Center of gravity

As a center of mass energy or invariant mass (with the Mandelstam variables ) is called in the particle in a collision process the total energy - that is, the sum of the rest energies and kinetic energies - all particles involved in terms of their common center of gravity coordinate system . It is only part of the total energy generated by the particle accelerator ; the rest is in the movement of the center of gravity that occurs in the laboratory system . Only the center of mass energy is available to be converted into excitation energy or into the mass of new particles. ${\displaystyle {\sqrt {s}}}$ ${\displaystyle s}$

The coincidence of the two terms -energy and mass is based on the equivalence of mass and energy , as they only differ by a constant conversion factor . In high-energy physics, this is often set equal to one and also in this article . ${\displaystyle c^{2}}$

The special case of the invariant mass of a single particle is its physical mass itself.

formula

When using natural units in particle physics, energy and mass have the same unit. The center of mass energy is then generally the square root of the total four-fold momentum :

${\displaystyle {\sqrt {s}}={\sqrt {\left(\sum _{i=1}^{n}{P_{i}}\right)^{2}}}}$,

where the square means the scalar product of the Minkowski metric:

${\displaystyle {\sqrt {s}}={\sqrt {\left(\sum _{i=1}^{n}{P_{\mu ,i}}\right)\cdot \left(\sum _{i=1}^{n}{P_{i}^{\mu }}\right)}}}$.

Here is

• ${\displaystyle n}$ the number of particles
• ${\displaystyle P_{i}}$ their four-pulse.

properties

• The center of mass energy is invariant under Lorentz transformations ; hence the name invariant mass . This follows from the fact that the sum of four vectors is a four vector and the square of a four vector is a Lorentz scalar ; H. a scalar that remains invariant under Lorentz transformations. Accordingly, the root of a Lorentz scalar is also a scalar.
• The center of mass energy of all particles before a collision is equal to their center of mass energy after the collision ( conservation quantity ).

Examples

Colliding beam experiment

In a colliding beam experiment, if two particles with identical masses and opposite pulses of the same size collide, the four-momentum is:

${\displaystyle p_{1}={\begin{pmatrix}E\\p_{x}\\p_{y}\\p_{z}\end{pmatrix}}}$and . ${\displaystyle p_{2}={\begin{pmatrix}E\\-p_{x}\\-p_{y}\\-p_{z}\end{pmatrix}}\,}$

Applied this results in:

${\displaystyle {\sqrt {s}}={\sqrt {\left({\begin{pmatrix}E\\p_{x}\\p_{y}\\p_{z}\end{pmatrix}}+{\begin{pmatrix}E\\-p_{x}\\-p_{y}\\-p_{z}\end{pmatrix}}\right)^{2}}}={\sqrt {{\begin{pmatrix}2\cdot E\\0\\0\\0\end{pmatrix}}^{2}}}={\sqrt {4\cdot E^{2}}}=2\cdot E\,}$.

This is the ideal limit case, which cannot be fully achieved in practice, in which the total energy of both particles can be converted. In this case, the center of gravity energy increases proportionally with the energy of each of the two particles. ${\displaystyle E}$

Target experiment

If, in a target experiment, a particle with mass meets a stationary particle of the same mass , the four-momentum is: ${\displaystyle m}$ ${\displaystyle m}$

${\displaystyle p_{1}={\begin{pmatrix}E\\p_{x}\\p_{y}\\p_{z}\end{pmatrix}}}$and . ${\displaystyle p_{2}={\begin{pmatrix}m\\0\\0\\0\end{pmatrix}}\,}$

Applied this results in:

${\displaystyle {\sqrt {s}}={\sqrt {\left({\begin{pmatrix}E\\p_{x}\\p_{y}\\p_{z}\end{pmatrix}}+{\begin{pmatrix}m\\0\\0\\0\end{pmatrix}}\right)^{2}}}={\sqrt {{\begin{pmatrix}E+m\\p_{x}\\p_{y}\\p_{z}\end{pmatrix}}^{2}}}={\sqrt {(E+m)^{2}-(p_{x}^{2}+p_{y}^{2}+p_{z}^{2})}}={\sqrt {E^{2}+2\cdot E\cdot m+m^{2}-{\boldsymbol {p}}^{2}}}\,}$.

With the relationship follows: ${\displaystyle {\boldsymbol {p}}^{2}=E^{2}-m^{2}}$

${\displaystyle {\sqrt {s}}={\sqrt {2\cdot E\cdot m+2\cdot m^{2}}}\,}$.

The center of gravity energy of a target experiment is therefore much smaller for the same energy of the accelerated particle than in a colliding beam experiment if the mass of the particles is small compared to their kinetic energy. In addition, it then only increases proportionally to the root of the energy applied by the accelerator. This shows the advantage of colliding beam experiments before target experiments.

literature

• Povh / Rith / Scholz / Zetsche: Particles and Cores . 8th edition. Springer, Berlin / Heidelberg 2009, ISBN 978-3-540-68075-8 .
• Christoph Berger: Elementary Particle Physics: From the Basics to Modern Experiments . 2nd Edition. Springer, Berlin / Heidelberg 2006, ISBN 3-540-23143-9 .