Bjorken scaling (after J. Bjorken , who introduced it in 1969) describes in physics a dependence of the structure functions in the case of deep inelastic scattering (e.g. of electron and proton ) on only one kinematic variable.
This behavior corresponds to an elastic scattering at point objects, which led to the development of the Parton model .
In the case of inelastic scattering, a dependency on two independent kinematic variables is actually expected; However, this does not occur due to the internal structure of the proton, since it is effectively scattered on individual quarks .
Mathematical formulation
For inelastic electron-proton scattering, the cross-section can generally be written with the structure functions as:
W.
1
,
W.
2
{\ displaystyle W_ {1}, W_ {2}}
d
2
σ
d
Ω
d
E.
′
=
(
d
σ
d
Ω
)
Mott
[
2
W.
1
(
Q
2
,
ν
)
tan
2
(
θ
/
2
)
+
W.
2
(
Q
2
,
ν
)
]
{\ displaystyle {\ frac {d ^ {2} \ sigma} {d \ Omega \, dE ^ {\ prime}}} = \ left ({\ frac {d \ sigma} {d \ Omega}} \ right) _ {\ text {Mott}} \ left [2W_ {1} (Q ^ {2}, \ nu) \, \ tan ^ {2} (\ theta / 2) + W_ {2} (Q ^ {2} , \ nu) \ right]}
.
It is
(
d
σ
d
Ω
)
Mott
{\ displaystyle \ left ({\ tfrac {d \ sigma} {d \ Omega}} \ right) _ {\ text {Mott}}}
the Mott cross section
Q
2
=
-
q
2
=
(
p
e
-
p
e
′
)
2
{\ displaystyle Q ^ {2} = - q ^ {2} = (p_ {e} -p_ {e} ^ {\ prime}) ^ {2}}
the ( four ) momentum transfer
p
e
{\ displaystyle p_ {e}}
the electron impulse
ν
=
P
q
M.
=
E.
laboratory
-
E.
laboratory
′
{\ displaystyle \ nu = {\ tfrac {Pq} {M}} = E _ {\ text {Labor}} - E _ {\ text {Labor}} ^ {\ prime}}
the energy transfer
P
{\ displaystyle P}
the quadruple momentum of the target (e.g. a proton)
M.
{\ displaystyle M}
the mass of the target
θ
{\ displaystyle \ theta}
the scattering angle .
In the elastic case
d
σ
d
Ω
=
(
d
σ
d
Ω
)
Mott
E.
′
E.
[
2
K
1
sin
2
(
θ
/
2
)
+
K
2
cos
2
(
θ
/
2
)
]
{\ displaystyle {\ frac {d \ sigma} {d \ Omega}} = \ left ({\ frac {d \ sigma} {d \ Omega}} \ right) _ {\ text {Mott}} {\ frac { E ^ {\ prime}} {E}} \ left [2K_ {1} \ sin ^ {2} (\ theta / 2) + K_ {2} \ cos ^ {2} (\ theta / 2) \ right] }
the structure functions only depend on one variable.
K
1
,
K
2
{\ displaystyle K_ {1}, K_ {2}}
The variable can be used instead of or as an independent variable. In the quark model, it indicates the momentum fraction of a quark in the proton.
x
Bjorken
≡
x
=
-
q
2
2
q
P
=
-
q
2
2
ν
M.
{\ displaystyle x _ {\ text {Bjorken}} \ equiv x = - {\ tfrac {q ^ {2}} {2qP}} = - {\ tfrac {q ^ {2}} {2 \ nu M}}}
ν
{\ displaystyle \ nu}
Q
2
{\ displaystyle Q ^ {2}}
x
P
{\ displaystyle xP}
James Bjorken predicted that at high energies the structure functions behave like
M.
W.
1
(
q
2
,
x
)
→
F.
1
(
x
)
{\ displaystyle MW_ {1} (q ^ {2}, x) \ rightarrow F_ {1} (x)}
-
q
2
2
M.
c
2
x
W.
2
(
q
2
,
x
)
→
F.
2
(
x
)
{\ displaystyle {\ frac {-q ^ {2}} {2Mc ^ {2} x}} W_ {2} (q ^ {2}, x) \ rightarrow F_ {2} (x)}
,
so only depend on one variable . This behavior, with the dependency on only one variable, is called Bjorken scaling.
x
=
x
Bjorken
{\ displaystyle x = x _ {\ text {Bjorken}}}
Scale violation
For extreme values by a function of the structure function occurs by scale injury to:
x
{\ displaystyle x}
F.
2
{\ displaystyle F_ {2}}
Q
2
{\ displaystyle Q ^ {2}}
with small increases with (increasing)
x
{\ displaystyle x}
F.
2
{\ displaystyle F_ {2}}
Q
2
{\ displaystyle Q ^ {2}}
with large falls with (increasing) .
x
{\ displaystyle x}
F.
2
{\ displaystyle F_ {2}}
Q
2
{\ displaystyle Q ^ {2}}
This is due to how the structural functions of the proton depend on the energy scale :
with small ones the relative proportion of sea quarks and gluons increases with large ones
x
{\ displaystyle x}
Q
2
{\ displaystyle Q ^ {2}}
for large , the relative proportion of valence quarks decreases for large .
x
{\ displaystyle x}
Q
2
{\ displaystyle Q ^ {2}}
literature
David Griffiths: "Introduction to elementary particles". Wiley-VCH Verlag, Weinheim (2004).
Bogdan Povh, et al .: "Particles and Cores", Springer Verlag Berlin (2014).
Web links
Individual evidence
^ J. Bjorken, Asymptotic Sum Rules at Infinite Momentum. Phys. Rev., Volume 179, 1969, pp. 1547-1553.
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