Labelle P. Supersymmetry DeMYSTiFied
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476
Supersymmetry Demystified
coordinates, superspace:
about, 240–242
supersymmetric transformations of, 244–248
coupling constants, 401
CPT invariance, 70–71
cutoff, 6, 7
D
D term (Wess-Zumino model), 289–290,
302–308
D-type supersymmetry breaking, 330,
340–344, 380
d’Alembertian operator, 82
dark matter, 379
density, lagrangian, 14
differential operators:
about, 108
supersymmetric charges as, 255–261
dimensional analysis (sypersymmetric
lagrangian), 90–91
Dirac adjoint, 14
Dirac delta, 181
Dirac equation, 14
Dirac lagrangian, 14, 25, 214
Dirac matrices, 14
Dirac spinors:
charge conjugation and antiparticles for,
68–70
massive, 74–75
and parity, 81–82
as reducible representation of Lorentz
group, 18
two-component spinors, 17
Weyl spinors vs., 17–22
dot products, spinor, 35–37, 54–55
dual field strength, 316
dynamical symmetry breaking, 350
E
effective field theories, 6–7
eigenstates, 18
eigenvalues, 110
Einstein, Albert, 16
electron mass, 8
Euler-Lagrange equation, 82–83
explicit supercharges, as quantum field
operators, 153–155
explicit supersymmetry breaking, 344–350
extended supersymmetry, 145, 157–158
F
F-type supersymmetry breaking, 329–331, 380
Fayet-Illiopoulos term, 210, 219, 232, 343, 344
fermions, 2, 3
massive, 73
in MSSM, 146
in standard model, 352–355
Feynman, Richard, 14
Feynman diagrams, 177, 180, 184, 204
Feynman rules, 177–178
field-strength superfield, 306–307
fine-tuning problem, 2
flat direction (of potential), 332
four-component spinors, 17
free supersymmetric abelian gauge theory,
206–209
component fields, abelian field-strength
superfield in terms of, 308–313
lagrangian of, in superfield notation, 306–308
superfield approach, free abelian
supersymmetric lagrangian from, 313–317
and supersymmetric QED, 317–322
G
gauge bosons, 8–9
kinetic terms of, in MSSM, 371
kinetic terms of, in standard model, 356
masses of, in MSSM, 387–391
gauge invariance, 2
gauge-mediated SUSY breaking, 350
gauge multiplet, 205 (See also vector multiplet)
gauge theories, supersymmetric (see free
supersymmetric abelian gauge theory;
supersymmetric gauge theories)
gauge vector superfields, in standard
model, 368
gauginos, 147, 206
Gell-Mann matrices, 212, 218
general relativity, 3
generators, 156
of symmetry, 101
of translation, 106
global supersymmetric transformations, 91–92
INDEX
477
gluinos, 147
goldstinos, 338
graded Lie algebra, 101
grand unified theories (GUT), 401 (See also
supersymmetric GUT)
Grassmann calculus, 248–255
Grassmann coordinates, 241
gravitinos, 147
gravity-mediated SUSY breaking
(SUGRA-mediated SUSY breaking), 350
GUT scale, 402
H
Haag-Lopuszanski-Sohnius theorem, 157–158
handedness, 18–19
harmonic oscillator, 108
Heisenberg uncertainty principle, 7
helicity, 18, 19, 22–24
helicity operator, 23
hermitian conjugate, 26–28, 38–39
hierarchy problem, 2
Higgs boson, 2, 9–11
Higgs fields:
kinetic term of, in standard model,
356–357
masses of, in MSSM, 392–398
in MSSM, 146, 368–370
Higgs potential, in standard model, 359
Higgsinos, 147, 367
holomorphic functions, 164, 280
I
index-free notation, 53–56
internal symmetry, 106
J
Jacobi identities, 211
K
Kähler potential, 295
Klein-Gordon equation, 82
Kronecker delta, 16
L
lagrangian density, 14
lagrangians:
supersymmetric (see supersymmetric
lagrangians)
Wess-Zumino, 173–176
Large Hadron Collider (LHC), 3, 158
left-chiral antiparticle spinor, 318
left-chiral multiplet, explicit interactions
between abelian gauge multiplet and,
303–306
left-chiral spinors, 18, 29, 35, 56–57, 69
left-chiral superfields, 269–295
component fields, SUSY transformations
of, 273–277
constructing SUSY invariants out of,
277–285
defined, 267
general expansion of, 269–273
holomorphic functions of, 280
of MSSM, 365–367
products of, 278
and superpotential, 285–288
Lehmann-Symanzik-Zimmermann reduction
formula, 178, 182
Lepage, Peter, 7
leptons:
kinetic energy of, in MSSM,
368–370
masses of, in MSSM, 398–400
Levi-Civita symbol, 16, 134, 208, 212
LHC (see Large Hadron Collider)
Lie algebra, 322
lightest supersymmetric particle
(LSP), 377
loop momenta, 7
Lorentz indices, 16
Lorentz invariance, 2
Lorentz invariants, 25–26
building, 51–53
from two left-chiral spinors, 56–57
from Weyl spinors, 28–31
Lorentz transformations, of Weyl spinors,
18, 19, 24–26
LSP (lightest supersymmetric particle),
377
478
Supersymmetry Demystified
M
Majorana fermions, propagators of, 182–185
Majorana spinors:
massive, 78–81
supersymmetric algebra in Majorana form,
150–151
Wess-Zumino lagrangian in Majorana form,
175–176
and Weyl spinors, 86–88
mass spectrum, 334–338
massive fermions, 73
massless multiplets:
and MSSM, 146–147
supersymmetric algebra with, 144–150
minimal supergravity (minimal SUGRA),
380–381
minimal supersymmetric standard model
(MSSM), 146, 215, 370–373, 379–412
and conservation of R-parity, 401
and coupling unification in supersymmetric
GUT, 401–412
finding a minimum in, 384–387
left-chiral superfields in, 365–367
masses of gauge bosons in, 387–391
masses of Higgs fields in, 392–398
masses of leptons and quarks in, 399–400
and massless SUSY multiplets, 146–147
scalar potential in, 381–384
superpotential of, 371–378
supersymmetry breaking in, 380–381
mostly minus convention, 15
mostly plus convention, 15
MSSM (see minimal supersymmetric standard
model)
multiplets:
chiral, 146, 222–236
explicit interactions between abelian gauge
multiplet and left-chiral, 303–306
gauge, 205
massless, 144–150
vector, 147, 205
N
natural units, 14
naturalness problem, 2, 6–9
neutralino, 379
no-go theorems, 156
nonabelian gauge theories, 211–217, 322–324
chiral multiplet, combination of nonabelian
gauge multiplet with, 233–236
free supersymmetric, 219–222
nonrenormalization theorems, 203–204
notation
[ symbol] notation for ±iσ
2
, 57–60
hermitian conjugate, 26–28
index-free notation, 53–56
for indices of σ
μ
and σ [¯σ ]
μ
, 60–65
in standard model, 364
for two- vs. four-component spinors, 17
van der Waerden notation, 34–35, 45–64
O
one-point function, 185–189
O’Raifeartaigh model, 331–338
P
parity, and Dirac spinors, 81–82
Pauli-Lubanski operator, 134, 135
Pauli matrices, 15–16
photinos, 147, 206
Planck scale, 158
Poincar´e generators, 156
Poincar´e group, 107, 134, 156
propagators, 182–185
of B field to one loop, 189–201
super-, 204
Q
QCD lagrangian:
about, 214–215
in terms of Weyl spinors, 217–219
QED lagrangian, in terms of Weyl spinors,
76–78
quantum chromodynamics (QCD), 7, 76, 370
quantum electrodynamics (QED), 8, 76
F-type SUSY breaking, 330
supersymmetric QED, 317–322
quantum field operators, 108, 109, 111, 153–155
quantum field theory, 5
calculations of processes in, 178–182
hermitian conjugate in, 26
INDEX
479
quarks:
kinetic energy of, in MSSM, 368–370
masses of, in MSSM, 399–400
R
R-parity, 376–377, 400–401
R-symmetry, 375–376
Rarita-Schwinger fields, 147
renormalization, 6
renormalization group equation, 401
right-chiral spinors, 18, 28–29, 69
right-chiral superfields, 267
S
scalar fields, masses of, 82–83
scalar potential, in MSSM, 381–384
selectrons, 146
sfermions, 146
sleptons, 146
SM (see standard model)
“softly broken” (term), 325
spacetime transformations, 3
spinor dot products, 35–37, 54–57
spinor fields, masses of, 83–86
spontaneous symmetry breaking (SSB),
325–330, 380
and MSSM, 352
in standard model, 360–364
spurious fields (spurions), 203
squared-mass matrix, 335
squarks, 146
SSB (see spontaneous symmetry breaking)
standard model (SM), 2, 351–378
fermions in, 352–355
and gauge vector superfields, 368
Higgs potential in, 359
interaction terms in, 357–359
kinetic term of Higgs field in, 356–357
kinetic terms of gauge bosons in, 356
and left-chiral superfields of MSSM, 365–367
and MSSM lagrangian, 368–371
notation in, 364
quick review of, 351–359
spontaneous symmetry breaking in,
360–364
and superpotential of MSSM, 371–378
standard particles, 374–375
states, effects of supercharges on, 140–143
stop (top squark), 146, 398
structure constants, 211
SU(3)
C
× SU(2)
L
×U (1)
Y
,2
SUGRA (supergravity), 3, 92
SUGRA-mediated SUSY breaking, 350
supercharges, 115, 140–143
superfield formalism, abelian gauge invariance
in, 297–303
superfields, 4, 241, 248–250
abelian field-strength superfield, in terms of
component fields, 308–313
component fields of, 250
constraints and, 261–268
field-strength, 306–307
and free abelian supersymmetric lagrangian,
313–317
left-chiral, 269–295
products of, 284–285
vector, 300
supergravity (see SUGRA)
superphysicists, 13
superpotential, 284–288
of the MSSM, 373–380
superpropagators, 204
superspace, 4, 110, 239–268
coordinates in, 240–242
supersymmetric transformations of
coordinates in, 244–248
translations in, 242–244
supersymmetric charges, 115–130
algebra of, 115–125
auxiliary field, introduction of,
128–130
and closure of algebra, 131
as differential operators, 255–261
and nonclosure of algebra for spinor field,
125–127
as quantum field operators, 155–157
supersymmetric gauge theories, 205–236
and auxiliary fields, 209–211
chiral multiplet, combination of abelian
vector multiplet with, 222–232
chiral multiplet, combination of nonabelian
gauge multiplet with, 233–236
and elimination of auxiliary fields,
232–233
480
Supersymmetry Demystified
supersymmetric gauge theories (Cont.):
free supersymmetric abelian gauge theory,
206–209
free supersymmetric nonabelian gauge
theories, 219–222
nonabelian gauge theories, 211–217, 322–324
and QCD lagrangian in terms of Weyl
spinors, 211–217
supersymmetric GUT, MSSM and coupling
unification in, 401–412
supersymmetric lagrangians, 89–99
dimensional analysis, review of, 90–91
with masses and interactions, 160–167
and transformation of the fields, 91–95
and transformation of the lagrangian, 95–98
supersymmetric transformations:
of component fields, 273–277
of coordinates in superspace, 244–248
supersymmetry (SUSY):
characteristics of, 2
difficulties in learning about, 3–4
and exact invariance, 2–3
extended, 145, 157–158
and standard model, 2
supersymmetry breaking, 325–350
D-type, 340–344
explicit, 344–350
F-type, 330–331
and mass spectrum, 334–338
in MSSM, 380–381
and O’Raifeartaigh model, 331–338
spontaneous, 326–330
and supertrace, 339–340
supertrace, 339–340
SUSY (see supersymmetry)
SUSY algebra, 115–125
classification of states using, 134–140
closure of, 131
and effects of supercharges on states,
140–143
finding, without explicit charges, 111–115
in Majorana form, 150–151
with massless SUSY multiplets,
144–150
nonclosure of, 125–127
symmetry, internal, 106
symmetry currents, obtaining charges from,
151–153
T
tadpole diagrams, 177, 201–202
Taylor expansions, 7
translations (in superspace), 242–244
two-component spinors, 17
V
van der Waerden, Bartel Leendert, 45n.
van der Waerden notation, 34–40,
45, 124
vector multiplet, 147, 205, 222–232
vector superfields, 300
W
Weinberg (weak) mixing angle, 362
Wess-Zumino gauge, 301
Wess-Zumino lagrangian:
full, 173–175
in Majorana form, 175–176
and superfields, 262
Wess-Zumino model:
D term in, 289–290, 302–308
free part of, 288–292
and full Wess-Zumino lagrangian,
173–175
and more general lagrangian, 167–173
one-point function of A field in,
185–189
and symmetric lagrangian with masses and
interactions, 160–167
and Wess-Zumino lagrangian in Majorana
form, 175–176
Weyl equations, 22
Weyl representation, 15
Weyl spinors, 17–40
Dirac spinors vs., 17–22
and helicity, 22–24
and invariants containing Lorentz indices,
32–33
as irreducible representation of Lorentz
group, 18
and Lorentz invariants, 25–26, 28–31,
51–53
Lorentz transformations of, 18, 19, 24–26
and Majorana spinors, 86–88
INDEX
481
Weyl spinors (Cont.):
massive, 73–74, 85–86
massless, 72
notational issues, 26–28, 34–40, 43–65
and propagators, 184–185
QCD lagrangian in terms of, 217–219
QED lagrangian in terms of, 76–78
winos, 147
Y
Yukawa coupling, 358
Yukawa terms, 352
Z
zinos, 147