Mandelstam variables

In this diagram, two particles come in with momenta p₁ and p₂, they interact in some fashion, and then two particles with different momentum (p₃ and p₄) leave.

In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy, momentum, and angles of particles in a scattering process in a Lorentz-invariant fashion. They are used for scattering processes of two particles to two particles. The Mandelstam variables were first introduced by physicist Stanley Mandelstam in 1958.

If the Minkowski metric is chosen to be Template:Math, the Mandelstam variables Template:Math are then defined by

s = (p_{1} + p_{2})^{2} c^{2} = (p_{3} + p_{4})^{2} c^{2},

t = (p_{1} - p_{3})^{2} c^{2} = (p_{4} - p_{2})^{2} c^{2},

u = (p_{1} - p_{4})^{2} c^{2} = (p_{3} - p_{2})^{2} c^{2},

where Template:Math and Template:Math are the four-momenta of the incoming particles and Template:Math and Template:Math are the four-momenta of the outgoing particles.

$s$ is also known as the square of the center-of-mass energy (invariant mass) and $t$ as the square of the four-momentum transfer.

Feynman diagrams

The letters Template:Math are also used in the terms s-channel (timelike channel), t-channel, and u-channel (both spacelike channels). These channels represent different Feynman diagrams or different possible scattering events where the interaction involves the exchange of an intermediate particle whose squared four-momentum equals Template:Math, respectively.


s-channel	t-channel	u-channel

For example, the s-channel corresponds to the particles 1,2 joining into an intermediate particle that eventually splits into 3,4: Template:Citation needed span The t-channel represents the process in which the particle 1 emits the intermediate particle and becomes the final particle 3, while the particle 2 absorbs the intermediate particle and becomes 4. The u-channel is the t-channel with the role of the particles 3,4 interchanged.

When evaluating a Feynman amplitude one often finds scalar products of the external four-momenta. One can use the Mandelstam variables to simplify these:

(p_{1} c) \cdot (p_{2} c) = \frac{1}{2} (s - {(m_{1} c^{2})}^{2} - {(m_{2} c^{2})}^{2}),

(p_{1} c) \cdot (p_{3} c) = \frac{1}{2} ({(m_{1} c^{2})}^{2} + {(m_{3} c^{2})}^{2} - t),

(p_{1} c) \cdot (p_{4} c) = \frac{1}{2} ({(m_{1} c^{2})}^{2} + {(m_{4} c^{2})}^{2} - u),

where $m_{i}$ is the mass of the particle with corresponding momentum $p_{i}$ .

Sum

Note that

s + t + u = {(m_{1} c^{2})}^{2} + {(m_{2} c^{2})}^{2} + {(m_{3} c^{2})}^{2} + {(m_{4} c^{2})}^{2}

where Template:Math is the mass of particle Template:Mvar.^[1]

Template:Collapse top To prove this, we need to use two facts: the square of a particle's four-momentum is the square of its mass,

p_{i}^{2} = (m_{i} c)^{2},

and conservation of four-momentum,

p_{1} + p_{2} = p_{3} + p_{4},

p_{1} = - p_{2} + p_{3} + p_{4} .

So, to begin,

s / c^{2} = (p_{1} + p_{2})^{2} = p_{1}^{2} + p_{2}^{2} + 2 p_{1} \cdot p_{2},

t / c^{2} = (p_{1} - p_{3})^{2} = p_{1}^{2} + p_{3}^{2} - 2 p_{1} \cdot p_{3},

u / c^{2} = (p_{1} - p_{4})^{2} = p_{1}^{2} + p_{4}^{2} - 2 p_{1} \cdot p_{4} .

Adding the three while inserting squared masses leads to

(s + t + u) / c^{2} = (m_{1} c)^{2} + (m_{2} c)^{2} + (m_{3} c)^{2} + (m_{4} c)^{2} + 2 p_{1}^{2} + 2 p_{1} \cdot p_{2} - 2 p_{1} \cdot p_{3} - 2 p_{1} \cdot p_{4} .

Then note that the last four terms add up to zero using conservation of four-momentum,

2 p_{1}^{2} + 2 p_{1} \cdot p_{2} - 2 p_{1} \cdot p_{3} - 2 p_{1} \cdot p_{4} = 2 p_{1} \cdot (p_{1} + p_{2} - p_{3} - p_{4}) = 0.

So finally,

(s + t + u) / c^{2} = (m_{1} c)^{2} + (m_{2} c)^{2} + (m_{3} c)^{2} + (m_{4} c)^{2} .

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Relativistic limit

In the relativistic limit, the momentum (speed) is large, so using the relativistic energy-momentum equation, the energy becomes essentially the momentum norm (e.g. $E^{2} = 𝐩 \cdot 𝐩 + m_{0}^{2}$ becomes $E^{2} \approx 𝐩 \cdot 𝐩$ ). The rest mass can also be neglected.

So, for example,

s / c^{2} = (p_{1} + p_{2})^{2} = p_{1}^{2} + p_{2}^{2} + 2 p_{1} \cdot p_{2} \approx 2 p_{1} \cdot p_{2},

because $p_{1}^{2} = (m_{1} c)^{2}$ and $p_{2}^{2} = (m_{2} c)^{2}$ . Thus,

$s / c^{2} \approx$	$2 p_{1} \cdot p_{2} \approx$	$2 p_{3} \cdot p_{4},$
$t / c^{2} \approx$	$- 2 p_{1} \cdot p_{3} \approx$	$- 2 p_{2} \cdot p_{4},$
$u / c^{2} \approx$	$- 2 p_{1} \cdot p_{4} \approx$	$- 2 p_{3} \cdot p_{2} .$

References

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[1] Template:Cite book

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Mandelstam variables

Contents

Feynman diagrams

Sum

Relativistic limit

See also

References

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Mandelstam variables

Feynman diagrams

Sum

Relativistic limit

See also

References

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