Editing Mandelstam variables

{{Short description|Variables used in scattering processes}}
{{for|other articles using the same surname|Mandelstam}}
[[Image:Mandelstam.svg|right|thumb|220px|In this diagram, two particles come in with momenta p<sub>1</sub> and p<sub>2</sub>, they interact in some fashion, and then two particles with different momentum (p<sub>3</sub> and p<sub>4</sub>) 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 symmetry|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 <math>\mathrm{diag}(1, -1,-1,-1)</math>, the Mandelstam variables <math>s,t,u</math> are then defined by
:*<math>s=(p_1+p_2)^2 c^2 =(p_3+p_4)^2 c^2</math>
:*<math>t=(p_1-p_3)^2 c^2 =(p_4-p_2)^2 c^2</math>
:*<math>u=(p_1-p_4)^2 c^2 =(p_3-p_2)^2 c^2</math>,
where ''p''<sub>1</sub> and ''p''<sub>2</sub> are the [[four-momentum|four-momenta]] of the incoming particles and ''p''<sub>3</sub> and ''p''<sub>4</sub> are the four-momenta of the outgoing particles.

<math>s</math> is also known as the square of the center-of-mass energy ([[invariant mass]]) and <math>t</math> as the square of the [[four-momentum]] transfer.

==Feynman diagrams==
The letters ''s,t,u'' are also used in the terms '''s-channel''' (timelike channel), '''t-channel''', and '''u-channel''' (both spacelike channels). These channels represent different [[Feynman diagram]]s or different possible scattering events where the interaction involves the exchange of an intermediate particle whose squared four-momentum equals ''s,t,u'', respectively.

::{|cellpadding="10"
|[[Image:S-channel.svg|150px]]
|[[Image:T-channel.svg|150px]]
|[[Image:U-channel.svg|150px]]
|-
|align="center"|'''s-channel'''
|align="center"|'''t-channel'''
|align="center"|'''u-channel'''
|}
For example, the s-channel corresponds to the particles 1,2 joining into an intermediate particle that eventually splits into 3,4: {{citation needed span|the s-channel is the only way that [[resonance (quantum field theory)|resonances]] and new [[unstable particle]]s may be discovered provided their lifetimes are long enough that they are directly detectable.|date=July 2014|reason=See 's-channel discovery' discussion on talk page}} 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:

<math>\left(p_1c\right)\cdot\left(p_2c\right) = \frac{1}{2}\left(s - \left(m_1c^2\right)^2 - \left(m_2c^2\right)^2\right)</math>  

<math>\left(p_1c\right)\cdot\left(p_3c\right) = \frac{1}{2}\left(\left(m_1c^2\right)^2 + \left(m_3c^2\right)^2 - t\right)</math>

<math>\left(p_1c\right)\cdot\left(p_4c\right) = \frac{1}{2}\left(\left(m_1c^2\right)^2 + \left(m_4c^2\right)^2 - u\right)</math>

Where <math>m_i</math> is the mass of the particle with corresponding momentum <math>p_i</math>.

==Sum==
Note that
:<math>s + t + u = \left(m_1c^2\right)^2 + \left(m_2c^2\right)^2 + \left(m_3c^2\right)^2 + \left(m_4c^2\right)^2</math>
where ''m''<sub>''i''</sub> is the mass of particle ''i''.<ref>{{cite book |last=Griffiths |first=David |author-link=David J. Griffiths |year=2008 |title=Introduction to Elementary Particles |edition=2nd |publisher=[[Wiley-VCH]] |isbn=978-3-527-40601-2 |page=113}}</ref>

{{Collapse top|title=Proof|collapse=no}}
To prove this, we need to use two facts:
:*The square of a particle's four momentum is the square of its mass,
::<math>p_i^2 = \left(m_ic\right)^2  \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (1)</math>
:*And conservation of four-momentum,
::<math>p_1 + p_2 = p_3 + p_4</math>
::<math>p_1 = -p_2 + p_3 + p_4 \quad \quad \quad \quad \quad \quad \,\, (2)</math>

So, to begin,
::<math>s /c^2 =(p_1+p_2)^2 =p_1^2 + p_2^2 + 2p_1 \cdot p_2</math>
::<math>t /c^2 =(p_1-p_3)^2=p_1^2 + p_3^2 - 2p_1 \cdot p_3</math>
::<math>u /c^2 =(p_1-p_4)^2=p_1^2 + p_4^2 - 2p_1 \cdot p_4</math>

Then adding the three while inserting squared masses leads to,
::<math>(s+t+u)/c^2=\left(m_1c\right)^2 + \left(m_2c\right)^2 + \left(m_3c\right)^2 + \left(m_4c\right)^2 + 2 p_1^2 + 2p_1 \cdot p_2 - 2p_1 \cdot p_3 - 2p_1 \cdot p_4</math>

Then note that the last four terms add up to zero using conservation of four-momentum,
::<math>2 p_1^2 + 2p_1 \cdot p_2 - 2p_1 \cdot p_3 - 2p_1 \cdot p_4 = 2p_1 \cdot (p_1 + p_2 - p_3 - p_4) = 0</math>

So finally,
:<math>(s+t+u)/c^2 = \left(m_1c\right)^2 + \left(m_2c\right)^2 + \left(m_3c\right)^2 + \left(m_4c\right)^2</math>.
{{Collapse bottom|}}

==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. <math>E^2= \mathbf{p} \cdot \mathbf{p} + {m_0}^2</math> becomes <math>E^2 \approx \mathbf{p} \cdot \mathbf{p}</math> ).  The rest mass can also be neglected.

So for example,
::<math>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</math>
because <math>p_1^2 = \left(m_1c\right)^2</math> and <math>p_2^2 = \left(m_2c\right)^2</math>.

Thus,
::{|
|align="right"|<math>s/c^2 \approx</math>
|align="right"|<math>2 p_1 \cdot p_2 \approx</math>
|align="right"|<math>2 p_3 \cdot p_4</math>
|-
|align="right"|<math>t/c^2 \approx</math>
|<math>-2 p_1 \cdot p_3 \approx</math>
|<math>-2 p_2 \cdot p_4</math>
|-
|align="right"|<math>u/c^2 \approx</math>
|<math>-2 p_1 \cdot p_4 \approx</math>
|<math>-2 p_3 \cdot p_2</math>
|}

==See also==
*[[Feynman diagrams]]
*[[Bhabha scattering]]
*[[Møller scattering]]
*[[Compton scattering]]

==References==
{{Reflist}}

*{{cite journal |last=Mandelstam |first=S. |authorlink=Stanley Mandelstam |year=1958 |title=Determination of the Pion-Nucleon Scattering Amplitude from Dispersion Relations and Unitarity |journal=[[Physical Review]]  |doi=10.1103/PhysRev.112.1344 |bibcode=1958PhRv..112.1344M |volume=112 |issue=4 |pages=1344 |url=http://dbserv.ihep.su/hist/owa/hw.part2?s_c=MANDELSTAM+1958 |url-status=dead |archive-url=https://web.archive.org/web/20000528212400/http://dbserv.ihep.su/hist/owa/hw.part2?s_c=MANDELSTAM+1958 |archive-date=2000-05-28|url-access=subscription }}
*{{cite book |last1=Halzen |first1=Francis |authorlink1=Francis Halzen |last2=Martin |first2=Alan |authorlink2=Alan Martin (physicist) |year=1984 |title=Quarks & Leptons: An Introductory Course in Modern Particle Physics |publisher=[[John Wiley & Sons]] |isbn=0-471-88741-2 |url=https://archive.org/details/quarksleptonsint0000halz |url-access=registration}}
*{{cite book |last=Perkins |first=Donald H. |year=2000 |title=Introduction to High Energy Physics |edition=4th |publisher=[[Cambridge University Press]] |isbn=0-521-62196-8}}

{{DEFAULTSORT:Mandelstam Variables}}
[[Category:Kinematics (particle physics)]]
[[Category:Scattering]]
[[Category:Quantum field theory]]
[[Category:Eponyms in physics]]