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Past exam of the mathematics course of the University of Cambridge
/
2019
/
iii
/
Paper 305
/
4
/
d
/
Solution
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Past exam of the mathematics course of the University of Cambridge
2019
iii
Paper 305
4
d
2026-10-03
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Put
L
=
(
2
π
)
−
1
lo
g
(
M
GUT
/
m
Z
)
. At the
gauge coupling unification
scale, write the common normalized
coupling
as
α
U
=
α
2
(
M
GUT
)
=
α
3
(
M
GUT
)
=
(
5/3
)
α
1
(
M
GUT
)
. Running downward gives
α
3
−
1
(
m
Z
)
=
α
U
−
1
−
β
3
L
,
α
2
−
1
(
m
Z
)
=
α
U
−
1
−
β
2
L
,
5
3
α
1
−
1
(
m
Z
)
=
α
U
−
1
−
5
3
β
1
L
.
(1)
Subtracting the
second
equation
from the third determines
L
=
β
2
−
5
3
β
1
5
3
α
1
−
1
(
m
Z
)
−
α
2
−
1
(
m
Z
)
.
(2)
Eliminating
L
from the
first
two
equations
yields
α
3
−
1
(
m
Z
)
=
α
2
−
1
(
m
Z
)
+
5
3
β
1
−
β
2
β
3
−
β
2
[
5
3
α
1
−
1
(
m
Z
)
−
α
2
−
1
(
m
Z
)
]
.
(3)
Ancestors
(11)
d
4
Paper 305
iii
2019
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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