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Chiral Dynamics How s and Why s

Chiral Dynamics How s and Why s. 1 st lecture: basic ideas. 23 rd Students’ Workshop, Bosen, 3-8.IX.2006. Martin Mojžiš, Comenius University. effective theories. a fundamental theory. an effective theory. derivation. calculations considerably simpler. valid in much wider range.

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Chiral Dynamics How s and Why s

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  1. Chiral DynamicsHows and Whys 1st lecture: basic ideas 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  2. effective theories a fundamental theory an effective theory derivation calculations considerably simpler valid in much wider range certain circumstances: the same results 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  3. some examples underlying theoryeffective theory general relativity Newtonian gravity kinetic theory hydrodynamics electroweak SM Fermi theory QCD ChPT quarks, gluons pions, kaons, nucleons, … 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  4. an effective theory of hadrons if possible at all, it has to be QFT Steven Weinberg: The QFT is the way it is because (aside from theories like string theory that have an infinite number of particle types) it is the only way to reconcilethe principles of quantum mechanics with those of special relativity. the most general relativistic Lagrangian should include all relativistic quantum physics 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  5. a simple example a scalar field φ(x) 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  6. why do some constants vanish • c1φ redefinition of fields • c5φ5 renormalizability • d1μμφ 4-divergence • d3φμμφ linear combination: d2 + 4-div • e1μμννφ renormalizability • e2μμφννφ renormalizability 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  7. exploited substantially in EFT relaxed completely in EFT the most important constrains symmetry renormalizability all the symmetries of QCD (not just the Lorentz one) are accounted for infinite # of parameters not an issue, if only finite # relevant in the range of validity 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  8. non-renormalizable  non-feasible • effective field theory needs some organizing principle • for any n (n) should contain finite # of terms • the higher is the n the less important should (n) be • non-renormalization may require higher and higher n • never mind they are less and less important 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  9. the organizing principle • the range of validity of EFT = the low-energy region • truncated Taylor expansions in powers of momenta • derivatives in   momenta in vertices • n = the number of derivatives 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  10. measurable quantities what is the relation between the low-energy expansion of the effective Lagrangian and low-energy expansion of measurable quantities? to which order one has to know the effective Lagrangian if one wants to calculate a scattering amplitude up to the Nth order in the low-energy expansion? 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  11. the answer for spinless massless particles (which will turn out to be relevant later on) for any Feynman diagram the amplitude is a homogeneous function of external momenta pipi Mfi Mfi • to dolist • prove this • show, how this answers the question NL# of loops NI# of internal lines d# of derivatives (in the vertex) Nd# of vertices with d derivatives 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  12. the proof for the tree diagrams • external momenta pi • internal momenta kj(fixed by the vertex -functions) • pipikjkj • propagator  -2 propagator (1/k2 1/2k2) • vertex with d derivatives  d vertex • amplitude   amplitude 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  13. the proof for the loop diagrams • amplitude   amplitude • dimensional regularization does not spoil the picture ( ln   0 ) 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  14. the consequences (the answer) • if for some reason (0) = (1) = 0 then to an amplitude of order  only (n) with n = d  can contribute • bonus: a systematic order-by-order renormalization 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  15. order-by-order renormalization • every loop increases order by 2 • 1-loop renormalization of (n) requires adjustment of parameters of (n+2) • 2-loop renormalization of (n) requires adjustment of parameters of (n+4), etc. • for the renormalization of the parameters of (m)only (n) with n < m relevant 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  16. how could this work? • c2 must vanish (for massless particles) • (0) must vanish (to get decent power counting) • (2) should contain only finite # of terms • (4), (6), ... as well • everything perhaps due to some symmetry 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  17. the role of symmetries • once the renormalizability is not an issue, the constrains come just from symmetry • one has to identify all the symmetries of QCD • one has to trace the fate of these symmetries • then one can start to construct the most general effective Lagrangian sharing all the symmetries of the underlying theory 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  18. the symmetries of QCD fundamental SU(3)-color symmetry no relevance for EFT (since hadrons are color singlets) various accidental approximate symmetries every relevance for EFT (ChPT is based on these symmetries) 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  19. the SU(2) isospin symmetry • Heisenberg (30’s) • in QCD this symmetry is present for mu= md • if so, the strong interactions do not distinguish between u and d quarks • consequently they do not distinguish some hadrons • clearly visible, works almost perfectly • conclusion: md - mu is small (in some relevant respect) 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  20. the SU(3) flavor symmetry • Gell-Mann (60’s) • in QCD this symmetry is present for mu= md = ms • if so, the strong interactions do not distinguish between u, d and s quarks • consequently they do not distinguish more hadrons • visible, works reasonably • conclusion: ms - md is larger, but still small enough 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  21. a friendly cheat • particle data booklet: mu  5 MeV md  10 MeV ms 175 MeV • isospin SU(2): md - mu  0 mu  md  0 • flavor SU(3): ms - md  0 mu  md  ms 0 • it seems quite reasonable to consider the limit mu = md =0 and even mu = md = ms =0 • this assumption leads to the chiral symmetry of the QCD 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

  22. why cheat? • for pedagocical purposes • because the logic is turned upside-down the quark masses are known due to the chiral symmetry, not the other way round • the chiral symmetry of the QCD is quite hidden • much more sophisticated than isospin or flavor • topic of the 2nd lecture 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

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