Pseudorapidity

Comparison of polar angle and pseudorapidity for some exemplary values. The forward direction is the angular range with large values ​​of . ${\displaystyle \theta }$ ${\displaystyle \eta }$
${\displaystyle \eta }$

The pseudorapidity (eta) is a spatial coordinate that is used in experimental particle physics to indicate the angle of a vector relative to the beam axis. It is preferred to specifying the polar angle , because in hadron-Hadron collisions the flow of the generated particles per pseudorapidity interval is approximately constant. ${\displaystyle \eta }$ ${\displaystyle \theta }$

The pseudorapidity is defined as

${\displaystyle \eta =-\ln \left[\tan \left({\frac {\theta }{2}}\right)\right].}$

For a particle with momentum (and ) this can be rewritten as: ${\displaystyle {\vec {p}}}$ ${\displaystyle \left|{\vec {p}}\right|=p}$

${\displaystyle \eta =\operatorname {artanh} (p_{L}/p)={\frac {1}{2}}\cdot \ln \left({\frac {p+p_{L}}{p-p_{L}}}\right),}$

wherein

• ${\displaystyle \operatorname {artanh} }$is the hyperbolic areatangent function and
• the longitudinal impulse is the impulse component along the beam axis. ${\displaystyle p_{L}}$

In the high energy approximation , i.e. H. for a particle with an energy whose mass is negligible compared to its momentum , the pseudorapidity is numerically roughly equal to the rapidity ${\displaystyle E}$ ${\displaystyle m}$ ${\displaystyle p}$ ${\displaystyle m\ll p\Rightarrow E\approx p}$

${\displaystyle \eta \approx y,}$

which in experimental particle physics is defined as

${\displaystyle y={\frac {1}{2}}\cdot \ln \left({\frac {E+p_{L}}{E-p_{L}}}\right).}$

For comparison: the original rapidity according to the special theory of relativity is

${\displaystyle \vartheta ={\frac {1}{2}}\cdot \ln \left({\frac {E+p}{E-p}}\right)={\frac {1}{2}}\cdot \ln \left({\frac {1+\beta }{1-\beta }}\right),}$

where is the ratio of particle speed to the speed of light . ${\displaystyle \beta =v/c}$ ${\displaystyle v}$ ${\displaystyle c}$

The shape of the differential cross section is invariant under a Lorentz boost . The same applies to a good approximation for the pseudorapidity, only this is easier to measure: It is not the mass of the particle that has to be determined, only its direction of flight through the detector . ${\displaystyle d\sigma /dy}$