Labelle P. Supersymmetry DeMYSTiFied
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356
Supersymmetry Demystified
Kinetic Terms of the Gauge Bosons
The gauge-invariant kinetic energy terms of the U (1)
Y
, SU(2)
L
, and SU(3)
C
gauge
fields are, respectively (see Section 10.3 for a review of nonabelian gauge theories),
L
GB
=−
1
4
F
μν
F
μν
−
1
2
Tr
W
μν
W
μν
−
1
2
Tr
G
μν
G
μν
where
F
μν
≡ ∂
μ
B
ν
− ∂
ν
B
μ
W
μν
≡ ∂
μ
W
ν
− ∂
ν
W
μ
−ig[W
μ
, W
ν
]
G
μν
≡ ∂
μ
G
ν
− ∂
ν
G
μ
−ig
s
[G
μ
, G
ν
]
The weak bosons and gluon fields appearing in those equations are, of course, the
matrix-valued quantities:
W
μ
≡ W
i
μ
τ
i
2
G
μ
≡ G
a
μ
λ
a
2
Unfortunately, we are using the same symbol, λ
a
, for the Gell-Mann matrices and for
the spin components of the fermion superpartners of the gauge fields, the gauginos
λ
a
, but the context is usually clear enough to avoid any confusion.
Kinetic Term of the Higgs Field
Consider now the Higgs field H , which has quantum numbers 1, 2, 1. It is a singlet
with respect to the color group, so it does not interact directly with the gluons (there
are, of course, indirect interactions, i.e., interactions through loop diagrams), and
it is an SU(2)
L
doublet that we will write as
H =
H
+
H
0
where each of the fields H
+
and H
0
are complex scalar fields, and the labels indicate
their electric charge. The kinetic term is given by
(D
μ
H)
†
D
μ
H (15.2)
CHAPTER 15 Introduction to the MSSM
357
where D
μ
is the gauge-covariant derivative corresponding to the quantum numbers
given above, namely,
D
μ
H =
∂
μ
+ig
τ
i
2
W
i
μ
+i
g
2
B
μ
H (15.3)
Interaction Terms
Let us now turn our attention to the interaction terms. The interactions between the
gauge bosons and the other fields (fermions and Higgs) all arise from the gauge-
covariant derivatives listed earlier. The self-interactions of the nonabelian gauge
bosons are all contained in their kinetic terms.
We’ll turn our attention now to the interactions between the fermions and the
Higgs field, which are all given by Yukawa terms. Let’s first restrict our attention
to the fermions of the first generation; the generalization to all three generations
will be straightforward.
The Higgs field is an SU(2) doublet, so it must be combined with another
SU(2) doublet to give a quantity invariant under SU(2) transformations. On the
other hand, it is a color singlet, so it must be combined with a color singlet (i.e.,
a color-invariant). Finally, it has a hypercharge of 1, so it must be combined with
a quantity with hypercharge −1. To summarize, the Higgs field must be coupled
with a quantity that has quantum numbers 1,
¯
2, −1. We therefore need to build a
quantity out of two fermion fields that have these quantum numbers.
Let’s first consider only the leptons. One possibility is the combination
L
e
R
(15.4)
This is an SU(2) antidoublet (because of the hermitian conjugation) and a color
singlet. Taking into account the fact the Dirac conjugate
L
=
†
L
γ
0
(15.5)
changes the sign of the hypercharge (because of the hermitian conjugate), we see
that Eq. (15.4) has hypercharge 1 − 2 =−1 and therefore has all the right quantum
numbers to be coupled with the Higgs field in a gauge-invariant manner. A gauge
invariant interaction is therefore
L
H
e
R
(15.6)
358
Supersymmetry Demystified
We must, of course, multiply it by an arbitrary constant that we will call y
e
,
the Yukawa coupling for the electron field (do not confuse Yukawa couplings,
represented by lower case y, with the hypercharge assignments, denoted by capital
Y ). This constant is dimensionless because Eq. (15.6) has dimension 3/2 + 1 +
3/2 = 4. We also must add to Eq. (15.6) its hermitian conjugate to get a real
contribution to the lagrangian of the standard model. The final result is that we
must include the following contribution to the standard model:
y
e
L
H
e
R
+ h.c.
On SSB, this term generates the electron mass.
There is also a Yukawa coupling between the Higgs field and the quarks. A quark
combination that has quantum numbers 1, 2, −1is
Q
1
d
R
, so we have a second
gauge-invariant Yukawa term to write:
y
d
Q
L
H
d
R
+ h.c. (15.7)
After SSB, this term will generate a mass for the down quark.
In order to obtain a mass for the up quark as well, we must find a way to couple
the Higgs field with
Q
L
and
u
R
. This works okay with respect to the SU(2) and
SU(3) quantum numbers, but we run into trouble with the hypercharge. A term
Q
L
H
u
R
has a total hypercharge of −1/3 + 1 + 4/3 = 2, so it is not invariant
under U (1)
Y
. The solution is to use the fact that the quantity i τ
2
H
†T
transforms
the same way as H itself under SU(2)
L
, i.e. as a doublet (see for example Section
12.1.3 of Ref. 2). This quantity therefore has quantum numbers 1, 2, −1 so the
interaction
¯
Q
L
iτ
2
H
†T
u
R
is gauge invariant.
We will have to combine SU(2) doublets using the matrix i τ
2
very often in
the SM and in the MSSM, so it would be nice to have a shorthand notation for
this product. Some authors write
Q
L
iτ
2
H
†T
simply as
Q
L
· H
†T
(or, rather, as
Q
L
· H
∗
) but this might be mistaken for the spinor dot product notation we have
introduced in chapter 2. Others actually leave out the iτ
2
altogether, which is even
more confusing.
To make the notation as clear as possible, we will introduce the following explicit
(but nonstandard) definition
Q
L
iτ
2
H
†T
≡
Q
L
◦ H
†T
From now on, whenever you encounter the symbol ◦, it will appear between two
SU(2) doublets and will mean that a matrix iτ
2
must be inserted there.
CHAPTER 15 Introduction to the MSSM
359
Using this notation, our third gauge invariant Yukawa term is
y
u
Q
L
◦ H
†T
u
R
+ h.c. (15.8)
Recall that by H
†T
, we mean the column vector whose components are the
Hermitian conjugates of the fields, i.e.,
H
†T
=
(H
+
)
†
(H
0
)
†
=
H
−
H
0†
The fact that we absolutely need H
†
as well as H to obtain masses for the quarks (af-
ter SSB) in the standard model will have profound consequences when we consider
its supersymmetric extension.
The next step is to generalize the Yukawa interactions to three generations:
L
Yu k
= y
ij
e
(
L
)
i
H(
e
R
)
j
+ y
ij
d
(
Q
L
)
i
H(
Q
R
)
j
+ y
ij
u
(
Q
L
)
i
◦ H
†T
(
Q
R
)
j
+ h.c.
(15.9)
where it is implicit that the subscripts i and j label the generations.
The Higgs Potential
The only thing missing is the Higgs potential. It is conventionally written as
V (H ) =−μ
2
H
†
H + λ(H
†
H)
2
(15.10)
and the constants μ
2
and λ are chosen to be positive in order to generate SSB. We
now turn our attention to SSB in the standard model. This is a very useful warm
up in preparation for the study of symmetry breaking in the MSSM, which is much
more involved than in the standard model.
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Supersymmetry Demystified
15.2 Spontaneous Symmetry Breaking
in the Standard Model
Spontaneous symmetry breaking in the standard model is very simple; one can see
by simple inspection that Eq. (15.10) has a minimum at the condition that both λ
and μ
2
are positive real numbers and that the minimum occurs when the fields have
a vacuum expectation value (vev) given by
H
†
H≡v
2
with v =
μ
√
2λ
We may always use an SU(2) transformation to bring the vev of H in the standard
form, i.e.,
H
min
=
0
v
In another popular normalization, the vev is instead defined as
H
†
H≡
˜v
2
2
with ˜v =
μ
√
λ
(15.11)
The next step is to define a shifted field H
0
, i.e.,
H
0
≡ H
0
− v
that has a zero vev, i.e., H
0
=0. We now replace H
0
by H
0
+ v in the kinetic energy
of the Higgs field, [Eq. (15.2)]. The gauge-covariant derivative D
μ
H becomes
D
μ
H = D
μ
H
+
H
0
+ D
μ
0
v
(15.12)
with D
μ
given in Eq. (15.3). To be explicit, the second term of Eq. (15.12) is
D
μ
0
v
=
iv
2
gW
1μ
−igW
2μ
−gW
3μ
+ g
B
μ
Inserting this into the kinetic energy term of the Higgs [Eq. (15.2)] and writing
down only the terms bilinear or quadratic in the gauge fields, which will contribute
CHAPTER 15 Introduction to the MSSM
361
to the masses of the gauge bosons, we get
L
MGB
=
v
2
4
g
2
W
2
1
+ W
2
2
+
v
2
4
(gW
3
− g
B)
2
(15.13)
We finally introduce the W
±
and Z gauge fields by defining
L
MGB
≡ m
2
W
W
+μ
W
−
μ
+
1
2
m
2
Z
Z
μ
Z
μ
with
m
2
W
W
+μ
W
+†
μ
≡
v
2
g
2
4
W
2
1
+ W
2
2
=
v
2
g
2
4
W
1μ
+iW
2μ
W
μ
1
−iW
μ
2
We define
W
±
μ
≡
1
√
2
(W
1μ
±iW
2μ
)
where the factor of 1/
√
2 is simply a normalizing factor. This identification then
implies the important relation
m
2
W
=
v
2
g
2
2
(15.14)
In addition, by considering weak interactions involving the W bosons at small
energies, one can relate the ratio g
2
/m
2
W
to Fermi’s constant G
F
. The explicit
relation is
g
2
8m
2
W
=
G
F
√
2
(15.15)
Let us turn our attention to the Z boson. We define
1
2
m
2
Z
Z
μ
Z
μ
≡
v
2
4
(gW
3
− g
B)
2
(15.16)
362
Supersymmetry Demystified
so that Z is proportional to the combination gW
3μ
− g
B
μ
. Normalizing, we there-
fore have
Z
μ
=
1
g
2
+ g
gW
3μ
− g
B
μ
Using this in Eq. (15.16), we get a relation for the mass of the Z:
m
2
Z
=
v
2
2
(g
2
+ g
) (15.17)
Obviously, the linear combination orthogonal to Z
μ
is massless. This corresponds
to the photon field:
A
μ
≡
1
g
2
+ g
2
g
W
3μ
+ gB
μ
It is conventional to introduce an angle relating the two coupling constants called
the Weinberg (or weak) mixing angle θ
W
through the relation
tan θ
W
≡
g
g
This implies
cos θ
W
=
g
g
2
+ g
sin θ
W
=
g
g
2
+ g
(15.18)
in which case the Z and photon fields take a nicer form:
Z
μ
= cos θ
W
W
3μ
− sin θ
W
B
μ
A
μ
= sin θ
W
W
3μ
+ cos θ
W
B
μ
Inverting those relations, we find
W
3μ
= cos θ
W
Z
μ
+ sin θ
W
A
μ
B
μ
= cos θ
W
A
μ
− sin θ
W
Z
μ
CHAPTER 15 Introduction to the MSSM
363
Particles couple to the B
μ
with a strength g
; therefore, they will couple to the
photon field A
μ
with a strength g
cos θ
w
, which, by definition, is the electromagnetic
coupling constant e, i.e.,
e = g
cos θ
W
(15.19)
Applying the same reasoning to the W
3μ
we also get
e = g sin θ
W
(15.20)
We also could have obtained Eq. (15.20) directly from Eq. (15.19) using the defi-
nition of the Weinberg angle.
Note that from Eqs. (15.17), (15.13), and (15.18), we get
m
W
= cos θ
W
m
Z
so these three parameters are not independent. This relation is often expressed in
terms of the so-called ρ parameter, defined as
ρ ≡
m
2
W
m
2
Z
cos
2
θ
W
(15.21)
which is therefore equal to 1 at tree level. This prediction of the standard model
is confirmed experimentally and is a key test of the gauge structure and pattern of
symmetry breaking of the theory. One of the challenges of theories “beyond the
standard model” is to reproduce this result.
Let us summarize the situation regarding SSB in the gauge sector of the standard
model. The lagrangian contains four parameters: g, g
,μ and λ. Physical results
depend only on g, g
, and the combination μ/
√
2λ ≡ v. These three parameters
can be expressed in terms of any three measurements among G
F
, e, M
Z
, M
W
, and
θ
W
. In using these relations, one must keep in mind that coupling constants run
with energy, so one must be careful to use all values evaluated at a common scale.
We will discuss the running of coupling constants and the measured values of the
standard model parameters more in Section 16.5. Here, let us simply mention that
experimental results can be used to calculate
v ≈ 174 GeV
g ≈ 0.651
g
≈ 0.357
364
Supersymmetry Demystified
where all values are evaluated at the Z mass. With the normalization (15.11), we
have ˜v ≈ 246 GeV, which you may have seen quoted in the literature.
Now we turn our attention to the MSSM. But before doing this, a short comment
about notation is in order.
15.3 Aside on Notation
In writing down the MSSM, we are going to use a notation that will be the obvious
generalization of the one introduced to describe supersymmetric quantum electro-
dynamics (QED) in Section 13.6. If you haven’t covered that section yet, or if it
has been quite a while, it would be a good idea to read (or reread) it before tackling
the MSSM. To summarize,
•
We will work only with left-chiral Weyl spinors. The right-chiral particle
states are traded off for the corresponding left-chiral antiparticle spinors.
•
We will use calligraphic capital letters for all the superfields (E, F, U, etc).
This will make it easier to distinguish superfields from “ordinary” fields (i.e.,
fields that are functions of spacetime only).
•
All the scalar fields appearing in left-chiral multiplets with one exception,
will be represented by φ (with labels indicating which multiplet it belongs
to). The exception will be for the Higgs scalar fields, which will be denoted
by H (with some labels). All the left-chiral Weyl spinors will be denoted by
χ, and all the auxiliary fields will be written as F, with appropriate labels.
•
The gauge fields will be denoted by the same symbols as in the standard
model: B
μ
for the hypercharge gauge field, A
a
μ
for the color fields, and W
i
μ
for
the weak SU(2) fields. Their supersymmetric left-chiral Weyl partners will be
denoted by λ with appropriate labels (see next section), and the corresponding
auxiliary fields will be written as D.
•
A bar over the name of a particle denotes the corresponding antiparticle. For
example, χ
¯
u
is the left-chiral state of the anti-up quark.
•
We will use a tilde over the symbols of supersymmetric partners of known
particles, which are all particles that haven’t been observed yet. For example,
˜
φ
¯
u
is the scalar SUSY partner of the left-chiral anti-up quark, and
˜
λ
a
C
will be
the eight spinor partners of the color gauge fields.
There is no universal standard for the notation used in presentation of the MSSM.
The notation introduced here has been chosen to maximize clarity, hopefully.
CHAPTER 15 Introduction to the MSSM
365
15.4 The Left-Chiral Superfields of the MSSM
We are now ready to construct the superfields of the MSSM. An important point
is that as we saw in Chapter 10, supersymmetry (SUSY) transformations commute
with gauge transformations. This implies that all the fields making up a SUSY
multiplet must have the same quantum numbers under SU (3)
C
× SU(2)
L
×U (1)
Y
.
This is a key observation.
As mentioned several times, we trade the right-chiral particle states for the left-
chiral antiparticle states. Let’s start with the left-chiral positron state χ
¯
e
, which has
quantum numbers 1, 1, 2. Note that the sign of the hypercharge is opposite to the
one of the left-chiral electron spinor χ
e
. The corresponding superfield is identical
to the one we used in the supersymmetric extension of QED (see Section 13.6),
with the difference that we will use a numeral to refer to the generation to which
the corresponding component fields belong, i.e.,
E
1
=
˜
φ
¯
e
+ θ ·χ
¯
e
+
1
2
θ ·θ F
¯
e
(15.22)
The corresponding superfields for the second and third generations will be written
as
E
2
≡
˜
φ
¯μ
+ θ ·χ
¯μ
+
1
2
θ ·θ F
¯μ
E
3
≡
˜
φ
¯τ
+ θ ·χ
¯τ
+
1
2
θ ·θ F
¯τ
In the MSSM, as in the standard model, there is no right-chiral neutrino field.
Consider now the left-chiral state of the electron. This state is paired up with the
electron neutrino left-chiral state to form the SU(2)
L
doublet of the first-generation
leptons familiar from the standard model:
χ
ν
e
χ
e
with quantum numbers 1, 2, −1. We will write the corresponding superfield as
L
1
≡
˜
φ
ν
e
˜
φ
e
+ θ ·
χ
ν
e
χ
e
+
1
2
θ ·θ
F
ν
e
F
e
where the index 1 on the lepton superfield refers to the generation. Note that in this
chapter and the next one, a calligraphic L with a numerical subscript will represent
a lepton superfield, not a lagrangian!