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

2. a brief reminder ChPT is the low-energy effective theory of the QCD it shares all the symmetries of the QCD: Lorentz invariance, space and time reflection, charge conjugation the chiral symmetry (a symmetry of QCD with massless quarks) the latter is broken both spontaneously and explicitly spontaneously broken symmetry: pseudo Goldstone bosons SU(2) (massless u, d) Goldstone bosons: pions SU(3) (massless u, d, s) Goldstone bosons: pions, kaons, eta 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

3. GBs transform according to a nonlinear realization of the chiral group which reduces to a linear representation when restricted to the unbroken subgroup more common: linear representations (in the Hilbert space) symmetry operators should obey superposition principle, which means linearity quantum fields - no such thing like the superposition principle nevertheless representations are quite common also here, but for different reason linearity means that a +  linear combination of a + operators i.e. the symmetry operators do not change the number of particles usually a desired feature, but not for Goldstone bosons symmetry operators generate Goldstone bosons, nonlinear realizations are called for non-linear realizations 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

4. how to construct invariants? there is an infinite # of nonlinear realizations, which one is the one? apparently a very important question any will do (an equivalence from the point of view of the S-matrix) certain particular choice may be of some (perhaps huge) practical advantage construction of invariants: a problem from the differential geometry of the manifold given by the chiral group factorized by the unbroken subgroup clever choice of convenient functions of fields simplifies life a lot for some standard choices invariants are simply traces of products of matrices 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

5. towards the convenient choice 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University a group elementgenerators

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

7. 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University the standard notation:

8. any Lagrangian in terms of φ can be rewritten in terms of U U is much more user friendly, it transforms in a simple way the invariance of the Lagrangian independent of variables used the effective Lagrangian is constructed in terms of the U matrix technical remark: once also non-GB fields are accounted for u becomes more appropriate than U another remark: the standard relation between u and φ contains some constant F, which is omitted here 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

9. the lowest order 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University  (0) (U) should not change under chiral transformations but starting from some particular value of U one can get any other value by appropriate transformations g R Ug L -1 reason: even for g L =1 the g R U covers the whole SU(2) conclusion:  (0) (U) has the same value for every U  (0) (U) = const and since the constant is irrelevant in  one can take  (0) (U) = 0 which is mandatory (see lecture 1)

10. the next order 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University after some algebra one obtains the coefficient ¼ is fixed by the kinetic term   φ   φ which appears after one expands U in terms of φ on top of the kinetic term,  (2) (U) contains higher powers of φ describing φ φ  φ φ, φ φ  φ φ φ φ, etc. for each of these processes we have a complete information about the threshold behavior  (2) (U) = ¼ Tr(   U †   U)

11. yet another one 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University after some more algebra one obtains  (4) (U) = a 1 ( Tr (   U †   U) ) 2 + a 2 Tr (   U †   U) Tr (  ν U †  ν U) the coefficients a 1, a 2 are the so-called low-energy constants in principle they are calculable from the QCD in real life they are not so they are treated as free parameters, fitted by data once they are pinned down,  (4) (U) provides lot of predictions

12. beyond the genuine GBs in terms of which fields is the ChPT formulated? the simplest version: Goldstone bosons (definitely the lightest) more realistic version: + external scalar field (mimics quark masses) more interesting version: + external vector and axial fields (EW inter.) even more interesting: + some heavier particles (e.g. nucleons) even more ambitious: + specific trick to cover virtual photons 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

13. a treatment of non-zero quark masses how is the explicit breakdown of the chiral symmetry accounted for? instead of the mass matrix M one considers an external matrix field s the transformation properties of s are given by invariance of the invariant L eff is constructed with the field s(x) included at the end of the day one sets s(x)  M + s ext (x) in the L eff this produces the mass term for Goldstone bosons + infinite number of other terms 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

14. the lowest order (in s) 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University in this way, quark masses enter the effective Lagrangian with s = M they become present explicitly in ChPT quark masses enter the results of pseudoGB masses one can calculate the former from the latter quark masses always multiplied by the LEC b which drops out from mass ratios ChPT gives just quark masses ratios (SU(3) quite illustrative)  (2) (U) = b Tr(U † s - s † U)

15. a treatment of electroweak interactions Gasser, Leutwyler: even pseudoscalar, vector and axial external fields the transformations of p, v, a are given by the invariance of the invariant L eff is constructed with the fields p, v, a included finally one sets v,a to external electroweak fields in L eff remark: in this way only external photons are accounted for to include virtual photons one replaces even the quark charge by some external field 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

16. local or global chiral symmetry? with the v μ, a μ present, the symmetry can be promoted to a local one one can use them to change derivatives to covariant derivatives would be nice: the local symmetry is stronger, i.e. more constraining however, what about Higgs mechanism with SB gauge symmetry? nothing, it only applies to dynamical fields, not to the external ones still, should ChPT be based on local or global chiral symmetry? Gasser and Leutwyler: it has to be the local one (beyond the scope here) 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

17. the chiral counting several small parameters in the game low energies, small quark masses, small EW couplings all of them are treated on (almost) the same footing the s-field gives the mass term  M  2 ; M  is of the low-energy order chiral order = # derivatives + 2 # s fields + 2 # p fields + # v fields + # a fields extension of the low-energy expansion with no serious problems namely the loop expansion with dimensional regularization fits well into the scheme 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

18. the chiral counting with non-GB hadrons two cornerstones: the chiral symmetry and the low-energy expansion symmetry: non-Goldstone hadrons slightly more complicated low-energy expansion: more serious complications the chiral expansion of L eff does not imply a simple expansion of scattering amplitudes massive particles (even in case of massless quarks) spoil the consistent chiral counting the new techniques are needed they are available remark: in this case L (1) does not vanish, nevertheless the chiral counting survives 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University

19. pion-nucleon effective Lagrangian nucleon (non-Goldstone) fields  pion (Goldstone) fields packed in u scalar and pseudoscalar external fields packed in  + external vector and axial fields are incorporated in both  and u μ g A and c 1 are free parameters (low-energy constants) we shall use the beast in the 4 th lecture 23 rd Students’ Workshop, Bosen, 3-8.IX.2006Martin Mojžiš, Comenius University