Stars as Laboratories for Fundamental Physics - MPP Theory Group

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294 Chapter 8 The dispersion relation for a single flavor in a medium was given by Eq. (6.107). For three flavors which mix according to Eq. (8.3) the Klein-Gordon equation in Fourier space is ⎧⎡ ⎛ ⎞⎤2 ⎪⎨ ⎢ ⎣ω − G 3Y e − 1 0 0 √ Fn B ⎜ ⎟⎥ ⎝ 0 Y e − 1 0 ⎠⎦ ⎪⎩ 2 0 0 Y e − 1 ⎛ m 2 ⎞ ⎫ ⎛ −k 2 1 0 0 ⎪⎬ ⎜ − U ⎝ 0 m 2 ⎟ 2 0 ⎠ U † ⎜ ⎝ 0 0 m 2 ⎪ ⎭ 3 ⎞ ν e ⎟ ν µ ν τ ⎠ = 0, (8.26) where a possible neutrino background was ignored and Y n = 1 − Y p = 1−Y e was used. Also, the higher-order difference between ν µ and ν τ was ignored. This equation has nonzero solutions only if det{. . .} vanishes, a condition that gives us the dispersion relation for the three normal modes. For unmixed neutrinos where U is the unit matrix one recovers Eq. (6.107) for each flavor. For all practical cases the neutrinos are highly relativistic so that one may linearize this equation, (ω − k − M 2 eff/2k) Ψ = 0 (8.27) where it is easy to read the matrix M 2 eff from Eq. (8.26). Because to lowest order ω = k one may equally use M 2 eff/2ω in order to derive the dispersion relation, depending on whether one wishes to write ω as a function of k or vice versa. For two-flavor mixing between ν e and ν µ or ν τ (mixing angle θ 0 ) the effective mass matrix may be written in the same form as in vacuum Meff/2ω 2 = b 0 − 1 B · σ, (8.28) 2 where b 0 = (m 2 1 + m 2 2)/4k + √ 2 G F n B (Y e − 1 2 ) and ⎛ ⎞ B = 2π sin 2θ ⎜ ⎝ 0 l osc cos 2θ ⎟ ⎠ = m2 2 − m 2 1 2ω ⎛ ⎜ ⎝ ⎞ sin 2θ 0 ⎟ 0 cos 2θ 0 ⎛ ⎞ ⎠ − √ 0 ⎜ ⎟ 2 G F n e ⎝ 0 ⎠ . 1 (8.29) This equation defines implicitly the mixing angle θ as well as the oscillation length l osc in the medium in terms of the masses, the vacuum mixing angle θ 0 , and the electron density n e .
Neutrino Oscillations 295 Fig. 8.8. Mixing angle, oscillation length, and neutrino dispersion relation as a function of the electron density. The medium was taken to have equal numbers of protons and neutrons (Y e = 1 2 ), the ratio of neutrino masses was taken to be m 1 : m 2 = 1 : 2, and sin 2 2θ 0 = 0.15.
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Georg G. Raffelt Stars as Laborator
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Table of Contents Preface (xiv) Ack
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Table of Contents vii 4.5 Neutrino
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Table of Contents ix 8. Neutrino Os
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Table of Contents xi 12. Radiative
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Table of Contents xiii 16.3 New Int
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Preface xv Second, particles from d
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Preface xvii search for proton deca
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Preface xix cuts of higher-order gr
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Preface xxi quoted “astrophysical
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Chapter 1 The Energy-Loss Argument
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The Energy-Loss Argument 3 example
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The Energy-Loss Argument 5 trinos,
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The Energy-Loss Argument 7 As a sim
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The Energy-Loss Argument 9 The most
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The Energy-Loss Argument 11 against
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The Energy-Loss Argument 13 ing M;
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The Energy-Loss Argument 15 M ′ (
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The Energy-Loss Argument 17 Table 1
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The Energy-Loss Argument 19 ing to
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The Energy-Loss Argument 21 scalars
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Chapter 2 Anomalous Stellar Energy
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Anomalous Stellar Energy Losses Bou
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Chapter 3 Particles Interacting wit
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Particles Interacting with Electron
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Chapter 4 Processes in a Nuclear Me
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Processes in a Nuclear Medium 119 t
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Processes in a Nuclear Medium 121 f
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Processes in a Nuclear Medium 123 F
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Processes in a Nuclear Medium 129 v
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Processes in a Nuclear Medium 131 4
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Processes in a Nuclear Medium 133 i
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Processes in a Nuclear Medium 135 H
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Processes in a Nuclear Medium 137 I
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Processes in a Nuclear Medium 143 i
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Processes in a Nuclear Medium 145 s
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Processes in a Nuclear Medium 147 E
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Processes in a Nuclear Medium 149 a
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Processes in a Nuclear Medium 151 T
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Processes in a Nuclear Medium 153 r
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Processes in a Nuclear Medium 155 4
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Processes in a Nuclear Medium 157 I
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Processes in a Nuclear Medium 159 t
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Processes in a Nuclear Medium 161 A
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Processes in a Nuclear Medium 163 R
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Chapter 5 Two-Photon Coupling of Lo
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Two-Photon Coupling of Low-Mass Bos
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Chapter 6 Particle Dispersion and D
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Particle Dispersion and Decays in M
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Page 263 and 264: Particle Dispersion and Decays in M
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Page 275 and 276: Nonstandard Neutrinos 255 kinematic
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Page 279 and 280: Nonstandard Neutrinos 259 Fig. 7.2.
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Page 297 and 298: Nonstandard Neutrinos 277 7.5.2 Spi
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Page 303 and 304: Neutrino Oscillations 283 As usual
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Page 307 and 308: Neutrino Oscillations 287 Fig. 8.2.
Page 309 and 310: Neutrino Oscillations 289 8.2.4 Exp
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Page 313: Neutrino Oscillations 293 Apparentl
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Solar Neutrinos 345 There exist vas
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Solar Neutrinos 347 10.2 Calculated
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Solar Neutrinos 349 Fig. 10.3. Norm
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Solar Neutrinos 351 a certain range
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Solar Neutrinos 353 While helium an
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Solar Neutrinos 355 Bahcall and Pin
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Solar Neutrinos 357 Xu et al. (1994
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Solar Neutrinos 359 Table 10.4. Spe
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Solar Neutrinos 361 Table 10.5. Pre
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Solar Neutrinos 363 rates attribute
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Solar Neutrinos 365 10.3.4 Water Ch
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Solar Neutrinos 367 Fig. 10.10. Dif
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Solar Neutrinos 369 Fig. 10.13. Mea
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Solar Neutrinos 371 Table 10.9. Cur
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Solar Neutrinos 373 of order the an
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Solar Neutrinos 375 Fig. 10.16. 37
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Solar Neutrinos 377 GALLEX, or Home
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Solar Neutrinos 379 Table 10.10. Me
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Solar Neutrinos 381 deficits in all
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Solar Neutrinos 383 The ν e surviv
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Solar Neutrinos 385 The allowed ran
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Solar Neutrinos 387 10.7 Spin and S
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Solar Neutrinos 389 10.8 Neutrino D
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Solar Neutrinos 391 The small- and
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Solar Neutrinos 393 flux above its
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Chapter 11 Supernova Neutrinos The
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Supernova Neutrinos 397 Fig. 11.1.
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Supernova Neutrinos 399 beyond the
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Supernova Neutrinos 401 ate neutrin
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Supernova Neutrinos 403 Convection
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Supernova Neutrinos 405 Hayes, and
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Supernova Neutrinos 407 11.2 Predic
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Supernova Neutrinos 409 must then b
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Supernova Neutrinos 411 Figure 11.6
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Supernova Neutrinos 413 signal (Sec
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Supernova Neutrinos 415 time) on 23
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Supernova Neutrinos 417 All of the
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Supernova Neutrinos 419 to multiple
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Supernova Neutrinos 421 the final-s
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Supernova Neutrinos 423 ter of 53 e
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Supernova Neutrinos 425 Given these
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Supernova Neutrinos 427 thorough st
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Supernova Neutrinos 429 The forward
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Supernova Neutrinos 431 Another int
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Supernova Neutrinos 433 ber of ν e
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Supernova Neutrinos 435 A “normal
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Supernova Neutrinos 437 Fig. 11.19.
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Supernova Neutrinos 439 The approxi
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Supernova Neutrinos 441 be taken to
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Supernova Neutrinos 443 sense that
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Supernova Neutrinos 445 presence of
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Supernova Neutrinos 447 An even mor
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Chapter 12 Radiative Particle Decay
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Radiative Particle Decays 451 one c
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Radiative Particle Decays 453 corre
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Radiative Particle Decays 455 is no
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Radiative Particle Decays 457 Fig.
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Radiative Particle Decays 461 12.3.
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Radiative Particle Decays 463 emiss
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Radiative Particle Decays 465 range
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Radiative Particle Decays 469 weak
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Radiative Particle Decays 471 value
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Radiative Particle Decays 473 ignor
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Radiative Particle Decays 479 Table
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Radiative Particle Decays 481 In or
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Radiative Particle Decays 485 while
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Radiative Particle Decays 487 h 100
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Radiative Particle Decays 491 still
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Chapter 13 What Have We Learned fro
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What Have We Learned from SN 1987A
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Axions 525 Table 14.1. Particle mag
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Axions 527 or axion decay constant.
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Axions 529 The Yukawa coupling h is
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Axions 531 14.2.3 Pseudoscalar vs.
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Axions 533 Notably, the Higgs field
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Axions 535 otic heavy quarks: only
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Axions 537 Various axion models dif
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Axions 539 Bounds on the Yukawa cou
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Axions 541 Fig. 14.5. Astrophysical
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Axions 543 decay into axions. This
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Chapter 15 Miscellaneous Exotica St
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Miscellaneous Exotica 547 Table 15.
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Miscellaneous Exotica 549 15.2.3 Pr
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Miscellaneous Exotica 551 Most rece
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Miscellaneous Exotica 553 Fig. 15.1
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Miscellaneous Exotica 555 A violati
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Miscellaneous Exotica 557 nuclei it
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Miscellaneous Exotica 559 mass, and
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Miscellaneous Exotica 561 backgroun
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Miscellaneous Exotica 563 SN 1987A
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Miscellaneous Exotica 565 which led
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Miscellaneous Exotica 567 For a nar
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Neutrinos: The Bottom Line 569 the
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Neutrinos: The Bottom Line 571 Neut
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Neutrinos: The Bottom Line 573 Apar
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Neutrinos: The Bottom Line 575 is d
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Neutrinos: The Bottom Line 577 of t
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Neutrinos: The Bottom Line 579 siti
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Units and Dimensions 581 In the ast
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Appendix B Neutrino Coupling Consta
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Appendix C Numerical Neutrino Energ
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Numerical Neutrino Energy-Loss Rate
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Numerical Neutrino Energy-Loss Rate
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Appendix D Characteristics of Stell
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Characteristics of Stellar Plasmas
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References 607 Akhmedov, E. Kh., an
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References 609 Bahcall, J. N., Kras
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References 611 Bilenky, S. M., and
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References 613 Cannon, R. D. 1970,
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References 615 Cooper-Sarkar, A. M.
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References 617 Dodelson, S., Friema
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References 619 Fukuda, Y., et al. 1
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References 621 Goyal, A., Dutta, S.
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References 623 Hoogeveen, F., and S
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References 625 Kawakami, H., et al.
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References 627 Landau, L. D., and P
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References 629 Mayle, R. W., and Wi
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References 631 Nieves, J. F., and P
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References 633 Pochoda, P., and Sch
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References 635 Ross, J. E., and All
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References 637 Shlyakhter, A. I. 19
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References 639 Totsuka, Y. 1993, Ne
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References 641 Wiezorek, C., et al.
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Acronyms 643 L l.h. l.h.s. ly LMC L
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Symbols 645 dp differential in phas
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Symbols 647 X j Peccei-Quinn charge
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Subject Index A α-Ori 189 A665, A1
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Subject Index 651 Coulomb gauge 204
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Subject Index 653 H Hayashi line 32
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Subject Index 655 multiple scatteri
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Subject Index 657 neutrino radiativ
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Subject Index 659 Primakoff process
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Subject Index 661 SN 1987A bounds (
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Subject Index 663 supernova core bi
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