Chiral DynamicsChiral DynamicsHowHowss and Why and Whyss

3rd lecture: the effective Lagrangian

Martin Mojžiš, Comenius University23rd Students’ Workshop, Bosen, 3-8.IX.2006

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: pseudoGoldstone bosons

• SU(2) (massless u, d) Goldstone bosons: pions

• SU(3) (massless u, d, s) Goldstone bosons: pions, kaons, eta

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

• 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

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

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

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

towards the convenient choice

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

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• 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 here23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

the lowest order

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

• (0)(U) should not change under chiral transformations1' LR gUgU

• but starting from some particular value of U one can get

any other value by appropriate transformations gRUgL-1

• reason: even for gL=1 the gRU 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)

the next order

23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin 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)

yet another one

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

• after some more algebra one obtains

(4)(U) = a1 (Tr (U† U))2 + a2 Tr (U† U) Tr (νU† νU)

• the coefficients a1 , a2 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

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

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

a treatment of non-zero quark masses

• how is the explicit breakdown of the chiral symmetry accounted for? ... qMDiqLQCD

...)(. qxsDiqL fieldextQCD

• instead of the mass matrix M one considers an external matrix field s

• the transformation properties of s are given by invariance

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fieldextQCDL .

• the invariant Leff is constructed with the field s(x) included

• at the end of the day one sets s(x) M + sext(x) in the Leff

this produces the mass term for Goldstone bosons + infinite number of other terms 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University

the lowest order (in s)

23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin 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)

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

fieldsextQCDL .

• the invariant Leff is constructed with the fields p, v, a included

• finally one sets v,a to external electroweak fields in Leff

• remark: in this way only external photons are accounted for

to include virtual photons one replaces even the quark charge by some external field

...55. qavipsDiqL fieldsextQCD

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

local or global chiral symmetry?

• with the vμ, aμ 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)

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

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 M2 ; 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

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

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 Leff 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

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

pion-nucleon effective Lagrangian

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fieldsexteffective

• 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 uμ

• gA and c1 are free parameters (low-energy constants)

• we shall use the beast in the 4th lecture23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University