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- Mandelstam variables - Physics Stack Exchange
- special relativity - Physical interpretation of Mandelstam variables ...
- quantum field theory - Mandelstam variables for 2 to 3 particle ...
- On physical interpretation of Mandelstam variables
- Mandelstam Variables for 2->2 Scattering with Equal Masses in …
- particle physics - Using Mandelstam Variables in Experimental …
- 2-body scattering and Mandelstam Variables - Physics Forums
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- Physical interpretation of Mandelstam variables - Physics Forums
- quantum field theory - Why Mandelstam variables in Minkowski …
mandelstam variables
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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
d
i
a
g
(
1
,
−
1
,
−
1
,
−
1
)
{\displaystyle \mathrm {diag} (1,-1,-1,-1)}
, the Mandelstam variables
s
,
t
,
u
{\displaystyle s,t,u}
are then defined by
s
=
(
p
1
+
p
2
)
2
c
2
=
(
p
3
+
p
4
)
2
c
2
{\displaystyle 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
{\displaystyle 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
{\displaystyle u=(p_{1}-p_{4})^{2}c^{2}=(p_{3}-p_{2})^{2}c^{2}}
,
where p1 and p2 are the four-momenta of the incoming particles and p3 and p4 are the four-momenta of the outgoing particles.
s
{\displaystyle s}
is also known as the square of the center-of-mass energy (invariant mass) and
t
{\displaystyle t}
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 diagrams or different possible scattering events where the interaction involves the exchange of an intermediate particle whose squared four-momentum equals s,t,u, respectively.
For example, the s-channel corresponds to the particles 1,2 joining into an intermediate particle that eventually splits into 3,4: the s-channel is the only way that resonances and new unstable particles may be discovered provided their lifetimes are long enough that they are directly detectable. 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
⋅
p
2
=
s
/
c
2
−
m
1
2
−
m
2
2
2
{\displaystyle p_{1}\cdot p_{2}={\frac {s/c^{2}-m_{1}^{2}-m_{2}^{2}}{2}}}
p
1
⋅
p
3
=
m
1
2
+
m
3
2
−
t
/
c
2
2
{\displaystyle p_{1}\cdot p_{3}={\frac {m_{1}^{2}+m_{3}^{2}-t/c^{2}}{2}}}
p
1
⋅
p
4
=
m
1
2
+
m
4
2
−
u
/
c
2
2
{\displaystyle p_{1}\cdot p_{4}={\frac {m_{1}^{2}+m_{4}^{2}-u/c^{2}}{2}}}
Where
m
i
{\displaystyle m_{i}}
is the mass of the particle with corresponding momentum
p
i
{\displaystyle p_{i}}
.
Sum
Note that
(
s
+
t
+
u
)
/
c
4
=
m
1
2
+
m
2
2
+
m
3
2
+
m
4
2
{\displaystyle (s+t+u)/c^{4}=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}}
where mi is the mass of particle i.
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
⋅
p
+
m
0
2
{\displaystyle E^{2}=\mathbf {p} \cdot \mathbf {p} +{m_{0}}^{2}}
becomes
E
2
≈
p
⋅
p
{\displaystyle E^{2}\approx \mathbf {p} \cdot \mathbf {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
⋅
p
2
≈
2
p
1
⋅
p
2
{\displaystyle s/c^{2}=(p_{1}+p_{2})^{2}=p_{1}^{2}+p_{2}^{2}+2p_{1}\cdot p_{2}\approx 2p_{1}\cdot p_{2}}
because
p
1
2
=
m
1
2
{\displaystyle p_{1}^{2}=m_{1}^{2}}
and
p
2
2
=
m
2
2
{\displaystyle p_{2}^{2}=m_{2}^{2}}
.
Thus,
See also
Feynman diagrams
Bhabha scattering
Møller scattering
Compton scattering
References
Mandelstam, S. (1958). "Determination of the Pion-Nucleon Scattering Amplitude from Dispersion Relations and Unitarity". Physical Review. 112 (4): 1344. Bibcode:1958PhRv..112.1344M. doi:10.1103/PhysRev.112.1344. Archived from the original on 2000-05-28.
Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.
Perkins, Donald H. (2000). Introduction to High Energy Physics (4th ed.). Cambridge University Press. ISBN 0-521-62196-8.
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Mandelstam variables - Physics Stack Exchange
Dec 14, 2024 · One can show that in addition to the particle masses, the so-called Mandelstam variables are sufficient to describe the kinematics of the scattering process. What is meant with kinematics? And how one shows that? Constrain? I don't understand the use of this word here. independent lorentz invariant quantity? Independent in what way?
special relativity - Physical interpretation of Mandelstam variables ...
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quantum field theory - Mandelstam variables for 2 to 3 particle ...
Nov 8, 2016 · I'm trying to work out the mandelstam variables for 2 particles scattering to produce 3 particles. Also each particle is massless. I think there must be 5 because all possible scalar products of the incoming and outgoing momentum must be expressible in terms of the mandelstam variables, and there are 6 possible scalar products.
On physical interpretation of Mandelstam variables
Jul 15, 2022 · The accepted answer to this post gives a nice physical explanation of the Mandelstam variables.. The only problem is that each variable seems to make sense only within a specific channel.
Mandelstam Variables for 2->2 Scattering with Equal Masses in …
Oct 21, 2016 · For ##2\\rightarrow 2## scattering with equal masses in the centre-of-mass frame, are all the four three-momenta equal to each other, or is it that the incoming three-momenta and the outgoing three-momenta sum to 0 separately?
particle physics - Using Mandelstam Variables in Experimental …
Aug 7, 2023 · Using Mandelstam Variables in Experimental Data. Ask Question Asked 1 year, 6 months ago.
2-body scattering and Mandelstam Variables - Physics Forums
May 24, 2017 · The Mandelstam variables are useful in theoretical calculations because they are invariant under Lorentz transformation. Demonstrate that in the centre of mass frame of A and B, the total CM energy, i.e., E total ≡ E A +E B = E C + E D, is equal to √ s. Homework Equations
quantum field theory - Multiparticle Mandlestam Variables …
So in 4D we have three Mandlestam variables for a 4-particle scattering process. ... We have 3 Mandelstam ...
Physical interpretation of Mandelstam variables - Physics Forums
Oct 13, 2016 · The Mandelstam variables a priori have nothing to do with what diagrams you are using. They appear in the propagator if you have an s-, t-, or u-channel diagram, but that is another matter. It is just a convenient way of parametrising the …
quantum field theory - Why Mandelstam variables in Minkowski …
Oct 20, 2020 · The Mandelstam variables correspond to particular Feynman diagrams where momentum in the propagator has ...