Chiral Dynamics How s and Why s 1 st lecture: basic ideas Martin Mojžiš, Comenius University23 rd Students’ Workshop, Bosen, 3-8.IX.2006

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

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

an effective theory of hadrons 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 reconcile the principles of quantum mechanics with those of special relativity. if possible at all, it has to be QFT 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University the most general relativistic Lagrangian should include all relativistic quantum physics

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

why do some constants vanish c 1 φredefinition of fields c 5 φ 5 renormalizability d 1  μ  μ φ4-divergence d 3 φ  μ  μ φlinear combination: d div e 1  μ  μ  ν  ν φrenormalizability e 2  μ  μ φ  ν  ν φrenormalizability 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

the most important constrains exploited substantially in EFT relaxed completely in EFT symmetryrenormalizability 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 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

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

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 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

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

the answer for spinless massless particles for any Feynman diagram the amplitude is a homogeneous function of external momenta p i  p i  M fi   M fi N L # of loops N I # of internal lines d # of derivatives (in the vertex) N d # of vertices with d derivatives to do list 1.prove this 2.show, how this answers the question (which will turn out to be relevant later on) 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

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

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

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

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 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

how could this work? c 2 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 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

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 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

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

the SU(2) isospin symmetry Heisenberg (30’s) in QCD this symmetry is present for m u = m d 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: m d - m u is small (in some relevant respect) 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

the SU(3) flavor symmetry Gell-Mann (60’s) in QCD this symmetry is present for m u = m d = m s 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: m s - m d is larger, but still small enough 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

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

why cheat? for pedagocical purposes becausethe 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 2 nd lecture 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University