93 the scalar and fermion lines in the loop are

loop graph shown in Fig. 9.3. The scalar and fermion lines in the loop
are messenger fields. Recall that the interaction vertices in Fig. 9.3
are of gauge coupling strength even though they do not involve gauge
bosons; compare Fig. 6.3g. In this way, gauge-mediation provides that
q, q messenger loops give masses to the gluino and the bino, and ,
messenger loops give masses to the wino and bino fields. By computing
the 1-loop diagrams one finds that the resulting MSSM gaugino masses
are given by
M
a
=
α
a
4 π
Λ , ( a = 1 , 2 , 3) , (9.40)
(in the normalization discussed in section 8.4) where we have introduced
a mass parameter
Λ ≡ F
S
/ S . (9.41)

222 9 Origins of supersymmetry breaking
Fig. 9.4. Contributions to MSSM scalar squared masses in gauge-mediated
supersymmetry breaking models arise in leading order from these two-loop
Feynman graphs.
(Note that if F
S
were 0, then Λ = 0 and the messenger scalars
would be degenerate with their fermionic superpartners and there would
be no contribution to the MSSM gaugino masses.) In contrast, the
corresponding MSSM gauge bosons cannot get a corresponding mass shift,
since they are protected by gauge invariance. So supersymmetry breaking
has been successfully communicated to the MSSM (“visible sector”). To
a good approximation, eq. (9.40) holds for the running gaugino masses at
an RG scale Q
0
corresponding to the average characteristic mass of the
heavy messenger particles, roughly of order M
mess
∼ y
i
S . The running
mass parameters can then be RG-evolved down to the electroweak scale
to predict the physical masses to be measured by future experiments.
The scalars of the MSSM do not get any radiative corrections to their
masses at one-loop order. The leading contribution to their masses
comes from the two-loop graphs shown in Fig. 9.4, with the messenger
fermions (heavy solid lines) and messenger scalars (heavy dashed lines)
and ordinary gauge bosons and gauginos running around the loops. By
computing these graphs, one finds that each MSSM scalar φ gets a (mass)
2
given by:
m
2
φ
= 2Λ
2
α
3
4 π
2
C
φ
3
+
α
2
4 π
2
C
φ
2
+
α
1
4 π
2
C
φ
1
. (9.42)
Here C
φ
a
are the quadratic Casimir group theory invariants for the scalar
φ for each gauge group. They are defined by C
φ
a
δ
j
i
= ( T
a
T
a
)
j
i
where the
T
a
are the group generators which act on the scalar φ . Explicitly, they

9.4 Gauge-mediated supersymmetry breaking models 223
are:
C
φ
3
=
⎧
⎨
⎩
4 / 3 for φ = Q
i
, u
i
, d
i
;
0 for φ = L
i
, e
i
, H
u
, H
d
(9.43)
C
φ
2
=
⎧
⎨
⎩
3 / 4 for φ = Q
i
, L
i
, H
u
, H
d
;
0 for φ = u
i
, d
i
, e
i
(9.44)
C
φ
1
= 3 Y
2
φ
/ 5 for each φ with weak hypercharge Y
φ
. (9.45)
The squared masses in eq. (9.42) are positive (fortunately!).
The terms a
u
, a
d
, a
e
arise first at two-loop order, and are suppressed
by an extra factor of α
a
/ (4 π ) compared to the gaugino masses. So, to a
very good approximation one has, at the messenger scale,
a
u
= a
d
= a
e
= 0 , (9.46)
a significantly stronger condition than eq. (8.16). Again, eqs. (9.42) and
(9.46) should be applied at an RG scale equal to the average mass of
the messenger fields running in the loops. However, after evolving the