{ensuremath{rightarrow}}{Q}|)}^{1ensuremath{-}B}P(E,stackrel{ensuremath{rightarrow}}{Q})]=mathrm{const}ifmmodetimeselsetexttimesfi{}{[ensuremath{Gamma}(1+B)]}^{ensuremath{-}1}B{E}^{Bensuremath{-}1}{I}_{0}(k)mathrm{exp}[F(k)]$, where $k=(E,Estackrel{^}{k})$. For ${e}^{+}ensuremath{-}{e}^{ensuremath{-}}$ annihilation at 3 on 3 Gev, $mathrm{exp}[F(k)]={E}^{ensuremath{-}B}mathrm{exp}[F(stackrel{^}{k})]$ produces a 30% angular modulation." @default.
- W1966255950 created "2016-06-24" @default.
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- W1966255950 date "1979-10-15" @default.
- W1966255950 modified "2023-09-27" @default.
- W1966255950 title "Energy and momentum spectral function of coherent bremsstrahlung radiation" @default.
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- W1966255950 doi "https://doi.org/10.1103/physrevd.20.2011" @default.
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