Problems
37
(ii) the splitting between the n = 2 and n = 1 states in the upsilon
series (bb) is 563 MeV, and
¯
m b = 5280 MeV.
(c) In positronium, the n = 1
3 S 1 and n = 1
1 S 0 states are split by the
hyperfine interaction, which has the form
7 α
4 m e σ 1 · σ 2 where m e
48
is the electron mass and σ 1 , σ 2 are the spin matrices for the e
−
and e
+ respectively. Calculate the expectation value of σ 1 · σ 2 in
the
3 S 1 and
1 S 0 states, and hence evaluate the splitting between
these levels (calculated in lowest order perturbation theory) in eV.
1
[Hint : the total spin S is given by S = (σ 1 + σ 2 ). So S
2 =
2
1 (σ 1
2 + σ
2
2 + 2σ 1 · σ 2 ). Hence the eigenvalues of σ 1 · σ 2 are directly
4
related to those of S
2 .]
(d) Suppose an analogous ‘strong’ hyperfine interaction existed in the c¯ c
system, and was responsible for the splitting between the n = 1
3 S 1
and n = 1
1 S 0 states, which is 116 MeV experimentally (i.e. replace
α by α s and m e by m c = 1870 MeV). Calculate the corresponding
value of α s .
1.5 The potential between a heavy quark Q and an antiquark ¯
Q is found
empirically to be well represented by
α s
V (r) = − + br
r
where α s ≈ 0.5 and b ≈ 0.18 GeV
2 . Indicate the origin of the first term in
V (r), and the significance of the second.
An estimate of the ground-state energy of the bound Q ¯
Q system may be
made as follows. For a given r, the total energy is
2
α s
p
E(r) = 2m −
+ br +
r
m
where m is the mass of the Q (or ¯
Q) and p is its momentum (assumed nonrelativistic). Explain why p may be roughly approximated by 1/r, and sketch
the resulting E(r) as a function of r. Hence show that, in this approximation,
the radius of the ground state, r 0 , is given by the solution of
2
α s
=
+ b.
3
2
mr
r
0
0
Taking m = 1.5 GeV as appropriate to the c¯ c system, verify that for this
system
(1/r 0 ) ≈ 0.67 GeV
and calculate the energy of the c¯ c ground state in GeV, according to this
model.
An excited c¯ c state at 3.686 GeV has a total width of 278 keV, and one
at 3.77 GeV has a total width of 24 MeV. Comment on the values of these
widths.
37
(ii) the splitting between the n = 2 and n = 1 states in the upsilon
series (bb) is 563 MeV, and
¯
m b = 5280 MeV.
(c) In positronium, the n = 1
3 S 1 and n = 1
1 S 0 states are split by the
hyperfine interaction, which has the form
7 α
4 m e σ 1 · σ 2 where m e
48
is the electron mass and σ 1 , σ 2 are the spin matrices for the e
−
and e
+ respectively. Calculate the expectation value of σ 1 · σ 2 in
the
3 S 1 and
1 S 0 states, and hence evaluate the splitting between
these levels (calculated in lowest order perturbation theory) in eV.
1
[Hint : the total spin S is given by S = (σ 1 + σ 2 ). So S
2 =
2
1 (σ 1
2 + σ
2
2 + 2σ 1 · σ 2 ). Hence the eigenvalues of σ 1 · σ 2 are directly
4
related to those of S
2 .]
(d) Suppose an analogous ‘strong’ hyperfine interaction existed in the c¯ c
system, and was responsible for the splitting between the n = 1
3 S 1
and n = 1
1 S 0 states, which is 116 MeV experimentally (i.e. replace
α by α s and m e by m c = 1870 MeV). Calculate the corresponding
value of α s .
1.5 The potential between a heavy quark Q and an antiquark ¯
Q is found
empirically to be well represented by
α s
V (r) = − + br
r
where α s ≈ 0.5 and b ≈ 0.18 GeV
2 . Indicate the origin of the first term in
V (r), and the significance of the second.
An estimate of the ground-state energy of the bound Q ¯
Q system may be
made as follows. For a given r, the total energy is
2
α s
p
E(r) = 2m −
+ br +
r
m
where m is the mass of the Q (or ¯
Q) and p is its momentum (assumed nonrelativistic). Explain why p may be roughly approximated by 1/r, and sketch
the resulting E(r) as a function of r. Hence show that, in this approximation,
the radius of the ground state, r 0 , is given by the solution of
2
α s
=
+ b.
3
2
mr
r
0
0
Taking m = 1.5 GeV as appropriate to the c¯ c system, verify that for this
system
(1/r 0 ) ≈ 0.67 GeV
and calculate the energy of the c¯ c ground state in GeV, according to this
model.
An excited c¯ c state at 3.686 GeV has a total width of 278 keV, and one
at 3.77 GeV has a total width of 24 MeV. Comment on the values of these
widths.