Aitchison I. Supersymmetry in Particle Physics: An Elementary Introduction
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12.2 Sparticle production processes 195
4-momentum and polarization of the Z
0
by m
Z
, k and λ. Evaluating the appropriate
matrix element of (12.59), we obtain the decay amplitude
i(2π)
4
δ
4
(k
i
− k
j
− k)W
ij
¯
u(k
j
, s
j
)γ
μ
(γ
5
)
θ
˜χ
0
i
+θ
˜χ
0
j
+1
u(k
i
, s
i
)
μ
(k,λ). (12.61)
For the decay rate we need to evaluate the contraction
N ≡ N
μν
P
μν
(12.62)
where (see, for example, equation (19.19) of [7])
P
μν
=
λ
μ
(k,λ)
∗
ν
(k,λ) =−g
μν
+ k
μ
k
ν
/m
2
Z
(12.63)
and
N
μν
=
1
2
s
i
,s
j
¯
u
j
γ
μ
(γ
5
)
θ
˜χ
0
i
+θ
˜χ
0
j
+1
u
i
¯
u
i
(−γ
5
)
θ
˜χ
0
i
+θ
˜χ
0
j
+1
γ
ν
u
j
=−
1
2
Tr
(/k
j
+ m
j
)γ
μ
(−/k
i
+ (−)
θ
˜χ
0
i
+θ
˜χ
0
j
m
i
)γ
ν
= 2
k
μ
i
k
ν
j
+ k
ν
i
k
μ
j
− g
μν
k
i
· k
j
− (−)
θ
˜χ
0
i
+θ
˜χ
0
j
2g
μν
m
i
m
j
. (12.64)
Performing the contraction (12.62) yields the result
N = m
2
˜χ
0
i
+ m
2
˜χ
0
j
− m
2
Z
+
m
2
˜χ
0
i
− m
2
˜χ
0
j
2
− m
4
Z
m
2
Z
+ 6(−)
θ
˜χ
0
i
+θ
˜χ
0
j
m
˜χ
0
i
m
˜χ
0
j
. (12.65)
The decay rate is then
˜χ
0
i
→ ˜χ
0
j
+ Z
0
=
1
8πm
2
˜χ
0
i
|W
ij
|
2
Nk
m
˜χ
0
j
, m
Z
, m
˜χ
0
i
. (12.66)
This differs from formula (B.61b) of [49] by a factor of 4.
12.2 Sparticle production processes
12.2.1 Squark pair production in qq collisions
We begin by considering the process
q
1
q
2
→ ˜q
1L
˜q
2R
, (12.67)
where q
1
and q
2
are non-identical quarks, belonging in practice to the first or second
generation. The relevant Feynman diagram is shown in Figure 12.4. The momenta,
spins and colour labels of the quarks are p
1
, s
1
, c
1
and p
2
, s
2
, c
2
, and the momenta
and colour labels of the squarks are k
1
, c
1
and k
2
, c
2
. The interaction which produces
196 Some simple tree-level calculations in the MSSM
k
1
,c
1
′
k
2
,c
2
′
q
2R
q
2
p
2
,s
2
,c
2
p
1
,s
1
,c
1
g
~
~
~
q
1
q
1L
Figure 12.4 Lowest-order diagram for the process q
1
q
2
→ ˜q
1L
˜q
2R
.
the ˜q
1L
is the hermitian conjugate of (12.15), namely
L
1L
=−
√
2g
s
(−i)
θ
˜g
˜
q
†
L
¯
˜
ga
M
P
L
1
2
λ
a
q
. (12.68)
Similarly, ˜q
2R
is produced by the hermitian conjugate of (12.22), namely
L
2R
=+
√
2g
s
(i)
θ
˜g
˜
q
†
R
¯
˜
ga
M
P
R
1
2
λ
a
q
. (12.69)
The amplitude for Figure 12.4 is then
˜q
1L
, k
1
, c
1
;˜q
2R
, k
2
, c
2
d
4
xd
4
yT[iL
1L
(x)iL
2R
(y)]
q
1
, p
1
, s
1
, c
1
;q
2
, p
2
, s
2
, c
2
.
(12.70)
After reducing the particles in the initial and final states, this becomes
2g
2
s
d
4
xd
4
y e
i(k
1
·x +k
2
·y−p
1
·x −p
2
·y)
ω
†
(c
1
)
λ
a
2
ω(c
1
)ω
†
(c
2
)
λ
b
2
ω(c
2
)
×
0
T
¯
˜
ga
Mα
(x)
¯
˜
gb
Mγ
(y)
0
P
Lαβ
u
β
( p
1
, s
1
)P
Rγδ
u
δ
( p
2
, s
2
), (12.71)
where we have indicated the spinor indices explicitly. We use (2.144) to write the
spinor part as
δ
ab
C
T
α
S
Fγ
(x − y)P
Lαβ
u
β
( p
1
, s
1
)P
Rγδ
u
δ
( p
2
, s
2
)
= δ
ab
(
CP
L
u( p
1
, s
1
)
)
T
S
F
(x − y)P
R
u( p
2
, s
2
). (12.72)
Now
(
CP
L
u
)
T
= u
T
P
L
C
T
= ¯vC
T
P
L
C
T
= ¯v P
L
(C
T
)
2
=−¯v P
L
, (12.73)
12.2 Sparticle production processes 197
where we have used Cγ
5
= γ
5
C, and equations (2.133) and (2.142). Writing
S
F
(x − y) in terms of its Fourier transform (2.138) allows us to perform the
integrals over x and y, and we obtain the amplitude
i(2π)
4
δ
4
( p
1
+ p
2
− k
1
− k
2
)M
q˜q
, (12.74)
where
M
q˜q
=−2g
2
s
ω
†
1
λ
a
2
ω
1
ω
†
2
λ
a
2
ω
2
¯v( p
1
, s
1
)P
L
1
/k
1
− /p
1
− m
˜g
P
R
u( p
2
, s
2
). (12.75)
For the cross section, we require the modulus squared of the amplitude, summed
over final state colours and averaged over intial state colours and spins, which we
denote by
|M
q˜q
|
2
. For the colour part, we note that
c
1
,c
1
ω
†
(c
1
)
λ
a
2
ω(c
1
)ω
†
(c
1
)
λ
b
2
ω(c
1
) =
1
4
Trλ
a
λ
b
=
1
2
δ
ab
, (12.76)
and hence the colour factor is
1
4
1
9
ab
δ
ab
δ
ab
=
2
9
. (12.77)
The spinor factor is
1
4
1
ˆ
t − m
2
˜g
2
Tr[/p
1
P
L
(/k
1
− /p
1
+ m
˜g
)P
R
/p
2
P
L
(/k
1
− /p
1
+ m
˜g
)P
R
]
=
1
ˆ
t − m
2
˜g
2
1
4
−
ˆ
t − m
2
˜q
L
ˆ
t − m
2
˜q
R
−
ˆ
s
ˆ
t
, (12.78)
where we are using the ‘hatted’ Mandelstam variables (conventional in parton
kinematics) defined by
ˆ
s = ( p
1
+ p
2
)
2
= (k
1
+ k
2
)
2
,
ˆ
t = ( p
1
− k
1
)
2
= ( p
2
− k
2
)
2
,
ˆ
u = ( p
1
− k
2
)
2
= ( p
2
− k
1
)
2
. (12.79)
The differential cross section in the centre of mass system is given in terms of
|M
q˜q
|
2
by (see, for example, [15] Section 6.3.4)
dσ
d
=
1
4 p
ˆ
W
p
16π
2
ˆ
W
2
|M
q˜q
|
2
, (12.80)
where
ˆ
W =
√
ˆ
s, and p, p
are the magnitudes of the initial and final state momenta,
in the CM system. The kinematics is simplified if we neglect the quark masses, and
198 Some simple tree-level calculations in the MSSM
q
L
q
q
L
p
2
,s
2
,c
2
p
1
,s
1
,c
1
k
1
,c
1
′
k
2
,c
2
′
q
g
~
~
~
Figure 12.5 Lowest-order exchange diagram for the process qq → ˜q
L
˜q
L
.
assume the final squarks have equal mass. Then
d
ˆ
t =
ˆ
Wp
d cos θ =
ˆ
Wp
d/2π, p =
ˆ
W/2. (12.81)
Putting all this together, we arrive at
dσ
d
ˆ
t
(q
1
q
2
→ ˜q
1L
˜q
2R
) =
2πα
2
s
9
ˆ
s
2
−
ˆ
t − m
2
˜q
L
ˆ
t − m
2
˜q
R
−
ˆ
s
ˆ
t
ˆ
t − m
2
˜g
2
(12.82)
in agreement with formula (A.7d) of [49].
Exercise 12.2 Show that the cross section for q
1
q
2
→ ˜q
1L
˜q
2L
is
dσ
d
ˆ
t
(q
1
q
2
→ ˜q
1L
˜q
2L
) =
2πα
2
s
9
m
2
˜g
ˆ
s
ˆ
t − m
2
˜g
2
(12.83)
in agreement with formula (A.7e) of [49].
A new complication arises when we consider the analogous calculations for the
case in which the initial state quarks are identical, q
1
= q
2
, for example
qq → ˜q
L
˜q
L
. (12.84)
There is now a ‘direct’ amplitude of the form (12.75), corresponding to Figure 12.4,
which is obtained straightforwardly as
M
(d)
q˜q
= 2g
2
s
ω
†
1
λ
a
2
ω
1
ω
†
2
λ
a
2
ω
2
¯v( p
1
, s
1
)P
L
1
/k
1
− /p
1
− m
˜g
P
L
u( p
2
, s
2
)(−i)
2θ
˜g
.
(12.85)
In addition, to ensure antisymmetry for identical fermions, we must subtract from
this the ‘exchange’ amplitude, corresponding to the diagram of Figure 12.5, given by
12.2 Sparticle production processes 199
the interchanges p
1
↔ p
2
, s
1
↔ s
2
, c
1
↔ c
2
, so that the total amplitude is M
(d)
q˜q
+
M
(e)
q˜q
where
M
(e)
q˜q
=−2g
2
s
ω
†
1
λ
a
2
ω
2
ω
†
2
λ
a
2
ω
1
¯v( p
2
, s
2
)P
L
1
/k
1
− /p
2
− m
˜g
P
L
u( p
1
, s
1
)(−i)
2θ
˜g
.
(12.86)
This result can, of course, also be obtained by evaluating the matrix element of the
relevant interaction. The cross section will therefore involve the sum of three terms:
the square of the direct amplitude
M
(d)
q˜q
2
, (12.87)
the square of the exchange amplitude
M
(e)
q˜q
2
, (12.88)
and the interference term
2Re
M
(d)
q˜q
M
(e)∗
q˜q
. (12.89)
The part of the cross section arising from (12.87) is given by (12.83) as before, and
the part from (12.88) is the same but with
ˆ
t replaced by
ˆ
u; we must also remember
to include an overall factor of 1/2 due to identical particles in the final state.
The evaluation of the interference contribution is more involved. Consider first
the colour factor, which (apart from the averaging factor 1/9) is
c
1
,c
2
,c
1
,c
2
,a,b
ω
†
1
λ
a
2
ω
1
ω
†
1
λ
b
2
ω
2
ω
†
2
λ
a
2
ω
2
ω
†
2
λ
b
2
ω
1
=
1
16
c
1
,a,b
ω
†
1
λ
a
λ
b
λ
a
λ
b
ω
1
=
1
16
a,b
Tr(λ
a
λ
b
λ
a
λ
b
). (12.90)
Nowwehave
λ
b
λ
a
= λ
a
λ
b
− 2i f
abc
λ
c
, (12.91)
where the coefficients f
abc
are as usual the structure constants of SU(3). The part
of (12.90) not involving the f s is then
Tr
a
[(λ
a
/2)(λ
a
/2)]
b
[(λ
b
/2)(λ
b
/2)] = Tr
4
3
4
3
I
3
= 16/3 (12.92)
where I
3
is the unit 3 × 3 matrix. In (12.92) we have used the fact that
a
[(λ
a
/2)(λ
a
/2)] is the Casimir operator C
2
of SU(3) (see [7], Section M.5),
200 Some simple tree-level calculations in the MSSM
having the value 4/3 I
3
in the representation 3. The remaining part of (12.90) is
2i
16
a,b,c
f
bac
Tr(λ
b
λ
a
λ
c
) =
i
16
a,b,c
f
bac
Tr([λ
b
,λ
a
]λ
c
)
=−
1
8
a,b,c,d
f
bac
f
bad
Tr(λ
d
λ
c
) =−
1
4
a,b,c
f
bac
f
bac
. (12.93)
To evaluate the product of f s, we note that the generators of SU(3) in the repres-
entation 8 are given by (see [7], equation (12.84))
G
(8)
a
bc
=−i f
abc
, (12.94)
and that the value of C
2
in this representation is 3I
8
, where I
8
is the unit 8 × 8
matrix, so that
a,b,c
G
(8)
a
bc
G
(8)
a
cb
= 24. (12.95)
Hence
a,b,c
f
abc
f
abc
= 24, (12.96)
and expression (12.93) is equal to −6. Combining this result with (12.92), we find
that (12.90) equals −2/3.
The spinor part of the interference term (12.89) is
−¯v(p
1
, s
1
)P
L
1
/k
1
− /p
1
− m
˜g
P
L
u( p
2
, s
2
)
¯
u( p
1
, s
1
)P
R
1
/k
1
− /p
2
− m
˜g
P
R
v( p
2
, s
2
),
(12.97)
which has to be summed over s
1
and s
2
(with a factor of 1/4 for the spin average). As
it stands, (12.97) is not in a suitable form for using the standard Trace techniques:
first of all, spinors referring to the same spin and momenta variables need to be
adjacent to each other, and secondly we need expressions of the form u
¯
u and v ¯v,
not ¯v
¯
u and uv. To deal with the first difficulty, we write out (12.97) including all
the spinor labels explicitly, and rearrange it as
−¯v
α
( p
1
, s
1
)
¯
u
λ
( p
1
, s
1
)(P
R
)
λμ
1
/k
1
− /p
2
− m
˜g
μν
(P
R
)
ντ
v
τ
( p
2
, s
2
)
×u
δ
( p
2
, s
2
)
P
T
L
δγ
1
/k
1
− /p
1
− m
˜g
T
γβ
P
T
L
βα
, (12.98)
where ‘
T
’ denotes the transpose. We now use
¯v
α
( p
1
, s
1
) = C
T
ασ
u
σ
( p
1
, s
1
), and v
τ
( p
2
, s
2
) = C
τη
¯
u
η
( p
2
, s
2
) (12.99)
12.2 Sparticle production processes 201
from (2.133), which enables the spin-averaged expression to be written as
−
1
4
Tr
C
T
/p
1
P
R
1
/k
1
− /p
2
− m
˜g
P
R
C/p
2
T
P
L
1
(/k
1
− /p
1
− m
˜g
)
T
P
L
(12.100)
where we have used γ
T
5
= γ
5
, and taken the quarks to be massless. We now note
that
Cγ
T
μ
=−γ
μ
C and CC
T
= 1, (12.101)
so that (12.100) becomes
−1
4
ˆ
u − m
2
˜g
ˆ
t − m
2
˜g
Tr[/p
1
P
R
(/k
1
− /p
2
+ m
˜g
)P
R
/p
2
P
L
(/k
1
− /p
1
− m
˜g
)P
L
]
=
m
2
˜g
4
ˆ
u − m
2
˜g
ˆ
t − m
2
˜g
Tr[/p
1
/p
2
P
L
] =
m
2
˜g
ˆ
s
4
ˆ
u − m
2
˜g
ˆ
t − m
2
˜g
. (12.102)
Remembering now the factor of 2 in (12.89), the result (12.83) (and the corres-
ponding one with
ˆ
t replaced by
ˆ
u), and the overall factor of 1/2, we find that the
cross section for qq → ˜q
L
˜q
L
is
dσ
d
ˆ
t
(qq → ˜q
L
˜q
L
) =
πα
2
s
m
2
˜g
9
ˆ
s
1
ˆ
t − m
2
˜g
2
+
1
ˆ
u − m
2
˜g
2
−
2/3
ˆ
t − m
2
˜g
ˆ
u − m
2
˜g
(12.103)
in agreement with formula (A.7i) of [49]. Similar manipulations are presented in
Appendix E of [45].
Exercise 12.3 Show that the interference term vanishes (in the limit of vanishing
quark mass) for the case qq → ˜q
L
˜q
R
, and hence that the cross section is
dσ
d
ˆ
t
(qq → ˜q
L
˜q
R
)
=
2πα
2
s
9
ˆ
s
2
−
ˆ
t − m
2
˜q
L
ˆ
t − m
2
˜q
R
−
ˆ
s
ˆ
t
ˆ
t − m
2
˜g
2
+
−
ˆ
u − m
2
˜q
L
ˆ
u − m
2
˜q
R
−
ˆ
s
ˆ
u
ˆ
u − m
2
˜g
2
(12.104)
in agreement with formula (A.7j) of [49].
The expressions for the cross sections of squark (or gluino) production in qq
collisions are very similar to those obtained from standard QCD tree graphs (see,
for example, [7] Section 14.3); the main qualitative difference is that the propagator
factor
ˆ
t
−2
for the massless gluon is replaced by (
ˆ
t − m
2
˜g
)
−2
for the massive gluino
(and similarly for the
ˆ
u-channel terms). For an order of magnitude estimate, we
202 Some simple tree-level calculations in the MSSM
2000
10
6
10
4
10
2
10
0
500
1000 1500
s = 14 TeV
m = +m
g
= m
q
CTEQ5L PDFs
gq
qq
~
~
~~
~~
~~
~
gg
s (fb)
m
g
(GeV)
Figure 12.6 Cross sections for squark and gluino production at the CERN LHC pp
collider for m
˜q
= m
˜g
(solid) and for m
˜q
= 2m
˜g
(dashed). [Figure reprinted with
permission from Weak Scale Supersymmetry by H. Baer and X. Tata (Cambridge:
Cambridge University Press, 2006), p. 318.]
may set
σ ∼
2πα
2
s
9
ˆ
s
∼ 250 fb (12.105)
for α
s
∼ 0.15 and
√
ˆ
s = 5 TeV. The initial state quarks are, of course, constituents
of hadrons, and so these parton-level cross sections must be convoluted with ap-
propriate parton distribution functions to obtain the cross sections for physical
production processes in hadron–hadron collisions; see, for example, [118]. As an
illustration of the predictions, we show in Figure 12.6 (taken from [49]) the cross
sections for squark and gluino production at the CERN LHC pp collider. For m
˜g
and m
˜q
less than about 1 TeV, and an integrated luminosity of 10 fb
−1
, one expects
some thousands of ˜g, ˜q events at the LHC.
We now turn to sparticle production via electroweak interactions.
12.2.2 Slepton and sneutrino pair production in q¯q collisions
We consider first the production of a charged slepton in association with its sneutrino
partner
d¯u →
˜
l
L
¯
˜ν
L
, (12.106)
12.2 Sparticle production processes 203
d
u
p
1
,s
1
p
2
,s
2
~
W
−
k
1
k
2
k
n
L
l
L
Figure 12.7 Lowest-order diagram for the process d¯u →
˜
l
L
¯
˜ν
L
.
proceeding via W
−
-exchange in the
ˆ
s-channel, as shown in Figure 12.7. The 4-
momenta and spins of the d and ¯u quarks are p
1
, s
1
and p
2
, s
2
, and the 4-momenta
of the slepton and sneutrino are k
1
and k
2
; colour labels are suppressed. The
SM interaction at the first vertex is contained in the quark analogue of (8.34),
and is
L
qW
=−
g
√
2
V
ud
¯
uL
γ
μ
dL
W
μ
, (12.107)
where (see, for example, [7] equation (22.25))
W
μ
= (W
1μ
− iW
2μ
)/
√
2 (12.108)
is the field which destroys the W
+
or creates the W
−
, and V
ud
is the appropriate
element of the CKM matrix. The interaction at the sparticle vertex is contained in
the SU(2) gauge invariant kinetic term (see (7.67))
D
μ
φ
˜
l
L
†
D
μ
φ
˜
l
L
(12.109)
where
φ
˜
l
L
=
˜ν
L
˜
l
L
, and D
μ
= ∂
μ
+ ig
τ
2
· W
μ
. (12.110)
The relevant term is
L
Wsl
=−
g
√
2
i(
˜
l
†
L
∂
ν
˜ν
L
− (∂
ν
˜
l
†
L
)˜ν
L
)W
†
ν
. (12.111)
Note that (12.111) is the same as the hermitian conjugate of (12.107) with the
fermionic current replaced by the corresponding bosonic one, and with V
ud
→ 1.
The amplitude for Figure 12.7 is then
˜
l
L
, k
1
;
¯
˜ν
L
, k
2
d
4
xd
4
yT
iL
q W
(x)iL
Wsl
(y)
d, p
1
, s
1
;¯u, p
2
, s
2
. (12.112)
204 Some simple tree-level calculations in the MSSM
Z
0
p
1
,s
1
p
2
,s
2
u
u
t
1
t
2
k
1
k
2
~
~
Figure 12.8 Z
0
exchange diagram for the process u¯u → ˜τ
1
¯
˜τ
2
.
The reduction of the final state sleptons leads to a factor i(k
ν
2
− k
ν
1
) from the deriva-
tives in L
Wsl
, and (12.112) becomes
(2π)
4
δ
4
( p
1
+ p
2
− k
1
− k
2
)iM
ch sl
, (12.113)
where
M
ch sl
=
g
2
2
V
ud
¯v( p
2
, s
2
)γ
μ
P
L
u( p
1
, s
1
)
−g
μν
+ k
μ
k
ν
/m
2
W
ˆ
s − m
2
W
+ im
W
W
k
ν
2
− k
ν
1
(12.114)
and k = k
1
+ k
2
= p
1
+ p
2
. The term
¯v( p
2
, s
2
)γ
μ
P
L
u( p
1
, s
1
)k
μ
= ¯v(p
2
, s
2
)(/p
1
+ /p
2
)P
L
u( p
1
, s
1
) (12.115)
vanishes in the massless quark limit. The spinor factor in the cross section is
1
12
Tr[/p
2
(/k
2
− /k
1
)P
L
/p
1
P
R
(/k
2
− /k
1
)] =
1
24
Tr[/p
2
(/k
2
− /k
1
)/p
1
(/k
2
− /k
1
)]
=
1
3
ˆ
t
ˆ
u − m
2
˜
l
m
2
˜ν
, (12.116)
where
ˆ
t = ( p
1
− k
1
)
2
,
ˆ
u = ( p
1
− k
2
)
2
, and the factor of 1/3 comes from the colour
average. The cross section is then
dσ
d
ˆ
t
(d¯u →
˜
l
L
¯
˜ν
L
) =
g
4
|V
ud
|
2
192π
ˆ
s
2
1
ˆ
s − m
2
W
2
+ m
2
W
2
W
ˆ
t
ˆ
u − m
2
˜
l
L
m
2
˜ν
L
(12.117)
in agreement with equation (A.14) of [49], setting V
ud
to unity.
An analogous neutral current process is
u¯u → ˜τ
1
¯
˜τ
2
(12.118)
proceeding via Z
0
exchange in the
ˆ
s channel, as shown in Figure 12.8. We take the
4-momenta and spins of the u and ¯utobe p
1
, s
1
and p
2
, s
2
, and the 4-momenta of