Labelle P. Supersymmetry DeMYSTiFied
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416
Supersymmetry Demystified
37. What are the F and D terms of the following expression:
α(x) · θ +
¯
β(x) ·
¯
θ + A(x)θ · θ + B(x)
¯
θ ·
¯
θ +χ(x) · θ
¯
θ ·
¯
θ +C(x)θ ·θ
¯
θ ·
¯
θ?
38. Write down the potential for the neutral Higgs fields in the MSSM, in the
absence of any soft SUSY breaking terms (you may consult Section 16.3).
Show that for this potential, both SUSY and the gauge symmetry are unbroken.
39. What is the R-parity of all the particles of the MSSM? You do not need to
list them all. Simply give the R-parity for the quarks, leptons, Higgs scalars
and gauge bosons, as well as for their supersymmetric partners, the squarks,
sleptons, Higgsinos and gauginos.
40. Consider the left-chiral superfields H
u
(quantum numbers 1, 2, 1),
H
d
(1, 2, −1), and E
i
(1, 1, 2). Write down the most general gauge invariant
lagrangian density containing up to four of these superfields.
41. What is a Fayet-Iliopoulos term? Why isn’t there such a term in the MSSM?
42. Calculate
∂
∂θ
1
∂
∂θ
1
(θ
2
θ
1
)
43. What are the chirality and helicity operators?
44. In Section 7.5 we had to prove that
Tr
(−1)
N
f
Q
a
, Q
†
b
= 0
where the trace is over all the fields of a SUSY supermultiplet and N
f
is equal
to 1 for a fermionic state and zero for a bosonic state. Prove this again, without
looking at Chapter 7!
45. Why does a small Higgs mass (in comparison with a GUT unification scale)
represent a naturalness problem in the standard model?
46. When evaluating Feynman diagrams with fermion loops, what is the difference
between diagrams containing Dirac fermion loops and diagrams containing
Majorana fermion loops?
47. Using the notation of dotted and undotted indices, prove (χ ·λ)
†
=
¯
λ · ¯χ.
48. What is the spin of the gauginos? In what representation of the gauge group
do they transform?
49. What is the definition of the supertrace? Why is it equal to zero when SUSY
is unbroken?
50. Consider a supersymmetric abelian gauge theory with a single charged left-
chiral superfield . What is the most general superpotential for such a theory?
References
1. Aitchison, I. J. R.: Supersymmetry in Particle Physics, Cambridge University
Press, New York, 2007.
2. Aitchison, I. J. R., and Hey, A. J. G.: Gauge Theories in Particle Physics,
Vol. I: From Relativistic Quantum Mechanics to QED, and Vol. II: QCD and
the Electroweak Theory, Taylor and Francis, London, 2004.
3. Alvarez-Gaum´e, A., and Hassan, S. F.: “Introduction to S-Duality and N =
2 Supersymmetric Gauge Theories (A Pedagogical Review of the Work of
Seiberg and Witten),” arXiv:hep-th/9701069v1; accessed January 1997.
4. Amsler, C., et al.: Phys. Lett. B667:1, 2008.
5. Baer, H., and Tata, X.: Weak Scale Supersymmetry, Cambridge University Press,
New York, 2006.
6. Bailin, D., and Love, A.: Supersymmetric Gauge Field Theory and String
Theory, Institute of Physics Publishing, Bristol, 1994.
7. Berezin, F. A.: The Method of Second Quantization, Academic Press, New
York, 1966.
8. Bin´etruy, P.: Supersymmetry: Theory, Experiment, and Cosmology, Oxford
University Press, New York, 2006.
9. Burgess, C.: “Introduction to Effective Field Theory,” arXiv:hep-th/0701053v2;
accessed January 2007.
418
Supersymmetry Demystified
10. Chung, D. J. H., Everett, L. L., Kane, G. L., et al.: “The Soft Supersymmetry-
Breaking Lagrangian: Theory and Applications,” arXiv:hep-ph/0312378v1; ac-
cessed December 2003.
11. Coleman, S., and Mandula, J.: Phys. Rev. 159:1251, 1967.
12. Dine, M.: Supersymmetry and String Theory, Cambridge University Press, New
York, 2007.
13. Dreiner, H. K., Haber, H. E., and Martin, S. P.: “Two-Component Spinor Tech-
niques and Feynman Rules for Quantum Field Theory and Supersymmetry,”
arXiv:hep-ph/0812.1594v1; accessed December 2008.
14. Fayet, P., and Iliopoulos, J.: Phys. Lett. 51B:461, 1974.
15. Ferrara, S. (ed.): Supersymmetry, North Holland/World Scientific, Amsterdam,
The Netherlands, 1987.
16. Gervais, J.-L., and Sakita, B.: Nucl. Phys. B34: 632, 1971. Reprinted in Ref. 40.
17. Gof’land, Y. A., and Likhtman, E. P.: JETPS Lett. 13:323, 1971. Reprinted in
Ref. 15.
18. Girardello, L., and Grisaru, M.: Nucl. Phys. B194:65, 1982.
19. Greiner, W., and Reinhardt, J.: Quantum Electrodynamics, Springer, New York,
2008; Greiner, W., Schramm, S., Stein, E., and Bromley, D. A.: Quantum
Chromodynamics, Springer, New York, 2007; Greiner, W., and M¨uller, B.,
Gauge Theory of Weak Interactions, Springer, New York, 2009.
20. Greiner, W., and Reinhardt, J.: Field Quantization, Springer-Verlag, Berlin,
1996.
21. Griffiths, D.: Introduction to Elementary Particles, Wiley-VCH, New York,
2008.
22. Grisaru, M., Siegel, W., and Rocek, M.: Nucl. Phys. B159:429, 1979.
23. Haag, R., Lopuszanski, J., and Sohnius, M.: Nucl. Phys. B88:257, 1975.
24. Hatfield, B.: Quantum Field Theory of Point Particles and Strings, Addison-
Wesley, Reading, Mass., 1992.
25. Jacob, M. (ed.): Supersymmetry and Supergravity: A Reprint Volume of Physics
Reports, North Holland/World Scientific, 1986.
26. Lepage, G. P.: “What Is Renormalization?” arXiv:hep-ph/0506330v1; accessed
June 2005.
27. Lykken, J. D.: “Introduction to Supersymmetry,” arXiv:hep-th/9612114v1;
accessed December 1996.
28. Maggiore, M.: A Modern Introduction to Quantum Field Theory, Oxford
University Press, New York, 2005.
29. Martin, S. P.: “A Supersymmetry Primer,” arXiv:hep-ph/9709356v5; accessed
December 2008.
30. McMahon, D.: Quantum Field Theory Demystified, McGraw-Hill, New York,
2008.
References
419
31. M¨uller-Kirsten, H. J. W., and Wiedemann, A.: Supersymmetry: An Introduction
with Conceptual and Calculational Details, World Scientific, Singapore, 1987.
32. Murayama, H.: “Supersymmetry Phenomenology,” arXiv:hep-ph/0002232v2;
accessed March 2000.
33. Neveu, A., and Schwarz, J. H.: Nucl. Phys. B31:86, 1971, and Phys. Rev.
D4:1109, 1971. Both reprinted in Ref. 40.
34. O’Raifeartaigh, L., Nucl. Phys. B96:331, 1975.
35. Paige, F. E.: “Supersymmetry Signatures at the CERN LHC,” arXiv:hep-
ph/9801254v1; accessed January 1998.
36. Peskin, M. E.: “Supersymmetry in Elementary Particle Physics,” arXiv:hep-
ph/0801.1928v1; accessed January 2008.
37. Peskin, M. E., and Schroeder, D. V.: An Introduction to Quantum Field Theory,
Addison-Wesley, Reading, Mass., 1995.
38. Polonsky, N.: “Supersymmetry: Structure and Phenomena,” arXiv:hep-
ph/0108236v1; accessed August 2001.
39. Ramond, P.: Phys. Rev. D3:2415, 1971. Reprinted in Ref. 40.
40. Schwarz, J. H. (ed.): Superstrings: The First 15 Years of Superstring Theory,
World Scientific, 1985.
41. Salam, A., and Strathdee, J.: Nucl. Phys. B76:477, 1974.
42. Seiberg, N.: Phys. Lett. B318:469, 1993 (arXiv:hep-ph/9309335).
43. Shirman, Y.: “TASI 2008 Lectures: Introduction to Supersymmetry and Super-
symmetry Breaking,” arXiv:hep-ph/0907.0039v1; July 2009.
44. Srednicki, M.: Quantum Field Theory, Cambridge University Press, New York,
2007.
45. t’ Hooft, G.: “Naturalness, Chiral Symmetry, and Spontaneous Chiral Symme-
try Breaking,” in Recent Developments in Gauge Theories (NATO ASI Series
B: Physics, Vol. 59), Plenum Press, New York, 1979.
46. Volkov, D. V., and Akulov, V. P.: Phys. Lett. 46B:109, 1973. Reprinted in
Ref. 15.
47. Weinberg, S.: The Quantum Theory of Fields, Vol. III: Supersymmetry,
Cambridge University Press, New York, 2000.
48. Weinberg, S.: The Quantum Theory of Fields, Vol. I: Foundations, and Vol. II:
Modern Applications, Cambridge University Press, New York, 2005.
49. Wess, J., and Bagger, J.: Supersymmetry and Supergravity, Princeton University
Press, Princeton, N.J., 1992.
50. Wess, J., and Zumino, B.: Nucl. Phys. B70:39, 1974, and Phys. Lett. 49B:52,
1974. Both articles are reprinted in Ref. 15.
51. West, P. C.: “Introduction to Rigid Supersymmetric Theories,” arXiv:hep-
th/9805055v1; accessed May 1998.
52. Wigner, E. P.: Annals of Mathematics 40:149, 1939.
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APPENDIX A
Useful Identities
Gamma Matrices and Trace Identities
In the following, a and b stand for arbitrary four-vectors. The following list is far
from being exhaustive; we are only providing the few identities needed in this book.
For a more complete set of rules, please consult one of the books on quantum field
theory listed in the References section.
(γ
5
)
2
= 1
γ
5
γ
μ
=−γ
μ
γ
5
Trγ
μ
= Tr /a = 0
Tr(/a /b) = 4a · b
Tr(γ
5
/aγ
5
/b) =−4a · b
Tr(γ
5
) = Tr(γ
5
/a) = 0
422
Supersymmetry Demystified
We also need the charge conjugation matrix
C ≡−iγ
2
γ
0
=
⎛
⎜
⎝
iσ
2
0
0 −iσ
2
⎞
⎟
⎠
(A.1)
which obeys the following identities:
C
2
= (C
T
)
2
=−1
γ
5
C
T
= C
T
γ
5
Identities Involving Pauli Matrices Only
σ
μ
≡ (1, σ) (A.2)
¯σ
μ
≡ (1, −σ) (A.3)
σ
i
σ
j
= δ
ij
1 +i
ijk
σ
k
(A.4)
σ
2
σ
μ
σ
2
= ¯σ
μT
(A.5)
σ
2
¯σ
μ
σ
2
= σ
μT
(A.6)
σ
2
σ
μT
σ
2
= ¯σ
μ
(A.7)
σ
2
¯σ
μT
σ
2
= σ
μ
(A.8)
Here are a few identities involving products of σ
μ
and ¯σ
μ
matrices:
σ
μ
¯σ
ν
+ σ
ν
¯σ
μ
= 2η
μν
1 (A.9)
¯σ
μ
σ
ν
+ ¯σ
ν
σ
μ
= 2η
μν
1 (A.10)
¯σ
ν
σ
μ
¯σ
ρ
= η
μν
¯σ
ρ
− η
νρ
¯σ
μ
+ η
μρ
¯σ
ν
−i
νμρδ
¯σ
δ
(A.11)
Tr(σ
μ
¯σ
ν
) = 2η
μν
(A.12)
Tr(σ
ν
¯σ
μ
σ
λ
¯σ
ρ
) = 2(η
νμ
η
λρ
+ η
μλ
η
νρ
− η
νλ
η
μρ
−i
νμλρ
) (A.13)
By definition, the matrices σ
μ
and ¯σ
μ
carry the following indices:
¯σ
μ
= ( ¯σ
μ
)
˙ab
(A.14)
σ
μ
= (σ
μ
)
a
˙
b
(A.15)
APPENDIX A Useful Identities
423
We also define
σ
μν
≡
i
4
(σ
μ
¯σ
ν
− σ
ν
¯σ
μ
) (A.16)
¯σ
μν
≡
i
4
( ¯σ
μ
σ
ν
− ¯σ
ν
σ
μ
) (A.17)
which carry the following indices
σ
μν
= (σ
μν
)
b
a
(A.18)
¯σ
μν
= ( ¯σ
μν
)
˙a
˙
b
(A.19)
Identities involving :
ab
= (iσ
2
)
ab
(A.20)
˙a
˙
b
=
ab
(A.21)
ab
= (−iσ
2
)
ab
(A.22)
˙a
˙
b
=
ab
(A.23)
ab
bc
=
cb
ba
= δ
a
c
(A.24)
˙a
˙
b
˙
b˙c
=
˙c
˙
b
˙
b ˙a
= δ
˙a
˙c
(A.25)
ca
˙
d
˙
b
(σ
μ
)
a
˙
b
= ( ¯σ
μ
)
˙
dc
(A.26)
cb
˙c ˙a
( ¯σ
μ
)
˙ab
= (σ
μ
)
c
˙
d
(A.27)
˙c
˙
b
˙
d ˙a
( ¯σ
μν
)
˙a
˙
b
= ( ¯σ
μν
)
˙c
˙
d
(A.28)
Definitions of the Various Components of
Weyl Spinors
In the following, the relations are written in terms of a left-chiral spinor χ ,but
they would apply as well to a right-chiral spinor. In all the following equations, the
barred spinor components always may be replaced by the hermitian conjugate of
424
Supersymmetry Demystified
the unbarred components using
¯χ
˙a
= χ
a†
(A.29)
¯χ
˙a
= χ
†
a
(A.30)
The matrix iσ
2
may be used to raise both dotted and undotted indices:
χ
a
= (iσ
2
)
ab
χ
b
=
ab
χ
b
¯χ
˙a
= (iσ
2
)
˙a
˙
b
¯χ
˙
b
=
˙a
˙
b
¯χ
˙
b
(A.31)
To be explicit,
χ
1
= χ
2
χ
2
=−χ
1
(A.32)
¯χ
˙
1
= ¯χ
˙
2
¯χ
˙
2
=−¯χ
˙
1
(A.33)
The matrix (−iσ
2
) may be used to lower either type of index:
¯χ
˙a
= (−iσ
2
)
˙a
˙
b
¯χ
˙
b
(A.34)
=
˙a
˙
b
¯χ
˙
b
(A.35)
Identities Involving Two Spinors
A general relation is (see Section 2.8)
α
T
A β =−β
T
A
T
α (A.36)
where α and β are arbitrary Weyl spinors.
The spinor dot products are defined by
χ ·λ = χ
T
(−iσ
2
)λ
¯χ ·
¯
λ = χ
†
iσ
2
λ
†T
(A.37)
APPENDIX A Useful Identities
425
Writing the indices explicitly, we have
χ ·λ =χ
a
λ
a
= χ
a
(−iσ
2
)
ab
λ
b
= χ
2
λ
1
− χ
1
λ
2
(A.38)
¯χ ·
¯
λ =¯χ
˙a
¯
λ
˙a
= ¯χ
˙a
(iσ
2
)
˙a
˙
b
¯
λ
˙
b
=−¯χ
˙
2
¯
λ
˙
1
+ ¯χ
˙
1
¯
λ
˙
2
(A.39)
Using the notation, we also may write the spinor dot products as
χ ·λ = χ
a
ab
λ
b
= χ
a
ab
λ
b
(A.40)
¯χ ·
¯
λ = ¯χ
˙a
˙a
˙
b
¯
λ
˙
b
= ¯χ
˙a
˙a
˙
b
¯
λ
˙
b
(A.41)
The order in the spinor dot products does not matter:
χ ·λ = λ · χ
¯χ ·
¯
λ =
¯
λ · ¯χ
(A.42)
Or, in terms of components,
χ
a
λ
a
= λ
a
χ
a
¯χ
˙a
¯
λ
˙a
=
¯
λ
˙a
¯χ
˙a
(A.43)
The effect of hermitian conjugation on the spinor dot products is
(λ · χ)
†
=
¯
λ · ¯χ
(
¯
λ · ¯χ)
†
= λ · χ
(A.44)
The product of two components of a spinor may be rewritten in terms of the spinor
dotted with itself using
χ
a
χ
b
=−
1
2
(iσ
2
)
ab
χ ·χ =
1
2
ab
χ ·χ (A.45)
¯χ
˙a
¯χ
˙
b
=
1
2
(iσ
2
)
˙a
˙
b
¯χ · ¯χ =−
1
2
˙a
˙
b
¯χ · ¯χ (A.46)
χ
a
χ
b
=−
1
2
(iσ
2
)
ab
χ ·χ =−
1
2
ab
χ ·χ (A.47)
¯χ
˙a
¯χ
˙
b
=
1
2
(iσ
2
)
˙a
˙
b
¯χ · ¯χ =
1
2
˙a
˙
b
¯χ · ¯χ (A.48)