Labelle P. Supersymmetry DeMYSTiFied
Подождите немного. Документ загружается.
406
Supersymmetry Demystified
Table 16.1 Quantum Numbers (Except Color) of the First-Generation
Fermions of the Standard Model
Particle YT
3
u
L
1/3 1/2
u
R
4/3 0
d
L
1/3 −1/2
d
R
−2/3 0
e
L
−1 −1/2
e
R
−20
v
L
−1 1/2
For the trace over the quarks, we must first remind ourselves of the definition of λ
3
:
λ
3
=
⎛
⎜
⎜
⎜
⎜
⎝
100
0 −10
000
⎞
⎟
⎟
⎟
⎟
⎠
We see that the eigenvalues of λ
3
of each color state are respectively 1, −1, and
0. The first generation contains four quarks (the left- and right-chiral up and down
quarks), so we get
Tr(λ
3
)
2
= 4(1
2
+ (−1)
2
+ 0
2
) = 8 (16.39)
Using Eqs. (16.37), (16.38), and (16.39) in Eq. (16.36), we finally get
c = 1 c
=
3
5
What this is telling us is that the generators of the grand unified theory must be
taken to be
λ
3
2
τ
3
2
and
3
5
Y
2
CHAPTER 16 Phenomenological Implications
407
Thus the coupling constants that we have to evolve with the RG equations are
g
s
g and
5
3
g
It’s important to emphasize that the angle θ
W
remains defined in terms of the
original, unscaled constants g and g
through tan θ
W
= g
/g. In addition, the rela-
tions
α
g
=
α
em
sin
2
θ
W
α
g
=
α
em
cos
2
θ
W
remain valid, so the numerical values obtained in Eqs. (16.32) and (16.33) are
unaffected by our rescaling of the hypercharge operator. The only change is that
instead of running α
g
, we must run 5α
g
/3. Let’s define
˜α ≡
5
3
α
g
of which the inverse at the Z mass is equal to
1
˜α(M
Z
)
=
3
5
1
α
g
(M
Z
)
= 59.0
The three RG equations then are
1
α
s
(E)
= 8.47 +
β
S
2π
ln
E
M
Z
1
α
g
(E)
= 29.6 +
β
g
2π
ln
E
M
Z
1
˜α(E)
= 59.0 +
˜
β
2π
ln
E
M
Z
(16.40)
We now need to evaluate the beta functions of the three groups. For SU(N )
gauge theories, the beta function to one loop is given by
β[SU(N )] =
11
3
N −
1
3
n
f
−
1
6
n
s
(16.41)
where n
f
and n
s
are the number of Weyl fermions and complex scalar fields in
the fundamental representation of the gauge group, respectively (see Ref. 37 for a
408
Supersymmetry Demystified
derivation of this result). In Eq. (16.41), the effects of the standard model gauge
bosons are already included in the first term.
For SU(3)
C
, we need only include the quark fields. Because we count the number
of quarks in the fundamental representation, we do not count the three color states
of a given quark as independent (the three color states of a given quark simply
represent the three states of a single fundamental representation). In the standard
model, the quark states are the left- and right-chiral up, down, charm, strange,
top, and bottom quarks, for a total of 12 fermion states. The only scalar particle,
the Higgs, is a color singlet, so n
s
= 0. Setting N = 3, n
f
= 12, and n
s
= 0in
Eq. (16.41), we find
β
s
= 7
For SU(2)
L
, we included only the left-chiral fermion states because the right-chiral
states are singlet under that group. The left-chiral electron and neutrino states belong
to a common fundamental representation of SU(2)
L
, so they count for only 1 in the
calculation of n
f
. The left-chiral up and down quark doublets count for 3 because
now states of different colors are different fields as far as SU(2)
L
is concerned.
So the fermions of the first generation contribute a total of 4 to n
f
, and therefore,
the three generations combined yield a total n
f
= 12. The Higs field is a complex
doublet, so it counts only as 1 for n
s
. We finally get, with N = 2, n
f
= 12, and
n
s
= 1,
β
g
=
19
6
For a U (1) group, the formula for the beta function to one loop is
β[U (1)] =−
1
20
2Y
2
f
+ Y
2
s
(16.42)
where the sum is over all the fermions and scalar fields with hypercharges Y
f
and
Y
s
, respectively, Again, the effects of the gauge bosons are already included.
Using the hypercharge assignments of the fermions given in Table 16.1, we get
(the overall factor of 3 accounts for the three generations, and there is an additional
factor of 3 for the three color states of each quark):
Y
2
f
= 3
3
1
3
2
+ 3
4
3
2
+ 3
1
3
2
+ 3
−
2
3
2
+ (−1)
2
+ (−2)
2
+ (−1)
2
= 40 (16.43)
CHAPTER 16 Phenomenological Implications
409
Recall that the Higgs has Y = 1. The Higgs field is an SU(2)
L
doublet of com-
plex fields. As far as the U(1)
Y
force is concerned, these two complex fields are
independent, which gives us
Y
2
s
= 2 (16.44)
Substituting Eqs. (16.43) and (16.44) in Eq. (16.42), we finally get
˜
β =−
41
10
To summarize, the three RG equations (16.40) are
1
α
s
(E)
= 8.47 + 7
ln
E
M
Z
2π
1
α
g
(E)
= 29.6 +
19
6
ln
E
M
Z
2π
1
˜α(E)
= 59.0 −
41
10
ln
E
M
Z
2π
To see if these three lines meet at a common point, we could plot them but a
simple analytical test can be made. If we have three straight lines with slopes m
1
,
m
2
, and m
3
and with y intercepts b
1
, b
2
, and b
3
, it is easy to show that the condition
for those three lines to meet at the same point is
b
1
− b
2
b
3
− b
2
m
3
− m
2
m
1
− m
2
= 1 (16.45)
Using the three slopes and y intercepts of our RG equations, the expression on
the left of Eq. (16.45) is found to be equal to 1.36, which clearly shows that the lines
do not meet at a single point. It is important to emphasize that our whole derivation
is only valid to one loop because the expressions for the beta functions and the RG
equations are only valid to that order. However, the three lines miss one another by
so much that there is no hope that higher-order corrections could cure the problem.
Let us now repeat the calculation for the MSSM. The values of the coupling
constants at the Z mass remain the same (of course, because they are measured
values!), but now the beta functions will be different because of the different particle
content. To start with, Eq. (16.41) must be modified because of the superpartners of
the gauge bosons that, it turns out, contribute −2N /3, so the SU(N ) beta function
410
Supersymmetry Demystified
now reads
β
SU(N )
=
9
3
N −
1
3
n
f
−
1
6
n
s
For SU(3)
C
, we have, as before, 12 quark states, but now in addition, we have the
12 gluinos, the scalar superpartners of the gluons. Using n
f
= n
s
= 12, we find for
the MSSM,
β
S
= 3
For SU(2)
L
, we have the 12 fermion states of the standard model, but in addition,
we have the fermionic partners of the Higgs fields, the higgsinos. There are two
Higgs doublets in the MSSM, so the total n
f
is equal to 14. All these fermions have
scalar partners, so n
s
= 14 as well. Putting all this together, we find
β
g
=−1
Note that β
g
has changed sign relative to its value in the standard model!
The formula for the U (1) beta function remains Eq. (16.42). To each standard
model fermion with a certain hypercharge assignment corresponds a scalar sfermion
with the same hypercharge. Therefore, excluding the Higgs fields and their super-
partners, we have
2Y
2
f
+ Y
2
s
= 80 + 40
= 120 (16.46)
The two Higgs doublets comprise four complex scalar fields with hypercharges ±1.
The corresponding four higgsinos have the same hypercharges, so for those fields,
we find
2Y
2
f
+ Y
2
s
= 12 (16.47)
Substituting Eqs. (16.46) and (16.47) into Eq. (16.42), we finally get
˜
β =−
33
5
CHAPTER 16 Phenomenological Implications
411
Thus, for the MSSM, Eq. (16.45) is replaced by
1
α
s
(E)
= 8.47 + 3
ln
E
M
Z
2π
1
α
g
(E)
= 29.6 −
ln
E
M
Z
2π
1
˜α(E)
= 59.0 −
33
5
ln
E
M
Z
2π
which leads to
b
1
− b
2
b
3
− b
2
m
3
− m
2
m
1
− m
2
= 1.006
This is an amazing result: The three coupling constants now do meet up at a certain
energy (taking into account the uncertainty in our experimental values)! We can,
of course, estimate this energy: We simply set equal two of the coupling constants
and solve for E. This yields
E ≈ e
33
M
Z
= 2 × 10
16
GeV
which is the unification scale in a grand unified extension of the MSSM.
A generic feature of grand unified theories is the decay of protons. You may have
heard that the SU(5) GUT is ruled out experimentally because it predicts too rapid
a decay. The MSSM gut fares better because its GUT scale is about twenty times
larger than the SU(5) grand unification scale, which raises the proton lifetime. A
factor of twenty does not seem like such a big deal, until one notices that the proton
lifetime is extremely sensitive to the GUT scale, being proportional to E
4
GU T
.An
increase by a factor of twenty in the GUT scale therefore translates into an increase
by over five orders of magnitude in the proton lifetime. This is sufficient to keep
the MSSM grand unified theory in the game.
At this energy, all three values of α
−1
are approximately 24, which means that
the coupling constant α itself is about
α(GUT scale) ≈ 0.041
412
Supersymmetry Demystified
Note that at this scale,
g =
5
3
g
which implies that
tan θ
W
≡
g
g
=
3
5
or
sin
2
θ
W
=
3
8
This is, of course, not an exact result. First, as pointed out earlier, our calculation
is only valid at one loop order. More crucial is the fact that the equations that we used
for the beta functions assume that no mass terms explicitly break the symmetries.
As we have seen, however, in the MSSM we have to include such explicit SUSY-
breaking masses. There is much more to say about supersymmetry and the MSSM
in particular but alas, it’s now time to end our introduction to the topic. We hope
you enjoyed the ride!
16.9 Quiz
1. How many parameters are needed to describe the most general soft SUSY-
breaking terms? How many parameters are needed in the minimal supergravity
model?
2. How many observable Higgs fields (i.e., not including the states that get
absorbed by the gauge bosons) are there in the MSSM?
3. In a supersymmetric GUT theory, which inverse coupling constants increase
with energy and which ones decrease? Is this the same behavior as in a GUT
extension of the standard model?
4. One of the Higgs fields of the MSSM is of particular interest from a phe-
nomenological point of view. Explain why.
5. How many parameters were necessary to describe the soft SUSY-breaking
terms of the Higgs sector?
Final Exam
1. What are the dimensions of the fields φ, χ, and F appearing in the Wess-
Zumnino model, as well as the dimension of the infinitesimal spinor ζ that
parametrizes SUSY transformations?
2. What is the result of the “integral”
dθ
1
dθ
2
θ
1
θ
2
?
3. Using the antisymmetric symbol, prove that ¯χ ·
¯
λ =
¯
λ · ¯χ.
4. What is the action of the supercharges Q
1
,Q
2
, Q
†
1
and Q
†
2
on a one-particle
state?
5. Write down the kinetic energy term for a left-chiral Weyl spinor and show
that it is a real quantity.
6. What was the motivation for introducing an auxiliary scalar field in the Wess-
Zumino model?
7. Why is it not possible to use a Fayet-Iliopoulos D term to break SUSY spon-
taneously in the MSSM?
8. Consider a supersymmetric theory with the potential for the scalar fields
depicted in Figure 1 on the next page. Is SUSY broken spontaneously in this
theory?
414
Supersymmetry Demystified
Scalar field
Potential
Figure 1 Potential for Question 8.
9. In a supersymmetric abelian gauge theory, write down the most general trans-
formation for the auxiliary field D that has the right dimension and correct
Lorentz transformation properties (without looking at Chapter 10!). It must
of course contain the SUSY parameter ζ and one of the other fields present
in the supermultiplet.
10. Consider the interactions and mass terms of the Wess-Zumino model. If we do
not impose invariance under supersymmetry and do not assume any relation
between the coupling constants and masses, how many parameters are there?
How many parameters are there when invariance under SUSY is imposed?
11. What constraint is obeyed by left-chiral superfields?
12. Write down the mass terms of a Dirac fermion and of a Majorana fermion in
terms of Weyl spinors.
13. Why can’t we take a superfield to depend only on the variables x
μ
and θ (and
not on
¯
θ)?
14. Write the components of a left-chiral spinor with upper dotted indices ¯χ
˙
1
and
¯χ
˙
2
in terms of the components with lower undotted indices χ
1
and χ
2
.
15. What is a gaugino?
16. What do we mean by the D term of a product of superfields?
17. What are helicity and chirality?
Final Exam
415
18. Write down the supersymmetric lagrangian of a free left-chiral superfield in
terms of the corresponding component fields.
19. What is the LSP?
20. If R-parity is a valid symmetry, what implication does this have for the LSP?
21. Could we add a term of the form c to the lagrangian of the MSSM, where
is any of the left-chiral superfield in the theory?
22. Why does the component field of a superfield that has the highest dimension
necessarily transform with a total derivative under SUSY?
23. Why did we need to introduce R-parity in the MSSM?
24. What is the value of the Weinberg (or weak) angle predicted at the unification
scale in grand unified theories?
25. A renormalizable lagrangian is of dimension 4, and left-chiral scalar super-
fields
i
have dimension 1, so why do we include only terms with up to three
left-chiral superfields in the superpotential?
26. Why is it crucial to include loop effects when considering the mass of the h
0
?
27. Why do you impose that the vacuum expectation values of the sleptons and
squarks be zero?
28. What is the mass term for a Majorana fermion?
29. How is the relation between the W and Z masses different from what we have
in the standard model?
30. There are three types of terms that break SUSY softly. What are they?
31. List the component fields of a left-chiral superfield (which is a scalar under
Lorentz transformation) and show that the bosonic and fermionic degrees of
freedom match both on-shell and off-shell.
32. What is the superpotential of the Wess-Zumino model in terms of the left-
chiral superfields
i
?
33. What is the most general supersymmetric lagrangian for an abelian gauge
theory with a set of left-chiral superfield
i
of charges q
i
? Write the answer
in terms of V, F
a
and
i
.
34. How does the field-strength superfield F
a
transform under Lorentz transfor-
mations?
35. Write χ ·λ,¯χ ·
¯
λ and ¯χ ¯σ
μ
λ with the indices explicitly shown (using dotted
and undotted indices in the appropriate positions).
36. Why do the coupling constants evolve differently in the standard model and in
the MSSM (give a physical explanation as opposed to a purely mathematical
one)?