### Abstract

A double spectral representation of Mandelstam type is obtained for the invariant amplitude T of the binary collision process AB → CD involving scalar particles, using unitarity to evaluate the contribution to Im T from a two-particle intermediate state (EF) and making one-pole approximations to the amplitudes for the processes AB → EF, CD → EF. For the case where the masses satisfy the inequalities (m_{A}+m_{B}) ≤ (m_{E}+m_{F}), m_{C}+m_{D}) ≤ (m_{E}+m_{F}) the above approximations are shown to give the same result as that obtained from the Feynman parametrization of the amplitude arising from the box diagram. The proof proceeds by directly evaluating the box diagram amplitude and in this way we obtain a spectral representation of the amplitude for all allowed mass configurations, including cases for which there are anomalous thresholds and cases for which a double spectral representation no longer exists.

Original language | English |
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Pages (from-to) | 503-544 |

Number of pages | 42 |

Journal | Annals of Physics |

Volume | 75 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1973 |

Externally published | Yes |

### Cite this

*Annals of Physics*,

*75*(2), 503-544. https://doi.org/10.1016/0003-4916(73)90079-1

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*Annals of Physics*, vol. 75, no. 2, pp. 503-544. https://doi.org/10.1016/0003-4916(73)90079-1

**The analytic properties of the box diagram amplitude : I.** / Frederiksen, J. S.; Woolcock, W. S.

Research output: Contribution to journal › Article › Other

TY - JOUR

T1 - The analytic properties of the box diagram amplitude

T2 - I

AU - Frederiksen, J. S.

AU - Woolcock, W. S.

PY - 1973

Y1 - 1973

N2 - A double spectral representation of Mandelstam type is obtained for the invariant amplitude T of the binary collision process AB → CD involving scalar particles, using unitarity to evaluate the contribution to Im T from a two-particle intermediate state (EF) and making one-pole approximations to the amplitudes for the processes AB → EF, CD → EF. For the case where the masses satisfy the inequalities (mA+mB) ≤ (mE+mF), mC+mD) ≤ (mE+mF) the above approximations are shown to give the same result as that obtained from the Feynman parametrization of the amplitude arising from the box diagram. The proof proceeds by directly evaluating the box diagram amplitude and in this way we obtain a spectral representation of the amplitude for all allowed mass configurations, including cases for which there are anomalous thresholds and cases for which a double spectral representation no longer exists.

AB - A double spectral representation of Mandelstam type is obtained for the invariant amplitude T of the binary collision process AB → CD involving scalar particles, using unitarity to evaluate the contribution to Im T from a two-particle intermediate state (EF) and making one-pole approximations to the amplitudes for the processes AB → EF, CD → EF. For the case where the masses satisfy the inequalities (mA+mB) ≤ (mE+mF), mC+mD) ≤ (mE+mF) the above approximations are shown to give the same result as that obtained from the Feynman parametrization of the amplitude arising from the box diagram. The proof proceeds by directly evaluating the box diagram amplitude and in this way we obtain a spectral representation of the amplitude for all allowed mass configurations, including cases for which there are anomalous thresholds and cases for which a double spectral representation no longer exists.

UR - http://www.scopus.com/inward/record.url?scp=49549169889&partnerID=8YFLogxK

U2 - 10.1016/0003-4916(73)90079-1

DO - 10.1016/0003-4916(73)90079-1

M3 - Article

VL - 75

SP - 503

EP - 544

JO - Annals of Physics

JF - Annals of Physics

SN - 0003-4916

IS - 2

ER -