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Introduzione alle teorie supersimmetriche
Giovanni Ridolfi - INFN Genova
Dottorato di Ricerca - Pavia, 6-8 Giugno 2007
Lay-out:
1. Why supersymmetry
2. General features of a supersymmetric theory
3. The minimal supersymmetric Standard Model
4. Breaking supersymmetry
5. Present experimental status
6. Conclusions and outlook
In many respects
we are happy with the Standard Model
• a unitary, renormalizable relativistic quantum field theory
• amazingly consistent with data.
The problem of spontaneous gauge symmetry breaking (short
range of weak interactions) is solved in a simple way by the Higgs
mechanism.
This also allows breaking of the flavour symmetry in a
phenomenologically consistent way (flavour breaking confined in
the charged-current interaction sector).
Achieving the same results without the Higgs mechanism is
extremely difficult.
At the same time,
we are unhappy with the Standard Model.
Many unsatisfactory aspects. Among others:
• What is the origin of flavour symmetry breaking?
• Is there a grand unification?
• Where is gravitation?
The last two points raise further problems:
• Hierarchy
• Naturalness
Two facts:
• The masses of all known particles (including the minimal
Standard Model Higgs boson) are not far from the weak scale,
∼ 200 GeV.
• Much larger energy scales become relevant at some point
A first question arises: why is the weak scale so much smaller than
the Planck scale, MP ∼ 1019 GeV, or the unification scale,
MGUT ∼ 1016 GeV?
This is usually referred to as the hierarchy problem.
Even worse: The Higgs mass (masses of scalar particles, in general)
is strongly sensitive to any large energy scale unless a fine tuning of
parameters is performed.
This is the so-called naturalness problem.
There is a very simple reason for this: scalar masses are not
naturally small, in the sense that no symmetry is recovered when
they are let go to zero.
Fermion and vector boson masses are naturally small: radiative
corrections are proportional to the masses themselves.
Naturalness and fine tuning in a simple example
Consider a theory of two real scalars fields:
L =1
2∂µφ∂µφ+
1
2∂µΦ ∂µΦ − V (φ,Φ)
with
V (φ,Φ) =m2
2φ2 +
M2
2Φ2 +
λ
4!φ4 +
σ
4!Φ4 +
δ
4φ2Φ2
Assume λ, σ, δ are all positive, small and comparable in
magnitude, and assume M 2 � m2 > 0.
Is the mass hierarchy m2 �M2 conserved at the quantum level?
Compute one-loop radiative corrections to m2 (e.g. by taking the
second derivatives of the effective potential at the minimum
φ = Φ = 0):
m2one loop = m2(µ2) +
λm2
32π2
(
logm2
µ2− 1
)
+δM2
32π2
(
logM2
µ2− 1
)
µ2 ∂m2
∂µ2=
1
32π2
(
λm2 + δM2)
Corrections proportional to M 2 appear at one loop. One can
choose µ2 ∼M2 in order to get rid of them, but they reappear
through the running of m2(µ2).
The mass hierarchy is preserved only if the parameters are such
that
λm2 ∼ δM2 → δ
λ∼ m2
M2
This is what we usually call a fine tuning of the parameters.
The same thing happens if m2 < 0, M2 �∣
∣m2∣
∣ > 0. In this case the tree-level
potential has a minimum at
Φ = 0, φ2 = −6m2/λ ≡ v2
and the symmetry φ → −φ is spontaneously broken. The degrees of freedom in
this case are Φ and φ′ ≡ φ − v, with
m2Φ = M2 m2
φ′ = −2m2 = λv2/3
At one loop, the minimization condition m2 + λv2/6 = 0 is replaced by
m2+
λv2
6= −
λ
32π2
(
m2+
λv2
2
)(
logm2 + λv2
2
µ2−1
)
−
δ
32π2
(
M2+
δv2
2
)(
logM2 + δv2
2
µ2−1
)
Following the same procedure as in the unbroken case one finds
m2φ′ =
λv2
3+
v2
32π2
[
λ2 logm2 + λ
2v2
µ2+ δ2 log
M2 + δ2v2
µ2
]
with v ∼ M without a suitable tuning of the parameters.
Naturalness: a closer look
The scalar potential in the Standard Model:
V (φ) = m2 |φ|2 + λ |φ|4
One-loop corrections to m2 due to fermionic (a) or bosonic (b)
degrees of freedom:
(a)
H
f
(b)
H
S
(
∆m2)
a=
|λf |216π2
[
−2Λ2 + 6m2f log
Λ
mf+ . . .
]
(
∆m2)
b=
λS
16π2
[
Λ2 − 2m2S log
Λ
mS+ . . .
]
Here Λ is an ultraviolet cut-off, to be identified with the energy
scale at which the SM is no longer reliable, and the dots stand for
terms that do not grow with Λ.
In dimensional regularization the Λ2 term would be absent, but
contributions proportional to m2f ,m
2S would still be there.
Even if the heavy degrees of freedom are not directly coupled to
the SM Higgs, it can be shown that similar contributions arise at
higher orders.
In the absence of very special cancellations, the Higgs boson
becomes as heavy as the heaviest degrees of freedom.
A symmetry that relates fermions to bosons would do the job, at
least at one loop. Suppose there are two scalars for each fermion:
(
∆m2)
a+b=λS − |λf |2
8π2Λ2 + . . .
For suitable values of the couplings the quadratic divergence
disappears.
No surprise: with bosons and fermions in the same multiplet,
scalar masses are protected by the same (chiral) symmetry that
protects fermion masses from large radiative corrections.
Clearly, more restrictions will be needed in order to guarantee that
the cancellation takes place at all orders.
Such a symmetry is called a supersymmetry:
Q|boson〉 = |fermion〉 Q|fermion〉 = |boson〉
The symmetry generator Q (and its hermitian conjugate Q†) carry
spin 1/2: it is a space-time symmetry.
The form of possible supersymmetry algebras is strongly
constrained on the basis of very general theorems in field theory.
For example, it is impossible with ordinary symmetry generators
(elements of a commutator algebra).
There is essentially one possibility:
{Q,Q†} = Pµ
{Q,Q} = {Q†, Q†} = 0
[Pµ, Q] = [Pµ, Q†] = 0
(more on this later). Further specifications:
• Q,Q† transform as spinors under the Lorentz group
• Q,Q† commute with gauge symmetry generators.
In principle, we may have more than one Q: Qi, i = 1, . . . , N
(extended supersymmetry).
Historical remark: symmetries that relate particles with different spin first
studied in the context of approximate symmetries of hadrons. Non relativistic
quark models have an approximate SU(6) symmetry (3 quark flavors with spin
1/2), which is observed hadron spectrum. It relates hadrons with the same
flavor content, but different spin (e.g. K ↔ K∗). Consequence of approximate
spin and flavor independence of quark-quark forces.
The Coleman-Mandula theorem tells us that this property cannot be
possessed by a relativistic theory: with some reasonable assumptions, the most
general Lie algebra of symmetry operators that commute with the S matrix
consists of Poincare generators Pµ and Jµν , plus ordinary internal symmetry
generators that act on one-particle states with matrices that are diagonal in,
and independent of, momentum and spin. Crucial point: the Poincare group is
non compact, it has no non-trivial unitary representations.
Rules out SU(6), but also Lie algebras of supersymmetry generators.
A few basic properties of a supersymmetric theory can already be
recognized:
• particles in the same supersymmetric multiplet (which we will
call a supermultiplet) have equal masses and equal gauge
transformation properties (electric charge, weak isospin and
color)
• within the same supermultiplet, there is an equal number of
bosonic and fermionic degrees of freedom (a proof on the next
slide)
A proof (taken from S. Martin):
(−1)2s|boson〉 = +|boson〉
(−1)2s|fermion〉 = −|fermion〉
}
⇒{
(−1)2s, Q}
={
(−1)2s, Q†}
= 0
Consider the subspace of states |i〉 within a supermultiplet with the same
eigenvalue pµ of the four-momentum operator P µ.∑
i|i〉〈i| = 1 within this
subspace of states.
∑
i
〈i|(−1)2sPµ|i〉 =∑
i
〈i|(−1)2sQQ†|i〉 +∑
i
〈i|(−1)2sQ†Q|i〉
=∑
i
〈i|(−1)2sQQ†|i〉 +∑
i
∑
j
〈i|(−1)2sQ†|j〉〈j|Q|i〉
=∑
i
〈i|(−1)2sQQ†|i〉 +∑
j
〈j|Q(−1)2sQ†|j〉
=∑
i
〈i|(−1)2sQQ†|i〉 −∑
j
〈j|(−1)2sQQ†|j〉
= 0.
Fermionic fields: a reminder
We use the Weyl representation of the Dirac matrices:
γµ =
0 σµ
σµ 0
σµ = (σ0, ~σ) σµ = (σ0,−~σ) γ5 =
−1 0
0 1
where ~σ are the three Pauli matrices
σ1 =
0 1
1 0
σ2 =
0 −ii 0
σ3 =
1 0
0 −1
and
σ0 =
1 0
0 1
The familiar lagrangian for a free, massive Dirac spinor is
L = ψ (i∂/−m)ψ → (i∂/−m)ψ = 0
Four-component Dirac spinors realize a reducible representation of
the Lorentz group:
ψ =
ξL
ξR
ψL =
ξL
0
=1
2(1 − γ5)ψ ψR =
0
ξR
=1
2(1 + γ5)ψ
the two-component spinors ξL, ξR transform independently under
Lorentz transformations.
In terms of ξL, ξR we have
L = iξ†L σµ∂µ ξL + iξ†R σ
µ ∂µξR −m(ξ†LξR + ξ†RξL)
It can be shown that the transformation law of the two-component
spinor ξR under Lorentz transformations are the same as those of
εξ∗L, where ε is the antisymmetric matrix
ε =
0 1
−1 0
.
It follows that any Dirac spinor may always be written in terms of
two left-handed Weyl spinors ξ, χ as
ψ =
ξ
−εχ∗
(the minus sign on the second Weyl spinor is conventional).
The free lagrangian for a massive Dirac spinor is
L = ψ (i∂/−m)ψ = iχT εTσµε ∂µχ∗ + iξ† σµ∂µξ −m(χT εξ − ξ†εχ∗)
= iχ† σµ ∂µχ+ iξ† σµ∂µξ −m(χT εξ − ξ†εχ∗)
Note that
χT εξ = ξT εχ
because of the anticommutation properties of fermion fields, and
that
(χT εξ)† = ξ†εTχ∗ = −ξ†εχ∗
The shorthand notation
χξ = ξχ = χT εξ ξ†χ† = χ†ξ† = −ξ†εχ∗
is often used.
Any theory involving spin-1/2 particles may be written as a
collection of left-handed Weyl spinors ψi.
Left-handed fermions in the Standard Model:
ui, di, ei, νi
ui, di, ei
where i = 1, 2, 3 is a generation index. For example, the Dirac
spinor for the up quark is written in terms of two left-handed Weyl
spinors u and u as
ψu ≡
uL
uR
=
u
−εu∗
(NB: the bar on u here has no special meaning, it is just part of
the name)
The Weyl notation is convenient, since left- and right-handed
components of Dirac fermions, which behave differently in weak
interactions, are treated separately.
The simplest (irreducible) representations of supersymmetry
contain Weyl fermions.
Massless supermultiplets of N = 1 supersymmetry are formed with
a state with helicity λmax and a state with helicity λmax − 1/2.
The simplest supermultiplet candidate: λmax = 1/2, λmin = 0
a Weyl fermion ψ and two real (or one complex) scalars φ1, φ2.
This is called a chiral or matter supermultiplet.
Next-to-simplest possibility: λmax = 1, λmin = 1/2
a massless gauge vector boson Aaµ and a Weyl fermion, called a
gaugino, λa
This is called a vector supermultiplet.
Gauge bosons belong to the adjoint representation of the gauge
group, so the same is true for their supersymmetric partners.
Thus,
gauginos are not chiral fermions
because the adjoint representation is equivalent to its conjugate.
The pair Aaµ, λ
a is called a gauge supermultiplet.
It turns out that any other representation of the supersymmetry
algebra can be reduced to a combination of chiral and gauge
supermultiplets, at least if there is only one supersymmetry
generator Q (N = 1 supersymmetry).
We will not consider extended (N > 1) supersymmetries, because
they cannot accommodate chiral fermions.
A word on extended supersymmetries
If λmax is the maximum helicity in a supermultiplet, then λmin = λmax − N/2.
Hence, in order to exclude states with |λ| > 2, we cannot allow N > 8.
N = 8 and N = 7 supergravity theories are entirely equivalent (same particle
content); N ≤ 6 are all different from each other.
Global supersymmetry (no spin 2 and 3/2 particles) is possible with N ≤ 4.
N = 4 and N = 3 global supersymmetries equivalent.
Fermions in extended supersymmetries are either partners of gauge bosons, or
have both helicity components in the same supermultiplet; impossible to
accommodate chiral fermions.
The particle content of a supersymmetric SM
• No supermultiplet can be formed out of standard particles
(e.g. γ, ν).
• Matter fermions must belong to chiral supermultiplets, because
left and right fermions transform differently under the weak
gauge group.
• Gauge bosons must go into gauge supermultiplets.
• At least two Higgs doublets, with their fermionic partners:
cancellation of the axial anomaly.
spin 0 spin 1/2 spin 1 SU(3)C SU(2)L U(1)Y
uL, dL uL, dL 3 2 + 13
uR uR 3 1 + 43
dR dR 3 1 − 23
ν, eL ν, eL 1 2 −1
eR eR 1 1 −2
H+u , H
0u h+
u , h0u 1 2 +1
H0d , H
−d h0
d, h−d 1 2 −1
g g 8 1 0
w±, w0 W±,W 0 1 3 0
b0 B0 1 1 0
Supersymmetric partners of standard particles (e.g. a scalar
electron) with the same masses would have been detected in
experiments. Since none of them has been observed so far, we
must conclude that
supersymmetry must be broken
in a realistic theory. Recall our formula for scalar mass corrections:
(
∆m2)
a+b=λS − |λf |2
8π2Λ2 + . . .
Supersymmetry forces λS = |λf |2 to all orders in perturbation
theory, so that quadratic divergences are systematically cancelled.
This feature must be preserved in the broken theory:
L = Lsupersymmetric + Lsoft
where Lsoft only contains mass terms and couplings with positive
mass dimension.
Supersymmetric partners of ordinary particles are so heavy that
they have escaped detection so far.
Is there a reason for that?
All ordinary particles, including the W,Z bosons and the top
quark, would be massless in the absence of spontaneous breaking
of the electroweak gauge symmetry.
The contrary is true for their partners: scalar masses are always
allowed by gauge symmetries, and gaugino can be massive because
they belong to a real representation of the gauge group.
The only exception is the Higgs boson.
A rough estimate of superparticle masses
Call msoft the largest mass scale present in Lsoft. Corrections to m2
arising from Lsoft must vanish as msoft → 0, so they cannot grow as
Λ2 (corrections proportional to msoftΛ are also forbidden: UV
divergences are either quadratic or logarithmic). Therefore
∆m2 ∼ λm2soft log
Λ
msoft
(λ a generic coupling in Lsoft). It follows (λ ∼ 1,Λ ∼MP ) that
msoft<∼ 1 TeV
in order to get the correct value of the Higgs v.e.v. Furthermore,
m2F −m2
B ∼ m2soft
within a supermultiplet: superpartners cannot be too heavy.
Building a supersymmetric theory
A simple example: one left-handed Weyl fermion
ψ ≡ PL ψ PL,R =1
2(1 ∓ γ5)
and one complex scalar φ, no masses, no interactions:
Lchiral = Lf + Ls
Lf = ψ i∂/ ψ
Ls = ∂µφ∗∂µφ
Is it possible to define a set of transformation rules on the fields,
such that scalars are transformed into fermions, under which Lchiral
remains unchanged?
Let us consider the scalar field first, and tentatively define
δφ = η ψ δφ∗ = ψ η
where η is a constant, infinitesimal, anticommuting spinor. Then
δLs = (∂µψ) (∂µφ) η + η (∂µφ∗) (∂µψ)
How does ψ tranform? δψ must be left-handed, linear in η, and
must contain one derivative. Only one possibility (up to a
proportionality factor):
δψ = −i(∂µφ) γµη δψ = iη γµ∂µφ∗
which tells us that η is right-handed, PRη = η, and gives
δLf = −η (∂µφ∗) γµγν∂ν ψ + ψ γνγµ∂ν (∂µφ) η
Using {γµ, γν} = 2gµν and ∂µ∂ν = ∂ν∂µ we get
δLf = −η (∂µφ∗) (∂µψ) − (∂µψ) (∂µφ) η
−η ∂ν [(∂µφ∗) γµγν ψ] + η ∂µ[(∂µφ∗)ψ] + ∂µ(ψ ∂µφ) η
The first row cancels against δLs, and the rest is a total derivative.
So,
δ
∫
d4xLchiral = 0
Not yet enough to declare that Lchiral is supersymmetric: we must
check that the commutator of two transformations is a symmetry
of the theory. For the scalar fields we find
(δη1δη2
− δη2δη1
)φ = −(η2 γµ η1 − η1 γ
µ η2) i∂µφ
Good news: i∂µ is nothing but the four-momentum operator Pµ,
the generator of space-time translations, a symmetry of
space-time.
Now consider the fermion fields. After some tedious algebra,∗ we
get
(δη1δη2
− δη2δη1
)ψ = −(η2 γµ η1 − η1 γ
µ η2) i∂µψ
+1
2(η2 γ
µ η1 − η1 γµ η2) iγµ∂/ψ
The first term is similar to what we got for φ, while the second
one vanishes on the mass shell,
∂/ψ = 0.
OK, but it might be useful to close the algebra even off shell.
∗ For those who wish to try: you need the Fierz identities (and some patience).
This can be done. Let us introduce an auxiliary scalar field F , with
the lagrangian
La = F ∗F
(note the unusual dimension: m2 instead of m). It is harmless: the
equation of motion is F = 0. Now assume
δF = −i η γµ ∂µψ δF ∗ = i ∂µψ γµ η
and modify the fermion transformation rules as follows:
δψ = PL [−i∂µφ γµη + Fη] δψ = [iη γµ∂µφ
∗ + ηF ∗]PR
(which amounts to nothing, because the auxiliary field vanishes on
shell).
Things start getting complicated:
1. We can no longer assume that PRη = η, we must promote it to a
full four-component spinor. In order not to increase the
number of independent parameters, we assume it is a Majorana
spinor:
η =
ηL
ηR
ηL = ε η∗R
2. We now have to write explicitly the projector PL in front of δψ,
in order to keep ψ left-handed.
Nevertheless, let us press on ...
... It is immediate to check that
Lchiral = Ls + Lf + La
is our first supersymmetric lagrangian (not very exciting: it’s a
non-interacting theory), invariant under the transformation
δφ = η ψ δφ∗ = ψ η
δψ = PL [−i∂µφ γµη + Fη] δψ = [iη γµ∂µφ
∗ + ηF ∗]PR
δF = −i η γµ ∂µψ δF ∗ = i ∂µψ γµ η
with
(δη1δη2
− δη2δη1
)X = −(η2 γµPR η1 − η1 γ
µPR η2) i∂µX
for X = φ, ψ, F , both on and off the mass shell. This structure can
be replicated for an arbitrary number of chiral supermultiplets.
The same results take a much simpler form when recast in Weyl
language. Define Weyl spinors through
ψ →
ψ
0
η →
η
−εη∗
Recalling the definitions of γ matrices, we get
δφ = η ψ
δψ = i(σµεη∗) ∂µφ+ Fη
δF = −i η† σµ ∂µψ
and
(δη1δη2
− δη2δη1
)X = −(η†1 σµ η2 − η†2 σ
µ η1) i∂µX
Why do we need auxiliary fields? Count fermionic and bosonic
degrees of freedom:
• on shell: ψ has two helicity states (it is a Weyl fermion): nf = 2
φ is a complex scalar: nb = 2
• off shell: ψ is a complex two-component spinor: nf = 4
φ is a complex scalar: nb = 2
F is a complex scalar: nb = 2
Without the auxiliary fields, the counting rule nf = nb would be
violated off the mass shell.
Ordinary symmetries: a reminder
Consider a lagrangian density
L(φ, ∂φ)
functions of a set of fields φi, i = 1, . . . , n. The transformation
δφi = iηATAijφj ;
(
TA)†
= TA;[
TA, TB]
= ifABCTC
is a symmetry transformation if
δL = ηA∂µKAµ ⇒ δS[φ] = δ
∫
d4xL = 0
As a consequence, a set of conserved currents exists:
∂µJAµ = 0; JA
µ = TAijφj
∂L∂∂µφi
−KAµ
QA(t) =
∫
d3x JA0 (t,x);
d
dtQA(t) = 0;
[
QA, QB]
= ifABCQC
The same procedure can be applied to our supersymmetry: a
conserved (spinor) current Jµ and a conserved charge
Q =√
2
∫
d3x J0(x)
can be built by tedious but straightforward calculations
The supersymmetry charge Q (a Weyl spinor!) obey the
anticommutator algebra
{
Q,Q†}
= 2σµPµ; {Q,Q} ={
Q†, Q†}
= 0
as anticipated.
Summary:
Lchiral = ψ† iσµ∂µ ψ + ∂µφ∗∂µφ+ F ∗F
(ψ left-handed fermion, φ and F complex scalars) is invariant under
the supersymmetric transformation
δφ = η ψ
δψ = i(σµεη∗) ∂µφ+ Fη
δF = −i η† σµ ∂µψ
The field F is auxiliary: F = 0 on shell. For all fields X = ψ, φ, F
(δη1δη2
− δη2δη1
)X = −(η†1 σµ η2 − η†2 σ
µ η1) i∂µX
Masses and non-gauge interactions
The most general renormalizable supersymmetric term has the
form (in Weyl notation)
Lng = −1
2Wij ψiψj +Wi Fi + h.c.
where
Wi =∂W
∂φiWij =
∂2W
∂φi∂φj
and W is the superpotential:
W =1
2Mij φiφj +
1
6yijk φiφjφk,
The constants Mij and yijk must be totally symmetric in their
indices.
A proof
Let us write in general
Lng = −1
2Wijψiψj +WiFi + h.c.
where Wij and Wi are some functions of the bosonic fields with
dimensions of (mass) and (mass)2 respectively. Wij is symmetric
under i↔ j.
No term which is a function of the scalar fields φi, φ∗i only can be
included in Lng: under a supersymmetry transformation it would
go into another function of the scalar fields only, multiplied by ηψi
or η†ψ†i, and with no spacetime derivatives or Fi, F∗i fields: nothing
of this form can possibly be cancelled by the supersymmetry
transformation of any other term in the lagrangian.
Renormalizability: each term has field content with mass
dimension ≤ 4. Hence
• Wij and Wi cannot be functions of the fermionic or auxiliary
fields
• Wi is a quadratic polynomial, and Wij is linear, in the fields φi
and φ∗i .
Now require supersymmetry invariance of Lng. We do it in steps:
1) Part which contains four spinors:
[δLng]4−spinor = −1
2
∂Wij
∂φk(ηψk)(ψiψj) −
1
2
∂Wij
∂φ∗k(η†ψ†
k)(ψiψj) + h.c.
The first term cannot cancel against any other. It vanishes if and
only if ∂Wij/∂φk totally symmetric in i, j, k, because of the Fierz
identity:
(ηψi)(ψjψk) + (ηψj)(ψkψi) + (ηψk)(ψiψj) = 0.
No such identity for the second term. Hence, supersymmetry
requires∂Wij
∂φ∗k= 0
An important feature: the superpotential W must be an analytic
function of the scalar fields. It is a function of φ, not of φ and φ∗.
A complex function f(z, z∗) = u(x, y) + iv(x, y) of a complex variable z = x + iy is
analytic if∂u
∂x=
∂v
∂y
∂u
∂y= −
∂v
∂x
This implies
∂f
∂z∗=
∂(u + iv)
∂x
∂x
∂z∗+
∂(u + iv)
∂y
∂y
∂z∗
=∂(u + iv)
∂x+ i
∂(u + iv)
∂y
=
(
∂u
∂x−
∂v
∂y
)
+ i
(
∂v
∂x+
∂u
∂y
)
= 0
It follows that
Wij = Mij + yijkφk
with Mij and yijk totally symmetric. We define the superpotential
W =1
2Mijφiφj +
1
6yijkφiφjφk
so that
Wij =∂
∂φi
∂
∂φjW
Mij fermion mass matrix, yijk Yukawa couplings.
2) Parts of δLng which contain a spacetime derivative:
[δLng]∂ = −iWij ψi∂µφjσµ ε η∗ − iWi η
† σµ ∂µψi + h.c.
= −i (Wij ψi∂µφj +Wi ∂µψi) σµ ε η∗ + h.c.
Using
Wij∂µφj = ∂µ∂W
∂φi
this becomes
[δLng]∂ = −i(
ψi ∂µδW
δφi+Wi ∂µψi
)
σµ ε η∗ + h.c.
which is a total derivative provided
Wi =∂W
∂φi= Mijφj +
1
2yijkφjφk
The remaining terms in δLng cancel, provided Wi and Wij are the
first and second derivatives of the same function W .
The field equations for the auxiliary fields Fi are now
F ∗i = −Wi = −Mij φj −
1
2yijk φjφk
and therefore
Lng + La = −1
2Wij ψiψj −
1
2W ∗
ij ψ†iψ
†j +WiFi +W ∗
i F∗i + Fi F
∗i
= −1
2Wij ψiψj −
1
2W ∗
ij ψ†iψ
†j −WiW
∗i
The scalar potential is entirely determined by the superpotential:
V (φ) = WiW∗i = Fi F
∗i
= MijM∗il φjφ
∗l scalar masses
+1
2Mijy
∗ilm φjφ
∗l φ
∗m +
1
2M∗
ilyijk φjφkφ∗l trilinear couplings
+1
4yijky
∗ilm φjφkφ
∗l φ
∗m quartic couplings
The remaining terms contain mass terms for the fermions and
Yukawa couplings:
−1
2Wij ψiψj −
1
2W ∗
ij ψ†iψ
†j = −1
2Mij ψiψj −
1
2M∗
ij ψ†iψ
†j
−1
2yijk φiψjψk − 1
2y∗ijk φ
∗iψ
†jψ
†k
Many interaction terms, but couplings related by supersymmetry.
A simple example: just one supermultiplet φ, ψ.
Lng + La = |M |2 |φ|2 +1
2M y∗ φφ∗φ∗ +
1
2M∗ y φφφ∗ +
1
4|y|2 |φ|4
−1
2M ψψ − 1
2M∗ ψ†ψ†
−1
2y φψψ − 1
2y∗ φ∗ ψ†ψ†
Some comments:
• The scalar potential is positive semi-definite: (almost) no
problems with the stability of the ground state.
• The quartic scalar self-coupling and the fermion-fermion-scalar
Yukawa coupling are related: λS = |λf |2.
• It is easy to check that scalars and fermions have the same
squared mass matrix
(M2)ij = MikM∗kj
which is symmetric and can be diagonalized.
• The analiticity of W requires the introduction of two Higgs
doublets in the supersymmetric version of the Standard Model.
In summary, we have the following
Recipe
for non-gauge interaction terms of a set of chiral supermultiplets
φi, ψi, Fi:
1. Give a superpotential
W =1
2Mij φiφj +
1
6yijk φiφjφk,
with Mij and yijk completely symmetric.
2. Build the interaction lagrangian as
Lng = −1
2Wij ψiψj +Wi Fi + h.c.
where
Wi =∂W
∂φiWij =
∂2W
∂φi∂φj
Gauge interactions
A gauge supermultiplet contains a vector field Aaµ and a Weyl
fermion λa. An infinitesimal gauge symmetry transformation with
parameter αa acts on Aaµ and λa as
δgaugeAaµ = g fabcAb
µαc − ∂µα
a
δgaugeλa = g fabcλbαc
color SU(3) a = 1, . . . , 8 fabc → fabcSU(3) g → gs
weak isospin SU(2) a = 1, . . . , 3 fabc → εabc g → g
weak hypercharge U(1) a = 1 fabc → 0 g → g′
Counting fermionic and bosonic degrees of freedom, we see that we
should add one real scalar auxiliary field, Da (off shell, nb = 3 from
Aµ and nf = 4 from λ). Then
Lgauge = −1
4F a
µνFaµν + iλa† σµDµ λ
a +1
2DaDa
where
Dµλa = ∂µλ
a − gfabcAbµλ
c
The corresponding supersymmetry transformation laws are easily
worked out, requiring that δAaµ be real, and δDa proportional to
the equation of motion of λa:
δAaµ = − 1√
2
(
η† σµ λa + λa† σµ η
)
δλa = − i
2√
2σµσνη F a
µν +1√2Daη
δDa = − i√2
[
η† σµDµ λa − (Dµλ
a)† σµ η]
With these definitions,
(δη1δη2
− δη2δη1
)X = −(η†1 σµ η2 − η†2 σ
µ η1) iDµX
on X = F aµν , λ
a, Da.
Back to the chiral supermultiplet X = φ, ψ, F . All of them must be
in the same representation of the gauge group:
δgaugeXi = i g αa (T a)ij Xj
where T a are the generators of the gauge group in the
representation of X. Ordinary derivatives must be replaced by
covariant derivatives everywhere:
∂µ → Dµ = ∂µ + i g Aaµ T
a
Is this enough to turn our simple model into a gauge-invariant
supersymmetric theory?
Not quite: there are other possible gauge invariant and
renormalizable terms that can be formed out of the fields in the
gauge and chiral supermultiplets:
φ∗ T a ψ λa (φ∗ T a φ)Da
They are not taken into account by the covariant derivatives,
because they involve the superpartners of the gauge fields.
They can be added to our supersymmetric lagrangian, with
appropriate coefficients, provided the transformation law of the
auxiliary field F is slightly modified:
δφ = η ψ
δψ = i(σµεη∗)Dµφ+ Fη
δF = −i η† σµ Dµψ + g√
2(T aφ)λa†η†
Then
L = Lgauge + Lchiral + g√
2(
φ∗ T a ψ λa + φT a λa† ψ†)
+ g(φ∗ T a φ)Da
is supersymmetric, provided the superpotential is gauge-invariant:
δgaugeW = Wi (T a)ijφj = 0
The equations of motion for Da are now
Da = −g(φ∗ T a φ)
and the scalar potential becomes
V (φ) = F ∗i Fi +
1
2DaDa = W ∗
i Wi +1
2g2(φ∗ T a φ)(φ∗ T a φ)
completely determined by the auxiliary fields. This fact is relevant
for the discussion of spontaneous supersymmetry breaking.
• The scalar potential is positive semidefinite: a sum of squares.
• A unique feature of supersymmetric theories: the scalar
potential is entirely determined by other interactions. F - terms
are fixed by Yukawa couplings, D- terms by gauge couplings
In particular, the quartic Higgs couplings are no longer free
parameters.
Superpotential couplings
i
k
j
(a) (b)
j
i
l
k
yijk yijmy∗klm
i
k
j
(a) (b)
i j
(c)
i j
M∗imyjkm Mij M∗
ikMkj
Gauge couplings
(a) (b) (c) (d)
(e) (f) (g) (h)
Comments
• Mij and yijk in general restricted by gauge invariance
• Pattern of Yukawa couplings required for the cancellation of
quadratic divergences
• Fermion mass terms flip chiality, scalar masses do not
Soft supersymmetry breaking
The most general soft supersymmetry breaking lagrangian is given
by
Lsoft = −m2ijφiφ
∗j −
(
1
2mλ λ
aλa +1
2bijφiφj +
1
6aijkφiφjφk + h.c.
)
−1
2cijkφ
∗iφjφk + h.c.
It can be shown that a theory with exact supersymmetry, plus
Lsoft, is free of quadratic divergences.
The last term is usually negligible in most supersymmetry
breaking scenarios.
A word on gaugino masses. A mass term for a left-handed Weyl spinor
ξT ε ξ + h.c.
is called a Majorana mass term. It is allowed by Lorentz invariance, but
generally forbidden if ξ belongs to a complex representation of a gauge group.
A simple example: a phase transformation δξ = iαξ. Matter fermions in the
Standard Model (with the possible exception of a right-handed neutrino)
cannot have a Majorana mass term.
The case of gauginos λa is different: they transform according to the adjoint
representation,
δgaugeλa = g fabcλbαc
which is self-conjugate. This gives
δgauge(λTa ελa) = g fabcαc(λ
Tb ελa + λT
a ελb) = 0
because fabc is completely antisymmetric.
The minimal supersymmetric Standard Model
With some specifications, there is a unique superpotential which is
allowed by the Standard Model gauge invariance and by
renormalizability:
WMSSM = u∗R yu qLHu − d∗R yd qLHd − e∗R yl˜LHd + µHuHd
(call it W from now on). The first term in detail:
u∗R yu qLHu = (u∗R)jf (yu)fg(qL)gj
α (Hu)βεαβ
j is an SU(3)color index, j = 1, 2, 3
α, β are SU(2)L indices, α = 1, 2, β = 1, 2
f, g are generation indices, f = 1, 2, 3, g = 1, 2, 3
If φ1 and φ2 are SU(2) doublets,
δφ1 = iαaσa φ1 δφ2 = iαaσa φ2
then
φT1 ε φ2, ε =
(
0 1
−1 0
)
≡ iσ2
is an SU(2) scalar:
δ(φT1 ε φ2) = iαaφT
1 (σTa ε + εσa) φ2 = 0
because σ1, σ3 are symmetric, σ2 antisymmetric, and {σa, σb} = 2δab. The
quantities
φ†1φ1 φ†
1φ2 φ†
2φ2
are also, obviously, SU(2) invariants, by they cannot enter the superpotential,
which is an analytic function of all scalar fields.
Some comments:
• It is now clear why we need two Higgs doublets: in the
Standard Model, H gives mass to up-type quarks, while
Hc = εH∗ gives mass to down quarks and lepons. Here, H and
H∗ cannot appear simultaneously in the superpotential.
• yu, yd, yl are 3 × 3 matrices in family space of dimensionless
Yukawa couplings. They contain fermion masses and
generation mixing. A good approximation:
y33u = yt, y
33b = yb, y
33l = yτ , all other entries equal to 0.
• No bilinear term other than µHuHd is allowed by gauge
symmetry. µ is the only dimensionful parameter in the
superpotential; in particular, mass terms for fermions are
forbidden.
The superpotential originates a large variety of interaction
vertices, with coupling constants related by supersymmetry.
Recall our recipe,
Lng = −WiW∗i − 1
2Wij ψiψj −
1
2W ∗
ij ψ†iψ
†j
where
Wi =∂W
∂φi,Wij =
∂2W
∂φi∂φj
We find e.g.
• Yukawa couplings, such as
∂W
∂u∗R∂uLuu = u yu uH
0u
as in the Standard Model.
• quark-squark-higgsino couplings,
∂W
∂u∗R∂H0u
uh0u = u yu qL h
0u
• Quartic scalar couplings:
∣
∣
∣
∣
∂W
∂u∗R
∣
∣
∣
∣
2
= |yu qLHu|2
The term µHuHd provides a higgsino mass term
−µ (h+u h
−d − h0
uh0d + h.c.)
and a scalar Higgs mass term
− |µ|2(
∣
∣H+u
∣
∣
2+∣
∣H0u
∣
∣
2+∣
∣H0d
∣
∣
2+∣
∣H−d
∣
∣
2)
This term cannot induce spontaneous breaking of the electroweak
gauge symmetry. We must rely upon soft breaking terms.
The value of µ is a problem: we expect it to be of the same order
(∼ 100 GeV) as the soft breaking terms, in order to provide the
correct value of the W and Z masses, but it has a completely
different origin.
Accidental symmetries and R-parity
Baryon and lepton number conservation in the Standard Model
follow from accidental symmetries: a consequence of
renormalizability.
This is no longer true in supersymmetry:
W∆L=1 =1
2λijk
˜iL
˜jL(ek
R)∗ +1
2λ′ijk
˜iLq
jL(dk
R)∗ + µ′i˜iLHu
W∆B=1 =1
2λ′′ijk (ui
R)∗(djR)∗(dk
R)∗
are allowed by renormalizability and gauge symmetry.
In order to implement B and L conservation we must impose some
extra symmetry (not B and L themselves: they are violated by
nonperturbative effects!).
Consider the so-called matter parity:
PM = (−1)3(B−L)
where B = 1/3 for quarks and squarks and 0 otherwise, L = 1 for
leptons and sleptons, 0 otherwise.
PM commutes with supersymmetry: all particles in the same
supermultiplet have the same PM (−1 for quarks and leptons, +1
for Higgs and gauge).
Requiring PM = +1 for all terms in the lagrangian (or in the
superpotential) eliminates B and L violating terms.
A different, but equivalent language: define R−parity as
PR = (−1)3(B−L)+2s
Conserved if PM is conserved, but does not commute with
supersymmetry:
PR = +1 for standard particles
PR = −1 for superpartners
R-parity conserved ⇔ matter parity conserved.
Useful for phenomenology: each vertex must contain an even
number of fields with PR = −1.
Consequences of R-parity conservation
• The lightest particle with PR = −1 is stable. It is called the
LSP. A candidate for dark matter.
• Supersymmetric particles decay into states with an odd
number of PR = −1 particles.
• Supersymmetric particles are always produced in even numbers
(usually 2).
Soft supersymmetry breaking in the MSSM
The most general soft supersymmetry breaking term for the
MSSM is
Lsoft = −1
2
(
M3gg +M2ww +M1bb+ h.c.)
−(
u∗R Au qL Hu − d∗RAd qL Hd − e∗RAe˜L Hd
)
+ h.c.
−q†Lm2Q qL − ˜†
Lm2L
˜L − u†R m
2U uR − d†Rm
2D dR − e†R m
2E eR
−m2Hu
H†uHu −m2
HdH†
d Hd − (BHuHd + h.c.)
An enormous number of new independent parameters (105).
All of them are expected to be of order msoft (to the appropriate
power), something between 100 GeV and 1 TeV.
Do we have any guiding principle that can help us reducing the
number of arbitrary parameters?
m2Q,m
2U ,m
2D,m
2E are arbitrary 3 × 3 matrices in flavour space; they
are subject from strong constraints from
• flavour changing neutral current phenomena (e.g. K0K0 mixing)
• individual leptonic number conservations (e.g. µ→ eγ)
• CP violation
The (unknown) mechanism that generates soft breaking terms
must respect these constraints.
All these constraints are satisfied with the simple assumption that
flavour mixing and CP violation are only originated by Yukawa
couplings. This is achieved by assuming that
• (m2Q)fg = m2
Qδfg (m2U )fg = m2
Uδfg (m2D)fg = m2
Dδfg
(m2E)fg = m2
Eδfg
• (Au)fg = Au(yu)fg (Ad)fg = Ad(yd)fg (Ae)fg = Ae(ye)
fg
• all soft breaking parameters are real.
This corresponds to assuming that the sector that generates
soft-breaking terms is flavor-blind.
This (or any other) structure of soft breaking terms results from
some underlying mechanism that breaks supersymmetry at a scale
Q0, presumably very large (e.g. the grand-unification scale).
Predictions at energy scales relevant for our experiments, mexp,
typically of order 100 to 1000 GeV, will therefore contain large
logarithms
logQ0
mexp
induced by radiative corrections. These must be resummed by
standard renormalization group techniques.
An encouraging fact: coupling constant unification works much
better in the MSSM than in the Standar Model. Define
g3 = gs
g2 = g =e
sin θW
g1 =
√
5
3g′ =
√
5
3
e
cos θW
(in order to have the same normalization for symmetry
generators). The corresponding RGEs at one loop are
µ2 dgi
dµ2=
bi8πg3
i ⇒ µ2 d
dµ2
1
αi= −bi
with αi = g2i /(4π).
0
10
20
30
40
50
60
10 5� 10 10 10 151µ (GeV)
α i-1(µ
)
SMWorld Average�α1
α2�
α3 αS(MZ)=0.117±0.005�
sin2ΘMS__=0.2317±0.0004�
0
10
20
30
40
50
60
10 5� 10 10 10 151µ (GeV)
α i-1(µ
)
MSSMWorld Average�68%�
CL�
M S=103.7±0.8 ±0.4� GeV M� U =1015.9±0.2� ±0.1�U.A.�W.d.B�H.F.�
α1
α2
α3
2 4 6 8 10 12 14 16 18Log10(Q/1 GeV)
0
10
20
30
40
50
60
α−1
α1−1
α2−1
α3−1
The mass spectrum
1. The Higgs sector
The Higgs potential of the MSSM is
VHiggs = (|µ|2 +m2Hd
)(|H0d |2 + |H−
d |2) + (|µ|2 +m2Hu
)(|H0u|2 + |H+
u |2)+B
(
H+u H
−d −H0
uH0d
)
+ h.c.
+1
8(g2 + g′
2)(|H0
u|2 + |H+u |2 − |H0
d |2 − |H−d |2)2
+1
2g2|H+
u H0∗d +H0
uH−∗d |2
No arbitrary quartic couplings: they are fixed by the D terms.
Minimization: We can take H+u = H−
d = 0 (good news: EM
unbroken) and H0u, H
0d real and positive at the minimum without
loss of generality. Then we restrict to the neutral sector:
VHiggs = (|µ|2 +m2Hd
)|H0d |2 + (|µ|2 +m2
Hu)|H0
u|2 − (BH0uH
0d + h.c.)
+1
8(g2 + g′
2)(|H0
u|2 − |H0d |2)2
B may also be taken to be real and positive: no CP violating
phases in the Higgs sector.
Perform an SU(2)L gauge rotation that makes 〈H+u 〉 = 0; then, the condition
∂V/∂H+u = 0 is only fulfilled if also 〈H−
d〉 = 0.
The only place where the phases of H0u, H0
dappear is the term
−BH0uH0
d + h.c.
We can make B real and positive by a suitable phase choice for H0u, H0
d.
Clearly, the potential has a minimum if also H0uH0
dis real and positive, so 〈H0
u〉
and 〈H0d〉 must have opposite phases, which can be rotated away by a U(1)Y
gauge transformation.
Conditions for electroweak spontaneous symmetry breaking:
1. The origin H0u = H0
d = 0 is not a minimum of the potential
2. The potential is bounded from below
Condition 1. leads to the constraint
(|µ|2 +m2Hd
)(|µ|2 +m2Hu
) < B2
In some cases, guaranteed by large negative radiative corrections
to m2Hu
.
Condition 2. is almost automatic: the quartic couplings are
positive definite. There are however D-flat directions∣
∣H0u
∣
∣ =∣
∣H0d
∣
∣.
We must require that the potential in these directions is stabilized
by quadratic terms:
2B < 2 |µ|2 +m2Hd
+m2Hu
Define vu = 〈H0u〉 and vd = 〈H0
d〉. We find
m2Z =
1
2(g2 + g′
2)(v2
u + v2d) ⇒ v2 ≡ v2
u + v2d ' (174 GeV)2
Following tradition, we also define an angle β through
tanβ =vu
vd
Clearly 0 ≤ β ≤ π/2. The minimization conditions are
|µ|2 +m2Hd
= B tanβ − m2Z
2cos 2β
|µ|2 +m2Hu
= B cotβ +m2
Z
2cos 2β
The µ problem is now evident.
Mass eigenstates:
1. Two CP-odd neutral scalars
G0
A0
=√
2
sinβ − cosβ
cosβ sinβ
ImH0u
ImH0d
2. Two charged scalars
G+
H+
=
sinβ − cosβ
cosβ sinβ
H+u
H+d
G0, G± are pseudo-Goldstone bosons: they can be removed from
the spectrum by an appropriate gauge transformation.
A0, H± are physical degrees of freedom.
3. Two CP-even neutral scalars
h0
H0
=√
2
cosα − sinα
sinα cosα
ReH0u − vu
ReH0d − vd
Finding the mass eigenvalues is a matter of algebra:
m2A =
2B
sin 2β
m2H± = m2
A +m2W
m2h,H =
1
2
(
m2A +m2
Z ∓√
(m2A +m2
Z)2 − 4m2Zm
2A cos2 2β
)
sin 2α
sin 2β= −m
2A +m2
Z
m2H −m2
h
cos 2α
cos 2β= −m
2A −m2
Z
m2H −m2
h
It is easy to show that
m2H± ≥ m2
A
mh ≤ mA ≤ mH
mh < |cos 2β|mZ
Radiative corrections weaken this bound. At one loop, and
neglectic squark mixing, one finds
∆m2h ' 3g2
8π2
m4t
m2W
logmt1
mt2
m2t
An extra term appears if mixing is taken into account:
(
∆m2h
)
mix' 3g2
8π2
m4t
m2W
(At − µ cotβ)2
m2t1−m2
t2
[
logm2
t1
m2t2
+ (At − µ cotβ)2 f(m2t1,m2
t2)
]
f(a, b) =1
a− b
[
1 − 1
2
a+ b
a− blog
a
b
]
Including all corrections, one obtains approximately
mh<∼ 130 GeV
still very interesting in view of LHC.
Yukawa couplings are related to fermion masses in the third
generation by
yt =g mt√
2mW sinβyb =
g mb√2mW cosβ
yτ =g mτ√
2mW cosβ
A difference with respect to the Standard Model: yb and yτ may be
as large as yt, if tanβ is large enough (tanβ < 1 disfavoured: yt
becomes nonperturbative for tanβ > 1.2).
The couplings of h,H,A to standard particles are the same as in
the Standard Model, rescaled by α− and β−dependent factors:
dd, ss, bb uu, cc, tt W+W−, ZZ
e+e−, µ+µ−, τ+τ−
h − sinα/ cosβ cosα/ sinβ sin(β − α)
H cosα/ cosβ sinα/ sinβ cos(β − α)
A −iγ5 tanβ −iγ5 cotβ 0
At tree level, the Higgs sector is described by two parameters (a
common choice: m2A and tanβ).
In addition, there are h(H)AA and h(H)AZ couplings.
2. Neutralinos and charginos
The supersymmetric partner of neutral gauge bosons, w0 and b,
and of Higgs scalars, h0u, h
0d, are not mass eigenstates. We find
Ln = −1
2ψT
0 Mn ψ0 + h.c.
where the matrix Mn in the basis b, w0, h0d, h
0u is given by
M1 0 −mZ cosβ sin θW mZ sinβ sin θW
0 M2 mZ cosβ cos θW −mZ sinβ cos θW
−mZ cosβ sin θW mZ cosβ cos θW 0 −µmZ sinβ sin θW −mZ sinβ cos θW −µ 0
The mass matrix Mn is symmetric, and can be diagonalized by a
unitary transformation N . The four mass eigenstates
χ0i = Nij ψ
0j ; Mn = NT Mn N
are called neutralinos. Their coupling are entirely determined by
the matrix N , and thus by the four parameters
M1,M2, µ, tanβ
The lightest neutralino is in most models (= for most of the
allowed parameter choices) the LSP; neutralinos play a crucial role
in phenomenology.
Charged gauginos and higgsinos
w+, h+u , w
−, h−d
mix in a similar way:
Lc = −1
2ψT
c Mc ψc + h.c.
Mc =
0 XT
X 0
X =
M2
√2mW sinβ
√2mW cosβ µ
The mass eigenstates (called charginos) are found by diagonalizing
X with a bi-unitary transformation:
χ+1
χ+2
= V
w+
h+u
χ−1
χ−2
= U
w−
h−d
X = UT
mχ±
1
0
0 mχ±
2
V
m2χ±
1,χ±
2
=1
2
[ (
|M2|2 + |µ|2 + 2m2W
)
∓√
(
|M2|2 + |µ|2 + 2m2W
)2
− 4 |µM2 −m2W sin 2β|2
]
As in the case of neutralinos, the matrices U, V contain all the
relevant information about chargino couplings.
It is quite natural to assume that gaugino masses have a common
value m1/2 at the reference scale Q0, where the (properly
normalized) gauge couplings take approximately the same value
αU ∼ 0.04.
It turns out that gaugino running masses obey the same
renormalization group equations. Therefore, with the above
assumption,
Mi(Q) =αi(Q)
αUm1/2
This assumption reduces (from 4 to 3) the number of arbitrary
parameters in the gaugino sector.
3. Squarks and sleptons
The scalar partners of matter fermions (squarks and sleptons) have
mass matrices with contributions from various terms:
• quartic Higgs-Higgs-squark-squark vertices
• trilinear Higgs-squark-squark couplings
• soft-breaking masses
• D-terms
In general, there is mixing among different families, and mixing
betweel scalar parners of left- and right-handed fermions. This can
be sizeable in the third generation.
Mass matrices for squarks and sleptons in the third generation, in
the simplified situation outlined above:
s− top : (t∗L t∗R)
m2Q +m2
t + ∆Q mt(At − µ cotβ)
mt(At − µ cotβ) m2U +m2
t + ∆U
tL
tR
s− bottom : (b∗L b∗R)
m2Q +m2
b + ∆Q mb(Ab − µ tanβ)
mb(Ab − µ tanβ) m2D +m2
b + ∆D
bL
bR
s − tau : (τ∗L τ∗R)
m2L +m2
τ + ∆L mτ (Aτ − µ tanβ)
mτ (Aτ − µ tanβ) m2E +m2
τ + ∆E
τL
τR
where
∆ = m2Z(T3 −Q sin2 θW ) cos 2β
Spontaneous supersymmetry breaking
How are soft supersymmetry-breaking terms generated? We need
a mechanism of spontaneous supersymmetry breaking (explicit
breaking not acceptable).
A symmetry is said to be spontaneously broken when the vacuum
state is not invariant under that symmetry:
Q|0〉 6= 0
Supersymmetry is no exception. This can happen only if some of
the auxiliary fields Fi or Da acquires a non-zero vacuum
expectation value.
Proof: the supersymmetry algebra implies
H = P 0 =1
4
(
Q1Q†1 +Q†
1Q1 +Q2Q†2 +Q†
2Q2
)
which implies
Qi|0〉 6= 0 ⇔ 〈0|H|0〉 > 0 ⇔ 〈0|V |0〉 > 0
for a translation-invariant vacuum state. But
V = FiF∗i +
1
2DaDa
so supersymmetry is spontaneously broken if and only if the
system
Fi = 0 Da = 0
has no solution.
Spontaneous breaking based on 〈Da〉 6= 0 for some Da is called a la
Fayet-Iliopoulos. If there is a U(1) factor in the gauge group, a
term
LD−breaking = kD
can be added to the lagrangian. Then
V =1
2D2 − kD + gD
∑
i
qiφiφ∗i
D = k − g∑
i
qiφiφ∗i
D is different from zero if for some reason all scalar fields are zero
at the minimum. This is not the case in the MSSM.
Not very promising, although not completely ruled out.
Models in which supersymmetry is spontaneously broken by
〈Fi〉 6= 0 for some Fi are called O’Raifeartaigh models. An explicit
example: three chiral superfields, and
W = −kφ1 +mφ2φ3 +y
2φ1φ
23
where φ1 must be a gauge singlet. Then
V = |F1|2 + |F2|2 + |F3|2
F1 = k − y
2φ∗3
2 F2 = −mφ∗3 F3 = −mφ∗2 − yφ∗1φ∗3
F1 = F2 = F3 = 0 has no solution. For m2 > yk, the minimum of V is
for φ2 = φ3 = 0, φ1 undetermined (flat direction).
The solution is F1 = k, V = k2 at the minimum. The flat direction is
no longer flat after radiative corrections, and the minimum is at
φ1 = 0. The spectrum: six scalars with masses
0 0 m2 m2 m2 − yk m2 + yk
and three fermions with masses
0 m m
The zero-mass fermion is the analog of a Goldstone boson in
broken ordinary symmetry: we call it a goldstino. It is the partner
of the auxiliary field F1, responsible for supersymmetry breaking.
A general conclusion: spontaneous breaking of supersymmetry
cannot take place within the MSSM. There is no gauge-singlet
supermultiplet whose auxiliary field can take a non-vanishing
vacuum expectation value.
Hard to do by simply extending the MSSM with new fields and
new renormalizable interactions:
• gaugino masses cannot arise from renormalizable couplings
• the spectrum must include light scalars:
TrM2real scalars = 2 TrM2
chiral fermions
even after supersymmetry breaking.
Mass terms are generated after spontaneous symmetry breaking
as, for example, quark masses in the standard model:
y φψψ → y 〈φ〉ψψ = mψψ
In supersymmetry, the same mechanism is at work. A gaugino
mass could be generated either by a coupling
φλλ→ 〈φ〉λλ
but such a term is absent in a supersymmetric theory, or by
Fλλ→ 〈F 〉λλ
which is not renormalizable.
Way out: assume that supersymmetry is broken in a hidden sector
of non-observable degrees of freedom, that communicate somehow
with the observable sector. Historically, two proposals:
supersymmetry breaking communicated to the observable sector by
1. gravitational interactions
2. gauge interactions
The differences between the two cases are due to a different size of
the supersymmetry breaking scale
〈0|F |0〉
In gravity mediated supersymmetry breaking, soft breaking terms
arise from (in general nonrenormalizable) interaction terms (of
gravitational strength) of observable fields with combinations of
fields in the hidden sector that acquire a non-zero vev. We expect
therefore
msoft ∼〈F 〉MP
on dimensional grounds: msoft must vanish as 〈F 〉 goes to zero (no
spontaneous breaking) and as MP → ∞ (no interaction between
hidden and observable sectors.
For msoft ∼ 1 TeV we get
√
〈F 〉 ∼ 1011 GeV
In the gauge mediated scenario, superymmetry breaking is
communicated to the observable sector by the ordinary gauge
interactions via loop effects due to particles called messengers.
A rough estimate of the relevant supersymmetry breaking scale
can be obtained. We have in this case
msoft ∼α
4π
〈F 〉M
where M is the typical messenger mass. This corresponds to a
much lower value for the supersymmetry order parameter:
√
〈F 〉 ∼ 104 − 105 GeV
if 〈F 〉 and M2 are assumed to be roughly of the same size.
A closer look at the goldstino
The goldstino field can be easily identified. The full fermion mass
matrix of a generic supersymmetric theory is
Mf =
0√
2ga(T a〈φ〉)i√2ga(〈φ∗〉T a)j 〈Wij〉
There is always one combination of fermion fields that corresponds
to zero mass:
G =
〈Da〉/√
2
〈Fi〉
is an eigenvector of Mf with zero eigenvalue. This is the goldstino.
The first component of Mf G is
√2〈ga(T aφ)iFi〉 = −
√2〈ga(T aφ)iW
∗i 〉 = −
√2〈∂W
∗
∂φ∗iδgaugeφ
∗i 〉 = 0
by gauge-invariance of the superpotential.
The second component is given by
〈ga(φ∗T a)jDa +WijFi〉
The scalar potential is
V = FiF∗i +
1
2DaDa = WiW
∗i +
1
2g2
a(φ∗T aφ)2
It follows that
〈 ∂V∂φk
〉 = 〈W ∗i Wik + g2
a(φ∗T aφ)(φ∗T a)k〉
= −〈FiWik + gaDa(φ∗T a)k〉 = 0
Let us consider the case of local supersymmetry: η = η(x). Then
(δη1δη2
− δη2δη1
)X = −(η†1 σµ η2 − η†2 σ
µ η1) i∂µX
is a local translation, i.e. a general coordinate transformation.
Gravitation is included in a natural way.
Local supersymmetry is called supergravity.
In this case, the so-called super-Higgs effect takes place: the
spin-3/2 partner of the graviton, the gravitino, acquires a mass.
The missing spin degrees of freedom are provided by the goldstino.
By dimensional analysis,
m3/2 ∼ 〈F 〉MP
(
m3/2 =〈F 〉√3MP
)
Clearly, the gravitino mass has very different values in
gravity-mediated and gauge mediated scenarios:
• gravity-mediated: the gravitino mass
m3/2 ∼ 〈F 〉MP
∼ msoft
and its interactions are gravitational. Almost no interest for
phenomenology at colliders.
• gauge-mediated:
m3/2 ∼ 〈F 〉MP
∼ 105
1019GeV
if M ∼ 〈F 〉 �MP . Its goldstino components may have
non-negligible couplings, and may therefore play a role in
collider physics (X → XG).
Gravity-mediated models: some details
The supergravity lagrangian is non-renormalizable (as expected).
Assume there is a hidden chiral superfield X whose auxiliary
component FX gets a non-zero vev. The lagrangian includes
L = − 1
MPFX
∑
a
1
2faλ
aλa + h.c.
+1
M2P
FXF∗Xkijφ
∗iφj
− 1
MPFX
(
1
6y′ijkφiφjφk +
1
2µ′
ijφiφj + h.c.
)
When FX → 〈FX〉, this gives precisely the soft terms we need.
In a minimal version of supergravity, there are considerable
simplifications:
fa = f kij = kδij y′ = αy µ′ = βµ
This leads to
M1 = M2 = M3 = m1/2
m2Q = m2
U = m2D = m2
L = m2E = m2
Hu= m2
Hd= m2
0
Au = A0yu Ad = A0yd Ae = A0ye
B = B0µ
Neutral flavour changing effects automatically suppressed.
Gauge-mediated models: some details
In the simplest version, the model contains four messenger
supermultiplets:
q ∼(
3,1,−2
3
)
q ∼(
3,1,+2
3
)
` ∼ (1,2,+1) ¯∼ (1,2,−1)
and a chiral multiplet S, whose scalar and auxiliary components
take a vev (e.g. by the O’Raifeartaigh mechanism). They interact
through a superpotential
Wmessengers = y2S`¯+ y3Sqq
which becomes effectively
Wmessengers = y2〈S〉`¯+ y3〈S〉qq
Gauginos get masses
Ma =αa
4π
〈FS〉〈S〉
through one-loop diagrams:
⟨ S ⟩
⟨ FS ⟩
B, W, g
Soft scalar masses and trilinear couplings arise at two-loops. One
gets
Au = Ad = Ae = 0
at the messenger scale.
General features of soft masses in gauge-mediated models:
• larger for strongly-interacting particles
• determined by gauge properties only
The second properties leads to a degeneracy in squark and slepton
masses, that suppresses FCNC effects.
Summary
• Supersymmetry provides a solution to the naturalness problem
without spoiling the good features of the Standard Model.
• It has a number of welcome side-effects:
– a better context for grand unification
– a natural candidate for cold dark matter.
• A large portion of the parameter space for supersymmetric
theories has already been explored.
• Supersymmetry breaking in realistic models is still an open
problem.
