Chiral Dynamics How s and Why s 3 rd lecture: the effective Lagrangian Martin Mojžiš, Comenius...
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Transcript of Chiral Dynamics How s and Why s 3 rd lecture: the effective Lagrangian Martin Mojžiš, Comenius...
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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• 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
of
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