Supersymmetry. Theory, Experiment, and Cosmology

A-terms 91, 149, 180, 488

abelian gauge supermultiplets, coupling of chiral supermultiplets to 47–48

A eck–Dine mechanism 330–331 age of the Universe 464 Aharonov–Bohm experiment 270

alignment, family symmetries 339–348 angular diameter, of a source in the

sky 462

annual modulation of WIMP signal 113–114

anomalies

axial 411–415

flavor problem 343–345 in MSSM 87

in ten dimensions 295–296 anomalous dimensions 486

anomalous U (1) symmetry see pseudoanomalous U (1) symmetry

anomaly mediation 179–181 anti-de Sitter space 127 antichiral superfields 436

antisymmetric tensor field 294–296 component of a linear supermultiplet

445

rˆole in anomaly cancellation 295 asymptotically free gauge theories

11–12

attractor mechanism, towards general relativity 315

auxiliary fields D field 47

F field 38–41

in supergravity 123

role in supersymmetry breaking 39, 41

axial anomaly, see anomalies axions

and pseudo-anomalous U (1) symmetries 302–304

duality with antisymmetric tensor 294

baryogenesis 328–331, 474–475 baryon number 102

and R-parity 90, 249 B-L 244, 247–249

in Standard Model 102

violation in proton decay 244–246 violations with dimension 5 opera-

tors 102, 245–246

beta decay see nuclear beta decay beta function 11, 229–231, 234–235, 485 Bianchi identity 294, 442

Big Bang scenario, hot 465–473 Bogomol’nyi bound 73, 76, 78 Bogomol’nyi condition 74, 75 bolometer detection of dark matter

111–114

boson–fermion states, pairing of 26–27 bottom–tau unification 241–242 boundary conditions

open strings 259 orbifold 274

supersymmetry breaking 305

510 Index

boundary conditions (Contd.) topological gauge symmetry

breaking 272

toroidal compactification 265, 268 BPS monopole 73–74

BPS state 65–80 brane 279, 280 breathing mode 264

Brown–Teitelboim mechanism 353–354 b to s gamma 158–161, 162, 163f, 336,

337t

Cabibbo matrix 389 Cabibbo–Kobayashi–Maskawa matrix

332, 395–396

Wolfenstein parametrization 332, 396 Cabibbo–Kobayashi–Maskawa phase 4,

332, 339, 342, 394 Calabi–Yau compactification 282–285,

286t, 287t, 296–297 Calabi–Yau manifolds 281, 283 Cartan classification, 226, 227t central charges 55, 61–62, 77, 217, 220 Chan–Paton degrees of freedom 259 charges

currents and 364–366

problem of the quantization of charge 3, 228, 371

charginos 97–98, 186, 328 Chern–Simons 3-form 295 chiral superfield 432–441

action 434–437

coupling to vector superfield 443–444, 445

definition 432–434

dual to a linear superfield 446 K¨ahler invariance 438–441 R-symmetry 437–438

chiral supermultiplets 37–45, 433 auxiliary fields 38–39

coupling to gauge supermultiplets 47–48, 49–50

coupling to supergravity 123–126 interactions 40–41

in the MSSM 88–89 N = 1 59–60

with Majorana spinor 37–42 with Weyl spinor 42–45

Christo el symbol 455

for a K¨ahler manifold 440

CKM see Cabibbo–Kobayashi–Maskawa Cli ord algebra 262, 426

closed strings 255, 256–258 superstring 261–262 T-duality 268–269 twisted 274

CMB see cosmic microwave background cold relics 469, 470 Coleman–Mandula no-go theorem

20–23, 53 color strings 258, 259

color triplet exchange, dimension-5 operators generated by 245, 246

compactification 263–274 Calabi–Yau 282–285, 286t, 287t,

296–297 Kaluza–Klein 264–267, 281 orbifold 272–274

string toroidal 267–269 conformal anomaly 263, 410–411

conformal symmetry and string theory 255, 262–263, 352

conformal Killing vector 406 conformal transformations

in two dimensions 257, 406 in four dimensions 405

in D dimensions 405–406

cosmic coincidence problem 351–352 cosmic microwave background (CMB)

468, 471–473

cosmic string 324–327, 481–483 D -term string 327

F -term string 324–326 global string 481

local string 482 superconducting string 325, 483

cosmological constant 349–362, 456, 458, 463

cosmological constant problem 350 cosmological distances, measure of

460–463

cosmological evolution, stages of 466t cosmology

Big Bang scenario 465–475 cosmic microwave background

471–473

cosmic strings 481–483 Friedmann–Robertson–Walker

universes 457–460 inflationary 475–481

Coulomb phase 206, 207 couplings

R-parity breaking 187–190

of fermions to the neutral Higgs scalars 97t

of matter to supergravity 123–126 of the pseudo-anomalous U (1) gauge supermultiplet 302–304

of the lightest neutrino 107, 110–111 CP symmetry and dyons 70

CP violation 333, 334–336, 338, 395–396

and baryogenesis 328, 474 CP-violating mass insertions, limits on

flavor-conserving 336, 337t covariant derivative

gauge 368, 369, 371 gravity 455

K¨ahler 440 supersymmetry 432

currents

and charges 364–366 supermultiplet of 55–57, 208

custodial SU (2) symmetry 155–157 CVC hypothesis 382

D -brane 280

D -flat directions 169, 171, 199, 202, 203–205

D -term breaking 47–48, 129 D -term inflation 321–324

D -term strings 327

D -terms 46, 92, 101, 199, 303–304 dark energy 354–362

pseudo Goldstone boson 361 quintessential problems 360–361 quintessence 355–359

Index 511

dark matter 102–116, 459 codes 116

direct detection 111–114 indirect detection 114–116 lightest supersymmetric particle

(LSP) 103–111, 183

de Sitter space 127, 352, 476–477 event-horizon 477 particle-horizon 476

decoupling, e ective theories and 208–209

deflation 359 dilatations 403, 404

and conformal transformations 405 and renormalization groups 410–411

dilaton 406–407

string, see string dilaton

and scalar-tensor theories 313–315 and Wess–Zumino term 411

dimensional reduction (DR, DR) 142, 222, 242, 488–489

dimension regularization 488

Dirac, time variation of fundamental constants 315–316

Dirac lagrangian 366

Dirac mass term 2, 345–346 Dirac quantization condition,

monopoles and the 69–70 Dirac spinors 421–423

contribution to beta function 230 Dirichlet boundary conditions 259 Dirichlet brane, see D-brane Dirichlet p-brane 279

discrete symmetries 90, 102, 297 distances, measure of 460–463 domain walls 65–67

doublet–triplet splitting problem 225, 238–239, 249–251

DR, see dimensional reduction dual models 258

duality

between string theories 278–281 electric–magnetic 213–215 Hodge 294, 446 Montonen–Olive 78–80

512 Index

duality (Contd.) Seiberg–Witten 218–220 S-duality 278–279

T -duality 268–269, 278–279, 280 dynamical supersymmetry breaking

31, 195–223 and quintessence 359

’t Hooft consistency condition 208 role of R-symmetry 209–210 Seiberg–Witten model 215–220 SU (Nc) with Nf flavors 209–215

dyons 70, 76, 77, 78–80, 220

E6 (exceptional gauge group of rank 6) 227t, 251–252

anomalies in E6 228

E8 × E8 heterotic string theory 276–277

duality relations 278

e ective supergravity action 289–290 lowest-lying states 278t

e ective action 408

Wilson e ective action 200–202 e ective potential 164–166, 322,

407–410

e ective supersymmetry 138 e ective theories 4–8

and decoupling 208–209 Einstein equations 349, 456 Einstein tensor 123, 456, 457 Einstein–K¨ahler manifold 149 electric dipole moment, neutron

191–192, 336, 337t electroweak precision tests 154–157,

397–403

energy, dark, see dark energy equilibrium distributions 465–467 Euler characteristics 288

event horizon (de Sitterspace) 477 relation with number of families 285

evolution, of the Universe 466t, 468

F -flat directions 169, 171, 202 F -term breaking 41–42, 48, 129 F -term inflation 320–321

F -term strings 324–326 F -terms 40–42, 44, 92

family symmetries 334, 339–348 Fayet, P. 86, 87, 90

Fayet–Iliopoulos term 46–47, 49, 204, 221, 303–304, 321–323

FCNC (flavor changing neutral currents) 137–138, 158–161, 179, 332–334, 393–395

fermion masses 1, 390–391 hierarchies 339–344

in SO(10) model 248–249 in SU (5) model 240–244

Fierz rearrangement formula 400n, 422 fine tuning 9, 144–147, 238, 350, 475 fixed point (renormalization group)

139–142, 206–208

flat directions 41, 76–77, 169–173, 174, 198–199, 202–205

flavor changing neutral currents, see FCNC

flavor problem 3, 332–349 alignment and family symmetries

339–348

minimal flavor violation 161, 338–339

neutrinos 345–348 supersymmetric CP problem

334–336

supersymmetry breaking versus flavor dynamics 336

flavor structure, Standard Model 393–397

flux 308, 354

focus points 142–144 free electric phase 206

free gauge theories, asymptotically 11–12

free magnetic phase 206

Friedmann equation 349, 458–460, 478 Friedmann–Lemaˆıtre universe 349 Friedmann–Robertson–Walker

universes 457–465 Froggat–Nielsen model 340–341 fundamental constants, time variation

of 315–316

Gamow–Teller transitions 382 gauge coupling unification 229–235

in string theory 300–302 naturalness and hierarchy problem

238–240 nonsupersymmetric case 232–234 supersymmetric case 234–235, 237

gauge coupling renormalization 229–231, 485

gauge groups, simple 226 gauge mediation 181–184 µ/Bµ problem 184

gangino mass 182f scalar mass 183f

gauge symmetry breaking spontaneous breaking 379–381 topological 271–272

gauge theories, phases of 205–206 gaugino 37, 87

gaugino condensation 174–177, 201–202, 304, 306–308, 356

gaugino masses 18, 91, 98 nonuniversal 178

renormalization group 135–137, 486 universal in anomaly mediation 179

gaugino–Higgsino sector 97–99 Gauss law constraint 73 Gell-Mann matrices 204, 258 general relativity 254

attractor mechanism towards 315 elements of 454–456

geodesics 455

Georgi–Glashow model 386n, 417–418 monopoles in the 71–73

GIM mechanism 394–395, 395f Giudice–Masiero mechanism 147–148 Glashow–Weinberg–Salam model

388–397 global string 481

global symmetry 366–368 B–L 244

spontaneous breaking 26, 372–379 gluino 97

Goldstino 26–27, 39, 46, 92, 128, 129, 132–133, 150, 202

Index 513

Goldstone boson 5, 39, 291, 374, 375, 406

Goldstone fermion, see Goldstino Goldstone realization 376 Goldstone theorem 375–377 graded Lie algebra 54n

grand desert hypothesis 234 grand unification 224–253, 328

E6 model 251–252

and monopoles 475–476

SU (5) model 225–226, 236–246 SO(10) model 246–251

Grassmann variables 430 gravitational coupling 122 gravitational lensing 463 gravitino 121, 123

and Goldstino field 132–133 degrees of freedom 132–133 LSP 106, 183

mass as criterion of supersymmetry breaking 128

gravitino problem 316–319 graviton 121–122, 255f , 256 degrees of freedom 133 gravity mediation 173–179

Green–Schwarz counterterm 295, 296f, 302, 323, 344

Green’s functions 165, 166, 376 407–409

GSO projection 262, 263f

Hausdor formula 431

Heavy Quark E ective Theory (HQET) 160

helicity 132–133

heterotic string theory 275, 276–277, 278t, 278–279, 282

hexagonal anomaly diagram 296f hidden sector 129–131, 173, 174f gauge coupling, evolution of

174–175

hierarchy problem 195, 238–240 Higgs doublets 156

Higgs mass 96, 164–168 naturalness 8–9 triviality 6–7

514 Index

Higgs mass (Contd.) unitarity 5–6 vacuum stability 7–8

Higgs mechanism 379–381 Higgs phase 206

Higgs sector (supersymmetric) 87–88, 164–169

gauge symmetry breaking and the 92–97

masses 94, 164–167 searching for 168–169

upper limit on the mass of the lightest Higgs 167–168

Higgs (Standard Model) 6, 387, 388– 390, 402–403

Higgsino 87, 97–101, 108, 161 Higgstrahlung process 169 Hodge duality relation 294, 446 Hodge number 285 holomorphic zeros 341

holomorphy 40, 196–198, 203–205 holonomy 283, 456 Hoˇrava–Witten supergravity 280

e ective theory 290–291 horizon

event horizon 477 particle horizon 471

horizontal (or family) symmetries 334, 339–348

Hosotani mechanism 298

hot Big Bang scenario 465–473 hot relics 470

Hubble parameter 349, 458 Hubble radius 477, 479 hybrid inflation 480 hypermultiplets 64–65

index formula 32, 325 inflation scenarios 480

D -term inflation 321–324 F -term inflation 320–321

inflationary cosmology 475–481 flatness problem 475

horizon problem 475 monopole problem 475–476 slow-roll parameters 478

infrared fixed point, top quark and the 139–142

intermediate vector boson 6, 384, 385f internal symmetry 364

isospin symmetry 366–368

Jacobi identity 54, 55

k -essence 359

Kalb–Ramond field, see antisymmetric tensor field

Kaˇc–Moody level 300, 301, 344 K¨ahler covariant derivatives 440 K¨ahler invariance 124, 175, 438–441 K¨ahler manifold 129, 282, 440, 448 K¨ahler metric 124, 439–440

K¨ahler potential 123, 125,

147, 199, 217, 438–439 generalized 126, 150–151

K¨ahler transformation 124, 175–176, 293

Kaluza–Klein compactification 264–267, 281

Kaluza–Klein modes 267, 267–269 kination 359

kink solution, and domain walls 65–68 Konishi anomaly 201

Landau pole 7, 11 LEP experiments

LSP mass limit 187f slepton search 185f stop search 186f

leptogenesis 328–330 nonthermal 330

lepton number 102, 245–246 Lie algebra 54n, 226–228, 227t

graded 54n

lightest supersymmetric particle (LSP) 90, 183

annihilation 107–109 and dark matter 102–116

interaction with matter 109–111 mass limit as a function of tan β 86,

187f

nature of 106–107, 388 as a WIMP 103–106

linear superfield 445–446 linear supermultiplet 291, 295 local gauge transformation

368–371

local supersymmetry 120–123 spontaneous breaking 126–129

local symmetry, spontaneous breaking 379–381

Lorentz factor 189 Lorentz condition 380

LSP see lightest supersymmetric particle (LSP)

M-theory 279f

magnetic operator 158, 159f , 192 magnetic moment, muon anomalous

161–162

Majorana mass term 2, 346, 423 Majorana spinors 23, 37, 110, 122, 423

in D dimensions 426 symplectic 63

Maki–Nakagawa–Sakata matrix 394 manifolds

Calabi–Yau 282–287

Einstein–K¨ahler 149

K¨ahler 124, 439–441 mass, problem of 3

mass insertion 335–336, 337t massive chiral supermultiplets 60 massive states

in the absence of central charges 60–61

in the presence of central charges 61–62

massless states, in the absence of central charges 58–59

matter

coupling to LSP 102–111 coupling to supergravity 123–126 dark, see dark matter

matter–antimatter asymmetry 328, 474 Maxwell equations 69

meson fields 205, 210 messenger 181–182, 306

Index 515

minimal coupling 368

minimal flavor violation 161, 338–339 minimal SU (5) model 225–226

supersymmetric 236–246

minimal subtraction (MS, MS) 242, 488–489

minimal supergravity model 120, 134, 138–139

constraints on 162–164 symmetries of 191t

minimal supersymmetric model (MSSM) 88, 89, 91–101

electroweak precision tests 154–162 minimal supersymmetric SU (5) model

236–246

fermion masses 240–244 gauge coupling unification 237

naturalness and hierarchy problem 238–240

proton decay 244–246 missing doublet mechanism 240,

249–251

mixing (maximal/no) 169 M¨obius band 289f

moduli (modulus) 67, 74–76, 199, 287t, 291–293, 296–297

comparison with Goldstone bosons 291

cosmological relevance 313–319 dilaton and 291–297

in string theory 264, 279 moduli problem 316–319 moduli space 67, 74–76, 198–199

monopoles (magnetic) 68–80, 220

and the Dirac quantization condition 68–71

BPS 73–74

cosmological problems 475–476

in the Georgi–Glashow model 71–73 Montonen–Olive duality 78–80

MS, see minimal subtraction µ parameter 88–89, 485 µ-problem 89, 147–148

muon anomalous magnetic moment 161–162

516 Index

N = 2 supersymmetry 59, 62–65, 215–220, 296

naturalness 8–11, 195

supersymmetry as a solution to the problem of 12–18

Neumann boundary conditions 259 neutral currents 387, 392–393, 398–402

flavor changing, see FCNC neutralinos 97, 99, 106–110, 186

annihilation 107–109, 114–116 capture rate 114

scalar and spin-dependent interactions 110f

neutrino masses 2–3, 249, 345–348, 391 neutron, electric dipole moment

191–192, 336, 337t

next lightest supersymmetric particle (NLSP) 183

next-to-minimal supersymmetric model (NMSSM) 117, 167–168

no-go theorem 20–22, 352–353 no-scale models 149–151, 293 Noether current or charge 20, 21, 45,

365, 412

nonabelian electric–magnetic duality 213–215

nonabelian gauge symmetries 48–50, 369–371, 444–445

nonminimal models 117 nonrenormalization theorem 200–205, 400

graphic method 14–16 nonuniversalities 139, 178–179 nuclear beta decay 381–382

one-loop contribution to 384f tree-level Fermi interaction 383f

oblique corrections 400–403 one-loop closed string diagram,

topology 288f open strings 258, 259, 280

and branes 280 boundary conditions 259

Ramond–Neveu–Schwarz model 262, 263t

T-duality 280

O’Raifeartaigh mechanism 41–42

orbifolds, compactification 272–274 orientifolds 275

p-brane 279

parallel transport 455 par sec 457

particle horizon 471 Pauli–Ljubanski operator 57

Peccei–Quinn symmetry 147, 176, 302 Pendleton–Ross fixed point 140–142 perturbative nonrenormalization

theorems 200–205

phases

Cabibbo–Kobayashi–Maskawa phase 4, 332, 395–396

supersymmetric 191–192 Planck length 255, 350

Planck mass MP 122, 254, 350 in D dimensions 266 reduced (mP ) 122

Poincar´e algebra 22, 405 Polonyi model 129–131, 139, 173 Polonyi problem 316 Prasad–Sommerfield limit 74 precision tests, see electroweak

precision tests problem of mass 3

problem of the quantization of charge 3

Proca equation 379–380 proton decay 244–246, 312

example of amplitude contributing to 245f, 246f

pseudo-anomalous U (1) symmetry 302, 323, 344

gauge supermultiplet, couplings of the 303–304

supersymmetry breaking 306–308 symmetries, axions and 302–304

pseudo-Goldstone boson and dark energy 361

QCD see quantum chromodynamics QED see quantum electrodynamics quadratic divergences 8–9, 12–18 quantization of charge 3, 228

quantum chromodynamics (QCD) 195–196, 224

quantum electrodynamics (QED) 1, 9–10, 368–369

quantum mechanics, supersymmetric 27–34

quark–squark alignment 138, 339–345 quarks

bottom-tau unification 241–242

first and second family unification 243 top quark and the infrared fixed point

139–142 quasifixed point 140–142 quintessence 355–356

quintessential inflation 359 quintessential problems 360–361

R-parity 89–91, 102 R-parity breaking 187–190

R-symmetry 55, 56–57, 175–176, 197, 208, 209–215, 297, 305, 437– 438

racetrack model 176

radiative breaking of SU (2) × U (1) 134–135

radiative corrections in the Standard Model 397–404

Ramond vacuum 262 Ramond–Neveu–Schwarz model 261–262

lowest-lying states for the open string 263t

Randall–Sundrum model 36 rare decay 158–161 Rarita–Schwinger action 123

Rarita–Schwinger equation 132–133 redshift parameter 461

reheating 477

relaxation mechanisms (vacuum energy) 352–354

relics 469–470 cold 469–470 hot 470

renormalization group, dilatations and 403–411

renormalization group equations 485–488

Index 517

A-terms 488

anomalous dimensions 486 gauge couplings 485 gaugino masses 486

mu parameter 485 soft scalar masses 487 Yukawa couplings 486

renormalization group fixed point see fixed point

renormalization schemes 488–489 representations of supersymmetry algebra 57–62

ρ parameter 155–157, 389, 399–403 Ricci tensor 455

K¨ahler manifold 440 Robertson–Walker metric 457 runaway quintessence 355–359

s-channel resonant production of sfermions 190t

S -duality 278–279

S-matrix, unitarity of the 5–6 scalar masses, see soft scalar masses scalar-tensor theories 313–315 scales in string theory 286–291 scaling dimension 403

scaling solutions of dark energy models 356–357

Scherk–Schwarz mechanism 305 screening theorem 403

seesaw mechanism 2–3, 249, 346, 391 Seiberg–Witten model 215–220 selectron mass limit 185

short supermultiplet 62 simple gauge groups 226–228 simple Lie algebras 226–228

Cartan classification 226, 227t sleptons 87t, 90t, 101, 184–185 sliding singlet 239–240

slowroll 478 parameters 478

smuon mass limits 184–185 sneutrino 106

SO (10) model 246–251

SO (32) heterotic string theory 276–278 soft scalar masses 16–17, 91, 487

518 Index

soft supersymmetry breaking 16–18 soft supersymmetry breaking terms 91,

136–138

boundary values 136, 138–139, 178 renormalization group evolution

486–488 solitons 65

spacetime, dimensions 258, 262, 263, 264–267, 275, 424–427

spacetime general coordinate transformation 121, 454–455

spacetime symmetry 364 spectral index 480 sphaleron processes 328, 329 spinor 419–428

Dirac 42, 230, 421–423

in four dimensions 419–423 in higher dimensions 424–427 relation with vectors 420–421

SO (2n), representations of 424–425 Van der Waerden notation 419–420 Weyl Spinon 42

split supersymmetry 348–349 spontaneous breaking

global symmetry 26, 372–379 local supersymmetry

126–129

local symmetry 379–381

global supersymmetry 25–27, 39 spurion 191, 334

squarks 90t, 100–101, 185–186 Standard Model 381–397

agreement with experimental data 1–3

chiral nature 1, 390

electroweak precision tests 155–157, 397–403

flavor structure 393–397 radiative corrections 397–403 problems of the 1–19 triangle anomalies 414, 415

spin connection 123, 284, 455, 456 stau mass limit 185

stop mass limit 185–186 string dilaton 258

and gaugino condensation 174–179

in four dimensions 291–296 stabilization 178

string coupling 287

transformation under S-duality 278 transformation under T-duality 269

string theory 174–175, 197, 254–311 closed 256–258, 273

color 258–259 diagram 260f duality 278–280

frame metric 287, 292–293 open 258, 259, 280 superstring 261–262

toroidal compactification 267–269 string, see cosmic string or string

theory

SU (2)×U (1) model, see Glashow– Weinberg–Salam model

super-Higgs mechanism 39, 41, 126–129 Super-Kamiokande Cherenkov detector

246 superfields 429–453

antichiral 436 chiral 432–441 conventions 453 linear 445–446

superspace and 429–432 vector 441–445

supergravity 26, 120–153 10-dimensional 287t, 289–290 11-dimensional 279–280, 290–291 couplings of matter to 123–126 minimal, see minimal supergravity

model supermultiplet 37–52

chiral 37–45, 432–441 current 55–57, 208 gravity 120–123

linear 291, 295, 445–446 short 62, 217

vector 45–50, 441–445 supernovae type Ia

best-fit in (ΩM, ΩΛ) plane 465f Hubble plot 464f

as standard candles 463